On the effective behavior of nonlinear inelastic composites: I. Incremental variational principles

A new method for determining the overall behavior of composite materials comprised of nonlinear inelastic constituents is presented. Upon use of an implicit time-discretization scheme, the evolution equations describing the constitutive behavior of the phases can be reduced to the minimization of an incremental energy function. This minimization problem is rigorously equivalent to a nonlinear thermoelastic problem with a transformation strain which is a nonuniform field (not even uniform within the phases). In this first part of the study the variational technique of Ponte Castañeda is used to approximate the nonuniform eigenstrains by piecewise uniform eigenstrains and to linearize the nonlinear thermoelastic problem. The resulting problem is amenable to simpler calculations and analytical results for appropriate microstructures can be obtained. The accuracy of the proposed scheme is assessed by comparison of the method with exact results.     

Estimates for the overall linear properties of pointwise heterogeneous solids with application to elasto-viscoplasticity

New estimates are derived for the overall properties of linear solids with pointwise heterogeneous local properties. The derivation relies on the use of ‘comparison solids’ which, unlike comparison solids considered previously, are themselves pointwise heterogeneous. The estimates are then exploited within an incremental homogenization scheme to determine the overall response of multiphase elasto-viscoplastic solids under arbitrary loading histories. By way of example, the scheme is applied to incompressible Maxwellian solids with power-law plastic dissipation; particularly simple estimates of the Hashin–Shtrikman type are obtained. Predictions are confronted with full-field simulations for particulate composites under cyclic and rotating loading conditions. Good agreement is found for all cases considered. In particular, elasto-plastic transitions, tension-compression asymmetries (Bauschinger effect) and stress-path distortions induced by material heterogeneity are all well-captured, thus improving significantly on commonly used elastic-plastic decoupled schemes.     

Elasto-viscoplastic behaviour of a polyvinylidene fluoride (PVDF) in tension

The present paper is concerned with the plasticity of a polyvinylidene fluoride (PVDF) in tension. Strain rate strongly influences the plastic behaviour, but the variation of the elastic properties is almost negligible within the range of strain rates considered in the study (from 1.6 × 10⁻⁴ s⁻¹ up to 1.6 × 10⁻¹ s⁻¹). In particular, the yield stress and the ultimate tensile strength are strongly rate-dependent. A one-dimensional elasto-viscoplastic phenomenological model is proposed and analysed. Despite the nonlinearity of the model equations, only one tensile test performed with variable strain rate is sufficient to identify all material parameters. Model predictions are compared with experiments showing good agreement.     

Analysis of the cyclic tensile behaviour of an elasto-viscoplastic polyamide

The present paper is concerned with the experimental and theoretical investigation of the progressive accumulation of inelastic deformation observed in cyclic tension tests performed on a particular polyamide. The elastic properties are not strongly affected by the strain rate, but the strain hardening induced by the plastic deformation is rate-dependent. Thus, the material behaviour is elasto-viscoplastic rather than viscoelastic or elasto-plastic. For the polymer studied in this paper, the kinematic hardening is much more significant than the isotropic hardening, and a negative plastic strain rate may occur even with a positive stress. The kinematic hardening is strongly dependent, not only on the accumulated plastic strain, but also on the loading rate. An elasto-viscoplastic mechanical model able to describe the cyclic inelastic behaviour for an arbitrary loading history is proposed. All parameters that arise in the theory are identified experimentally. The preliminary theoretical results concerning the modelling of cyclic load-unload tests are in good agreement with the experiments.     

Temperature and variable strain rate sensitivity in recycled HDPE

The recycling of post-consumer plastics and their utilization as raw materials to develop value-added products has become an important goal worldwide. The present work is concerned with the thermo-mechanical analysis of recycled high-density polyethylene (HDPE) under uniaxial tensile loading. The main focus is to propose a one-dimensional phenomenological model able to describe the influence of temperature and strain rate on the mechanical behavior. Tensile tests were performed over a wide range of temperatures (from 25°C to 100°C). Each experiment was performed under controlled strain rate varying from 7.25 × 10⁻⁵ s⁻¹ to 7.25 × 10⁻³ s⁻¹ in steps. It is shown that only one tensile test performed at three different temperatures is necessary to fully identify experimentally all material parameters that arise in the theory. Thus, with this experimental procedure, the number of tests used to evaluate the mechanical properties of recycled HDPE is significantly reduced. The experiments are compared with the model predictions and show good agreement.     

Proper Generalized Decomposition (PGD) for the numerical simulation of polycrystalline aggregates under cyclic loading

The numerical modelling of the behaviour of materials at the microstructural scale has been greatly developed over the last two decades. Unfortunately, conventional resolution methods cannot simulate polycrystalline aggregates beyond tens of loading cycles, and they do not remain quantitative due to the plasticity behaviour. This work presents the development of a numerical solver for the resolution of the Finite Element modelling of polycrystalline aggregates subjected to cyclic mechanical loading. The method is based on two concepts. The first one consists in maintaining a constant stiffness matrix. The second uses a time/space model reduction method. In order to analyse the applicability and the performance of the use of a space–time separated representation, the simulations are carried out on a three-dimensional polycrystalline aggregate under cyclic loading. Different numbers of elements per grain and two time increments per cycle are investigated. The results show a significant CPU time saving while maintaining good precision. Moreover, increasing the number of elements and the number of time increments per cycle, the model reduction method is faster than the standard solver.     

On the effective behavior of nonlinear inelastic composites: II: A second-order procedure

A new method for determining the overall behavior of composite materials comprising nonlinear viscoelastic and elasto-viscoplastic constituents is presented. Part I of this work showed that upon use of an implicit time-discretization scheme, the evolution equations describing the constitutive behavior of the phases can be reduced to the minimization of an incremental energy function. This minimization problem is rigorously equivalent to a nonlinear thermoelastic problem with a transformation strain which is a nonuniform field (not even uniform within the phases). In part I of this paper the nonlinearity was handled using a variational (or secant) technique. In this second part of the study, a proper modification of the second-order procedure of Ponte Castañeda is proposed and leads to replacing, at each time-step, the actual nonlinear viscoelastic composite by a linear viscoelastic one. The linearized problem is even further simplified by using an “effective internal variable” in each individual phase. The resulting predictions are in good agreement with exact results and improve on the predictions of the secant model proposed in part I of this paper.     

Mean-field homogenization of elasto-viscoplastic composites based on a general incrementally affine linearization method

We propose a general formulation – which we believe to be new – for the mean-field homogenization of inclusion-reinforced elasto-viscoplastic composites assuming small strains. Our proposal is based on an interplay between constitutive equations and numerical algorithms, and the key ideas behind it are the following. The evolution equations for inelastic strain and internal variables at the beginning of each time interval are linearized around the ending time of the same interval. The linearized equations are then numerically integrated using a fully implicit backward Euler scheme. The obtained algebraic equations lead to an incrementally affine stress–strain relation which involves two important terms. The first one is the algorithmic tangent operator, obtained by consistent linearization of the time discretized constitutive equations. The second term is a new one and called an affine strain increment. The proposal leads to thermoelastic-like relations directly in the time domain, and not in the Laplace–Carson (L–C) one. There is no need for viscoelastic-type integral rewriting of the evolution equations, for L–C transforms, or for numerical inversion back from L–C to time domains. The proposed method can be readily applied to sophisticated elasto-viscoplastic models with an arbitrary set of scalar or tensor internal variables, and is valid for multi-axial, non-monotonic and non-proportional loading histories. The theory is applied in detail to a well-known constitutive model, and verified against finite element simulations of representative volume elements or unit cells, for a number of composite materials.