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

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

The Marlow model differs from most hyperelastic models in that it does not 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.

## Model Files

## 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

**Property Material****Values**- Density
- 1 x 10
^{-9}tonnes/mm^{3} - Poisson's ratio
- 0.499

## 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.

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.

`nu`or

`D`

`1`to have constant volumetric response throughout the deformation.

### Results

## MATHE Compression

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

`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

## MATTHE

`nu`(or

`D`

`1`) 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

## 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:

- $\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:

Usually, the relaxation function is expressed by a Prony series containing $N$ relaxation times ${\tau}_{i}$ and coefficients ${g}_{i}$ as parameters.

${g}_{1}$ | ${\tau}_{1}$ 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 |

_{0}. 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 U

_{0 }does not have the physical meaning of the high frequency limit of the real material behavior. Nevertheless, U

_{0 }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.

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

## MATTHE+VE

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

### Results

^{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