{"id":151720,"date":"2023-11-02T11:29:56","date_gmt":"2023-11-02T15:29:56","guid":{"rendered":"https:\/\/effectsofanxiety.net\/?p=151720"},"modified":"2023-11-01T19:57:37","modified_gmt":"2023-11-01T23:57:37","slug":"ductility-toughness-and-resilience","status":"publish","type":"post","link":"https:\/\/effectsofanxiety.net\/archives\/151720","title":{"rendered":"Ductility, toughness and resilience"},"content":{"rendered":"<div class=\"jetpack-video-wrapper\"><iframe loading=\"lazy\" title=\"Ductility, toughness and resilience\" width=\"660\" height=\"371\" src=\"https:\/\/www.youtube.com\/embed\/CMuBpazobwE?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe><\/div>\r\n\r\n&nbsp;\r\n\r\nThere are\r\na few other things that we can determine\r\nfrom the stress-strain curve of metal that\r\nis quite interesting. And I’d like to just spend\r\na moment looking at that. So again, we’ve got\r\nthe stress here, on the vertical axis,\r\nstrain– typical metal. We have a curve that\r\nlooks something like this. It comes up looking linear\r\nelastic, plastic, ultimate tensile strength,\r\nand then fracture. And so we’ve determined\r\nall of the strengths. We have the yield strength,\r\nthe ultimate tensile strength, and the fracture strength. But what other properties\r\ncan we determine? Well, first of all, one of them\r\nthat you may have heard about is the ductility. So ductility has a usage\r\nin common language. You might say, well,\r\nwhat’s ductility all about? If something’s very ductile,\r\nyou might say, well, it describes how much you\r\ncan stretch something. But of course, we\r\nknow that that’s not accurate enough. Stretch, is that referring\r\nto elastic or plastic? So we’ve got to be\r\nbetter than that.\r\n\r\n&nbsp;\r\n\r\nAnd in fact, I’ll\r\ntell you ductility is a measure of a plastic strain. So we know\r\nit’s a strain quantity, and it refers to plastic\r\ndeformation, only plastic strain to fracture. Now we’ve got something we\r\ncan work with– plastic strain to fracture. So let’s see well, this\r\nis the point of fracture. That’s a fracture. So that fracture,\r\nif we unload we’d have a value here\r\nfor total strain. Let me write that in there\r\nfor you, total strain. If we unloaded– if we\r\ntook the total strain there at fracture, just a moment\r\nbefore it fractured, that would be our total strain. But what if we unloaded it? Somehow you knew just\r\ninfinitesimally before it was going to fracture– well,\r\nwe know that Young’s modulus is structure-independent.\r\n\r\n&nbsp;\r\n\r\nSo it won’t change. So we would have unloaded\r\nthat same modulus. This means we come back down\r\nhere to a value on the strain axis, a finite value\r\ncorresponding to zero stress. It’s unloaded, there’s\r\nno stress on it, but there’s still some\r\npersistent strain. That strain has to be plastic. That’s a plastic strain, which\r\nmeans that this strain here is elastic. That’s elastic, and that makes\r\nsense because what is that? That’s the strain underneath\r\nthis linear unloading portion.\r\n\r\n&nbsp;\r\n\r\nAnd the unloading\r\nportion, if it’s linear, is governed by Hooke’s law. And we know that’s elastic\r\nbecause Hooke’s law refers to elastic behavior. So if we unload down and\r\nwe get plastic strain, that plastic strain has\r\ngot to be the ductility. So ductility, you\r\nunload at fracture. And the remaining\r\nstrain is the ductility. Another interesting property\r\nthat we can determine from this stress-strain\r\nbehavior for a metal, for other material classes as\r\nwell, is called toughness.