{"id":150032,"date":"2022-07-16T03:59:49","date_gmt":"2022-07-16T07:59:49","guid":{"rendered":"https:\/\/effectsofanxiety.net\/?p=150032"},"modified":"2022-07-15T13:28:52","modified_gmt":"2022-07-15T17:28:52","slug":"lec-3-stress-acting-at-a-point-stress-tensor","status":"publish","type":"post","link":"https:\/\/effectsofanxiety.net\/archives\/150032","title":{"rendered":"Lec 3 : Stress acting at a point- Stress tensor"},"content":{"rendered":"<div class=\"jetpack-video-wrapper\"><iframe loading=\"lazy\" title=\"Lec 3 : Stress acting at a point- Stress tensor\" width=\"660\" height=\"371\" src=\"https:\/\/www.youtube.com\/embed\/lDDoYIuVk3I?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, welcome back, this is the next lecture\r\non stress tensors. So in the last lecture, we have categorically seen\r\nwhat is Cauchy’s stress, sigma.\u00a0\u00a0 And we have seen that the definition of sigma\r\nindicates what is the internal force that gets developed within a plane or a body\r\nat a point due to some action of external forces.\u00a0\u00a0 Now, what is sigma?\r\n\r\n&nbsp;\r\n\r\nIn\r\nfact sigma is a stress tensor. So, Cauchy stress can be considered a\r\ntensor. So, now we are going to define a new term what is known as a tensor. If you want to study or\r\nif you want to do modeling in continuum mechanics,\u00a0\u00a0 as the complexity of the problem increases, it\r\nis always convenient to define what is known as a tensor. And we have already stated stress\r\nis a tensor quantity. Now, what is a tensor?\u00a0\u00a0 We know what are scalars, and we know what are\r\nvectors. So tensor is also a similar kind of quantity. So why tensor, because it is very\r\nconvenient to express stress as a tensor.\u00a0 In short stress itself is a tensor. In\r\nsimple terms, we can say that tensor can be defined as a quantity with\r\nmagnitude and multiple spatial directions.\u00a0\u00a0 So, possibly you will think like what is the\r\ndifference between a tensor and a vector.\u00a0\u00a0 Vector also has a magnitude and a direction,\r\nbut we will see that vector has magnitude,\u00a0\u00a0 but it will have only one direction whereas in\r\nthe case of tensor multiple directions are there.\u00a0 So that is the essential difference\r\nand tensor is a more general term.\u00a0\u00a0 And the subsets of the tensor are scalar, vector, and\r\nany other tensor of higher order.\r\n\r\n&nbsp;\r\n\r\nSo different tensors which are popularly used are yes, that is\r\nwhat I told the first one is scalar the simplest tensor is scalar and it is called zero-order\r\ntensor. A quantity which has only magnitude and zero direction, scalar we all of us know that it\r\ndoes not have any direction it has only magnitude.\u00a0 So we call it zero direction. And zero order\r\ntensor which is a scalar that has three raised to zero, where zero represents the number of\r\ndirections. So three raises to zero are equal to one element and that is true, it is merely a number\r\nthat shows the magnitude. Scalar is a zero-order tensor.\r\n\r\n&nbsp;\r\n\r\nThe second one is a vector, which is the\r\nfirst-order tensor. A vector is a quantity,\u00a0\u00a0 which has magnitude and one direction you\r\ncan see that vector has only one direction.\u00a0 Accordingly, the number of elements will be three\r\nraised to one which is equal to three elements.\u00a0\u00a0 So if you have an x y z axis you have a vector in\r\nthree different directions. So that is possible.\u00a0\u00a0 So that is what it means it has three\r\nelements. So it has one direction every vector is associated with only one direction. And it\r\nhas three raised to one which is three elements and specifically, Cauchy stress is\r\nknown as a second order tensor.\u00a0 Why does Cauchy stress sigma have magnitude and two\r\ndirections? Now, what are these two directions?\u00a0\u00a0 Now we will see specifically how these two\r\ndirections come into the picture when you define a stress component and it is very easy also if you\r\nremember Cauchy stress, we represent it as sigma xx or sigma xy.\r\n\r\n&nbsp;\r\n\r\nSo there are two symbols\r\nassociated and that is why it is always associated with two directions it is associated\r\nwith which plane it acts, which means the normal to that particular plane.\r\nIt is also dependent on, which direction that particular traction acts. So we will\r\ndiscuss that a bit later, only to specify here is Cauchy stress is a second-order tensor.