On the incremental plastic work and related aspects of invariance – Part I

Summary Taking for granted that the free energy function is invariant under a change of a finite strain measure and/or the reference configuration, Hill's transformation rules for selected fundamental constitutive quantities (such as tangent elastic modulus, plastic increments of total strain and work conjugate stress, the work of work-conjugate stress, the work expended in the plastic part of incremental strain etc.) are derived in a manner different from that of Hill. On this background distinguished by Hill [6] subtle aspects of invariance in mechanics of elastic plastic solids are discussed. It is shown that the plastic part of the increment of elastic strain energy (when taken with reverse sign) defines the true invariant incremental plastic work which in general is not equal to the work expended in the plastic part of the strain increment. It plays the role of a potential for the plastic part of the increment of work-conjugate stress. This fundamental fact has not found proper account in the literature. The analytical interrelations between two apparently different theoretical frameworks, Hill-Rice (fixed reference configuration) and Eckart-Mandel (mobile unloaded configuration) are discussed showing their equivalence. Since the transformation rules are complex in the general 3D case, the first part of the paper illustrates instructively the discussed aspects in a 1D situation (simple tension or simple extension).     

The strain path dependence of multiaxial cyclic hardening behaviour

The small-strain, isotropic deformation theory is used in incremental form to model an additional cyclic hardening for any arbitrary loading path. The theory is of the unified type and does not employ yield or loading/ unloading criteria. The scalar-valued functions involved in the tensorial constitutive equations as well a growth law for these functions are identified based on idea of the equivalent state. Definitions of equivalent stress and equivalent strain have been developed to correlate step by step loading programmes taking the history of deformation into account. Use is made of the total work increment together with an interpolation method for tensor functions to generalize the simple state to a multiaxial behaviour in the strain space for a given strain increment. For the demonstration of model capability, the numerical simulation is undertaken on cyclic nonproportional paths in two-dimensional axial-shear strain space. The results are verified for stainless steel and brass by comparison with the material response experimentally obtained in the stress space.     

Numerical finite element formulation of the Schapery non‐linear viscoelastic material model

This study presents a numerical integration method for the non‐linear viscoelastic behaviour of isotropic materials and structures. The Schapery's three‐dimensional (3D) non‐linear viscoelastic material model is integrated within a displacement‐based finite element (FE) environment. The deviatoric and volumetric responses are decoupled and the strain vector is decomposed into instantaneous and hereditary parts. The hereditary strains are updated at the end of each time increment using a recursive formulation. The constitutive equations are expressed in an incremental form for each time step, assuming a constant incremental strain rate. A new iterative procedure with predictor–corrector type steps is combined with the recursive integration method. A general polynomial form for the parameters of the non‐linear Schapery model is proposed. The consistent algorithmic tangent stiffness matrix is realized and used to enhance convergence and help achieve a correct convergent state. Verifications of the proposed numerical formulation are performed and compared with a previous work using experimental data for a glassy amorphous polymer PMMA. Copyright © 2003 John Wiley & Sons, Ltd.     

A semi‐implicit integration scheme for a combined viscoelastic‐damage model of plastic bonded explosives

This paper presents a new implementation of a constitutive model commonly used to represent plastic bonded explosives in finite element simulations of thermomechanical response. The constitutive model, viscoSCRAM, combines linear viscoelasticity with isotropic damage evolution. The original implementation was focused on short duration transient events; thus, an explicit update scheme was used. For longer duration simulations that employ significantly larger time step sizes, the explicit update scheme is inadequate. This work presents a new semi‐implicit update scheme suitable for simulations using relatively large time steps. The algorithm solves a nonlinear system of equations to ensure that the stress, damaged state, and internal stresses are in agreement with implicit update equations at the end of each increment. The crack growth is advanced in time using a sub‐incremental explicit scheme; thus, the entire implementation is semi‐implicit. The theory is briefly discussed along with previous explicit integration schemes. The new integration algorithm and its implementation into the finite element code, Abaqus, are detailed. Finally, the new and old algorithms are compared via simulations of uniaxial compression and beam bending. The semi‐implicit scheme has been demonstrated to provide higher accuracy for a given allocated computational time for the quasistatic cases considered here. Published 2014. This article is a US Government work and is in the public domain in the USA.     

