Mullins Effect

Mullins Effect

Hyperelastic materials are described in terms of a “strain energy potential” function $U\left(F\right)$ that defines the strain energy stored in the material per unit reference volume (volume in the initial configuration). The quantity $F$ is the deformation gradient tensor. To account for the Mullins effect, Ogden and Roxburgh propose a material description that is based on an energy function of the form $U(F,\eta )$ , where the additional scalar variable, $\eta$ , represents damage in the material. The damage variable controls the material properties in the sense that it enables the material response to be governed by an energy function on unloading and subsequent submaximal reloading different from that on the primary (initial) loading path from a virgin state. Because of the above interpretation of $\eta$ , it is no longer appropriate to think of $U$ as the stored elastic energy potential. Part of the energy is stored as strain energy, while the rest is dissipated as a result of damage.

In preparation for writing the constitutive equations for Mullins effect, it is useful to separate the deviatoric and the volumetric parts of the total strain energy density as

$U$ , $U_{dev}$ , and $U_{vol}$ are the total, deviatoric, and volumetric parts of the strain energy density, respectively. All the hyperelasticity models described in Isotropic Hyperelasticity have strain energy potential functions that are separated into deviatoric and volumetric parts. For example, the polynomial models use a strain energy potential of the form where the symbols have the usual interpretations. The first term on the right represents the deviatoric part of the elastic strain energy density function, and the second term represents the volumetric part.

The Mullins effect is accounted for by using an augmented energy function of the form

where ${\tilde{U}}_{dev}\left({\overline{\lambda }}_i\right)$ is the deviatoric part of the strain energy density of the primary hyperelastic behavior, defined, for example, by the first term on the right-hand-side of the polynomial strain energy function given above; ${\tilde{U}}_{vol}\left(J^{el}\right)$ is the volumetric part of the strain energy density, defined, for example, by the second term on the right-hand-side of the polynomial strain energy function given above; ${\overline{\lambda }}_i(i=1,2)$ represent the deviatoric principal stretches; and $J^{el}$ represents the elastic volume ratio. The function $\varphi \left(\eta \right)$ is a continuous function of the damage variable $\eta$ and is referred to as the damage function. When the deformation state of the material is on a point on the curve that represents the primary hyperelastic behavior, $\eta =1$ , $\varphi \left(\eta \right)=0$ , $U({\overline{\lambda }}_i,1)={\tilde{U}}_{dev}\left({\overline{\lambda }}_i\right)+{\tilde{U}}_{vol}\left(J^{el}\right)$ and the augmented energy function reduces to the strain energy density function of the primary hyperelastic behavior. The damage variable varies continuously during the deformation and always satisfies $0<\eta \le 1$ .

The primary hyperelastic behavior is defined by using the hyperelastic material model (see Isotropic Hyperelasticity). The Mullins effect model is defined by specifying the Mullins effect parameters directly.

With the above modification to the energy function, the stresses are given by

where $\tilde{S}$ is the deviatoric stress corresponding to the primary hyperelastic behavior at the current deviatoric deformation level ${\overline{\lambda }}_i$ and $\tilde{p}$ is the hydrostatic pressure of the primary hyperelastic behavior at the current volumetric deformation level $J^{el}$ . Thus, the deviatoric stress as a result of Mullins effect is obtained by simply scaling the deviatoric stress of the primary hyperelastic behavior with the damage variable $\eta$ . The pressure stress is the same as that of the primary behavior. The model predicts loading/unloading along a single curve (that is different, in general, from the primary hyperelastic behavior) from any given strain level that passes through the origin of the stress-strain plot. It cannot capture permanent strains upon removal of load. The model also predicts that a purely volumetric deformation will not have any damage or Mullins effect associated with it.

The damage variable, $\eta$ , varies with the deformation according to

where $U_{dev}^m$ is the maximum value of ${\tilde{U}}_{dev}$ at a material point during its deformation history; and $r$ , $\beta$ , and $m$ are material parameters.

The parameters $r$ , $\beta$ , and $m$ do not have direct physical interpretations in general. The parameter m controls whether damage occurs at low strain levels. If $m=0$ , there is a significant amount of damage at low strain levels. On the other hand, a nonzero m leads to little or no damage at low strain levels.

Parameter	Description
r	Value of the $r$ coefficient in the Mullins effect model. $r$ must be greater than 1.
m	Value of the $m$ coefficient in the Mullins effect model. $m$ must be greater than or equal to zero, and the values of $m$ and $\beta$ cannot both be zero.
β	Value of the $\beta$ coefficient in the Mullins effect model. $\beta$ must be greater than or equal to zero, and the values of $m$ and $\beta$ cannot both be zero.
Use temperature-dependent data	Specify material parameters that depend on temperature. A Temperature field appears in the data table.