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When using Finite Element Analysis for Fluids we solve the Navier Stokes Equation and continuity equation, when solving for temperature we solve the heat equation and fouriers law, when dealing with diffusion we solve Ficks law. But when dealing with solid mechanics what are the PDE's the usual hookes law may relate stress to strain but where are we getting the partial time term? Is it becz $$ F=ma=m \frac{d^2x}{dt^2}~? $$

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After going on Comsol's documenation I found that the governing equations are:

  • Newton's Second Law: $\boldsymbol{\nabla}\cdot\boldsymbol{\sigma} + \mathbf{F} = \rho\ddot{\mathbf{u}}$;

  • Hookes law: $ \boldsymbol{\sigma} = \mathsf{C}:\boldsymbol{\varepsilon}$; and

  • Linearized strain: $ \boldsymbol{\varepsilon} =\tfrac{1}{2} \left[\boldsymbol{\nabla}\mathbf{u}+(\boldsymbol{\nabla}\mathbf{u})^\mathrm{T}\right] \, .$

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  • $\begingroup$ Why linearized strain, for large displacement problems, full non-linear Green-SantVenant strain tensor is ued! $\endgroup$
    – Davius
    Commented Jul 5, 2022 at 11:43

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