{"id":18374,"date":"2018-12-18T18:53:16","date_gmt":"2018-12-18T18:53:16","guid":{"rendered":"https:\/\/www.simscale.com\/?page_id=18374"},"modified":"2021-09-24T10:07:55","modified_gmt":"2021-09-24T10:07:55","slug":"hertzian-contact-between-two-spheres","status":"publish","type":"page","link":"https:\/\/www.simscale.com\/docs\/validation-cases\/hertzian-contact-between-two-spheres\/","title":{"rendered":"Validation Case: Hertzian Contact Between Two Spheres"},"content":{"rendered":"\n\n\n\n<p class=\"wp-block-paragraph\">This validation case belongs to structural dynamics. The aim of this test case is to validate the following parameters at the point of the Hertzian contact between two spheres:<\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>\\(\u03c3_{zz}\\) stress using a frictionless penalty contact.<\/li><li>\\(\u03c3_{zz}\\) stress using a frictionless augmented Lagrange contact.<\/li><\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">The simulation results of SimScale were compared to the results presented in [1]. <\/p>\n\n\n\n<div class=\"hw-block hw-btnWrapper hw-btnWrapper--alignCenter \">\n    <a href=\"https:\/\/www.simscale.com\/workbench\/?pid=281850436977849370\" 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\">Only one-eighth of each of the two spheres (with a radius of 50 \\(mm\\) ) is used for the analysis due to the symmetry of the case study.<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large is-resized\"><a href=\"https:\/\/frontend-assets.simscale.com\/media\/2020\/07\/Geometry-4.png\" rel=\"noopener\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/frontend-assets.simscale.com\/media\/2020\/07\/Geometry-4.png\" alt=\"spheres in contact geometry cad model\" class=\"wp-image-31892\" width=\"521\" height=\"547\" srcset=\"https:\/\/frontend-assets.simscale.com\/media\/2020\/07\/Geometry-4.png 694w, https:\/\/frontend-assets.simscale.com\/media\/2020\/07\/Geometry-4-286x300.png 286w\" sizes=\"auto, (max-width: 521px) 100vw, 521px\" \/><\/a><figcaption>Figure 1: Geometry of the two spheres in contact<\/figcaption><\/figure><\/div>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"analysis-type-and-domain\" >Analysis Type and Domain<\/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>: Static nonlinear<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Type of contact: <\/strong>Physical<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Mesh and element types<\/strong>: The meshes were created with the standard meshing algorithm on the SimScale platform. While a single region refinement is used in the meshes (A) and (C), the meshes in (B) and (D) were created with an additional region refinement around the contact region.<\/p>\n\n\n\n<figure class=\"wp-block-table\"><table><thead><tr><th>Case<\/th><th class=\"has-text-align-center\" data-align=\"center\">Element Type<\/th><th class=\"has-text-align-center\" data-align=\"center\">Number of Nodes<\/th><th class=\"has-text-align-center\" data-align=\"center\">Element Technology<\/th><\/tr><\/thead><tbody><tr><td>(A)<\/td><td class=\"has-text-align-center\" data-align=\"center\">1st Order Tetrahedral<\/td><td class=\"has-text-align-center\" data-align=\"center\">1559<\/td><td class=\"has-text-align-center\" data-align=\"center\">Standard<\/td><\/tr><tr><td>(B)<\/td><td class=\"has-text-align-center\" data-align=\"center\">1st Order Tetrahedral<\/td><td class=\"has-text-align-center\" data-align=\"center\">8229<\/td><td class=\"has-text-align-center\" data-align=\"center\">Standard<\/td><\/tr><tr><td>(C)<\/td><td class=\"has-text-align-center\" data-align=\"center\">2nd Order Tetrahedral<\/td><td class=\"has-text-align-center\" data-align=\"center\">34320<\/td><td class=\"has-text-align-center\" data-align=\"center\">Reduced Integration<\/td><\/tr><tr><td>(D)<\/td><td class=\"has-text-align-center\" data-align=\"center\">2nd Order Tetrahedral<\/td><td class=\"has-text-align-center\" data-align=\"center\">58973<\/td><td class=\"has-text-align-center\" data-align=\"center\">Reduced Integration <\/td><\/tr><\/tbody><\/table><figcaption>Table 1: The final mesh details for all