{"id":172,"date":"2014-11-28T01:39:42","date_gmt":"2014-11-28T01:39:42","guid":{"rendered":"http:\/\/box5393.temp.domains\/~quranive\/\/?p=172"},"modified":"2022-01-24T15:55:00","modified_gmt":"2022-01-24T15:55:00","slug":"expansion-and-stiffness-of-thin-glue-or-rubber-layers","status":"publish","type":"post","link":"https:\/\/mechanicsandmachines.com\/?p=172","title":{"rendered":"Expansion and Stiffness of Thin Adhesive or Rubber Layers"},"content":{"rendered":"


\n<\/a><\/p>\n

Thin layers of adhesive, plastic, or rubber are often employed in precision machines for joining, shimming, and sealing. These layers are often the most compliant and most dimensionally unstable elements of an assembly, so it is important to understand their behavior.<\/p>\n

\"thinlayer\"<\/a><\/p>\n

<\/p>\n

Consider a thin layer of relatively compliant material sandwiched between two rigid parts. If the layer is thin compared to its width, we can think of the strains \"\epsilon_x\" and \"\epsilon_y\" as being restrained to zero except near the edges. This is a state of \"\emph{uniaxial strain}\" in the \"z\" direction. As we show below, the stiffness and thermal expansion in uniaxial strain are often considerably larger than they are for uniaxial stress.<\/p>\n

Hooke’s law for an isotropic solid is written as<\/p>\n

  <\/span>   <\/span>\"\begin{eqnarray*}<\/p>\n

The longitudinal strains \"\epsilon_x\", \"\epsilon_y\", and \"\epsilon_z\" are related to the longitudinal stresses \"\sigma_x\", \"\sigma_y\", and \"\sigma_z\" by the Young’s modulus \"E\" and Poisson’s ratio \"\nu\". The strain \"\epsilon_T\" could arise from temperature changes, absorption or desorption of humidity, or changes in molecular structure of an adhesive during curing.<\/p>\n

If \"\epsilon_T\" arises only from temperature change, it is common to replace \"\epsilon_T\" by a linear coefficient of thermal expansion (CTE) times a temperature change. But many polymers as well as low-expansion materials (for example Invar, ULE, or Zerodur) have nonlinear thermal expansion relationships, so we prefer to retain the more general \"\epsilon_T\" in our models.<\/p>\n

Setting \"\epsilon_x\" and \"\epsilon_y\" to zero in Hooke’s law and combining the equations to eliminate \"\sigma_x\" and \"\sigma_y\", we obtain Hooke’s law for uniaxial strain in the form
\n<\/a><\/p>\n

(1) <\/span>   <\/span>\"\begin{equation*}<\/p>\n

We can think of the leading coefficient as an effective modulus for uniaxial strain:<\/p>\n

(2) <\/span>   <\/span>\"\begin{equation*}<\/p>\n

The difference between the stiffnesses for uniaxial strain and uniaxial stress is usually significant. A Poisson’s ratio of 0.4 is common for plastics and adhesives, for which \"\hat{E} \approx 2.1 E\".<\/p>\n

Elastomers (rubbers) are nearly incompressible, so that \"\nu\" is nearly 0.5. If \"\nu\" is exactly 0.5, \"\hat{E}\" goes to infinity, but we can get a more realistic estimate by setting \"\nu = 0.5-\nu^*\" and expanding for small \"\nu^*\" to obtain<\/p>\n

  <\/span>   <\/span>\"\[<\/p>\n

For example, if \"\nu = 0.49\", then \"\nu^* =0.01\" and \"\hat{E} \approx 17 E\". Thus a thin sheet of rubber is much stiffer in the normal direction than it is in shear, and this can be used to advantage in designing a class of flexural bearings. (For such applications, the value of \"\nu^*\" is very important, and quoted values of \"\nu\" may not be accurate enough. It is better for such cases to work from the bulk modulus if it is available.)<\/p>\n

The effects of material expansion \"\epsilon_T\" are also amplified by the Poisson’s ratio. For example if, \"\nu =0.4\", the coefficient of the strain \"\epsilon_T\" in Eq.~(1<\/a>) is about 2.3, so that thermal expansion in the thin layer constrained in \"x\" and \"y\" results in more than twice as much thickness change than would be obtained if the thin layer were not restrained.<\/p>\n","protected":false},"excerpt":{"rendered":"

Thin layers of adhesive, plastic, or rubber are often employed in precision machines for joining, shimming, and sealing. These layers are often the most compliant and most dimensionally unstable elements of an assembly, so it is important to understand their behavior.<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"nf_dc_page":"","jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_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","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[8],"tags":[4,11,10,3,2],"series":[7],"class_list":["post-172","post","type-post","status-publish","format-standard","hentry","category-fundamentals","tag-cte","tag-glue-expansion","tag-glue-shrink","tag-thermal-expansion","tag-thermal-strain","series-elasticity"],"aioseo_notices":[],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p5f9h7-2M","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/mechanicsandmachines.com\/index.php?rest_route=\/wp\/v2\/posts\/172","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mechanicsandmachines.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mechanicsandmachines.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mechanicsandmachines.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mechanicsandmachines.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=172"}],"version-history":[{"count":38,"href":"https:\/\/mechanicsandmachines.com\/index.php?rest_route=\/wp\/v2\/posts\/172\/revisions"}],"predecessor-version":[{"id":216,"href":"https:\/\/mechanicsandmachines.com\/index.php?rest_route=\/wp\/v2\/posts\/172\/revisions\/216"}],"wp:attachment":[{"href":"https:\/\/mechanicsandmachines.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=172"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mechanicsandmachines.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=172"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mechanicsandmachines.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=172"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/mechanicsandmachines.com\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=172"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}