Given the generalized Hooke’s law for linear isotropic materials (on the equation sheet), derive the simplified equations for plane stress and plane strain conditions. Material Property Relations Poisson’s ratio v= Elat Elong Generalized Hooke’s Law -=[\u03c3-\u03bd(\u03c3 +\u03c3)] y = [-1(x+2)] \u20ac =[2-(x+y)] 1 1 Yxy=xyYyzyzYzx=x where E 2(1+1)<\/p>\n\n\n\n
The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n For a linear isotropic material under 3D stress, the strain components are expressed as:<\/p>\n\n\n\n [ where ( E ) is Young’s modulus, ( \\nu ) is Poisson’s ratio, and ( G = \\frac{E}{2(1 + \\nu)} ) is the shear modulus.<\/p>\n\n\n\n Plane stress assumes that out-of-plane stresses are zero (( \\sigma_z = \\tau_{yz} = \\tau_{zx} = 0 )). Substituting ( \\sigma_z = 0 ) into the generalized equations:<\/p>\n\n\n\n [ Here, ( \\varepsilon_z ) is a secondary strain caused by the in-plane stresses due to Poisson’s effect.<\/p>\n\n\n\n Plane strain assumes that out-of-plane strains are zero (( \\varepsilon_z = \\gamma_{yz} = \\gamma_{zx} = 0 )). From the strain equation for ( \\varepsilon_z ):<\/p>\n\n\n\n [ Solving for ( \\sigma_z ): Substitute ( \\sigma_z = \\nu (\\sigma_x + \\sigma_y) ) into the in-plane strain equations: Simplify: The derivation shows how 3D stress-strain relations reduce to simplified equations under specific assumptions. In plane stress<\/strong>, the absence of out-of-plane stress simplifies the strain directly, while in plane strain<\/strong>, out-of-plane constraints induce an equivalent stress (( \\sigma_z )). These relations are critical for structural analysis where one dimension is negligible (plane stress) or constrained (plane strain).<\/p>\n","protected":false},"excerpt":{"rendered":" Given the generalized Hooke’s law for linear isotropic materials (on the equation sheet), derive the simplified equations for plane stress and plane strain conditions. Material Property Relations Poisson’s ratio v= Elat Elong Generalized Hooke’s Law -=[\u03c3-\u03bd(\u03c3 +\u03c3)] y = [-1(x+2)] \u20ac =[2-(x+y)] 1 1 Yxy=xyYyzyzYzx=x where E 2(1+1) The Correct Answer and Explanation is : […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-185257","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/185257","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=185257"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/185257\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=185257"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=185257"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=185257"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}Derivation of Simplified Equations for Plane Stress and Plane Strain Conditions<\/h3>\n\n\n\n
Generalized Hooke’s Law:<\/h4>\n\n\n\n
\\varepsilon_x = \\frac{1}{E} \\left[\\sigma_x – \\nu (\\sigma_y + \\sigma_z) \\right]
]
[
\\varepsilon_y = \\frac{1}{E} \\left[\\sigma_y – \\nu (\\sigma_x + \\sigma_z) \\right]
]
[
\\varepsilon_z = \\frac{1}{E} \\left[\\sigma_z – \\nu (\\sigma_x + \\sigma_y) \\right]
]
[
\\gamma_{xy} = \\frac{\\tau_{xy}}{G}, \\quad \\gamma_{yz} = \\frac{\\tau_{yz}}{G}, \\quad \\gamma_{zx} = \\frac{\\tau_{zx}}{G}
]<\/p>\n\n\n\n
\n\n\n\nPlane Stress Conditions:<\/h3>\n\n\n\n
\\varepsilon_x = \\frac{1}{E} \\left[\\sigma_x – \\nu \\sigma_y \\right]
]
[
\\varepsilon_y = \\frac{1}{E} \\left[\\sigma_y – \\nu \\sigma_x \\right]
]
[
\\varepsilon_z = \\frac{-\\nu}{E} (\\sigma_x + \\sigma_y)
]<\/p>\n\n\n\n
\n\n\n\nPlane Strain Conditions:<\/h3>\n\n\n\n
\\varepsilon_z = \\frac{1}{E} \\left[\\sigma_z – \\nu (\\sigma_x + \\sigma_y) \\right] = 0
]<\/p>\n\n\n\n
[
\\sigma_z = \\nu (\\sigma_x + \\sigma_y)
]<\/p>\n\n\n\n
[
\\varepsilon_x = \\frac{1}{E} \\left[\\sigma_x – \\nu (\\sigma_y + \\nu (\\sigma_x + \\sigma_y)) \\right]
]
[
\\varepsilon_y = \\frac{1}{E} \\left[\\sigma_y – \\nu (\\sigma_x + \\nu (\\sigma_x + \\sigma_y)) \\right]
]<\/p>\n\n\n\n
[
\\varepsilon_x = \\frac{1 – \\nu^2}{E} \\sigma_x – \\frac{\\nu (1 + \\nu)}{E} \\sigma_y
]
[
\\varepsilon_y = \\frac{1 – \\nu^2}{E} \\sigma_y – \\frac{\\nu (1 + \\nu)}{E} \\sigma_x
]<\/p>\n\n\n\n
\n\n\n\nExplanation:<\/h3>\n\n\n\n