Heuristic search for a predictive strain-energy function in nonlinear elasticity

In this work, a new, quasi-structural model – bootstrapped eight-chain model – is proposed as a modification to the strain energy of eight-chain model [Arruda, E.M., Boyce, M.C., 1993. A three-dimensional constitutive model for the large stretch behaviour of rubber elastic materials. J. Mech. Phys. Solids 41, 389—412] that invokes the Langevin chain statistics. This development has been led to by our heuristic search into how the strain energy of eight-chain model may be adapted in order to account better for the mechanical behaviour of elastomeric materials in both linear and nonlinear elastic regimes [Treloar, L.R.G., 1944. Stress–strain data for vulcanised rubber under various types of deformation. Trans. Faraday Soc. 40, 59–70]. The eight-chain model appears to produce very similar results in predicting biaxial stress to those of a first stretch-invariant model that gives a good fit in uniaxial extension and, thus, it is shown that the former can not be significantly enhanced within the limitation of the latter. Evaluation of predictive capability for an additive invariant-separated form of strain energy shows that an explicit inclusion of a second stretch-invariant function would not work and that any thus added term ought to be dependent on both the first and second stretch-invariants of deformation tensor, and hints that an improvement is possibly needed at low strain. The composite and filament models [Miroshnychenko, D., Green, W.A., Turner, D.M., 2005. Composite and filament models for the mechanical behaviour of elastomeric materials. J. Mech. Phys. Solids 53 (4), 748–770] have their strain-energy functions in that suggested form and cope very well with predicting the experimental data of Treloar (1944). We use the form of strain energy for the filament model, that proved to be successful, to bootstrap the strain energy of eight-chain model in order to improve the performance of the latter at low strain. Thus, we derive a new model – bootstrapped eight-chain model – that requires only two material parameters – a rubber modulus and a limiting chain extensibility. The proposed model is quasi-structural due to bootstrapping and it retains the best traits and corrects the faults of the eight-chain model, conforming more closely to the classical experimental data of Treloar (1944).     

Elastica-based strain energy functions for soft biological tissue

Continuum strain energy density functions are developed for soft biological tissues that possess slender, fibrillar components. The treatment is based on the model of an elastica, which is our fine scale model, and is homogenized in a simple fashion to obtain a continuum strain energy density function. Notably, we avoid solving the exact, fourth-order, non-linear, partial differential equation for deformation of the elastica by resorting to other assumptions, kinematic and energetic, on the response of individual, elastica-like fibrils. The formulation, discussion of responses of different models and comparison with experiment are presented.     

On the mechanical modeling of anisotropic biological soft tissue and iterative parallel solution strategies

Biological soft tissues appearing in arterial walls are characterized by a nearly incompressible, anisotropic, hyperelastic material behavior in the physiological range of deformations. For the representation of such materials we apply a polyconvex strain energy function in order to ensure the existence of minimizers and in order to satisfy the Legendre–Hadamard condition automatically. The 3D discretization results in a large system of equations; therefore, a parallel algorithm is applied to solve the equilibrium problem. Domain decomposition methods like the Dual-Primal Finite Element Tearing and Interconnecting (FETI-DP) method are designed to solve large linear systems of equations, that arise from the discretization of partial differential equations, on parallel computers. Their numerical and parallel scalability, as well as their robustness, also in the incompressible limit, has been shown theoretically and in numerical simulations. We are using a dual-primal FETI method to solve nonlinear, anisotropic elasticity problems for 3D models of arterial walls and present some preliminary numerical results.     

A composites-based hyperelastic constitutive model for soft tissue with application to the human annulus fibrosus

This paper presents a composites-based hyperelastic constitutive model for soft tissue. Well organized soft tissue is treated as a composite in which the matrix material is embedded with a single family of aligned fibers. The fiber is modeled as a generalized neo-Hookean material in which the stiffness depends on fiber stretch. The deformation gradient is decomposed multiplicatively into two parts: a uniaxial deformation along the fiber direction and a subsequent shear deformation. This permits the fiber-matrix interaction caused by inhomogeneous deformation to be estimated by using effective properties from conventional composites theory based on small strain linear elasticity and suitably generalized to the present large deformation case. A transversely isotropic hyperelastic model is proposed to describe the mechanical behavior of fiber-reinforced soft tissue. This model is then applied to the human annulus fibrosus. Because of the layered anatomical structure of the annulus fibrosus, an orthotropic hyperelastic model of the annulus fibrosus is developed. Simulations show that the model reproduces the stress-strain response of the human annulus fibrosus accurately. We also show that the expression for the fiber-matrix shear interaction energy used in a previous phenomenological model is compatible with that derived in the present paper.     

