Product: ABAQUS/Standard
Material 1:
Polynomial coefficients (N=1):  = 8.,  = 2. |
Compressible case:  = 0.1. |
Prony series coefficients (N=1):  = 0.,  = 0.5,  = 3. |
Material 2:
| Polynomial coefficients (N=1): = 8., = 2. |
| Compressible case: = 0.1. |
| Prony series coefficients (N=1): = 0.5, = 0., = 3. |
| Heat transfer properties for coupled analysis: conductivity = 0.01, density = 1., |
| specific heat = 1. |
Material 3:
| Polynomial coefficients (N=1): = 1.5 × 106, = 0.5 × 106. |
| Compressible case: = 1. × 10–7. |
| Prony series coefficients (N=2): |
| = 0.5, = 0., = 0.2. |
 = 0.49,  = 0.,  = 0.5. |
Material 4:
| Polynomial coefficients (N=1): = 27.02, = 1.42. |
| Compressible case: = 0.000001. |
| Prony series coefficients (N=2): |
| = 0.25, = 0.25, = 5. |
| = 0.25, = 0.25, = 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): = 8., = 2. |
| Compressible case: = 0.001. |
| Prony series coefficients (N=2): |
| = 0.5, = 0., = 1. |
| = 0.49, = 0., = 2. |
Material 6:
| Polynomial coefficients (N=1): = 550.53, = –275.265. |
| Compressible case: = 7. × 10–7. |
| Prony series coefficients (N=6): |
| = 0.1986, = 0., = 0.281 × 10–7. |
| = 0.1828, = 0., = 0.281 × 10–5. |
 = 0.1388,  = 0.,  = 0.281 × 10–3. |
 = 0.2499,  = 0.,  = 0.281 × 10–1. |
 = 0.1703,  = 0.,  = 0.281 × 101. |
 = 0.0593,  = 0.,  = 0.281 × 103. |
Material 7:
Ogden coefficients (N=2):  = 16.,  = 2.,  = 4.,  = –2. |
| Prony series coefficients (N=1): = 0.5, = 0., = 3. |
Material 8:
Material 9:
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.
Compressible, volumetric compression, CPS4 elements.
Compressible, volumetric compression, CPS4 elements; Prony series parameters calibrated from frequency-dependent moduli.
Compressible, volumetric compression, CPS4 elements; steady-state dynamic, frequency-dependent moduli data derived from specified Prony series parameters.
Compressible, volumetric compression, CPS4 elements; steady-state dynamic, direct specification of frequency-dependent moduli data.
Tabulated frequency-dependent moduli data used in mvhcdo2sr2.inp and mvhcdo2ss2.inp as an *INCLUDE file.
Compressible, volumetric compression, CPE4 elements.
Incompressible, uniaxial tension, coupled analysis, CPE4HT elements.
Incompressible, relaxation in uniaxial tension, CPS4 elements.
Incompressible, uniaxial tension, CPE4H elements.
Incompressible, triaxial, CPE4H elements.
Compressible, relaxation in uniaxial tension, CPE4 elements.
Compressible, uniaxial tension, static and relaxation, CPE4H elements.
Creep and relaxation test data, uniaxial tension, static and relaxation, 2 CPE4RH elements.
Compressible, uniaxial tension, static and relaxation, 2 CPE4RH elements; Prony series parameters calibrated from frequency-dependent moduli.
Creep and relaxation test data, compressible, uniaxial tension, steady-state dynamic, 2 CPE4RH elements.
Compressible, uniaxial tension, steady-state dynamic, 2 CPE4RH elements; direct specification of Prony series parameters calibrated in mvhtdo3ssd.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.
Tabulated frequency-dependent moduli data used in mvhtdo3ss3.inp as an *INCLUDE file.
Combined test data, uniaxial tension, static and relaxation, 2 CPE4RH elements.
Compressible, uniaxial tension and rotation, static and relaxation, CPS4 elements.
Compressible, uniaxial tension and rotation, static and relaxation with static linear perturbation steps containing *LOAD CASE, CPS4R elements.
Compressible, relaxation in uniaxial tension, M3D4 elements.
Compressible, biaxial compression tension, CAX4R elements.
Compressible, biaxial compression tension, CAX4R elements; Prony series parameters calibrated from frequency-dependent moduli.
