Low-density, highly compressible elastomeric foams are widely used in the automotive industry as energy absorbing materials. Foam padding is used in many passive safety systems, such as behind headliners for head impact protection, in door trims for pelvis and thorax protection, etc. Energy absorbing foams are also commonly used in packaging of hand-held and other electronic devices.

The low-density foam material model in Abaqus/Explicit is intended to capture the highly strain-rate sensitive behavior of these materials. The model uses a pseudo visco-hyperelastic formulation whereby the strain energy potential is constructed numerically as a function of principal stretches and a set of internal variables associated with strain rate. The model is based on the assumption that the Poisson's ratio of the material is zero. With this assumption, the evaluation of the stress-strain response becomes uncoupled along the principal deformation directions.

The model requires as input the stress-strain response of the material for both uniaxial tension and uniaxial compression tests. The tests can be performed at different strain rates. For each test the strain data should be given in nominal strain values (change in length per unit of original length), and the stress data should be given in nominal stress values (force per unit of original cross-sectional area). Uniaxial tension and compression curves are specified separately, and the stress and strain data are given in absolute values (positive in both tension and compression). Rate-dependent behavior is specified by providing the uniaxial stress-strain curves for different values of nominal strain rates.

Both loading and unloading rate-dependent curves can be specified to better characterize the hysteretic behavior and energy absorption properties of the material during cyclic loading. Use positive values of nominal strain rates for loading curves and negative values for the unloading curves. Currently this option is available only with linear strain rate regularization (see “Regularization of strain-rate-dependent data” in “Material data definition,” Section 18.1.2). When the unloading behavior is not specified directly, the model assumes that unloading occurs along the loading curve associated with the smallest deformation rate. A representative schematic of typical rate-dependent uniaxial compression data is shown in Figure 19.9.1–1 with both loading and unloading curves. It is important that the specified rate-dependent stress-strain curves do not intersect. Otherwise, the material is unstable, and Abaqus issues an error message if an intersection between curves is found.

During the analysis, the stress along each principal deformation direction is evaluated by interpolating the specified loading/unloading stress-strain curves using the corresponding values of principal nominal strain and strain rate. The representative response of the model for a uniaxial compression cycle is shown in Figure 19.9.1–1.

Unphysical jumps in stress due to sudden changes in the deformation rate are prevented using a technique based on viscous regularization. This technique also models stress relaxation effects in a very simplistic manner. In the case of a uniaxial test, for example, the relaxation time is given as where , , and are material parameters and is the stretch. is a linear viscosity parameter that controls the relaxation time when , and typically small values of this parameter should be used. is a nonlinear viscosity parameter that controls the relaxation time at higher values of deformation. The smaller this value, the shorter the relaxation time. controls the sensitivity of the relaxation speed to the stretch. The default values of these parameters are (time units), (time units), and .

*LOW DENSITY FOAM
*UNIAXIAL TEST DATA, DIRECTION=TENSION
*UNIAXIAL TEST DATA, DIRECTION=COMPRESSION

*LOW DENSITY FOAM, STRAIN RATE=VOLUMETRIC

*LOW DENSITY FOAM, STRAIN RATE=PRINCIPAL

*LOW DENSITY FOAM, RATE EXTRAPOLATION=YES

Low-density foams have limited strength in tension and can easily rupture under excessive tensile loading. The model in Abaqus/Explicit provides the option to specify a cutoff value for the maximum principal tensile stress that the material can sustain. The maximum principal stresses computed by the program will stay at or below this value. You can also activate deletion (removal) of the element from the simulation when the tension cutoff value is reached, which provides a simple method for modeling rupture.

*LOW DENSITY FOAM, TENSION CUTOFF=value

*LOW DENSITY FOAM, TENSION CUTOFF=value, FAIL=YES

Only isotropic thermal expansion is permitted with the low-density foam material model.

The elastic volume ratio, , relates the total volume ratio (current volume/reference volume), J, and the thermal volume ratio, , via the simple relationship:

The Drucker stability condition for a compressible material requires that the change in the Kirchhoff stress, , following from an infinitesimal change in the logarithmic strain, , satisfies the inequality

For an isotropic elastic formulation the inequality can be represented in terms of the principal stresses and strains

Thus, the relation between changes in the stress and changes in the strain can be obtained in the form of the matrix equation

You should be careful defining the input data for the low-density foam model to ensure stable material response. If an instability is found, Abaqus issues a warning message and prints the lowest value of strain for which the instability is observed. Ideally, no instability should occur. If instabilities are observed at strain levels that are likely to occur in the analysis, it is strongly recommended that you carefully examine and revise the material input data.