{"id":1175,"date":"2020-12-30T18:35:35","date_gmt":"2020-12-30T18:35:35","guid":{"rendered":"https:\/\/pickedshares.com\/?p=1175"},"modified":"2021-05-08T12:43:22","modified_gmt":"2021-05-08T12:43:22","slug":"engineering-mechanics-ii-exercise-6-invariants-and-principal-stresses","status":"publish","type":"post","link":"https:\/\/pickedshares.com\/en\/engineering-mechanics-ii-exercise-6-invariants-and-principal-stresses\/","title":{"rendered":"Invariants and principal stresses"},"content":{"rendered":"\n<p>This exercise is about the invariants and principal stresses of a stress tensor and addresses the following questions:<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>How to calculate the invariants of a stress tensor?<\/li><li>How to calculate the determinant of a stress tensor?<\/li><li>How to calculate the principal stresses of a stress tensor?<\/li><li>How to calculate the principal stress directions of a stress tensor?<\/li><li>How to calculate a stress vector?<\/li><\/ul>\n\n\n\n<h2 class=\"wp-block-heading\">Task<\/h2>\n\n\n\n<script src=\"https:\/\/cdnjs.cloudflare.com\/ajax\/libs\/mathjax\/2.7.7\/MathJax.js?config=TeX-AMS_HTML\" async=\"async\">  \/\/ A comment that hinders wxWidgets from optimizing this tag too much.\n<\/script>\n\\[ \\newcommand{\\myvec}[1]{{\\begin{bmatrix}#1\\end{bmatrix}}} \\]\n<p>The following stress tensor was calculated for a point in space:<\/p>\n<p>\\[ S = \\myvec{1 &amp; 1 &amp; 3\\\\1 &amp; 5 &amp; 1\\\\3 &amp; 1 &amp; 1} \\, Nmm^{-2} \\]<\/p>\n<p>Calculate:<\/p>\n<p>a) the three invariants of S<\/p>\n<p>b) the three principal stresses \u03c3<sub>1<\/sub>, \u03c3<sub>2<\/sub> and \u03c3<sub>3<\/sub><\/p>\n<p>c) the three main directions of stress (main axis system) and <\/p>\n<p>d) the stress vector on the plane with<\/p>\n<p>\\[ \\vec{n} = \\frac{1}{13} \\, \\myvec{3\\\\4\\\\12} \\]<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Solution<\/h2>\n\n\n\n<script src=\"https:\/\/cdnjs.cloudflare.com\/ajax\/libs\/mathjax\/2.7.7\/MathJax.js?config=TeX-AMS_HTML\" async=\"async\">  \/\/ A comment that hinders wxWidgets from optimizing this tag too much.\n<\/script>\n\\[ \\newcommand{\\myvec}[1]{{\\begin{bmatrix}#1\\end{bmatrix}}} \\]\n<p>The stress tensor is noted in the general form beforehand in order to capture the correct values \u200b\u200bwhen calculating the invariants.