\r\nIt has magnitude and it has two directions.\u00a0\u00a0 So this is a second-order tensor and it has three\r\nsquares equal to nine elements which we have already seen in the Cauchy stress tensor.\r\n\r\n&nbsp;\r\n\r\nThere\r\nare nine elements and the second order tensor linearly maps to vectors that also we have seen.\r\nWe have seen that t is sigma transpose of n. So it linearly transforms\r\nto vectors, that is Cauchy’s formula. Now, some aspects of tensor to be very specific,\r\nmay not be useful, but then this is important to understand the tensor. Let us say two, there\r\nare two vectors u and v a tensor T is a second order tensor if it linearly maps vector v to u\r\nas can be shown here and the second order tensor satisfies the properties of linear transformation.\r\nSo this is what has been written, t maps v to u\u00a0\u00a0 or there is a linear mapping of v to u.\r\nIf you compare this with Cauchy’s formula, it is more or less the same thing that is how we define\r\nThe Cauchy stress tensor is a second-order tensor.\u00a0\u00a0 Having said that, now the next job is to interpret\r\nthe components of the Cauchy stress tensor.\r\n\r\n&nbsp;\r\n\r\nWe know that there are nine elements. Now, what are\r\nthese nine elements? What does it represent? So, for that, we need to define the Cartesian\r\ncoordinate. So you have a Cartesian coordinate x y\u00a0\u00a0 and z. And to make it simple a\r\ncontrol volume is also shown,\u00a0\u00a0 control volume is a very common terminology that\r\nis used in continuum mechanics or any other form of mechanics. The con this control volume\r\nis not required, but to make things simple and for one to understand it has been shown.\r\nSo we have a Cartesian coordinate x y z,\u00a0\u00a0 this is positive x direction, this is positive\r\ny and this is positive z which is also important here.\r\n\r\n&nbsp;\r\n\r\nSo you can consider a positive x plane, now\r\nx is an axis which is meant by x plane. It means the plane on which x direction is the\r\nnormal to that plane that is what is written here the plane whose normal is in the positive\r\nx direction. So it is called the positive x plane.\u00a0 So what will be the negative x plane?\r\nThe negative x plane will be here because the outward normal to this plane is in the\r\nnegative x direction. So this is the negative x plane.\u00a0\u00a0 So you need to understand this very\r\ncarefully. Consider positive x planes,\u00a0\u00a0 so we are talking about this particular plane.\r\nThat is a positive x plane because why because normal to this particular plane x this is the y z plane,\r\nthis is y, this is z.\r\n\r\n&nbsp;\r\n\r\nSo this plane is y z plane.\u00a0 Now for this y z plane, the normal is in the\r\ndirection of x. So that is what it means.\u00a0\u00a0 So x plane means, positive x plane means, y\r\nz plane which is shown here so, positive x,\u00a0\u00a0 negative x plane both are there. Now we will\r\ncome back to Cauchy’s formula and cauchys stress sigma. So now, the normal vector to x\r\nplane\u00ac this positive x direction.\u00a0\u00a0 Please understand the normal vector\r\nto x plane\u00ac. So this is the x plane.\u00a0 The normal is x. So you can easily write\r\nwhat is the normal vector. So this is the normal vector. So for the x\r\ndirection, it is one zero.\u00a0\u00a0 So n T is given in this manner and similarly, for y\r\nit will be zero one zero, and for z zero one.\u00a0\u00a0 So normal vector to x plane is\r\ndefined that is n transpose is given.\u00a0\u00a0 Now, what are the components of traction vector\r\ntx, ty, and tz?\r\n\r\n&nbsp;\r\n\r\nYou already have this to be,\u00a0\u00a0 that is tx, ty, tz is equal to sigma and n.\r\nSo if you substitute the value of n that is for positive x plane one zero here, so it will\r\nbe one zero, and do the matrix operation,\u00a0\u00a0 you will see that t x is equal to sigma xx, ty\r\nwill be equal to sigma xy. So this is sigma xx,\u00a0\u00a0 sigma xi. So ty will be equal to sigma x y and\r\nt z is equal to sigma x z. So what does it mean,\u00a0\u00a0 it means that the components which are present\r\nin cauchys stress tensor are components of traction vector in a given direction.\r\nSo\u00a0\u00a0 if you see, you can see that sigma xx is the x\r\ncomponent of traction vector on x plane.\r\n\r\n&nbsp;\r\n\r\nSo there are two references which are coming and that\r\nis why we said that there are two directions.