{"group":{"group":{"id":59112,"name":"Materials Science I","lockable":false,"created_at":"2022-12-30T10:27:32.000Z","updated_at":"2025-12-14T01:33:56.000Z","description":"A collection of problems relating to materials science.\n\n(Illustration of materials paradigm by Dhatfield at Wikipedia, released into the public domain.)","is_default":false,"created_by":332395,"badge_id":62,"featured":false,"trending":false,"solution_count_in_trending_period":31,"trending_last_calculated":"2025-12-14T00:00:00.000Z","image_id":5417,"published":true,"community_created":true,"status_id":2,"is_default_group_for_player":false,"deleted_by":null,"deleted_at":null,"restored_by":null,"restored_at":null,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eA collection of problems relating to materials science.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e(\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://commons.wikimedia.org/wiki/File:Materials_science_tetrahedron;structure,_processing,_performance,_and_proprerties.svg\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eIllustration of materials paradigm\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e by Dhatfield at Wikipedia, released into the public domain.)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.44px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none solid rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 102px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 289.5px 51px; transform-origin: 289.5px 51px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 266.5px 10.5px; text-align: left; transform-origin: 266.5px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; 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display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e by Dhatfield at Wikipedia, released into the public domain.)\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","published_at":"2022-12-30T18:55:10.000Z"},"current_player":null},"problems":[{"id":8048,"title":"Stress-Strain Properties - 1","description":"This is the first in a series of problems regarding mechanics of materials, in particular, material properties drawn from stress-strain responses. A simplified typical stress-strain response is illustrated below (from quora.com):\r\n\r\nThe yield stress is the pressure required to start deformation of the material being tested. The yield point is the point along the response indicated by the yield stress (vertical axis) and the yield strain (horizontal axis). The response of the material up to this point is elastic, meaning that all deformation is reversible. The elastic modulus (E, also known as modulus of elasticity or Young's modulus) is the slope of this line. Write a function to calculate the elastic modulus for a material, provided the elastic strain and yield stress (yield point).\r\nNext problem: 2 - resilience.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.440001px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none; white-space: normal; \"\u003e\u003cdiv style=\"block-size: 541px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 332px 270.5px; transform-origin: 332px 270.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 309px 31.5px; text-align: left; transform-origin: 309px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eThis is the first in a series of problems regarding mechanics of materials, in particular, material properties drawn from stress-strain responses. A simplified typical stress-strain response is illustrated below (from\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e quora.com):\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 304px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 309px 152px; text-align: center; transform-origin: 309px 152px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cimg class=\"imageNode\" style=\"vertical-align: baseline\" src=\"https://qph.cf2.quoracdn.net/main-qimg-b2693f4b9ea8430af25df920757e0b29-lq\" data-image-state=\"image-loaded\"\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 126px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 309px 63px; text-align: left; transform-origin: 309px 63px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eThe yield stress is the pressure required to start deformation of the material being tested. The yield point is the point along the response indicated by the yield stress (vertical axis) and the yield strain (horizontal axis). The response of the material up to this point is elastic, meaning that all deformation is reversible. The elastic modulus (E, also known as modulus of elasticity or Young's modulus) is the slope of this line. Write a function to calculate the elastic modulus for a material, provided the elastic strain and yield stress (yield point).\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 309px 10.5px; text-align: left; transform-origin: 309px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eNext problem: 2 -\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/8049-stress-strain-properties-2\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eresilience\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function [E] = stress_strain1(S_y,e_y)\r\n\r\nE = 1;\r\n\r\nend\r\n","test_suite":"%% Note\r\n% The following properties are measured at room temperature and are tensile \r\n% in a single direction. Some materials, such as metals are generally\r\n% isotropic, whereas others, like composite are highly anisotropic\r\n% (different properties in different directions). Also, property values can\r\n% range depending on the material grade. Finally, thermal or environmental\r\n% changes can alter these properties, sometimes drastically.\r\n\r\n%% steel alloy (ASTM A36)\r\nS_y = 250e6; %Pa\r\nS_u = 400e6; %Pa\r\ne_y = 0.00125;\r\ne_u = 0.35;\r\nnu = 0.26;\r\nG = 79.3e9; %Pa\r\nE = 200e9; %Pa\r\ndensity = 7.85; %g/cm^3\r\nsh_exp = 0.14; %strain-hardening exponent\r\nsh_coeff = 0.463; %strain-hardening coefficient\r\nassert(abs(stress_strain1(S_y,e_y)-E)\u003c1e9)\r\n\r\n%% titanium (Ti-6Al-4V)\r\nS_y = 830e6; %Pa\r\nS_u = 900e6; %Pa\r\ne_y = 0.00728;\r\ne_u = 0.14;\r\nnu = 0.342;\r\nG = 44e9; %Pa\r\nE = 114e9; %Pa\r\ndensity = 4.51; %g/cm^3\r\nsh_exp = 0.04; %strain-hardening exponent\r\nsh_coeff = 0.974; %strain-hardening coefficient\r\nassert(abs(stress_strain1(S_y,e_y)-E)\u003c1e9)\r\n\r\n%% Inconel 718\r\nS_y = 1172e6; %Pa\r\nS_u = 1407e6; %Pa\r\ne_y = 0.00563;\r\ne_u = 0.027;\r\nnu = 0.29;\r\nG = 11.6e9; %Pa\r\nE = 208e9; %Pa\r\ndensity = 8.19; %g/cm^3\r\nsh_exp = 0.075; %strain-hardening exponent\r\nsh_coeff = 1.845; %strain-hardening coefficient\r\nassert(abs(stress_strain1(S_y,e_y)-E)\u003c1e9)\r\n\r\n%% aluminum alloy (6061-T6)%^\u0026\r\nS_y = 241e6; %Pa\r\nS_u = 300e6; %Pa\r\ne_y = 0.0035;\r\ne_u = 0.15;\r\nnu = 0.33;\r\nG = 26e9; %Pa\r\nE = 68.9e9; %Pa\r\ndensity = 2.7; %g/cm^3\r\nsh_exp = 0.042; %strain-hardening exponent\r\nsh_coeff = 0.325; %strain-hardening coefficient\r\nassert(abs(stress_strain1(S_y,e_y)-E)\u003c1e9)\r\n\r\n%% copper\r\nS_y = 70e6; %Pa\r\nS_u = 220e6; %Pa\r\ne_y = 0.00054;\r\ne_u = 0.48;\r\nnu = 0.34;\r\nG = 48e9; %Pa\r\nE = 130e9; %Pa\r\ndensity = 8.92; %g/cm^3\r\nsh_exp = 0.44; %strain-hardening exponent\r\nsh_coeff = 0.304; %strain-hardening coefficient 530MPa\r\nassert(abs(stress_strain1(S_y,e_y)-E)\u003c1e9)\r\n\r\n%% rhenium\r\nS_y = 317e6; %Pa\r\nS_u = 1130e6; %Pa\r\ne_y = 0.000685;\r\ne_u = 0.24;\r\nnu = 0.3;\r\nG = 178e9; %Pa\r\nE = 463e9; %Pa\r\ndensity = 21.02; %g/cm^3\r\nsh_exp = 0.353; %strain-hardening exponent\r\nsh_coeff = 1.870; %strain-hardening coefficient\r\nassert(abs(stress_strain1(S_y,e_y)-E)\u003c1e9)\r\n\r\n%% polymer (nylon, 6/6)\r\nS_y = 82e6; %Pa\r\nS_u = 82e6; %Pa\r\ne_y = 0.0265;\r\ne_u = 0.45;\r\nnu = 0.41;\r\nG = 2.8e9; %Pa\r\nE = 3.1e9; %Pa\r\ndensity = 1.14; %g/cm^3\r\nassert(abs(stress_strain1(S_y,e_y)-E)\u003c1e9)\r\n\r\n%% polymer (nylon, 6/6) reinforced with 45wt.% glass fiber\r\nS_y = 230e6; %Pa\r\nS_u = 230e6; %Pa\r\ne_y = 0.016;\r\ne_u = 0.016;\r\nnu = 0.35;\r\nG = 13.0e9; %Pa\r\nE = 14.5e9; %Pa\r\ndensity = 1.51; %g/cm^3\r\nassert(abs(stress_strain1(S_y,e_y)-E)\u003c1e9)\r\n\r\n%% diamond\r\nS_y = 1200e6; %Pa\r\nS_u = 1200e6; %Pa\r\ne_y = 0.001;\r\ne_u = 0.001;\r\nnu = 0.20;\r\nG = 478e9; %Pa\r\nE = 1200e9; %Pa\r\ndensity = 3.51; %g/cm^3\r\nassert(abs(stress_strain1(S_y,e_y)-E)\u003c1e9)\r\n","published":true,"deleted":false,"likes_count":1,"comments_count":3,"created_by":26769,"edited_by":26769,"edited_at":"2024-03-27T17:40:39.000Z","deleted_by":null,"deleted_at":null,"solvers_count":324,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2015-03-30T18:09:31.000Z","updated_at":"2026-03-31T10:50:52.000Z","published_at":"2015-03-30T18:09:58.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThis is the first in a series of problems regarding mechanics of materials, in particular, material properties drawn from stress-strain responses. A simplified typical stress-strain response is illustrated below (from\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e quora.com):\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"center\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe yield stress is the pressure required to start deformation of the material being tested. The yield point is the point along the response indicated by the yield stress (vertical axis) and the yield strain (horizontal axis). The response of the material up to this point is elastic, meaning that all deformation is reversible. The elastic modulus (E, also known as modulus of elasticity or Young's modulus) is the slope of this line. Write a function to calculate the elastic modulus for a material, provided the elastic strain and yield stress (yield point).\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eNext problem: 2 -\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/8049-stress-strain-properties-2\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eresilience\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/image\",\"target\":\"/media/image1.net/main-qimg-b2693f4b9ea8430af25df920757e0b29-lq\",\"relationshipId\":\"rId1\"}]},{\"partUri\":\"/media/image1.net/main-qimg-b2693f4b9ea8430af25df920757e0b29-lq\",\"contentType\":\"image/net/main-qimg-b2693f4b9ea8430af25df920757e0b29-lq\",\"content\":\"https://qph.cf2.quoracdn.net/main-qimg-b2693f4b9ea8430af25df920757e0b29-lq\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":8049,"title":"Stress-Strain Properties - 2","description":"The resilience of a material is its ability to resist permanent (or plastic) deformation. The resilience coincides with the elastic region in the figure below and is calculated as the area under the stress-strain curve up to the yield point. Given that the elastic region is presumed to be entirely linear, this area is a triangle. Write a function to calculate the resilience of a material provided its elastic strain and yield stress (yield strength).\r\n\r\n(from quora.com)\r\nPrevious problem: 1 - elastic modulus. Next problem: 3 - qualitative measure of brittleness.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.440001px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none; white-space: normal; \"\u003e\u003cdiv style=\"block-size: 478px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 332px 239px; transform-origin: 332px 239px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 105px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 309px 52.5px; text-align: left; transform-origin: 309px 52.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eThe resilience of a material is its ability to resist permanent (or plastic) deformation. The resilience coincides with the elastic region in the figure below and is calculated as the area under the stress-strain curve up to the yield point. Given that the elastic region is presumed to be entirely linear, this area is a triangle. Write a function to calculate the resilience of a material provided its elastic strain and yield stress (yield strength).\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 304px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 309px 152px; text-align: center; transform-origin: 309px 152px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cimg class=\"imageNode\" style=\"vertical-align: baseline\" src=\"https://qph.cf2.quoracdn.net/main-qimg-b2693f4b9ea8430af25df920757e0b29-lq\" data-image-state=\"image-loaded\"\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 309px 10.5px; text-align: center; transform-origin: 309px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e(from quora.com)\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 309px 10.5px; text-align: left; transform-origin: 309px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003ePrevious problem: 1 - \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/8048-stress-strain-properties-1\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eelastic modulus\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e. Next problem: 3 -\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"/#null\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003equalitative measure of brittleness\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function [R] = stress_strain2(S_y,e_y)\r\n\r\nR = 1;\r\n\r\nend\r\n","test_suite":"%% Note\r\n% The following properties are measured at room temperature and are tensile\r\n% in a single direction. Some materials, such as metals are generally\r\n% isotropic, whereas others, like composite are highly anisotropic\r\n% (different properties in different directions). Also, property values can\r\n% range depending on the material grade. Finally, thermal or environmental\r\n% changes can alter these properties, sometimes drastically.\r\n\r\n%% steel alloy (ASTM A36)\r\nS_y = 250e6; %Pa\r\nS_u = 400e6; %Pa\r\ne_y = 0.00125;\r\ne_u = 0.35;\r\nnu = 0.26;\r\nG = 79.3e9; %Pa\r\nE = 200e9; %Pa\r\ndensity = 7.85; %g/cm^3\r\nsh_exp = 0.14; %strain-hardening exponent\r\nsh_coeff = 0.463; %strain-hardening coefficient\r\nassert(abs(stress_strain2(S_y,e_y)-1.5625e5)/1.5625e5\u003c5e-2)\r\n\r\n%% titanium (Ti-6Al-4V)\r\nS_y = 830e6; %Pa\r\nS_u = 900e6; %Pa\r\ne_y = 0.00728;\r\ne_u = 0.14;\r\nnu = 0.342;\r\nG = 44e9; %Pa\r\nE = 114e9; %Pa\r\ndensity = 4.51; %g/cm^3\r\nsh_exp = 0.04; %strain-hardening exponent\r\nsh_coeff = 0.974; %strain-hardening coefficient\r\nassert(abs(stress_strain2(S_y,e_y)-3.0212e6)/3.0212e6\u003c5e-2)\r\n\r\n%% Inconel 718\r\nS_y = 1172e6; %Pa\r\nS_u = 1407e6; %Pa\r\ne_y = 0.00563;\r\ne_u = 0.027;\r\nnu = 0.29;\r\nG = 11.6e9; %Pa\r\nE = 208e9; %Pa\r\ndensity = 8.19; %g/cm^3\r\nsh_exp = 0.075; %strain-hardening exponent\r\nsh_coeff = 1.845; %strain-hardening coefficient\r\nassert(abs(stress_strain2(S_y,e_y)-3.29918e6)/3.29918e6\u003c5e-2)\r\n\r\n%% aluminum alloy (6061-T6)%^\u0026\r\nS_y = 241e6; %Pa\r\nS_u = 300e6; %Pa\r\ne_y = 0.0035;\r\ne_u = 0.15;\r\nnu = 0.33;\r\nG = 26e9; %Pa\r\nE = 68.9e9; %Pa\r\ndensity = 2.7; %g/cm^3\r\nsh_exp = 0.042; %strain-hardening exponent\r\nsh_coeff = 0.325; %strain-hardening coefficient\r\nassert(abs(stress_strain2(S_y,e_y)-4.2175e5)/4.2175e5\u003c5e-2)\r\n\r\n%% copper\r\nS_y = 70e6; %Pa\r\nS_u = 220e6; %Pa\r\ne_y = 0.00054;\r\ne_u = 0.48;\r\nnu = 0.34;\r\nG = 48e9; %Pa\r\nE = 130e9; %Pa\r\ndensity = 8.92; %g/cm^3\r\nsh_exp = 0.44; %strain-hardening exponent\r\nsh_coeff = 0.304; %strain-hardening coefficient 530MPa\r\nassert(abs(stress_strain2(S_y,e_y)-1.89e4)/1.89e4\u003c5e-2)\r\n\r\n%% rhenium\r\nS_y = 317e6; %Pa\r\nS_u = 1130e6; %Pa\r\ne_y = 0.000685;\r\ne_u = 0.24;\r\nnu = 0.3;\r\nG = 178e9; %Pa\r\nE = 463e9; %Pa\r\ndensity = 21.02; %g/cm^3\r\nsh_exp = 0.353; %strain-hardening exponent\r\nsh_coeff = 1.870; %strain-hardening coefficient\r\nassert(abs(stress_strain2(S_y,e_y)-1.085725e5)/1.085725e5\u003c5e-2)\r\n\r\n%% polymer (nylon, 6/6)\r\nS_y = 82e6; %Pa\r\nS_u = 82e6; %Pa\r\ne_y = 0.0265;\r\ne_u = 0.45;\r\nnu = 0.41;\r\nG = 2.8e9; %Pa\r\nE = 3.5e-2; %Pa\r\ndensity = 1.14; %g/cm^3\r\nassert(abs(stress_strain2(S_y,e_y)-1.0865e6)/1.0865e6\u003c5e-2)\r\n\r\n%% polymer (nylon, 6/6) reinforced with 45wt.