Hyperelastic Formulation Research Articles

Three-dimensional nonlinear open-cell foams with large deformations

In this article, a hyperelastic formulation for light and compliant foams which accounts for nonlinear effects at material and kinematic levels is introduced. This theory is applicable to a large number of 2- and 3-D irregular open-cell structures. An expression for the strain-energy function is proposed which includes bending and stretching contributions. Although this description allows for irregularity in the structure at local or cell level, it also assumes that the macro structure is homogeneous, i.e. built by repetition of the same irregular unit cell. This micromechanical formulation has explicit correlation with the foam structure, and therefore it preserves in the constitutive relation the symmetries or directional properties of the corresponding structures, including the cases of re-entrant foams which exhibit negative Poisson’s ratio effects. Due to the introduction of nonlinear kinematics, the evolution of the structure during the loading process and its effects on the constitutive behavior can be traced, including the cases where configurational transformations are present leading to non-convex strain-energy functions. Several examples of the stress–strain behavior for arbitrary large homogeneous deformations in a diamond-like structure are presented. The effect of the structure reorientation on the transition of local deformation mechanisms is clearly shown. The development of texture and anisotropy induced by the deformation process is also demonstrated. Finally, the role of the deformation mechanisms on the relation between foam stiffness and foam density is analyzed.

Estimation of in situ elastic properties of biphasic cartilage based on a transversely isotropic hypo-elastic model.

Articular cartilage is known to behave nonlinearly for large deformations. Mechanical properties derived from small strain experiments yield excessively large deformations in finite element models used in the study of severe blunt impact to joints. In this manuscript, a method is presented to determine the nonlinear elastic properties of biphasic cartilage based on a transversely isotropic hypo-elastic model. The elastic properties were estimated by fitting two force-displacement curves (in rapid loading and at equilibrium) obtained from large deformation indentation relaxation tests on cartilage using a nonporous spherical indentor. The solid skeleton of the cartilage was modeled as a transversely isotropic hypo-elastic material and a commercial finite element program was employed to solve the problem of a layer indented by a rigid sphere. Components of the hypo-elasticity tensor were made dependent on deformation according to the variations defined by a transversely isotropic hyperelastic formulation given earlier by others. Material incompressibility was assumed during the initial stage of rapid loading. The analysis was utilized for the determination of in situ properties of rabbit retropatellar cartilage at large deformations. The model was able to fit the material response to rapid loading and equilibrium indentation test data to approximately 50 percent strain. This material model suggested even higher percentage of stress supported by the fluid phase of cartilage than given earlier by small deformation theories of biphasic cartilage.

Self-consistent Eulerian rate type elasto-plasticity models based upon the logarithmic stress rate

The objective of this article is to suggest new Eulerian rate type constitutive models for isotropic finite deformation elastoplasticity with isotropic hardening, kinematic hardening and combined isotropic-kinematic hardening etc. The main novelty of the suggested models is the use of the newly discovered logarithmic stress rate and the incorporation of a simple, natural explicit integrable-exactly rate type formulation of general hyperelasticity. Each new model is thus subjected to no incompatibility of rate type formulation for elastic behaviour with the notion of elasticity, as encountered by any other existing Eulerian rate type model for elastoplasticity or hypoelasticity. As particular cases, new Prandtl-Reuss equations for elastic-perfect plasticity and elastoplasticity with isotropic hardening, kinematic hardening and combined isotropic-kinematic hardening, respectively, are presented for computational and practical purposes. Of them, the equations for kinematic hardening and combined isotropic–kinematic hardening are, respectively, reduced to three uncoupled equations with respect to the spherical stress component, the shifted stress and the back-stress. The effects of finite rotation on the current strain and stress and hardening behaviour are indicated in a clear and direct manner. As illustrations, finite simple shear responses for the proposed models are studied by means of numerical integration. Further, it is proved that, among all possible (infinitely many) objective Eulerian rate type models, the proposed models are not only the first, but unique, self-consistent models of their kinds, in the sense that the rate type equation used to represent elastic behaviour is exactly integrable to really deliver an elastic relation. ©

A Natural Generalization of Hypoelasticity and Eulerian Rate Type Formulation of Hyperelasticity