<\/p>\n<p>\\[ S = \\myvec{\\sigma_x &amp; \\tau_{xy} &amp; \\tau_{xz}\\\\\\tau_{yx} &amp; \\sigma_y &amp; \\tau_{yz}\\\\\\tau_{zx} &amp; \\tau_{zy} &amp; \\sigma_z}  \\]<\/p>\n<p><strong>reg. a)<\/strong><\/p>\n<p><\/p><h3>Invariant 1, Trace of the stress tensor<\/h3><p><\/p>\n<p>The general solution for the trace of the stress tensor is<\/p>\n<p>\\[ \\tag{1} I_1 = \\sigma_x + \\sigma_y + \\sigma_z \\]<\/p>\n<p>\\[ \\tag{2} I_1 = (1 + 5 + 1) \\, Nmm^{-2} \\]<\/p>\n<p>\\[ \\tag{3} I_1 = 7 \\, Nmm^{-2} \\]<\/p>\n<p><\/p><h3>Invariant 2<\/h3>\n<p>The general solution for invariant 2 of the stress tensor is<\/p>\n\n<div style=\"overflow:auto;\">\n<p>\\[ \\tag{4} I_2 = \\sigma_x \\cdot \\sigma_y + \\sigma_y \\cdot \\sigma_z + \\sigma_x \\cdot \\sigma_z -\\tau_{xy}^2 - \\tau_{xz}^2 - \\tau_{yz}^2\\]<\/p>\n<p>\\[ \\tag{5} I_2 = (1 \\cdot 5 + 5 \\cdot 1 + 1 \\cdot 1 -1^2 - 3^2 - 1^2) \\, N^2mm^{-4} \\]<\/p>\n<\/div>\n<p>\\[ \\tag{6} I_2 = 0  \\]<\/p>\n<p><\/p><h3>Invariant 3, Determinant of the stress tensor<\/h3><p><\/p>\n<p>The general solution for the determinant of the stress tensor is<\/p>\n\n<div style=\"overflow:auto;\">\n<p>\\[ \\tag{7} I_3 = det \\,S = \\sigma_x \\cdot \\sigma_y \\cdot \\sigma_z + \\tau_{xy} \\cdot \\tau_{yz} \\cdot \\tau_{zx} + \\tau_{xz} \\cdot \\tau_{yx} \\cdot \\tau_{zy} \\, ...\\]<\/p>\n<p>\\[ ... \\, - \\sigma_x \\cdot \\tau_{yz} \\cdot \\tau_{zy} - \\tau_{xy} \\cdot \\tau_{yx} \\cdot \\sigma_z - \\tau_{xz} \\cdot \\sigma_y \\cdot \\tau_{zx} \\]<\/p>\n<p>\\[ \\tag{8} I_3 = (1 \\cdot 5 \\cdot 1 + 1 \\cdot 1 \\cdot 3 + 3 \\cdot 1 \\cdot 1 \\, ...\\]<\/p>\n<p>\\[ ... \\, - 1 \\cdot 1 \\cdot 1 - 1 \\cdot 1 \\cdot 1 - 3 \\cdot 5 \\cdot 3)\\, N^3mm^{-6} \\]<\/p>\n<\/div>\n<p>\\[\\tag{9} I_3 = -36\\, N^3mm^{-6} \\]<\/p>\n\n<p><strong>reg. b)<\/strong><\/p>\n<p><\/p><h3>Calculation of the 3 principal stresses<\/h3><p><\/p>\n<p>The characteristic equation of the stress tensor is solved to calculate the three principal stresses. This is:<\/p>\n\n<div style=\"overflow:auto;\">\n<p>\\[\\tag{10} \\sigma^3 - I_1 \\cdot \\sigma^2 - I_2 \\cdot \\sigma - I_3 = 0 \\]<\/p>\n<p>\\[\\tag{11} \\sigma^3 - 7\\,Nmm^{-2} \\cdot \\sigma^2  + 36\\,N^3mm^{-6} = 0 \\]<\/p>\n<\/div>\n<p>The solution of this equation leads to the principal stresses<\/p>\n<p>\\[ \\tag{12} \\sigma_1 = -2\\, Nmm^{-2}\\]<\/p>\n<p>\\[ \\tag{13} \\sigma_2 = 6\\, Nmm^{-2}\\]<\/p>\n<p>\\[ \\tag{14} \\sigma_3 = 3\\, Nmm^{-2}\\]<\/p>\n\n\n<p><strong>reg. c)<\/strong><\/p>\n<p><\/p><h3>Calculation of the main stress directions<\/h3><p><\/p>\n<p>The general system of equations for calculating a stress direction is, with n as the component of the normal vector and \u03c3 as the principal stress:<\/p>\n\n<div style=\"overflow:auto;\">\n<p>\\[ \\tag{15.I} (\\sigma_x-\\sigma) \\cdot n_x + \\tau_{xy} \\cdot n_y + \\tau_{xz} \\cdot n_z = 0 \\]<\/p>\n<p>\\[ \\tag{15.II} \\tau_{yx} \\cdot n_x + (\\sigma_y-\\sigma) \\cdot n_y + \\tau_{yz} \\cdot n_z = 0 \\]<\/p>\n<p>\\[ \\tag{15.III} \\tau_{zx} \\cdot n_x + \\tau_{zy} \\cdot n_y + (\\sigma_z - \\sigma) \\cdot n_z = 0 \\]<\/p>\n<\/div>\n<p><\/p><h4>Establishing the systems of equations<\/h4><p><\/p>\n<p>The unit Nmm<sup>-2<\/sup> is not included in the following equation systems in order to improve the readability of the equations.<\/p>\n<p><strong>Principal stress \u03c3<sub>1<\/sub><\/strong><\/p>\n<p>The system of equations is<\/p>\n\n<div style=\"overflow:auto;\">\n<p>\\[ \\tag{16.I} (1+2) \\cdot n_{x1} + 1 \\cdot n_{y1} + 3 \\cdot n_{z1} = 0 \\]<\/p>\n<p>\\[ \\tag{16.II} 1 \\cdot n_{x1} + (5+2) \\cdot n_{y1} + 1 \\cdot n_{z1} = 0 \\]<\/p>\n<p>\\[ \\tag{16.III} 3 \\cdot n_{x1} + 1 \\cdot n_{y1} + (1 +2) \\cdot n_{z1} = 0 \\]<\/p>\n<\/div>\n<p>From this it follows immediately<\/p>\n<p>\\[ \\tag{17} n_{y1} = 0 \\]<\/p>\n<p>and from equation (16.II)<\/p>\n<p>\\[ \\tag{18} n_{x1} + n_{z1} = 0 \\]<\/p>\n<p>\\[ \\tag{19} n_{x1} = - n_{z1} \\]<\/p>\n<p>The absolut value of the normal vector is 1, therefore applies:<\/p>\n<p>\\[ \\tag{20} | \\vec{n_1}| = 1  \\]<\/p>\n<p>\\[ \\tag{21}  1 = \\sqrt{n_{x1}^2+n_{y1}^2+n_{z1}^2} \\]<\/p>\n<p>or with the previously calculated n<sub>y1<\/sub> and n<sub>z1<\/sub><\/p>\n<p>\\[ \\tag{21}  2 \\cdot n_{x1}^2 = 1 \\]<\/p>\n<p>\\[ \\tag{22}  n_{x1} = \\frac{1}{\\sqrt{2}} \\]<\/p>\n<p>\\[ \\tag{23}  n_{z1} = - \\frac{1}{\\sqrt{2}} \\]<\/p>\n<p>The normal vector for the principal stress 1 thus is<\/p>\n<p>\\[ \\tag{24} \\vec{n_1} = \\frac{1}{\\sqrt{2}}\\myvec{1\\\\0\\\\-1}  \\]<\/p>\n\n<p><strong>Principal stress \u03c3<sub>2<\/sub><\/strong><\/p>\n<p>The system of equations is<\/p>\n\n<div style=\"overflow:auto;\">\n<p>\\[ \\tag{25.I} -5 \\cdot n_{x2} + 1 \\cdot n_{y2} + 3 \\cdot n_{z2} = 0 \\]<\/p>\n<p>\\[ \\tag{25.II} 1 \\cdot n_{x2}  -1 \\cdot n_{y2} + 1 \\cdot n_{z2} = 0 \\]<\/p>\n<p>\\[ \\tag{25.III} 3 \\cdot n_{x2} + 1 \\cdot n_{y2}  -5 \\cdot n_{z2} = 0 \\]<\/p>\n<\/div>\n<p>The procedure for solving the problem is the same as for calculating the first principal stress, so it is not listed again here. The second normal vector is<\/p>\n<p>\\[ \\tag{26} \\vec{n_2} = \\frac{1}{\\sqrt{6}}\\myvec{1\\\\2\\\\1}  \\]<\/p>\n\n<p><strong>Principal stress \u03c3<sub>3<\/sub><\/strong><\/p>\n<p>The system of equations is<\/p>\n\n<div style=\"overflow:auto;\">\n<p>\\[ \\tag{27.I} -2 \\cdot n_{x3} + 1 \\cdot n_{y3} + 3 \\cdot n_{z3} = 0 \\]<\/p>\n<p>\\[ \\tag{27.II} 1 \\cdot n_{x3} + 2 \\cdot n_{y3} + 1 \\cdot n_{z3} = 