{"id":9262,"date":"2021-08-15T12:37:13","date_gmt":"2021-08-15T16:37:13","guid":{"rendered":"https:\/\/www.purdue.edu\/freeform\/me323\/?page_id=9262"},"modified":"2022-11-30T10:13:44","modified_gmt":"2022-11-30T15:13:44","slug":"failure-boundaries-and-mohrs-circle","status":"publish","type":"page","link":"https:\/\/www.purdue.edu\/freeform\/me323\/animations-and-demonstrations\/failure-boundaries-and-mohrs-circle\/","title":{"rendered":"Failure boundaries and Mohr&#8217;s circle"},"content":{"rendered":"<p><span style=\"font-size: 14pt\"><em><strong>DUCTILE MATERIAL<\/strong><\/em><\/span><br \/>\nOne theory that is used for the prediction of failure in a ductile material is the maximum shear stress theory (MSS). The basis of this theory is that failure will occur when the maximum shear stress exceeds the maximum shear stress that exists for yielding in a uni-axial test. Consequences:<\/p>\n<ul>\n<li>The failure boundary corresponds to|\u03c4|<span style=\"font-size: 8pt\">max,abs<\/span> = <span style=\"font-size: 12pt\">\u03c3<span style=\"font-size: 8pt\">Y<\/span>\/2.<\/span><\/li>\n<li>In the principal stress plane (\u03c3<span style=\"font-size: 8pt\">P1<\/span> vs. \u03c3<span style=\"font-size: 8pt\">P2<\/span> plane), this failure boundary is the hexagonal region shown to the right.<\/li>\n<\/ul>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-9264 aligncenter\" src=\"https:\/\/www.purdue.edu\/freeform\/me323\/wp-content\/uploads\/sites\/2\/2021\/08\/Untitled-13.jpg\" alt=\"\" width=\"259\" height=\"213\" \/><\/p>\n<p>The hexagonal-shaped boundary above is the locus of points in the principal stress plane that corresponds to constant values of \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0|\u03c4|<span style=\"font-size: 8pt\">max,abs<\/span> . How does this locus of points link back to the Mohr\u2019s circle plane? The animation below shows how the Mohr\u2019s circle changes as we move around on this hexagonal-shaped boundary between safe and failure.<\/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\/trace03.gif\" width=\"776\" height=\"357\" \/><\/p>\n<p>&nbsp;<\/p>\n<p><em><strong>Observations<\/strong><\/em><\/p>\n<ul>\n<li>In quadrant 3 of the principal stress plane, Mohr&#8217;s circle is in the left-half plane with the failure boundary prescribed by\u00a0\u03c3<span style=\"font-size: 8pt\">P2 <span style=\"font-size: 12pt\">=<\/span>\u00a0<span style=\"font-size: 12pt\">&#8211;<\/span><\/span>\u03c3<span style=\"font-size: 8pt\">Y<span style=\"font-size: 12pt\"> .<\/span><\/span><\/li>\n<li>In quadrant 1 of the principal stress plane, Mohr&#8217;s circle is in the right-half plane with the failure boundary prescribed by\u00a0\u03c3<span style=\"font-size: 8pt\">P1\u00a0<span style=\"font-size: 12pt\">=<\/span>\u00a0<\/span>\u03c3<span style=\"font-size: 8pt\">Y<span style=\"font-size: 12pt\"> .<\/span><\/span><\/li>\n<li>In quadrant 4 of the principal stress plane, Mohr&#8217;s circle has a constant radius <em>R<\/em> = \u03c3<span style=\"font-size: 8pt\">Y<span style=\"font-size: 12pt\">\/2 .<\/span><\/span><\/li>\n<li>At the beginning and end points of the above animation, the failure boundary is located at\u00a0\u03c3<span style=\"font-size: 8pt\">P2 <span style=\"font-size: 12pt\">=<\/span>\u00a0<\/span>\u03c3<span style=\"font-size: 8pt\">P1<span style=\"font-size: 12pt\">. As discussed earlier in the course, this corresponds to a state of &#8220;hydrostatic&#8221; stress. This corresponds to a Mohr&#8217;s circle with a zero radius. Such a state of stress has <span style=\"text-decoration: underline\"><em>zero in-plane shear stress<\/em><\/span>.