A comparative study of two constitutive models within an inverse approach to determine the spatial stiffness distribution in soft materials

A comparative study is presented to solve the inverse problem in elasticity for the shear modulus (stiffness) distribution utilizing two constitutive equations: (1) linear elasticity assuming small strain theory, and (2) finite elasticity with a hyperelastic neo-Hookean material model. Displacement data was experimentally measured using a digital image correlation system on a silicone composite sample having two stiff inclusions of different sizes. We observe that the mapped stiffness contrast between inclusion and background acquired from the linear elasticity model is larger than that from the neo-Hookean finite elasticity model. A similar trend is observed when the inverse problem is solved using simulated experiments, where measured displacements were simulated using a neo-Hookean solid. In addition, we also observe that the solution to the inverse problem is inclusion size-sensitive. We then introduce a 1-D model to explain why these phenomena occur. This 1-D analysis also reveals that by using a linear elastic approach, the overestimation of the stiffness contrast between inclusion and background increases with the increase of external loads and target stiffness contrast. This investigation provides valuable information on the assumption of linear elasticity in solving inverse problems for soft solids undergoing large deformations.164