% glass fiber\r\nS_y = 230e6; %Pa\r\nS_u = 230e6; %Pa\r\ne_y = 0.016;\r\ne_u = 0.016;\r\nnu = 0.35;\r\nG = 13.0e9; %Pa\r\nE = 14.5e9; %Pa\r\ndensity = 1.51; %g/cm^3\r\nassert(abs(stress_strain2(S_y,e_y)-1.84e6)/1.84e6\u003c5e-2)\r\n\r\n%% diamond\r\nS_y = 1200e6; %Pa\r\nS_u = 1200e6; %Pa\r\ne_y = 0.001;\r\ne_u = 0.001;\r\nnu = 0.20;\r\nG = 478e9; %Pa\r\nE = 1200e9; %Pa\r\ndensity = 3.51; %g/cm^3\r\nassert(abs(stress_strain2(S_y,e_y)-6e5)/6e5\u003c5e-2)\r\n\r\n%%\r\nind = randi(4);\r\nswitch ind\r\n\tcase 1\r\n\t\tS_y = 250e6; %Pa\r\n\t\te_y = 0.00125;\r\n\t\tassert(abs(stress_strain2(S_y,e_y)-1.5625e5)/1.5625e5\u003c5e-2)\r\n\tcase 2\r\n\t\tS_y = 82e6; %Pa\r\n\t\te_y = 0.0265;\r\n\t\tassert(abs(stress_strain2(S_y,e_y)-1.0865e6)/1.0865e6\u003c5e-2)\r\n\tcase 3\r\n\t\tS_y = 241e6; %Pa\r\n\t\te_y = 0.0035;\r\n\t\tassert(abs(stress_strain2(S_y,e_y)-4.2175e5)/4.2175e5\u003c5e-2)\r\n\tcase 4\r\n\t\tS_y = 1172e6; %Pa\r\n\t\te_y = 0.00563;\r\n\t\tassert(abs(stress_strain2(S_y,e_y)-3.29918e6)/3.29918e6\u003c5e-2)\r\nend\r\n\r\n%%\r\nind = randi(4);\r\nswitch ind\r\n\tcase 1\r\n\t\tS_y = 1200e6; %Pa\r\n\t\te_y = 0.001;\r\n\t\tassert(abs(stress_strain2(S_y,e_y)-6e5)/6e5\u003c5e-2)\r\n\tcase 2\r\n\t\tS_y = 1172e6; %Pa\r\n\t\te_y = 0.00563;\r\n\t\tassert(abs(stress_strain2(S_y,e_y)-3.29918e6)/3.29918e6\u003c5e-2)\r\n\tcase 3\r\n\t\tS_y = 230e6; %Pa\r\n\t\te_y = 0.016;\r\n\t\tassert(abs(stress_strain2(S_y,e_y)-1.84e6)/1.84e6\u003c5e-2)\r\n\tcase 4\r\n\t\tS_y = 250e6; %Pa\r\n\t\te_y = 0.00125;\r\n\t\tassert(abs(stress_strain2(S_y,e_y)-1.5625e5)/1.5625e5\u003c5e-2)\r\nend\r\n\r\n%%\r\nind = randi(4);\r\nswitch ind\r\n\tcase 1\r\n\t\tS_y = 830e6; %Pa\r\n\t\te_y = 0.00728;\r\n\t\tassert(abs(stress_strain2(S_y,e_y)-3.0212e6)/3.0212e6\u003c5e-2)\r\n\tcase 2\r\n\t\tS_y = 230e6; %Pa\r\n\t\te_y = 0.016;\r\n\t\tassert(abs(stress_strain2(S_y,e_y)-1.84e6)/1.84e6\u003c5e-2)\r\n\tcase 3\r\n\t\tS_y = 70e6; %Pa\r\n\t\te_y = 0.00054;\r\n\t\tassert(abs(stress_strain2(S_y,e_y)-1.89e4)/1.89e4\u003c5e-2)\r\n\tcase 4\r\n\t\tS_y = 317e6; %Pa\r\n\t\te_y = 0.000685;\r\n\t\tassert(abs(stress_strain2(S_y,e_y)-1.085725e5)/1.085725e5\u003c5e-2)\r\nend\r\n","published":true,"deleted":false,"likes_count":2,"comments_count":0,"created_by":26769,"edited_by":26769,"edited_at":"2024-03-27T17:41:09.000Z","deleted_by":null,"deleted_at":null,"solvers_count":264,"test_suite_updated_at":"2015-03-30T18:44:33.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2015-03-30T18:27:49.000Z","updated_at":"2026-03-31T10:53:49.000Z","published_at":"2015-03-30T18:27:49.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe resilience of a material is its ability to resist permanent (or plastic) deformation. The resilience coincides with the elastic region in the figure below and is calculated as the area under the stress-strain curve up to the yield point. Given that the elastic region is presumed to be entirely linear, this area is a triangle. Write a function to calculate the resilience of a material provided its elastic strain and yield stress (yield strength).\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"center\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"center\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e(from quora.com)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ePrevious problem: 1 - \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/8048-stress-strain-properties-1\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eelastic modulus\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e. Next problem: 3 -\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"\\\"\u003e\u003cw:r\u003e\u003cw:t\u003equalitative measure of brittleness\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/image\",\"target\":\"/media/image1.net/main-qimg-b2693f4b9ea8430af25df920757e0b29-lq\",\"relationshipId\":\"rId1\"}]},{\"partUri\":\"/media/image1.net/main-qimg-b2693f4b9ea8430af25df920757e0b29-lq\",\"contentType\":\"image/net/main-qimg-b2693f4b9ea8430af25df920757e0b29-lq\",\"content\":\"https://qph.cf2.quoracdn.net/main-qimg-b2693f4b9ea8430af25df920757e0b29-lq\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":8050,"title":"Stress-Strain Properties - 3","description":"A brittle material will not exhibit a yield point. In other words, the yield point and failure point coincide. In such cases, the yield strain and failure strain (also known as ultimate strain or elongation) are the same value. On the other hand, ductile materials have a failure strain that is significantly greater than the elastic strain, as shown in the figure below.\r\n\r\n(from quora.com)\r\nWrite a function to determine the qualitative brittleness of the material by calculating the ratio of elastic strain to failure strain. A ratio of one indicates complete brittleness, whereas a ratio close to zero indicates essentially no brittleness.\r\nPrevious problem: 2 - resilience. Next problem: 4 - strength-to-weight ratio.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.440001px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none; white-space: normal; \"\u003e\u003cdiv style=\"block-size: 529px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 332px 264.5px; transform-origin: 332px 264.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 84px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 309px 42px; text-align: left; transform-origin: 309px 42px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eA brittle material will not exhibit a yield point. In other words, the yield point and failure point coincide. In such cases, the yield strain and failure strain (also known as ultimate strain or elongation) are the same value. On the other hand, ductile materials have a failure strain that is significantly greater than the elastic strain, as shown in the figure below.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 304px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 309px 152px; text-align: center; transform-origin: 309px 152px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cimg class=\"imageNode\" style=\"vertical-align: baseline\" src=\"https://qph.cf2.quoracdn.net/main-qimg-b2693f4b9ea8430af25df920757e0b29-lq\" data-image-state=\"image-loaded\"\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 309px 10.5px; text-align: center; transform-origin: 309px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e(from quora.com)\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 309px 31.5px; text-align: left; transform-origin: 309px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eWrite a function to determine the qualitative brittleness of the material by calculating the ratio of elastic strain to failure strain. A ratio of one indicates complete brittleness, whereas a ratio close to zero indicates essentially no brittleness.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 309px 10.5px; text-align: left; transform-origin: 309px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003ePrevious problem: 2 -\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/8049-stress-strain-properties-2\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eresilience\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e. Next problem: 4 -\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"/#null\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003estrength-to-weight ratio\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function [BR] = stress_strain3(e_y,e_u)\r\n\r\nBR = 1;\r\n\r\nend\r\n","test_suite":"%% Note\r\n% The following properties are measured at room temperature and are tensile\r\n% in a single direction. Some materials, such as metals are generally\r\n% isotropic, whereas others, like composite are highly anisotropic\r\n% (different properties in different directions). Also, property values can\r\n% range depending on the material grade. Finally, thermal or environmental\r\n% changes can alter these properties, sometimes drastically.\r\n\r\n%% steel alloy (ASTM A36)\r\nS_y = 250e6; %Pa\r\nS_u = 400e6; %Pa\r\ne_y = 0.00125;\r\ne_u = 0.35;\r\nnu = 0.26;\r\nG = 79.3e9; %Pa\r\nE = 200e9; %Pa\r\ndensity = 7.85; %g/cm^3\r\nsh_exp = 0.14; %strain-hardening exponent\r\nsh_coeff = 0.463; %strain-hardening coefficient\r\nBR_corr = 0.003571;\r\nassert(abs(stress_strain3(e_y,e_u)-BR_corr)/BR_corr\u003c1e-2)\r\n\r\n%% titanium (Ti-6Al-4V)\r\nS_y = 830e6; %Pa\r\nS_u = 900e6; %Pa\r\ne_y = 0.00728;\r\ne_u = 0.14;\r\nnu = 0.342;\r\nG = 44e9; %Pa\r\nE = 114e9; %Pa\r\ndensity = 4.51; %g/cm^3\r\nsh_exp = 0.04; %strain-hardening exponent\r\nsh_coeff = 0.974; %strain-hardening coefficient\r\nBR_corr = 0.052;\r\nassert(abs(stress_strain3(e_y,e_u)-BR_corr)/BR_corr\u003c1e-2)\r\n\r\n%% Inconel 718\r\nS_y = 1172e6; %Pa\r\nS_u = 1407e6; %Pa\r\ne_y = 0.00563;\r\ne_u = 0.027;\r\nnu = 0.29;\r\nG = 11.6e9; %Pa\r\nE = 208e9; %Pa\r\ndensity = 8.19; %g/cm^3\r\nsh_exp = 0.075; %strain-hardening exponent\r\nsh_coeff = 1.845; %strain-hardening coefficient\r\nBR_corr = 0.2085;\r\nassert(abs(stress_strain3(e_y,e_u)-BR_corr)/BR_corr\u003c1e-2)\r\n\r\n%% aluminum alloy (6061-T6)%^\u0026\r\nS_y = 241e6; %Pa\r\nS_u = 300e6; %Pa\r\ne_y = 0.0035;\r\ne_u = 0.15;\r\nnu = 0.33;\r\nG = 26e9; %Pa\r\nE = 68.9e9; %Pa\r\ndensity = 2.7; %g/cm^3\r\nsh_exp = 0.042; %strain-hardening exponent\r\nsh_coeff = 0.325; %strain-hardening coefficient\r\nBR_corr = 0.02333;\r\nassert(abs(stress_strain3(e_y,e_u)-BR_corr)/BR_corr\u003c1e-2)\r\n\r\n%% copper\r\nS_y = 70e6; %Pa\r\nS_u = 220e6; %Pa\r\ne_y = 0.00054;\r\ne_u = 0.48;\r\nnu = 0.34;\r\nG = 48e9; %Pa\r\nE = 130e9; %Pa\r\ndensity = 8.92; %g/cm^3\r\nsh_exp = 0.44; %strain-hardening exponent\r\nsh_coeff = 0.304; %strain-hardening coefficient\r\nBR_corr = 0.001125;\r\nassert(abs(stress_strain3(e_y,e_u)-BR_corr)/BR_corr\u003c1e-2)\r\n\r\n%% rhenium\r\nS_y = 317e6; %Pa\r\nS_u = 1130e6; %Pa\r\ne_y = 0.000685;\r\ne_u = 0.24;\r\nnu = 0.3;\r\nG = 178e9; %Pa\r\nE = 463e9; %Pa\r\ndensity = 21.02; %g/cm^3\r\nsh_exp = 0.353; %strain-hardening exponent\r\nsh_coeff = 1.870; %strain-hardening coefficient\r\nBR_corr = 0.002854;\r\nassert(abs(stress_strain3(e_y,e_u)-BR_corr)/BR_corr\u003c1e-2)\r\n\r\n%% polymer (nylon, 6/6)\r\nS_y = 82e6; %Pa\r\nS_u = 82e6; %Pa\r\ne_y = 0.0265;\r\ne_u = 0.45;\r\nnu = 0.41;\r\nG = 2.8e9; %Pa\r\nE = 3.1e-2; %Pa\r\ndensity = 1.14; %g/cm^3\r\nBR_corr = 0.058889;\r\nassert(abs(stress_strain3(e_y,e_u)-BR_corr)/BR_corr\u003c1e-2)\r\n\r\n%% polymer (nylon, 6/6) reinforced with 45wt.% glass fiber\r\nS_y = 230e6; %Pa\r\nS_u = 230e6; %Pa\r\ne_y = 0.016;\r\ne_u = 0.016;\r\nnu = 0.35;\r\nG = 13.0e9; %Pa\r\nE = 14.5e9; %Pa\r\ndensity = 1.51; %g/cm^3\r\nBR_corr = 1.0;\r\nassert(abs(stress_strain3(e_y,e_u)-BR_corr)/BR_corr\u003c1e-2)\r\n\r\n%% diamond\r\nS_y = 1200e6; %Pa\r\nS_u = 1200e6; %Pa\r\ne_y = 0.001;\r\ne_u = 0.001;\r\nnu = 0.20;\r\nG = 478e9; %Pa\r\nE = 1200e9; %Pa\r\ndensity = 3.51; %g/cm^3\r\nBR_corr = 1.0;\r\nassert(abs(stress_strain3(e_y,e_u)-BR_corr)/BR_corr\u003c1e-2)\r\n\r\n%%\r\nind = randi(4);\r\nswitch ind\r\n\tcase 1\r\n\t\te_y = 0.00125;\r\n\t\te_u = 0.35;\r\n\t\tBR_corr = 0.003571;\r\n\tcase 2\r\n\t\te_y = 0.00054;\r\n\t\te_u = 0.48;\r\n\t\tBR_corr = 0.001125;\r\n\tcase 3\r\n\t\te_y = 0.0035;\r\n\t\te_u = 0.15;\r\n\t\tBR_corr = 0.02333;\r\n\tcase 4\r\n\t\te_y = 0.00054;\r\n\t\te_u = 0.48;\r\n\t\tBR_corr = 0.001125;\r\nend\r\nassert(abs(stress_strain3(e_y,e_u)-BR_corr)/BR_corr\u003c1e-2)\r\n\r\n%%\r\nind = randi(4);\r\nswitch ind\r\n\tcase 1\r\n\t\te_y = 0.0265;\r\n\t\te_u = 0.45;\r\n\t\tBR_corr = 0.058889;\r\n\tcase 2\r\n\t\te_y = 0.00728;\r\n\t\te_u = 0.14;\r\n\t\tBR_corr = 0.052;\r\n\tcase 3\r\n\t\te_y = 0.00563;\r\n\t\te_u = 0.027;\r\n\t\tBR_corr = 0.2085;\r\n\tcase 4\r\n\t\te_y = 0.016;\r\n\t\te_u = 0.016;\r\n\t\tBR_corr = 1.0;\r\nend\r\nassert(abs(stress_strain3(e_y,e_u)-BR_corr)/BR_corr\u003c1e-2)\r\n\r\n%%\r\nind = randi(4);\r\nswitch ind\r\n\tcase 1\r\n\t\te_y = 0.00125;\r\n\t\te_u = 0.35;\r\n\t\tBR_corr = 0.003571;\r\n\tcase 2\r\n\t\te_y = 0.00563;\r\n\t\te_u = 0.027;\r\n\t\tBR_corr = 0.2085;\r\n\tcase 3\r\n\t\te_y = 0.00728;\r\n\t\te_u = 0.14;\r\n\t\tBR_corr = 0.052;\r\n\tcase 4\r\n\t\te_y = 0.00054;\r\n\t\te_u = 0.48;\r\n\t\tBR_corr = 0.001125;\r\nend\r\nassert(abs(stress_strain3(e_y,e_u)-BR_corr)/BR_corr\u003c1e-2)\r\n","published":true,"deleted":false,"likes_count":4,"comments_count":1,"created_by":26769,"edited_by":26769,"edited_at":"2024-03-27T17:42:29.000Z","deleted_by":null,"deleted_at":null,"solvers_count":241,"test_suite_updated_at":"2015-03-30T18:54:51.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2015-03-30T18:53:23.000Z","updated_at":"2026-03-31T10:56:32.000Z","published_at":"2015-03-30T18:53:23.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eA brittle material will not exhibit a yield point. In other words, the yield point and failure point coincide. In such cases, the yield strain and failure strain (also known as ultimate strain or elongation) are the same value. On the other hand, ductile materials have a failure strain that is significantly greater than the elastic strain, as shown in the figure below.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"center\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"center\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e(from quora.com)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to determine the qualitative brittleness of the material by calculating the ratio of elastic strain to failure strain. A ratio of one indicates complete brittleness, whereas a ratio close to zero indicates essentially no brittleness.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ePrevious problem: 2 -\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/8049-stress-strain-properties-2\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eresilience\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e. Next problem: 4 -\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"\\\"\u003e\u003cw:r\u003e\u003cw:t\u003estrength-to-weight ratio\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/image\",\"target\":\"/media/image1.net/main-qimg-b2693f4b9ea8430af25df920757e0b29-lq\",\"relationshipId\":\"rId1\"}]},{\"partUri\":\"/media/image1.net/main-qimg-b2693f4b9ea8430af25df920757e0b29-lq\",\"contentType\":\"image/net/main-qimg-b2693f4b9ea8430af25df920757e0b29-lq\",\"content\":\"https://qph.cf2.quoracdn.net/main-qimg-b2693f4b9ea8430af25df920757e0b29-lq\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":8051,"title":"Stress-Strain Properties - 4","description":"A common measure of the ability of a material to carry load per unit mass is termed strength-to-weight ratio and is calculated by dividing the ultimate tensile strength of the material by its density. This property is key in weight-critical applications, such as aerospace, where many materials with high strength-to-weight ratios are used (e.g., Ni-based superalloys, Ti-based alloys, Al-based alloys, and composites).\r\n\r\nWrite a function to calculate the strength-to-weight ratio for a given material provided its ultimate tensile strength and density.\r\n\r\nPrevious problem: 3 - \u003chttp://www.mathworks.com/matlabcentral/cody/problems/8050-stress-strain-properties-3 qualitative measure of brittleness\u003e. Next problem: 5 - \u003chttp://www.mathworks.com/matlabcentral/cody/problems/8052-stress-strain-properties-5 stiffness-to-weight ratio\u003e.","description_html":"\u003cp\u003eA common measure of the ability of a material to carry load per unit mass is termed strength-to-weight ratio and is calculated by dividing the ultimate tensile strength of the material by its density. This property is key in weight-critical applications, such as aerospace, where many materials with high strength-to-weight ratios are used (e.g., Ni-based superalloys, Ti-based alloys, Al-based alloys, and composites).\u003c/p\u003e\u003cp\u003eWrite a function to calculate the strength-to-weight ratio for a given material provided its ultimate tensile strength and density.