According to the classical hypoelasticity theory, the hypoelasticity tensor, i.e. the fourth order Eulerian constitutive tensor, characterizing the linear relationship between the stretching and an objective stress rate, is dependent on the current stress and must be isotropic. Although the classical hypoelasticity in this sense includes as a particular case the isotropic elasticity, it fails to incorporate any given type of anisotropic elasticity. This implies that one can formulate the isotropic elasticity as an integrable-exactly classical hypoelastic relation, whereas one can in no way do the same for any given type of anisotropic elasticity. A generalization of classical theory is available, which assumes that the material time derivative of the rotated stress is dependent on the rotated Cauchy stress, the rotated stretching and a Lagrangean spin, linear and of the first degree in the latter two. As compared with the original idea of classical hypoelasticity, perhaps the just-mentioned generalization might be somewhat drastic. In this article, we show that, merely replacing the isotropy property of the aforementioned stress-dependent hypoelasticity tensor with the invariance property of the latter under an R-rotating material symmetry group R⋆ G 0, one may establish a natural generalization of classical theory, which includes all of elasticity. Here R is the rotation tensor in the polar decomposition of the deformation gradient and G 0 any given initial material symmetry group. In particular, the classical case is recovered whenever the material symmetry is assumed to be isotropic. With the new generalization it is demonstrated that any two non-integrable hypoelastic relations based on any two objective stress rates predict quite different path-dependent responses in nature and hence can in no sense be equivalent. Thus, the non-integrable hypoelastic relations based on any given objective stress rate constitute an independent constitutive class in its own right which is disjoint with and hence distinguishes itself from any class based on another objective stress rate. Only for elasticity, equivalent hypoelastic formulations based on different stress rates may be established. Moreover, universal integrability conditions are derived for all kinds of objective corotational stress rates and for all types of material symmetry. Explicit, simple, integrable-exactly hypoelastic relations based on the newly discovered logarithmic stress rate are presented to characterize hyperelasticity with any given type of material symmetry. It is shown that, to achieve the latter goal, the logarithmic stress rate is the only choice among all infinitely many objective corotational stress rates.

Application of Reproducing Kernel Particle Method to Large Deformation Contact Analysis of Elastomers

Abstract A meshless formulation based on the Reproducing Kernel Particle Method (RKPM) is applied to the large deformation and contact analysis of elastomeric components. In this approach, a kernel estimate that ensures the completeness requirement of a linear field is introduced to construct the global Reproducing Kernel shape functions of the displacements. The Galerkin approximation of the variational equation is formulated using the Reproducing Kernel shape functions, and the discretization can be performed without the need of a structured mesh. In this paper, a RKPM formulation for rubber materials including frictional contact conditions is presented. The contact constraint equations and essential boundary conditions are formulated in the nodal coordinate using a direct transformation method. The global Reproducing Kernel shape functions expressed in a material description are used in the total Lagrangian formulation of hyperelasticity. By the use of the smooth Reproducing Kernel shape functions, the method is effective in dealing with large deformation and contact conditions in elastomeric applications.

Large-strain 3D-analysis of fibre-reinforced composites using rebar elements: hyperelastic formulations for cords

The well-known finite element representation of reinforcing bars by means of overlay (“rebar”) elements is recast in the context of finite strain analyses of cord-reinforced composite materials. The variational formulation including the linearized forms is presented on the basis of hyperelasticity. Three material laws including two variants of the Neo-Hooekean model and the quadratic logarithmic model are investigated. An explicit formulation for uniaxial stress states is given for the Neo-Hooekean model. A comparative evaluation with regards to computational efficiency and physical plausibility shows that the logarithmic model is optimally suited for this class of problems and for moderately large strains. The rebar-element concept in conjunction with an incompressible finite element formulation for the representation of a rubber matrix material is applied to comparative finite strain FE-analyses of a cord-reinforced rubber sandwich panel, with different hyperelastic models used for modelling of the ply material.

The mechanical behaviour of bony endplate and annulus in prolapsed disc configuration

The mechanical response of intervertebral joints is deeply influenced by disc degeneration. The phenomenon is expressed in terms of variations in the biomechanical properties of the material, whose compressibility characteristics change because of the liquid content loss in the tissue and, what is even more important, to prolapse. In this work, the problem is investigated by means of a computational mechanics approach; a coupled material and geometric non-linear model is developed, representing vertebra, annulus and nucleus submitted to an axial load. A transversely isotropic law is assumed for cortical bone in the vertebral body and an isotropic law for the cancellous portion; a hyperelastic formulation is assumed for the disc, allowing effective interpretation of the mechanical characteristics of degeneration. The results obtained are reported with regard to bony endplate and annulus behaviour; interaction phenomena between bony endplate and nucleus are emphasized.

Analytical Solutions for Circular Bars Subjected to Large Strain Plastic Torsion

The inelastic behavior of solid circular bars twisted to arbitrarily large strains is considered. Various phenomenological constitutive laws currently employed to model finite strain inelastic behavior are shown to lead to closed-form analytical solutions for torsion. These include rate-independent elastic-plastic isotropic hardening J2 flow theory of plasticity, various kinematic hardening models of flow theory, and both hypoelastic and hyperelastic formulations of J2 deformation theory. Certain rate-dependent inelastic laws, including creep and strain-rate sensitivity models, also permit the development of closed-form solutions. The derivation of these solutions is presented as well as numerous applications to a wide variety of time-independent and rate-dependent plastic constitutive laws.