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The present paper is devoted to the construction and comparative study of upwind methods as applied to the system of one-dimensional non-linear elasticity equations with particular attention to robustness and accurate resolution of delicate features such as linearly degenerate fields. Copyright © 2007 John Wiley & Sons, Ltd.

The present work is devoted to the damped Newton method applied for solving a class of non-linear elasticity problems. Following the approach suggested in earlier related publications, we consider a two-level procedure which involves (i) solving the non-linear problem on a coarse mesh, (ii) interpolating the coarse-mesh solution to the fine mesh, (iii) performing non-linear iterations on the fine mesh. Numerical experiments suggest that in the case when one is interested in the minimization of the L2-norm of the error rather than in the minimization of the residual norm the coarse-mesh solution gives sufficiently accurate approximation to the displacement field on the fine mesh, and only a few (or even just one) of the costly non-linear iterations on the fine mesh are needed to achieve an acceptable accuracy of the solution (the accuracy which is of the same order as the accuracy of the Galerkin solution on the fine mesh). Copyright © 2000 John Wiley & Sons, Ltd.

In the present contribution, an innovative stabilization technique for two-dimensional low-order finite elements is presented. The new approach results in an element formulation that is much simpler than the recently proposed enhanced strain element formulation, yet which gives results of at least the same quality. An important feature in the regime of large deformations is the stability of the element, which is addressed in detail. The main advantages of the new formulation are, besides its simplicity, its computational efficiency and robust behaviour. Only three history variables have to be stored, making this stabilization concept particularly interesting for large-scale problems. Copyright © 1999 John Wiley & Sons, Ltd.

An empirically established rule of Wroth1 for the dependence of the shear modulus on the mean effective pressure and the overconsolidation ratio in clays is investigated within the framework of non-linear elasticity. The resulting isotropic-deviatoric coupling is derived and compared to experiments.

All fields have piecewise linear approximants over triangles except pressure, which is taken to be constant over matched pairs of triangles. The code is used in an investigation of the tilting mode of deformation of bonded rubber blocks

Key points The force in the half-sarcomere (hs), the functional unit of muscle, is due to the contributions of individual myosin motors arranged in parallel in the half-myosin filament and pulling on the opposing actin filament. According to a linear hs model, during an isometric contraction the force rises to its maximal steady value (T0) in proportion to the number of actin-attached motors, while the hs strain rises with a slope that depends on the compliance of the myofilaments. We measured the hs stiffness, superimposing small 4 kHz length oscillations on the development of isometric contraction, and found an elastic element in parallel to the myosin motors with a constant stiffness ∼1/20th that of the motor array at T0. The results support a structural model in which myosin motors are distributed in multiple substates, of which only the first ones are occupied during isometric force generation, causing a motor strain of ∼1.7 nm. Abstract Force in striated muscle is due to attachment of the heads of the myosin, the molecular motors extending from the myosin filament, to the actin filament in each half-sarcomere, the functional unit where myosin motors act in parallel. Mechanical and X-ray structural evidence indicates that at the plateau of isometric contraction (force T0), less than half of the elastic strain of the half-sarcomere is due to the strain in the array of myosin motors (s), with the remainder being accounted for by the compliance of filaments acting as linear elastic elements in series with the motor array. Early during the development of isometric force, however, the half-sarcomere compliance has been found to be less than that expected from the linear elastic model assumed above, and this non-linearity may affect the estimate of s. This question is investigated here by applying nanometre–microsecond-resolution mechanics to single intact fibres from frog skeletal muscle at 4°C, to record the mechanical properties of the half-sarcomere throughout the development of force in isometric contraction. The results are interpreted with mechanical models to estimate the compliance of the myosin motors. Our conclusions are as follows: (i) early during the development of an isometric tetanus, an elastic element is present in parallel with the myosin motors, with a compliance of ∼200 nm MPa−1 (∼20 times larger than the compliance of the motor array at T0); and (ii) during isometric contraction, s is 1.66 ± 0.05 nm, which is not significantly different from the value estimated with the linear elastic model.

