Viscoelastic materials

This problem contains basic test cases for one or more Abaqus elements and features.

This page discusses:

Large-strain time domain viscoelasticity with hyperelasticity
Large-strain time domain viscoelasticity with a hyperfoam material
Small-strain time domain viscoelasticity with linear elasticity
Small-strain time domain viscoelasticity with anisotropic elasticity
Small-strain time domain viscoelasticity with traction-separation elasticity
Frequency domain viscoelasticity
Frequency domain viscoelasticity defined directly in terms of storage and loss moduli

ProductsAbaqus/StandardAbaqus/Explicit

Large-strain time domain viscoelasticity with hyperelasticity

Elements tested

B31

CAX4R

CPE4

CPE4H

CPE4HT

CPE4RH

CPS4

CPS4R

C3D8RH

C3D8RHT

M3D4

Problem description

Material 1


Polynomial coefficients (N=1): $C_{10}$ = 8., $C_{01}$ = 2.
Compressible case: $D_{1}$ = 0.1.
Prony series coefficients (N=1): ${\bar{g}}_{1}^{P}$ = 0., ${\bar{k}}_{1}^{P}$ = 0.5, $τ_{1}$ = 3.

Material 2


Polynomial coefficients (N=1): $C_{10}$ = 8., $C_{01}$ = 2.
Compressible case: $D_{1}$ = 0.1.
Prony series coefficients (N=1): ${\bar{g}}_{1}^{P}$ = 0.5, ${\bar{k}}_{1}^{P}$ = 0., $τ_{1}$ = 3.
Heat transfer properties for coupled analysis: conductivity = 0.01, density = 1.,
specific heat = 1.

Material 3


Polynomial coefficients (N=1): $C_{10}$ = 1.5 × 10⁶, $C_{01}$ = 0.5 × 10⁶.
Compressible case: $D_{1}$ = 1. × 10⁻⁷.
Prony series coefficients (N=2):
${\bar{g}}_{1}^{P}$ = 0.5, ${\bar{k}}_{1}^{P}$ = 0., $τ_{1}$ = 0.2.
${\bar{g}}_{2}^{P}$ = 0.49, ${\bar{k}}_{2}^{P}$ = 0., $τ_{2}$ = 0.5.

Material 4


Polynomial coefficients (N=1): $C_{10}$ = 27.02, $C_{01}$ = 1.42.
Compressible case: $D_{1}$ = 0.000001.
Prony series coefficients (N=2):
${\bar{g}}_{1}^{P}$ = 0.25, ${\bar{k}}_{1}^{P}$ = 0.25, $τ_{1}$ = 5.
${\bar{g}}_{2}^{P}$ = 0.25, ${\bar{k}}_{2}^{P}$ = 0.25, $τ_{2}$ = 10.
Creep compliance test data generated from Prony series above.
Stress relaxation test data generated from Prony series above.

Material 5


Polynomial coefficients (N=1): $C_{10}$ = 8., $C_{01}$ = 2.
Compressible case: $D_{1}$ = 0.001.
Prony series coefficients (N=2):
${\bar{g}}_{1}^{P}$ = 0.5, ${\bar{k}}_{1}^{P}$ = 0., $τ_{1}$ = 1.
${\bar{g}}_{2}^{P}$ = 0.49, ${\bar{k}}_{2}^{P}$ = 0., $τ_{2}$ = 2.

Material 6


Polynomial coefficients (N=1): $C_{10}$ = 550.53, $C_{01}$ = −275.265.
Compressible case: $D_{1}$ = 7. × 10⁻⁷.
Prony series coefficients (N=6):
${\bar{g}}_{1}^{P}$ = 0.1986, ${\bar{k}}_{1}^{P}$ = 0., $τ_{1}$ = 0.281 × 10⁻⁷.
${\bar{g}}_{2}^{P}$ = 0.1828, ${\bar{k}}_{2}^{P}$ = 0., $τ_{2}$ = 0.281 × 10⁻⁵.
${\bar{g}}_{3}^{P}$ = 0.1388, ${\bar{k}}_{3}^{P}$ = 0., $τ_{3}$ = 0.281 × 10⁻³.
${\bar{g}}_{4}^{P}$ = 0.2499, ${\bar{k}}_{4}^{P}$ = 0., $τ_{4}$ = 0.281 × 10⁻¹.
${\bar{g}}_{5}^{P}$ = 0.1703, ${\bar{k}}_{5}^{P}$ = 0., $τ_{5}$ = 0.281 × 10¹.
${\bar{g}}_{6}^{P}$ = 0.0593, ${\bar{k}}_{6}^{P}$ = 0., $τ_{6}$ = 0.281 × 10³.

