Mooney Material

Constitutive Law

This MPM Material is an isotropic, elastic material in large strains using a hyperelastic formulation. In hyperelasticity, the existence of a stored-energy W as a function of a deformation gradient tensor (F), is postulated and the constitutive law is derived from that strain energy function according to the inequality of Clausius-Duheim. Regarding to the objectivity conditions and using the representation theorem, the strain energy function is a function of invariants of a strain tensor. Using the left Cauchy-Green strain tensor (B = FF^T), the Cauchy stress is given by:

      [math]\displaystyle{ \mathbf{\sigma} =2 {\delta W \over {\delta \mathbf{B}}} \mathbf{B} }[/math]

In a Mooney-Rivlin material, stored stain energy is give by the expression:

      [math]\displaystyle{ W =U(J) + {G_{1} \over 2 } (\bar I_{1}-3) + {G_{2} \over 2 }(\bar I_{2}-3) }[/math]

where J (= det F) is relative volume change, G₁ and G₂ are shear material properties, and [math]\displaystyle{ \bar I_{1} }[/math] and [math]\displaystyle{ \bar I_{2} }[/math] are the strain invarients:

      [math]\displaystyle{ \bar I_{1} ={B_{xx}+B_{yy}+B_{zz} \over J^{2/3}} \qquad {\rm and} \qquad \bar I_{2} ={1 \over 2} (\bar I_{1}^2-{{B_{xx}^2+B_{yy}^2+B_{zz}^2+2B_{xy}^2+2B_{xz}^2+2B_{yz}^2} \over J^{2/3}}) }[/math]

This material allows three options for the volumetric energy term:

      [math]\displaystyle{ U(J) ={\kappa\over 2 }({1\over 2 }(J^2-1)-\ln J) \qquad({\rm UJOption}=0) }[/math]

      [math]\displaystyle{ U(J) ={\kappa\over 2 }(J-1)^2 \qquad\qquad\qquad({\rm UJOption}=1) }[/math]

      [math]\displaystyle{ U(J) ={\kappa\over 2 }(\ln J)^2 \qquad\qquad\qquad({\rm UJOption}=2) }[/math]

where [math]\displaystyle{ \kappa }[/math] is bulk modulus. At low strains, this material is equivalent to a linear elastic, isotropic material with shear modulus G = G₁ + G₂ and bulk modulus [math]\displaystyle{ \kappa }[/math]. If G₂ = 0, the material is a Neo-Hookean material. Some hyperelastic rubber models assume incompressible materials, which corresponds to [math]\displaystyle{ \kappa\to\infty }[/math]; such models do not work in dynamic code (because the dilational wave speed is infinite).

The Cauchy stress tensor is determined by differentiating the strain energy function. It is calculated by the addition of pressure and deviatoric stress, [math]\displaystyle{ \mathbf{\sigma} = -p \mathbf{I} + \bar \mathbf{\sigma} }[/math] resulting in:

      [math]\displaystyle{ \mathbf{\sigma} ={\kappa\over 2}\left(J-{1\over J}\right)\mathbf{I} + {G_{1} \over J^{5/3} } (\mathbf{B}-{I_{1} \over3}\mathbf{I}) + {G_{2} \over J^{7/3}} (I_{1} \mathbf{B}-\mathbf{B^2}-{2I_{2} \over3}\mathbf{I}) }[/math]

where [math]\displaystyle{ I_{1} = J^{2/3} \bar I_{1} }[/math] and [math]\displaystyle{ I_{2} = J^{4/3} \bar I_{2} }[/math]. The above stress assumes the default UJOption=0. For the other options, the pressure term changes to

      [math]\displaystyle{ p = -\kappa(J-1) \qquad\qquad({\rm UJOption}=1) }[/math]

      [math]\displaystyle{ p = -\kappa {\ln J\over J} \qquad\qquad({\rm UJOption}=2) }[/math]

Note that UJOption=0 is the default option because it is the only pressure that correctly becomes infinite for both J approaching 0 and J approaching infinity.

Residual Stress

In the presence of temperature or concentration changes, this material accounts for residual stresses by replacing F with F/λ_res, where λ_res is the free residual expansion in one direction (and all three directions are the same for this isotropic material). The net effect is B and J are replaced with

      [math]\displaystyle{ \mathbf{B}_{eff} = {\mathbf{B}\over \lambda_{res}^2} \qquad {\rm and} \qquad J_{eff} = {J\over J_{res}} }[/math]

where J_res = λ_res³ is the ratio of the free expansion volume to the reference volume. Thus J_eff is the ratio of the current volume to the free expansion volume while J is the ratio of current volume to the initial volume. The Cauchy stress becomes:

      [math]\displaystyle{ \mathbf{\sigma} = -p(J_{eff})\mathbf{I} + {G_{1} \over J_{eff}J^{2/3} } (\mathbf{B}-{I_{1} \over3}\mathbf{I}) + {G_{2} \over J_{eff}J^{4/3}} (I_{1} \mathbf{B}-\mathbf{B^2}-{2I_{2} \over3}\mathbf{I}) }[/math]

where p(J_eff) is one of the above pressure options evaluated using J_eff instead of J. Finally, the Kirchhoff stress used in the MPM time step multiplies by actual J to get:

      [math]\displaystyle{ \mathbf{\tau} = -Jp(J_{eff})\mathbf{I} + {G_{1} J^{1/3}\over J_{eff} } (\mathbf{B}-{I_{1} \over3}\mathbf{I}) + {G_{2} \over J_{eff}J^{1/3}} (I_{1} \mathbf{B}-\mathbf{B^2}-{2I_{2} \over3}\mathbf{I}) }[/math]

Ideal Rubber

This material provides an acceptable material model for rubbers or elastomers, but it will not get the thermodynamics correct unless you indicate the material is an ideal rubber (using the IdealRubber property). An ideal rubber means that during isothermal loading, there is no change in internal energy. In other words, all the work is turned into heat. In adiabatic loading, this heat causes the temperature to rise. For the thermal calculations to be correct, the material must specify a valid heat capacity.

The entropy for this material is the net effect of heat flow by external heating, by conduction, by internal mechanisms, or by work when it is set to be an ideal rubber. For example, when an ideal rubber is loaded isothermally in tension in the x direction, the force is related to entropy by

      [math]\displaystyle{ F = A\sigma_{xx} = -T \left({\partial S\over\partial L}\right)_T }[/math]

where A is cross-sectional area and L in length. An ideal rubber in NairnMPM correctly follows this law.

Material Properties

The material properties are entered using:

Property	Description	Units	Default
K	Bulk modulus	MPa	none
G1	The G1 shear modulus	MPa	none
G2	The G2 shear modulus	MPa	0
UJOption	Set to 0, 1, or 2, to select the energy term from above.	none	0
alpha	Thermal expansion coefficient	ppm/M	0
IdealRubber	Set to 0 to be an elastic material and to 1 to be an ideal rubber.	none	0
(other)	Properties common to all materials	varies	varies

See these relations to covert other properties (such as modulus and Poisson's ratio) to bulk and shear moduli.

History Variables

This material uses history #1 to store the volumetric strain (i.e., the determinant of the deformation gradient). The total strain, which is elastic, is stored in the elastic strain variable, while the plastic strain stores the left Cauchy Green tensor.

Examples

These commands model polymer as an isotropic, hyperelastic material with G1=G2 =G/2 (using scripted or XML commands):

Material "polymer","Ideal Elastomer","Mooney"
  K 3000
  G1 10
  nu .48
  alpha 60
  rho 1.2
  IdealRubber 1
  Cv 1200
Done