A consistent equilibrium in a cross-section of an elastic–plastic beam

A phenomenon of inequality of equilibrium and constitutive internal forces in a cross-section of elastic–plastic beams is common to many finite element formulations. It is here discussed in a rate-independent, elastic–plastic beam context, and a possible treatment is presented. The starting point of our discussion is Reissners finite-strain beam theory, and its finite element implementation. The questions of the consistency of interpolations for displacements and rotations, and the related locking phenomena are fully avoided by considering the rotation function of the centroid axis of a beam as the only unknown function of the problem. Approximate equilibrium equations are derived by the use of the distribution theory in conjunction with the collocation method. The novelty of our formulation is an inclusion of a balance function that measures the error between the equilibrium and constitutive bending moments in a cross-section. An advantage of the present approach is that the locations, where the balance of equilibrium and constitutive moments should be satisfied, can be prescribed in advance. In order to minimize the error, explicit analytical expressions are used for the constitutive forces; for a rectangular cross-section and bilinear constitutive law, they are given in Appendix A. The comparison between the results of the two finite element formulations, the one using consistent, and the other inconsistent equilibrium in a cross-section, is shown for a cantilever beam subjected to a point load. The problem of high curvature gradients in a plastified region is also discussed and solved by using an adapted collocation method, in which the coordinate system is transformed such to follow high gradients of curvature.     

Shakedown theorems for elastic–plastic solids in the framework of gradient plasticity

Static and kinematic shakedown theorems are given for a class of generalized standard materials endowed with a hardening saturation surface in the framework of strain gradient plasticity. The so-called residual-based gradient plasticity theory is employed. The hardening law admits a hardening potential, which is a C¹-continuous function of a set of kinematic internal variables and of their spatial gradients, and is required to satisfy a global sign restriction (but not to be necessarily convex). The totally produced, the accumulated and the freely moving dislocations per unit volume, distinguished as statistically stored and geometrically necessary ones, are in this way accounted for. Like for a generalized standard material, the shakedown safety factor is found to depend on the (generally size dependent) yield and saturation limits, but not on the particular differential-type hardening law of the material.     

The surface-forming energy release rate based fracture criterion for elastic–plastic crack propagation

The J-integral based criterion is widely used in elastic–plastic fracture mechanics. However, it is not rigorously applicable when plastic unloading appears during crack propagation. One difficulty is that the energy density with plastic unloading in the J-integral cannot be defined unambiguously. In this paper, we alternatively start from the analysis on the power balance, and propose a surface-forming energy release rate (ERR), which represents the energy available for separating the crack surfaces during the crack propagation and excludes the loading-mode-dependent plastic dissipation. Therefore the surface-forming ERR based fracture criterion has wider applicability, including elastic–plastic crack propagation problems. Several formulae are derived for calculating the surface-forming ERR. From the most concise formula, it is interesting to note that the surface-forming ERR can be computed using only the stress and deformation of the current moment, and the definition of the energy density or work density is avoided. When an infinitesimal contour is chosen, the expression can be further simplified. For any fracture behaviors, the surface-forming ERR is proven to be path-independent, and the path-independence of its constituent term, so-called J_s-integral, is also investigated. The physical meanings and applicability of the proposed surface-forming ERR, traditional ERR, J_s-integral and J-integral are compared and discussed. Besides, we give an interpretation of Rice paradox by comparing the cohesive fracture model and the surface-forming ERR based fracture criterion.     

Effect of large elastic strains on cavitation instability predictions for elastic–plastic solids

For an infinite solid containing a void, the cavitation instability limit is defined as the remote stress–and strain state, at which the void grows without bound, driven by the elastic energy stored in the surrounding material. Such cavitation limits have been analysed by a number of authors for metal plasticity as well as for nonlinear elastic solids. The analyses for elastic–plastic solids are here extended to consider the effect of a large initial yield strain, and it is shown how the critical stress value decays for increasing value of the yield strain. Analyses are carried out for remote hydrostatic tension as well as for more general axisymmetric remote stress field, with an initially spherical void. Different levels of strain hardening are considered.     