Material 7


Ogden coefficients (N=2): $μ_{1}$ = 16., $α_{1}$ = 2., $μ_{2}$ = 4., $α_{2}$ = −2.
Prony series coefficients (N=1): ${\bar{g}}_{1}^{P}$ = 0.5, ${\bar{k}}_{1}^{P}$ = 0., $τ_{1}$ = 3.

Material 8


Arruda-Boyce coefficients: $μ$ = 20., $λ_{m}$ = 7.
Prony series coefficients (N=1): ${\bar{g}}_{1}^{P}$ = 0.5, ${\bar{k}}_{1}^{P}$ = 0., $τ_{1}$ = 3.

Material 9


Van der Waals coefficients: $μ$ = 20., $λ_{m}$ = 10., a = 0.1, $β$ = 0.02.
Prony series coefficients (N=1): ${\bar{g}}_{1}^{P}$ = 0.5, ${\bar{k}}_{1}^{P}$ = 0., $τ_{1}$ = 3.

Material 10


Neo-Hookean coefficient: $c_{10}$ = 1, $D$ = 0.1.
Prony series coefficients (N=1): ${\bar{g}}_{1}^{P}$ = 0.5, ${\bar{k}}_{1}^{P}$ = 0, $τ_{1}$ = 0.1.

Material 11


Ogden coefficients (N=3): $μ_{1}$ = 64.26, $α_{1}$ = 1.8, $μ_{2}$ = 25., $α_{2}$ = −2., $μ_{3}$ = 18.76, $α_{3}$ = 7.
Prony series coefficients (N=1): ${\bar{g}}_{1}^{P}$ = 0.72, ${\bar{k}}_{1}^{P}$ = 0., $τ_{1}$ = 17.5.
Heat transfer properties for coupled analysis: conductivity = 1 × 10⁻⁶, density = 7800, specific heat = 10, inelastic heat fraction = 0.8.

Results and discussion

The results agree well with exact analytical or approximate solutions.

Calibration of Prony series parameters from frequency-dependent moduli and vice versa has been tested for Materials 1, 4, and 6 in various relaxation and steady-state dynamic analyses. The data conversion is performed automatically in Abaqus. In the tests described below some of the time domain analyses are repeated using frequency-dependent moduli data and some of the frequency domain (steady-state dynamic) analyses are repeated using time-dependent moduli data. The results of the repeated analyses are in good agreement with those of the original.

Input files

Material 1:

mvhcdo2ahc.inp: Compressible, volumetric compression, CPS4 elements.
mvhcdo2sr2.inp: Compressible, volumetric compression, CPS4 elements; Prony series parameters calibrated from frequency-dependent moduli.
mvhcdo2ssd.inp: Compressible, volumetric compression, CPS4 elements; steady-state dynamic, frequency-dependent moduli data derived from specified Prony series parameters.
mvhcdo2ss2.inp: Compressible, volumetric compression, CPS4 elements; steady-state dynamic, direct specification of frequency-dependent moduli data.
mvhcdo2zzz.inp: Tabulated frequency-dependent moduli data included in mvhcdo2sr2.inp and mvhcdo2ss2.inp.
mvhcdo3ahc.inp: Compressible, volumetric compression, CPE4 elements.

