# OS-V: 0820 Marlow Hyperelastic with Viscoelasticity Material Model

The combination of hyperelastic material models with viscoelasticity allows to model the strain rate dependent large strain response.

The Marlow model differs from most hyperelastic models by the concept not to use a small number of model parameters, but a scalar function to define the mechanical properties. It can be defined conveniently by providing the stress–stretch (stretch = engineering strain+1) curve without needs for parameter calibration. The coupling of the Marlow model and viscoelasticity is an approach to create a strain rate-dependent hyperelastic model which has good accuracy and is convenient to use. In this combination, the Marlow model requires to specify the stress–stretch curve for the instantaneous or long-term material response, while experimental data can be obtained only at finite strain rates.

## Benchmark Model

The single CHEXA8 element model has an edge length of 1.0 mm. The Marlow model is derived from experimental data only using a single set of data. The test data in the form of uniaxial tension, uniaxial compression, equi-biaxial, or planar test is used. Deviatoric behavior depends on the 1st stretch invariant only and it is independent of the 2nd invariant.

## Materials

The property materials used for the Marlow model are:
Property Material
Values
Density
1 x 10-9 tonnes/mm3
Poisson's ratio
0.499

## Model Files

Refer to Access the Model Files to download the required model file(s).

The model files used in this problem include:
• MATHE-Marlow_tension.fem
• MATHE-Marlow_compression.fem
• MATTHE-Marlow.fem
• MATHE-VE-Marlow.fem
• MATTHE-VE-Marlow.fem

## MATHE Tension

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.

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.

Marlow’s model uses scalar functions instead of scalar parameters to define the material behavior. The model assumes that the strain energy density is independent of the second deviatoric strain invariant and can be decomposed into a deviatoric and a volumetric part. The volumetric part becomes relevant in confined compression.(1)
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.

### Results

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

## MATHE Compression

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.

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.

## MATTHE

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.
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.

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.

## MATHE+VE

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.

Considering a shear deformation, an integral formulation of linear small strain viscoelasticity is given by:(2) $\tau \left(t\right)=\underset{0}{\overset{t}{\int }}G\left(t-s\right)\frac{d\gamma \left(s\right)}{dt}ds$
Where,
$\tau$
Shear stress
$\gamma$
Shear strain
$G\left(t\right)$
Relaxation function
This can be expressed using the instantaneous shear modulus ${G}_{0}$ and a relative relaxation function $g\left(t\right)$ by:(3) $\tau \left(t\right)=G\underset{0}{\overset{t}{\int }}g\left(t-s\right)\frac{d\gamma \left(s\right)}{dt}ds$
Usually, the relaxation function is expressed by a Prony series containing $N$ relaxation times ${\tau }_{i}$ and coefficients ${g}_{i}$ as parameters.(4) $g\left(t\right)=1-\sum _{i=1}^{N}{g}_{i}\left(1-{e}^{-\frac{t}{{\tau }_{i}}}\right)$
The Prony series parameters can be obtained from a relaxation test, where a constant strain is applied instantly and the relaxation of the stress over time is measured. A straightforward evaluation of this test is to interpret the normalized stress over time as the normalized relaxation function and to fit it using the Prony series. This approach assumes that the decrease of the normalized stress over time is not significantly influenced by the nonlinearity of the hyperelastic part of the material model.
Table 1. Prony series parameters and bulk modulus of the example
${g}_{1}$ ${\tau }_{1}$ in s ${g}_{2}$ ${\tau }_{2}$ in s ${g}_{3}$ ${\tau }_{3}$ in s ${g}_{4}$ ${\tau }_{4}$ in s ${g}_{5}$ ${\tau }_{5}$ in s K of GPa
0.3539 0.08124 0.07458 1.692 0.05052 35.23 0.04117 733.5 0.04575 15275 2.5
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.
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.
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.

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.

## MATTHE+VE

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.
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.
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.

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.
1 Transformation of Test Data for the Specification of a Viscoelastic Marlow Model. Olaf Hesebeck, Fraunhofer Institute for Manufacturing Technology and Advanced Materials IFAM, Wiener Straße 12, 28359 Bremen, Germany.
2 Parameter Identification Methods for Hyperelastic and Hyper-Viscoelastic Models. Yifeng Wu, Hao Wang, and Aiqun Li. School of Civil Engineering, Southeast University, Nanjing 210096, China, Beijing University of Civil Engineering and Architecture, Beijing 100044, China