Thermo-mechanical coupling strategies in elastic–plastic problems

Modeling strategies aimed at thermo-mechanical coupled problems has been developed for a wide range of engineering applications. Staggered-type coupling procedures have been largely used in materials processing operations, especially in commercial codes, owing to their simplicity and flexibility. The present work shows that, in thermo-plastic problems, the classical implementation of the most common coupling procedure may present accuracy issues and time-stepping dependency. Numerical experiments indicate that an iterative coupling scheme constitutes a viable and simple approach to this class of problems.     

Orthotropic elastic–plastic material model for paper materials

Paper and paperboard generally exhibit anisotropic and non-linear mechanical material behaviour. In this work, the development of an orthotropic elastic–plastic constitutive model, suitable for modelling of the material behaviour of paper is presented. The anisotropic material behaviour is introduced into the model by orthotropic elasticity and an isotropic plasticity equivalent transformation tensor. A parabolic stress–strain relation is adopted to describe the hardening of the material. The experimental and numerical procedures for evaluation of the required material parameters for the model are described. Uniaxial tensile testing in three different inplane material directions provides the calibration of the material parameters under plane stress conditions. The numerical implementation of the material model is presented and the model is shown to perform well in agreement with experimentally observed mechanical behaviour of paper.     

Spherical impression of thin elastic films on elastic–plastic substrates

The mechanical behavior of thin elastic films deposited onto structural alloys plays a critical role in determining film durability. This paper presents analysis of an impression experiment designed to evaluate some of the relevant properties of these films. The modeling provides quantitative strain information which can be used to estimate the fracture toughness of the film, the static friction coefficient of the surface and the constitutive behavior of the substrate. Results are presented for radial and circumferential strain distributions in the film relevant to the interpretation of cracking patterns. Additionally, load-displacement curves are provided that may be used to evaluate the plastic properties of the substrate. To facilitate estimates of the film cracking strain through correlation with experiments, the radial strain distributions are presented as functions of impression depth, yield strain and hardening exponent.     

Shakedown kinematic theorem for elastic–perfectly plastic bodies

The classical shakedown kinematic theorem due to Koiter for elastic–perfectly plastic bodies is re-examined and divided into separated shakedown and nonshakedown theorems. While the shakedown theorem is based on the set of Koiter's plastic strain rate cycles, the non-shakedown one involves a broader set of admissible plastic strain rate cycles, the end-cycle accumulated strains of which are deviatoric parts of compatible strain fields. For certain broad classes of practical problems the two statements are unified to yield the unique theorem in Koiter's sense.     

Dual finite element methods in homogenization for elastic–plastic fibrous composite material

The problem of homogenization for a periodic, elastic–perfectly plastic, fiber reinforced, composite material is considered. The overall mechanical behavior of the material is described using the anisotropic model of elastic–plastic body with kinematic hardening. The appropriate initial–boundary value problem, set for one repeatable cell of the composite, is solved in order to find the effective constitutive relations. The cell problem is solved using the finite element method formulated in two dual forms: in displacements and in stresses. Stress functions are used in the latter formulation.     

Effect of deviatoric nonassociativity on the failure prediction for elastic–plastic materials

This paper presents a theoretical study of the effect of nonassociativity of the plastic flow rule on the critical plastic modulus for discontinuous bifurcation in an elastic–plastic material. Nonassociativity in both the spherical and the deviatoric spaces are considered, with an emphasis on the effect of nonassociativity in the deviatoric space. A particular form of nonassociativity in the deviatoric space is introduced, where the projections of the plastic flow direction and the normal to the yield surface are assumed to have the same length but the projection of plastic flow direction is allowed to lag that of the normal by an angle. It is shown that even for the simple yield surface of von Mises, nonassociativity in the deviatoric space can lead to a bifurcation for a load parameter significantly lower than the value predicted with an associated flow rule.     