Material 2:

mvccoo3hut.inp: Incompressible, uniaxial tension, coupled analysis, CPE4HT elements.
mvhcoo2rre.inp: Incompressible, relaxation in uniaxial tension, CPS4 elements.
mvhcoo3hut.inp: Incompressible, uniaxial tension, CPE4H elements.
mvhcoo3ltr.inp: Incompressible, triaxial, CPE4H elements.

Material 3:

mvhcdo3rre.inp: Compressible, relaxation in uniaxial tension, CPE4 elements.

Material 4:

mvhcdo3srs.inp: Compressible, uniaxial tension, static and relaxation, CPE4H elements.
mvhtdo3srs.inp: Creep and relaxation test data, uniaxial tension, static and relaxation, 2 CPE4RH elements.
mvhtdo3sr2.inp: Compressible, uniaxial tension, static and relaxation, 2 CPE4RH elements; Prony series parameters calibrated from frequency-dependent moduli.
mvhtdo3ssd.inp: Creep and relaxation test data, compressible, uniaxial tension, steady-state dynamic, 2 CPE4RH elements.
mvhtdo3ss2.inp: Compressible, uniaxial tension, steady-state dynamic, 2 CPE4RH elements; direct specification of Prony series parameters calibrated in mvhtdo3ssd.inp.
mvhtdo3ss3.inp: Compressible, uniaxial tension, steady-state dynamic, 2 CPE4RH elements; frequency-dependent moduli data derived from Prony series parameters calibrated from shear relaxation and creep test data as used in mvhtdo3ssd.inp.
mvhtdo3zzz.inp: Tabulated frequency-dependent moduli data included in mvhtdo3ss3.inp.
mvhtdo3srs1.inp: Combined test data, uniaxial tension, static and relaxation, 2 CPE4RH elements.
mvhcdo2srs.inp: Compressible, uniaxial tension and rotation, static and relaxation, CPS4 elements.
mvhcdo2vlp.inp: Compressible, uniaxial tension and rotation, static and relaxation with static linear perturbation steps containing LOAD CASE, CPS4R elements.

Material 5:

mvhcdo2rre.inp: Compressible, relaxation in uniaxial tension, M3D4 elements.

Material 6:

mvhcdo3kct.inp: Compressible, biaxial compression tension, CAX4R elements.
mvhcdo3kc2.inp: Compressible, biaxial compression tension, CAX4R elements; Prony series parameters calibrated from frequency-dependent moduli.
mvhcdo3ssd.inp: Compressible, biaxial compression tension, CAX4R elements; steady-state dynamic, frequency-dependent moduli data derived from specified Prony series parameters.
mvhcdo3ss2.inp: Compressible, biaxial compression tension, CAX4R elements; steady-state dynamic, direct specification of frequency-dependent moduli data.
mvhcdo3zzz.inp: Tabulated frequency-dependent moduli data used in mvhcdo3kc2.inp and mvhcdo3ss2.inp as an INCLUDE file.

Material 7:

mvhcoo3rre.inp: Incompressible, relaxation in uniaxial tension, Ogden model, CPE4H elements.
mvhcoo3vlp.inp: Incompressible, uniaxial tension with static linear perturbation steps, Ogden model, CPE4H elements.

Material 8:

mvacoo3rre.inp: Incompressible, relaxation in uniaxial tension, Arruda-Boyce model, CPE4H elements.
mvacoo3vlp.inp: Incompressible, uniaxial tension with static linear perturbation steps containing LOAD CASE, Arruda-Boyce model, CPE4H elements.

Material 9:

mvvcoo3rre.inp: Incompressible, relaxation in uniaxial tension, Van der Waals model, CPE4H elements.
mvvcoo3vlp.inp: Incompressible, uniaxial tension with static linear perturbation steps, Van der Waals model, CPE4H elements.

Material 10:

neoh_ve_unicyclic_b31.inp: Incompressible, uniaxial cyclic test with neo-Hookean model, B31 and C3D8RH elements.
neoh_ve_creep_b31.inp: Creep test with neo-Hookean model (compressible and incompressible), B31 and C3D8RH elements.
neoh_ve_relax_b31.inp: Relaxation test with neo-Hookean model (compressible and incompressible), B31 and C3D8RH elements.

