Efficient evaluation of the material response of tissues reinforced by statistically oriented fibres

Abstract

For several classes of soft biological tissues, modelling complexity is in part due to the arrangement of the collagen fibres. In general, the arrangement of the fibres can be described by defining, at each point in the tissue, the structure tensor (i.e., the tensor product of the unit vector of the local fibre arrangement by itself) and a probability distribution of orientation. In this approach, assuming that the fibres do not interact with each other, the overall contribution of the collagen fibres to a given mechanical property of the tissue can be estimated by means of an averaging integral of the constitutive function describing the mechanical property at study over the set of all possible directions in space. Except for the particular case of fibre constitutive functions that are polynomial in the transversely isotropic invariants of the deformation, the averaging integral cannot be evaluated directly, in a single calculation because, in general, the integrand depends both on deformation and on fibre orientation in a non-separable way. The problem is thus, in a sense, analogous to that of solving the integral of a function of two variables, which cannot be split up into the product of two functions, each depending only on one of the variables. Although numerical schemes can be used to evaluate the integral at each deformation increment, this is computationally expensive. With the purpose of containing computational costs, this work proposes approximation methods that are based on the direct integrability of polynomial functions and that do not require the step-by-step evaluation of the averaging integrals. Three different methods are proposed: a) a Taylor expansion of the fibre constitutive function in the transversely isotropic invariants of the deformation; b) a Taylor expansion of the fibre constitutive function in the structure tensor; c) for the case of a fibre constitutive function having a polynomial argument, an approximation in which the directional average of the constitutive function is replaced by the constitutive function evaluated at the directional average of the argument.Each of the proposed methods approximates the averaged constitutive function in such a way that it is multiplicatively decomposed into the product of a function of the deformation only and a function of the structure tensors only. In order to assess the accuracy of these methods, we evaluate the constitutive functions of the elastic potential and the Cauchy stress, for a biaxial test, under different conditions, i.e., different fibre distributions and different ratios of the nominal strains in the two directions. The results are then compared against those obtained for an averaging method available in the literature, as well as against the integration made at each increment of deformation.

A stochastic model of stroma: interweaving variability and compressed fibril exclusion

“…The computational challenge of describing the microstructural properties in a reliable manner motivated the development of several approximations. [10][11][12][13][14][15][16][17][18] The present study departs from the second order approximation of the strain energy density ψ f * of the collagen fiber distribution illustrated in Pandolfi and Vasta. 5 The approximation is based on the second order Taylor expansion of the fiber strain energy density about the fourth pseudo-invariant I 4 = C : A , where A = a ⊗ a is the structure tensor associated to the mean direction a of the fiber distribution, i.e., ψ :…”

On fibre dispersion modelling of soft biological tissues: a review

“…The more recent paper of Melnik et al [9] employed a deformation-dependent structure tensor in order to exclude compressed fibres, which the authors applied to uniaxial extension with two families of fibres. The study of Hashlamoun et al [46] also used a Heaviside function to exclude compressed fibres, and the authors compared the results for three different polynomial expansions of the energy function about I 4 = 1, and they also performed comparisons of the Cauchy stresses σ 11 and σ 22 as functions of stretch for equibiaxial and biaxial deformations. Note that the comparison in the exponential case was performed with the same parameters c 1 = k 1 and for c 2 = k 2 = 1 for each case.…”

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

“…The complex, in some cases unaffordable and computationally demanding, description of the microstructure has pushed for approximations based on the homogenization of the microstructure by means of parameters of the collagen distribution (i.e., average and higher order statistics) in the strain energy density [24][25][26]. Strong homogenization techniques, though, may cancel out the features of the microstructure and compromise the predictive properties of the model at the macroscale [27]. Among countless research papers discussing stochastic models of fiber reinforced materials, only a few contributions have been trying to characterize analytically the probability distribution functions (PDF) by means of statistical descriptors [9,12,[28][29][30].…”

“…Fig.2(c) visualizes condition(27), which results as a combination of uniaxial and biaxial conditions since it is expressed in the principal stretch reference system. In particular, the w(I 4 ) spans over the whole integration range with lower bound I 4 = 1 and upper boundI 4 = λ 21 .…”