Three dimensional combined fracture–plastic material model for concrete

This paper describes a combined fracture–plastic model for concrete. Tension is handled by a fracture model, based on the classical orthotropic smeared crack formulation and the crack band approach. It employs the Rankine failure criterion, exponential softening, and it can be used as a rotated or a fixed crack model. The plasticity model for concrete in compression is based on the Menétrey–Willam failure surface, the plastic volumetric strain as a hardening/softening parameter and a non-associated flow rule based on a nonlinear plastic potential function. Both models use a return-mapping algorithm for the integration of constitutive equations. Special attention is given to the development of an algorithm for the combination of the two models. The suggested combination algorithm is based on a recursive substitution, and it allows for the two models to be developed and formulated separately. The algorithm can handle cases when failure surfaces of both models are active, but also when physical changes such as crack closure occur. The model can be used to simulate concrete cracking, crushing under high confinement and crack closure due to crushing in other material directions. The model is integrated in a general finite element package ATENA and its performance is evaluated by comparisons with various experimental results from the literature.     

Determination of the effective elastic–plastic response of metal–ceramic composites

The mechanical response of metal–ceramic composites is analysed through a homogenization model accounting for the mechanical behaviour of the constituent materials. In order to achieve this purpose a nonlinear homogenization method based on the phase field approach has been suitably implemented into a numerical code. A prescribed homogenized strain state is applied to a unit volume element of a metal–ceramic composite with proportional loading in which all components of the strain tensor are proportional to one scalar parameter. The mechanical response of the material has been modeled by considering a von Mises plasticity model for the metal phase and a Drucker–Prager associative elastic–plastic material model for the ceramic phase. A two stages plasticity has been obtained in which inelastic strain develops in the metal phase followed by a fully plastic response. A comparison with a finite element model of the stress–strain response of an axisymmetric unit cell has been carried out with the purpose to validate the homogenization based modeling presented in the paper. Plastic parameters of a Drucker–Prager yield surface for the homogenized composite have been calculated at different materials compositions. Associative Drucker–Prager plasticity has been found to be accurate for high ceramic content.     

Steady crack growth in elastic–plastic fluid-saturated porous media

An asymptotic solution is obtained for stress and pore pressure fields near the tip of a crack steadily propagating in an elastic–plastic fluid-saturated porous material displaying linear isotropic hardening. Quasi-static crack growth is considered under plane strain and Mode I loading conditions. In particular, the effective stress is assumed to obey the Drucker–Prager yield condition with associative or non-associative flow-rule and linear isotropic hardening is adopted. Both permeable and impermeable crack faces are considered. As for the problem of crack propagation in poroelastic media, the behavior is asymptotically drained at the crack-tip. Plastic dilatancy is observed to have a strong effect on the distribution and intensity of pore water pressure and to increase its flux towards the crack-tip.     

The elastic–plastic behaviour of foam under shock loading

An experimental investigation of the elastic–plastic nature of shock wave propagation in foams was undertaken. The study involved experimental blast wave and shock tube loading of three foams, two polyurethane open-cell foams and a low-density polyethylene closed-cell foam. Evidence of precursor waves was observed in all three foam samples under various compressive wave loadings. Experiments with an impermeable membrane are used to determine if the precursor wave in an open-cell foam is a result of gas filtration or an elastic response of the foam. The differences between quasi-static and shock compression of foams is discussed in terms of their compressive strain histories and the implications for the energy absorption capacity of foam in both loading scenarios. Through a comparison of shock tube and blast wave loading techniques, suggestions are made concerning the accurate measurements of the principal shock Hugoniot in foams.     

An elastic–plastic constitutive law for the description of uniaxial and multiaxial ratchetting

An elastic–plastic constitutive model is proposed to describe 1-D and 2-D ratchetting. The model is also able to give correct results for 2-D ratchetting when only uniaxial identification is used, while no special threshold or parameter is used for the case of non-proportional loading. The original feature of this model consist in the introduction of a ratchetting stress (material characteristic) along with the maximal stress supported in the history of loading and the plastic strain at the last unloading. In this paper uniaxial and 3-D formulations have been described based on a numerical implementation in the software Code_Aster. Uniaxial and also multiaxial identifications have been used. Simulations have been realized for proportional and non-proportional homogeneous cases, as well as for structures under anisothermal thermomechanical loading. The results of a benchmark on a structure, comparing experiment, simulations by this model and some other phenomenological models, and a polycrystalline model are presented. An analysis of error margin due to the choice of Mises criterion is exposed.     