Material 11:

ogden_ve_ssh_cyclic.inp: Coupled temperature-displacement analysis with viscous dissipation as a heat source, incompressible, cyclic simple shear test, Ogden model, C3D8RHT element.

Large-strain time domain viscoelasticity with a hyperfoam material

Elements tested

CPE4

Problem description

Material 1


Hyperfoam coefficients (N=3):
$μ_{1}$ = −17.4, $α_{1}$ = −1.22, $μ_{2}$ = 548.2, $α_{2}$ = 17.3, $μ_{3}$ = 10.47, $α_{3}$ = −1.775, $ν_{1}$ = $ν_{2}$ = $ν_{3}$ = 0.
Prony series coefficients (N=1): ${\bar{g}}_{1}^{P}$ = 0.5, ${\bar{k}}_{1}^{P}$ = 0., $τ_{1}$ = 3.

Material 2


Hyperfoam coefficients (N=3): Uniaxial test data, compression, Poisson's ratio = 0.
Prony series coefficients (N=1): ${\bar{g}}_{1}^{P}$ = 0.5, ${\bar{k}}_{1}^{P}$ = 0.5, $τ_{1}$ = 3.

Results and discussion

The results agree well with exact analytical or approximate solutions.

Input files

Material 1 (Coefficient input):

mvfcdo3rre.inp: Compressible, relaxation in uniaxial tension, CPE4 elements.
mvfcdo3vlp.inp: Compressible, uniaxial tension with static linear perturbation steps containing LOAD CASE, CPE4 elements.

Material 2 (Test data input):

mvftdo3rre.inp: Compressible, relaxation in uniaxial tension, CPE4 elements.
mvftdo3rre_stbil_adap.inp: Compressible, relaxation in uniaxial tension, CPE4 elements; with adaptive stabilization.

Small-strain time domain viscoelasticity with linear elasticity

Elements tested

CPS4

Problem description

Material:


Young's modulus = 30.
Poisson's ratio = 0.4.
Prony series coefficients (N=2):
${\bar{g}}_{1}^{P}$ = 0.25, ${\bar{k}}_{1}^{P}$ = 0.25, $τ_{1}$ = 5.
${\bar{g}}_{2}^{P}$ = 0.25, ${\bar{k}}_{2}^{P}$ = 0.25, $τ_{2}$ = 10.
Creep compliance test data generated from Prony series above.
Stress relaxation test data generated from Prony series above.

Results and discussion

The results agree well with exact analytical or approximate solutions.

Input files

mvliso2srs.inp: Time domain viscoelasticity, elastic, CPS4 elements.

Small-strain time domain viscoelasticity with anisotropic elasticity

Elements tested

C3D8R

CPE4R

CPS4R

S4R

M3D4R

Problem description

The verification tests in this section consist of one-element relaxation tests with viscoelastic materials. The elements are loaded in tension or shear, followed by relaxation at constant strain.

Results and discussion

The results agree well with exact analytical or approximate solutions.

Input files

visco_ortho_relax.inp: Time-domain viscoelasticity with orthotropic elasticity.
visco_ortho_creep.inp: Time-domain viscoelasticity with orthotropic elasticity.
visco_aniso_relax.inp: Time-domain viscoelasticity with anisotropic elasticity.
visco_aniso_creep.inp: Time-domain viscoelasticity with anisotropic elasticity.

Small-strain time domain viscoelasticity with traction-separation elasticity

Elements tested

COH2D4

COH3D8

Problem description

This section includes verification tests for time domain viscoelasticity in combination with cohesive elements with traction-separation elasticity. One set of verification tests consists of relaxation tests in which the cohesive elements are loaded in the normal or shear directions, followed by relaxation at constant separation. Another set of verification tests is included for the combination of viscoelasticity with traction-separation elasticity and progressive damage. In these tests the cohesive elements are loaded monotonically up to the point of failure.

Results and discussion

The results agree well with exact analytical or approximate solutions.

