MATVE

Format A: Prony Series (Model = PRONY)

Format B: Bergström-Boyce (Model = BBOYCE)

Example A: Prony Series (Model = PRONY)

Example B: Bergström-Boyce (Model = BBOYCE)

Definitions

Comments

1. The CHEXA, CTETRA, CPENTA, and CPYRA elements are currently supported.
2. The instantaneous or long-term material property can be provided by MAT1, MAT9 or MATHE entries, which should have the same MID as the MATVE entry.
3. The linear viscoelastic material (Model = PRONY) is represented by the generalized Maxwell model. The material response is given by the following convolution representation, for deviatoric deformation.(1)
   $$\sigma ={\displaystyle {\int }_0^{t}G\left(t-s\right){\dot{\sigma }}_{0}ds}$$
   If the MTIME field on MAT1/MAT9/MATHE entries is set to LONG (default), then the input material property is considered as the long-term material deviatoric input ( $G_{\infty }$ ) and the following equation is used for calculation of the material property incorporating relaxation:(2)

   $$g\left(t\right)=G_{\infty }+{\displaystyle \sum _i{G}_i{e}^{-\frac{t}{{\tau }_i}}}$$
   If the MTIME field on the MAT1/MAT9/MATHE entries is set to INSTANT, then the input material property is considered as the instantaneous material input ( $G_0$ ) and the following equation is used for calculation of the material property incorporating relaxation:(3)

   $$g\left(t\right)=G_0-{\displaystyle \sum _i{G}_i\left[1-e^{-\frac{t}{{\tau }_i}}\right]}$$

   The subscript $i$ indicates the $i$ -th component in the Prony series. A maximum of 5 components are allowed.

   Where,

   $G_i$

   Deviatoric modulus ratio.

   ${\tau }_i$

   Relaxation time.

   ${\sigma }_0$

   Instantaneous stress response.

   The above equation(s) can be written in analogous form for the bulk material modulus.

4. For the isotropic model, the deviatoric and bulk responses can be specified separately. For the anisotropic model, only gDi and tDi are used and the bulk specifications are ignored.
5. The material relaxation response is controlled by the card VISCO. For example, if the user wants to simulate a physical relaxation test, the first subcase can omit the VISCO card so that material response is only the instantaneous elasticity in this subcase. In the next subcase, the user can add a VISCO card so that the material response is viscoelastic.
6. For Implicit Nonlinear Analysis, MATVE is supported for small displacement and large displacement nonlinear analysis.
7. The nonlinear viscoelastic material (Model = BBOYCE) is supported only for solid elements in Nonlinear Explicit Analysis.

   The response of the material can be represented using an equilibrium hyperelastic network A, and a time-dependent hyperelastic - nonlinear viscoelastic network B. The Hyperelastic material models for network A and B can be selected from existing MATHE card.

   The deformation gradient tensor, $F$ is assumed to act on both networks and is decomposed into elastic ( $F_B^e$ ) and inelastic ( $F_B^{cr}$ ) parts in network B as:(4)

   $$F=F_A=F_B^e.F_B^{cr}$$
   The evolution of inelastic deformation gradient on network B is governed by:(5)

   $$F_B^e.{\dot{F}}_B^{cr}.F_B^{cr-1}.F_B^{e-1}={\dot{\epsilon }}_B^v\frac{S_B}{{\overline{\sigma }}_B}$$
   The Bergström-Boyce hardening formulation is given by:(6)

   $${\dot{\epsilon }}_B^v=A{(\tilde{\lambda }-1+E)}^c{\overline{\sigma }}_B^m$$
   Where,

   ${\overline{\sigma }}_B=\sqrt{S_B:S_B}$

   $\tilde{\lambda }=\sqrt{\frac{1}{3}I:\left(F_B^{cr}.{\left(F_B^{cr}\right)}^T\right)}$

   $S_B$

   Deviatoric part of the Cauchy stress tensor in network B.

   $F_B^{cr}$

   Inelastic deformation gradient tensor in network B.