Structural shakedown for elastic–plastic materials with hardening saturation surface

An elastic–plastic material model with internal variables and thermodynamic potential, not admitting hardening states out of a saturation surface, is assumed as a basis to formulate a statical Melan-type shakedown theorem. Grounding on the optimality conditions relative to the shakedown load multiplier problem for a structure subjected to cyclic loads, the impending inadaptation collapse mechanism at the shakedown limit state is analyzed and discussed. It is shown that the adopted model is able to catch ratchetting collapse mode at a structural level. Numerical results for a simple structure are finally reported.     

On the plastic wave speeds in rate-independent elastic–plastic materials with anisotropic elasticity

This paper presents a theoretical study of the speeds of plastic waves in rate-independent elastic–plastic materials with anisotropic elasticity. It is shown that for a given propagation direction the plastic wave speeds are equal to or lower than the corresponding elastic speeds, and a simple expression is provided for the bound on the difference between the elastic and the plastic wave speeds. The bound is given as a function of the plastic modulus and the magnitude of a vector defined by the current stress state and the propagation direction. For elastic–plastic materials with cubic symmetry and with tetragonal symmetry, the upper and lower bounds on the plastic wave speeds are obtained without numerically solving an eigenvalue problem. Numerical examples of materials with cubic symmetry (copper) and with tetragonal symmetry (tin) are presented as a validation of the proposed bounds. The lower bound proposed here on the minimum plastic wave speed may also be used as an efficient alternative to the bifurcation analysis at early stages of plastic deformation for the determination of the loss of ellipticity.     

Evolution of plastic zones in dynamically loaded plates using different elastic–viscoplastic laws

In the framework of viscoplastic theory many different laws were developed, accounting for material behaviors like creep, relaxation or evolution of overstresses. Though each model is able to predict in uni-axial material tests the values of stresses depending on plastic strains and plastic strain rates the question is if solutions of simulations are still realistic if the viscoplastic law is applied on structural deformations. In the present study strain rate sensitive metal plates are subjected to shock waves. The purpose is to compare simulation results obtained with different elastic–viscoplastic laws to experiments in order to determine the most appropriate material model. By subjecting circular metal plates experimentally to shock wave loadings realistic deformation histories are measured. The measurements are compared to simulation results obtained with different viscoplastic laws. The aim is to find out the accuracy of each model concerning the predictions of displacements, shape formings, spread of plastic zones and evolutions of inner bending moments.     

Energy relaxation of non-convex incremental stress potentials in a strain-softening elastic–plastic bar

We propose a fundamentally new concept to the treatment of material instabilities and localization phenomena based on energy minimization principles in a strain-softening elastic–plastic bar. The basis is a recently developed incremental variational formulation of the local constitutive response for generalized standard media. It provides a quasi-hyperelastic stress potential that is obtained from a local minimization of the incremental energy density with respect to the internal variables. The existence of this variational formulation induces the definition of the material stability of inelastic solids based on convexity properties in analogy to treatments in elasticity. Furthermore, localization phenomena are understood as micro-structure development associated with a non-convex incremental stress potential in analogy to phase decomposition problems in elasticity. For the one-dimensional bar considered the two-phase micro-structure can analytically be resolved by the construction of a sequentially weakly lower semicontinuous energy functional that envelops the not well-posed original problem. This relaxation procedure requires the solution of a local energy minimization problem with two variables which define the one-dimensional micro-structure developing: the volume fraction and the intensity of the micro-bifurcation. The relaxation analysis yields a well-posed boundary-value problem for an objective post-critical localization analysis. The performance of the proposed method is demonstrated for different discretizations of the elastic–plastic bar which document on the mesh-independence of the results.