Theory Manual Version 3.4
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Subsection 5.2.1: Isotropic Elasticity Up Section 5.2: Compressible Materials Subsection 5.2.3: Neo-Hookean Hyperelasticity

### 5.2.2 Orthotropic Elasticity

An extension of the St. Venant-Kirchhoff model [23] to orthotropic symmetry is provided in FEBio, referred to as orthotropic elasticity. This model is objective for large strains and rotations. It can be derived from the following hyperelastic strain-energy function: where is the structural tensor corresponding to one of the three mutually orthogonal planes of symmetry whose unit outward normal is (). The material constants are the three shear moduli and six moduli , where . They may be related to the Young's moduli , shear moduli and Poisson's ratios via The second Piola-Kirchhoff stress can be derived from this strain energy density function: Note that these equations are similar to the corresponding equations in the linear orthotropic elastic case, except that the small strain tensor is replaced by the Green-Lagrange elasticity tensor . The material elasticity tensor is then given by, It is important to note that although this model is objective, it should only be used for small strains. For large strains, the response can be somewhat strange if not completely unrealistic. For example, it can be shown that under uni-axial tension the stress becomes infinite and the volume tends to zero for finite strains. Therefore, for large strains it is highly recommended to avoid this material and instead use one of the other non-linear material models described below. The Cauchy stress is where and the spatial elasticity tensor is
Subsection 5.2.1: Isotropic Elasticity Up Section 5.2: Compressible Materials Subsection 5.2.3: Neo-Hookean Hyperelasticity