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Subsubsection 4.1.1.2: Orthotropic Materials Up Section 4.1: Elastic Solids Subsubsection 4.1.2.1: Arruda-Boyce

### 4.1.2 Uncoupled Materials

Uncoupled, nearly-incompressible hyperelastic materials are described by a strain energy function that features an additive decomposition of the hyperelastic strain energy into deviatoric and dilational parts [48]: where and is the deviatoric part of the deformation gradient. The resulting 2 Piola-Kirchhoff stress is given by where and and is the deviatoric operator in the material frame.
The corresponding Cauchy stress is given by where and is the deviatoric operator in the spatial frame.
For these materials, the entire bulk (volumetric) behavior is determined by the function , and represents the entire hydrostatic stress. The function is constructed to have a value of 0 for 1 and to have a positive value for all other values of 0.
All of these materials make use of the three-field element described by Simo and Taylor [48]. This element uses a trilinear interpolation of the displacement field and piecewise-constant interpolations for the pressure and volume ratio.
The uncoupled materials and the associated three-field element are very effective for representing nearly incompressible material behavior. Fully incompressible behavior can be obtained using an augmented Lagrangian method. To use the augmented Lagrangian method for enforcement of the incompressibility constraint to a user-defined tolerance, the user must define two additional material parameters:
 Parameter Description Default Turn augmented Lagrangian on for this material or off (1) 0 (off) Augmentation tolerance (2) 0.01
1. A value of 1 (one) turns augmentation on, where a value of 0 (zero) turns it off.
2. The augmentation tolerance determines the convergence condition that is used for the augmented Lagrangian method: convergence is reached when the relative ratio of the lagrange multiplier norm of the previous augmentation to the current one is less than the specified value:
Thus, a value of 0.01 implies that the change in norm between the previous augmentation loop and the current loop is less than 1%.
The augmented Lagrangian method for incompressibility enforcement is available for all materials that are based on an uncoupled hyperelastic strain energy function.
Example:
<material id="1" type="Mooney-Rivlin">
<c1>5</c1>
<c2>0.4</c2>
<k>10000</k>
<laugon>1</laugon>  turns on augmented Lagrangian iterations
<atol>0.05</atol>  sets the augmentation tolerance
</material>


Subsubsection 4.1.1.2: Orthotropic Materials Up Section 4.1: Elastic Solids Subsubsection 4.1.2.1: Arruda-Boyce