Two-dimensional linear elasticity by the boundary node method

This paper presents a further development of the Boundary Node Method (BNM) for 2-D linear elasticity. In this work, the Boundary Integral Equations (BIE) for linear elasticity have been coupled with Moving Least Square (MLS) interpolants; this procedure exploits the mesh-less attributes of the MLS and the dimensionality advantages of the BIE. As a result, the BNM requires only a nodal data structure on the bounding surface of a body. A cell structure is employed only on the boundary in order to carry out numerical integration. In addition, the MLS interpolants have been suitably truncated at corners in order to avoid some of the oscillations observed while solving potential problems by the BNM (Mukherjee and Mukherjee, 1997a) . Numerical results presented in this paper, including those for the solution of the Lamé and Kirsch problems, show good agreement with analytical solutions.     

A three-dimensional nonlinear model for dissipative response of soft tissue

A set of three-dimensional constitutive equations is proposed for modeling the nonlinear dissipative response of soft tissue. These constitutive equations are phenomenological in nature and they model a number of physical features that have been observed in soft tissue. The equations model the tissue as a composite of a purely elastic component and a dissipative component, both of which experience the same total dilatation and distortion. The stress response of the purely elastic component depends on dilatation, distortion and the stretch of material fibers, whereas the stress response of the dissipative component depends on distortional deformation only. The equations are hyperelastic in the sense that the stress is obtained by derivatives of a strain energy function, and they are properly invariant under superposed rigid body motions. In contrast with standard viscoelastic models of tissues, the proposed constitutive model includes the total deformation rate in evolution equations that can reproduce the observed physical feature that the hysteresis loops of most biological soft tissues are nearly independent of strain rate (Biomechanics, Mechanical Properties of Living Tissues, second ed. (1993)). Material constants are determined which produce good agreement with uniaxial stress experiments on superficial musculoaponeurotic system and facial skin.     

Dependence of linear elasticity solutions on the elastic constants

Necessary and sufficient conditions on the assigned data for the displacement and/or the stress to be independent of the shear modulus in the standard boundary-value problems of three-dimensional classical elastostatics are determined.     

Dependence of linear elasticity solutions on the elastic constants

Necessary and sufficient conditions for the displacement and/or the stress to be independent of Poisson's ratio or the shear modulus or the mass density in the standard boundary-initial-value problems of three-dimensional classical elastodynamics are determined.

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Résumé On détermine des conditions nécessaires et suffisantes pour que les déplacements et (ou) les contraintes soient indépendants du coefficient de Poisson ou du module de rigidité ou du densité de masse dans les problémes classiques de l'élasticité tri-dimensionnelle dynamique.

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Department of Theoretical and Applied Mechanics, University of Illinois at Urbana     

Solving the three-dimensional equations of the linear theory of elasticity

The three-dimensional Lamé equations are solved using Cartesian and curvilinear orthogonal coordinates. It is proved that the solution includes only three independent harmonic functions. The general solution of equations of elasticity for stresses is found. The stress tensor is expressed in both coordinate systems in terms of three harmonic functions. The general solution of the problem of elasticity in cylindrical coordinates is presented as an example. The three-dimensional stress–strain state of an elastic cylinder subjected, on the lateral surface, to arbitrary forces represented by a series of eigenfunctions is determined. An axisymmetric problem for a finite cylinder is solved numerically     

Dependence of linear elasticity solutions on the elastic constants

Necessary and sufficient conditions on the dilatation for the displacement and/or the stress to be independent of Poisson's ratio in the standard boundary-value problems of three-dimensional classical elastostatics are determined. Formulas for the volume averages of the changes induced in the stress and strain by a variation of Poisson's ratio are also given.

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Résumé On détermine des conditions nécessaires et suffisantes sur le changement de volume pour que les déplacements et (ou) les contraintes soient indépendants du coefficient de Poisson dans les problèmes des conditions des limites classiques de l'élasticité tri-dimensionnelle. On donne également des formules pour les moyennes voluminiques des changements subis par les contraintes et les déformations lors d'une variation du coefficient de Poisson.

     

The representation problem of constrained linear elasticity

The title problem is given the following explicit solution: = _D S|D, where the elasticity tensor in the constrained case is the restriction to the constraint subspace D of a corresponding unconstrained elasticity tensor S, followed by composition with the orthogonal projection _D on D.     

