Energetically stable curve fitting to hyperelastic models based on uniaxial and biaxial tensile tests

Abstract

Hyperelastic constitutive laws in biomechanics are used to model soft tissues, and material model parameters are often determined by performing curve fitting on data from uniaxial or biaxial tensile tests. The strain energy function of the applied constitutive law must to be energetically stable; however, this condition is not inherently provided by most currently available models. This study provides a procedure to determine stable strain energy functions in a biaxial strain space based on either uniaxial or biaxial tensile tests. Instead of conservative, strain-independent conditions, a stability region is defined in the strain space based on the sample's tensile tests, thus allowing optimisation within a wider parameter space, resulting in better approximations. An extension of the Levenberg-Marquardt algorithm incorporating user-defined stability constraints is proposed, and the constrained optimisation algorithm is applied to isotropic and anisotropic models. The uniqueness of solutions of the Fung model is also discussed. The material model parameters of stable solutions for soft tissue measurements from various literature sources are determined to demonstrate the proposed procedure. Applying appropriate constraints in the optimisation algorithm resulted in stable and physically permissible constrained solutions for the strain energy function, in contrast to the results of most unconstrained optimisation cases.

Keywords: Constrained optimisation; Convexity; Hyperelastic material; Stability; Strain energy function; Uniaxial and biaxial test.