\u00a0\u00a0 It is the x component that is the traction vector\r\nin the x direction and it is acting on the x plane.\u00a0\u00a0 So there are two things which are coming.\r\nSimilarly, you have sigma xy. Sigma XY is the y component of the traction vector acting on the x plane.\r\nSimilarly, you have the z component of the traction vector acting on the x plane. Similarly, other components of\r\nCauchy stress tensor can be identified based on\u00a0\u00a0 Cauchy’s formula. So that is what is the meaning\r\nof each of the terms which are present in the\u00a0\u00a0 Cauchy stress, it is nothing but the components\r\nof the traction vector acting in a specific direction. So, the component of the Cauchy stress\r\ntensor, in general, is sigma ij, it is the j component of the traction vector it can be\r\nx, it can be y, and it can be z. So it is a j\u00a0\u00a0 component of traction vector\r\nacting on the ith plane. So the first index I show which plane it is associated\r\nwith. Which plane means, which is the normal and j\u00a0\u00a0 is the direction of that component, direction\r\nof the component of traction vector.\u00a0 So i is the plane on which traction is considered,\r\nand j is the direction in which the traction component is considered.\r\n\r\n&nbsp;\r\n\r\n<img decoding=\"async\" src=\"https:\/\/images.pexels.com\/photos\/7278855\/pexels-photo-7278855.jpeg?auto=compress&amp;cs=tinysrgb&amp;h=650&amp;w=940\" \/>\r\n\r\n&nbsp;\r\n\r\nSo we can see the overall\r\nrepresentation of the Cauchy stress tensor.\u00a0\u00a0 So first is stresses acting on x\r\nplane. Now which is the x plane,\u00a0\u00a0 this is the x plane. So there are two\r\nx planes, this is negative x and this is positive x. So what are the\r\nstresses which are acting,\u00a0\u00a0 we have sigma xx in the direction of x?\r\nSo, all of them are acting on x plane.\u00a0 Then we have sigma XY and sigma xz. Similarly,\r\non the other side of another plane which is a negative x\u00a0\u00a0 plane we have sigma XY, sigma xz, and sigma xx.\r\nIt is identical but it is on the other side.\u00a0\u00a0 Then we have stressed on y plane.\r\n\r\n&nbsp;\r\n\r\nNow what is\r\nmeant by y plane, a plane with y direction as the normal. So you are talking about this and this.\r\nSo you have positive y and this is negative y.\u00a0 Similarly, in this, you have\r\nsigma yy, sigma yy, which is the direction, in the direction\r\nof y and you have sigma y x, sigma yz,\u00a0\u00a0 similarly sigma y x and sigma yz. Then\r\nwe have stresses acting on the z plane,\u00a0\u00a0 what are the stresses acting on the z plane, and\r\nwhat is the z plane, this is the positive z plane and this is the negative z plane and\r\nthis stress is acting as sigma zz,\u00a0\u00a0 sigma zx, sigma zy. Similarly, here also\r\nyou have sigma zz, sigma zx, and sigma zy.\u00a0\u00a0 So these are the representation of the components\r\nof Cauchy stress on a given control volume.\u00a0 So, all these stress components are acting at\r\na point. Now we need to keep in mind that I\u00a0\u00a0 have shown a control volume in the figure, and\r\nthat is only for understanding how the stresses are oriented. Otherwise, it does not serve\r\nany purpose. We need to still understand that whatever stress components are there in\r\nthe Cauchy stress tensor, it is acting at a point and the control volume, the cuboid is shown only\r\nto indicate the plane on which it is acting.\u00a0 So that\u2019s that notion we should not forget.\r\nSo it is stress acting at a point.\u00a0\u00a0 Now having said that, we need to now define some\r\nsort of sign convention of the Cauchy stress tensor.\r\n\r\n&nbsp;\r\n\r\nSo the given sign conventions are the traction\r\ncomponents on the positive plane. So now we have already marked what is a positive plane.\r\nSo the traction component on the positive plane acting in the positive direction means the\r\ndirection of x y z which is in the positive direction, so is positive. So you have a positive\r\nplane and the traction component is acting in the positive direction, so it is positive.\r\nSimilarly, if you have a positive plane and the traction component is acting in\r\na negative direction, so it is negative.\u00a0 For the negative plane, if the plane is negative\r\nand the traction component is acting in a negative direction, so it is positive.