A comparative study of two constitutive models within an inverse approach to determine the spatial stiffness distribution in soft materials. (May 2018) - Mei, Y.; Stover, B.; Afsar Kazerooni, N.; Srinivasa, A.; Hajhashemkhani, M.; Hematiyan, M.R.; Goenezen, S.
Highlights: Digital Image Correlation (DIC) System to collect displacement data at high resolutions. Data collection at small and large overall strains on a silicone composite material having two stiff inclusions. Mapping the material property distribution by solving an inverse problem using a nonlinear hyperelastic material model and a linear elastic model. Comparison of results and accuracy from the linear model and nonlinear model on finite strains. Analyzing observations with analytical mathematical models and explaining observations in the results. Abstract: 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. Assuming that a material undergoes large deformations and material nonlinearity is assumed negligible, the inverse solution using (2) is anticipated to yield better results than (1). Given the fact that solving a linear elastic model is significantly faster than a nonlinear model and more robust numerically, we posed the following question: How accurately could we map the shear modulus distribution with a linear elastic model using small strain theory for a specimen undergoing large deformations? To this end, experimental displacement data of a silicone composite sample containing two stiff inclusions of different sizes under uniaxial displacement controlled extension were acquired using a digital image correlation system. The silicone based composite was modeled both as a linear elastic solid under infinitesimal strains and as a neo-Hookean hyperelastic solid that takes into account geometrically nonlinear finite deformations. We observed that the mapped shear modulus contrast, determined by solving an inverse problem, between inclusion and background was higher for the linear elastic model as compared to that of the hyperelastic one. A similar trend was observed for simulated experiments, where synthetically computed displacement data were produced and the inverse problem solved using both, the linear elastic model and the neo-Hookean material model. In addition, it was observed that the inverse problem solution was inclusion size-sensitive. Consequently, an 1-D model was introduced to broaden our understanding of this issue. This 1-D analysis revealed that by using a linear elastic approach, the overestimation of the shear modulus contrast between inclusion and background increases with the increase of external loads and target shear modulus contrast. Finally, this investigation provides valuable information on the validity of the assumption for utilizing linear elasticity in solving inverse problems for the spatial distribution of shear modulus associated with soft solids undergoing large deformations. Thus, this work could be of importance to characterize mechanical property variations of polymer based materials such as rubbers or in elasticity imaging of tissues for pathology. Graphical abstract: 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. Image, graphical abstract