### 17.8.1 Hysteresis in elastomers Products: ABAQUS/Standard  ABAQUS/CAE

### Overview The hysteresis material model:

### Strain-rate-dependent material behavior for elastomers Nonlinear strain-rate dependence of elastomers is modeled by decomposing the mechanical response into that of an equilibrium network (A) corresponding to the state that is approached in long-time stress relaxation tests and that of a time-dependent network (B) that captures the nonlinear rate-dependent deviation from the equilibrium state. The total stress is assumed to be the sum of the stresses in the two networks. The deformation gradient, , is assumed to act on both networks and is decomposed into elastic and inelastic parts in network B according to the multiplicative decomposition The nonlinear rate-dependent material model is capable of reproducing the hysteretic behavior of elastomers subjected to repeated cyclic loading. It does not model “Mullins effect”—the initial softening of an elastomer when it is first subjected to a load.

The material model is defined completely by:

• a hyperelastic material model that characterizes the elastic response of the model;

• a stress scaling factor, S, that defines the ratio of the stress carried by network B to the stress carried by network A under instantaneous loading; i.e., identical elastic stretching in both networks;

• a positive exponent, m, generally greater than 1, characterizing the effective stress dependence of the effective creep strain rate in network B;

• an exponent, C, restricted to lie in , characterizing the creep strain dependence of the effective creep strain rate in network B;

• a nonnegative constant, A, in the expression for the effective creep strain rate—this constant also maintains dimensional consistency in the equation; and

• a constant, E, in the expression for the effective creep strain rate—this constant regularizes the creep strain rate near the undeformed state.

The effective creep strain rate in network B is given by the expression where is the effective creep strain rate in network B, is the nominal creep strain in network B, and is the effective stress in network B. The chain stretch in network B, , is defined as where . The effective stress in network B is defined as , where is the deviatoric Cauchy stress tensor.

### Defining strain-rate-dependent material behavior for elastomers The elasticity of the model is defined by a hyperelastic material model. You input the stress scaling factor and the creep parameters for network B directly when you define the hysteresis material model. Typical values of the material parameters for a common elastomer are , (sec)–1(MPa)m, , , and (Bergstrom and Boyce, 1998; 2001).

 Input File Usage: Use both of the following options within the same material data block: ```*HYSTERESIS *HYPERELASTIC```

 ABAQUS/CAE Usage: Property module: material editor: Mechanical Elasticity Hyperelastic: Suboptions Hysteresis The input of the parameter is not supported in ABAQUS/CAE.

### Elements The use of the hysteresis material model is restricted to elements that can be used with hyperelastic materials (Hyperelastic behavior of rubberlike materials, Section 17.5.1). In addition, this model cannot be used with elements based on the plane stress assumption (shell, membrane, and continuum plane stress elements). Hybrid elements can be used with this model only when the accompanying hyperelasticity definition is completely incompressible. When this model is used with reduced-integration elements, the instantaneous elastic moduli are used to calculate the default hourglass stiffness.

### Output In addition to the standard output identifiers available in ABAQUS/Standard (ABAQUS/Standard output variable identifiers, Section 4.2.1), the following variables have special meaning if hysteretic behavior is defined:

 EE Elastic strain corresponding to the stress state at time t and the instantaneous elastic material properties.
 CE Equivalent creep strain defined as the difference between the total strain and the elastic strain.

These strain measures are used to approximate the strain energy, SENER, and the viscous dissipation, CENER. These approximations may lead to underestimation of the strain energy and overestimation of the viscous dissipation since the effects of internal stresses on these energy quantities are neglected. This inaccuracies may be particularly noticeable in the case of nonmonotonic loading.

#### Additional references • Bergstrom,  J. S., and M. C. Boyce, Constitutive Modeling of the Large Strain Time-Dependent Behavior of Elastomers,Journal of the Mechanics and Physics of Solids, vol. 46, no.5, pp. 931–954, May 1998.

• Bergstrom,  J. S., and M. C. Boyce, Constitutive Modeling of the Time-Dependent and Cyclic Loading of Elastomers and Application to Soft Biological Tissues,Mechanics of Materials, vol. 33, no.5, pp. 523–530, 2001.