# MATFVE

Bulk Data Entry Defines material properties for frequency-dependent viscoelastic materials.

## Format 1 (TYPE=FORMULA)

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
MATFVE MID FORMULA
R1 I1 a R1k I1k b

## Format 2 (TYPE=TABLE)

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
MATFVE MID TABLE
TID1 TID2 TID3 TID4

## Format 3 (TYPE=PRONY)

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

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
EUNIAXI
El Es f $\text{ε}$
etc.
EVOLUME
Kl Ks f J
etc.

## Format 5 (TYPE=RTEST)

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

## Format 6 (TYPE=CTEST)

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

## Example

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

## Definition

Field Contents SI Unit Example
MID Unique material identification number.

No default (Integer > 0)

TYPE Frequency-dependent visco-elastic material model type.
FORMULA 4
TABLE 5
PRONY 6
RTEST
CTEST

No default

R1 Real part of the deviatoric part for the FORMULA type.

Default = blank (Real)

I1 Imaginary part of the deviatoric part for the FORMULA type.

Default = blank (Real)

a A real number for the deviatoric part for the FORMULA type.

Default = blank (Real)

R1k Real part of the bulk part for the FORMULA type.

Default = blank (Real)

I1k Imaginary part of the bulk part for the FORMULA type.

Default = blank (Real)

b A real number for the bulk part for the FORMULA type.

Default = blank (Real)

TID1 Table ID (TABLEDi only) to specify the real part of the deviatoric part.

Default = blank (Integer > 0)

TID2 Table ID (TABLEDi only) to specify the Imaginary part of the deviatoric part.

Default = blank (Integer > 0)

TID3 Table ID (TABLEDi only) to specify the real part of the bulk part.

Default = blank (Integer > 0)

TID4 Table ID (TABLEDi only) to specify the Imaginary part of the bulk part.

Default = blank (Integer>0)

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

No default (Real > 0.0)

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

No default (Real > 0.0)

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

No default (Real > 0.0)

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

No default (Real > 0.0)

EUNIAXI Continuation line to indicate the storage and loss moduli from uniaxial tests when TYPE = PRELOAD.
EVOLUME Continuation line to indicate the storage and loss moduli from volumetric tests when TYPE = PRELOAD.
El Uniaxial loss modulus.

No default (Real)

Es Uniaxial storage modulus.

No default (Real)

f Frequency.

No default (Real ≥ 0.0)

$\text{ε}$ Uniaxial nominal strain.

No default (Real)

Kl Bulk loss modulus.

No default (Real)

Ks Bulk storage modulus.

No default (Real)

J Volume ratio between current and original volumes.

No default (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 long-term response is given by an elastic material description, which needs to have the same MID. MATHE/MAT1/MAT9 are supported.
2. CHEXA, CTETRA, CPENTA, CPYRA elements are supported.
3. A function $g*\left(t\right)$ of deviatoric relaxation in the time-domain is considered.

If the MTIME field on MAT1/MAT9/MATHE entries is set to LONG (default), the input material property is considered as the long-term material deviatoric input modulus ( ${G}_{\infty }$ ) and Equation 1 is used for calculation of the material property incorporating relaxation.

(1)
$g*\left(t\right)=\sum _{i}\frac{{G}_{i}}{{G}_{\infty }}{e}^{-\frac{t}{{\tau }_{i}}}=\sum _{i}\frac{{g}_{i}}{{g}_{\infty }}{e}^{-\frac{t}{{\tau }_{i}}}$

The subscript $i$ indicates the $i$ -th term in the Prony series.

This equation is equivalent to the equation on the MATVE entry. The equation on MATVE is rewritten by using: (2)
$g\left(t\right)={g}_{\infty }\left(g*\left(t\right)+1\right)$
If the MTIME field on the MAT1/MAT9/MATHE entries is set to INSTANT, the input material property is considered as the instantaneous material input ( ${G}_{0}$ ) and Equation 2 for $g*\left(t\right)$ can be rewritten by substituting ${g}_{\infty }=1-\sum _{i}{g}_{i}$ for calculation of the material property incorporating relaxation.
${G}_{\infty }={g}_{\infty }*{G}_{0}$
Long-term deviatoric modulus.
Where, ${g}_{\infty }$ is the normalized long term deviatoric modulus.
${G}_{i}={g}_{i}*{G}_{0}$
Prony series deviatoric parameter.
Where, ${g}_{i}$ are the Prony series deviatoric parameters.
${\tau }_{i}$
Relaxation time.
Where,
$g$
Indicates the normalized modulus.
$G$
Indicates the modulus for relaxation.
$j$
Indicates the normalized compliance.
$J$
Indicates the compliance for creep.
See Comment 8.

The example equations in this comment show the calculation for shear relaxation. These can be similarly written in an analogous manner for bulk relaxation.

