On the use of non‐linear transformations for the evaluation of anisotropic rotationally symmetric directional integrals. Application to the stress analysis in fibred soft tissues

Abstract

SUMMARYMicrosphere-based constitutive models are a helpful tool in the modelling of materials with a microstructure composed of contributing elements directionally arranged. This is the case, for instance, for fibred soft tissues. In these models, the macroscopic mechanical behaviour is obtained from the integration of the micro-structural contribution of each component (e.g. each fibre) over the surface of an underlying microsphere, which allows incorporating the mechanical features of the micro-constituents to the macroscopic response. The combination of this sort of models and the associated numerical techniques constitutes a powerful modelling tool for which an efficient integration scheme is required. In this regard, the unit sphere discretizations proposed by Bažant and Oh (ZAMM-J Appl Math Mech Z Angew Math Mech 1986; 66(1):37-49) have been used for the integration of the microscopic contributions in isotropic materials. Nevertheless, the inclusion of anisotropy has important implications with regard to the integration scheme, since very fine discretizations are needed to perform the integration accurately, causing the integration process to be very costly. In addition, the storage of internal variables at each integration direction of every integration point is required for constitutive models based on the use of internal variables at the micro-structural level, which renders this approach rather complex and memory demanding. In order to reduce the number of necessary integration directions, several non-linear transformations for the integration of rotationally symmetric functions over the surface of the unit sphere are here presented. Their accuracy in the integration of the von Mises orientation distribution function is evaluated. Furthermore, a hyperelastic microsphere-based constitutive law for the modelling of soft biological tissues is used in order to check the accuracy and computational efficiency of the proposed transformations within a Finite Element context in inhomogeneous deformation problems. Simulation results show the suitability of the * Correspondence to: V. Alastrué, Mechanical Engineering Department, University of Zaragoza, Agustín de Betancourt Building, María de Luna, s/n. E-50018 Zaragoza, Spain. USE OF NON-LINEAR TRANSFORMATIONS 475proposed methodology in order to accurately approximate the value of the integrals within reasonable computational costs.

Modelling non-symmetric collagen fibre dispersion in arterial walls

“…A PDF based on the von Mises distribution taking account of a non-rotationally symmetric fibre dispersion and based on the micro-sphere model was suggested in [30], and an ellipsoidal distribution with a power-law strain-energy function, based on the AI formulation, was employed in [31] and applied to cartilage. Based on the AI approach with the von Mises distribution Raghupathy & Barocas [32] derived a closedform solution for a simple exponential fibre stress-strain law and applied their model to planar biaxial extension of a bioartificial tissue.…”

Anisotropic microsphere-based approach to damage in soft fibered tissue

“…Moreover, for the damage problem only ID → ∞ leads to optimal results. Alastrué et al (2009b) presented a non-linear transformation in order to reduce the number of integration directions and increase accuracy for highly anisotropic cases. Although the discrete integration performed can give rise to disadvantages, it does produce a new view of micro-structural behavior, as pointed out previously.…”

“…Therefore, an increase of the integration directions leads to more accurate results. Another possible choice would be to consider another integration technique such as that used by Hardin and Sloane (1996) or the non-linear transformation, proposed by Alastrué et al (2009b), in order to adjust the distribution of the integration directions to the statistical distribution function. The discretization and associated peaks may be considered as lacking accuracy.…”

An anisotropic discrete fibre model based on a generalised strain invariant with application to soft biological tissues

“…Therefore, the fibre weights were not pure measures of the anisotropy of the soft tissue being simulated. The issue of undesirable anisotropy as a result of numerical integration over a sphere has been reported extensively (Alastrue, Martinez, Menzel, & Doblare, 2009;Bazant & Oh, 1986;Itskov, Ehret, & Dargazany, 2010;Miehe & Goktepe, 2005;Miehe, Goktepe, & Lulei, 2004) with application (Alastrue et al, 2009) to finite deformations of tissues using modifications of the structural model of the type proposed in Lanir (1983), and with applications to rubber-like materials (Itskov et al, 2010;Miehe & Goktepe, 2005;Miehe et al, 2004). There is a need for a constitutive model that gives an unambiguous interpretation of the anisotropy when the weights are not equal and a clear understanding of the effect of the undulation distribution on the fibre bundle.…”