Inverse Finite Elastostatics for Isotropic Models Formulated in Principal Stretches

Inverse finite strain elastostatics has been mainly used for hyperelastic materials based on invariants such as, for instance, neo‐Hookean and Mooney–Rivlin models. However, isotropic hyperelasticity can also be formulated with models based on principal stretches. The main objective of this paper is to show how these latter can be used in inverse elastostatics as well. This task is simply accomplished by rewriting the inverse constitutive relation in the form of a classical state law. Then, with the use of spectral decompositions, we provide explicit stress functions and tangent moduli that can be used within a finite element implementation for commonly used models, including Ogden‐type models and the classical Hencky model, the latter based on logarithmic stretches. Furthermore, and independently of the specific hyperelastic law, we detail in this paper the consideration of self‐weight in the case of compressible materials, since it contributes to the tangent matrix during the iterative solution procedure. The effectiveness of the present framework is then demonstrated through a set of numerical simulations.