\r\n\r\n&nbsp;\r\n\r\nAnd the toughness is sometimes\r\nnot such an intuitive quantity. You can understand strength,\r\nit’s force over area. You got a sense for that. It’s pressure, if you will. Even modulus you can kind\r\nof get a bit of an intuitive sense for it because\r\nit’s how hard is it to bend\r\nsomething elastically. It’s a little harder, but\r\nthe toughness is– toughness, I’ll tell you what the tough is. Toughness is the energy–\r\nit’s an energy term. And it’s energy\r\nabsorbed to fracture. What we can do\r\nis integrate. And that is to take the\r\narea under the curve. So if we take the area\r\nunder this curve here, it would be this area here, all\r\nthis area here under the curve is the toughness.\r\n\r\n&nbsp;\r\n\r\n<img decoding=\"async\" src=\"https:\/\/images.pexels.com\/photos\/3772618\/pexels-photo-3772618.jpeg?auto=compress&amp;cs=tinysrgb&amp;h=650&amp;w=940\" \/>\r\n\r\nAnd how do we know that? We could look at\r\nit dimensionally. If we’re taking a product\r\nof stress and strain and looking at the dimensions,\r\nstress has units of Pascals. And what’s a Pascal? A Pascal is a Newton\r\nper square meter. Well, I can go on living\r\nmy life multiplying whatever I want by 1 and just\r\nmultiply this screen by 1. You didn’t even notice. So here we go, where I multiply\r\nNewton per square meter by 1, meter over meter, and I\r\nend up with a familiar term in the numerator– Newton meter. And of course in the denominator\r\nI’ve got volume units. But what’s the Newton meter? A Newton meter is\r\nnothing more than a joule.\r\n\r\n&nbsp;\r\n\r\nSo we’ve now got\r\njoules per volume as units when we\r\nintegrate under this. And that’s great because\r\nwe want an energy unit. So if we integrate under the\r\nentire curve up to fracture, it tells us how much energy\r\nwent into fracturing that. And that includes elastic\r\nand plastic deformation. The final thing that we\r\ncan obtain from the stress curve, is\r\nanother energy unit and it’s quite useful– it’s\r\na stored energy unit this time. We’ve got stress and strain. We’ve got our linear elastic\r\nregion, plastic deformation, and fracture– is\r\nthe resilience. Thus resilience\r\nis a measure of the stored elastic strain\r\nenergy at the yield strength.\r\n\r\n&nbsp;\r\n\r\nSo again, we know if it’s going\r\nto be an energy term, energy per volume for a given\r\nvolume of material, we’re going to have to\r\nintegrate under the curve. And where are we\r\ngoing to do it from? Well, we’ll go to\r\nthe yield strength. And we go down from there. And if we unloaded at\r\nthe yield strength– I’m going to be a little\r\ncareful about something– if we unloaded at the\r\nyield strength, you’d find that you have\r\na little sliver of permanent or plastic strain. Maybe it’s close to\r\nthe 0.2% offset strain. You’d probably have some plastic\r\nstrain accumulated when we had yield. For practical purposes, we say\r\nit’s elastic before yielding and it’s plastic after. You might have a sliver. So we’re not going to\r\ninclude that if we’re going to be strict with\r\nour definition here. And so that area there\r\nis the resilience. And that area is\r\njust an area of a triangle. And we know that the\r\narea of a triangle is 1\/2 base times\r\nheight, which in our case is 1\/2 of– well,\r\nwhat’s the base? The base is the elastic strain.\r\n\r\n&nbsp;\r\n\r\nAnd that’s good because we’re\r\nafter the stored elastic strain energy. So we’ve got strain elastic. And what’s the height? Well, the height is\r\nthe yield strength. But we can, again,\r\ndo better than this. Because if it’s elastic,\r\nit’s the area under this– or it’s the strain underneath\r\nthis linear unloading portion. And the linear\r\nunloading portion, we have a mathematical\r\nequation for. We have stress equals\r\nE times strain. It’s a straight line. So that means that\r\nthe strain is going to be equal to sigma over\r\nE. And we fire that in here, and we find that\r\nthe resilience is– I should erase that–\r\nthe resilience is going to be 1\/2 of sigma,\r\nand this is the sigma yield. That’s what we’re using here. So that’s sigma yield\r\nover E times sigma yield.