This paper presents the formulation and a partial analysis of a class of discontinuous Galerkin methods for quasistatic non-linear elasticity problems. These methods are endowed with several salient features. The equations that define the numerical scheme are the Euler–Lagrange equations of a one-field variational principle, a trait that provides an elegant and simple derivation of the method. In consonance with general discontinuous Galerkin formulations, it is possible within this framework to choose different numerical fluxes. Numerical evidence suggests the absence of locking at near-incompressible conditions in the finite deformations regime when piecewise linear elements are adopted. Finally, a conceivable surprising characteristic is that, as demonstrated with numerical examples, these methods provide a given accuracy level for a comparable, and often lower, computational cost than conforming formulations. Stabilization is occasionally needed for discontinuous Galerkin methods in linear elliptic problems. In this paper we propose a sufficient condition for the stability of each linearized non-linear elastic problem that naturally includes material and geometric parameters; the latter needed to account for buckling. We then prove that when a similar condition is satisfied by the discrete problem, the method provides stable linearized deformed configurations upon the addition of a standard stabilization term. We conclude by discussing the complexity of the implementation, and propose a computationally efficient approach that avoids looping over both elements and element faces. Several numerical examples are then presented in two and three dimensions that illustrate the performance of a selected discontinuous Galerkin method within the class. Copyright © 2006 John Wiley & Sons, Ltd.

An iterative solver is proposed to solve the family of linear equations arising from the numerical computation of non-linear problems. This solver relies on two quantities coming from previous steps of the computations: the preconditioning matrix is a matrix that has been factorized at an earlier step and previously computed vectors yield a reduced basis. The principle is to define an increment in two sub-steps. In the first sub-step, only the projection of the unknown on a reduced subspace is incremented and the projection of the equation on the reduced subspace is satisfied exactly. In the second sub-step, the full equation is solved approximately with the help of the preconditioner. Last, the convergence of the sequences is accelerated by a well-known method, the modified minimal polynomial extrapolation. This algorithm assessed by classical benchmarks coming from shell buckling analysis. Finally, its insertion in path following techniques is discussed. This leads to non-linear solvers with few matrix factorizations and few iterations. Copyright © 2009 John Wiley & Sons, Ltd.

A large strain analysis of undrained expansion of a spherical/cylindrical cavity in a soil modelled as non-linear elastic modified Cam clay material is presented. The stress–strain response of the soil is assumed to obey non-linear elasticity until yielding. A power-law characteristic or a hyperbolic stress–strain curve is used to describe the gradual reduction of soil stiffness with shear strain. It is assumed that, after yielding, the elasto-plastic behaviour of the soil can be described by the modified Cam clay model. Based on a closed-form stress–strain response in undrained condition, a numerical solution is obtained with the aid of simple numerical integration technique. The results show that the stresses and the pore pressure in the soil around an expanded cavity are significantly affected by the non-linear elasticity, especially if the soil is overconsolidated. The difference between large strain and small strain solutions in the elastic zone is not significant. The stresses and the pore pressure at the cavity wall can be expressed as an approximate closed-form solution. Copyright © 2001 John Wiley & Sons, Ltd.

The debonding of pressure-sensitive-adhesives from a rigid surface often occurs with the formation of a fibrillar structure bridging the surface and the adhesive. This fibrillar structure is responsible for the large (of the order of 1 kJ/m 2 ) adhesion energy, which is measured, for this type of adhesive despite the very low level of applied stress. One way to account for this dissipated energy is viscous flow in the elongation process of the fibrils. However, for certain types of adhesives, the extension of the fibrils is essentially a quasi reversible elastic process and the energy is only dissipated rather rapidly when the fibrils are detached from the surface. It is therefore essential to characterize the elastic properties of theses adhesives not only in the linear viscoelastic regime but also in the large strain non-linear elastic domain. We have therefore performed tensile tests of model adhesive films based on block copolymers. The respective roles played by the linear viscoelasticity and non-linear elasticity in controlling the properties will be discussed.

An approximate Riemann solver is developed for the equations of non-linear elasticity in a heterogeneous medium, where each grid cell has an associated density and stress-strain relation. The non-linear flux function is spatially varying and a wave decomposition of the flux difference across a cell interface is used to approximate the wave structure of the Riemann solution. This solver is used in conjunction with a high-resolution finite-volume method using the CLAWPACK software. As a test problem, elastic waves in a periodic layered medium are studied. Dispersive effects from the heterogeneity, combined with the non-linearity, lead to solitary wave solutions that are well captured by the numerical method.

We investigate the characteristics and performance of goal-oriented a posteriori error measures for a class of non-linear elasticity models, while restriction is made to small strain theory. The chosen error measure of the displacement field can be global or local (probing the chosen quantity in a specific spatial point). The error is computable with the aid of the solution of a dual problem whose data depend on the error measure. The main thrust of the paper is to evaluate the performance of a few different approximation strategies for computing the dual solution. The chosen strategies are compared in terms of accuracy, ease of implementation, reliability and cost-efficiency. A well-known numerical example, the Cook's membrane, is used for the numerical evaluations.