Compressible, biaxial compression tension, CAX4R elements; steady-state dynamic, frequency-dependent moduli data derived from specified Prony series parameters.
Compressible, biaxial compression tension, CAX4R elements; steady-state dynamic, direct specification of frequency-dependent moduli data.
Tabulated frequency-dependent moduli data used in mvhcdo3kc2.inp and mvhcdo3ss2.inp as an *INCLUDE file.
Incompressible, relaxation in uniaxial tension, Ogden model, CPE4H elements.
Incompressible, uniaxial tension with static linear perturbation steps, Ogden model, CPE4H elements.
Incompressible, relaxation in uniaxial tension, Arruda-Boyce model, CPE4H elements.
Incompressible, uniaxial tension with static linear perturbation steps containing *LOAD CASE, Arruda-Boyce model, CPE4H elements.
Incompressible, relaxation in uniaxial tension, Van der Waals model, CPE4H elements.
Incompressible, uniaxial tension with static linear perturbation steps, Van der Waals model, CPE4H elements.
Material 1:
| Hyperfoam coefficients (N=3): |
= –17.4, = –1.22, = 548.2, = 17.3,  = 10.47,  = –1.775,  =  =  = 0. |
| Prony series coefficients (N=1): = 0.5, = 0., = 3. |
Material 2:
Compressible, relaxation in uniaxial tension, CPE4 elements.
Compressible, uniaxial tension with static linear perturbation steps containing *LOAD CASE, CPE4 elements.
Compressible, relaxation in uniaxial tension, CPE4 elements.
Material 1:
| Young's modulus = 200 GPa. |
| Poisson's ratio = 0.3. |
| Density = 8000 kg/m3. |
| Fourier transform coefficients (tabular): |
 = 1.161 × 10–2,  = –3.21 × 10–2,  = 0,  = 0,  = 1. |
 = 7.849 × 10–3,  = –2.222 × 10–2,  = 0,  = 0,  = 15.8. |
 = 5.354 × 10–3,  = –1.533 × 10–2,  = 0,  = 0,  = 25.1. |
 = 3.639 × 10–3,  = –1.062 × 10–2,  = 0,  = 0,  = 39.8. |
 = 2.543 × 10–3,  = –7.382 × 10–3,  = 0,  = 0,  = 63.1. |
 = 1.775 × 10–3,  = –5.116 × 10–3,  = 0,  = 0,  = 100. |
Material 2:
| Young's modulus = 200 GPa. |
| Poisson's ratio = 0.3. |
| Density = 8000 kg/m3. |
| Fourier transform coefficients (formula): |
 = 2.3508 × 10–3,  = 6.5001 × 10–3, a = 1.38366,  =  = b = 0. |
Material 3:
| Polynomial coefficients (N=1): = 33.333333 × 109, = 0, = 12.0 × 10–12. |
| Fourier transform coefficients (tabular): |
| = 1.161 × 10–2, = –3.21 × 10–2, = 0, = 0, = 1. |
| = 7.849 × 10–3, = –2.222 × 10–2, = 0, = 0, = 15.8. |
| = 5.354 × 10–3, = –1.533 × 10–2, = 0, = 0, = 25.1. |
| = 3.639 × 10–3, = –1.062 × 10–2, = 0, = 0, = 39.8. |
| = 2.543 × 10–3, = –7.382 × 10–3, = 0, = 0, = 63.1. |
| = 1.775 × 10–3, = –5.116 × 10–3, = 0, = 0, = 100. |
Material 4:
| Polynomial coefficients (N=1): = 33.333333 × 109, = 0, = 12.0 × 10–12. |
| Fourier transform coefficients (formula): |
| = 2.3508 × 10–3, = 6.5001 × 10–3, a = 1.38366, = = b = 0. |
Material 5:
| Polynomial coefficients (N=1): = 33.333333 × 109, = 0, = 12.0 × 10–12. |
| Fourier transform coefficients (formula): |
| = 2.3508 × 10–3, = 6.5001 × 10–3, a = 0, = = b = 0. |
Material 6:
Arruda-Boyce coefficients:  = 66.6666 × 109,  = 5. , D = 12.0 × 10–12. |
| Fourier transform coefficients (tabular): |
| = 1.161 × 10–2, = –3.21 × 10–2, = 0, = 0, = 1. |
| = 7.849 × 10–3, = –2.222 × 10–2, = 0, = 0, = 15.8. |
| = 5.354 × 10–3, = –1.533 × 10–2, = 0, = 0, = 25.1. |
| = 3.639 × 10–3, = –1.062 × 10–2, = 0, = 0, = 39.8. |
| = 2.543 × 10–3, = –7.382 × 10–3, = 0, = 0, = 63.1. |
| = 1.775 × 10–3, = –5.116 × 10–3, = 0, = 0, = 100. |
Material 7:
| Arruda-Boyce coefficients: = 66.6666 × 109, = 5. , D = 12.0 × 10–12. |
