Nonlinear elasticity of biological tissues with statistical fibre orientation

Abstract

The elastic strain energy potential for nonlinear fibre-reinforced materials is customarily obtained by superposition of the potentials of the matrix and of each family of fibres. Composites with statistically oriented fibres, such as biological tissues, can be seen as being reinforced by a continuous infinity of fibre families, the orientation of which can be represented by means of a probability density function defined on the unit sphere (i.e. the solid angle). In this case, the superposition procedure gives rise to an integral form of the elastic potential such that the deformation features in the integral, which therefore cannot be calculated a priori. As a consequence, an analytical use of this potential is impossible. In this paper, we implemented this integral form of the elastic potential into a numerical procedure that evaluates the potential, the stress and the elasticity tensor at each deformation step. The numerical integration over the unit sphere is performed by means of the method of spherical designs, in which the result of the integral is approximated by a suitable sum over a discrete subset of the unit sphere. As an example of application, we modelled the collagen fibre distribution in articular cartilage, and used it in simulating displacementcontrolled tests: the unconfined compression of a cylindrical sample and the contact problem in the hip joint.

Statistical characterization of the anisotropic strain energy in soft materials with distributed fibers

“…Lanir defined the strain energy density as the integral of the strain energy density of single fibers, spatially oriented according to a statistical distribution. Extensions and particular applications of this approach have been discussed in subsequent research (Holzapfel et al, 2000;Rodríquez et al, 2006;Alastrué et al, 2007;Federico and Gasser, 2010;Gizzi et al, 2014).…”

An affine continuum mechanical model for cross-linked F-actin networks with compliant linker proteins

“…Many of these models are based on the microstructure of the network and adopt the idea of integrating the single filament response into a network [9]. For example, an algorithmic treatment for affine and non-affine networks, the micro-sphere model, was introduced by Miehe et al [10] and the affine network model was adopted for modeling the mechanics of collagen in arteries [11,12]. Applications of the filament-to-network approach to the elasticity of actin networks have also been proposed [13,14].…”

On the contribution of Angelo Luongo to Mechanics: in honor of his 60th birthday

“…Recently, an increasing attention has been attracted by phenomena which govern the growth of living tissues and, in particular, the reconstructed bones (see, e.g., [115,116]). These phenomena have a relevant mechanical content, and it seems that they can be described by means of suitable mathematical models having a similar structure as those used in disparate fields, for instance, in plasticity.…”