Low-density foams

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. By default the Poisson's ratio of the material is assumed to be zero. With this assumption, the evaluation of the stress-strain response becomes uncoupled along the principal deformation directions. Optionally, nonzero Poisson effects can be specified to include coupling along the principal directions.

The model requires as input the stress-strain response of the material for both uniaxial tension and uniaxial compression tests. Poisson effects can be included by also specifying lateral strain data for each of these 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. The uniaxial stress and strain data are given in absolute values (positive in both tension and compression). On the other hand, when specified, the lateral strain data must be negative in tension and positive in compression, corresponding to a positive Poisson's effect. The model does not support negative Poisson's effect. 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 Abaqus/Explicit). 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 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.

Figure 1. Rate-dependent loading/unloading stress-strain curves.

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 stress is then corrected by a coupling term if non-zero Poisson effects are included. The representative response of the model for a uniaxial compression cycle is shown in Figure 1.

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

LOW DENSITY FOAM,LATERAL STRAIN DATA=NO (default)
LOW DENSITY FOAM, LATERAL STRAIN DATA=YES 

UNIAXIAL TEST DATA, DIRECTION=TENSION
UNIAXIAL TEST DATA, DIRECTION=COMPRESSION

Property module: material editor: MechanicalElasticityLow Density Foam: Uniaxial Test DataUniaxial Tension Test Data, Uniaxial Test DataUniaxial Compression Test Data

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

Property module: material editor: MechanicalElasticityLow Density Foam: toggle on Maximum allowable principal tensile stress: value

Property module: material editor: MechanicalElasticityLow Density Foam: toggle on Remove elements exceeding maximum 

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

The elastic volume ratio, $J^{e\mathrm{\ell }}$, relates the total volume ratio (current volume/reference volume), J, and the thermal volume ratio, $J^{th}$, via the simple relationship:

$J^{th}$ is given by

where ${\epsilon }^{th}$ is the linear thermal expansion strain that is obtained from the temperature and the isotropic thermal expansion coefficient (Thermal expansion).

The Drucker stability condition for a compressible material requires that the change in the Kirchhoff stress, $d\mathit{\tau }$, following from an infinitesimal change in the logarithmic strain, $d\mathit{\epsilon }$, satisfies the inequality

where the Kirchhoff stress $\mathit{\tau }=J\mathit{\sigma }$. Using $d\mathit{\tau }=\mathbf{D}:d\mathit{\epsilon }$, the inequality becomes

This restriction requires that the tangential material stiffness $\mathbf{D}$ be positive definite.

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

where $D_{ij}=\delta {\tau }_i/\delta {\epsilon }_j$. Since $\mathbf{D}$ must be positive definite, it is necessary that

When Poisson's ratio is zero, the off diagonal terms of $\mathbf{D}$ become zero. In that case the necessary conditions for a positive definite matrix reduce to $D_{ii}>0$; that is, the slope of the specified uniaxial stress-strain curves in the space of Kirchhoff stress versus logarithmic strain must be positive.

You should be careful defining the input data for the low-density foam model to ensure stable material response for all strain rates. 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. When nonzero Poisson effects are defined, it is highly recommended that you provide uniaxial test data in tension and compression for the same range of strain rates.