\u003c/p\u003e\u003cp\u003ePrevious problem: 3 - \u003ca href = \"http://www.mathworks.com/matlabcentral/cody/problems/8050-stress-strain-properties-3\"\u003equalitative measure of brittleness\u003c/a\u003e. Next problem: 5 - \u003ca href = \"http://www.mathworks.com/matlabcentral/cody/problems/8052-stress-strain-properties-5\"\u003estiffness-to-weight ratio\u003c/a\u003e.\u003c/p\u003e","function_template":"function [StWR] = stress_strain4(S_u,density)\r\n\r\nStWR = 1;\r\n\r\nend\r\n","test_suite":"%% Note\r\n% The following properties are measured at room temperature and are tensile\r\n% in a single direction. Some materials, such as metals are generally\r\n% isotropic, whereas others, like composite are highly anisotropic\r\n% (different properties in different directions). Also, property values can\r\n% range depending on the material grade. Finally, thermal or environmental\r\n% changes can alter these properties, sometimes drastically.\r\n\r\n%% steel alloy (ASTM A36)\r\nS_y = 250e6; %Pa\r\nS_u = 400e6; %Pa\r\ne_y = 0.00125;\r\ne_u = 0.35;\r\nnu = 0.26;\r\nG = 79.3e9; %Pa\r\nE = 200e9; %Pa\r\ndensity = 7.85; %g/cm^3\r\nsh_exp = 0.14; %strain-hardening exponent\r\nsh_coeff = 0.463; %strain-hardening coefficient\r\nStWR_corr = 5.096e7;\r\nassert(abs(stress_strain4(S_u,density)-StWR_corr)/StWR_corr\u003c1e-2)\r\n\r\n%% titanium (Ti-6Al-4V)\r\nS_y = 830e6; %Pa\r\nS_u = 900e6; %Pa\r\ne_y = 0.00728;\r\ne_u = 0.14;\r\nnu = 0.342;\r\nG = 44e9; %Pa\r\nE = 114e9; %Pa\r\ndensity = 4.51; %g/cm^3\r\nsh_exp = 0.04; %strain-hardening exponent\r\nsh_coeff = 0.974; %strain-hardening coefficient\r\nStWR_corr = 19.96e7;\r\nassert(abs(stress_strain4(S_u,density)-StWR_corr)/StWR_corr\u003c1e-2)\r\n\r\n%% Inconel 718\r\nS_y = 1172e6; %Pa\r\nS_u = 1407e6; %Pa\r\ne_y = 0.00563;\r\ne_u = 0.027;\r\nnu = 0.29;\r\nG = 11.6e9; %Pa\r\nE = 208e9; %Pa\r\ndensity = 8.19; %g/cm^3\r\nsh_exp = 0.075; %strain-hardening exponent\r\nsh_coeff = 1.845; %strain-hardening coefficient\r\nStWR_corr = 17.18e7;\r\nassert(abs(stress_strain4(S_u,density)-StWR_corr)/StWR_corr\u003c1e-2)\r\n\r\n%% aluminum alloy (6061-T6)%^\u0026\r\nS_y = 241e6; %Pa\r\nS_u = 300e6; %Pa\r\ne_y = 0.0035;\r\ne_u = 0.15;\r\nnu = 0.33;\r\nG = 26e9; %Pa\r\nE = 68.9e9; %Pa\r\ndensity = 2.7; %g/cm^3\r\nsh_exp = 0.042; %strain-hardening exponent\r\nsh_coeff = 0.325; %strain-hardening coefficient\r\nStWR_corr = 11.11e7;\r\nassert(abs(stress_strain4(S_u,density)-StWR_corr)/StWR_corr\u003c1e-2)\r\n\r\n%% copper\r\nS_y = 70e6; %Pa\r\nS_u = 220e6; %Pa\r\ne_y = 0.00054;\r\ne_u = 0.48;\r\nnu = 0.34;\r\nG = 48e9; %Pa\r\nE = 130e9; %Pa\r\ndensity = 8.92; %g/cm^3\r\nsh_exp = 0.44; %strain-hardening exponent\r\nsh_coeff = 0.304; %strain-hardening coefficient\r\nStWR_corr = 2.466e7;\r\nassert(abs(stress_strain4(S_u,density)-StWR_corr)/StWR_corr\u003c1e-2)\r\n\r\n%% rhenium\r\nS_y = 317e6; %Pa\r\nS_u = 1130e6; %Pa\r\ne_y = 0.000685;\r\ne_u = 0.24;\r\nnu = 0.3;\r\nG = 178e9; %Pa\r\nE = 463e9; %Pa\r\ndensity = 21.02; %g/cm^3\r\nsh_exp = 0.353; %strain-hardening exponent\r\nsh_coeff = 1.870; %strain-hardening coefficient\r\nStWR_corr = 5.376e7;\r\nassert(abs(stress_strain4(S_u,density)-StWR_corr)/StWR_corr\u003c1e-2)\r\n\r\n%% polymer (nylon, 6/6)\r\nS_y = 82e6; %Pa\r\nS_u = 82e6; %Pa\r\ne_y = 0.0265;\r\ne_u = 0.45;\r\nnu = 0.41;\r\nG = 2.8e9; %Pa\r\nE = 3.1e-2; %Pa\r\ndensity = 1.14; %g/cm^3\r\nStWR_corr = 7.193e7;\r\nassert(abs(stress_strain4(S_u,density)-StWR_corr)/StWR_corr\u003c1e-2)\r\n\r\n%% polymer (nylon, 6/6) reinforced with 45wt.% glass fiber\r\nS_y = 230e6; %Pa\r\nS_u = 230e6; %Pa\r\ne_y = 0.016;\r\ne_u = 0.016;\r\nnu = 0.35;\r\nG = 13.0e9; %Pa\r\nE = 14.5e9; %Pa\r\ndensity = 1.51; %g/cm^3\r\nStWR_corr = 15.23e7;\r\nassert(abs(stress_strain4(S_u,density)-StWR_corr)/StWR_corr\u003c1e-2)\r\n\r\n%% diamond\r\nS_y = 1200e6; %Pa\r\nS_u = 1200e6; %Pa\r\ne_y = 0.001;\r\ne_u = 0.001;\r\nnu = 0.20;\r\nG = 478e9; %Pa\r\nE = 1200e9; %Pa\r\ndensity = 3.51; %g/cm^3\r\nStWR_corr = 34.19e7;\r\nassert(abs(stress_strain4(S_u,density)-StWR_corr)/StWR_corr\u003c1e-2)\r\n\r\n%%\r\nind = randi(4);\r\nswitch ind\r\n\tcase 1\r\n\t\tS_u = 400e6; %Pa\r\n\t\tdensity = 7.85; %g/cm^3\r\n\t\tStWR_corr = 5.096e7;\r\n\tcase 2\r\n\t\tS_u = 230e6; %Pa\r\n\t\tdensity = 1.51; %g/cm^3\r\n\t\tStWR_corr = 15.23e7;\r\n\tcase 3\r\n\t\tS_u = 1130e6; %Pa\r\n\t\tdensity = 21.02; %g/cm^3\r\n\t\tStWR_corr = 5.376e7;\r\n\tcase 4\r\n\t\tS_u = 1200e6; %Pa\r\n\t\tdensity = 3.51; %g/cm^3\r\n\t\tStWR_corr = 34.19e7;\r\nend\r\nassert(abs(stress_strain4(S_u,density)-StWR_corr)/StWR_corr\u003c1e-2)\r\n\r\n%%\r\nind = randi(4);\r\nswitch ind\r\n\tcase 1\r\n\t\tS_u = 300e6; %Pa\r\n\t\tdensity = 2.7; %g/cm^3\r\n\t\tStWR_corr = 11.11e7;\r\n\tcase 2\r\n\t\tS_u = 900e6; %Pa\r\n\t\tdensity = 4.51; %g/cm^3\r\n\t\tStWR_corr = 19.96e7;\r\n\tcase 3\r\n\t\tS_u = 220e6; %Pa\r\n\t\tdensity = 8.92; %g/cm^3\r\n\t\tStWR_corr = 2.466e7;\r\n\tcase 4\r\n\t\tS_u = 230e6; %Pa\r\n\t\tdensity = 1.51; %g/cm^3\r\n\t\tStWR_corr = 15.23e7;\r\nend\r\nassert(abs(stress_strain4(S_u,density)-StWR_corr)/StWR_corr\u003c1e-2)\r\n\r\n%%\r\nind = randi(4);\r\nswitch ind\r\n\tcase 1\r\n\t\tS_u = 300e6; %Pa\r\n\t\tdensity = 2.7; %g/cm^3\r\n\t\tStWR_corr = 11.11e7;\r\n\tcase 2\r\n\t\tS_u = 1200e6; %Pa\r\n\t\tdensity = 3.51; %g/cm^3\r\n\t\tStWR_corr = 34.19e7;\r\n\tcase 3\r\n\t\tS_u = 82e6; %Pa\r\n\t\tdensity = 1.14; %g/cm^3\r\n\t\tStWR_corr = 7.193e7;\r\n\tcase 4\r\n\t\tS_u = 900e6; %Pa\r\n\t\tdensity = 4.51; %g/cm^3\r\n\t\tStWR_corr = 19.96e7;\r\nend\r\nassert(abs(stress_strain4(S_u,density)-StWR_corr)/StWR_corr\u003c1e-2)\r\n","published":true,"deleted":false,"likes_count":4,"comments_count":0,"created_by":26769,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":222,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2015-03-30T19:24:41.000Z","updated_at":"2026-03-10T20:20:32.000Z","published_at":"2015-03-30T19:24:41.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eA common measure of the ability of a material to carry load per unit mass is termed strength-to-weight ratio and is calculated by dividing the ultimate tensile strength of the material by its density. This property is key in weight-critical applications, such as aerospace, where many materials with high strength-to-weight ratios are used (e.g., Ni-based superalloys, Ti-based alloys, Al-based alloys, and composites).\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to calculate the strength-to-weight ratio for a given material provided its ultimate tensile strength and density.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ePrevious problem: 3 -\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://www.mathworks.com/matlabcentral/cody/problems/8050-stress-strain-properties-3\\\"\u003e\u003cw:r\u003e\u003cw:t\u003equalitative measure of brittleness\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e. Next problem: 5 -\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://www.mathworks.com/matlabcentral/cody/problems/8052-stress-strain-properties-5\\\"\u003e\u003cw:r\u003e\u003cw:t\u003estiffness-to-weight ratio\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":8052,"title":"Stress-Strain Properties - 5","description":"Similar to the previous problem, materials may be characterized by their stiffness-to-weight ratio, which is the elastic modulus divided by density. Write a function to calculate this ratio for a material provided its elastic modulus and density.\r\n\r\nPrevious problem: 4 - \u003chttp://www.mathworks.com/matlabcentral/cody/problems/8051-stress-strain-properties-4 strength-to-weight ratio\u003e. Next problem: 6 - \u003chttp://www.mathworks.com/matlabcentral/cody/problems/8053-stress-strain-properties-6 absorbed strain energy\u003e.","description_html":"\u003cp\u003eSimilar to the previous problem, materials may be characterized by their stiffness-to-weight ratio, which is the elastic modulus divided by density. Write a function to calculate this ratio for a material provided its elastic modulus and density.\u003c/p\u003e\u003cp\u003ePrevious problem: 4 - \u003ca href = \"http://www.mathworks.com/matlabcentral/cody/problems/8051-stress-strain-properties-4\"\u003estrength-to-weight ratio\u003c/a\u003e. Next problem: 6 - \u003ca href = \"http://www.mathworks.com/matlabcentral/cody/problems/8053-stress-strain-properties-6\"\u003eabsorbed strain energy\u003c/a\u003e.\u003c/p\u003e","function_template":"function [EtWR] = stress_strain5(E,density)\r\n\r\nEtWR = 1\r\n\r\nend\r\n","test_suite":"%% Note\r\n% The following properties are measured at room temperature and are tensile\r\n% in a single direction. Some materials, such as metals are generally\r\n% isotropic, whereas others, like composite are highly anisotropic\r\n% (different properties in different directions). Also, property values can\r\n% range depending on the material grade. Finally, thermal or environmental\r\n% changes can alter these properties, sometimes drastically.\r\n\r\n%% steel alloy (ASTM A36)\r\nS_y = 250e6; %Pa\r\nS_u = 400e6; %Pa\r\ne_y = 0.00125;\r\ne_u = 0.35;\r\nnu = 0.26;\r\nG = 79.3e9; %Pa\r\nE = 200e9; %Pa\r\ndensity = 7.85; %g/cm^3\r\nsh_exp = 0.14; %strain-hardening exponent\r\nsh_coeff = 0.463; %strain-hardening coefficient\r\nEtWR_corr = 2.548e10;\r\nassert(abs(stress_strain5(E,density)-EtWR_corr)/EtWR_corr\u003c1e-2)\r\n\r\n%% titanium (Ti-6Al-4V)\r\nS_y = 830e6; %Pa\r\nS_u = 900e6; %Pa\r\ne_y = 0.00728;\r\ne_u = 0.14;\r\nnu = 0.342;\r\nG = 44e9; %Pa\r\nE = 114e9; %Pa\r\ndensity = 4.51; %g/cm^3\r\nsh_exp = 0.04; %strain-hardening exponent\r\nsh_coeff = 0.974; %strain-hardening coefficient\r\nEtWR_corr = 2.528e10;\r\nassert(abs(stress_strain5(E,density)-EtWR_corr)/EtWR_corr\u003c1e-2)\r\n\r\n%% Inconel 718\r\nS_y = 1172e6; %Pa\r\nS_u = 1407e6; %Pa\r\ne_y = 0.00563;\r\ne_u = 0.027;\r\nnu = 0.29;\r\nG = 11.6e9; %Pa\r\nE = 208e9; %Pa\r\ndensity = 8.19; %g/cm^3\r\nsh_exp = 0.075; %strain-hardening exponent\r\nsh_coeff = 1.845; %strain-hardening coefficient\r\nEtWR_corr = 2.540e10;\r\nassert(abs(stress_strain5(E,density)-EtWR_corr)/EtWR_corr\u003c1e-2)\r\n\r\n%% aluminum alloy (6061-T6)%^\u0026\r\nS_y = 241e6; %Pa\r\nS_u = 300e6; %Pa\r\ne_y = 0.0035;\r\ne_u = 0.15;\r\nnu = 0.33;\r\nG = 26e9; %Pa\r\nE = 68.9e9; %Pa\r\ndensity = 2.7; %g/cm^3\r\nsh_exp = 0.042; %strain-hardening exponent\r\nsh_coeff = 0.325; %strain-hardening coefficient\r\nEtWR_corr = 2.552e10;\r\nassert(abs(stress_strain5(E,density)-EtWR_corr)/EtWR_corr\u003c1e-2)\r\n\r\n%% copper\r\nS_y = 70e6; %Pa\r\nS_u = 220e6; %Pa\r\ne_y = 0.00054;\r\ne_u = 0.48;\r\nnu = 0.34;\r\nG = 48e9; %Pa\r\nE = 130e9; %Pa\r\ndensity = 8.92; %g/cm^3\r\nsh_exp = 0.44; %strain-hardening exponent\r\nsh_coeff = 0.304; %strain-hardening coefficient\r\nEtWR_corr = 1.457e10;\r\nassert(abs(stress_strain5(E,density)-EtWR_corr)/EtWR_corr\u003c1e-2)\r\n\r\n%% rhenium\r\nS_y = 317e6; %Pa\r\nS_u = 1130e6; %Pa\r\ne_y = 0.000685;\r\ne_u = 0.24;\r\nnu = 0.3;\r\nG = 178e9; %Pa\r\nE = 463e9; %Pa\r\ndensity = 21.02; %g/cm^3\r\nsh_exp = 0.353; %strain-hardening exponent\r\nsh_coeff = 1.870; %strain-hardening coefficient\r\nEtWR_corr = 2.203e10;\r\nassert(abs(stress_strain5(E,density)-EtWR_corr)/EtWR_corr\u003c1e-2)\r\n\r\n%% polymer (nylon, 6/6)\r\nS_y = 82e6; %Pa\r\nS_u = 82e6; %Pa\r\ne_y = 0.0265;\r\ne_u = 0.45;\r\nnu = 0.41;\r\nG = 2.8e9; %Pa\r\nE = 3.1e9; %Pa\r\ndensity = 1.14; %g/cm^3\r\nEtWR_corr = 0.272e10;\r\nassert(abs(stress_strain5(E,density)-EtWR_corr)/EtWR_corr\u003c1e-2)\r\n\r\n%% polymer (nylon, 6/6) reinforced with 45wt.% glass fiber\r\nS_y = 230e6; %Pa\r\nS_u = 230e6; %Pa\r\ne_y = 0.016;\r\ne_u = 0.016;\r\nnu = 0.35;\r\nG = 13.0e9; %Pa\r\nE = 14.5e9; %Pa\r\ndensity = 1.51; %g/cm^3\r\nEtWR_corr = 0.960e10;\r\nassert(abs(stress_strain5(E,density)-EtWR_corr)/EtWR_corr\u003c1e-2)\r\n\r\n%% diamond\r\nS_y = 1200e6; %Pa\r\nS_u = 1200e6; %Pa\r\ne_y = 0.001;\r\ne_u = 0.001;\r\nnu = 0.20;\r\nG = 478e9; %Pa\r\nE = 1200e9; %Pa\r\ndensity = 3.51; %g/cm^3\r\nEtWR_corr = 34.19e10;\r\nassert(abs(stress_strain5(E,density)-EtWR_corr)/EtWR_corr\u003c1e-2)\r\n\r\n%%\r\nind = randi(4);\r\nswitch ind\r\n\tcase 1\r\n\t\tE = 114e9; %Pa\r\n\t\tdensity = 4.51; %g/cm^3\r\n\t\tEtWR_corr = 2.528e10;\r\n\tcase 2\r\n\t\tE = 68.9e9; %Pa\r\n\t\tdensity = 2.7; %g/cm^3\r\n\t\tEtWR_corr = 2.552e10;\r\n\tcase 3\r\n\t\tE = 200e9; %Pa\r\n\t\tdensity = 7.85; %g/cm^3\r\n\t\tEtWR_corr = 2.548e10;\r\n\tcase 4\r\n\t\tE = 1200e9; %Pa\r\n\t\tdensity = 3.51; %g/cm^3\r\n\t\tEtWR_corr = 34.19e10;\r\nend\r\nassert(abs(stress_strain5(E,density)-EtWR_corr)/EtWR_corr\u003c1e-2)\r\n\r\n%%\r\nind = randi(4);\r\nswitch ind\r\n\tcase 1\r\n\t\tE = 68.9e9; %Pa\r\n\t\tdensity = 2.7; %g/cm^3\r\n\t\tEtWR_corr = 2.552e10;\r\n\tcase 2\r\n\t\tE = 3.1e9; %Pa\r\n\t\tdensity = 1.14; %g/cm^3\r\n\t\tEtWR_corr = 0.272e10;\r\n\tcase 3\r\n\t\tE = 14.5e9; %Pa\r\n\t\tdensity = 1.51; %g/cm^3\r\n\t\tEtWR_corr = 0.960e10;\r\n\tcase 4\r\n\t\tE = 208e9; %Pa\r\n\t\tdensity = 8.19; %g/cm^3\r\n\t\tEtWR_corr = 2.540e10;\r\nend\r\nassert(abs(stress_strain5(E,density)-EtWR_corr)/EtWR_corr\u003c1e-2)\r\n\r\n%%\r\nind = randi(4);\r\nswitch ind\r\n\tcase 1\r\n\t\tE = 208e9; %Pa\r\n\t\tdensity = 8.19; %g/cm^3\r\n\t\tEtWR_corr = 2.540e10;\r\n\tcase 2\r\n\t\tE = 463e9; %Pa\r\n\t\tdensity = 21.02; %g/cm^3\r\n\t\tEtWR_corr = 2.203e10;\r\n\tcase 3\r\n\t\tE = 130e9; %Pa\r\n\t\tdensity = 8.92; %g/cm^3\r\n\t\tEtWR_corr = 1.457e10;\r\n\tcase 4\r\n\t\tE = 3.1e9; %Pa\r\n\t\tdensity = 1.14; %g/cm^3\r\n\t\tEtWR_corr = 0.272e10;\r\nend\r\nassert(abs(stress_strain5(E,density)-EtWR_corr)/EtWR_corr\u003c1e-2)\r\n","published":true,"deleted":false,"likes_count":2,"comments_count":0,"created_by":26769,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":212,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2015-03-30T19:40:12.000Z","updated_at":"2026-03-10T20:42:38.000Z","published_at":"2015-03-30T19:40:12.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eSimilar to the previous problem, materials may be characterized by their stiffness-to-weight ratio, which is the elastic modulus divided by density. Write a function to calculate this ratio for a material provided its elastic modulus and density.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ePrevious problem: 4 -\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://www.mathworks.com/matlabcentral/cody/problems/8051-stress-strain-properties-4\\\"\u003e\u003cw:r\u003e\u003cw:t\u003estrength-to-weight ratio\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e. Next problem: 6 -\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://www.mathworks.com/matlabcentral/cody/problems/8053-stress-strain-properties-6\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eabsorbed strain energy\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":8053,"title":"Stress-Strain Properties - 6","description":"The total energy absorbed by a material up to failure in a tensile test is termed the absorbed strain energy. With respect to the figure below, it is the total area of the elastic and plastic regions and can be calculated by integrating the stress-strain curve. As a first approximation, many stress-strain responses can be approximated by:\r\n\r\nwhere K is a strength coefficient, eps_p is the plastic strain, and n is the hardening exponent. Stress as a function of strain can be calculated by creating a strain vector from zero to the ultimate strain and integrating the stress values in that vector.\r\n\r\n(from quora.com)\r\nWrite a function to return the absorbed strain energy for a material provided K and n. If the material does not strain harden, then K and n will be set equal to zero. In these cases, the absorbed strain energy is equal to the resilience (triangular area up to yield point) and any absorbed plastic energy, if applicable, which can be approximated by a rectangle from the yield point to the failure point with those stresses being equal. If the ultimate strain equals the yield strain, that rectangular area is zero.