The Fourier transform of this $g*\left(t\right)$ function is:(3)
$\stackrel{˜}{g}*\left(\omega \right)=F\left(g*\left(t\right)\right)=R+iI$
Where, $F$ is the Fourier transformation. Equation 3 is for the frequency-dependent deviatoric relaxation. Similarly, the equation for the frequency-dependent bulk relaxation can be written in an analogous manner. (4)
$\stackrel{˜}{k}*\left(\omega \right)=F\left(k*\left(t\right)\right)={R}_{k}+i{I}_{k}$

After Fourier transformation, the Real and Imaginary parts $R$ , $I$ and ${R}_{k}$ , ${I}_{k}$ are frequency-dependent and can be input in four different ways (FORMULA, TABLE, PRONY, or PRELOAD).

4. When TYPE=FORMULA: (5)
$R=R1*{f}^{-a}$
(6)
$I=I1*{f}^{-a}$
(7)
${R}_{k}=R1k*{f}^{-b}$
(8)
${I}_{k}=I1k*{f}^{-b}$
Where,
$f$
Frequency.
$R$ , $I$
Real and imaginary parts of the Fourier transform of the deviatoric relaxation function.
${R}_{k}$ , ${I}_{k}$
Real and imaginary parts of the Fourier transform of the bulk relaxation function.
5. When TYPE=TABLE:
TID1
Specify $\omega R$ as a function of the frequency $f$ .
TID2
Specify $\omega I$ as a function of the frequency $f$ .
TID3
Specify $\omega {R}_{k}$ as a function of the frequency $f$ .
TID4
Specify $\omega {I}_{k}$ as a function of the frequency $f$ .
Where,
$\omega$
Angular frequency.
$R$ , $I$
Real and imaginary parts of the Fourier transform of the deviatoric relaxation function.
${R}_{k}$ , ${I}_{k}$
Real and imaginary parts of the Fourier transform of the bulk relaxation function.
6. When TYPE=PRONY, the data inputs have the same meaning as the MATVE card. The storage and loss moduli are determined as follows, after a Fourier transform of the time-domain Prony series. (9)
${G}_{s}=\left({G}_{\infty }+\sum _{i}\frac{{G}_{i}{\tau }_{i}^{2}{\omega }^{2}}{1+{\tau }_{i}^{2}{\omega }^{2}}\right)$
(10)
${K}_{s}=\left({K}_{\infty }+\sum _{i}\frac{{K}_{i}{\tau }_{i}^{2}{\omega }^{2}}{1+{\tau }_{i}^{2}{\omega }^{2}}\right)$
(11)
${G}_{l}=\left(\sum _{i}\frac{{G}_{i}{\tau }_{i}\omega }{1+{\tau }_{i}^{2}{\omega }^{2}}\right)$
(12)
${K}_{l}=\left(\sum _{i}\frac{{K}_{i}{\tau }_{i}\omega }{1+{\tau }_{i}^{2}{\omega }^{2}}\right)$
Where,
${G}_{s}$ , ${G}_{l}$
Deviatoric storage and loss moduli.
${K}_{s}$ , ${K}_{l}$
Bulk storage and loss moduli.
${G}_{\infty }$ , ${K}_{\infty }$
Long-term material modulus.
${G}_{i}={g}_{i}*{G}_{0}$ , ${K}_{i}={k}_{i}*{K}_{0}$
Where, ${g}_{i}$ and ${k}_{i}$ are Prony series parameters for shear and bulk moduli, respectively.
${\tau }_{i}$
Relaxation time.
$\omega$
Angular frequency.
(13)
$\omega I=1-\frac{{G}_{s}}{{G}_{\infty }}$
(14)
$\omega R=\frac{{G}_{l}}{{G}_{\infty }}$
(15)
$\omega {I}_{k}=1-\frac{{K}_{s}}{{K}_{\infty }}$
(16)
$\omega {R}_{k}=\frac{{K}_{l}}{{K}_{\infty }}$
Where,
$\omega$
Angular frequency.
$R$ , $I$
Real and imaginary parts of the Fourier transform of the deviatoric relaxation function.
${R}_{k}$ , ${I}_{k}$
Real and imaginary parts of the Fourier transform of the bulk relaxation function.
EUNIAXI
Test data to specify uniaxial loss modulus, uniaxial storage modulus, the frequency $f$ , uniaxial nominal strain.
EVOLUME
Test data to specify bulk loss modulus, bulk storage modulus, the frequency $f$ , volume ratio (current volume/initial volume).
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,(17)
$\underset{0}{\overset{t}{\int }}g\left(s\right)j\left(t-s\right)ds=t$
If the Laplace transform, $L$ , is written as:(18)
$\stackrel{^}{f}\left(s\right)=L\left(f\left(t\right)\right)=\underset{0}{\overset{\infty }{\int }}f\left(t\right){e}^{-st}dt$
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.

9. For all types, deviatoric component and bulk component specifications cannot be blank at the same time.
10. Only Direct Frequency Response Analysis is supported.
11. A nonlinear LGDISP analysis needs to precede the direct frequency response analysis which establishes the base state for the direct frequency analysis. STATSUB(PRELOAD) in the Subcase Information section for the direct frequency analysis needs to point to this nonlinear LGDISP analysis. A dummy nonlinear LGDISP analysis without loadings will mean that the base state is the initial configuration.
12. The time domain and the frequency domain viscoelastic behavior cannot be specified together. That is, MATVE and MATFVE cannot be used together.