Mikhail Itskov

Elastic constants and their admissible values for incompressible and slightly compressible anisotropic materials

SummaryConstitutive relations for incompressible (slightly compressible) anisotropic materials cannot (could hardly) be obtained through the inversion of the generalized Hookes law since the corresponding compliance tensor becomes singular (ill-conditioned) in this case. This is due to the fact that the incompressibility (slight compressibility) condition imposes some additional constraints on the elastic constants. The problem requires a special procedure discussed in the present paper. The idea of this procedure is based on the spectral decomposition of the compliance tensor but leads to a closed formula for the elasticity tensor without explicit using the eigenvalue problem solution. The condition of nonnegative (positive) definiteness of the material tensors restricts the elastic constants to belong to an admissible value domain. For orthotropic and transversely isotropic incompressible as well as isotropically compressible materials the corresponding domains are illustrated graphically.

The Derivative with respect to a Tensor: some Theoretical Aspects and Applications

The object of the paper is the derivative with respect to a second-order tensor. Basis-free as well as basis-related definitions of the derivative are addressed and compared. Special attention is focused on the derivative of a tensor function defined on a subset of all linear mappings within the real vector space, symmetric second-order tensors representing a most important example of such a subset. Due to a special definition of the simple contraction of fourth- and second-order tensors the well-known product rule of differentiation valid for scalar functions is proved to hold also for second-order tensor functions. Introducing a new tensor product of second-order tensors, another tensor differentiation rules as well as some useful tensor algebra operations are formulated. Applying this formalism the derivative of non-symmetric tensor power series is obtained in a closed form. This closed-formula solution is finally illustrated by some examples of continuum mechanics. As such, simple expressions for the derivative of the stretch and rotation tensor with respect to the deformation gradient and for the stresses conjugate to the logarithmic and more general Hills strains are presented.

Impact of transmural heterogeneities on arterial adaptation

Recent experimental and computational studies have shown that transmurally heterogeneous material properties through the arterial wall are critical to understanding the heterogeneous expressions of constituent degrading molecules. Given that expression of such molecules is thought to be intimately linked to local magnitudes of stress, modelling the transmural stress distribution is critical to understanding arterial adaption during disease. The aim of this study was to develop an arterial growth and remodelling framework that can incorporate both transmurally heterogeneous constituent distributions and residual stresses, into a 3-D finite element model. As an illustrative example, we model the development of a fusiform aneurysm and investigate the effects of elastinous and collagenous heterogeneities on the stress distribution during evolution. It is observed that the adaptive processes of growth and remodelling exhibit transmural variations. For physiological heterogeneous constituent distributions, a stress peak appears in the media towards the intima, and a stress plateau occurs towards the adventitia. These features can be primarily attributed to the underlying heterogeneity of elastinous constituents. During arterial adaption, the collagen strain is regulated to remain in its homoeostatic level; consequently, the partial stress of collagen has less influence on the total stress than the elastin. However, following significant elastin degradation, collagen plays the dominant role for the transmural stress profile and a marked stress peak occurs towards the adventitia. We conclude that to improve our understanding of the arterial adaption and the aetiology of arterial disease, there is a need to: quantify transmural constituent distributions during histopathological examinations, understand and model the role of the evolving transmural stress distribution.

Computation of the exponential and other isotropic tensor functions and their derivatives

Abstract In the present paper we focus on numerical aspects of the computation of isotropic tensor functions and their derivative. In the general case of non-symmetric tensor arguments only two numerical algorithms appear to be appropriate. The first one represents a recurrent procedure resulting from the Taylor power series expansion of an isotropic tensor function. The second algorithm is based on a recently proposed closed-form representation which can be obtained from the definition of an isotropic tensor function either by the tensor power series or by the Dunford–Taylor integral. To improve the accuracy in the case of nearly equal eigenvalues a series expansion of this closed formula is proposed. Both algorithms are finally illustrated by an example of the exponential tensor function where emphasis is placed on the precision issue.