\r\nPrevious problem: 5 - stiffness-to-weight ratio. Next problem: 7 - toughness.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.440001px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none; white-space: normal; \"\u003e\u003cdiv style=\"block-size: 702px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 332px 351px; transform-origin: 332px 351px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 84px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 309px 42px; text-align: left; transform-origin: 309px 42px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eThe total energy absorbed by a material up to failure in a tensile test is termed the absorbed strain energy. With respect to the figure below, it is the total area of the elastic and plastic regions and can be calculated by integrating the stress-strain curve. As a first approximation, many stress-strain responses can be approximated by:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 29px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 309px 14.5px; text-align: center; transform-origin: 309px 14.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cimg class=\"imageNode\" width=\"73\" height=\"23\" style=\"vertical-align: baseline;width: 73px;height: 23px\" src=\"data:image/png;base64,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\" data-image-state=\"image-loaded\"\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 309px 31.5px; text-align: left; transform-origin: 309px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003ewhere K is a strength coefficient, eps_p is the plastic strain, and n is the hardening exponent. Stress as a function of strain can be calculated by creating a strain vector from zero to the ultimate strain and integrating the stress values in that vector.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 304px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 309px 152px; text-align: center; transform-origin: 309px 152px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cimg class=\"imageNode\" style=\"vertical-align: baseline\" src=\"https://qph.cf2.quoracdn.net/main-qimg-b2693f4b9ea8430af25df920757e0b29-lq\" data-image-state=\"image-loaded\"\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 309px 10.5px; text-align: center; transform-origin: 309px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e(from quora.com)\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 126px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 309px 63px; text-align: left; transform-origin: 309px 63px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eWrite a function to return the absorbed strain energy for a material provided K and n. If the material does not strain harden, then K and n will be set equal to zero. In these cases, the absorbed strain energy is equal to the resilience (triangular area up to yield point) and any absorbed plastic energy, if applicable, which can be approximated by a rectangle from the yield point to the failure point with those stresses being equal. If the ultimate strain equals the yield strain, that rectangular area is zero.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 309px 10.5px; text-align: left; transform-origin: 309px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003ePrevious problem: 5 -\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"/#null\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003estiffness-to-weight ratio\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e. Next problem: 7 -\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"/#null\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003etoughness\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function [ASE] = stress_strain6(e_u,sh_exp,sh_coeff,S_y,e_y)\r\n\r\nASE = 1;\r\n\r\nend\r\n","test_suite":"%% Note\r\n% The following properties are measured at room temperature and are tensile\r\n% in a single direction. Some materials, such as metals are generally\r\n% isotropic, whereas others, like composite are highly anisotropic\r\n% (different properties in different directions). Also, property values can\r\n% range depending on the material grade. Finally, thermal or environmental\r\n% changes can alter these properties, sometimes drastically.\r\n\r\n%% steel alloy (ASTM A36)\r\nS_y = 250e6; %Pa\r\nS_u = 400e6; %Pa\r\ne_y = 0.00125;\r\ne_u = 0.35;\r\nnu = 0.26;\r\nG = 79.3e9; %Pa\r\nE = 200e9; %Pa\r\ndensity = 7.85; %g/cm^3\r\nsh_exp = 0.14; %strain-hardening exponent\r\nsh_coeff = 463e6; %strain-hardening coefficient\r\nASE_corr = 12.28e7;\r\nassert(abs(stress_strain6(e_u,sh_exp,sh_coeff,S_y,e_y)-ASE_corr)/ASE_corr\u003c1e-2)\r\n\r\n%% titanium (Ti-6Al-4V)\r\nS_y = 830e6; %Pa\r\nS_u = 900e6; %Pa\r\ne_y = 0.00728;\r\ne_u = 0.14;\r\nnu = 0.342;\r\nG = 44e9; %Pa\r\nE = 114e9; %Pa\r\ndensity = 4.51; %g/cm^3\r\nsh_exp = 0.04; %strain-hardening exponent\r\nsh_coeff = 974e6; %strain-hardening coefficient\r\nASE_corr = 12.12e7;\r\nassert(abs(stress_strain6(e_u,sh_exp,sh_coeff,S_y,e_y)-ASE_corr)/ASE_corr\u003c1e-2)\r\n\r\n%% Inconel 718\r\nS_y = 1172e6; %Pa\r\nS_u = 1407e6; %Pa\r\ne_y = 0.00563;\r\ne_u = 0.027;\r\nnu = 0.29;\r\nG = 11.6e9; %Pa\r\nE = 208e9; %Pa\r\ndensity = 8.19; %g/cm^3\r\nsh_exp = 0.075; %strain-hardening exponent\r\nsh_coeff = 1845e6; %strain-hardening coefficient\r\nASE_corr = 3.535e7;\r\nassert(abs(stress_strain6(e_u,sh_exp,sh_coeff,S_y,e_y)-ASE_corr)/ASE_corr\u003c1e-2)\r\n\r\n%% aluminum alloy (6061-T6)%^\u0026\r\nS_y = 241e6; %Pa\r\nS_u = 300e6; %Pa\r\ne_y = 0.0035;\r\ne_u = 0.15;\r\nnu = 0.33;\r\nG = 26e9; %Pa\r\nE = 68.9e9; %Pa\r\ndensity = 2.7; %g/cm^3\r\nsh_exp = 0.042; %strain-hardening exponent\r\nsh_coeff = 325e6; %strain-hardening coefficient\r\nASE_corr = 4.321e7;\r\nassert(abs(stress_strain6(e_u,sh_exp,sh_coeff,S_y,e_y)-ASE_corr)/ASE_corr\u003c1e-2)\r\n\r\n%% copper\r\nS_y = 70e6; %Pa\r\nS_u = 220e6; %Pa\r\ne_y = 0.00054;\r\ne_u = 0.48;\r\nnu = 0.34;\r\nG = 48e9; %Pa\r\nE = 130e9; %Pa\r\ndensity = 8.92; %g/cm^3\r\nsh_exp = 0.44; %strain-hardening exponent\r\nsh_coeff = 304e6; %strain-hardening coefficient\r\nASE_corr = 7.342e7;\r\nassert(abs(stress_strain6(e_u,sh_exp,sh_coeff,S_y,e_y)-ASE_corr)/ASE_corr\u003c1e-2)\r\n\r\n%% rhenium\r\nS_y = 317e6; %Pa\r\nS_u = 1130e6; %Pa\r\ne_y = 0.000685;\r\ne_u = 0.24;\r\nnu = 0.3;\r\nG = 178e9; %Pa\r\nE = 463e9; %Pa\r\ndensity = 21.02; %g/cm^3\r\nsh_exp = 0.353; %strain-hardening exponent\r\nsh_coeff = 1870e6; %strain-hardening coefficient\r\nASE_corr = 20.06e7;\r\nassert(abs(stress_strain6(e_u,sh_exp,sh_coeff,S_y,e_y)-ASE_corr)/ASE_corr\u003c1e-2)\r\n\r\n%% polymer (nylon, 6/6)\r\nS_y = 82e6; %Pa\r\nS_u = 82e6; %Pa\r\ne_y = 0.0265;\r\ne_u = 0.45;\r\nnu = 0.41;\r\nG = 2.8e9; %Pa\r\nE = 3.1e9; %Pa\r\ndensity = 1.14; %g/cm^3\r\nsh_exp = 0; %strain-hardening exponent\r\nsh_coeff = 0; %strain-hardening coefficient\r\nASE_corr = 3.581e7;\r\nassert(abs(stress_strain6(e_u,sh_exp,sh_coeff,S_y,e_y)-ASE_corr)/ASE_corr\u003c1e-2)\r\n\r\n%% polymer (nylon, 6/6) reinforced with 45wt.% glass fiber\r\nS_y = 230e6; %Pa\r\nS_u = 230e6; %Pa\r\ne_y = 0.016;\r\ne_u = 0.016;\r\nnu = 0.35;\r\nG = 13.0e9; %Pa\r\nE = 14.5e9; %Pa\r\ndensity = 1.51; %g/cm^3\r\nsh_exp = 0; %strain-hardening exponent\r\nsh_coeff = 0; %strain-hardening coefficient\r\nASE_corr = 0.184e7;\r\nassert(abs(stress_strain6(e_u,sh_exp,sh_coeff,S_y,e_y)-ASE_corr)/ASE_corr\u003c1e-2)\r\n\r\n%% diamond\r\nS_y = 1200e6; %Pa\r\nS_u = 1200e6; %Pa\r\ne_y = 0.001;\r\ne_u = 0.001;\r\nnu = 0.20;\r\nG = 478e9; %Pa\r\nE = 1200e9; %Pa\r\ndensity = 3.51; %g/cm^3\r\nsh_exp = 0; %strain-hardening exponent\r\nsh_coeff = 0; %strain-hardening coefficient\r\nASE_corr = 0.06e7;\r\nassert(abs(stress_strain6(e_u,sh_exp,sh_coeff,S_y,e_y)-ASE_corr)/ASE_corr\u003c1e-2)\r\n\r\n%%\r\nind = randi(4);\r\nswitch ind\r\n\tcase 1\r\n\t\tS_y = 250e6; %Pa\r\n\t\te_y = 0.00125;\r\n\t\te_u = 0.35;\r\n\t\tsh_exp = 0.14; %strain-hardening exponent\r\n\t\tsh_coeff = 463e6; %strain-hardening coefficient\r\n\t\tASE_corr = 12.28e7;\r\n\tcase 2\r\n\t\tS_y = 82e6; %Pa\r\n\t\te_y = 0.0265;\r\n\t\te_u = 0.45;\r\n\t\tsh_exp = 0; %strain-hardening exponent\r\n\t\tsh_coeff = 0; %strain-hardening coefficient\r\n\t\tASE_corr = 3.581e7;\r\n\tcase 3\r\n\t\tS_y = 241e6; %Pa\r\n\t\te_y = 0.0035;\r\n\t\te_u = 0.15;\r\n\t\tsh_exp = 0.042; %strain-hardening exponent\r\n\t\tsh_coeff = 325e6; %strain-hardening coefficient\r\n\t\tASE_corr = 4.321e7;\r\n\tcase 4\r\n\t\tS_y = 317e6; %Pa\r\n\t\te_y = 0.000685;\r\n\t\te_u = 0.24;\r\n\t\tsh_exp = 0.353; %strain-hardening exponent\r\n\t\tsh_coeff = 1870e6; %strain-hardening coefficient\r\n\t\tASE_corr = 20.06e7;\r\nend\r\nassert(abs(stress_strain6(e_u,sh_exp,sh_coeff,S_y,e_y)-ASE_corr)/ASE_corr\u003c1e-2)\r\n\r\n%%\r\nind = randi(4);\r\nswitch ind\r\n\tcase 1\r\n\t\tS_y = 830e6; %Pa\r\n\t\te_y = 0.00728;\r\n\t\te_u = 0.14;\r\n\t\tsh_exp = 0.04; %strain-hardening exponent\r\n\t\tsh_coeff = 974e6; %strain-hardening coefficient\r\n\t\tASE_corr = 12.12e7;\r\n\tcase 2\r\n\t\tS_y = 241e6; %Pa\r\n\t\te_y = 0.0035;\r\n\t\te_u = 0.15;\r\n\t\tsh_exp = 0.042; %strain-hardening exponent\r\n\t\tsh_coeff = 325e6; %strain-hardening coefficient\r\n\t\tASE_corr = 4.321e7;\r\n\tcase 3\r\n\t\tS_y = 250e6; %Pa\r\n\t\te_y = 0.00125;\r\n\t\te_u = 0.35;\r\n\t\tsh_exp = 0.14; %strain-hardening exponent\r\n\t\tsh_coeff = 463e6; %strain-hardening coefficient\r\n\t\tASE_corr = 12.28e7;\r\n\tcase 4\r\n\t\tS_y = 70e6; %Pa\r\n\t\te_y = 0.00054;\r\n\t\te_u = 0.48;\r\n\t\tsh_exp = 0.44; %strain-hardening exponent\r\n\t\tsh_coeff = 304e6; %strain-hardening coefficient\r\n\t\tASE_corr = 7.342e7;\r\nend\r\nassert(abs(stress_strain6(e_u,sh_exp,sh_coeff,S_y,e_y)-ASE_corr)/ASE_corr\u003c1e-2)\r\n\r\n%%\r\nind = randi(4);\r\nswitch ind\r\n\tcase 1\r\n\t\tS_y = 1200e6; %Pa\r\n\t\te_y = 0.001;\r\n\t\te_u = 0.001;\r\n\t\tsh_exp = 0; %strain-hardening exponent\r\n\t\tsh_coeff = 0; %strain-hardening coefficient\r\n\t\tASE_corr = 0.06e7;\r\n\tcase 2\r\n\t\tS_y = 250e6; %Pa\r\n\t\te_y = 0.00125;\r\n\t\te_u = 0.35;\r\n\t\tsh_exp = 0.14; %strain-hardening exponent\r\n\t\tsh_coeff = 463e6; %strain-hardening coefficient\r\n\t\tASE_corr = 12.28e7;\r\n\tcase 3\r\n\t\tS_y = 230e6; %Pa\r\n\t\te_y = 0.016;\r\n\t\te_u = 0.016;\r\n\t\tsh_exp = 0; %strain-hardening exponent\r\n\t\tsh_coeff = 0; %strain-hardening coefficient\r\n\t\tASE_corr = 0.184e7;\r\n\tcase 4\r\n\t\tS_y = 1172e6; %Pa\r\n\t\te_y = 0.00563;\r\n\t\te_u = 0.027;\r\n\t\tsh_exp = 0.075; %strain-hardening exponent\r\n\t\tsh_coeff = 1845e6; %strain-hardening coefficient\r\n\t\tASE_corr = 3.535e7;\r\nend\r\nassert(abs(stress_strain6(e_u,sh_exp,sh_coeff,S_y,e_y)-ASE_corr)/ASE_corr\u003c1e-2)\r\n","published":true,"deleted":false,"likes_count":0,"comments_count":7,"created_by":26769,"edited_by":26769,"edited_at":"2024-03-27T17:39:31.000Z","deleted_by":null,"deleted_at":null,"solvers_count":92,"test_suite_updated_at":"2015-03-30T21:25:46.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2015-03-30T20:58:35.000Z","updated_at":"2026-02-19T09:44:40.000Z","published_at":"2015-03-30T21:25:46.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe total energy absorbed by a material up to failure in a tensile test is termed the absorbed strain energy. With respect to the figure below, it is the total area of the elastic and plastic regions and can be calculated by integrating the stress-strain curve. As a first approximation, many stress-strain responses can be approximated by:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"center\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"23\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"73\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ewhere K is a strength coefficient, eps_p is the plastic strain, and n is the hardening exponent. Stress as a function of strain can be calculated by creating a strain vector from zero to the ultimate strain and integrating the stress values in that vector.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"center\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"298\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"420\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId2\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"center\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e(from quora.com)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to return the absorbed strain energy for a material provided K and n. If the material does not strain harden, then K and n will be set equal to zero. In these cases, the absorbed strain energy is equal to the resilience (triangular area up to yield point) and any absorbed plastic energy, if applicable, which can be approximated by a rectangle from the yield point to the failure point with those stresses being equal. If the ultimate strain equals the yield strain, that rectangular area is zero.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ePrevious problem: 5 -\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"\\\"\u003e\u003cw:r\u003e\u003cw:t\u003estiffness-to-weight ratio\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e. Next problem: 7 -\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"\\\"\u003e\u003cw:r\u003e\u003cw:t\u003etoughness\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/image\",\"target\":\"/media/image1.png\",\"relationshipId\":\"rId1\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/image\",\"target\":\"/media/image1.net/main-qimg-b2693f4b9ea8430af25df920757e0b29-lq\",\"relationshipId\":\"rId2\"}]},{\"partUri\":\"/media/image1.png\",\"contentType\":\"image/png\",\"content\":\"data:image/png;base64,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\",\"relationship\":null},{\"partUri\":\"/media/image1.net/main-qimg-b2693f4b9ea8430af25df920757e0b29-lq\",\"contentType\":\"image/net/main-qimg-b2693f4b9ea8430af25df920757e0b29-lq\",\"content\":\"https://qph.cf2.quoracdn.net/main-qimg-b2693f4b9ea8430af25df920757e0b29-lq\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":8054,"title":"Stress-Strain Properties - 7","description":"The toughness of a material is technically defined as the plastic strain energy absorbed by the material (the plastic region in the figure below). Practically speaking, it's a measure of how much deformation a material can undergo (or energy it can absorb) before failure.\r\n\r\nWrite a function to calculate the toughness of a material—the absorbed strain energy minus the resilience. This can be accomplished by combining the code written in problem 2 (resilience) and problem 6 (absorbed strain energy). Also, return the fraction of absorbed strain energy that the toughness represents.\r\nPrevious problem: 6 - absorbed strain energy. Next problem: 8 - material properties list.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.440001px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none; white-space: normal; \"\u003e\u003cdiv style=\"block-size: 499px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 332px 249.5px; transform-origin: 332px 249.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 309px 31.5px; text-align: left; transform-origin: 309px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eThe toughness of a material is technically defined as the plastic strain energy absorbed by the material (the plastic region in the figure below). Practically speaking, it's a measure of how much deformation a material can undergo (or energy it can absorb) before failure.