Input files

visco_coh2d.inp: Time domain viscoelasticity with traction-separation elasticity, COH2D4 elements.
visco_coh3d.inp: Time domain viscoelasticity with traction-separation elasticity, COH3D8 elements.
visco_dmg_coh2d.inp: Time domain viscoelasticity with traction-separation elasticity and damage, COH2D4 elements.
visco_dmg_coh3d.inp: Time domain viscoelasticity with traction-separation elasticity and damage, COH3D8 elements.

Frequency domain viscoelasticity

Elements tested

C3D8

CPS4

Problem description

Material 1


Young's modulus = 200 GPa.
Poisson's ratio = 0.3.
Density = 8000 kg/m³.
Fourier transform coefficients (tabular):
$ℜ_{1} (w g)$ = 1.161 × 10⁻², $ℑ_{1} (w g)$ = −3.21 × 10⁻², $ℜ_{1} (w k)$ = 0, $ℑ_{1} (w k)$ = 0, $f_{1}$ = 1.
$ℜ_{2} (w g)$ = 7.849 × 10⁻³, $ℑ_{2} (w g)$ = −2.222 × 10⁻², $ℜ_{2} (w k)$ = 0, $ℑ_{2} (w k)$ = 0, $f_{2}$ = 15.8.
$ℜ_{3} (w g)$ = 5.354 × 10⁻³, $ℑ_{3} (w g)$ = −1.533 × 10⁻², $ℜ_{3} (w k)$ = 0, $ℑ_{3} (w k)$ = 0, $f_{3}$ = 25.1.
$ℜ_{4} (w g)$ = 3.639 × 10⁻³, $ℑ_{4} (w g)$ = −1.062 × 10⁻², $ℜ_{4} (w k)$ = 0, $ℑ_{4} (w k)$ = 0, $f_{4}$ = 39.8.
$ℜ_{5} (w g)$ = 2.543 × 10⁻³, $ℑ_{5} (w g)$ = −7.382 × 10⁻³, $ℜ_{5} (w k)$ = 0, $ℑ_{5} (w k)$ = 0, $f_{5}$ = 63.1.
$ℜ_{6} (w g)$ = 1.775 × 10⁻³, $ℑ_{6} (w g)$ = −5.116 × 10⁻³, $ℜ_{6} (w k)$ = 0, $ℑ_{6} (w k)$ = 0, $f_{6}$ = 100.

Material 2


Young's modulus = 200 GPa.
Poisson's ratio = 0.3.
Density = 8000 kg/m³.
Fourier transform coefficients (formula):
$ℜ (g_{1})$ = 2.3508 × 10⁻³, $ℑ (g_{1})$ = 6.5001 × 10⁻³, a = 1.38366, $ℜ (k_{1})$ = $ℑ (k_{1})$ = b = 0.

Material 3


Polynomial coefficients (N=1): $C_{10}$ = 33.333333 × 10⁹, $C_{01}$ = 0, $D_{1}$ = 12.0 × 10⁻¹².
Fourier transform coefficients (tabular):
$ℜ_{1} (w g)$ = 1.161 × 10⁻², $ℑ_{1} (w g)$ = −3.21 × 10⁻², $ℜ_{1} (w k)$ = 0, $ℑ_{1} (w k)$ = 0, $f_{1}$ = 1.
$ℜ_{2} (w g)$ = 7.849 × 10⁻³, $ℑ_{2} (w g)$ = −2.222 × 10⁻², $ℜ_{2} (w k)$ = 0, $ℑ_{2} (w k)$ = 0, $f_{2}$ = 15.8.
$ℜ_{3} (w g)$ = 5.354 × 10⁻³, $ℑ_{3} (w g)$ = −1.533 × 10⁻², $ℜ_{3} (w k)$ = 0, $ℑ_{3} (w k)$ = 0, $f_{3}$ = 25.1.
$ℜ_{4} (w g)$ = 3.639 × 10⁻³, $ℑ_{4} (w g)$ = −1.062 × 10⁻², $ℜ_{4} (w k)$ = 0, $ℑ_{4} (w k)$ = 0, $f_{4}$ = 39.8.
$ℜ_{5} (w g)$ = 2.543 × 10⁻³, $ℑ_{5} (w g)$ = −7.382 × 10⁻³, $ℜ_{5} (w k)$ = 0, $ℑ_{5} (w k)$ = 0, $f_{5}$ = 63.1.
$ℜ_{6} (w g)$ = 1.775 × 10⁻³, $ℑ_{6} (w g)$ = −5.116 × 10⁻³, $ℜ_{6} (w k)$ = 0, $ℑ_{6} (w k)$ = 0, $f_{6}$ = 100.