An integration scheme for Prandtl-Reuss elastoplastic constitutive equations

We investigate the Generalized Midpoint Rule for the time integration of elastoplastic constitutive equations for pressure-independent yield criteria. The incremental equations are divided into one scalar hydrostatic pressure/dilation rate equation, and a stress deviator/strain rate deviator tensorial equation, the solution of which reduces to one single scalar equation in the plastic multiplier. The existence and uniqueness of an incremental solution is discussed. The pressure/deviator decomposition is the basis for reduced integration of the pressure term in the Principle of Virtual Work, in order to avoid locking and spurious pressure oscillations. It is also shown that an optimal choice of the parameter of the Midpoint Rule can be computed by reference to the analytical solution of the equations assuming no work hardening. A benchmark test shows that this choice allows increased time steps. This formulation is applied to two classical problems: bulging of a tube under internal pressure and tension test on a notched specimen, and a comparison with the analytical solution is performed. Finally, the hypothesis which sustains these formulations of elastoplasticity (constant strain rate during an increment) is discussed with reference to elastic unloading and residual stress computation.     

Substructuring in the implicit simulation of single point incremental sheet forming

This paper presents a direct substructuring method to reduce the computing time of implicit simulations of single point incremental forming (SPIF). Substructuring is used to divide the finite element (FE) mesh into several non-overlapping parts. Based on the hypothesis that plastic deformation is localized, the substructures are categorized into two groups: the plastic—nonlinear—substructures and the elastic—pseudo-linear—substructures. The plastic substructures assemble a part of the FE mesh that is in contact with the forming tool; they are iteratively updated respecting all nonlinearities. The elastic substructures model the elastic deformation of the rest of the FE mesh. For these substructures, the geometrical and the material behaviour are assumed linear within the increment. The stiffness matrices and the internal force vectors are calculated at the beginning of each increment then they are statically condensed to eliminate the internal degrees of freedom (DOF). In the iteration process the condensed stiffness matrices for the elastic substructures are kept constant. The condensed internal force vectors are updated by the multiplication of the condensed stiffness matrices and the displacement increments. After convergence, any geometrical and material nonlinearity for the elastic substructures are nonlinearly updated. The categorization of substructures in plastic and elastic domains is adapted during the simulation to capture the tool motion. The resulting, plastic and condensed elastic, set of equations is solved on a single processor. In an example with 1600 shell elements, the presented substructuring of the SPIF implicit simulation is 2.4 times faster than the classical implicit simulation.     

Effects of desiccation on mechanical behaviour of concrete

The objective of this work is experimental characterisation and numerical modelling of coupled behaviours between drying shrinkage and plastic damage in concrete. In the first part, we present an original experimental study on an ordinary concrete in order to determine material damage induced by drying shrinkage. Uniaxial compression tests are performed on samples dried for different periods. It is shown that material damage can be caused by drying process. Mechanical behaviour becomes more brittle with higher damage kinetics when concrete is dried. In the second part, a constitutive model is proposed in order to describe coupled hydro-mechanical behaviour of partially saturated concrete. This model takes into account induced damage, mechanical and capillary plastic deformations. Numerical simulations of experimental tests are presented, and show a qualitatively good agreement with experimental data. The results are relevant with respect to the importance of drying process in the durability study of concrete structures.     

A meshless grain element for micromechanical analysis with crystal plasticity

This study concerns the development of a 2‐D meshless grain element for elasto‐plastic deformation and intergranular damage initiation and propagation in polycrystalline fcc metals under static loading. The crystallographic material behaviour of the grains is represented by a rate‐independent single‐crystal plasticity model while including material orthotropy. The two slip planes are arbitrarily located with respect to the crystallographic axis of the grain. A non‐linear constitutive model known as the cohesive zone model is employed to represent the inelastic interaction between the grain boundaries, thus permitting grain boundary opening and sliding. The cohesive model describes the deformation characteristics of the grain boundaries through a non‐linear relation between the effective grain boundary tractions and displacements. Because of the presence of non‐linear material behaviour both inside the grain and along the cohesive grain boundaries, the method utilizes the principle of virtual work in conjunction with the meshless formulation in the derivation of the system of non‐linear incremental equilibrium equations. The solution is obtained via an incremental procedure based on the Taylor series expansion about the current equilibrium configuration. The fidelity of the present approach is verified by considering simple polycrystalline metals of only a few grains. Copyright © 2006 John Wiley & Sons, Ltd.     

