{"id":151723,"date":"2023-11-02T02:29:55","date_gmt":"2023-11-02T06:29:55","guid":{"rendered":"https:\/\/effectsofanxiety.net\/?p=151723"},"modified":"2023-11-01T19:51:54","modified_gmt":"2023-11-01T23:51:54","slug":"true-stress-true-strain-and-work-hardening","status":"publish","type":"post","link":"https:\/\/effectsofanxiety.net\/archives\/151723","title":{"rendered":"True stress, true strain and work hardening"},"content":{"rendered":"<div class=\"jetpack-video-wrapper\"><iframe loading=\"lazy\" title=\"True stress, true strain and work hardening\" width=\"660\" height=\"371\" src=\"https:\/\/www.youtube.com\/embed\/33TUpTbn1O8?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\nSo we’ve sketched stress-strain. A curve like this\r\nfor typical metal. And we know an equation within\r\nthe linear elastic region. That’s before the\r\nproportional limit. And that’s, of course,\r\nHooke’s Law– sigma equals E times epsilon. But what do we have after\r\nplastic deformation? How can we perform calculations\r\nafter plastic deformation? If, for example,\r\nwe had something that– a bolt in\r\nthe ceiling– oh, that’s a horrendous drawing. Let’s fix that. Let’s fix that\r\nbefore I lose my job. So here’s– that’s even worse. Oh, my goodness. OK. This is good. No, that’s still awful. But there’s a hook\r\nand, I don’t know, something is hanging\u00a0from it. I was going to draw\r\na force so I don’t have to draw something else. There’s a hook hanging\r\nfrom the ceiling and you apply a\r\nforce to it and you want to know,\r\nwell, how much does that tie– that’s\r\nwhat that is– it’s called intention– that\r\ntensile tie– elongate? What’s its distance? What’s its length? And, well, you can only, at this point, do calculations\r\nif the force results in a stress that’s\r\nless than the yield strength.\r\n\r\n&nbsp;\r\n\r\nIf it’s more than that, while\r\nit’s classically deformed, we couldn’t deal\r\nwith it unless we had some kind of\r\nway of describing the shape of the\r\ncurve after it leaves the linear elastic region. And it’s nice. We do have an\r\nequation that fits the curve. But it’s going– we can’t\r\nuse engineering stress. We have to use what’s\r\ncalled true stress. So that’s what I’d like to\r\nintroduce to you right now.\r\n\r\n&nbsp;\r\n\r\nSo again, we’re going to take\r\na look at a generalized sample here. And the idea– the\r\ngoal– is of course that we’re going to go to\r\nbe able to this– calculate and understand\r\nplastic deformation. So here’s our sample with its\r\ninitial cross-sectional area as we’ve discussed before. And we said, well,\r\nwhen you load it, it gets longer\r\nand gets narrower. And it’s that reduction in\r\nthe cross-sectional area that we’re interested in. This area here now, we could\r\ncall it an instantaneous area. Whereas this area, the white\r\none, was the initial area. The initial area was\r\nwhat we started with. But then while the load\r\nis applied– sorry, let me draw the force in. While the load is applied,\r\nthe cross-sectional area has decreased. So the first thing\r\nyou could do is we could say, all right, well, that\r\nmeans that this material itself is experiencing a\r\nforce over a smaller area.\r\n\r\n&nbsp;\r\n\r\nSo we could define\r\nthe true stress as the force over that\r\nactual cross-sectional area. This is the true stress. This is the stress that\r\nthe material itself is feeling. OK? And that subscript\r\nI am telling us this is the instantaneous\r\ncross-sectional area. Instantaneous. I can’t do more than one thing\r\nat the same time. Instantaneous– I missed a\r\nU– cross-sectional area. Instantaneous cross-sectional\u00a0area. And I’ll show you\r\nwhat the plot would look like in just a moment. We could also do the same\r\nthing for the strain, although that’s going to be\r\njust a little bit– require a little bit more thinking. The true strain has to\r\naccount for the fact that what we’re doing is we’re\r\napplying a change in length. Right? We’re elongating it\r\nover a certain length. The very first little\r\nbit of elongation is elongation over l0. But then, after\r\nthat, the elongation is elongated over the\r\nprevious length which was l0 plus that little delta l.