{"id":18120,"date":"2018-12-16T16:53:19","date_gmt":"2018-12-16T16:53:19","guid":{"rendered":"https:\/\/www.simscale.com\/?page_id=18120"},"modified":"2021-05-19T15:46:45","modified_gmt":"2021-05-19T15:46:45","slug":"bimetallic-strip-under-thermal-load","status":"publish","type":"page","link":"https:\/\/www.simscale.com\/docs\/validation-cases\/bimetallic-strip-under-thermal-load\/","title":{"rendered":"Validation Case: Bimetallic Strip Under Thermal Load"},"content":{"rendered":"\n\n\n\n<p class=\"wp-block-paragraph\">This validation case belongs to thermomechanics, with the case of a bimetallic strip under thermal load. The aim of this test case is to validate the following parameters:<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>Thermomechanical solver<\/li><li>Multiple materials and bonded contact<\/li><\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">The simulation results of SimScale were compared to theoretical computations derived from [Roark&#8217;s]\\(^1\\). <\/p>\n\n\n\n<div class=\"hw-block hw-btnWrapper hw-btnWrapper--alignCenter \">\n    <a href=\"https:\/\/www.simscale.com\/workbench\/?pid=7126248482952358115\" class=\"hw-btn    \" rel=\"noopener \" target=\"_blank\"    >\n        View Project    <\/a>\n<\/div>\n\n\n\n\n<h2 class=\"wp-block-heading\" id=\"geometry\" >Geometry<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">The geometry used for the case is as follows: <\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><a href=\"https:\/\/frontend-assets.simscale.com\/media\/2020\/07\/image-101.png\"><img loading=\"lazy\" decoding=\"async\" width=\"770\" height=\"617\" src=\"https:\/\/frontend-assets.simscale.com\/media\/2020\/07\/image-101.png\" alt=\"geometry model bimetallic strip thermomechanical validation case\" class=\"wp-image-31800\" srcset=\"https:\/\/frontend-assets.simscale.com\/media\/2020\/07\/image-101.png 770w, https:\/\/frontend-assets.simscale.com\/media\/2020\/07\/image-101-300x240.png 300w, https:\/\/frontend-assets.simscale.com\/media\/2020\/07\/image-101-768x615.png 768w\" sizes=\"auto, (max-width: 770px) 100vw, 770px\" \/><\/a><figcaption>Figure 1: The strip external faces are split to create the needed points for boundary conditions.<\/figcaption><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">It represents a strip with length \\(l\\) of 10 \\(m\\), width \\(w\\) of 1 \\(m\\) and thickness \\(t\\) of 0.1 \\(m\\), composed of two strips each with thickness \\(t_a , t_b\\) of 0.05 \\(m\\). Nodes N1 and N3 are located at mid-thickness and node N2 is located at the bottom surface as shown in figure 1.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"analysis-type-and-mesh\" >Analysis Type and Mesh<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Tool Type<\/strong>: Code_Aster<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Analysis Type<\/strong>: Thermomechanical steady state with static inertia effect<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Mesh and Element Types<\/strong>:<\/p>\n\n\n\n<figure class=\"wp-block-table aligncenter is-style-stripes\"><table><thead><tr><th class=\"has-text-align-center\" data-align=\"center\"><strong>Mesh Type<\/strong><\/th><th class=\"has-text-align-center\" data-align=\"center\"><strong>Number of <\/strong><br><strong>Nodes<\/strong><\/th><th class=\"has-text-align-center\" data-align=\"center\"><strong>Element Type<\/strong><\/th><\/tr><\/thead><tbody><tr><td class=\"has-text-align-center\" data-align=\"center\">2nd order hexahedral<\/td><td class=\"has-text-align-center\" data-align=\"center\">3652<\/td><td class=\"has-text-align-center\" data-align=\"center\">Standard<\/td><\/tr><\/tbody><\/table><figcaption>Table 1: Mesh details<\/figcaption><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">The hexahedral mesh was computed locally and uploaded into the simulation project.