0 \\]<\/p>\n<p>\\[ \\tag{27.III} 3 \\cdot n_{x3} + 1 \\cdot n_{y3} -2 \\cdot n_{z3} = 0 \\]<\/p>\n<\/div>\n<p>Here, too, no further demonstration of the solution is required, since it is analogous to main stress direction 1. The normal vector for the third principal stress is<\/p>\n<p>\\[ \\tag{28} \\vec{n_3} = \\frac{1}{\\sqrt{3}}\\myvec{1\\\\-1\\\\1}  \\]<\/p>\n\n<p><strong>reg. d)<\/strong><\/p>\n<p><\/p><h3>Calculation of the stress vector<\/h3><p><\/p>\n<p>The stress vector is calculated using Cauchy's stress equation:<\/p>\n<p>\\[ \\tag{29} \\vec{S} = S \\cdot \\vec{n} \\]<\/p>\n\n<div style=\"overflow:auto;\">\n<p>\\[ \\tag{30} \\vec{S} = \\frac{1}{13} \\cdot \\myvec{1 &amp; 1 &amp; 3\\\\1 &amp; 5 &amp; 1\\\\3 &amp; 1 &amp; 1}  \\cdot  \\, \\myvec{3\\\\4\\\\12} \\, Nmm^{-2}   \\]<\/p>\n<p>\\[ \\tag{31} \\vec{S} = \\frac{1}{13} \\cdot \\myvec{1\\cdot3+1\\cdot4+3\\cdot12\\\\1\\cdot3+5\\cdot4+1\\cdot12\\\\3\\cdot3+1\\cdot4+1\\cdot12}  Nmm^{-2} \\]<\/p>\n<p>\\[ \\tag{32} \\vec{S} = \\frac{1}{13} \\cdot \\myvec{43\\\\35\\\\25}  Nmm^{-2} \\]<\/p>\n<\/div>\n\n\n\n<p>We have come to the end, all of the above questions have been answered.<\/p>\n\n\n\n<p>We have some interesting links in German language about the calculation of tensors and especially invariants. Maybe the language isn't so important in mathematical expressions: the <a href=\"https:\/\/de.wikipedia.org\/wiki\/Formelsammlung_Tensoralgebra#Invarianten\" target=\"_blank\" rel=\"noreferrer noopener\">article about tensor algebra at Wikipedia<\/a> and our <a href=\"https:\/\/pickedshares.com\/aufgabensammlungen-zur-technischen-mechanik-mit-loesungen\/\" target=\"_blank\" rel=\"noreferrer noopener\">compiled sources for exercises and solution regarding engineering mechanics<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p> ... <a title=\"Invariants and principal stresses\" class=\"read-more\" href=\"https:\/\/pickedshares.com\/en\/engineering-mechanics-ii-exercise-6-invariants-and-principal-stresses\/\" aria-label=\"Read more about Invariants and principal stresses\">Read more<\/a><\/p>\n","protected":false},"author":1,"featured_media":1167,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[52,60],"tags":[],"class_list":["post-1175","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-engineering-mechanics-ii","category-exercises","infinite-scroll-item","masonry-post","generate-columns","tablet-grid-50","mobile-grid-100","grid-parent","grid-33"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Invariants and principal stresses &#8226; 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