Material 4


Polynomial coefficients (N=1): $C_{10}$ = 33.333333 × 10⁹, $C_{01}$ = 0, $D_{1}$ = 12.0 × 10⁻¹².
Fourier transform coefficients (formula):
$ℜ (g_{1})$ = 2.3508 × 10⁻³, $ℑ (g_{1})$ = 6.5001 × 10⁻³, a = 1.38366, $ℜ (k_{1})$ = $ℑ (k_{1})$ = b = 0.

Material 5


Polynomial coefficients (N=1): $C_{10}$ = 33.333333 × 10⁹, $C_{01}$ = 0, $D_{1}$ = 12.0 × 10⁻¹².
Fourier transform coefficients (formula):
$ℜ (g_{1})$ = 2.3508 × 10⁻³, $ℑ (g_{1})$ = 6.5001 × 10⁻³, a = 0, $ℜ (k_{1})$ = $ℑ (k_{1})$ = b = 0.

Material 6


Arruda-Boyce coefficients: $μ$ = 66.6666 × 10⁹, $λ_{m}$ = 5. , D = 12.0 × 10⁻¹².
Fourier transform coefficients (tabular):
$ℜ_{1} (w g)$ = 1.161 × 10⁻², $ℑ_{1} (w g)$ = −3.21 × 10⁻², $ℜ_{1} (w k)$ = 0, $ℑ_{1} (w k)$ = 0, $f_{1}$ = 1.
$ℜ_{2} (w g)$ = 7.849 × 10⁻³, $ℑ_{2} (w g)$ = −2.222 × 10⁻², $ℜ_{2} (w k)$ = 0, $ℑ_{2} (w k)$ = 0, $f_{2}$ = 15.8.
$ℜ_{3} (w g)$ = 5.354 × 10⁻³, $ℑ_{3} (w g)$ = −1.533 × 10⁻², $ℜ_{3} (w k)$ = 0, $ℑ_{3} (w k)$ = 0, $f_{3}$ = 25.1.
$ℜ_{4} (w g)$ = 3.639 × 10⁻³, $ℑ_{4} (w g)$ = −1.062 × 10⁻², $ℜ_{4} (w k)$ = 0, $ℑ_{4} (w k)$ = 0, $f_{4}$ = 39.8.
$ℜ_{5} (w g)$ = 2.543 × 10⁻³, $ℑ_{5} (w g)$ = −7.382 × 10⁻³, $ℜ_{5} (w k)$ = 0, $ℑ_{5} (w k)$ = 0, $f_{5}$ = 63.1.
$ℜ_{6} (w g)$ = 1.775 × 10⁻³, $ℑ_{6} (w g)$ = −5.116 × 10⁻³, $ℜ_{6} (w k)$ = 0, $ℑ_{6} (w k)$ = 0, $f_{6}$ = 100.

Material 7


Arruda-Boyce coefficients: $μ$ = 66.6666 × 10⁹, $λ_{m}$ = 5. , D = 12.0 × 10⁻¹².
Fourier transform coefficients (formula):
$ℜ (g_{1})$ = 2.3508 × 10⁻³, $ℑ (g_{1})$ = 6.5001 × 10⁻³, a = 1.38366, $ℜ (k_{1})$ = $ℑ (k_{1})$ = b = 0.