cases<\/figcaption><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">Below the 1st order standard mesh for case A is visualized:<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-full is-resized\"><a href=\"https:\/\/frontend-assets.simscale.com\/media\/2021\/08\/coarse-mesh-hertzian-contact.jpg\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/frontend-assets.simscale.com\/media\/2021\/08\/coarse-mesh-hertzian-contact.jpg\" alt=\"mesh standard first order tetrahedral elements\" class=\"wp-image-47159\" width=\"462\" height=\"537\" srcset=\"https:\/\/frontend-assets.simscale.com\/media\/2021\/08\/coarse-mesh-hertzian-contact.jpg 616w, https:\/\/frontend-assets.simscale.com\/media\/2021\/08\/coarse-mesh-hertzian-contact-258x300.jpg 258w\" sizes=\"auto, (max-width: 462px) 100vw, 462px\" \/><\/a><figcaption> Figure 2:  A global refinement is applied in the case A mesh.<\/figcaption><\/figure><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">And the mesh (case B), which has a region refinement around the contact region is presented below:<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-full is-resized\"><a href=\"https:\/\/frontend-assets.simscale.com\/media\/2021\/08\/refined-mesh-hertzian-contact.jpg\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/frontend-assets.simscale.com\/media\/2021\/08\/refined-mesh-hertzian-contact.jpg\" alt=\"standard mesh with a local region refinement\" class=\"wp-image-47160\" width=\"456\" height=\"542\" srcset=\"https:\/\/frontend-assets.simscale.com\/media\/2021\/08\/refined-mesh-hertzian-contact.jpg 608w, https:\/\/frontend-assets.simscale.com\/media\/2021\/08\/refined-mesh-hertzian-contact-252x300.jpg 252w\" sizes=\"auto, (max-width: 456px) 100vw, 456px\" \/><\/a><figcaption>Figure 3: The mesh used for case B, created with an extra local refinement around the contact region<\/figcaption><\/figure><\/div>\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\/Solid:<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>Isotropic: <ul><li>\\(E\\) = 20 \\(GPa\\),<\/li><li>\\(\u03bd\\) &nbsp;= 0.3<\/li><\/ul><\/li><\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Constraints:<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>Faces ACD and A\u2019C\u2019D: zero x-displacement<\/li><li>Faces ABD and A\u2019B\u2019D: zero y-displacement<\/li><li>Face ABC: displacement of 2 \\(mm\\) in the z-direction<\/li><li>Face A\u2019B\u2019C\u2019 displacement of -2 \\(mm\\) in the z-direction<\/li><\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Physical Contacts: <\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\"><li>Augmented Lagrange:<ul><li>Contact smoothing enabled for linear elements and disabled for quadratic elements<\/li><li>Frictionless<\/li><li>Augmentation coefficient = 100<\/li><\/ul><\/li><li>Penalty:<ul><li>Contact smoothing enabled for linear elements and disabled for quadratic elements<\/li><li>Frictionless<\/li><li>Penalty coefficient = 10\\(^{15}\\)<\/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\">$$\\sigma_{zz} = \\frac{-E}{\\pi}\\frac{1}{1-{\\nu}^2}\\sqrt{\\frac{2h}{R}} \\tag{1}$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$h= 2 mm\u2212(\u22122 mm ) = 4mm\\tag{2} $$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">With equation&nbsp;(1)&nbsp;and equation&nbsp;(2)  the stress at point D results in:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$\\sigma_{zz} = \u22122798.3\\ MPa$$<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"results\" >Results<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Comparison of the stress&nbsp;\\(\u03c3_{zz}\\)&nbsp;at point D of the Hertzian contact obtained with SimScale with the analytical result of the reference solution&nbsp;[SSNV104_A]\\(^1\\):<\/p>\n\n\n\n<figure class=\"wp-block-table\"><table><thead><tr><th class=\"has-text-align-center\" data-align=\"center\">Case<\/th><th class=\"has-text-align-center\" data-align=\"center\">Physical Contact<\/th><th class=\"has-text-align-center\" data-align=\"center\">[SSNV104_A] (MPa)<\/th><th class=\"has-text-align-center\" data-align=\"center\">SimScale (MPa)<\/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\">(A)<\/td><td