\r\n\r\n&nbsp;\r\n\r\nAnd the final case\r\nis negative plane traction component direction is a positive direction, it is negative. So this is\r\none sign convention, you can see that numerous sign conventions are available, and uh one may use them at his convenience, but if you follow one sign convention,\r\nyou need to follow it throughout.\u00a0 So this is one convenient way of uh defining sign\r\nconvention, there are assigned conventions that are available based on movement also, and sometimes\r\nit may be difficult to understand. So this is very easy and very easy to define as well, one example\r\nis given here.\r\n\r\n&nbsp;\r\n\r\nSo this is the positive x plane,\u00a0\u00a0 and the stresses acting are sigma xx, sigma\r\nXY and sigma xz. If you consider sigma xx,\u00a0\u00a0 this is acting on a positive x plane.\r\nAnd sigma xx is in acting in the positive x direction. So that is why it is positive,\r\nsimilarly to sigma XY and sigma xz. Now consider the case of the negative x plane, if you consider sigma\r\nXY, this is a negative plane, negative x plane whereas, this is acting in the positive y direction.\r\nSo negative plane positive y direction,\u00a0\u00a0 so it is negative.\r\n\r\n&nbsp;\r\n\r\nSimilarly, all the stress\r\ncomponent signs can be assigned. So this is the sign convention of the Cauchy stress tensor. So what\r\nis the summary that we have understood till now? There are three normal\r\ncomponents or normal stresses,\u00a0\u00a0 sigma xx, sigma yy, and sigma oz. You can\r\nsee that in this figure, you have sigma x x,\u00a0\u00a0 sigma yy, and sigma zz, these are acting in\r\nthe same direction as that normal. So there are three normal components or normal\r\nstresses sigma xx, sigma yy, sigma zz or it is merely stated as sigma x, sigma\r\ny, sigma z which is a common terminology,\u00a0\u00a0 which we normally use in mechanics.\r\nThere are six shear components or shear stresses to be very specific all indices\r\nwere I not equal to j, here it is I equal to,\u00a0\u00a0 here I not equal to j.\r\n\r\n&nbsp;\r\n\r\nSo these are shear\r\ncomponents of traction or shear stresses,\u00a0\u00a0 it is written either in sigma form or in tau\r\nform. Cauchy stress tensor a second-order tensor quantifies the internal force distribution in a\r\nbody at a given position and time corresponding to a given deformation.\r\nWhy time is important is because we are considering the condition\r\ncorresponding to a given deformation.\u00a0\u00a0 And internal forces, which that gets developed\r\nfollowed the basic laws of mechanics. Now one particular aspect of why stress\r\nat a point that information is needed is to, define the equilibrium equation. So it is an\r\napplication of why you need to know stress at a point.\r\n\r\n&nbsp;\r\n\r\nNow stress at a point is\r\nvery important to define the equilibrium equation as we have seen in the beginning, you have seen\r\nthat certain requirements need to be satisfied like the equilibrium condition,\r\nthe compatibility condition, and so on.\u00a0 Now for defining the equilibrium condition, we\r\nneed to specify the equilibrium equation. I will not go into the derivation of this equilibrium\r\nequation it is very basic and is mostly seen by most of you. So by considering a given\r\ncontrol volume, the equilibrium equation can be represented as follows. And you can see that the\r\ncomponents of equilibrium equations are the stress tensor components, the only new term is gamma.\r\nWhere gamma is a self-weight of the gravity stress which acts in the z-direction or the vertical\r\ndirection and to be very specific stresses are in terms of total stresses in this particular\r\nequation, it is invariably necessary to know stress at a point for defining\r\nequilibrium condition.\r\n\r\n&nbsp;\r\n\r\nNow based on equilibrium,\u00a0\u00a0 we can say that tau yx is equal to tau XY, tau\r\nyz is equal to zy and tau zx is equal to tau xz.\u00a0 Therefore, the stress tensor is represented\r\nby six independent stress components,\u00a0\u00a0 there are nine components in the Cauchy\r\nstress tensor just because of this condition,\u00a0\u00a0 we have six independent stress components, and\r\nthey are three normal stresses, sigma x, sigma y,\u00a0\u00a0 sigma z, and three shear stresses tau XY, tau\r\nyz and tau Zx, where tau XY is equal to tau yx.