<\/span><\/span><\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n<p><span style=\"font-size: 14pt\"><em><strong>BRITTLE MATERIAL<\/strong><\/em><\/span><br \/>\nRecall that one failure theory that is used for brittle material is the Mohr&#8217;s criterion. With this theory, the brittle material is assumed to fail in normal stress, and that the ultimate strength of the material in compression, \u03c3<sub>UC<\/sub>, exceeds its ultimate strength in tension, \u03c3<sub>UT<\/sub>; i.e., \u03c3<sub>UC <\/sub>&gt; \u03c3<sub>UT<\/sub>. The failure boundary in the \u03c3<sub>P1<\/sub>\u00a0vs. \u03c3<sub>P2<\/sub>\u00a0 plane is given by: \u00a0 \u03c3<sub>P1<\/sub>\/ \u03c3<sub>UT<\/sub>\u00a0 &#8211; \u00a0\u03c3<sub>P2<\/sub>\u00a0\/\u03c3<sub>UC<\/sub>\u00a0= 1.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-11354\" src=\"https:\/\/www.purdue.edu\/freeform\/me323\/wp-content\/uploads\/sites\/2\/2022\/11\/Screen-Shot-2022-11-30-at-10.02.26-AM-300x139.jpg\" alt=\"\" width=\"604\" height=\"280\" srcset=\"https:\/\/www.purdue.edu\/freeform\/me323\/wp-content\/uploads\/sites\/2\/2022\/11\/Screen-Shot-2022-11-30-at-10.02.26-AM-300x139.jpg 300w, https:\/\/www.purdue.edu\/freeform\/me323\/wp-content\/uploads\/sites\/2\/2022\/11\/Screen-Shot-2022-11-30-at-10.02.26-AM-1024x473.jpg 1024w, https:\/\/www.purdue.edu\/freeform\/me323\/wp-content\/uploads\/sites\/2\/2022\/11\/Screen-Shot-2022-11-30-at-10.02.26-AM-768x355.jpg 768w, https:\/\/www.purdue.edu\/freeform\/me323\/wp-content\/uploads\/sites\/2\/2022\/11\/Screen-Shot-2022-11-30-at-10.02.26-AM.jpg 1516w\" sizes=\"auto, (max-width: 604px) 100vw, 604px\" \/><\/p>\n<p>The animation below shows how Mohr&#8217;s circle changes in position and size as we move around the perimeter of Mohr&#8217;s failure boundary. We can make observations here similar to those above for the maximum shear stress theory for ductile materials. In this case, note how the allowable maximum shear stress on the failure boundary decreases as we move through the 4th quadrant of the \u03c3<sub>P1<\/sub>\u00a0vs. \u03c3<sub>P2<\/sub>\u00a0 plane.<\/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\/2022\/11\/mohr_trace.gif\" width=\"686\" height=\"340\" \/><\/p>\n","protected":false},"excerpt":{"rendered":"<p>DUCTILE MATERIAL One theory that is used for the prediction of failure in a ductile material is the maximum shear stress theory (MSS). The basis of this theory is that failure will occur when the maximum shear stress exceeds the maximum shear stress that exists for yielding in a uni-axial test. Consequences: The failure boundary &hellip; <a href=\"https:\/\/www.purdue.edu\/freeform\/me323\/animations-and-demonstrations\/failure-boundaries-and-mohrs-circle\/\" class=\"more-link\">Continue reading <span class=\"screen-reader-text\">Failure boundaries and Mohr&#8217;s circle<\/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-9262","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/www.purdue.edu\/freeform\/me323\/wp-json\/wp\/v2\/pages\/9262","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=9262"}],"version-history":[{"count":7,"href":"https:\/\/www.purdue.edu\/freeform\/me323\/wp-json\/wp\/v2\/pages\/9262\/revisions"}],"predecessor-version":[{"id":11357,"href":"https:\/\/www.purdue.edu\/freeform\/me323\/wp-json\/wp\/v2\/pages\/9262\/revisions\/11357"}],"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=9262"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}