Plastic buckling with residual stresses—1. Postbifurcation behaviour

The influence of residual stresses on the bifurcation load and initial postbifurcation behaviour of elastic—plastic structures is studied. The applied load is supposed to be proportional to a monotonically increasing parameter and the 'perfect' version of the structure is assumed to display an 'abrupt' bifurcation, which takes place when the material is in the plastic range. A strain-hardening flow theory of plasticity for solids characterized by a smooth yield surface is employed as constitutive relation. Symmetric self-equilibrated residual stress distributions are onsidered and the analysis is based on Hill's incremental approach to bifurcation and on its extension into the initial postbifurcation range as developed by Hutchinson. The proposed method consists of an investigation of the effect of residual stresses on the lowest possible bifurcation load and on the first two terms of an asymptotically exact postbifurcation expansion. It is also possible to obtain estimates of the maximum load supported by structures containing residual stresses. An illustration of the theory is provided by two problems, namely a continuous spring model and an axially compressed cylindrical column. The analytical results are compared with fully numerical solutions.     

含随机参数的散粒体增量型内时本构关系

散粒体材料的力学性能具有较强的随机性,研究其对本构关系的影响很有必要。在散粒体增量型内时本构关系的基础上,考虑材料参数的随机性,利用小参数摄动法,分析了应力增量和材料参数的统计特征值之间的关系,探讨了各参数的随机性对应力的影响程度。首先推导了散粒体增量型内时本构关系的摄动表达式,然后对式中的各参数进行了确定,最后由材料参数的均值和方差得到了应力增量的均值和方差,并以一个算例验证了方法的可行性。     

Finite element simulation of plasticity induced crack closure with different material constitutive models

The study presented in this paper analyses the mechanical effects of material constitutive modelling on the numerical prediction of plasticity induced crack closure. With this aim, an elastoplastic stress analysis of a MT specimen was conducted using an implicit three dimensional finite element program. Two materials were studied: an Aluminium Alloy and a High Strength Steel. Several constitutive models were used to describe their cyclic behaviour, ranging from pure isotropic hardening or pure kinematic hardening models to combined isotropic plus kinematic hardening models. Numerical results showed clear differences in plastic behaviour and crack closure predictions for the different types of mechanical models used to describe the mechanical behaviour of the materials. The mechanisms of opening stress stabilization, usually observed in numerical simulations, are explained in this work by analysing the evolution of plastic deformation along the crack flanks. The same type of plastic deformation stabilization behaviour was observed independently of the hardening model in use.     

Comparison of tensile properties in the pre-yield region of metallocene-catalyzed and Ziegler-Natta-catalyzed linear polyethylenes

The mechanical nonlinear behaviour of metallocene- and Ziegler-Natta catalyzed polyethylenes with various contents of short chain branching was investigated using a nonlinear constitutive equation in which the plastic deformation and the anharmonicity of elastic response are taken into account. It is suggested that the mechanical behaviour is governed by the plastic deformation for the Ziegler-Natta catalyzed polyethylenes, whereas the anharmonicity strongly affects the mechanical behaviour for metallocene-catalyzed polyethylenes.     

A two-phase flow model to simulate mold filling and saturation in Resin Transfer Molding

This paper addresses the numerical simulation of void formation and transport during mold filling in Resin Transfer Molding (RTM). The saturation equation, based on a two-phase flow model resin/air, is coupled with Darcy’s law and mass conservation to simulate the unsaturated filling flow that takes place in a RTM mold when resin is injected through the fiber bed. These equations lead to a system composed of an advection–diffusion equation for saturation including capillary effects and an elliptic equation for pressure taking into account the effect of air residual saturation. The model introduces the relative permeability as a function of resin saturation. When capillary effects are omitted, the hyperbolic nature of the saturation equation and its strong coupling with Darcy equation through relative permeability represent a challenging numerical issue. The combination of the constitutive physical laws relating permeability to saturation with the coupled system of the pressure and saturation equations allows predicting the saturation profiles. The model was validated by comparison with experimental data obtained for a fiberglass reinforcement injected in a RTM mold at constant flow rate. The saturation measured as a function of time during the resin impregnation of the fiber bed compared very well with numerical predictions.     

