Theory Manual Version 3.4
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Subsection 2.3.3: Stress Up Chapter 2: Continuum Mechanics Subsection 2.4.1: Isotropic Hyperelasticity

## 2.4 Hyperelasticity

When the constitutive behavior is only a function of the current state of deformation, the material is elastic. In the special case when the work done by the stresses during a deformation is only dependent on the initial state and the final state, the material is termed hyperelastic and its behavior is path-independent. As a consequence of the path-independence a strain energy function per unit undeformed volume can be defined as the work done by the stresses from the initial to the final configuration: The rate of change of the potential is then given by Or alternatively, Comparing (2.4-2) with (2.4-3) reveals that This general constitutive equation can be further simplified by observing that, as a consequence of the objectivity requirement, may only depend on through the stretch tensor and must be independent of the rotation component . For convenience, however, is often expressed as a function of . Noting that is work conjugate to the second Piola-Kirchhoff stress , establishes the following general relationships for hyperelastic materials:
Subsection 2.3.3: Stress Up Chapter 2: Continuum Mechanics Subsection 2.4.1: Isotropic Hyperelasticity