MATVE

Format A: Prony Series (Model=PRONY)

Format B: Bergström-Boyce (Model=BBOYCE)

Format C (Model=RTEST)

Format D (Model=CTEST)

Format E (Model=UPRN)

Example A: Prony Series (Model=PRONY)

Example B: Bergström-Boyce (Model=BBOYCE)

Example E: Unlimited Prony Series (Model=UPRN)

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 Bulk Data Entries, which should have the same MID as the MATVE Bulk Data Entry.
3. The linear viscoelastic material (Model=PRONY) is represented by the generalized Maxwell model. The material response, indicated by stress ( $\sigma$ ) here is given by the following convolution representation, for deviatoric deformation.
   $$\sigma ={\displaystyle {\int }_0^{t}g\left(t-s\right){\dot{\sigma }}_{0}ds}$$
   Where,

   $g\left(t\right)=\frac{G\left(t\right)}{G_0}$

   Normalized modulus

   $G_0$

   Instantaneous modulus

   $G\left(t\right)$

   Time-dependent modulus

   Similarly,

   $J\left(t\right)$

   Time-dependent compliance

   $j\left(t\right)$

   Normalized compliance

   They satisfy $j\left(t\right)=G_{0}J\left(t\right)$ .

   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 modulus ( $G_{\infty }$ ) and the following equation is used for calculation of the material property incorporating relaxation:

   $$g\left(t\right)=g_{\infty }+{\displaystyle \sum _i{g}_i{e}^{-\frac{t}{{\tau }_i}}}$$

   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:

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

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

   Where,

   $g_i$

   Prony material parameters.

   ${\tau }_i$

   Relaxation time.

   $g_i$ and ${\tau }_i$

   Values determined from curve fitting, if RTEST is given or they can be directly input via Model=PRONY.

   $$\begin{array}{l}{\dot{\sigma }}_0=G_0\dot{\epsilon }\\ \dot{\epsilon }=\frac{d\epsilon }{dt}\end{array}$$
   Where,

   ${\sigma }_0$

   Instantaneous stress response.

   $\text{ε}$

   Strain as a function of time.

   $g$

   Indicates the normalized modulus.

   $G$

   Indicates the modulus for relaxation.

   $j$

   Indicates the normalized compliance.

   $J$

   Indicates the compliance for creep.

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 you 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, you 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:

   $$F=F_A=F_B^e.F_B^{cr}$$

   The evolution of inelastic deformation gradient on network B is governed by:

   $$F_B^e.{\dot{F}}_B^{cr}.F_B^{cr-1}.F_B^{e-1}={\dot{\epsilon }}_B^v\frac{S_B}{{\overline{\sigma }}_B}$$

   The Bergström-Boyce hardening formulation is given by:

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

8. When MODEL=RTEST/CTEST:

   Relaxation (RTEST) or Creep (CTEST) test data can be input using these two types. This test data will internally be used to calibrate a Prony series.

   If creep test data are used, then the creep test will be first converted to the relaxation test using the convolution integration,

   $${\displaystyle \underset{0}{\overset{t}{\int }}g\left(s\right)j\left(t-s\right)ds}=t$$

   If the Laplace transform, $L$ , is written as:

   $$\widehat{f}\left(s\right)=L\left(f\left(t\right)\right)={\displaystyle \underset{0}{\overset{\infty }{\int }}f\left(t\right)e^{-st}dt}$$
   The Laplace transforms of the functions $g$ and $f$ satisfy $\widehat{g}\left(s\right)\widehat{j}\left(s\right)=\frac{1}{s^2}$ , Then the calibration to a Prony series will be carried out based on the relaxation test.

   $g$

   Normalized modulus

   $j$

   Normalized compliance

   You can input shear test data or volumetric test data, respectively, using the continuation lines SHEAR or BULK. The continuation line COMB will allow both shear and volumetric test data together.

9. If the number of Prony series is greater than 5, model UPRN, which means unlimited prony series, needs to be used. The first two columns are for the deviatoric part and the next two columns are for the bulk part. The deviatoric part and the bulk part can have different number of Prony series (just using blanks to fill the unused positions).