\r\n\r\n&nbsp;\r\n\r\nSo at the end of the day, the\r\nresilience, which we often use this– I’ll introduce\r\nthis symbol here. The full name for this\r\nis the modulus of resilience. And modulus is just a fancy\r\nword for a special number. So our special number here\r\nis the modulus of resilience. And we use the\r\nuppercase letter U. Is 1\/2 sigma yield\r\nsquared upon E– and that’s an interesting\r\nlittle equation. It tells you the stored\r\nstrain energy for a material. So if you’re going to make\r\na material for a spring, you’d look for something with\r\na high modulus of resilience. And again, the units\r\nhere, the dimensions here of modulus of\r\nresilience are going to be joules per cubic meter.\r\n\r\n<a href=\"https:\/\/www.youtube.com\/watch?v=CMuBpazobwE\" target=\"_blank\" rel=\"noopener\">As found on YouTube<\/a>\r\n\r\n<a href=\"https:\/\/jvz4.com\/c\/335985\/319641\">Explaindio Agency Edition<\/a> FREE Training How to Create Explainer Videos &amp; SELL or RENT them! Join this FREE webinar | Work Less &amp; Earn More With <a href=\"https:\/\/jvz4.com\/c\/335985\/319641\">Explaindio AGENCY EDITION<\/a>\r\n\r\n<a href=\"https:\/\/jvz4.com\/c\/335985\/319641\" target=\"_blank\" rel=\"noopener\"><img decoding=\"async\" src=\"https:\/\/i.ibb.co\/HNmg3Ln\/OIP-48.jpg\" alt=\"OIP-48\" border=\"0\" \/><\/a>","protected":false},"excerpt":{"rendered":"<p>&nbsp; There are a few other things that we can determine from the stress-strain curve of metal that is quite interesting. And I’d like to just spend a moment looking at that. So again, we’ve got the stress here, on the vertical axis, strain– typical metal. We have a curve that looks something like this. &hellip; <a href=\"https:\/\/effectsofanxiety.net\/archives\/151720\" class=\"more-link\">Continue reading <span class=\"screen-reader-text\">Ductility, toughness and resilience<\/span><\/a><\/p>\n","protected":false},"author":3,"featured_media":151719,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","enabled":false},"version":2}},"categories":[483,100],"tags":[61050,61041,61063,61045,61034,61053,61056,61059,61037,61065,35524,61040,61048,61038],"class_list":["post-151720","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-media","category-politics","tag-condensed-matter-physics-field-of-study","tag-ductility","tag-engineering-industry","tag-khan-style-video","tag-ksv-khan-style-videos","tag-material","tag-materials-engineering","tag-materials-science","tag-materials-science-field-of-study","tag-physical-chemistry-field-of-study","tag-resilience","tag-solid-phase-of-matter","tag-solid-state-physics-field-of-study","tag-toughness"],"jetpack_publicize_connections":[],"jetpack_sharing_enabled":true,"jetpack_featured_media_url":"https:\/\/effectsofanxiety.net\/wp-content\/uploads\/2023\/11\/htmlF_IMG_6542e02302011.jpg","_links":{"self":[{"href":"https:\/\/effectsofanxiety.net\/wp-json\/wp\/v2\/posts\/151720","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/effectsofanxiety.net\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/effectsofanxiety.net\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/effectsofanxiety.net\/wp-json\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/effectsofanxiety.net\/wp-json\/wp\/v2\/comments?post=151720"}],"version-history":[{"count":2,"href":"https:\/\/effectsofanxiety.net\/wp-json\/wp\/v2\/posts\/151720\/revisions"}],"predecessor-version":[{"id":151740,"href":"https:\/\/effectsofanxiety.net\/wp-json\/wp\/v2\/posts\/151720\/revisions\/151740"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/effectsofanxiety.net\/wp-json\/wp\/v2\/media\/151719"}],"wp:attachment":[{"href":"https:\/\/effectsofanxiety.net\/wp-json\/wp\/v2\/media?parent=151720"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/effectsofanxiety.net\/wp-json\/wp\/v2\/categories?post=151720"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/effectsofanxiety.net\/wp-json\/wp\/v2\/tags?post=151720"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}