Improving the sensitivity to map nonlinear parameters for hyperelastic problems

The mechanical properties of tissues are important indicators of tissue “health”. Diseased tissues due to cancer and other causes tend to stiffen with increase in strain, exhibiting a nonlinear stress-strain behavior. The hyperelastic mechanical behavior of soft tissues can be characterized by an exponential model with two material parameters, namely the shear modulus μ and a nonlinearity parameter γ. A variety of methods and techniques have been developed to solve inverse problems in nonlinear elasticity to determine these properties given the mechanical response of the tissues. Reconstruction of the nonlinear parameter from noisy measurements of displacement response is a difficult problem, and obtaining a well-recovered solution is challenging. This article is directed towards the improvement in the reconstruction of the nonlinear parameter, γ, by introducing a new parameter, which is a combination of γ and the first invariant of the Cauchy-Green deformation tensor. Comparative study is carried out between reconstructions of γ directly from previously existing formulations and the reconstruction of γ from the new parameter, for 2D problems. The new method is compared with previous methods using numerical experiments in terms of shape of the stiff regions, the contrast in γ and robustness under different loading conditions. We find that the new method is a considerable improvement to previous methods and could be a valuable tool in biomedical applications.