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 304px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 309px 152px; text-align: center; transform-origin: 309px 152px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cimg class=\"imageNode\" style=\"vertical-align: baseline\" src=\"https://qph.cf2.quoracdn.net/main-qimg-b2693f4b9ea8430af25df920757e0b29-lq\" data-image-state=\"image-loaded\"\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 84px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 309px 42px; text-align: left; transform-origin: 309px 42px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eWrite a function to calculate the toughness of a material—the absorbed strain energy minus the resilience. This can be accomplished by combining the code written in\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/8049-stress-strain-properties-2\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eproblem 2\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e (resilience) and\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"/#null\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eproblem 6\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e (absorbed strain energy). Also, return the fraction of absorbed strain energy that the toughness represents.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 309px 10.5px; text-align: left; transform-origin: 309px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003ePrevious problem: 6 -\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"/#null\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eabsorbed strain energy\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e. Next problem: 8 -\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"/#null\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003ematerial properties list\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function [T,frac] = stress_strain7(e_y,e_u,S_y,sh_exp,sh_coeff)\r\n\r\nT = 1;\r\n\r\nfrac = 0.5;\r\n\r\nend\r\n","test_suite":"%% Note\r\n% The following properties are measured at room temperature and are tensile\r\n% in a single direction. Some materials, such as metals, are generally\r\n% isotropic, whereas others, like composites, are highly anisotropic\r\n% (different properties in different directions). Also, property values can\r\n% range depending on the material grade. Finally, thermal or environmental\r\n% changes can alter these properties, sometimes drastically.\r\n\r\n%% steel alloy (ASTM A36)\r\nS_y = 250e6; %Pa\r\nS_u = 400e6; %Pa\r\ne_y = 0.00125;\r\ne_u = 0.35;\r\nnu = 0.26;\r\nG = 79.3e9; %Pa\r\nE = 200e9; %Pa\r\ndensity = 7.85; %g/cm^3\r\nsh_exp = 0.14; %strain-hardening exponent\r\nsh_coeff = 463e6; %strain-hardening coefficient\r\nT_corr = 12.26e7;\r\nfrac_corr = 0.9987;\r\n[T,frac] = stress_strain7(e_y,e_u,S_y,sh_exp,sh_coeff);\r\nassert(abs(T-T_corr)/T_corr\u003c1e-2)\r\nassert(abs(frac-frac_corr)/frac_corr\u003c1e-2)\r\n\r\n%% titanium (Ti-6Al-4V)\r\nS_y = 830e6; %Pa\r\nS_u = 900e6; %Pa\r\ne_y = 0.00728;\r\ne_u = 0.14;\r\nnu = 0.342;\r\nG = 44e9; %Pa\r\nE = 114e9; %Pa\r\ndensity = 4.51; %g/cm^3\r\nsh_exp = 0.04; %strain-hardening exponent\r\nsh_coeff = 974e6; %strain-hardening coefficient\r\nT_corr = 11.82e7;\r\nfrac_corr = 0.9751;\r\n[T,frac] = stress_strain7(e_y,e_u,S_y,sh_exp,sh_coeff);\r\nassert(abs(T-T_corr)/T_corr\u003c1e-2)\r\nassert(abs(frac-frac_corr)/frac_corr\u003c1e-2)\r\n\r\n%% Inconel 718\r\nS_y = 1172e6; %Pa\r\nS_u = 1407e6; %Pa\r\ne_y = 0.00563;\r\ne_u = 0.027;\r\nnu = 0.29;\r\nG = 11.6e9; %Pa\r\nE = 208e9; %Pa\r\ndensity = 8.19; %g/cm^3\r\nsh_exp = 0.075; %strain-hardening exponent\r\nsh_coeff = 1845e6; %strain-hardening coefficient\r\nT_corr = 3.205e7;\r\nfrac_corr = 0.9067;\r\n[T,frac] = stress_strain7(e_y,e_u,S_y,sh_exp,sh_coeff);\r\nassert(abs(T-T_corr)/T_corr\u003c1e-2)\r\nassert(abs(frac-frac_corr)/frac_corr\u003c1e-2)\r\n\r\n%% aluminum alloy (6061-T6)\r\nS_y = 241e6; %Pa\r\nS_u = 300e6; %Pa\r\ne_y = 0.0035;\r\ne_u = 0.15;\r\nnu = 0.33;\r\nG = 26e9; %Pa\r\nE = 68.9e9; %Pa\r\ndensity = 2.7; %g/cm^3\r\nsh_exp = 0.042; %strain-hardening exponent\r\nsh_coeff = 325e6; %strain-hardening coefficient\r\nT_corr = 4.279e7;\r\nfrac_corr = 0.9902;\r\n[T,frac] = stress_strain7(e_y,e_u,S_y,sh_exp,sh_coeff);\r\nassert(abs(T-T_corr)/T_corr\u003c1e-2)\r\nassert(abs(frac-frac_corr)/frac_corr\u003c1e-2)\r\n\r\n%% copper\r\nS_y = 70e6; %Pa\r\nS_u = 220e6; %Pa\r\ne_y = 0.00054;\r\ne_u = 0.48;\r\nnu = 0.34;\r\nG = 48e9; %Pa\r\nE = 130e9; %Pa\r\ndensity = 8.92; %g/cm^3\r\nsh_exp = 0.44; %strain-hardening exponent\r\nsh_coeff = 304e6; %strain-hardening coefficient\r\nT_corr = 7.340e7;\r\nfrac_corr = 0.9997;\r\n[T,frac] = stress_strain7(e_y,e_u,S_y,sh_exp,sh_coeff);\r\nassert(abs(T-T_corr)/T_corr\u003c1e-2)\r\nassert(abs(frac-frac_corr)/frac_corr\u003c1e-2)\r\n\r\n%% rhenium\r\nS_y = 317e6; %Pa\r\nS_u = 1130e6; %Pa\r\ne_y = 0.000685;\r\ne_u = 0.24;\r\nnu = 0.3;\r\nG = 178e9; %Pa\r\nE = 463e9; %Pa\r\ndensity = 21.02; %g/cm^3\r\nsh_exp = 0.353; %strain-hardening exponent\r\nsh_coeff = 1870e6; %strain-hardening coefficient\r\nT_corr = 20.05e7;\r\nfrac_corr = 0.9995;\r\n[T,frac] = stress_strain7(e_y,e_u,S_y,sh_exp,sh_coeff);\r\nassert(abs(T-T_corr)/T_corr\u003c1e-2)\r\nassert(abs(frac-frac_corr)/frac_corr\u003c1e-2)\r\n\r\n%% polymer (nylon, 6/6)\r\nS_y = 82e6; %Pa\r\nS_u = 82e6; %Pa\r\ne_y = 0.0265;\r\ne_u = 0.45;\r\nnu = 0.41;\r\nG = 2.8e9; %Pa\r\nE = 3.1e9; %Pa\r\ndensity = 1.14; %g/cm^3\r\nsh_exp = 0; %strain-hardening exponent\r\nsh_coeff = 0; %strain-hardening coefficient\r\nT_corr = 3.473e7;\r\nfrac_corr = 0.9697;\r\n[T,frac] = stress_strain7(e_y,e_u,S_y,sh_exp,sh_coeff);\r\nassert(abs(T-T_corr)/T_corr\u003c1e-2)\r\nassert(abs(frac-frac_corr)/frac_corr\u003c1e-2)\r\n\r\n%% polymer (nylon, 6/6) reinforced with 45wt.% glass fiber\r\nS_y = 230e6; %Pa\r\nS_u = 230e6; %Pa\r\ne_y = 0.016;\r\ne_u = 0.016;\r\nnu = 0.35;\r\nG = 13.0e9; %Pa\r\nE = 14.5e9; %Pa\r\ndensity = 1.51; %g/cm^3\r\nsh_exp = 0; %strain-hardening exponent\r\nsh_coeff = 0; %strain-hardening coefficient\r\nT_corr = 0;\r\nfrac_corr = 0;\r\n[T,frac] = stress_strain7(e_y,e_u,S_y,sh_exp,sh_coeff);\r\nassert(isequal(T,T_corr))\r\nassert(isequal(frac,frac_corr))\r\n\r\n%% diamond\r\nS_y = 1200e6; %Pa\r\nS_u = 1200e6; %Pa\r\ne_y = 0.001;\r\ne_u = 0.001;\r\nnu = 0.20;\r\nG = 478e9; %Pa\r\nE = 1200e9; %Pa\r\ndensity = 3.51; %g/cm^3\r\nsh_exp = 0; %strain-hardening exponent\r\nsh_coeff = 0; %strain-hardening coefficient\r\nT_corr = 0;\r\nfrac_corr = 0;\r\n[T,frac] = stress_strain7(e_y,e_u,S_y,sh_exp,sh_coeff);\r\nassert(isequal(T,T_corr))\r\nassert(isequal(frac,frac_corr))\r\n\r\n%%\r\nfor i = 1:30\r\nind = randi(8);\r\nswitch ind\r\n\tcase 1\r\n\t\tS_y = 250e6; %Pa\r\n\t\te_y = 0.00125;\r\n\t\te_u = 0.35;\r\n\t\tsh_exp = 0.14; %strain-hardening exponent\r\n\t\tsh_coeff = 463e6; %strain-hardening coefficient\r\n\t\tT_corr = 12.26e7;\r\n\t\tfrac_corr = 0.9987;\r\n\t\t[T,frac] = stress_strain7(e_y,e_u,S_y,sh_exp,sh_coeff);\r\n\t\tassert(abs(T-T_corr)/T_corr\u003c1e-2)\r\n\t\tassert(abs(frac-frac_corr)/frac_corr\u003c1e-2)\r\n\tcase 2\r\n\t\tS_y = 830e6; %Pa\r\n\t\te_y = 0.00728;\r\n\t\te_u = 0.14;\r\n\t\tsh_exp = 0.04; %strain-hardening exponent\r\n\t\tsh_coeff = 974e6; %strain-hardening coefficient\r\n\t\tT_corr = 11.82e7;\r\n\t\tfrac_corr = 0.9751;\r\n\t\t[T,frac] = stress_strain7(e_y,e_u,S_y,sh_exp,sh_coeff);\r\n\t\tassert(abs(T-T_corr)/T_corr\u003c1e-2)\r\n\t\tassert(abs(frac-frac_corr)/frac_corr\u003c1e-2)\r\n\tcase 3\r\n\t\tS_y = 230e6; %Pa\r\n\t\te_y = 0.016;\r\n\t\te_u = 0.016;\r\n\t\tsh_exp = 0; %strain-hardening exponent\r\n\t\tsh_coeff = 0; %strain-hardening coefficient\r\n\t\tT_corr = 0;\r\n\t\tfrac_corr = 0;\r\n\t\t[T,frac] = stress_strain7(e_y,e_u,S_y,sh_exp,sh_coeff);\r\n\t\tassert(isequal(T,T_corr))\r\n\t\tassert(isequal(frac,frac_corr))\r\n\tcase 4\r\n\t\tS_y = 317e6; %Pa\r\n\t\te_y = 0.000685;\r\n\t\te_u = 0.24;\r\n\t\tsh_exp = 0.353; %strain-hardening exponent\r\n\t\tsh_coeff = 1870e6; %strain-hardening coefficient\r\n\t\tT_corr = 20.05e7;\r\n\t\tfrac_corr = 0.9995;\r\n\t\t[T,frac] = stress_strain7(e_y,e_u,S_y,sh_exp,sh_coeff);\r\n\t\tassert(abs(T-T_corr)/T_corr\u003c1e-2)\r\n\t\tassert(abs(frac-frac_corr)/frac_corr\u003c1e-2)\r\n\tcase 5\r\n\t\tS_y = 70e6; %Pa\r\n\t\te_y = 0.00054;\r\n\t\te_u = 0.48;\r\n\t\tsh_exp = 0.44; %strain-hardening exponent\r\n\t\tsh_coeff = 304e6; %strain-hardening coefficient\r\n\t\tT_corr = 7.340e7;\r\n\t\tfrac_corr = 0.9997;\r\n\tcase 6\r\n\t\tS_y = 1172e6; %Pa\r\n\t\te_y = 0.00563;\r\n\t\te_u = 0.027;\r\n\t\tsh_exp = 0.075; %strain-hardening exponent\r\n\t\tsh_coeff = 1845e6; %strain-hardening coefficient\r\n\t\tT_corr = 3.205e7;\r\n\t\tfrac_corr = 0.9067;\r\n\tcase 7\r\n\t\tS_y = 82e6; %Pa\r\n\t\te_y = 0.0265;\r\n\t\te_u = 0.45;\r\n\t\tsh_exp = 0; %strain-hardening exponent\r\n\t\tsh_coeff = 0; %strain-hardening coefficient\r\n\t\tT_corr = 3.473e7;\r\n\t\tfrac_corr = 0.9697;\r\n\tcase 8\r\n\t\tS_y = 241e6; %Pa\r\n\t\te_y = 0.0035;\r\n\t\te_u = 0.15;\r\n\t\tsh_exp = 0.042; %strain-hardening exponent\r\n\t\tsh_coeff = 325e6; %strain-hardening coefficient\r\n\t\tT_corr = 4.279e7;\r\n\t\tfrac_corr = 0.9902;\r\nend\r\nend % for i = 1:30\r\n","published":true,"deleted":false,"likes_count":1,"comments_count":2,"created_by":26769,"edited_by":26769,"edited_at":"2024-03-27T17:44:32.000Z","deleted_by":null,"deleted_at":null,"solvers_count":88,"test_suite_updated_at":"2021-08-03T17:04:10.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2015-03-30T22:03:11.000Z","updated_at":"2026-02-19T09:46:19.000Z","published_at":"2015-03-30T22:03:11.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe toughness of a material is technically defined as the plastic strain energy absorbed by the material (the plastic region in the figure below). Practically speaking, it's a measure of how much deformation a material can undergo (or energy it can absorb) before failure.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"center\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"298\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"420\\\"/\u003e\u003cw:attr w:name=\\\"verticalAlign\\\" w:val=\\\"baseline\\\"/\u003e\u003cw:attr w:name=\\\"altText\\\" w:val=\\\"\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to calculate the toughness of a material—the absorbed strain energy minus the resilience. This can be accomplished by combining the code written in\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/8049-stress-strain-properties-2\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eproblem 2\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e (resilience) and\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eproblem 6\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e (absorbed strain energy). Also, return the fraction of absorbed strain energy that the toughness represents.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ePrevious problem: 6 -\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eabsorbed strain energy\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e. Next problem: 8 -\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"\\\"\u003e\u003cw:r\u003e\u003cw:t\u003ematerial properties list\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/image\",\"target\":\"/media/image1.net/main-qimg-b2693f4b9ea8430af25df920757e0b29-lq\",\"relationshipId\":\"rId1\"}]},{\"partUri\":\"/media/image1.net/main-qimg-b2693f4b9ea8430af25df920757e0b29-lq\",\"contentType\":\"image/net/main-qimg-b2693f4b9ea8430af25df920757e0b29-lq\",\"content\":\"https://qph.cf2.quoracdn.net/main-qimg-b2693f4b9ea8430af25df920757e0b29-lq\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":8055,"title":"Stress-Strain Properties - 8","description":"Up to this point, you've calculated some material properties based on tensile stress-strain data. For this problem, you are tasked with writing a function to calculate all of these properties and gather them, along with supplied properties, such as strain values, into an array. You'll be provided a cell array of strings in the function template; you must return an accompanying numerical array that contains all the specified properties. Below is the list of properties for a material, both supplied and calculated, that make up the array (with variable names that have been used):\r\n\r\n* Yield Strength (S_y)\r\n* Yield Strain (e_y)\r\n* Ultimate Strength (S_u)\r\n* Failure Strain (e_u)\r\n* Poisson's Ratio (nu)\r\n* Shear Modulus (G)\r\n* Elastic Modulus (E)\r\n* Density\r\n* Strain-hardening Exponent (sh_exp)\r\n* Strain-hardening Coefficient (sh_coeff)\r\n* Resilience (R)\r\n* Strength-to-weight Ratio (StWR)\r\n* Stiffness-to-weight Ratio (EtWR)\r\n* Absorbed Strain Energy (ASE)\r\n* Toughness (T)\r\n\r\nPrevious problem: 7 - \u003chttp://www.mathworks.com/matlabcentral/cody/problems/8054-stress-strain-properties-7 toughness\u003e.","description_html":"\u003cp\u003eUp to this point, you've calculated some material properties based on tensile stress-strain data. For this problem, you are tasked with writing a function to calculate all of these properties and gather them, along with supplied properties, such as strain values, into an array. You'll be provided a cell array of strings in the function template; you must return an accompanying numerical array that contains all the specified properties. Below is the list of properties for a material, both supplied and calculated, that make up the array (with variable names that have been used):\u003c/p\u003e\u003cul\u003e\u003cli\u003eYield Strength (S_y)\u003c/li\u003e\u003cli\u003eYield Strain (e_y)\u003c/li\u003e\u003cli\u003eUltimate Strength (S_u)\u003c/li\u003e\u003cli\u003eFailure Strain (e_u)\u003c/li\u003e\u003cli\u003ePoisson's Ratio (nu)\u003c/li\u003e\u003cli\u003eShear Modulus (G)\u003c/li\u003e\u003cli\u003eElastic Modulus (E)\u003c/li\u003e\u003cli\u003eDensity\u003c/li\u003e\u003cli\u003eStrain-hardening Exponent (sh_exp)\u003c/li\u003e\u003cli\u003eStrain-hardening Coefficient (sh_coeff)\u003c/li\u003e\u003cli\u003eResilience (R)\u003c/li\u003e\u003cli\u003eStrength-to-weight Ratio (StWR)\u003c/li\u003e\u003cli\u003eStiffness-to-weight Ratio (EtWR)\u003c/li\u003e\u003cli\u003eAbsorbed Strain Energy (ASE)\u003c/li\u003e\u003cli\u003eToughness (T)\u003c/li\u003e\u003c/ul\u003e\u003cp\u003ePrevious problem: 7 - \u003ca href = \"http://www.mathworks.com/matlabcentral/cody/problems/8054-stress-strain-properties-7\"\u003etoughness\u003c/a\u003e.\u003c/p\u003e","function_template":"function [arr_vals] = stress_strain8(S_y,S_u,e_y,e_u,nu,G,E,density,sh_exp,sh_coeff)\r\n\r\narr_descr = {\r\n\t'Yield Strength','Yield Strain','Ultimate Strength','Failure Strain',...\r\n\t'Poisson''s Ratio','Shear Modulus','Elastic Modulus','Density',...\r\n\t'Strain-hardening Exponent','Strain-hardening Coefficient',...\r\n\t'Resilience','Strength-to-weight Ratio','Stiffness-to-weight Ratio',...\r\n\t'Absorbed Strain Energy','Toughness';\r\n};\r\n\r\narr_vals = [\r\n\t1;\r\n];\r\n\r\nend\r\n","test_suite":"%% Note\r\n% The following properties are measured at room temperature and are tensile\r\n% in a single direction. Some materials, such as metals are generally\r\n% isotropic, whereas others, like composite are highly anisotropic\r\n% (different properties in different directions). Also, property values can\r\n% range depending on the material grade. Finally, thermal or environmental\r\n% changes can alter these properties, sometimes drastically.\r\n\r\n%% steel alloy (ASTM A36)\r\nS_y = 250e6; %Pa\r\nS_u = 400e6; %Pa\r\ne_y = 0.00125;\r\ne_u = 0.35;\r\nnu = 0.26; %Poisson's ratio\r\nG = 79.3e9; %Pa (shear modulus)\r\nE = 200e9; %Pa  (elastic modulus)\r\ndensity = 7.85; %g/cm^3\r\nsh_exp = 0.14; %strain-hardening exponent\r\nsh_coeff = 463e6; %strain-hardening coefficient\r\n[arr_vals] = stress_strain8(S_y,S_u,e_y,e_u,nu,G,E,density,sh_exp,sh_coeff);\r\narr_vals_corr = [S_y, e_y, S_u, e_u, nu, G, E, density, sh_exp, sh_coeff,...\r\n\t1.5625e5, 5.096e7, 2.548e10, 12.28e7, 12.26e7];\r\ndiffs = abs(arr_vals-arr_vals_corr)./arr_vals_corr;\r\nfor i = 1:numel(diffs)\r\n\tassert(diffs(i)\u003c1e-2)\r\nend\r\n\r\n%% titanium (Ti-6Al-4V)\r\nS_y = 830e6; %Pa\r\nS_u = 900e6; %Pa\r\ne_y = 0.00728;\r\ne_u = 0.14;\r\nnu = 0.342;\r\nG = 44e9; %Pa\r\nE = 114e9; %Pa\r\ndensity = 4.51; %g/cm^3\r\nsh_exp = 0.04; %strain-hardening exponent\r\nsh_coeff = 974e6; %strain-hardening coefficient\r\n[arr_vals] = stress_strain8(S_y,S_u,e_y,e_u,nu,G,E,density,sh_exp,sh_coeff);\r\narr_vals_corr = [S_y, e_y, S_u, e_u, nu, G, E, density, sh_exp, sh_coeff,...