{"id":8822,"date":"2021-06-18T02:22:17","date_gmt":"2021-06-18T06:22:17","guid":{"rendered":"https:\/\/www.purdue.edu\/freeform\/me323\/?page_id=8822"},"modified":"2024-08-20T08:23:13","modified_gmt":"2024-08-20T12:23:13","slug":"general-state-of-stress-and-strain","status":"publish","type":"page","link":"https:\/\/www.purdue.edu\/freeform\/me323\/animations-and-demonstrations\/general-state-of-stress-and-strain\/","title":{"rendered":"Poisson&#8217;s ratio"},"content":{"rendered":"<p><em><strong><span style=\"font-size: 18pt\">Uniaxial loading<\/span><\/strong><\/em><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-9132 aligncenter\" src=\"https:\/\/www.purdue.edu\/freeform\/me323\/wp-content\/uploads\/sites\/2\/2021\/08\/positive_poisson-1-249x300.jpg\" alt=\"\" width=\"249\" height=\"300\" srcset=\"https:\/\/www.purdue.edu\/freeform\/me323\/wp-content\/uploads\/sites\/2\/2021\/08\/positive_poisson-1-249x300.jpg 249w, https:\/\/www.purdue.edu\/freeform\/me323\/wp-content\/uploads\/sites\/2\/2021\/08\/positive_poisson-1.jpg 382w\" sizes=\"auto, (max-width: 249px) 100vw, 249px\" \/><\/p>\n<ul>\n<li>Tensile\/compressive stress in the <em>x<\/em>-direction produces tensile\/compressive strain in the <em>x<\/em>-direction, with the stress and strain related through Young&#8217;s modulus <em>E<\/em>:<\/li>\n<li>Strain in the <em>x<\/em>-direction produces strains the <em>y<\/em>&#8211; and <em>z<\/em>-directions. The strains in the <em>y<\/em>&#8211; and <em>z<\/em>-directions are proportional to the strain in the x-direction through the negative of the Poisson&#8217;s ratio.<\/li>\n<\/ul>\n<p><em><strong><span style=\"font-size: 14pt\">Positive Poisson&#8217;s ratio<\/span><\/strong><\/em><br \/>\nIf the Poisson&#8217;s ratio of the material if positive, then tensile\/compressive strains in the x-direction produce compressive\/tensile strains in the y- and z-directions, as shown below.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" style=\"border: 1px solid #000000\" src=\"https:\/\/www.purdue.edu\/freeform\/me323\/wp-content\/uploads\/sites\/2\/2021\/06\/threed_stress_strain.gif\" width=\"265\" height=\"245\" \/><\/p>\n<p><em><strong><span style=\"font-size: 14pt\">Negative Poisson&#8217;s ratio<\/span><\/strong><\/em><br \/>\nIf the Poisson&#8217;s ratio of the material if positive, then tensile\/compressive strains in the x-direction produce tensile\/compressive strains in the y- and z-directions, just the opposite of that for a positive Poisson&#8217;s ratio.<\/p>\n<p><em><u>QUESTION<\/u><\/em>: Can a material possess a negative Poisson\u2019s ratio? The answer is \u201cyes\u201d. Such materials are known as \u201c<strong><em>auxetic materials<\/em><\/strong>\u201d. From above, we see than an auxetic material will expand in the transverse directions for a tensile axial load. Consider the animation below of such a material, and study how a negative value for Poisson&#8217;s ratio is possible with this material. As can be seen here, a compressive axial load produces contraction in the transverse direction, and a tensile axial load produces expansion in the transverse direction.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" style=\"border: 1px solid #000000\" src=\"https:\/\/www.purdue.edu\/freeform\/me323\/wp-content\/uploads\/sites\/2\/2021\/08\/negative_poissons_ratio.gif\" width=\"674\" height=\"415\" \/><\/p>\n<p>Some examples of auxetic materials <a href=\"https:\/\/en.wikipedia.org\/wiki\/Auxetics\">include<\/a>:<\/p>\n<ul>\n<li style=\"list-style-type: none\">\n<ul style=\"font-weight: 400\">\n<li>Auxetic\u00a0polyurethane\u00a0foam.<\/li>\n<li>Kevlar woven composite materials used for body armor.<\/li>\n<li>Nuclei of mouse embryonic stem cells in exiting pluripotent state.<\/li>\n<li>Certain states of crystalline materials: Li, Na, K, Cu, Rb, Ag, Fe, Ni, Co, Cs, Au, Be, Ca, Zn, Sr, Sb, MoS<sub>2<\/sub>, BAsO<sub>4<\/sub>, and others.<\/li>\n<li>Certain rocks and minerals.<\/li>\n<li>Graphene, which can be made auxetic through the introduction of vacancy defects.<\/li>\n<li>Carbon diamond-like phases.<\/li>\n<li>Noncarbon nanotubes.<\/li>\n<li>Living bone tissue (although this is only suspected).<\/li>\n<li>Tendons within their normal range of motion.<\/li>\n<li>Specific variants of\u00a0polytetrafluorethylene\u00a0polymers such as\u00a0Gore-Tex.<\/li>\n<li>Several types of origami folds like the Diamond-Folding-Structure (RFS), the\u00a0herringbone-fold-structure (FFS) or the\u00a0miura fold,\u00a0and other periodic patterns derived from it.<\/li>\n<li>Tailored structures designed to exhibit special designed Poisson&#8217;s ratios.<\/li>\n<li>Chain organic molecules. Recent researches revealed that organic crystals like n-paraffins\u00a0and similar to them may demonstrate an auxetic behavior.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<ul>\n<li style=\"list-style-type: none\"><\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Uniaxial loading Tensile\/compressive stress in the x-direction produces tensile\/compressive strain in the x-direction, with the stress and strain related through Young&#8217;s modulus E: Strain in the x-direction produces strains the y&#8211; and z-directions. The strains in the y&#8211; and z-directions are proportional to the strain in the x-direction through the negative of the Poisson&#8217;s ratio. &hellip; <a href=\"https:\/\/www.purdue.edu\/freeform\/me323\/animations-and-demonstrations\/general-state-of-stress-and-strain\/\" class=\"more-link\">Continue reading <span class=\"screen-reader-text\">Poisson&#8217;s ratio<\/span> <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":10,"featured_media":0,"parent":5074,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"footnotes":""},"class_list":["post-8822","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/www.purdue.edu\/freeform\/me323\/wp-json\/wp\/v2\/pages\/8822","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.purdue.edu\/freeform\/me323\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/www.purdue.edu\/freeform\/me323\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/www.purdue.edu\/freeform\/me323\/wp-json\/wp\/v2\/users\/10"}],"replies":[{"embeddable":true,"href":"https:\/\/www.purdue.edu\/freeform\/me323\/wp-json\/wp\/v2\/comments?post=8822"}],"version-history":[{"count":19,"href":"https:\/\/www.purdue.edu\/freeform\/me323\/wp-json\/wp\/v2\/pages\/8822\/revisions"}],"predecessor-version":[{"id":14652,"href":"https:\/\/www.purdue.edu\/freeform\/me323\/wp-json\/wp\/v2\/pages\/8822\/revisions\/14652"}],"up":[{"embeddable":true,"href":"https:\/\/www.purdue.edu\/freeform\/me323\/wp-json\/wp\/v2\/pages\/5074"}],"wp:attachment":[{"href":"https:\/\/www.purdue.edu\/freeform\/me323\/wp-json\/wp\/v2\/media?parent=8822"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}