Method of integro-differential relations in linear elasticity

Boundary-value problems in linear elasticity can be solved by a method based on introducing integral relations between the components of the stress and strain tensors. The original problem is reduced to the minimization problem for a nonnegative functional of the unknown displacement and stress functions under some differential constraints. We state and justify a variational principle that implies the minimum principles for the potential and additional energy under certain boundary conditions and obtain two-sided energy estimates for the exact solutions. We use the proposed approach to develop a numerical analytic algorithm for determining piecewise polynomial approximations to the functions under study. For the problems on the extension of a free plate made of two different materials and bending of a clamped rectangular plate on an elastic support, we carry out numerical simulation and analyze the results obtained by the method of integro-differential relations.     

The minimal error method for the Cauchy problem in linear elasticity. Numerical implementation for two-dimensional homogeneous isotropic linear elasticity

In this paper, yet another iterative procedure, namely the minimal error method (MEM), for solving stably the Cauchy problem in linear elasticity is introduced and investigated. Furthermore, this method is compared with another two iterative algorithms, i.e. the conjugate gradient (CGM) and Landweber–Fridman methods (LFM), previously proposed by Marin et al. [Marin, L., Háo, D.N., Lesnic, D., 2002b. Conjugate gradient-boundary element method for the Cauchy problem in elasticity. Quarterly Journal of Mechanics and Applied Mathematics 55, 227–247] and Marin and Lesnic [Marin, L., Lesnic, D., 2005. Boundary element-Landweber method for the Cauchy problem in linear elasticity. IMA Journal of Applied Mathematics 18, 817–825], respectively, in the case of two-dimensional homogeneous isotropic linear elasticity. The inverse problem analysed in this paper is regularized by providing an efficient stopping criterion that ceases the iterative process in order to retrieve stable numerical solutions. The numerical implementation of the aforementioned iterative algorithms is realized by employing the boundary element method (BEM) for two-dimensional homogeneous isotropic linear elastic materials.     

Measurement of rheological properties of soft biological tissue with a novel torsional resonator device

A new device for measuring the rheological properties of soft biological tissues is presented. The mechanical response is characterized for harmonic shear deformations at high frequencies (up to 10 kHz) and small strains (up to 0.2%). Experiments are performed using a cylindrical rod excited to torsional resonance. One extremity of the rod is in contact with the soft tissue and adherence is ensured by vacuum clamping. The damping characteristics and the resonance frequency of the vibrating system are inferred from the control variables of a phase stabilization loop. Due to the contact with the soft tissue, and depending on the rheological properties of the tissue, changes occur in the Q-factor and in the resonance frequency of the system. The shear modulus of the soft tissue is determined from the experimental results with an analytical model. The reliability of the proposed technique is evaluated through repeatability tests and comparative measurements with synthetic materials. The results of measurements on bovine organs demonstrate the suitability of the experimental procedure for the characterization of biological tissues and provide some insight in their rheological properties at frequencies in the range 1–10 kHz.     

Affine transformations of the equations of the linear theory of elasticity

This paper deals with the general formulas of affine transformations that preserve invariance of the static equations of the linear theory of elasticity in the case of arbitrary anisotropic materials. The invariance of the equations with respect to affine transformations allows one to model a given anisotropic material by another material. All anisotropic materials are divided into classes of mutually congruent materials. The congruency conditions are obtained for orthotropic and isotropic materials and for orthotropic and transversely isotropic materials. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 4, pp. 124–134, July–August, 2006.     

Optimization of the strain energy density in linear anisotropic elasticity

The problem considered here is that of extremizing the strain energy density of a linear anisotropic material by varying the relative orientation between a fixed stress state and a fixed material symmetry. It is shown that the principal axes of stress must coincide with the principal axes of strain in order to minimize or maximize the strain energy density in this situation. Specific conditions for maxima and minima are obtained. These conditions involve the stress state and the elastic constants. It is shown that the symmetry coordinate system of cubic symmetry is the only situation in linear anisotropic elasticity for which a strain energy density extremum can exist for all stress states. The conditions for the extrema of the strain energy density for transversely isotropic and orthotropic materials with respect to uniaxial normal stress states are obtained and illustrated with data on the elastic constants of some composite materials. Not surprisingly, the results show that a uniaxial normal stress in the grain direction in wood minimizes the strain energy in the set of all uniaxial stress states. These extrema are of interest in structural and material optimization.