\r\n\t3.0212e6, 19.96e7, 2.528e10, 12.12e7, 11.82e7];\r\ndiffs = abs(arr_vals-arr_vals_corr)./arr_vals_corr;\r\nfor i = 1:numel(diffs)\r\n\tassert(diffs(i)\u003c1e-2)\r\nend\r\n\r\n%% Inconel 718\r\nS_y = 1172e6; %Pa\r\nS_u = 1407e6; %Pa\r\ne_y = 0.00563;\r\ne_u = 0.027;\r\nnu = 0.29;\r\nG = 11.6e9; %Pa\r\nE = 208e9; %Pa\r\ndensity = 8.19; %g/cm^3\r\nsh_exp = 0.075; %strain-hardening exponent\r\nsh_coeff = 1845e6; %strain-hardening coefficient\r\n[arr_vals] = stress_strain8(S_y,S_u,e_y,e_u,nu,G,E,density,sh_exp,sh_coeff);\r\narr_vals_corr = [S_y, e_y, S_u, e_u, nu, G, E, density, sh_exp, sh_coeff,...\r\n\t3.29918e6, 17.18e7, 2.540e10, 3.535e7, 3.205e7];\r\ndiffs = abs(arr_vals-arr_vals_corr)./arr_vals_corr;\r\nfor i = 1:numel(diffs)\r\n\tassert(diffs(i)\u003c1e-2)\r\nend\r\n\r\n%% aluminum alloy (6061-T6)%^\u0026\r\nS_y = 241e6; %Pa\r\nS_u = 300e6; %Pa\r\ne_y = 0.0035;\r\ne_u = 0.15;\r\nnu = 0.33;\r\nG = 26e9; %Pa\r\nE = 68.9e9; %Pa\r\ndensity = 2.7; %g/cm^3\r\nsh_exp = 0.042; %strain-hardening exponent\r\nsh_coeff = 325e6; %strain-hardening coefficient\r\n[arr_vals] = stress_strain8(S_y,S_u,e_y,e_u,nu,G,E,density,sh_exp,sh_coeff);\r\narr_vals_corr = [S_y, e_y, S_u, e_u, nu, G, E, density, sh_exp, sh_coeff,...\r\n\t4.2175e5, 11.11e7, 2.552e10, 4.321e7, 4.279e7];\r\ndiffs = abs(arr_vals-arr_vals_corr)./arr_vals_corr;\r\nfor i = 1:numel(diffs)\r\n\tassert(diffs(i)\u003c1e-2)\r\nend\r\n\r\n%% copper\r\nS_y = 70e6; %Pa\r\nS_u = 220e6; %Pa\r\ne_y = 0.00054;\r\ne_u = 0.48;\r\nnu = 0.34;\r\nG = 48e9; %Pa\r\nE = 130e9; %Pa\r\ndensity = 8.92; %g/cm^3\r\nsh_exp = 0.44; %strain-hardening exponent\r\nsh_coeff = 304e6; %strain-hardening coefficient\r\n[arr_vals] = stress_strain8(S_y,S_u,e_y,e_u,nu,G,E,density,sh_exp,sh_coeff);\r\narr_vals_corr = [S_y, e_y, S_u, e_u, nu, G, E, density, sh_exp, sh_coeff,...\r\n\t1.89e4, 2.466e7, 1.457e10, 7.342e7, 7.340e7];\r\ndiffs = abs(arr_vals-arr_vals_corr)./arr_vals_corr;\r\nfor i = 1:numel(diffs)\r\n\tassert(diffs(i)\u003c1e-2)\r\nend\r\n\r\n%% rhenium\r\nS_y = 317e6; %Pa\r\nS_u = 1130e6; %Pa\r\ne_y = 0.000685;\r\ne_u = 0.24;\r\nnu = 0.3;\r\nG = 178e9; %Pa\r\nE = 463e9; %Pa\r\ndensity = 21.02; %g/cm^3\r\nsh_exp = 0.353; %strain-hardening exponent\r\nsh_coeff = 1870e6; %strain-hardening coefficient\r\n[arr_vals] = stress_strain8(S_y,S_u,e_y,e_u,nu,G,E,density,sh_exp,sh_coeff);\r\narr_vals_corr = [S_y, e_y, S_u, e_u, nu, G, E, density, sh_exp, sh_coeff,...\r\n\t1.085725e5, 5.376e7, 2.203e10, 20.06e7, 20.05e7];\r\ndiffs = abs(arr_vals-arr_vals_corr)./arr_vals_corr;\r\nfor i = 1:numel(diffs)\r\n\tassert(diffs(i)\u003c1e-2)\r\nend\r\n\r\n%% polymer (nylon, 6/6)\r\nS_y = 82e6; %Pa\r\nS_u = 82e6; %Pa\r\ne_y = 0.0265;\r\ne_u = 0.45;\r\nnu = 0.41;\r\nG = 2.8e9; %Pa\r\nE = 3.1e9; %Pa\r\ndensity = 1.14; %g/cm^3\r\nsh_exp = 0; %strain-hardening exponent\r\nsh_coeff = 0; %strain-hardening coefficient\r\n[arr_vals] = stress_strain8(S_y,S_u,e_y,e_u,nu,G,E,density,sh_exp,sh_coeff);\r\narr_vals_corr = [S_y, e_y, S_u, e_u, nu, G, E, density, sh_exp, sh_coeff,...\r\n\t1.0865e6, 7.193e7, 0.272e10, 3.581e7, 3.473e7];\r\ndiffs = abs(arr_vals-arr_vals_corr)./arr_vals_corr;\r\ndiffs(isnan(diffs)) = 0;\r\nfor i = 1:numel(diffs)\r\n\tassert(diffs(i)\u003c1e-2)\r\nend\r\n\r\n%% polymer (nylon, 6/6) reinforced with 45wt.% glass fiber\r\nS_y = 230e6; %Pa\r\nS_u = 230e6; %Pa\r\ne_y = 0.016;\r\ne_u = 0.016;\r\nnu = 0.35;\r\nG = 13.0e9; %Pa\r\nE = 14.5e9; %Pa\r\ndensity = 1.51; %g/cm^3\r\nsh_exp = 0; %strain-hardening exponent\r\nsh_coeff = 0; %strain-hardening coefficient\r\n[arr_vals] = stress_strain8(S_y,S_u,e_y,e_u,nu,G,E,density,sh_exp,sh_coeff);\r\narr_vals_corr = [S_y, e_y, S_u, e_u, nu, G, E, density, sh_exp, sh_coeff,...\r\n\t1.84e6, 15.23e7, 0.960e10, 0.184e7, 0];\r\ndiffs = abs(arr_vals-arr_vals_corr)./arr_vals_corr;\r\ndiffs(isnan(diffs)) = 0;\r\nfor i = 1:numel(diffs)\r\n\tassert(diffs(i)\u003c1e-2)\r\nend\r\n\r\n%% diamond\r\nS_y = 1200e6; %Pa\r\nS_u = 1200e6; %Pa\r\ne_y = 0.001;\r\ne_u = 0.001;\r\nnu = 0.20;\r\nG = 478e9; %Pa\r\nE = 1200e9; %Pa\r\ndensity = 3.51; %g/cm^3\r\nsh_exp = 0; %strain-hardening exponent\r\nsh_coeff = 0; %strain-hardening coefficient\r\n[arr_vals] = stress_strain8(S_y,S_u,e_y,e_u,nu,G,E,density,sh_exp,sh_coeff);\r\narr_vals_corr = [S_y, e_y, S_u, e_u, nu, G, E, density, sh_exp, sh_coeff,...\r\n\t6e5, 34.19e7, 34.19e10, 0.06e7, 0];\r\ndiffs = abs(arr_vals-arr_vals_corr)./arr_vals_corr;\r\ndiffs(isnan(diffs)) = 0;\r\nfor i = 1:numel(diffs)\r\n\tassert(diffs(i)\u003c1e-2)\r\nend\r\n","published":true,"deleted":false,"likes_count":0,"comments_count":0,"created_by":26769,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":48,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2015-03-31T02:36:58.000Z","updated_at":"2026-02-19T09:49:12.000Z","published_at":"2015-03-31T02:36:58.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eUp to this point, you've calculated some material properties based on tensile stress-strain data. For this problem, you are tasked with writing a function to calculate all of these properties and gather them, along with supplied properties, such as strain values, into an array. You'll be provided a cell array of strings in the function template; you must return an accompanying numerical array that contains all the specified properties. Below is the list of properties for a material, both supplied and calculated, that make up the array (with variable names that have been used):\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eYield Strength (S_y)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eYield Strain (e_y)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eUltimate Strength (S_u)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFailure Strain (e_u)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ePoisson's Ratio (nu)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eShear Modulus (G)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eElastic Modulus (E)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eDensity\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eStrain-hardening Exponent (sh_exp)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eStrain-hardening Coefficient (sh_coeff)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eResilience (R)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eStrength-to-weight Ratio (StWR)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eStiffness-to-weight Ratio (EtWR)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eAbsorbed Strain Energy (ASE)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eToughness (T)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ePrevious problem: 7 -\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://www.mathworks.com/matlabcentral/cody/problems/8054-stress-strain-properties-7\\\"\u003e\u003cw:r\u003e\u003cw:t\u003etoughness\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":2775,"title":"Rule of mixtures (composites) - lower bound","description":"The \u003chttp://en.wikipedia.org/wiki/Rule_of_mixtures rule of mixtures\u003e is used in the mechanical design of composite structures to estimate the elastic modulus (Ec) based on the following properties:\r\n\r\n* Ef: elastic modulus of the fiber material\r\n* Em: elastic modulus of the matrix material\r\n* ff or fm: volume fraction of the fiber or matrix material (ff = 1 – fm)\r\n\r\nBased on these values, the lower-bound estimate of elastic modulus is calculated by:\r\n\r\nEc = 1 / (ff / Ef + (1 – ff) / Em).\r\n\r\nWrite a function to calculate this bound for various values.","description_html":"\u003cp\u003eThe \u003ca href = \"http://en.wikipedia.org/wiki/Rule_of_mixtures\"\u003erule of mixtures\u003c/a\u003e is used in the mechanical design of composite structures to estimate the elastic modulus (Ec) based on the following properties:\u003c/p\u003e\u003cul\u003e\u003cli\u003eEf: elastic modulus of the fiber material\u003c/li\u003e\u003cli\u003eEm: elastic modulus of the matrix material\u003c/li\u003e\u003cli\u003eff or fm: volume fraction of the fiber or matrix material (ff = 1 – fm)\u003c/li\u003e\u003c/ul\u003e\u003cp\u003eBased on these values, the lower-bound estimate of elastic modulus is calculated by:\u003c/p\u003e\u003cp\u003eEc = 1 / (ff / Ef + (1 – ff) / Em).\u003c/p\u003e\u003cp\u003eWrite a function to calculate this bound for various values.\u003c/p\u003e","function_template":"function Ec = rule_of_mixtures_lower_bound(Ef,Em,ff)\r\n Ec = 1;\r\nend","test_suite":"%%\r\nEf = 100;\r\nEm = 10;\r\nff = 0.30;\r\nassert(abs(rule_of_mixtures_lower_bound(Ef,Em,ff) - 13.6986) \u003c 1e-4)\r\n\r\n%%\r\nEf = 100;\r\nEm = 10;\r\nff = 0.22;\r\nassert(abs(rule_of_mixtures_lower_bound(Ef,Em,ff) - 12.4688) \u003c 1e-4)\r\n\r\n\r\n%%\r\nEf = 100;\r\nEm = 10;\r\nff = 0.15;\r\nassert(abs(rule_of_mixtures_lower_bound(Ef,Em,ff) - 11.5607) \u003c 1e-4)\r\n\r\n%%\r\nEf = 1000;\r\nEm = 10;\r\nff = 0.30;\r\nassert(abs(rule_of_mixtures_lower_bound(Ef,Em,ff) - 14.2248) \u003c 1e-4)\r\n\r\n%%\r\nEf = 1000;\r\nEm = 10;\r\nff = 0.22;\r\nassert(abs(rule_of_mixtures_lower_bound(Ef,Em,ff) - 12.7845) \u003c 1e-4)\r\n\r\n%%\r\nEf = 1000;\r\nEm = 10;\r\nff = 0.15;\r\nassert(abs(rule_of_mixtures_lower_bound(Ef,Em,ff) - 11.7440) \u003c 1e-4)\r\n","published":true,"deleted":false,"likes_count":2,"comments_count":0,"created_by":26769,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":183,"test_suite_updated_at":"2014-12-16T22:51:58.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2014-12-14T04:09:25.000Z","updated_at":"2026-02-13T03:46:16.000Z","published_at":"2014-12-14T04:14:32.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://en.wikipedia.org/wiki/Rule_of_mixtures\\\"\u003e\u003cw:r\u003e\u003cw:t\u003erule of mixtures\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e is used in the mechanical design of composite structures to estimate the elastic modulus (Ec) based on the following properties:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eEf: elastic modulus of the fiber material\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eEm: elastic modulus of the matrix material\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eff or fm: volume fraction of the fiber or matrix material (ff = 1 – fm)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eBased on these values, the lower-bound estimate of elastic modulus is calculated by:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eEc = 1 / (ff / Ef + (1 – ff) / Em).\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to calculate this bound for various values.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":2774,"title":"Rule of mixtures (composites) - upper bound","description":"The \u003chttp://en.wikipedia.org/wiki/Rule_of_mixtures rule of mixtures\u003e is used in the mechanical design of composite structures to estimate the elastic modulus (Ec) based on the following properties:\r\n\r\n* Ef: elastic modulus of the fiber material\r\n* Em: elastic modulus of the matrix material\r\n* ff or fm: volume fraction of the fiber or matrix material (ff = 1 – fm)\r\n\r\nBased on these values, the upper-bound (linear) estimate of elastic modulus is calculated by:\r\n\r\nEc = ff * Ef + (1 – ff) * Em.\r\n\r\nWrite a function to calculate this bound for various values.","description_html":"\u003cp\u003eThe \u003ca href = \"http://en.wikipedia.org/wiki/Rule_of_mixtures\"\u003erule of mixtures\u003c/a\u003e is used in the mechanical design of composite structures to estimate the elastic modulus (Ec) based on the following properties:\u003c/p\u003e\u003cul\u003e\u003cli\u003eEf: elastic modulus of the fiber material\u003c/li\u003e\u003cli\u003eEm: elastic modulus of the matrix material\u003c/li\u003e\u003cli\u003eff or fm: volume fraction of the fiber or matrix material (ff = 1 – fm)\u003c/li\u003e\u003c/ul\u003e\u003cp\u003eBased on these values, the upper-bound (linear) estimate of elastic modulus is calculated by:\u003c/p\u003e\u003cp\u003eEc = ff * Ef + (1 – ff) * Em.\u003c/p\u003e\u003cp\u003eWrite a function to calculate this bound for various values.\u003c/p\u003e","function_template":"function Ec = rule_of_mixtures_upper_bound(Ef,Em,ff)\r\n Ec = 1;\r\nend","test_suite":"%%\r\nEf = 100;\r\nEm = 10;\r\nff = 0.30;\r\nassert(isequal(rule_of_mixtures_upper_bound(Ef,Em,ff),37))\r\n\r\n%%\r\nEf = 100;\r\nEm = 10;\r\nff = 0.22;\r\nassert(isequal(rule_of_mixtures_upper_bound(Ef,Em,ff),29.8))\r\n\r\n%%\r\nEf = 100;\r\nEm = 10;\r\nff = 0.15;\r\nassert(isequal(rule_of_mixtures_upper_bound(Ef,Em,ff),23.5))\r\n\r\n%%\r\nEf = 1000;\r\nEm = 10;\r\nff = 0.30;\r\nassert(isequal(rule_of_mixtures_upper_bound(Ef,Em,ff),307))\r\n\r\n%%\r\nEf = 1000;\r\nEm = 10;\r\nff = 0.22;\r\nassert(isequal(rule_of_mixtures_upper_bound(Ef,Em,ff),227.8))\r\n\r\n%%\r\nEf = 1000;\r\nEm = 10;\r\nff = 0.15;\r\nassert(isequal(rule_of_mixtures_upper_bound(Ef,Em,ff),158.5))\r\n","published":true,"deleted":false,"likes_count":3,"comments_count":0,"created_by":26769,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":129,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2014-12-14T04:06:34.000Z","updated_at":"2026-02-13T03:46:57.000Z","published_at":"2014-12-14T04:06:34.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://en.wikipedia.org/wiki/Rule_of_mixtures\\\"\u003e\u003cw:r\u003e\u003cw:t\u003erule of mixtures\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e is used in the mechanical design of composite structures to estimate the elastic modulus (Ec) based on the following properties:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eEf: elastic modulus of the fiber material\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eEm: elastic modulus of the matrix material\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eff or fm: volume fraction of the fiber or matrix material (ff = 1 – fm)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eBased on these values, the upper-bound (linear) estimate of elastic modulus is calculated by:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eEc = ff * Ef + (1 – ff) * Em.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to calculate this bound for various values.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":2776,"title":"Rule of mixtures (composites) - lower and upper bounds","description":"The \u003chttp://en.wikipedia.org/wiki/Rule_of_mixtures rule of mixtures\u003e is used in the mechanical design of composite structures to estimate the elastic modulus (Ec) based on the following properties:\r\n\r\n    Ef: elastic modulus of the fiber material\r\n    Em: elastic modulus of the matrix material\r\n    ff or fm: volume fraction of the fiber or matrix material (ff = 1 – fm)\r\n\r\nBased on these values, the lower-bound estimate of elastic modulus is calculated by:\r\n\r\nEc = 1 / (ff / Ef + (1 – ff) / Em).\r\n\r\nOn the other hand, the upper-bound (linear) estimate of elastic modulus is calculated by:\r\n\r\nEc = ff * Ef + (1 – ff) * Em.\r\n\r\nWrite a function to calculate both bounds for various values. Also, return the difference between the modulus estimates and the ratios of this difference to both moduli estimates.","description_html":"\u003cp\u003eThe \u003ca href = \"http://en.wikipedia.org/wiki/Rule_of_mixtures\"\u003erule of mixtures\u003c/a\u003e is used in the mechanical design of composite structures to estimate the elastic modulus (Ec) based on the following properties:\u003c/p\u003e\u003cpre\u003e    Ef: elastic modulus of the fiber material\r\n    Em: elastic modulus of the matrix material\r\n    ff or fm: volume fraction of the fiber or matrix material (ff = 1 – fm)\u003c/pre\u003e\u003cp\u003eBased on these values, the lower-bound estimate of elastic modulus is calculated by:\u003c/p\u003e\u003cp\u003eEc = 1 / (ff / Ef + (1 – ff) / Em).\u003c/p\u003e\u003cp\u003eOn the other hand, the upper-bound (linear) estimate of elastic modulus is calculated by:\u003c/p\u003e\u003cp\u003eEc = ff * Ef + (1 – ff) * Em.\u003c/p\u003e\u003cp\u003eWrite a function to calculate both bounds for various values. Also, return the difference between the modulus estimates and the ratios of this difference to both moduli estimates.