\u00a0\u00a0 So this is what it is. So that is how it boils\r\ndown to six independent stress components. So the final summary of what\r\nwe learned in this particular lecture is Cauchy stress, sigma is a second-order\r\ntensor. The element of stress tensor represents components of traction acting on three orthogonal\r\nplanes according to a given Cartesian coordinate.\u00a0\u00a0 Sigma I j means j component of traction vector\r\nacting on the ith plane. Stress tensor sigma has three normal stress components and six shear\r\nstress components.\r\n\r\n&nbsp;\r\n\r\nBut based on equilibrium,\u00a0\u00a0 there are six independent stresses\r\nthree normal and three shear stresses.\u00a0 All the stress components are acting at a\r\npoint that is very relevant and which is very important. The components\r\nof sigma depend on the coordinate axis,\u00a0\u00a0 please note here as such sigma\r\nis not dependent on the coordinate axis,\u00a0\u00a0 but the components of sigma, I mean to say\r\nsigma x, sigma xy those are the components or the traction vector components, they\r\nare dependent on the coordinate axis. So there is a distinction that needs to be\r\nvery clear, one should not get confused with sigma as a whole and the components of sigma.\r\nSigma as a whole is not dependent on any axis,\u00a0\u00a0 but the components of sigma keep changing, but\r\nthe overall sigma representation of internal force remains the same depending on the reference axis,\r\nand the component’s magnitude value keeps changing.\u00a0\u00a0 Stress tensor sigma at any point in the body\r\ndefines the internal force distribution of a body.\u00a0\u00a0 So this is all about this particular\r\nlecture, which we will see in the next lecture.\r\n\r\n\r\n<a href=\"https:\/\/www.youtube.com\/watch?v=lDDoYIuVk3I\" target=\"_blank\" rel=\"noopener\">As found on YouTube<\/a>\r\n\r\n<a href=\"https:\/\/jvz4.com\/c\/335985\/306988\">AnimationStudio \ua19b☣\ua415 Be The “Middle Man” And Profit With AnimationStudio Agency License. Here’s How You Can Earn $100, $200, or even $300 For Every Video You Create With AnimationStudio… Activate Your Profit Machine With The Agency License … $197\/month For Just $67 One Time Payment<\/a>","protected":false},"excerpt":{"rendered":"<p>&nbsp; So, welcome back, this is the next lecture on stress tensors. So in the last lecture, we have categorically seen what is Cauchy’s stress, sigma.\u00a0\u00a0 And we have seen that the definition of sigma indicates what is the internal force that gets developed within a plane or a body at a point due to &hellip; <a href=\"https:\/\/effectsofanxiety.net\/archives\/150032\" class=\"more-link\">Continue reading <span class=\"screen-reader-text\">Lec 3 : Stress acting at a point- Stress tensor<\/span><\/a><\/p>\n","protected":false},"author":3,"featured_media":150031,"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":false,"jetpack_social_options":{"image_generator_settings":{"template":"highway","enabled":false},"version":2}},"categories":[3672,532,138,3328,3355,3665,446,6,4792],"tags":[59742,59750,59748,59747,59745],"class_list":["post-150032","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-christmas","category-climate-change","category-community","category-golf","category-ice-hockey","category-natural-phenomena","category-parents","category-reducing-triggers","category-sports-recreation","tag-cauchy-stress","tag-continuum-mechanics","tag-sign-convention-of-cauchy-stress","tag-stress-tensor","tag-tensor"],"jetpack_publicize_connections":[],"jetpack_sharing_enabled":true,"jetpack_featured_media_url":"https:\/\/effectsofanxiety.net\/wp-content\/uploads\/2022\/07\/htmlF_IMG_62d1730104385.jpg","_links":{"self":[{"href":"https:\/\/effectsofanxiety.net\/wp-json\/wp\/v2\/posts\/150032"}],"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=150032"}],"version-history":[{"count":1,"href":"https:\/\/effectsofanxiety.net\/wp-json\/wp\/v2\/posts\/150032\/revisions"}],"predecessor-version":[{"id":150046,"href":"https:\/\/effectsofanxiety.net\/wp-json\/wp\/v2\/posts\/150032\/revisions\/150046"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/effectsofanxiety.net\/wp-json\/wp\/v2\/media\/150031"}],"wp:attachment":[{"href":"https:\/\/effectsofanxiety.net\/wp-json\/wp\/v2\/media?parent=150032"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/effectsofanxiety.net\/wp-json\/wp\/v2\/categories?post=150032"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/effectsofanxiety.net\/wp-json\/wp\/v2\/tags?post=150032"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}