<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"941\" height=\"702\" src=\"https:\/\/frontend-assets.simscale.com\/media\/2020\/07\/image-103.png\" alt=\"hexahedral mesh bimetallic strip thermomechanical validation case\" class=\"wp-image-31854\" srcset=\"https:\/\/frontend-assets.simscale.com\/media\/2020\/07\/image-103.png 941w, https:\/\/frontend-assets.simscale.com\/media\/2020\/07\/image-103-300x224.png 300w, https:\/\/frontend-assets.simscale.com\/media\/2020\/07\/image-103-768x573.png 768w\" sizes=\"auto, (max-width: 941px) 100vw, 941px\" \/><figcaption>Figure 2: Finite elements hexahedral mesh used for the simulation (closer view on the left)<\/figcaption><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"simulation-setup\" >Simulation Setup<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Material<\/strong>:<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>Top Strip:<ul><li>Elastic Modulus \\(E_a = \\) 200 \\(GPa\\)<\/li><li>Poison&#8217;s ratio \\(\\nu_a = \\) 0<\/li><li>Density \\( \\rho_a = \\) 7870 \\( kg\/m^3 \\)<\/li><li>Thermal conductivity \\( \\kappa_a&nbsp;= \\) 60 \\( W\/(mK) \\)<\/li><li>Expansion ratio \\( \\gamma_a = \\) 1e-5 \\( 1\/K \\)<\/li><li>Reference temperature 300 \\(K\\)<\/li><li>Specific heat \\(C_{pa} = \\) 480 \\( J\/(kgK) \\)<\/li><\/ul><\/li><li>Bottom Strip:<ul><li>Elastic Modulus \\(E_b = \\) 200 \\(GPa\\)<\/li><li>Poison&#8217;s ratio \\(\\nu_b = \\) 0<\/li><li>Density \\( \\rho_b = \\) 7870 \\( kg\/m^3 \\)<\/li><li>Thermal conductivity \\( \\kappa_b&nbsp;= \\) 60 \\( W\/(mK) \\)<\/li><li>Expansion ratio \\( \\gamma_b = \\) 2e-5 \\( 1\/K \\)<\/li><li>Reference temperature 300 \\(K\\)<\/li><li>Specific heat \\(C_{pb} = \\) 480 \\( J\/(kgK) \\)<\/li><\/ul><\/li><\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Boundary Conditions<\/strong>:<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>Constraints:<ul><li>N1 restrained with \\( d_x = d_y = d_z = \\) 0 \\(m\\)<\/li><li>N2 restrained with \\( d_x = d_y = \\) 0 \\(m\\)<\/li><li>N3 restrained with \\( d_y = \\) 0 \\(m\\)<\/li><li>Fixed temperature \\(T = \\) 400 \\(K\\) applied on top and bottom surfaces<\/li><\/ul><\/li><li>Contacts:<ul><li>Bonded contact between the strips <\/li><\/ul><\/li><\/ul>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"reference-solution\" >Reference Solution<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">The reference solution is of the analytical type, as presented in [Roark&#8217;s]\\(^1\\). It is given in terms of the displacements of the free end of the strip and the stress at the bottom surface:<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\">$$ d_x = l (T &#8211; T_0) \\frac{\\gamma_a + \\gamma_b}{2} $$<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\">$$ d_z = \\frac{ 3 l^2 (\\gamma_b &#8211; \\gamma_a) (T &#8211; T_0)(t_a + t_b) }{ t_b^2 K_1} $$<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\">$$ \\sigma_{bottom} = \\frac{ (\\gamma_b &#8211; \\gamma_a)(T &#8211; T_0) E_b }{ K_1 } \\Big[ 3 \\frac{t_a}{t_b} + 2 &#8211; \\frac{E_a}{E_b} \\Big( \\frac{t_a}{t_b} \\Big)^3 \\Big] $$<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\">$$ K_1 = 4 + 6 \\frac{t_a}{t_b} + 4 \\Big( \\frac{t_a}{t_b} \\Big)^2  + \\frac{E_a}{E_b} \\Big( \\frac{t_a}{t_b} \\Big)^3  + \\frac{E_b}{E_a} \\frac{t_b}{t_a} $$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The computed solutions are:<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\"> \\(d_x= 0.015\\ m\\) <\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\"> \\(d_z = 0.75\\ m\\)<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\"> \\(\\sigma_{bottom} = 50\\ MPa\\) <\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"result-comparison\" >Result Comparison<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">A comparison of displacements at point N3 and stress \\(\\sigma_{XX}\\) at point N2 with theoretical solution is presented below:<\/p>\n\n\n\n<figure class=\"wp-block-table is-style-stripes\"><table><thead><tr><th class=\"has-text-align-center\" data-align=\"center\">POINT<\/th><th class=\"has-text-align-center\" data-align=\"center\">FIELD<\/th><th class=\"has-text-align-center\" data-align=\"center\">COMPUTED<\/th><th class=\"has-text-align-center\" data-align=\"center\">REF<\/th><th class=\"has-text-align-center\" data-align=\"center\">ERROR<\/th><\/tr><\/thead><tbody><tr><td class=\"has-text-align-center\" data-align=\"center\">N3<\/td><td