\u003c/p\u003e","function_template":"function [Ec_l,Ec_u,E_diff,r_l,r_u] = rule_of_mixtures_l_and_u_bound(Ef,Em,ff)\r\n Ec_l = 1;\r\n Ec_u = 1;\r\n E_diff = 1;\r\n r_l = 1;\r\n r_u = 1;\r\nend","test_suite":"%%\r\nEf = 100;\r\nEm = 10;\r\nff = 0.30;\r\n[Ec_l,Ec_u,E_diff,r_l,r_u] = rule_of_mixtures_l_and_u_bound(Ef,Em,ff);\r\nassert(abs(Ec_l - 13.6986) \u003c 1e-4)\r\nassert(abs(Ec_u - 37) \u003c 1e-4)\r\nassert(abs(E_diff - 23.3014) \u003c 1e-4)\r\nassert(abs(r_l - 1.7010) \u003c 1e-4)\r\nassert(abs(r_u - .6298) \u003c 1e-4)\r\n\r\n\r\n%%\r\nEf = 100;\r\nEm = 10;\r\nff = 0.22;\r\n[Ec_l,Ec_u,E_diff,r_l,r_u] = rule_of_mixtures_l_and_u_bound(Ef,Em,ff);\r\nassert(abs(Ec_l - 12.4688) \u003c 1e-4)\r\nassert(abs(Ec_u - 29.8) \u003c 1e-4)\r\nassert(abs(E_diff - 17.3312) \u003c 1e-4)\r\nassert(abs(r_l - 1.3900) \u003c 1e-4)\r\nassert(abs(r_u - 0.5816) \u003c 1e-4)\r\n\r\n\r\n%%\r\nEf = 100;\r\nEm = 10;\r\nff = 0.15;\r\n[Ec_l,Ec_u,E_diff,r_l,r_u] = rule_of_mixtures_l_and_u_bound(Ef,Em,ff);\r\nassert(abs(Ec_l - 11.5607) \u003c 1e-4)\r\nassert(abs(Ec_u - 23.5) \u003c 1e-4)\r\nassert(abs(E_diff - 11.9393) \u003c 1e-4)\r\nassert(abs(r_l - 1.0327) \u003c 1e-4)\r\nassert(abs(r_u - 0.5081) \u003c 1e-4)\r\n\r\n%%\r\nEf = 1000;\r\nEm = 10;\r\nff = 0.30;\r\n[Ec_l,Ec_u,E_diff,r_l,r_u] = rule_of_mixtures_l_and_u_bound(Ef,Em,ff);\r\nassert(abs(Ec_l - 14.2248) \u003c 1e-4)\r\nassert(abs(Ec_u - 307) \u003c 1e-4)\r\nassert(abs(E_diff - 292.7752) \u003c 1e-4)\r\nassert(abs(r_l - 20.5821) \u003c 1e-4)\r\nassert(abs(r_u - 0.9537) \u003c 1e-4)\r\n%%\r\nEf = 1000;\r\nEm = 10;\r\nff = 0.22;\r\n[Ec_l,Ec_u,E_diff,r_l,r_u] = rule_of_mixtures_l_and_u_bound(Ef,Em,ff);\r\nassert(abs(Ec_l - 12.7845) \u003c 1e-4)\r\nassert(abs(Ec_u - 227.8) \u003c 1e-4)\r\nassert(abs(E_diff - 215.0155) \u003c 1e-4)\r\nassert(abs(r_l - 16.8185) \u003c 1e-4)\r\nassert(abs(r_u - 0.9439) \u003c 1e-4)\r\n\r\n%%\r\nEf = 1000;\r\nEm = 10;\r\nff = 0.15;\r\n[Ec_l,Ec_u,E_diff,r_l,r_u] = rule_of_mixtures_l_and_u_bound(Ef,Em,ff);\r\nassert(abs(Ec_l - 11.7440) \u003c 1e-4)\r\nassert(abs(Ec_u - 158.5) \u003c 1e-4)\r\nassert(abs(E_diff - 146.756) \u003c 1e-4)\r\nassert(abs(r_l - 12.4963) \u003c 1e-4)\r\nassert(abs(r_u - 0.9259) \u003c 1e-4)\r\n","published":true,"deleted":false,"likes_count":1,"comments_count":0,"created_by":26769,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":94,"test_suite_updated_at":"2014-12-16T22:53:00.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2014-12-14T04:43:12.000Z","updated_at":"2026-02-13T03:51:00.000Z","published_at":"2014-12-14T04:43:12.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://en.wikipedia.org/wiki/Rule_of_mixtures\\\"\u003e\u003cw:r\u003e\u003cw:t\u003erule of mixtures\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e is used in the mechanical design of composite structures to estimate the elastic modulus (Ec) based on the following properties:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[    Ef: elastic modulus of the fiber material\\n    Em: elastic modulus of the matrix material\\n    ff or fm: volume fraction of the fiber or matrix material (ff = 1 – fm)]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eBased on these values, the lower-bound estimate of elastic modulus is calculated by:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eEc = 1 / (ff / Ef + (1 – ff) / Em).\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eOn the other hand, the upper-bound (linear) estimate of elastic modulus is calculated by:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eEc = ff * Ef + (1 – ff) * Em.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to calculate both bounds for various values. Also, return the difference between the modulus estimates and the ratios of this difference to both moduli estimates.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":2777,"title":"Rule of mixtures (composites) - either bound","description":"The \u003chttp://en.wikipedia.org/wiki/Rule_of_mixtures rule of mixtures\u003e is used in the mechanical design of composite structures to estimate the elastic modulus (Ec) based on the following properties:\r\n\r\n    Ef: elastic modulus of the fiber material\r\n    Em: elastic modulus of the matrix material\r\n    ff or fm: volume fraction of the fiber or matrix material (ff = 1 – fm)\r\n\r\nBased on these values, the lower-bound estimate of elastic modulus is calculated by:\r\n\r\nEc = 1 / (ff / Ef + (1 – ff) / Em).\r\n\r\nOn the other hand, the upper-bound (linear) estimate of elastic modulus is calculated by:\r\n\r\nEc = ff * Ef + (1 – ff) * Em.\r\n\r\nWrite a function to calculate either bound, depending on which bound is requested in the input string.\r\n","description_html":"\u003cp\u003eThe \u003ca href = \"http://en.wikipedia.org/wiki/Rule_of_mixtures\"\u003erule of mixtures\u003c/a\u003e is used in the mechanical design of composite structures to estimate the elastic modulus (Ec) based on the following properties:\u003c/p\u003e\u003cpre\u003e    Ef: elastic modulus of the fiber material\r\n    Em: elastic modulus of the matrix material\r\n    ff or fm: volume fraction of the fiber or matrix material (ff = 1 – fm)\u003c/pre\u003e\u003cp\u003eBased on these values, the lower-bound estimate of elastic modulus is calculated by:\u003c/p\u003e\u003cp\u003eEc = 1 / (ff / Ef + (1 – ff) / Em).\u003c/p\u003e\u003cp\u003eOn the other hand, the upper-bound (linear) estimate of elastic modulus is calculated by:\u003c/p\u003e\u003cp\u003eEc = ff * Ef + (1 – ff) * Em.\u003c/p\u003e\u003cp\u003eWrite a function to calculate either bound, depending on which bound is requested in the input string.\u003c/p\u003e","function_template":"function [Ec] = rule_of_mixtures_either_bound(b_str,Ef,Em,ff)\r\n Ec = 1;\r\nend","test_suite":"%%\r\nEf = 100;\r\nEm = 10;\r\nff = 0.30;\r\nb_str = 'lower';\r\n[Ec] = rule_of_mixtures_either_bound(b_str,Ef,Em,ff);\r\nassert(abs(Ec - 13.6986) \u003c 1e-4)\r\n\r\n%%\r\nEf = 100;\r\nEm = 10;\r\nff = 0.22;\r\nb_str = 'L';\r\n[Ec] = rule_of_mixtures_either_bound(b_str,Ef,Em,ff);\r\nassert(abs(Ec - 12.4688) \u003c 1e-4)\r\n\r\n\r\n%%\r\nEf = 100;\r\nEm = 10;\r\nff = 0.15;\r\nb_str = 'Upper';\r\n[Ec] = rule_of_mixtures_either_bound(b_str,Ef,Em,ff);\r\nassert(abs(Ec - 23.5) \u003c 1e-4)\r\n\r\n%%\r\nEf = 1000;\r\nEm = 10;\r\nff = 0.30;\r\nb_str = 'Lower';\r\n[Ec] = rule_of_mixtures_either_bound(b_str,Ef,Em,ff);\r\nassert(abs(Ec - 14.2248) \u003c 1e-4)\r\n\r\n%%\r\nEf = 1000;\r\nEm = 10;\r\nff = 0.22;\r\nb_str = 'u';\r\n[Ec] = rule_of_mixtures_either_bound(b_str,Ef,Em,ff);\r\nassert(abs(Ec - 227.8) \u003c 1e-4)\r\n\r\n%%\r\nEf = 1000;\r\nEm = 10;\r\nff = 0.15;\r\nb_str = 'upper';\r\n[Ec] = rule_of_mixtures_either_bound(b_str,Ef,Em,ff);\r\nassert(abs(Ec - 158.5) \u003c 1e-4)\r\n","published":true,"deleted":false,"likes_count":2,"comments_count":0,"created_by":26769,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":78,"test_suite_updated_at":"2014-12-16T23:04:03.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2014-12-14T04:54:34.000Z","updated_at":"2026-02-13T03:58:28.000Z","published_at":"2014-12-14T04:54:34.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://en.wikipedia.org/wiki/Rule_of_mixtures\\\"\u003e\u003cw:r\u003e\u003cw:t\u003erule of mixtures\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e is used in the mechanical design of composite structures to estimate the elastic modulus (Ec) based on the following properties:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[    Ef: elastic modulus of the fiber material\\n    Em: elastic modulus of the matrix material\\n    ff or fm: volume fraction of the fiber or matrix material (ff = 1 – fm)]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eBased on these values, the lower-bound estimate of elastic modulus is calculated by:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eEc = 1 / (ff / Ef + (1 – ff) / Em).\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eOn the other hand, the upper-bound (linear) estimate of elastic modulus is calculated by:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eEc = ff * Ef + (1 – ff) * Em.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to calculate either bound, depending on which bound is requested in the input string.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":2778,"title":"Rule of mixtures (composites) - lower and upper bounds (volumes)","description":"The \u003chttp://en.wikipedia.org/wiki/Rule_of_mixtures rule of mixtures\u003e is used in the mechanical design of composite structures to estimate the elastic modulus (Ec) based on the following properties:\r\n\r\n    Ef: elastic modulus of the fiber material\r\n    Em: elastic modulus of the matrix material\r\n    ff or fm: volume fraction of the fiber or matrix material (ff = 1 – fm)\r\n\r\nSometimes, the volume fraction is not known and must be calculated from known volumes (Vf and Vm). In this case, ff = Vf / (Vf + Vm).\r\n\r\nBased on these values, the lower-bound estimate of elastic modulus is calculated by:\r\n\r\nEc = 1 / (ff / Ef + (1 – ff) / Em).\r\n\r\nOn the other hand, the upper-bound (linear) estimate of elastic modulus is calculated by:\r\n\r\nEc = ff * Ef + (1 – ff) * Em.\r\n\r\nWrite a function to calculate either bound, depending on which bound is requested in the input string and using the known volumes.\r\n","description_html":"\u003cp\u003eThe \u003ca href = \"http://en.wikipedia.org/wiki/Rule_of_mixtures\"\u003erule of mixtures\u003c/a\u003e is used in the mechanical design of composite structures to estimate the elastic modulus (Ec) based on the following properties:\u003c/p\u003e\u003cpre\u003e    Ef: elastic modulus of the fiber material\r\n    Em: elastic modulus of the matrix material\r\n    ff or fm: volume fraction of the fiber or matrix material (ff = 1 – fm)\u003c/pre\u003e\u003cp\u003eSometimes, the volume fraction is not known and must be calculated from known volumes (Vf and Vm). In this case, ff = Vf / (Vf + Vm).\u003c/p\u003e\u003cp\u003eBased on these values, the lower-bound estimate of elastic modulus is calculated by:\u003c/p\u003e\u003cp\u003eEc = 1 / (ff / Ef + (1 – ff) / Em).\u003c/p\u003e\u003cp\u003eOn the other hand, the upper-bound (linear) estimate of elastic modulus is calculated by:\u003c/p\u003e\u003cp\u003eEc = ff * Ef + (1 – ff) * Em.\u003c/p\u003e\u003cp\u003eWrite a function to calculate either bound, depending on which bound is requested in the input string and using the known volumes.\u003c/p\u003e","function_template":"function [Ec] = rule_of_mixtures_either_bound_vol(b_str,Ef,Em,Vf,Vm)\r\n Ec = 1;\r\nend","test_suite":"%%\r\nEf = 100;\r\nEm = 10;\r\nVf = 30;\r\nVm = 70;\r\nb_str = 'lower';\r\n[Ec] = rule_of_mixtures_either_bound_vol(b_str,Ef,Em,Vf,Vm);\r\nassert(abs(Ec - 13.6986) \u003c 1e-4)\r\n\r\n%%\r\nEf = 100;\r\nEm = 10;\r\nVf = 11;\r\nVm = 39;\r\nb_str = 'L';\r\n[Ec] = rule_of_mixtures_either_bound_vol(b_str,Ef,Em,Vf,Vm);\r\nassert(abs(Ec - 12.4688) \u003c 1e-4)\r\n\r\n\r\n%%\r\nEf = 100;\r\nEm = 10;\r\nVf = 150;\r\nVm = 850;\r\nb_str = 'Lower';\r\n[Ec] = rule_of_mixtures_either_bound_vol(b_str,Ef,Em,Vf,Vm);\r\nassert(abs(Ec - 11.5607) \u003c 1e-4)\r\n\r\n%%\r\nEf = 1000;\r\nEm = 10;\r\nVf = 3;\r\nVm = 7;\r\nb_str = 'U';\r\n[Ec] = rule_of_mixtures_either_bound_vol(b_str,Ef,Em,Vf,Vm);\r\nassert(abs(Ec - 307) \u003c 1e-4)\r\n\r\n%%\r\nEf = 1000;\r\nEm = 10;\r\nVf = 2.2;\r\nVm = 7.8;\r\nb_str = 'lower bound';\r\n[Ec] = rule_of_mixtures_either_bound_vol(b_str,Ef,Em,Vf,Vm);\r\nassert(abs(Ec - 12.7845) \u003c 1e-4)\r\n\r\n%%\r\nEf = 1000;\r\nEm = 10;\r\nVf = 30;\r\nVm = 170;\r\nb_str = 'U bound';\r\n[Ec] = rule_of_mixtures_either_bound_vol(b_str,Ef,Em,Vf,Vm);\r\nassert(abs(Ec - 158.5) \u003c 1e-4)\r\n\r\n%%\r\nEf = 57;\r\nEm = 3.9;\r\nVf = 1.27;\r\nVm = 9;\r\nb_str = 'Lower';\r\n[Ec] = rule_of_mixtures_either_bound_vol(b_str,Ef,Em,Vf,Vm);\r\nassert(abs(Ec - 4.4078) \u003c 1e-4)\r\n","published":true,"deleted":false,"likes_count":0,"comments_count":1,"created_by":26769,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":73,"test_suite_updated_at":"2014-12-16T23:05:00.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2014-12-14T05:06:25.000Z","updated_at":"2026-02-13T04:00:39.000Z","published_at":"2014-12-14T05:06:25.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://en.wikipedia.org/wiki/Rule_of_mixtures\\\"\u003e\u003cw:r\u003e\u003cw:t\u003erule of mixtures\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e is used in the mechanical design of composite structures to estimate the elastic modulus (Ec) based on the following properties:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[    Ef: elastic modulus of the fiber material\\n    Em: elastic modulus of the matrix material\\n    ff or fm: volume fraction of the fiber or matrix material (ff = 1 – fm)]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eSometimes, the volume fraction is not known and must be calculated from known volumes (Vf and Vm). In this case, ff = Vf / (Vf + Vm).\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eBased on these values, the lower-bound estimate of elastic modulus is calculated by:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eEc = 1 / (ff / Ef + (1 – ff) / Em).\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eOn the other hand, the upper-bound (linear) estimate of elastic modulus is calculated by:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eEc = ff * Ef + (1 – ff) * Em.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to calculate either bound, depending on which bound is requested in the input string and using the known volumes.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":2779,"title":"Rule of mixtures (composites) - weighted bound","description":"The \u003chttp://en.wikipedia.org/wiki/Rule_of_mixtures rule of mixtures\u003e is used in the mechanical design of composite structures to estimate the elastic modulus (Ec) based on the following properties:\r\n\r\n* Ef: elastic modulus of the fiber material\r\n* Em: elastic modulus of the matrix material\r\n* ff or fm: volume fraction of the fiber or matrix material (ff = 1 – fm)\r\n\r\nBased on these values, the lower-bound estimate of elastic modulus is calculated by:\r\n\r\nEc = 1 / (ff / Ef + (1 – ff) / Em).\r\n\r\nOn the other hand, the upper-bound (linear) estimate of elastic modulus is calculated by:\r\n\r\nEc = ff * Ef + (1 – ff) * Em.\r\n\r\nWrite a function to calculate the weighted Ec between both bounds, based on the provided weighting (wt) of the upper bound. (The lower bound will have the remainder of the weighting, or 1–wt.)\r\n","description_html":"\u003cp\u003eThe \u003ca href = \"http://en.wikipedia.org/wiki/Rule_of_mixtures\"\u003erule of mixtures\u003c/a\u003e is used in the mechanical design of composite structures to estimate the elastic modulus (Ec) based on the following properties:\u003c/p\u003e\u003cul\u003e\u003cli\u003eEf: elastic modulus of the fiber material\u003c/li\u003e\u003cli\u003eEm: elastic modulus of the matrix material\u003c/li\u003e\u003cli\u003eff or fm: volume fraction of the fiber or matrix material (ff = 1 – fm)\u003c/li\u003e\u003c/ul\u003e\u003cp\u003eBased on these values, the lower-bound estimate of elastic modulus is calculated by:\u003c/p\u003e\u003cp\u003eEc = 1 / (ff / Ef + (1 – ff) / Em).\u003c/p\u003e\u003cp\u003eOn the other hand, the upper-bound (linear) estimate of elastic modulus is calculated by:\u003c/p\u003e\u003cp\u003eEc = ff * Ef + (1 – ff) * Em.\u003c/p\u003e\u003cp\u003eWrite a function to calculate the weighted Ec between both bounds, based on the provided weighting (wt) of the upper bound. (The lower bound will have the remainder of the weighting, or 1–wt.)