OS-V: 0820 Marlow Hyperelastic with Viscoelasticity Material Model

Hyperelastic material models describe the nonlinear elastic behavior by formulating the strain energy density as a function of the deformation state. This elastic potential is expressed as a function of either the strain invariants ($$I1$$, $$I2$$, $$J$$) or the principal stretches ($$\lambda 1$$, $$\lambda 2$$, $$\lambda 3$$). Generally, hyperelastic model can be specified either with material constant or experimental data.

For Marlow model, only experimental data in table format can be specified, as shown in Figure 3.

Figure 2. CHEXA element model with applied tensile forces

Figure 3. Experimental uniaxial tensile stress-stretch curve

Tensile input data covers engineering stress up to 0.46 MPa and since the design load corresponds to nominal stress of 0.5 MPa, extrapolation is used at the end of the NLSTAT LGDISP load step.

In case of incompressible deformation, the elastic potential is essentially defined by a single scalar function $$Udev$$. This function is uniquely determined by the stress–strain response measured in a single test, for example, a uniaxial tensile test. MATHEs can be compressible or nearly incompressible, specify nu or D1 to have constant volumetric response throughout the deformation.

Figure 4. Poisson’s ratio, and density of material and reference to the table

Results

The peak deformation in uniaxial tensile test of hyperelastic material with CHEXA element is 13.09 mm.

Figure 5. Deformation of CHEXA element due to tensile forces

Figure 6. CHEXA element model with compressive forces

The nominal compressive stress of 0.05 MPa is applied with forces. The model is simply supported.

The volumetric part $$Uvol$$ is considered. Since free lateral expansion is allowed, it has less significance. The volumetric part of the potential of a compressible model can be derived from a stress–strain curve of hydrostatic compression tests. This is defined in TABD where volumetric ratio is specified as a function of pressure. Once $$Uvol$$ has been determined, $$Udev$$ can be obtained from standard compression test considering the volumetric behavior. This approach enables a user-friendly implementation of the Marlow model. You will define the deviatoric and volumetric behavior in tabular form: the test data of a uniaxial, biaxial, or planar test; and for compressible models, additionally, the test data of a volumetric test. In these examples TABD is not used; instead, Poisson’s ratio is specified. Deformation level independent volumetric behavior is internally derived from the initial shear stiffness (TAB1) and Poisson’s ratio.

Figure 7. Experimental compressive stress-stretch curve

In compression, stretch can be defined as $$0<\lambda \le 1$$ and the deformation of the model stays in the specified range.

Results

Hyperelastic Marlow model of CHEXA element shows peak displacement of 0.8185 mm in x-direction.

Figure 8. Deformation of CHEXA element due to compression forces

MATTHE is used to define temperature dependent hyperelastic materials. MATTHE is supported in implicit LGDISP analysis. Experimental engineering stress-stretch data at each temperature is presented in tabular format in ascending order.

Figure 9. Experimental stress-stretch curve at each temperature

Figure 10. Specify the MATTHE Marlow material model

While defining Marlow material model in OptiStruct, when volumetric behavior is temperature independent, the 1st column corresponds to nominal stress, 2nd to stretch and 3rd to temperature. In this case, either nu (or D1) defines the constant volumetric response throughout the deformation.

Figure 11. Mechanical load (left) and Thermal load (right) applied on CHEXA element

The example model contains two TLOAD1s that define mechanical and thermal load profiles that evolve in different phases and those are combined with DLOAD.

Results

Hyperelastic MATTHE Marlow model of CHEXA element shows peak displacement of 4.636 mm in x-direction.

Figure 12. Deformation of CHEXA element. displacement in loading direction

A combination of the hyperelastic Marlow model with viscoelasticity provides the capability to describe the strain-rate-dependent material behavior while preserving the advantages of Marlow’s approach, the convenient definition using test data directly and the exact modeling of the test data. Viscoelastic models allow to describe relaxation and strain-rate-dependent elastic properties.

When viscoelasticity is considered, moduli temporal property needs to be specified in the continuation line of MATHE card. By default, this is considered as long-term relaxed input. The other options are to specify it as instantaneous input. MATVE card has the same ID as MATHE.

Figure 13. MATHE + MATVE card specification with same ID for visco-hyperelastic model

According to the definition of the visco-hyperelastic model, it requires to specify the parameters of the Prony series as well as to specify the temporal (here instantaneous) hyperelastic response by the potential U₀. It is the response that the model shows on a timescale much smaller than the lowest considered relaxation time. This timescale lies outside the validity range of the model, so U₀ does not have the physical meaning of the high frequency limit of the real material behavior. Nevertheless, U₀ must be identified from experimental data to specify the model. It is basically a transformation from a uniaxial stress–strain curve at constant strain rate to the curve describing the instantaneous hyperelastic response.

Figure 14. Instantaneous stress-strain curve (left) and the corresponding stress-stretch curve for the MATHE card (right)

The red dashed curve is the instantaneous stress–strain curve of the viscoelastic Marlow model. The example model takes the stress-stretch data as instantaneous definition for MATHE and uses Prony series to define viscoelastic behavior.

Figure 15. Two loadsteps with different loads and loading rates

Single element model is uniaxially loaded from zero to nominal stress. First loadstep having 10.0 MPa of nominal stress ramped within 0.02s (left) and second loadstep with 5.5 MPa of nominal stress ramped within 200s (right).

Results

Both load steps return the same displacement of 2 mm at the end of the nonlinear transient analysis. Nominal strain matches with reference result.

Figure 16. Deformation of CHEXA element from unloaded state to nominal 10.0 MPa within 0.02s

Figure 17. Deformation of CHEXA element from unloaded state to nominal 5.5 MPa within 200s

Figure 18. Reference results

The model contains three cubes represented with CTETRA, CPENTA and CHEXA elements. Models are loaded with different strain rates up to 30% deformation level and then hold so that the total analysis time in nonlinear transient is 147s. Analysis environment is T=100°C and input data is given at T=80°C and T=120°C, respectively.

Figure 19. CTETRA, CPENTA and CHEXA model elements

Temperature dependent Marlow model with viscoelasticity combines previously described material definitions. Temperature dependent MATTHE is defined in tabular format. Rate-dependency described in MATVE is common for all environments.

Figure 20. Temperature dependent MATTHE Marlow model

The example models take the widely referred Treloar’s stress-stretch data as reference and this is considered to return the baseline solution at T=100°C. Stress values are multiplied by factors 0.8 and 1.2 to obtain stress-stretch data at two extrapolated environments.

Figure 21. Stress-stretch data of the material

Figure 22. Reference result with 3s, 6s and 30s Tload

The uniaxial tension data at the two extrapolated environments is used as the input data.

Results

MATTHE with MARLOW + VE returns a good match for stress relaxation against the published results. In the reference study OGDEN MATHE was used (single environment). OptiStruct MATHE(OGDEN) +VE perfectly matches the simulation results shown in the reference.

Figure 23. Elemental stresses with 3s, 6s, and 30s Tload