Material 8


Van der Waals coefficients: $μ$ = 66.6666 × 10⁹, $λ_{m}$ = 10. , a = 0.1, $β$ = 0., D = 12.0 × 10⁻¹².
Fourier transform coefficients (tabular):
$ℜ_{1} (w g)$ = 1.161 × 10⁻², $ℑ_{1} (w g)$ = −3.21 × 10⁻², $ℜ_{1} (w k)$ = 0, $ℑ_{1} (w k)$ = 0, $f_{1}$ = 1.
$ℜ_{2} (w g)$ = 7.849 × 10⁻³, $ℑ_{2} (w g)$ = −2.222 × 10⁻², $ℜ_{2} (w k)$ = 0, $ℑ_{2} (w k)$ = 0, $f_{2}$ = 15.8.
$ℜ_{3} (w g)$ = 5.354 × 10⁻³, $ℑ_{3} (w g)$ = −1.533 × 10⁻², $ℜ_{3} (w k)$ = 0, $ℑ_{3} (w k)$ = 0, $f_{3}$ = 25.1.
$ℜ_{4} (w g)$ = 3.639 × 10⁻³, $ℑ_{4} (w g)$ = −1.062 × 10⁻², $ℜ_{4} (w k)$ = 0, $ℑ_{4} (w k)$ = 0, $f_{4}$ = 39.8.
$ℜ_{5} (w g)$ = 2.543 × 10⁻³, $ℑ_{5} (w g)$ = −7.382 × 10⁻³, $ℜ_{5} (w k)$ = 0, $ℑ_{5} (w k)$ = 0, $f_{5}$ = 63.1.
$ℜ_{6} (w g)$ = 1.775 × 10⁻³, $ℑ_{6} (w g)$ = −5.116 × 10⁻³, $ℜ_{6} (w k)$ = 0, $ℑ_{6} (w k)$ = 0, $f_{6}$ = 100.

Material 9


Van der Waals coefficients: $μ$ = 66.6666 × 10⁹, $λ_{m}$ = 10. , a = 0.1, $β$ = 0. , D = 12.0 × 10⁻¹².
Fourier transform coefficients (formula):
$ℜ (g_{1})$ = 2.3508 × 10⁻³, $ℑ (g_{1})$ = 6.5001 × 10⁻³, a = 1.38366, $ℜ (k_{1})$ = $ℑ (k_{1})$ = b = 0.

Results and discussion

The problem involves a direct-integration steady-state dynamic procedure in which a harmonic pressure of amplitude 1.0 GPa is applied to the top surface of a cantilevered beam. Several subspace-based steady-state dynamic procedures follow. The results of most interest are the vertical displacement at the tip of the cantilever and the phase angles of the displacements for the specified frequencies.

Input files

Material 1:

mveft02the.inp: Tabular frequency domain viscoelasticity, elastic, CPS4 elements.
mveft03the.inp: Tabular frequency domain viscoelasticity, elastic, C3D8 elements.

Material 2:

mveff02the.inp: Formula frequency domain viscoelasticity, elastic, CPS4 elements.
mveff03the.inp: Formula frequency domain viscoelasticity, elastic, C3D8 elements.

Material 3:

mvyft02the.inp: Tabular frequency domain viscoelasticity, hyperelastic, CPS4 elements.
mvyft03the.inp: Tabular frequency domain viscoelasticity, hyperelastic, C3D8 elements.

Material 4:

mvyff02the.inp: Formula frequency domain viscoelasticity, hyperelastic, CPS4 elements.
mvyff03the.inp: Formula frequency domain viscoelasticity, hyperelastic, C3D8 elements.

Material 5:

mvyfn02the.inp: Formula frequency domain viscoelasticity, hyperelastic, CPS4 elements.
mvyfn03the.inp: Formula frequency domain viscoelasticity, hyperelastic, C3D8 elements.

Material 6:

mvxft02the.inp: Tabular frequency domain viscoelasticity, hyperelastic, CPS4 elements.
mvxft03the.inp: Tabular frequency domain viscoelasticity, hyperelastic, C3D8 elements.

Material 7:

mvxfn02the.inp: Formula frequency domain viscoelasticity, hyperelastic, CPS4 elements.
mvxfn03the.inp: Formula frequency domain viscoelasticity, hyperelastic, C3D8 elements.

