Hypoelastic Relation Research Articles

Family of Continuous Strain-Consistent Convective Tensor Rates and Its Application in Hooke-Like Isotropic Hypoelasticity

This paper presents a new family of objective (Lagrangian and Eulerian) continuous strain-consistent convective (corotational and non-corotational) tensor rates. Objective strain-consistent convective tensor rates are defined as having the property that there exist objective (Lagrangian and Eulerian) strain tensors from the Hill family such that the considered rates of these strain tensors are equal to the rotated/standard (Lagrangian/Eulerian) stretching (strain rate) tensors. The family of such Eulerian strain-consistent convective tensor rates was introduced by Bruhns et al. (Proc. R. Soc. Lond. A 460:909–928, 2004). On the one hand, the family of continuous strain-consistent convective tensor rates presented in this paper extends the Bruhns et al. family by including Lagrangian tensor rates and, on the other hand, it is narrower than the Bruhns et al. family due to the continuity requirement imposed on tensor rates, which is necessary for applications. The theorem that any strain tensors only from the Doyle–Ericksen family (which is a subfamily of the Hill family) provide sufficient continuity conditions for strain-consistent convective tensor rates was formulated and proved. The expressions obtained by proving this theorem for convector tensors show that each tensor from the Doyle–Ericksen family is associated with a single strain-consistent convective tensor rate (in the Bruhns et al. family, any strain tensor from the Hill family generates infinitely many convector tensors associated with infinitely many strain-consistent convective tensor rates). In addition, a new family of Hooke-like isotropic hypoelastic material models based on objective continuous strain-consistent convective rates of the rotated/standard (Lagrangian and Eulerian) Kirchhoff stress tensors was constructed. The theorem that any material model from this family is self-consistent provided that the Lame parameter $\lambda =0$ and/or the deformation of the body is isochoric was formulated and proved. By self-consistent hypoelastic material models are meant those models for which constitutive hypoelastic relations are counterparts of constitutive relations for Cauchy/Green elasticity. Some material models from the new family were tested by solving the simple shear problem. Both new and well-known solutions of this problem for material models were obtained using an approach that takes into account the self-consistency property of material models from the new family.

On instability of constitutive models for isotropic elastic-perfectly plastic material behaviour at finite deformations

The main goal of this work is to test the possibility of a newly introduced constitutive law to model the behaviour of the isotropic elastic-perfectly plastic material which is exposed to large elastoplastic deformations. The proposed constitutive relation is based on the hypo-elastic relation and the inelastic INTERATOM model. The verification of the model is done by its implementation into the commercial software ABAQUS/Standard via the user subroutine UMAT. For that purpose, the large simple shear problem is studied where selected objective corotational rates, i.e. the logarithmic rate, the Jaumann rate and the Green-Naghdi rate, are individually implemented in the aforementioned constitutive relations. The obtained results are compared mutually and with the relevant literature. The proposed constitutive model is also used to test the behaviour of the part of a real engineering structure, i.e. a seismic isolator, in order to obtain the correct input data for further analysis of superstructure behaviour due to seismic excitation.

An elasto-viscoplastic model to describe the ratcheting behavior of articular cartilage.

In the present work, a constitutive model for articular cartilage is proposed in finite elasto-viscoplasticity. For simplification, articular cartilage is supposed to be a typical composite composed of a soft basis and a fiber assembly. The stress tensor and free energy function are hence accordingly divided into two components. The high nonlinear stress-strain response is assumed to be mainly related to the fiber assembly and described by an exponential-type hypoelastic relation. Ratcheting is considered according to the viscoplasticity, the evolution rule of which is deduced from the dissipative inequality by the co-directionality hypotheses. Then, the capability of the proposed model is validated by comparing its predictions with related experimental observations. Results show that the ratcheting behavior and stress-strain hysteresis loops are reasonably captured by the proposed model.