class=\"has-text-align-center\" data-align=\"center\">DX \\([m]\\)<\/td><td class=\"has-text-align-center\" data-align=\"center\">0.015<\/td><td class=\"has-text-align-center\" data-align=\"center\">0.015<\/td><td class=\"has-text-align-center\" data-align=\"center\">0.00 %<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">N3<\/td><td class=\"has-text-align-center\" data-align=\"center\">DZ \\([m]\\)<\/td><td class=\"has-text-align-center\" data-align=\"center\">0.7479<\/td><td class=\"has-text-align-center\" data-align=\"center\">0.75<\/td><td class=\"has-text-align-center\" data-align=\"center\">-0.28 %<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">N2<\/td><td class=\"has-text-align-center\" data-align=\"center\">SIXX \\([MPa]\\)<\/td><td class=\"has-text-align-center\" data-align=\"center\">48.7631<\/td><td class=\"has-text-align-center\" data-align=\"center\">50<\/td><td class=\"has-text-align-center\" data-align=\"center\">-2.47 %<\/td><\/tr><\/tbody><\/table><figcaption>Table 2: Results comparison and computed errors<\/figcaption><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">Illustration of the deformed shape and stress distribution on the bimetallic strip below:<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large\"><a href=\"https:\/\/frontend-assets.simscale.com\/media\/2020\/07\/image-102.png\"><img decoding=\"async\" src=\"https:\/\/frontend-assets.simscale.com\/media\/2020\/07\/image-102-1024x541.png\" alt=\"stress contours developed due to deformation in simscale postprocessor thermomechanical\" class=\"wp-image-31804\"\/><\/a><figcaption>Figure 3: Deformed shape and stress contour on the bimetallic strip<\/figcaption><\/figure><\/div>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"related-tutorials\" >Related Tutorials<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\"><a href=\"https:\/\/www.simscale.com\/docs\/tutorials\/heat-transfer-engine-piston\/\">Advanced Tutorial: Thermomechanical Analysis of an Engine Piston<\/a><\/p>\n\n\n\n\n<div class='hw-block hw-references hw-references'>\n    <p class='hw-references__title'>References<\/p>\n    <ul class='hw-references__list'>\n\n        <li><cite>(2011)\u201dRoark\u2019s Formulas For Stress And Strain, Eighth Edition\u201d, W. C. Young, R. G. Budynas, A. M. Sadegh<\/cite><\/li>\n    <\/ul>\n<\/div>\n\n\n\n<div class=\"hw-block hw-note hw-note--info hw-note\">\n    <div class=\"hw-note__title\">\n        <p class=\"hw-note__titleText\"><i class=\"fa fa-exclamation-circle\" aria-hidden=\"true\"><\/i>Note<\/p>\n    <\/div>\n    <div class=\"hw-note__body\">\n        <p>If you still encounter problems validating you simulation, then please post the issue on our <a href=\"https:\/\/www.simscale.com\/forum\/\">forum<\/a> or <a href=\"mailto:support@simscale.com\">contact us<\/a>.<\/p>\n    <\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>This validation case belongs to thermomechanics, with the case of a bimetallic strip under thermal load. The aim of this...","protected":false},"author":94,"featured_media":0,"parent":17191,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"templates\/template-documentation.php","meta":{"_acf_changed":false,"_crdt_document":"","inline_featured_image":false,"footnotes":""},"class_list":["post-18120","page","type-page","status-publish","hentry"],"acf":[],"_links":{"self":[{"href":"https:\/\/www.simscale.com\/wp-json\/wp\/v2\/pages\/18120","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.simscale.com\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/www.simscale.com\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/www.simscale.com\/wp-json\/wp\/v2\/users\/94"}],"replies":[{"embeddable":true,"href":"https:\/\/www.simscale.com\/wp-json\/wp\/v2\/comments?post=18120"}],"version-history":[{"count":0,"href":"https:\/\/www.simscale.com\/wp-json\/wp\/v2\/pages\/18120\/revisions"}],"up":[{"embeddable":true,"href":"https:\/\/www.simscale.com\/wp-json\/wp\/v2\/pages\/17191"}],"wp:attachment":[{"href":"https:\/\/www.simscale.com\/wp-json\/wp\/v2\/media?parent=18120"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}