Elastic Deformation

Elastic Deformation

Revision for “Elastic Deformation” created on November 28, 2021 @ 11:05:12

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Elastic Deformation
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<h2 style="text-align: justify;">Definition</h2> <p style="text-align: justify;">Elastic deformation is a change of the shape of the body as a reaction to applied stress. This deformation is only temporary and once the stress is released, the undeformed shape of the body is restored, as shown in the figure below. In tribology, elastic deformation largely affects <a href="http://www.tribonet.org/wiki/elastohydrodynamic-lubrication-ehl/" target="_blank" rel="noopener noreferrer">elastohydrodynamic film thickness</a> build up, real contact area, etc. A problem of determination of elastic deformation in various scenarios is considered in the classical (linear elasticity) <a href="https://docs.google.com/file/d/0Bw8MfqmgWLS4NTlFNF9VZzBZdWs/view" target="_blank" rel="noopener noreferrer">Elasticity Theory book by Timoshenko and Goodier</a> [1].</p> <p style="text-align: justify;"><img class="aligncenter wp-image-1936 size-large" src="http://www.tribonet.org/wp-content/uploads/2017/02/Elastic-Deformation-300x234.jpg" alt="Elastic Deformation" width="300" height="234" data-wp-pid="1936" /></p> <h2 style="text-align: justify;">Elastic deformation explained</h2> [embed]https://www.youtube.com/watch?v=DLE-ieOVFjI&feature=youtu.be[/embed] <h2 style="text-align: justify;">Application</h2> <p style="text-align: justify;">In the field of tribology, the most commonly studied configuration of contact is the contact of a sphere or a cylinder with a flat (the contact of two spheres or two cylinders can be reduced to the contact on flat). In this case, a half-space approximation is applicable and the full system of Elasticity Theory equations can be solved analytically to link the elastic deflection of the surface to the applied pressure on the surface [1]. The resultant equation is given in integral form:</p> <p style="text-align: justify;">[math] \begin{eqnarray} \label{complete_sys1}</p> <p style="text-align: justify;">h_e(x,y) = \frac{2\pi}{E'} \int\int \frac{p(x',y')}{\sqrt{(x-x')^2+(y-y')^2}}dx'dy' \\</p> <p style="text-align: justify;">\end{eqnarray} [/math]</p> <p style="text-align: justify;">Here [math] h_e [/math] is the elastic deflection, [math] 1/{E'}= {1 - {\nu_1}^2}/{E_1} + {{\nu_2}^2}/{E_2} [/math] is the reduced elastic modulus, [math] {\nu_1}, {E_1},{\nu_2}, {E_2} [/math] are the Poisson's ratio and Young's modulus of the bodies, [math] p(x,y) [/math] is the contact pressure. This equation is used in most of tribological problems, including <a href="http://www.tribonet.org/wiki/elastohydrodynamic-lubrication-ehl/" target="_blank" rel="noopener noreferrer">EHL</a> problems, but also in contact mechanics, friction and wear simulation. Further information regarding the application of the equation can be found in the <a href="http://link.springer.com/article/10.1007/s11249-015-0536-z">reference</a> [2].</p> <h2 style="text-align: justify;">Calculating elastic deformation online</h2> <p style="text-align: justify;">There are several online tools to calculate elastic deformation for special cases.</p> <blockquote><a href="https://www.tribonet.org/rigid-flat-punch-calculator/">Online Calculator - Rigid Flat Punch on Flat</a> <a href="https://www.tribonet.org/online-calculator-rigid-cone-on-elastic-flat/">Online Calculator - Rigid Cone on Elastic Flat</a> <a href="https://www.tribonet.org/online-calculator-indentation-by-a-flat-elliptical-punch/">Online Calculator - Indentation by a Flat Elliptical Punch</a></blockquote> <p style="text-align: justify;">Here is <a href="https://www.tribonet.org/online-tribology-calculators/">a full list of available calculators</a>.</p> <h2>References</h2> <p style="text-align: justify;">[1] <a href="https://docs.google.com/file/d/0Bw8MfqmgWLS4NTlFNF9VZzBZdWs/view">Theory of Elasticity</a>, Timoshenko, S.P., Goodier, J.N., 1970.</p> <p class="ArticleTitle" lang="en" style="text-align: justify;">[2] On a Model for the Prediction of the Friction Coefficient in Mixed Lubrication Based on a Load-Sharing Concept with Measured Surface  Roughness, <span class="authors__name">Aydar Akchurin,</span><span class="authors__name">Rob Bosman, </span><span class="authors__name">Piet M. Lugt, </span><span class="authors__name">Mark van Drogen, <a href="https://doi.org/10.1007/s11249-015-0536-z">https://doi.org/10.1007/s11249-015-0536-z</a>. </span></p> <p lang="en">[3] LINCOLN, B. Elastic Deformation and the Laws of Friction. <i>Nature</i> <b>172, </b>169–170 (1953). <a class="vglnk" href="https://doi.org/10.1038/172169b0" rel="nofollow">https://doi.org/10.1038/172169b0</a></p> <p style="text-align: justify;"><span style="border-radius: 2px; text-indent: 20px; width: auto; padding: 0px 4px 0px 0px; text-align: center; font: bold 11px/20px 'Helvetica Neue',Helvetica,sans-serif; color: #ffffff; background: no-repeat scroll 3px 50% / 14px 14px #bd081c; position: absolute; opacity: 1; z-index: 8675309; display: none; cursor: pointer; top: 208px; left: 368px;">Save</span></p> <p style="text-align: justify;"><span style="border-radius: 2px; text-indent: 20px; width: auto; padding: 0px 4px 0px 0px; text-align: center; font: bold 11px/20px 'Helvetica Neue',Helvetica,sans-serif; color: #ffffff; background: no-repeat scroll 3px 50% / 14px 14px #bd081c; position: absolute; opacity: 1; z-index: 8675309; display: none; cursor: pointer; top: 208px; left: 368px;">Save</span></p>
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