Material 8:

mvzft02the.inp: Tabular frequency domain viscoelasticity, hyperelastic, CPS4 elements.
mvzft03the.inp: Tabular frequency domain viscoelasticity, hyperelastic, C3D8 elements.

Material 9:

mvzfn02the.inp: Formula frequency domain viscoelasticity, hyperelastic, CPS4 elements.
mvzfn03the.inp: Formula frequency domain viscoelasticity, hyperelastic, C3D8 elements.

Frequency domain viscoelasticity defined directly in terms of storage and loss moduli

Elements tested

C3D8R

C3D8RH

Problem description

In addition to the approach adopted in the verification problems of the earlier subsection, Abaqus allows definition of viscoelastic behavior in the frequency domain directly in terms of storage and loss moduli (as opposed to defining the viscoelastic behavior in terms of ratios that involve the long-term elastic shear and bulk moduli). The viscoelastic behavior can be defined using storage and loss moduli data obtained directly from a uniaxial tension test. Volumetric relaxation, if important, can also be defined in terms of bulk storage and loss moduli, obtained directly from a volumetric test. In both cases the viscoelastic properties can be defined in tabular forms as functions of frequency and level of preload. The problems described in this subsection use this approach.

The basic test setup consists of a reference element and a test element. For the reference element the viscoelastic behavior is defined using the approach used in the previous subsection (i.e., in terms of ratios that involve the long-term elastic modulus). For the test element the viscoelastic behavior is defined directly in terms of uniaxial storage and loss moduli (and in some cases, bulk storage and loss moduli). However, in the latter case the values of the uniaxial (and bulk) storage/loss moduli are hand-calculated based on the ratios specified for the reference element and the (preload-dependent) long-term elastic modulus. In computing the storage and loss moduli for the test case, it is assumed that the ratios specified for the reference case are independent of the level of preload. Since the purpose of the problems in this section is simply to verify that the implementation is correct, the aforementioned assumption should not be viewed as a limitation. Both the reference elements and the test elements are subjected to displacement-based harmonic excitations about an unloaded state as well as several levels of uniaxial and volumetric prestrain. The steady-state dynamic response is obtained in each case.

Results and discussion

By design, the reference elements and the test elements are expected to result in identical real and imaginary stresses. This acts as a verification for the implementation of the current approach.

Input files

frq_visco_prldu_ab.inp: Only uniaxial viscoelastic data specified, long-term elastic behavior defined using the Arruda-Boyce hyperelasticity model.
frq_visco_prldu_marlow.inp: Only uniaxial viscoelastic data specified, long-term elastic behavior defined using the Marlow hyperelasticity model.
frq_visco_prldu_poly1.inp: Only uniaxial viscoelastic data specified, long-term elastic behavior defined using the Mooney-Rivlin hyperelasticity model.
frq_visco_prldu_ogden.inp: Only uniaxial viscoelastic data specified, long-term elastic behavior defined using the third-order Ogden hyperelasticity model.
frq_visco_prldu_poly3.inp: Only uniaxial viscoelastic data specified, long-term elastic behavior defined using the third-order polynomial hyperelasticity model.
frq_visco_prldu_vdw.inp: Only uniaxial viscoelastic data specified, long-term elastic behavior defined using the Van der Waals hyperelasticity model.
frq_visco_prldu_hfoam.inp: Only uniaxial viscoelastic data specified, long-term elastic behavior defined using the second-order hyperfoam model.
frq_visco_prlduv_poly1.inp: Both uniaxial and volumetric viscoelastic data specified, long-term elastic behavior defined using the Mooney-Rivlin hyperelasticity model.
frq_visco_prlduv_poly3.inp: Both uniaxial and volumetric viscoelastic data specified, long-term elastic behavior defined using the third-order polynomial hyperelasticity model.
frq_visco_prlduv_hfoam.inp: Both uniaxial and volumetric viscoelastic data specified, long-term elastic behavior defined using the second-order hyperfoam model.
frq_visco_interp.inp: A basic test for interpolation of material properties.