class=\"has-text-align-center\" data-align=\"center\">Penalty<\/td><td class=\"has-text-align-center\" data-align=\"center\">-2798.3<\/td><td class=\"has-text-align-center\" data-align=\"center\">-2834.34<\/td><td class=\"has-text-align-center\" data-align=\"center\">1.288%<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">(A)<\/td><td class=\"has-text-align-center\" data-align=\"center\">Augmented Lagrange<\/td><td class=\"has-text-align-center\" data-align=\"center\">-2798.3<\/td><td class=\"has-text-align-center\" data-align=\"center\">-2836.32<\/td><td class=\"has-text-align-center\" data-align=\"center\">1.359%<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">(B)<\/td><td class=\"has-text-align-center\" data-align=\"center\">Penalty<\/td><td class=\"has-text-align-center\" data-align=\"center\">-2798.3<\/td><td class=\"has-text-align-center\" data-align=\"center\">-2834.97<\/td><td class=\"has-text-align-center\" data-align=\"center\">1.310%<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">(B)<\/td><td class=\"has-text-align-center\" data-align=\"center\">Augmented Lagrange<\/td><td class=\"has-text-align-center\" data-align=\"center\">-2798.3<\/td><td class=\"has-text-align-center\" data-align=\"center\">-2830.21<\/td><td class=\"has-text-align-center\" data-align=\"center\">1.140%<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">(C)<\/td><td class=\"has-text-align-center\" data-align=\"center\">Penalty<\/td><td class=\"has-text-align-center\" data-align=\"center\">-2798.3<\/td><td class=\"has-text-align-center\" data-align=\"center\">-2875.45<\/td><td class=\"has-text-align-center\" data-align=\"center\">2.757%<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">(D)<\/td><td class=\"has-text-align-center\" data-align=\"center\">Penalty<\/td><td class=\"has-text-align-center\" data-align=\"center\">-2798.3<\/td><td class=\"has-text-align-center\" data-align=\"center\">-2841.42<\/td><td class=\"has-text-align-center\" data-align=\"center\">1.541%<\/td><\/tr><\/tbody><\/table><figcaption>Table 2: The \\(\u03c3_{zz}\\) results&#8217; comparison for all cases A through D<\/figcaption><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">It is obvious from the table above that the best results were obtained with SimScale&#8217;s 1st order mesh (case B), using the Augmented Lagrange contact:<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-full\"><a href=\"https:\/\/frontend-assets.simscale.com\/media\/2021\/08\/Case-A-Penalty-Contact-Results-1.jpg\"><img loading=\"lazy\" decoding=\"async\" width=\"466\" height=\"749\" src=\"https:\/\/frontend-assets.simscale.com\/media\/2021\/08\/Case-A-Penalty-Contact-Results-1.jpg\" alt=\"hertzian contact two spheres cauchy stress \u03c3zz penalty contact\" class=\"wp-image-47161\" srcset=\"https:\/\/frontend-assets.simscale.com\/media\/2021\/08\/Case-A-Penalty-Contact-Results-1.jpg 466w, https:\/\/frontend-assets.simscale.com\/media\/2021\/08\/Case-A-Penalty-Contact-Results-1-187x300.jpg 187w\" sizes=\"auto, (max-width: 466px) 100vw, 466px\" \/><\/a><figcaption>Figure 4: The \\(\u03c3_{zz}\\) results on the spheres for case B with an Augmented Lagrange contact method<\/figcaption><\/figure><\/div>\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><a href=\"https:\/\/www.code-aster.org\/V2\/doc\/default\/fr\/man_v\/v6\/v6.04.104.pdf\" target=\"_blank\" rel=\"nofollow noopener\">[SSNV104] (1, 2, 3) SSNV104 &#8211; Contact de deux sph\u00e8res &#8211; Code_Aster validation<\/a>   <\/cite><\/li>\n    <\/ul>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>This validation case belongs to structural dynamics. The aim of this test case is to validate the following parameters at...","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-18374","page","type-page","status-publish","hentry"],"acf":[],"_links":{"self":[{"href":"https:\/\/www.simscale.com\/wp-json\/wp\/v2\/pages\/18374","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=18374"}],"version-history":[{"count":0,"href":"https:\/\/www.simscale.com\/wp-json\/wp\/v2\/pages\/18374\/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=18374"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}