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I'm working with the equilibrium equations in linear elasticity, which I have not worked with in the past. My engineering colleagues seem to "know" that the maximum Von Mises stress occurs on the boundary of the domain (assume the domain experiences no body force, and is loaded on part of its boundary while satisfying Dirichlet boundary conditions on the other part). To prove this rigorously via the maximum principle would involve deriving a scalar, second-order PDE for the Von Mises stress. Does anyone know how this is done? (Or can confirm or deny that the claimed result holds?) Many thanks for your help.

(Another way of asking this question perhaps is: suppose $T$ is a divergence-free symmetric tensor. Does any scalar function of the eigenvalues of $T$ satisfy a maximum principle? Of course in linear elasticity, $T = A\cdot ( \nabla u + \nabla^\top u)$ where $u$ is the displacement of the domain and $A$ is the elasticity tensor which we can assume to be homogeneous and isotropic.)

Adrian Butscher

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That depends on the elastic model that you deal with. Is it linear (infinitesimal displacements) or non-linear ? Is it isotropic or not ?

In general, because elasticity is a system, not an equation, there is no such scalar quantity that would obey a maximum principle. However, a linear isotropic model is governed by the system $$\lambda\Delta u+\mu\nabla{\rm div}u=0,$$ for some coefficients $\lambda,\mu>0$. Then, taking the divergence, we obtain $(\lambda+\mu)\Delta{\rm div}u=0$. This shows that ${\rm div}u$ obeys a maximum principle in this simple case.

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  • $\begingroup$ Brilliant! That answers my question. It's simple too. Yes, we're in linear elasticity with homogeneous, isotropic medium. Thanks. $\endgroup$
    – user2698883
    Commented Dec 5, 2014 at 15:24
  • $\begingroup$ I do not exclude that some principle occurs for a linear model with intermediate symmetry group, but this is beyond my expertise. $\endgroup$
    – Denis Serre
    Commented Dec 5, 2014 at 15:42
  • $\begingroup$ Understood. Actually one of my colleagues handed me an interesting paper giving examples and counterexamples of maximum principles in elasticity theory: Wheeler, L. T. "Maximum principles in classical elasticity." SERIES ON ADVANCES IN MATHEMATICS FOR APPLIED SCIENCES 38 (1996): 157-185. Thanks again for your help. $\endgroup$
    – user2698883
    Commented Dec 5, 2014 at 18:01
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In hertzian contacts (the archetype for linear elastic contact mechanics) the maximum Von Mises stress is subsurface (not on the boundary).

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