\u003c/p\u003e","function_template":"function [Ec] = rule_of_mixtures_wt_bound(Ef,Em,ff,wt)\r\n Ec = 1;\r\nend","test_suite":"%%\r\nEf = 100;\r\nEm = 10;\r\nff = 0.30;\r\nwt = 0.25;\r\n[Ec] = rule_of_mixtures_wt_bound(Ef,Em,ff,wt);\r\nassert(abs(Ec - 19.5240) \u003c 1e-4)\r\n\r\n\r\n%%\r\nEf = 100;\r\nEm = 10;\r\nff = 0.30;\r\nwt = 0.50;\r\n[Ec] = rule_of_mixtures_wt_bound(Ef,Em,ff,wt);\r\nassert(abs(Ec - 25.3493) \u003c 1e-4)\r\n\r\n\r\n%%\r\nEf = 100;\r\nEm = 10;\r\nff = 0.30;\r\nwt = 0.75;\r\n[Ec] = rule_of_mixtures_wt_bound(Ef,Em,ff,wt);\r\nassert(abs(Ec - 31.1747) \u003c 1e-4)\r\n\r\n\r\n%%\r\nEf = 100;\r\nEm = 10;\r\nff = 0.15;\r\nwt = 0.25;\r\n[Ec] = rule_of_mixtures_wt_bound(Ef,Em,ff,wt);\r\nassert(abs(Ec - 14.5455) \u003c 1e-4)\r\n\r\n%%\r\nEf = 100;\r\nEm = 10;\r\nff = 0.15;\r\nwt = 0.50;\r\n[Ec] = rule_of_mixtures_wt_bound(Ef,Em,ff,wt);\r\nassert(abs(Ec - 17.5303) \u003c 1e-4)\r\n\r\n%%\r\nEf = 100;\r\nEm = 10;\r\nff = 0.15;\r\nwt = 0.75;\r\n[Ec] = rule_of_mixtures_wt_bound(Ef,Em,ff,wt);\r\nassert(abs(Ec - 20.5152) \u003c 1e-4)\r\n\r\n%%\r\nEf = 1000;\r\nEm = 10;\r\nff = 0.30;\r\nwt = 0.25;\r\n[Ec] = rule_of_mixtures_wt_bound(Ef,Em,ff,wt);\r\nassert(abs(Ec - 87.4186) \u003c 1e-4)\r\n\r\n%%\r\nEf = 1000;\r\nEm = 10;\r\nff = 0.30;\r\nwt = 0.50;\r\n[Ec] = rule_of_mixtures_wt_bound(Ef,Em,ff,wt);\r\nassert(abs(Ec - 160.6124) \u003c 1e-4)\r\n\r\n%%\r\nEf = 1000;\r\nEm = 10;\r\nff = 0.30;\r\nwt = 0.75;\r\n[Ec] = rule_of_mixtures_wt_bound(Ef,Em,ff,wt);\r\nassert(abs(Ec - 233.8062) \u003c 1e-4)\r\n\r\n%%\r\nEf = 1000;\r\nEm = 10;\r\nff = 0.15;\r\nwt = 0.25;\r\n[Ec] = rule_of_mixtures_wt_bound(Ef,Em,ff,wt);\r\nassert(abs(Ec - 48.4330) \u003c 1e-4)\r\n\r\n%%\r\nEf = 1000;\r\nEm = 10;\r\nff = 0.15;\r\nwt = 0.50;\r\n[Ec] = rule_of_mixtures_wt_bound(Ef,Em,ff,wt);\r\nassert(abs(Ec - 85.1220) \u003c 1e-4)\r\n\r\n%%\r\nEf = 1000;\r\nEm = 10;\r\nff = 0.15;\r\nwt = 0.75;\r\n[Ec] = rule_of_mixtures_wt_bound(Ef,Em,ff,wt);\r\nassert(abs(Ec - 121.8110) \u003c 1e-4)\r\n","published":true,"deleted":false,"likes_count":2,"comments_count":0,"created_by":26769,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":94,"test_suite_updated_at":"2014-12-16T23:05:34.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2014-12-16T04:22:51.000Z","updated_at":"2026-02-13T07:13:31.000Z","published_at":"2014-12-16T04:22:51.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://en.wikipedia.org/wiki/Rule_of_mixtures\\\"\u003e\u003cw:r\u003e\u003cw:t\u003erule of mixtures\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e is used in the mechanical design of composite structures to estimate the elastic modulus (Ec) based on the following properties:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eEf: elastic modulus of the fiber material\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eEm: elastic modulus of the matrix material\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eff or fm: volume fraction of the fiber or matrix material (ff = 1 – fm)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eBased on these values, the lower-bound estimate of elastic modulus is calculated by:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eEc = 1 / (ff / Ef + (1 – ff) / Em).\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eOn the other hand, the upper-bound (linear) estimate of elastic modulus is calculated by:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eEc = ff * Ef + (1 – ff) * Em.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to calculate the weighted Ec between both bounds, based on the provided weighting (wt) of the upper bound. (The lower bound will have the remainder of the weighting, or 1–wt.)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":2780,"title":"Rule of mixtures (composites) - other bound","description":"The \u003chttp://en.wikipedia.org/wiki/Rule_of_mixtures rule of mixtures\u003e is used in the mechanical design of composite structures to estimate the elastic modulus (Ec) based on the following properties:\r\n\r\n* Ef: elastic modulus of the fiber material\r\n* Em: elastic modulus of the matrix material\r\n* ff or fm: volume fraction of the fiber or matrix material (ff = 1 – fm)\r\n\r\nYou are provided with the estimated elastic modulus at one bound. Determine if it is the lower or upper bound, based on the provided material properties, and return which bound was provided (0 = lower, 1 = upper) and the value for the other bound.\r\n\r\nThe lower-bound estimate of elastic modulus is calculated by:\r\n\r\nEc = 1 / (ff / Ef + (1 – ff) / Em).\r\n\r\nThe upper-bound (linear) estimate of elastic modulus is calculated by:\r\n\r\nEc = ff * Ef + (1 – ff) * Em.\r\n","description_html":"\u003cp\u003eThe \u003ca href = \"http://en.wikipedia.org/wiki/Rule_of_mixtures\"\u003erule of mixtures\u003c/a\u003e is used in the mechanical design of composite structures to estimate the elastic modulus (Ec) based on the following properties:\u003c/p\u003e\u003cul\u003e\u003cli\u003eEf: elastic modulus of the fiber material\u003c/li\u003e\u003cli\u003eEm: elastic modulus of the matrix material\u003c/li\u003e\u003cli\u003eff or fm: volume fraction of the fiber or matrix material (ff = 1 – fm)\u003c/li\u003e\u003c/ul\u003e\u003cp\u003eYou are provided with the estimated elastic modulus at one bound. Determine if it is the lower or upper bound, based on the provided material properties, and return which bound was provided (0 = lower, 1 = upper) and the value for the other bound.\u003c/p\u003e\u003cp\u003eThe lower-bound estimate of elastic modulus is calculated by:\u003c/p\u003e\u003cp\u003eEc = 1 / (ff / Ef + (1 – ff) / Em).\u003c/p\u003e\u003cp\u003eThe upper-bound (linear) estimate of elastic modulus is calculated by:\u003c/p\u003e\u003cp\u003eEc = ff * Ef + (1 – ff) * Em.\u003c/p\u003e","function_template":"function [Ec_other,u_or_l] = rule_of_mixtures_opp_bound(Ec,Ef,Em,ff)\r\n Ec_other = 1;\r\nend","test_suite":"%%\r\nEc = 37;\r\nEf = 100;\r\nEm = 10;\r\nff = 0.30;\r\n[Ec_other,u_or_l] = rule_of_mixtures_opp_bound(Ec,Ef,Em,ff);\r\nassert(abs(Ec_other - 13.6986) \u003c 1e-4)\r\nassert(u_or_l == 1)\r\n\r\n%%\r\nEc = 13.6986;\r\nEf = 100;\r\nEm = 10;\r\nff = 0.30;\r\n[Ec_other,u_or_l] = rule_of_mixtures_opp_bound(Ec,Ef,Em,ff);\r\nassert(abs(Ec_other - 37) \u003c 1e-4)\r\nassert(u_or_l == 0)\r\n\r\n%%\r\nEc = 23.5;\r\nEf = 100;\r\nEm = 10;\r\nff = 0.15;\r\n[Ec_other,u_or_l] = rule_of_mixtures_opp_bound(Ec,Ef,Em,ff);\r\nassert(abs(Ec_other - 11.5607) \u003c 1e-4)\r\nassert(u_or_l == 1)\r\n\r\n%%\r\nEc = 11.5607;\r\nEf = 100;\r\nEm = 10;\r\nff = 0.15;\r\n[Ec_other,u_or_l] = rule_of_mixtures_opp_bound(Ec,Ef,Em,ff);\r\nassert(abs(Ec_other - 23.5) \u003c 1e-4)\r\nassert(u_or_l == 0)\r\n\r\n%%\r\nEc = 11.7440;\r\nEf = 1000;\r\nEm = 10;\r\nff = 0.15;\r\n[Ec_other,u_or_l] = rule_of_mixtures_opp_bound(Ec,Ef,Em,ff);\r\nassert(abs(Ec_other - 158.5) \u003c 1e-4)\r\nassert(u_or_l == 0)\r\n\r\n%%\r\nEc = 158.5;\r\nEf = 1000;\r\nEm = 10;\r\nff = 0.15;\r\n[Ec_other,u_or_l] = rule_of_mixtures_opp_bound(Ec,Ef,Em,ff);\r\nassert(abs(Ec_other - 11.7440) \u003c 1e-4)\r\nassert(u_or_l == 1)\r\n","published":true,"deleted":false,"likes_count":3,"comments_count":0,"created_by":26769,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":72,"test_suite_updated_at":"2014-12-16T23:06:11.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2014-12-16T04:42:12.000Z","updated_at":"2026-02-13T07:37:07.000Z","published_at":"2014-12-16T04:42:12.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://en.wikipedia.org/wiki/Rule_of_mixtures\\\"\u003e\u003cw:r\u003e\u003cw:t\u003erule of mixtures\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e is used in the mechanical design of composite structures to estimate the elastic modulus (Ec) based on the following properties:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eEf: elastic modulus of the fiber material\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eEm: elastic modulus of the matrix material\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eff or fm: volume fraction of the fiber or matrix material (ff = 1 – fm)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eYou are provided with the estimated elastic modulus at one bound. Determine if it is the lower or upper bound, based on the provided material properties, and return which bound was provided (0 = lower, 1 = upper) and the value for the other bound.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe lower-bound estimate of elastic modulus is calculated by:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eEc = 1 / (ff / Ef + (1 – ff) / Em).\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe upper-bound (linear) estimate of elastic modulus is calculated by:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eEc = ff * Ef + (1 – ff) * Em.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"},{"id":2781,"title":"Rule of mixtures (composites) - reverse engineering","description":"The \u003chttp://en.wikipedia.org/wiki/Rule_of_mixtures rule of mixtures\u003e is used in the mechanical design of composite structures to estimate the elastic modulus (Ec) based on the following properties:\r\n\r\n* Ef: elastic modulus of the fiber material\r\n* Em: elastic modulus of the matrix material\r\n* ff or fm: volume fraction of the fiber or matrix material (ff = 1 – fm)\r\n\r\nSuppose that you are reverse engineering a composite article. You've tested coupons from the article to determine Ec and taken sections to determine the material properties and volume fraction for the fiber material. Using the known values, calculate modulus estimates for the matrix material based on both weightings of upper- and lower-bounds. Provide values for (upper-bound) weightings of 0, 1/4, 1/2, 3/4, and 1 (weighting is not related to fiber fraction or material volume).\r\n\r\nThe lower-bound estimate of elastic modulus is calculated by:\r\n\r\nEc = 1 / (ff / Ef + (1 – ff) / Em)   [eq.1]\r\n\r\nThe upper-bound (linear) estimate of elastic modulus is calculated by:\r\n\r\nEc = ff * Ef + (1 – ff) * Em   [eq.2]\r\n\r\nFor example, for each test case you will calculate Em_l, the lower-bound matrix elastic modulus extrapolated by eq.1, and Em_u, the upper-bound matrix elastic modulus extrapolated by eq.2. The weighted matrix elastic modulus is given by Em = wt * Em_u + (1-wt) * Em_l. For a weighting of 1/4, the weighted modulus is therefore given by Em = 0.25 * Em_u + 0.75 * Em_l.\r\n","description_html":"\u003cp\u003eThe \u003ca href = \"http://en.wikipedia.org/wiki/Rule_of_mixtures\"\u003erule of mixtures\u003c/a\u003e is used in the mechanical design of composite structures to estimate the elastic modulus (Ec) based on the following properties:\u003c/p\u003e\u003cul\u003e\u003cli\u003eEf: elastic modulus of the fiber material\u003c/li\u003e\u003cli\u003eEm: elastic modulus of the matrix material\u003c/li\u003e\u003cli\u003eff or fm: volume fraction of the fiber or matrix material (ff = 1 – fm)\u003c/li\u003e\u003c/ul\u003e\u003cp\u003eSuppose that you are reverse engineering a composite article. You've tested coupons from the article to determine Ec and taken sections to determine the material properties and volume fraction for the fiber material. Using the known values, calculate modulus estimates for the matrix material based on both weightings of upper- and lower-bounds. Provide values for (upper-bound) weightings of 0, 1/4, 1/2, 3/4, and 1 (weighting is not related to fiber fraction or material volume).\u003c/p\u003e\u003cp\u003eThe lower-bound estimate of elastic modulus is calculated by:\u003c/p\u003e\u003cp\u003eEc = 1 / (ff / Ef + (1 – ff) / Em)   [eq.1]\u003c/p\u003e\u003cp\u003eThe upper-bound (linear) estimate of elastic modulus is calculated by:\u003c/p\u003e\u003cp\u003eEc = ff * Ef + (1 – ff) * Em   [eq.2]\u003c/p\u003e\u003cp\u003eFor example, for each test case you will calculate Em_l, the lower-bound matrix elastic modulus extrapolated by eq.1, and Em_u, the upper-bound matrix elastic modulus extrapolated by eq.2. The weighted matrix elastic modulus is given by Em = wt * Em_u + (1-wt) * Em_l. For a weighting of 1/4, the weighted modulus is therefore given by Em = 0.25 * Em_u + 0.75 * Em_l.\u003c/p\u003e","function_template":"function [Em] = rule_of_mixtures_rev_eng(Ec,Ef,ff)\r\n Em = ones(5,1);\r\nend","test_suite":"%%\r\nEc = 35.4;\r\nEf = 100;\r\nff = 0.30;\r\n[Em] = rule_of_mixtures_rev_eng(Ec,Ef,ff);\r\nassert(sum(abs(sort(Em) - [7.7143   12.7168   17.7193   22.7218   27.7243])) \u003c 1e-2)\r\n\r\n%%\r\nEc = 85.1;\r\nEf = 250;\r\nff = 0.20;\r\n[Em] = rule_of_mixtures_rev_eng(Ec,Ef,ff);\r\nassert(sum(abs(sort(Em) - [43.8750   51.1696   58.4642   65.7589   73.0535])) \u003c 1e-2)\r\n\r\n%%\r\nEc = 155.5;\r\nEf = 1000;\r\nff = 0.05;\r\n[Em] = rule_of_mixtures_rev_eng(Ec,Ef,ff);\r\nassert(sum(abs(sort(Em) - [111.0526  120.5101  129.9676  139.4251  148.8826])) \u003c 1e-2)\r\n\r\n%%\r\nEc = 27.6;\r\nEf = 100;\r\nff = 0.10;\r\n[Em] = rule_of_mixtures_rev_eng(Ec,Ef,ff);\r\nassert(sum(abs(sort(Em) - [19.5556   21.0529   22.5503   24.0477   25.5450])) \u003c 1e-2)\r\n\r\n%%\r\nEc = 204.9;\r\nEf = 1000;\r\nff = 0.15;\r\n[Em] = rule_of_mixtures_rev_eng(Ec,Ef,ff);\r\nassert(sum(abs(sort(Em) - [64.5882   93.3631  122.1380  150.9128  179.6877])) \u003c 1e-2)\r\n","published":true,"deleted":false,"likes_count":1,"comments_count":6,"created_by":26769,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":54,"test_suite_updated_at":"2014-12-16T23:18:59.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2014-12-16T05:33:10.000Z","updated_at":"2026-02-13T10:45:04.000Z","published_at":"2014-12-16T05:33:10.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://en.wikipedia.org/wiki/Rule_of_mixtures\\\"\u003e\u003cw:r\u003e\u003cw:t\u003erule of mixtures\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e is used in the mechanical design of composite structures to estimate the elastic modulus (Ec) based on the following properties:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eEf: elastic modulus of the fiber material\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eEm: elastic modulus of the matrix material\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"1\\\"/\u003e\u003c/w:numPr\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eff or fm: volume fraction of the fiber or matrix material (ff = 1 – fm)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eSuppose that you are reverse engineering a composite article. You've tested coupons from the article to determine Ec and taken sections to determine the material properties and volume fraction for the fiber material. Using the known values, calculate modulus estimates for the matrix material based on both weightings of upper- and lower-bounds. Provide values for (upper-bound) weightings of 0, 1/4, 1/2, 3/4, and 1 (weighting is not related to fiber fraction or material volume).\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe lower-bound estimate of elastic modulus is calculated by:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eEc = 1 / (ff / Ef + (1 – ff) / Em) [eq.1]\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe upper-bound (linear) estimate of elastic modulus is calculated by:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eEc = ff * Ef + (1 – ff) * Em [eq.2]\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor example, for each test case you will calculate Em_l, the lower-bound matrix elastic modulus extrapolated by eq.1, and Em_u, the upper-bound matrix elastic modulus extrapolated by eq.2. The weighted matrix elastic modulus is given by Em = wt * Em_u + (1-wt) * Em_l. For a weighting of 1/4, the weighted modulus is therefore given by Em = 0.25 * Em_u + 0.75 * Em_l.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"}],"no_progress_badge":{"id":53,"name":"Unknown","symbol":"unknown","description":"Partially completed groups","description_html":null,"image_location":"/images/responsive/supporting/matlabcentral/cody/badges/problem_groups_unknown_2.png","bonus":null,"players_count":0,"active":false,"created_by":null,"updated_by":null,"deleted_by":null,"deleted_at":null,"restored_by":null,"restored_at":null,"created_at":"2018-01-10T23:20:29.000Z","updated_at":"2018-01-10T23:20:29.000Z","community_badge_id":null,"award_multiples":false}}