{"id":2918,"date":"2021-02-08T14:01:55","date_gmt":"2021-02-08T14:01:55","guid":{"rendered":"https:\/\/pickedshares.com\/?p=2918"},"modified":"2021-05-08T13:07:00","modified_gmt":"2021-05-08T13:07:00","slug":"engineering-mechanics-ii-exercise-compression-of-an-elastic-body","status":"publish","type":"post","link":"https:\/\/pickedshares.com\/en\/engineering-mechanics-ii-exercise-compression-of-an-elastic-body\/","title":{"rendered":"Compression of an elastic body"},"content":{"rendered":"\n<p>In this exercise the compression of an elastic body under different external constraints is considered.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Task<\/h2>\n\n\n\n<p>An elastic body is embedded in a rigid <a href=\"https:\/\/en.wikipedia.org\/wiki\/Die_(manufacturing)\" target=\"_blank\" rel=\"noreferrer noopener\">die<\/a> without clearance and without friction. The body is loaded with the force F via a rigid plunger.<\/p>\n\n\n\n<p><strong>Given parameters: <\/strong><\/p>\n\n\n\n<p>Dimensions a, b, c, Young's modulus E, Poisson's ratio \u03bd, strain \u03b8<sub>z<\/sub><\/p>\n\n\n\n<p><strong>To be determined:<\/strong><\/p>\n\n\n\n<p>a) Compression force F to compress the body in the z-direction by \u03b8<sub>z<\/sub>.<\/p>\n\n\n\n<p>b) Pressure on the surfaces of the die.<\/p>\n\n\n\n<p>c) Change of the contour length under the force F when the die is removed.<\/p>\n\n\n\n<p>d) Compression force F * with a compression by \u03b8<sub>z<\/sub> without a die.<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"801\" src=\"https:\/\/pickedshares.com\/wp-content\/uploads\/2021\/02\/tm2-14-1024x801.png\" alt=\"Compression of an elastic body\" class=\"wp-image-2904\" srcset=\"https:\/\/pickedshares.com\/wp-content\/uploads\/2021\/02\/tm2-14-1024x801.png 1024w, https:\/\/pickedshares.com\/wp-content\/uploads\/2021\/02\/tm2-14-300x235.png 300w, https:\/\/pickedshares.com\/wp-content\/uploads\/2021\/02\/tm2-14-768x601.png 768w, https:\/\/pickedshares.com\/wp-content\/uploads\/2021\/02\/tm2-14-400x313.png 400w, https:\/\/pickedshares.com\/wp-content\/uploads\/2021\/02\/tm2-14-800x626.png 800w, https:\/\/pickedshares.com\/wp-content\/uploads\/2021\/02\/tm2-14.png 1141w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><figcaption>Compression of an elastic body<\/figcaption><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\">Solution<\/h2>\n\n\n\n<script src=\"https:\/\/cdnjs.cloudflare.com\/ajax\/libs\/mathjax\/2.7.7\/MathJax.js?config=TeX-AMS_HTML\" async=\"async\">  \/\/ A comment that hinders wxWidgets from optimizing this tag too much.\n<\/script>\n<p>Hooke's law is:<\/p>\n<p>\\[ \\tag{1} \\epsilon_x = \\frac{1}{E} \\left[ \\sigma_x - \\nu \\left( \\sigma_y + \\sigma_z  \\right) \\right]  \\]<\/p>\n<p>\\[ \\tag{2} \\epsilon_y = \\frac{1}{E} \\left[ \\sigma_y - \\nu \\left( \\sigma_x + \\sigma_z  \\right) \\right]  \\]<\/p>\n<p>\\[ \\tag{3} \\epsilon_z = \\frac{1}{E} \\left[ \\sigma_z - \\nu \\left( \\sigma_x + \\sigma_y  \\right) \\right]  \\]<\/p>\n\n<p><strong>reg. a)<\/strong><\/p>\n<p>The strains in direction x and y are Zero.<\/p>\n<p>\\[ \\tag{4} \\epsilon_x = 0 \\]<\/p>\n<p>\\[ \\tag{5} \\epsilon_y = 0 \\]<\/p>\n<p>The strain in z-direction is<\/p>\n<p>\\[ \\tag{5} \\epsilon_z = - \\frac{\\theta_z}{c} \\]<\/p>\n<p>The tension in z-direction is<\/p>\n<p>\\[ \\tag{6} \\sigma_z = -\\frac{F}{ab} \\]<\/p>\n\n<p>With this Hooke's law leads to<\/p>\n<p>\\[ \\tag{7} F = \\frac{\\theta_z a b E}{c \\left( 1 - \\frac{2 \\nu^2}{1-\\nu} \\right)} \\]<\/p>\n\n<p><strong>reg. b)<\/strong><\/p>\n<p>The pressure at the surfaces of the die is<\/p>\n<p>\\[ \\tag{8} \\sigma_x = \\sigma_y = -\\frac{F \\nu}{ab \\left( 1 - \\nu \\right)} \\]<\/p>\n\n<p><strong>reg. c)<\/strong><\/p>\n<p>When the die is removed, the tensions are<\/p>\n<p>\\[ \\tag{9} \\sigma_x = 0 \\]<\/p>\n<p>\\[ \\tag{10} \\sigma_y = 0 \\]<\/p>\n<p>\\[ \\tag{11} \\sigma_z = -\\frac{F}{ab} \\]<\/p>\n<p>from which the following changes in the contour lengths result<\/p>\n\n<p>\\[ \\tag{12} \\Delta b = \\epsilon_x b = \\frac{\\nu F}{Ea} \\]<\/p>\n<p>\\[ \\tag{13} \\Delta a = \\epsilon_y a = \\frac{\\nu F}{Eb} \\]<\/p>\n<p>\\[ \\tag{14} \\Delta c = \\epsilon_z c = -\\frac{F c}{abE} \\]<\/p>\n\n<p><strong>reg. d)<\/strong><\/p>\n<p>The stresses \u03c3<sub>x<\/sub> and \u03c3<sub>y<\/sub> are zero. The strain is<\/p>\n\n<div style=\"overflow:auto;\">\n<p>\\[ \\tag{15} \\epsilon_z = -\\frac{\\theta_z}{c} = \\frac{\\sigma_z}{E} = - \\frac{F^*}{abE}  \\]<\/p>\n<\/div>\n<p>\\[ \\tag{16} F^* = \\frac{\\theta_z E a b}{c} \\]<\/p>\n","protected":false},"excerpt":{"rendered":"<p> ... <a title=\"Compression of an elastic body\" class=\"read-more\" href=\"https:\/\/pickedshares.com\/en\/engineering-mechanics-ii-exercise-compression-of-an-elastic-body\/\" aria-label=\"Read more about Compression of an elastic body\">Read more<\/a><\/p>\n","protected":false},"author":1,"featured_media":2906,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[52,60],"tags":[],"class_list":["post-2918","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-engineering-mechanics-ii","category-exercises","infinite-scroll-item","masonry-post","generate-columns","tablet-grid-50","mobile-grid-100","grid-parent","grid-33"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Compression of an elastic body &#8226; 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