A Lagrangian model for hardening behaviour of materials at finite deformation based on the right plastic stretch tensor

In this paper a modified multiplicative decomposition of the right stretch tensor is proposed and used for finite deformation elastoplastic analysis of hardening materials. The total symmetric right stretch tensor is decomposed into a symmetric elastic stretch tensor and a non-symmetric plastic deformation tensor. The plastic deformation tensor is further decomposed into an orthogonal transformation and a symmetric plastic stretch tensor. This plastic stretch tensor and its corresponding Hencky’s plastic strain measure are then used for the evolution of the plastic internal variables. Furthermore, a new evolution equation for the back stress tensor is introduced based on the Hencky plastic strain. The proposed constitutive model is integrated on the Lagrangian axis of the plastic stretch tensor and does not make reference to any objective rate of stress. The classic problem of simple shear is solved using the proposed model. Results obtained for the problem of simple shear are identical to those of the self-consistent Eulerian rate model based on the logarithmic rate of stress. Furthermore, extension of the proposed model to the mixed nonlinear isotropic/kinematic hardening behaviour is presented. The model is used to predict the nonlinear hardening behaviour of SUS 304 stainless steel under fixed end finite torsional loading. Results obtained are in good agreement with the available experimental results reported for this material under fixed end finite torsional loading.     

CRACK TIP BEHAVIOUR AND CRACK PROPAGATION IN DUCTILE MATERIALS

Dilatational plastic equations, which can include the effects of ductile damage, are derived based on the equivalency in expressions for dissipated plastic work. Void damage developed internally at the large-strain stage is represented by an effective continuum being strain-softened and plastically dilated. Accumulation of this local damage leads to progressive failure in materials. With regard to this microstructural background, the constitutive parameters included for characterizing material behaviour have the sense of internal variables. They are not able to be determined explicitly by macroscopic testing but rather through computer simulation of experimental curves and data. Application of this constitutive model to mode-I cracking examples demonstrates that a huge strain concentration accompanied by a substantial drop of stress does occur near the crack tip. Eventually, crack propagation is simulated by using finite elements in computations. Two numerical examples show good accordance with experimental data. The whole procedure of study serves as a justification of the constitutive formulation proposed in the text.     

Hybrid-stress models for elastic–plastic analysis by the initial-stress approach

Two alternative numerical methods are presented, based on the assumed-stress hybrid finite element model and the initial-stress approach, for the elastic–plastic small-deflection analysis of structures under static loading. The use of the initial-stress approach results in a set of simultaneous linear algebraic incremental equations to be solved at each loading step, with the elastic stiffness matrix remaining unchanged throughout the loading process and the effects of plasticity included as equivalent element loads. The derivation of these alternative methods differs in the assigning of the incremental stress which satisfies equilibrium; in one case it is the actual stress increment while in the other it is a fictitious stress increment. An equilibrium imbalance correction is included in each of these methods to prevent drifting of the solution during the incremental process. Example solutions are presented which demonstrate the accuracy of the two methods and permit comparisons of the relative efficiencies of the two methods.     

Efficient numerical implementation of pressure,time, and temperature superposition for elasto‐visco‐plastic material model by using a symbolic approach

This article is concerned with the finite element implementation of an elasto‐visco‐plastic constitutive model using a symbolic approach. The model combines the Knauss–Emri (KE) pressure, temperature, and time superposition principle in the implicit finite element scheme. The equation development and code generation was performed using the symbolic tool AceGen. The same symbolic system was applied to derive analytical sensitivities of the numerical model with respect to the material and shape parameters. To enable efficient numerical implementation of the KE model the convolution integrals were transformed into their respective incremental forms, so that radical improvements of code efficiency and computer storage requirements were achieved. The numerical examples derived for polyethylene terephthalate (PET) polymers demonstrate that symbolic systems can be applied to develop complex constitutive models capable of simulating material responses that are in good agreement with experimental measurements over a wide range of strain rates, temperatures, and loading conditions. Copyright © 2010 John Wiley & Sons, Ltd.     

Comparison of experimental results and finite element simulation of strain localization scheme under cyclic loading

The plastic behaviour of FCC materials is studied under cyclic tensile–compression loading at room temperature. The material is an oxygen-free high conductivity copper. The purpose of the work is to model the onset of plasticity, then the cycle by cycle evolution of the localized strain, at grain scale and at mesoscopic scale.A polycrystalline aggregate taking into account the material microstructure is developped to perform finite element simulations corresponding to the experiments. Finite element calculations are carried out on this mesh, using a constitutive law which takes into account the crystallographic orientation of each grain. An analysis of the localisation scheme is performed at different steps of the cyclic loading.