| Fourier transform coefficients (formula): |
| = 2.3508 × 10–3, = 6.5001 × 10–3, a = 1.38366, = = b = 0. |
Material 8:
Van der Waals coefficients: = 66.6666 × 109, = 10. , a = 0.1,  = 0., D = 12.0 × 10–12. |
| Fourier transform coefficients (tabular): |
| = 1.161 × 10–2, = –3.21 × 10–2, = 0, = 0, = 1. |
| = 7.849 × 10–3, = –2.222 × 10–2, = 0, = 0, = 15.8. |
| = 5.354 × 10–3, = –1.533 × 10–2, = 0, = 0, = 25.1. |
| = 3.639 × 10–3, = –1.062 × 10–2, = 0, = 0, = 39.8. |
| = 2.543 × 10–3, = –7.382 × 10–3, = 0, = 0, = 63.1. |
| = 1.775 × 10–3, = –5.116 × 10–3, = 0, = 0, = 100. |
Material 9:
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 to test several parameters on the *STEADY STATE DYNAMICS option. 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.
Tabular frequency domain viscoelasticity, elastic, CPS4 elements.
Tabular frequency domain viscoelasticity, elastic, C3D8 elements.
Formula frequency domain viscoelasticity, elastic, CPS4 elements.
Formula frequency domain viscoelasticity, elastic, C3D8 elements.
Tabular frequency domain viscoelasticity, hyperelastic, CPS4 elements.
Tabular frequency domain viscoelasticity, hyperelastic, C3D8 elements.
Formula frequency domain viscoelasticity, hyperelastic, CPS4 elements.
Formula frequency domain viscoelasticity, hyperelastic, C3D8 elements.
Formula frequency domain viscoelasticity, hyperelastic, CPS4 elements.
Formula frequency domain viscoelasticity, hyperelastic, C3D8 elements.
Tabular frequency domain viscoelasticity, hyperelastic, CPS4 elements.
Tabular frequency domain viscoelasticity, hyperelastic, C3D8 elements.
Formula frequency domain viscoelasticity, hyperelastic, CPS4 elements.
Formula frequency domain viscoelasticity, hyperelastic, C3D8 elements.
Tabular frequency domain viscoelasticity, hyperelastic, CPS4 elements.
Tabular frequency domain viscoelasticity, hyperelastic, C3D8 elements.
Formula frequency domain viscoelasticity, hyperelastic, CPS4 elements.
Formula frequency domain viscoelasticity, hyperelastic, C3D8 elements.
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.
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.
Only uniaxial viscoelastic data specified, long-term elastic behavior defined using the Arruda-Boyce hyperelasticity model.
Only uniaxial viscoelastic data specified, long-term elastic behavior defined using the Marlow hyperelasticity model.
Only uniaxial viscoelastic data specified, long-term elastic behavior defined using the Mooney-Rivlin hyperelasticity model.
Only uniaxial viscoelastic data specified, long-term elastic behavior defined using the third-order Ogden hyperelasticity model.
Only uniaxial viscoelastic data specified, long-term elastic behavior defined using the third-order polynomial hyperelasticity model.
Only uniaxial viscoelastic data specified, long-term elastic behavior defined using the Van der Waals hyperelasticity model.
Only uniaxial viscoelastic data specified, long-term elastic behavior defined using the second-order hyperfoam model.
Both uniaxial and volumetric viscoelastic data specified, long-term elastic behavior defined using the Mooney-Rivlin hyperelasticity model.
Both uniaxial and volumetric viscoelastic data specified, long-term elastic behavior defined using the third-order polynomial hyperelasticity model.
Both uniaxial and volumetric viscoelastic data specified, long-term elastic behavior defined using the second-order hyperfoam model.
A basic test for interpolation of material properties.