A spectral decomposition approach for the mechanical statistical characterization of distributed fiber-reinforced tissues

Abstract

We discuss a spectral decomposition formulation for the mechanical statistical characterization of the anisotropic strain energy density of soft hyperelastic materials embedded with distributed fibers. We consider a generalized angular probability density function (PDF) of the reinforcement built upon the local eigenvalue and eigenvector system of the Cauchy-Green deformation tensor. We focus our analysis to material models dependent on the fourth pseudo-invariant of the deformation, I 4 , and to exponential forms of the fiber strain energy function. Within such a spectral reference system, we derive the closed-form expression of the PDF for I 4 generalizing the multi-value random variable transformation procedure recently developed in . Our formulation bypasses the cumbersome extension-contraction switch, commonly adopted for shutting down the contribution of contracted fibers in models based on generalized structure tensors. Accordingly, we identify analytically the support of the fibers in pure extension for significant loading conditions. We can readily compute any statistics of the fourth pseudo-invariant and we can derive the direct definition of the average second Piola-Kirchhoff stress tensor according to the second order approximation.

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“…One reason that our model did not yield these different scenarios is our oversimplification of the real elastic lamella structure: at this stage, our computational model only includes a minimal set of essential features required for reproducing the stepwise progressive tearing behaviour, illustrating the possibility of reproducing the experimentally observed stepwise increases of pressure by the presence of radial struts [22]. Nonetheless, in future works, more detailed models may be developed, for instance, by accounting for the angular distribution of the orientation of fibres [68] or introducing actual heterogeneities in mural structure and properties [69], which may be inferred from imaging data, or by computationally accounting for tissue degeneration [70].…”

A three-dimensional finite-element model of gluteus medius muscle incorporating inverse-dynamics-based optimization for simulation of non-uniform muscle contraction

“…Constitutive models accounting for the effect of fibre stretches and velocity on the level of muscle activation and derived specifically for modelling of muscles (e.g. coupling of active and passive stresses) [37][38][39] will also be a consideration in future studies. Although the modelling and optimization procedures are based on a validated algorithm [26,40], the GM model itself was not validated due to the challenge in experimental measurements of the biomechanical behaviour of GM.…”

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“…While the above‐mentioned formulations have been used successfully to describe the passive constitutive behavior of various fibrous soft biological tissues, their applicability to gastric tissue still needs to be carefully evaluated. For this purpose, suitable experimental data from multiaxial layer‐specific tensile tests are required.…”

“…where the index i indicates the ith fiber family out of a total of N fiber families; k 1, i > 0 are stress-like material parameters, k 2, i > 0 are dimensionless material parameters and i is the elastic stretch of the material in direction of the ith fiber family. While the above-mentioned formulations have been used successfully to describe the passive constitutive behavior of various fibrous soft biological tissues, [192][193][194][195] their applicability to gastric tissue still needs to be carefully evaluated. For this purpose, suitable experimental data from multiaxial layer-specific tensile tests are required.…”