Stochastic hyperelastic constitutive laws and identification procedure for soft biological tissues with intrinsic variability

Abstract

In this work, we address the constitutive modeling, in a probabilistic framework, of the hyperelastic response of soft biological tissues. The aim is on the one hand to mimic the mean behavior and variability that are typically encountered in the experimental characterization of such materials, and on the other hand to derive mathematical models that are almost surely consistent with the theory of nonlinear elasticity. Towards this goal, we invoke information theory and discuss a stochastic model relying on a low-dimensional parametrization. We subsequently propose a two-step methodology allowing for the calibration of the model using standard data, such as mean and standard deviation values along a given loading path. The framework is finally applied and benchmarked on three experimental databases proposed elsewhere in the literature. It is shown that the stochastic model allows experiments to be accurately reproduced, regardless of the tissue under consideration.

Stochastic isotropic hyperelastic materials: constitutive calibration and model selection

“…Here, we explain in detail the calibration of models from table 1 to experimental mean values and standard deviations of the nonlinear shear modulus µ(a) of (2.13), for small shear superposed on finite axial stretch. The calibration to data values of the elastic (shear) stress [25] or of the nonlinear shear modulus µ(a) of (2.12), for simple shear, can then be performed analogously. Henceforth, the following notation is used: a quantity with an overbar denotes a value appearing in the theory of linear elasticity (e.g., µ); an underlined quantity denotes the mean value of that quantity (e.g., µ, µ, µ).…”

“…Next, following [23][24][25][26][27], for the random nonlinear shear modulus µ(a 0 ), defined by (3.7), we set the mathematical expectations: 12) where, by the constraint (3.11), the mean value µ(a 0 ) is fixed and greater than zero, and the logarithmic constraint (3.12) implies that both µ(a 0 ) and µ(a 0 ) −1 are second-order random variables (i.e. they have finite mean and finite variance).…”

“…Critically, equations (3.11) and (3.12) imply that µ(a 0 ) follows a Gamma distribution (the maximum entropy distribution) [72,73], 13) where the mean value µ(a 0 ) is obtained at step 1. For the random vector (R 1 (a 0 ), · · · , Rn(a 0 )), applying the constraints [23,25] 15) this vector follows a Dirichlet distribution [41,74], D (ξ 1 (a 0 ), · · · , ξn(a 0 )). Then, every random variable Rp(a 0 ), p = 1, · · · , n, follows a standard Beta distribution [40,41], B(ξp(a 0 ), ψp(a 0 )), with ξp(a 0 ) > 0 and ψp(a 0 ) = n q=1,q =p ξq(a 0 ) > 0 satisfying 17) and is calculated from the mean values obtained at step 1.…”

“…Recently, stochastic strategies based on information theory, which aids with the calibration of hyperelastic models for isotropic elastic solids, were proposed in [23][24][25]. Specifically, stochastichyperelastic models were identified from experimental data consisting of the mean values and standard deviations of elastic stresses under finite strain deformations.…”

“…The measure of entropy (or uncertainty) of a discrete probability distribution was first defined by Shannon (1948) [31,32] in the context of information theory. In [25], the stochastic approach was employed for the calibration of Ogden-type models to brain, liver, and spinal cord data representing mean values and standard deviation of the first Piola-Kirchhoff stress under finite compression tests. Ogden-type strain-energy functions and their extension to compressible materials in a stochastic framework were formulated originally in [23,24].…”