A closed-form representation for the derivative of non-symmetric tensor power series

In the present paper a closed-form representation for the derivative of non-symmetric tensor power series is proposed. Particular attention is focused on the special case of repeated eigenvalues. In this case, a non-symmetric tensor can possess no spectral decomposition (in diagonal form) such that the well-known solutions in terms of eigenprojections as well as basis-free representations for isotropic functions of symmetric tensor arguments cannot be used. Thus, our representation seems to be the only possibility to calculate the derivative of non-symmetric tensor power series in a closed form. Finally, this closed formula is illustrated by an example being of special importance in large strain anisotropic elasto-plasticity. As such, we consider the exponential function of the velocity gradient under simple shear. Right in this loading case the velocity gradient has a triple defective eigenvalue excluding the application of any other solutions based on the spectral decomposition.

Taylor expansion of the inverse function with application to the Langevin function

A Taylor power series is a powerful mathematical tool, which can be used to express an inverse function especially if it is given in an implicit form. This is for example the case for the inverse Langevin function, which is an indispensable ingredient of full-network rubber models. In the present paper, we propose a simple recurrence procedure for calculating Taylor series coefficients of the inverse function. This procedure is based on the Taylor series expansion of the original function and results in a simple recurrence formula. This formula is further applied to the inverse Langevin function. The convergence radius of the resulting series is evaluated. Within this convergence radius the obtained approximation of the inverse Langevin function demonstrates better agreement with the exact solution in comparison to different Padé approximants.

A novel experimental procedure based on pure shear testing of dermatome-cut samples applied to porcine skin

This paper communicates a novel and robust method for the mechanical testing of thin layers of soft biological tissues with particular application to porcine skin. The key features include the use of a surgical dermatome and the highly defined deformation kinematics achieved by pure shear testing. Thin specimens of accurate thickness were prepared using a dermatome and were subjected to different quasi-static and dynamic loading protocols. Although simple in its experimental realisation, pure shear testing provides a number of advantages over other classic uni- and biaxial testing procedures. The preparation of thin specimens of porcine dermis, the mechanical tests as well as first representative results are described and discussed in detail. The results indicate a pronounced anisotropy between the directions along and across the cleavage lines and a strain rate-dependent response.

Consistent formulation of the growth process at the kinematic and constitutive level for soft tissues composed of multiple constituents

Previous studies have investigated the possibilities of modelling the change in volume and change in density of biomaterials. This can be modelled at the constitutive or the kinematic level. This work introduces a consistent formulation at the kinematic and constitutive level for growth processes. Most biomaterials consist of many constituents and can be approximated as being incompressible. These two conditions (many constituents and incompressibility) suggest a straightforward implementation in the context of the finite element (FE) method which could now be validated more easily against histological measurements. Its key characteristic variable is the normalised partial mass change. Using the concept of homeostatic equilibrium, we suggest two complementary growth laws in which the evolution of the normalised partial mass change is governed by an ordinary differential equation in terms of either the Piola–Kirchhoff stress or the Green–Lagrange strain. We combine this approach with the classical incompatibility condition and illustrate its algorithmic implementation within a fully nonlinear FE approach. This approach is first illustrated for a simple uniaxial tension and extension test for pure volume change and pure density change and is validated against previous numerical results. Finally, a physiologically based example of a two-phase model is presented which is a combination of volume and density changes. It can be concluded that the effect of hyper-restoration may be due to the systemic effect of degradation and adaptation of given constituents.

A Full-Network Rubber Elasticity Model based on Analytical Integration

Full-network rubber elasticity models generally require numerical integration over the unit sphere. In the present paper, a procedure for analytical integration of power series in terms of stretch square is proposed instead. This procedure is applied both to the inverse Langevin function and its rounded Padé approximation. The integrated power series demonstrates fast convergence to the analytical solution so far as it is available or to the numerical one based on a high resolution integration scheme. Good agreement with experimental data on silicone rubber is obtained as well. The integration procedure is also implemented to average the stretch on the basis of a q-root operator. This operator is usually applied in order to introduce a non-affine relation between micro and macro stretches into a network model.

A simple and accurate approximation of the inverse Langevin function

The inverse Langevin function cannot be represented in an explicit form and requires an approximation by a series, a non-rational or a rational function as for example by a Padé approximation. In the current paper, an analytical method based on the Padé technique and the multiple point interpolation is presented for the inverse Langevin function. Thus, a new simple and accurate approximation of the inverse Langevin function is obtained. It might be advantageous, for example, for non-Gaussian statistical theory of rubber elasticity where the inverse Langevin function plays an important role.