iMechanica - finite elasticity
https://www.imechanica.org/taxonomy/term/6719
enSolving incompressible finite elasticity without tears
https://www.imechanica.org/node/23070
<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/6719">finite elasticity</a></div><div class="field-item odd"><a href="/taxonomy/term/3255">linear solvers</a></div><div class="field-item even"><a href="/taxonomy/term/6340"># Finite Element modeling</a></div><div class="field-item odd"><a href="/taxonomy/term/12267">computaitonal fluid dynamics</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>Solving incompressible elasticity has been quite challenging numerically. The conventional approach for handling incompressibility is the so-called penalty method. A volumetric energy term enters into the strain energy and penalizes the volumetric deformation. One straightforward issue is that the penalty parameter goes directly into the tangent matrix. The bigger the penalty parameter, the worse the condition number of the matrix. This is really a manifestation of the ill-posedness of theories based on the Helmholtz free energy, in my opinion [3]. When the problem size is not too big, direct solvers can be used, and we probably do not realize the ill-conditioning. This is not the case when the problem size gets bigger and gets coupled in the multiphysics setting (e.g. fluid-structure interaction). This drives us to think of things in a different way. Mathematically, the penalty formulation can be transformed into a saddle-point problem, just like what has been done in fluid mechanics. In thermodynamics, this is achieved through the Legendre transformation between potentials [3]. For a saddle-point problem, there are many ways of solving the equations [1]. The trick is to devise a good approximation for the Schur complement. In a recent JCP paper [2], we developed an effective way of solving the finite-strain elasticity, using an iterative algorithm for sparse matrices. <strong>The algorithm can drive the residual to machine tolerance within a few iterations; it works well for compressible and incompressible, soft and hard, isotropic and anisotropic materials; it is scalable on parallel machines.</strong> Additionally, for the thermomechanical foundation of this framework, readers are also referred to [3].</p>
<p>[1] Michele Benzi, Gene H. Golub, and Jörg Liesen. "Numerical solution of saddle point problems." Acta Numerica 14 (2005): 1-137.</p>
<p>[2] Ju Liu and Alison L. Marsden. "A robust and efficient iterative method for hyper-elastodynamics with nested blockpreconditioning." Journal of Computational Physics 383 (2019): 72-93.</p>
<p><a href="https://authors.elsevier.com/a/1YXBH508HiFTO" target="_blank" data-saferedirecturl="https://www.google.com/url?q=https://authors.elsevier.com/a/1YXBH508HiFTO&source=gmail&ust=1549654013317000&usg=AFQjCNG7f7htjltNqyjMgRgzijjXiBGDRA">https://authors.elsevier.com/a/1YXBH508HiFTO</a></p>
<p><a href="https://www.sciencedirect.com/science/article/pii/S0021999119300440">https://www.sciencedirect.com/science/article/pii/S0021999119300440</a></p>
<p>[3] Ju Liu and Alison L. Marsden. "A unified continuum and variational multiscale formulation for fluids, solids, and fluid-structure interaction." Computer Methods in Applied Mechanics and Engineering 337 (2018): 549-597.</p>
<p><a href="https://www.sciencedirect.com/science/article/pii/S0045782518301701">https://www.sciencedirect.com/science/article/pii/S0045782518301701</a></p>
</div></div></div>Thu, 07 Feb 2019 20:21:51 +0000Ju Liu23070 at https://www.imechanica.orghttps://www.imechanica.org/node/23070#commentshttps://www.imechanica.org/crss/node/23070Cauchy's first law of motion
https://www.imechanica.org/node/11243
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>
Hi All
</p>
<p>
I'm a bioengineering PhD student, I just started reading on the finite elasticity theory and have a question regarding to the governing equation.
</p>
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As I understand, Cauchy's first law of motion is the governing equation for finite elasticity. For steady-state (no acceleration), the equation is:
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<p>
<span>d</span><strong>σ</strong><span>/d<strong>x=F</strong></span>
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<p>
where the <span>d</span><strong>σ</strong><span>/d<strong>x </strong></span>is the tress gradient and is the <span><strong>F </strong></span>body force. This equation suggests that the stress shall be spatially constant with the absense of body force. However, I'm not sure where does the boundary forces (surface tractions) comes in? I know the stress may not be constant in the presence of the external force, but I just can not see the relationship between the surface tractionsand the governing equation.
</p>
<p>
Another question is, I know stress and strain is a relative measure, and I have often seen people to assume the stress is measured from an unstrained reference configuration such that zero stress occurs at zero strain. Why is this a common assumption? Is it just for convenience? or is it more to do with the fact that reference configuration shall be considered as a state in which the material is free of all forces (both bodyand surface forces), hence zero stress shall be present?
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