# MATVE

Bulk Data Entry Defines material properties for nonlinear viscoelastic materials.

## Format A: Prony Series (Model = PRONY)

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
MATVE MID Model     gD1 tD1 gB1 tB1
gD2 tD2 gD3 tD3 gD4 tD4 gD5 tD5
gB2 tB2 gB3 tB3 gB4 tB4 gB5 tB5

## Format B: Bergström-Boyce (Model = BBOYCE)

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
MATVE MID Model   Sb A C m E

## Format C (Model=RTEST)

Format for Separate Shear and Volumetric test data for relaxation:
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
MATVE MID Model etol npmax
SHEAR slong
gs(t) t
etc
BULK blong
gk(t) t
etc
Format for Combined Shear and Volumetric test data for relaxation:
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
MATVE MID Model etol npmax
COMB slong blong
gs(t) gk(t) t
etc

## Format D (Model=CTEST)

Format for Separate Shear and Volumetric test data for creep:
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
MATVE MID Model etol npmax
SHEAR slong
js(t) t
etc
BULK blong
jk(t) t
etc
Format for Combined Shear and Volumetric test data for creep:
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
MATVE MID Model etol npmax
COMB slong blong
js(t) jk(t) t
etc

## Example A: Prony Series (Model = PRONY)

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
MATVE 2 PRONY     0.25 5e-2 0.25 5e-2

## Example B: Bergström-Boyce (Model = BBOYCE)

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
MATVE 2 BBOYCE   2.0 0.1 -0.7 5.0 0.01

## Definitions

Field Contents SI Unit Example
MID Unique material identification number.

No default (Integer > 0)

Model Viscoelastic material model type.
PRONY (Default)
Linear viscoelastic model based on Prony series.
BBOYCE
Bergström-Boyce model.
RTEST
Relaxation test data for Prony series.
CTEST
Creep test data for Prony series.

gDi Modulus ratio for the $i$ -th deviatoric Prony series.

Default = Blank (Real > 0.0)

tDi Relaxation time for the $i$ -th deviatoric Prony series.

Default = Blank (Real > 0.0)

gBi Modulus ratio for the $i$ -th bulk Prony series.

Default = Blank (Real > 0.0)

tBi Relaxation time for the $i$ -th bulk Prony series.

Default = Blank (Real > 0.0)

Sb Stress scaling factor that defines the ratio of the stress carried by network B to that carried by network A under identical elastic stretching. 7

No default (Real > 0.0)

A Effective creep strain rate. 7

No default (Real > 0.0)

C Negative exponent characterizes the creep strain dependence of the effective creep strain rate in network B. 7

No default (-1.0 ≤ Real ≤ 0.0)

m Positive exponent characterizes the effective stress dependence of the effective creep strain rate in network B. 7

No default (Real ≥1.0)

E Material parameter to regularize the creep strain rate in the vicinity of the undeformed state. 7

Default = 0.01 (Real ≥0.0)

SHEAR Continuation line to indicate test data from shear relaxation/creep tests are to follow.
BULK Continuation line to indicate test data from volumetric relaxation/creep tests are to follow.
COMB Continuation line to indicate test data from both shear and volumetric relaxation/creep tests are to follow.
t Time; should be specified in an ascending order.

No default (Real > 0.0)

gs(t) Normalized shear modulus.

No default (0.0 ≤ Real ≤ 1.0)

gk(t) Normalized bulk modulus.

No default (0.0 ≤ Real ≤ 1.0)

js(t) Normalized shear compliance.

No default (1.0 ≤ Real)

jk(t) Normalized bulk compliance.

No default (1.0 ≤ Real)

etol Error tolerance for CTEST/RTEST material calibration.
0.0
Implies that the tolerance is automatically controlled.

Default = 0.0 (0.0 ≤ Real)

npmax Maximum number of terms in the Prony series for CTEST/RTEST material calibration.

Default = 5 (1 ≤ Integer ≤ 5)

slong Long term normalized Shear modulus for RTEST.

Default = blank (0.0 < Real < 1.0)

Long term normalized Shear compliance for CTEST.

Default = blank (1.0 < Real)

blong Long term normalized Bulk modulus for RTEST.

Default = blank (0.0 < Real < 1.0)

Long term normalized Bulk compliance for CTEST.

Default = blank (1.0 < Real)

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, indicated by stress ( $\sigma$ ) here is given by the following convolution representation, for deviatoric deformation.(1)
$\sigma ={\int }_{0}^{t}g\left(t-s\right){\stackrel{˙}{\sigma }}_{0}ds$
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:(2)
$g\left(t\right)={g}_{\infty }+\sum _{i}{g}_{i}{e}^{-\frac{t}{{\tau }_{i}}}$
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)
$g\left(t\right)=1-\sum _{i}{g}_{i}\left[1-{e}^{-\frac{t}{{\tau }_{i}}}\right]$

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.
(4)
$\begin{array}{l}{\stackrel{˙}{\sigma }}_{0}={G}_{0}\stackrel{˙}{\epsilon }\\ \stackrel{˙}{\epsilon }=\frac{d\epsilon }{dt}\end{array}$
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 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:(5)
$F={F}_{A}={F}_{B}^{e}.{F}_{B}^{cr}$
The evolution of inelastic deformation gradient on network B is governed by:(6)
${F}_{B}^{e}.{\stackrel{˙}{F}}_{B}^{cr}.{F}_{B}^{cr-1}.{F}_{B}^{e-1}={\stackrel{˙}{\epsilon }}_{B}^{v}\frac{{S}_{B}}{{\overline{\sigma }}_{B}}$
The Bergström-Boyce hardening formulation is given by:(7)
${\stackrel{˙}{\epsilon }}_{B}^{v}=A{\left(\stackrel{˜}{\lambda }-1+E\right)}^{c}{\overline{\sigma }}_{B}^{m}$
Where,
${\overline{\sigma }}_{B}=\sqrt{{S}_{B}:{S}_{B}}$
$\stackrel{˜}{\lambda }=\sqrt{\frac{1}{3}I:\left({F}_{B}^{cr}.{\left({F}_{B}^{cr}\right)}^{T}\right)}$
${S}_{B}$
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,(8)
$\underset{0}{\overset{t}{\int }}g\left(s\right)j\left(t-s\right)ds=t$
If the Laplace transform, $L$ , is written as:(9)
$\stackrel{^}{f}\left(s\right)=L\left(f\left(t\right)\right)=\underset{0}{\overset{\infty }{\int }}f\left(t\right){e}^{-st}dt$
The Laplace transforms of the functions $g$ and $f$ satisfy $\stackrel{^}{g}\left(s\right)\stackrel{^}{j}\left(s\right)=\frac{1}{{s}^{2}}$ , 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.