Logarithmic rate based elasto-viscoplastic cyclic constitutive model for soft biological tissues

Based on the logarithmic rate and piecewise linearization theory, a thermodynamically consistent elasto-viscoplastic constitutive model is developed in the framework of finite deformations to describe the nonlinear time-dependent biomechanical performances of soft biological tissues, such as nonlinear anisotropic monotonic stress–strain responses, stress relaxation, creep and ratchetting. In the proposed model, the soft biological tissue is assumed as a typical composites consisting of an isotropic matrix and anisotropic fiber aggregation. Accordingly, the free energy function and stress tensor are divided into two parts related to the matrix and fiber aggregation, respectively. The nonlinear biomechanical responses of the tissues are described by the piecewise linearization theory with hypo-elastic relations of fiber aggregation. The evolution equations of viscoplasticity are formulated from the dissipation inequalities by the co-directionality hypotheses. The anisotropy is considered in the hypo-elastic relations and viscoplastic flow rules by introducing some material parameters dependent on the loading direction. Then the capability of the proposed model to describe the nonlinear time-dependent deformation of soft biological tissues is verified by comparing the predictions with the corresponding experimental results of three tissues. It is seen that the predicted monotonic stress–strain responses, stress relaxation, creep and ratchetting of soft biological tissues are in good agreement with the corresponding experimental ones.

Particle-based interaction simulation of fluids and deformable bodies

In this paper, a novel particle model for simulating incompressible inviscid fluids and deformable solids is presented. In this model both fluids and solids are described by the Smoothed Particle Hydrodynamics: fluid dynamics is studied in the weakly compressible approximation, while solid dynamics is simulated through an incremental hypoelastic relation. Fluids and solids are characterized by the same formulations, thus the need to define an interface for coupling disparate fluids and solids models is eliminated, and the interaction between them can be achieved naturally without the complex coupling interface. The model has been tested with several examples, and the experimental results show the realistic aspects of coupling fluids and solid motion.

Bifurcation of elastoplastic pressure-sensitive spheres

Instability of a full sphere, under uniform compression or tension, is investigated within the framework of linear bifurcation theory. Material response is modeled by a Hookean-type hypoelastic relation with pressure dependent instantaneous moduli. Exploiting a formal analogy with Navier equations of linear elasticity we obtain exact solution for the bifurcated field. Two families of eigenmodes and eigenvalues are identified and discussed. We examine, in particular, surface instabilities in compression, and twisting modes with absence of radial velocity. The results are further specified for pressure-sensitive plastic solids with an elliptic yield surface. The deformation theory prediction of bifurcation loads for that material are much lower than those obtained from the flow theory version.

Numerical simulation of fluid–structure interaction by SPH

A Lagrangian model for the numerical simulation of fluid–structure interaction problems is proposed in the present paper. In the method both fluid and solid phases are described by smoothing particle hydrodynamics: fluid dynamics is studied in the inviscid approximation, while solid dynamics is simulated through an incremental hypoelastic relation. The interface condition between fluid and solid is enforced by a suitable term, obtained by an approximate SPH evaluation of a surface integral of fluid pressure. The method is validated by comparing numerical results with laboratory experiments where an elastic plate is deformed under the effect of a rapidly varying fluid flow.

Objective stress rates in repeated elastic deformation cycles

AbstractThe Eulerian elastoplastic description of initially isotropic materials essentially depends on the accurate formulation of the elastic deformation. This is often represented by Truesdell's hypoelastic equation, which involves an objective stress rate. Many objective stress rates have been presented in the past. With Bernstein's integrability conditions it has been shown that only a particular rate, namely the logarithmic stress rate, makes the hypoelastic relation exactly integrable.Here, we propose a simple, one‐parameter closed deformation cycle to demonstrate the error which will considerably accumulate over several cycles, whenever a non‐appropriate stress rate is in use. For this, several objective stress rates are compared in this cycle and the residual stresses at the end of the cycles are examined. By such a simple procedure the mathematical results of non‐integrability are evidenced by example. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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.

A Finite Deformation Theory of Viscoplasticity Based on Overstress: Part I—Constitutive Equations

The viscoplasticity theory based on overstress (VBO) is extended to finite deformation (FVBO). Yield surfaces and loading/unloading conditions are not part of this theory which represents creep, relaxation, and rate sensitivity in a “unified” way. Additive decomposition of the rate of deformation into the elastic and the inelastic parts is assumed. For the elastic part, the hypoelastic relation is used. For the inelastic part, the flow law of VBO is augmented by a term quadratic in the overstress together with a modified Jaumann stress rate which jointly or separately allow the modeling of second-order effects.

Hypoelastic form of equations in nonlinear elasticity theory

It is shown that the complete system of equations of elasticity theory for an isotropic medium admits a unique representation in the hypoelastic form (the tensor of the rate of change of stresses is a linear function of the tensor of strain rates with coefficients depending on the invariants of the stress tensor). It is necessary to this end that the hypothesis be satisfied on the determination of strains by stresses which are unknown. Any arbitrariness in the choice of the coefficients of the hypoelastic relation may result in the thermodynamic identity being infringed.