iMechanica - Thermo-mechanical coupled VUMAT
https://www.imechanica.org/taxonomy/term/6291
enImplementation of heat diffusion (Laplacian term) in Thermo-mechanical coupled VUMAT
https://www.imechanica.org/node/10305
<div class="field field-name-taxonomy-vocabulary-6 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/76">research</a></div></div></div><div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/5142">laplacian</a></div><div class="field-item odd"><a href="/taxonomy/term/6291">Thermo-mechanical coupled VUMAT</a></div><div class="field-item even"><a href="/taxonomy/term/6292">Heat Diffusion</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><span class="Apple-style-span"><span class="Apple-style-span"></span></span></p>
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<span class="apple-style-span"><span>Hi everyone, I am trying to implement a thermo-mechanical coupled constitutive model of bulk metallic glass into ABAQUS by VUMAT. The rate of temperature change is a fuction of current temperature and rate of plastic strain in the material with mathematic expression </span></span>
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<span class="apple-style-span"><span> dT/dt = A Laplacian(T) + B dP/dt<span> </span></span></span>
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<span>Where T</span><span><span class="Apple-converted-space"> </span></span><span>is</span><span><span class="Apple-converted-space"> </span></span><span>temperature, Laplacian(T) = d^2(T)/(dx)^2 + d^2(T)/(dy)^2, A and B</span><span><span class="Apple-converted-space"> </span></span><span>are material coefficient, and<span class="apple-style-span"><span><span class="Apple-converted-space"> </span>dP/dt</span></span><span class="Apple-converted-space"> </span>is plastic strain rate. In other words, the temperature change of the metallic glass are caused by two mechanisms, one is diffusion of heat (Laplacian), and another is heat generation due to plastic dissipation. In order to implement it into VUMAT, I convert the PDE above into forward Euler incremental equation as </span><span><span> </span></span>
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<span> dT= A(dt)<span class="Apple-converted-space"> </span><span class="apple-style-span"><span>Laplacian(T) + B dP</span></span></span><span> </span>
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<span>So far I manage to determine value of A, B and dP. If I could find the value of <span class="apple-style-span"><span>Laplacian(T) on each material point, I can solve the 2nd equation to determine value of dT as dt will be provided by ABAQUS. Then, I can use the dT value to update temperature at each material point by equation </span></span></span>
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<span><span class="apple-style-span"><span> TempNew = TempOld + dt</span></span></span>
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<p><span>However, my problem is I don’t know how to implement the Laplacian into VUMAT. It is because by using <span>finite difference method, Laplacian of a function takes the form as </span><span> </span><span> </span></span><span> </span></p>
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<span> Laplacian ( f(x,y) ) = ( f(x-h,y) + f(x+h,y) + f(x,y-h) + f(x,y+h) - 4f(x,y) ) / h^2</span>
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<span>which means in order to determine value of <span class="apple-style-span"><span>Laplacian(T) </span></span>in a material point, I need to call back temperature value of 4 nearby material points, and the h value, which is the distance between the material point I want to study and its neighbor material point. Is anyone has experience in solving this kind of problems? Or my problem can simply be solved by using some built in function in ABAQUS? Any opinion or suggestion on how to dealing with Laplacian term using VUMAT are welcomed, and that would be great if can provide the example solution for this kind of problem. </span>
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<span>Thank you.<span> </span></span>
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<span><span>P.S. This problem is actually identical with the problem discussed in blog below, and applogize to Zhang and Alexander as didn't reply on time.</span></span>
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<span><span><a href="node/10077">node/10077</a></span></span>
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</div></div></div>Tue, 17 May 2011 20:46:45 +0000gnoij10305 at https://www.imechanica.orghttps://www.imechanica.org/node/10305#commentshttps://www.imechanica.org/crss/node/10305