Weak formulation of dynamic plasticity

variational formulation
1

Hi,

I successfully used the variational formulation with dissipation [Simo and Hughes, p.162] for finite deformation rate-independent pasticity, i.e.

\Pi(u_{n+1}, C^p_{n+1}, \alpha_{n+1}, \gamma) = \int \psi^{el}(u_{n+1}, C^p_{n+1}) \ \mathrm{d} \Omega + \int \psi^h (\alpha_{n+1}) \ \mathrm{d} \Omega + P^{ext}(u_{n+1}) + \Delta t \int \Big( D^p - \gamma f \big( \tilde{\tau}(u_{n+1}, C^p_{n+1}), \alpha_{n+1} \big) \Big) \ \mathrm{d} \Omega + \frac{c}{2} \int \left\langle \gamma \right\rangle_{-}^2 \ \mathrm{d} \Omega

where

  • u is the displacement,

  • C^p is the plastic right Cauchy-Green tensor,

  • \alpha is the hardening parameter,

  • \gamma is the consistency parameter,

  • \psi^{el} is the elastic strain energy density (I used Hencky’s logarithmic strain),

  • \psi^h(\alpha) = \frac{1}{2} H \alpha^2,

  • P^{ext} is the external load,

  • f is the yield surface,

  • \tilde{\tau} is the equivalent Kirchhoff stress,

  • c is a penalty factor,

  • and \left\langle x \right\rangle_{-} = (x - |x|)/2.

The dissipation D^p is defined as

D^p=\tau : \ell^p - \frac{\partial \psi^h}{\partial \alpha} \dot{\alpha}

with

\ell^p = \frac{1}{2} F C^{p-1} \dot{C}^P F^{-1}

\dot{\alpha}, \dot{C}^p were discretized using backward Euler.

The weak form was then obtained using the symbolic differentiation of FEniCS, i.e.

form = derivative(Pi, u_cp_alp_gam, v)

(it was solved with legacy FEniCS).

My question is how can I change it to a dynamic setting?

Is it true to refer to the variational derivative of \Pi with respect to each of the variables as the appropriate stiffness term?

And how should I treat \dot{\alpha} and \dot{C}^p when differentiating with respect to \alpha and C^p?

Thanks,
Yaron

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