Hysteresis in 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, $\mathbf{F}$, is assumed to act on both networks and is decomposed into elastic and inelastic parts in network B according to the multiplicative decomposition $\mathbf{F}={\mathbf{F}}_B^e\cdot {\mathbf{F}}_B^{cr}.$ 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; that is, 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 $\left[-1,0\right]$, 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 ${\dot{ϵ}}_B^{cr}$ is the effective creep strain rate in network B, ${\lambda }_B^{cr}-1$ is the nominal creep strain in network B, and ${\sigma }_B$ is the effective stress in network B. The chain stretch in network B, ${\lambda }_B^{cr}$, is defined as

where ${\mathbf{C}}_B^{cr}=\mathbf{F}_B^{cr}{}^T\cdot {\mathbf{F}}_B^{cr}$. The effective stress in network B is defined as ${\sigma }_B=\sqrt{\frac{3}{2}{\mathbf{S}}_B:{\mathbf{S}}_B}$, where ${\mathbf{S}}_B$ is the deviatoric Cauchy stress tensor.

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 $S=1.6$, $A=\frac{5}{{\left(\sqrt{3}\right)}^m}$(sec)⁻¹(MPa)^−m, $m=4$, $C=-1.0$, and $E=0.01$ (Bergstrom and Boyce, 1998; 2001).