\r\n\r\n&nbsp;\r\n\r\n<img decoding=\"async\" src=\"https:\/\/images.pexels.com\/photos\/6837648\/pexels-photo-6837648.jpeg?auto=compress&amp;cs=tinysrgb&amp;h=650&amp;w=940\" \/>\r\n\r\n&nbsp;\r\n\r\nAnd so if you do that\r\nfor infinitely small– infinitesimally small\r\nchanges in length, the way we would write that\r\nis we’d have to say, the true strain, what we’re\r\ndoing is we’re integrating. We’re integrating those\r\ninfinitesimally small changes in length– that’s dl– by l\r\nfrom l0– the initial length– to the instantaneous length. And so if we do that,\r\nyou find that you have ln of l instantaneous\r\nminus length 0, which is ln of l instantaneous over l0. So we have another\r\nequation there. I’ll put a box around that. So this is the true strain. True strain. And if we take that true\r\nstress and we plot it against the true strain,\r\nI’ll show you what we get. Let me just\r\nplot stress and strain and I’ll show you what\r\nwe’ve already seen. That’s the engineering\r\nstress-strain curve. And then what I’ll do is I’ll\r\nplot for you– after it starts to plastically deform the–\r\nran out of space there– the true stress– so this one\r\nhere– continues to increase.\r\n\r\n&nbsp;\r\n\r\nIt doesn’t have that\r\ndecrease at the UTS. That’s the true stress\r\ntrue strain curve. And this one is, of\r\ncourse, engineering. The nice thing\r\nabout this plot is once you’ve got true\r\nstress and true strain, we can fit that data\r\nquite nicely for most metals with a simple equation. And that is true stress\r\nis equal to this coefficient times the true strain\r\nraised to the power n. So that’s an equation that fits\r\nthat true stress true strain data quite nicely. And what’s useful about\r\nthis is these are constants. That’s the constant n– I’ll\r\ndefine it for you in a moment– and this K is also a constant. Those are material properties. We can look those up in\r\nan engineering handbook. So n is called the–\r\nwell, this equation is called the\r\nstrain-hardening equation.\r\n\r\n&nbsp;\r\n\r\nStrain hardening equation. And strain hardening– hardening\r\ncorrelates to– hardness correlates to strength. So really this is\r\nthe equation that’s telling us that we’re\r\nstrengthening the material and we’ve got the\r\nstrain hardening exponent and the strain\r\nhardening coefficient K.\r\n\r\n<a href=\"https:\/\/www.youtube.com\/watch?v=33TUpTbn1O8\" 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; So we’ve sketched stress-strain. A curve like this for typical metal. And we know an equation within the linear elastic region. That’s before the proportional limit. And that’s, of course, Hooke’s Law– sigma equals E times epsilon. But what do we have after plastic deformation? How can we perform calculations after plastic deformation? If, &hellip; <a href=\"https:\/\/effectsofanxiety.net\/archives\/151723\" class=\"more-link\">Continue reading <span class=\"screen-reader-text\">True stress, true strain and work hardening<\/span><\/a><\/p>\n","protected":false},"author":3,"featured_media":151721,"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":[138,483],"tags":[61071,61074,61052,61043,61045,61037,61035,61040,61062,35676,40917,61069,5696,61057,61055,61066],"class_list":["post-151723","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-community","category-media","tag-deformation","tag-hardening","tag-holloman-equation","tag-khan","tag-khan-style-video","tag-materials-science-field-of-study","tag-mechanical-behaviour-of-solids","tag-solid-phase-of-matter","tag-stain-hardening","tag-strain","tag-strength","tag-strengthening-of-metals","tag-stress","tag-true-strain","tag-true-stress","tag-work-hardening"],"jetpack_publicize_connections":[],"jetpack_sharing_enabled":true,"jetpack_featured_media_url":"https:\/\/effectsofanxiety.net\/wp-content\/uploads\/2023\/11\/htmlF_IMG_6542e036264ba.jpg","_links":{"self":[{"href":"https:\/\/effectsofanxiety.net\/wp-json\/wp\/v2\/posts\/151723"}],"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=151723"}],"version-history":[{"count":2,"href":"https:\/\/effectsofanxiety.net\/wp-json\/wp\/v2\/posts\/151723\/revisions"}],"predecessor-version":[{"id":151734,"href":"https:\/\/effectsofanxiety.net\/wp-json\/wp\/v2\/posts\/151723\/revisions\/151734"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/effectsofanxiety.net\/wp-json\/wp\/v2\/media\/151721"}],"wp:attachment":[{"href":"https:\/\/effectsofanxiety.net\/wp-json\/wp\/v2\/media?parent=151723"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/effectsofanxiety.net\/wp-json\/wp\/v2\/categories?post=151723"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/effectsofanxiety.net\/wp-json\/wp\/v2\/tags?post=151723"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}