The paper presents a formulation for geometrically non-linear anisotropic elastic material behaviour. To this end the space of symmetric second order tensors related to the undeformed reference configuration is decomposed into invariant subspaces characterizing the symmetry conditions of the specific material. The subspaces are equipped with tensor bases, which have been determined in the literature for several material classes, such as cubic symmetry, transverse isotropy and complete isotropy. Upon formulating a free energy function in terms of generalized Seth–Hill strain tensors projected onto the subspaces, the symmetry conditions characterizing the specific material are satisfied a priori. From standard results of continuum mechanics the corresponding Lagrangian stress tensor and elastic moduli are obtained by straightforward application of the chain rule. Representative examples in one-dimensional loading illustrate the material behaviour of cubic symmetry for elastic materials. Copyright © 2005 John Wiley & Sons, Ltd.

A class of static (elliptic) optimal control problems in non-linear elasticity is considered. A new direct finite element formulation is used which results in a sequence of quadratic programming (QP) problems. The latter are solved using a standard QP algorithm. The method is applied to the problems of minimizing the lateral deflection of a membrane on a nonlinear elastic foundation and minimizing the in-plane large deformation of a hyperelastic plate.

In elastomechanics there exist besides of the classical variational principles so-called generalized (extended, mixed) ones which are used mainly for the discretization of boundary and eigenvalue problems. These principles can be constructed together with their incremental versions systematically from the canonical basic equations characterized by four special operators (statics, kinematics, material, loading), and the boundary- and (eventually) transition conditions. Under certain assumptions they can be strengthened to complementary extremum principles. The proposed abstract and unified formulation holds true for all kinds of elastic structures and, additionally, for related problems and facilitates the construction of the corresponding variational principles. At last the relevance of drilling degrees of freedom is discussed.

In this paper, the relation between the incremental right stretch tensor by the total Lagrangian (T.L.) description and that by the updated Lagrangian (U.L.) description is obtained. The incremental polar decomposition method without drilling degrees of freedom is developed. The non-linear shell elements using the Biot strain tensor as the strain measure are established. The non-linear finite element formulation based on the polar composition theorem is developed with respect to the U.L. description. As a by-product the material time derivative of the right stretch tensor is given by a new form suitable for numerical evaluations. Two examples demonstrate the feasibility of the proposed incremental polar decomposition method.

This paper proposes and analyzes an adaptive stabilization strategy for enhanced strain (ES) methods applied to quasistatic non-linear elasticity problems. The approach is formulated for any type of enhancements or material models, and it is distinguished by the fact that the stabilization term is solution dependent. The stabilization strategy is first constructed for general linearized elasticity problems, and then extended to the non-linear elastic regime via an incremental variational principle. A heuristic choice of the stabilization parameters is proposed, which in the numerical examples proved to provide stable approximations for a large range of deformations, different problems and material models. We also provide explicit lower bounds for the stabilization parameters that guarantee that the method will be stable. These are not advocated, since they are generally larger than the ones based on heuristics, and hence prone to deteriorate the locking-free behavior of ES methods. Numerical examples with two different non-linear elastic models in thin geometries and incompressible situations show that the method remains stable and locking free over a large range of deformations. Finally, the method is strongly based on earlier developments for discontinuous Galerkin methods, and hence throughout the paper we offer a perspective about the similarities between the two.

Using Ball's approach to non-linear elasticity, and in particular his concept of polyconvexity, we treat a unilateral three-dimensional contact problem for a hyperelastic body under volume and surface forces. Here the unilateral constraint is described by a sublinear function which can model the contact with a rigid convex cone. We obtain a solution to this generally non-convex, semicoercive Signorinin problem as a limit of solutions of related energy minimization problems involving friction normal to the contact surface where the friction coefficient goes to infinity. Thus we extend an approximation result of Duvaut and Lions for linear-elastic unilateral contact problems to finite deformations and to a class of non-linear elastic materials including the material models of Ogden and of Mooney-Rivlin for rubberlike materials. Moreover, the underlying penalty method is shown to be exact, that is a sufficiently large friction coefficient in the auxiliary energy minimization problems suffices to produce a solution of the original unilateral problem, provided a Lagrange multiplier to the unilateral constraint exists.

Starting from the general framework of isotropic nonlinear elasticity, a simple isotropic nonlinear elastic model has been developed to describe the behaviour of granular materials immediately after stress reversal. Its three parameters can be determined from the same drained triaxial test. Two parameters follow from the behaviour in isotropic unloading; the third parameter is the ratio of the shear modulus in deviatoric unloading and the bulk modulus for isotropic unloading. This shear modulus is measured most easily in deviatoric unloading at constant isotropic stress. (Author/TRRL)