Elastoplastic Formulation Research Articles

Thermodynamic laws and consistent Eulerian formulation of finite elastoplasticity with thermal effects

Recently it has been demonstrated that, on the basis of the separation D = D e + D p arising from the split of the stress power and two consistency criteria for objective Eulerian rate formulations, it is possible to establish a consistent Eulerian rate formulation of finite elastoplasticity in terms of the Kirchhoff stress and the stretching, without involving additional deformation-like variables labelled “elastic” or “plastic”. It has further been demonstrated that this consistent formulation leads to a simple essential structure implied by the work postulate, namely, both the normality rule for plastic flow D p and the convexity of the yield surface in Kirchhoff stress space. Here, we attempt to place such an Eulerian formulation on the thermodynamic grounds by extending it to a general case with thermal effects, where the consistency requirements are treated in a twofold sense. First, we propose a general constitutive formulation based on the foregoing separation as well as the two consistency criteria. This is accomplished by employing the corotational logarithmic rate and by incorporating an exactly integrable Eulerian rate equation for D e for thermo-elastic behaviour. Then, we study the consistency of the formulation with thermodynamic laws. Towards this goal, simple forms of restrictions are derived, and consequences are discussed. It is shown that the proposed Eulerian formulation is free in the sense of thermodynamic consistency. Namely, a Helmholtz free energy function in explicit form may be found such that the restrictions from the thermodynamic laws can be fulfilled with positive internal dissipation for arbitrary forms of constitutive functions included in the constitutive formulation. In particular, that is the case for the foregoing essential constitutive structure in the purely mechanical case. These results eventually lead to a complete, explicit constitutive theory for coupled fields of deformation, stress and temperature in thermo-elastoplastic solids at finite deformations.

A formulation of anisotropic continuum elastoplasticity at finite strains. Part I: Modelling

A constitutive model for anisotropic elastoplasticity at finite strains is developed together with its numerical implementation. An anisotropic elastic constitutive law is described in an invariant setting by use of structural tensors and the elastic strain measure C e . The elastic strain tensor as well as the structural tensors are assumed to be invariant in relation to superimposed rigid body rotations. An anisotropic Hill-type yield criterion, described by a non-symmetric Eshelby-like stress tensor and further structural tensors, is developed, where use is made of representation theorems for functions with non-symmetric arguments. The model also considers non-linear isotropic hardening. Explicit results for the specific case of orthotropic anisotropy are given. The associative flow rule is employed and the features of the inelastic flow rule are discussed in full. It is shown that the classical definition of the plastic material spin is meaningless in conjunction with the present formulation. Instead, the study motivates an alternative definition, which is based on the demand that such a quantity must be dissipation-free, as the plastic material spin is in the case of isotropy. Equivalent spatial formulations are presented too. The full numerical treatment is considered in Part II.

The integrability criterion in finite elastoplasticity and its constitutive implications

In traditional Eulerian rate formulation of finite elastoplasticity, an Eulerian rate equation of hypoelastic type is used as one of the basic constituents to relate the elastic part of the stretching to a stress rate. On account of the fact that the elastic-like behavior should be expected prior to yielding, the foregoing basic elastic rate equation should be exactly integrable to deliver a conventional hyperelastic stress-strain relation. Physically, it requires a consistent combination of hypoelasticity and hyperelasticity into a single Eulerian rate equation. Since this criterion is purely a physical consistency requirement and since the basic elastic rate equation involves no strain concept and allows for any stress rate in its own right, from a physical standpoint it may be of both interest and significance to know what consequences it will imply concerning the stress rate and the finite strain measure. By a simple, straightforward procedure we demonstrate in a general sense that both Hencky strain and the logarithmic rate emerge naturally as direct consequences of the foregoing criterion. This result may be regarded to reveal the physical essence behind Hencky strain and the logarithmic rate in connection with a basic physical consistency requirement in finite elastoplasticity. Constitutive implications are discussed in a few relevant respects concerning representative formulations of finite elastoplasticity.

Geometry Method for Localization Analysis in Gradient-Dependent J2 Plasticity

In this work the geometrical method for the assessment of discontinuous bifurcation conditions is extended to encompass gradient-dependent plasticity. To this end, the gradient-dependent localization condition is cast in the form of an elliptical envelope condition in the coordinates of Mohr. The results in this work demonstrate the capability of thermodynamically consistent gradient-dependent elastoplastic model formulations to suppress the localized failure modes of the classical plasticity that take place when the hardening/softening modulus H¯ equals the critical value for localization H¯c, provided the characteristic length l remains positive.

A convergent adaptive finite element method for the primal problem of elastoplasticity

AbstractThe boundary value problem representing one time step of the primal formulation of elastoplasticity with positive hardening leads to a variational inequality of the second kind with some nondifferentiable functional. This paper establishes an adaptive finite element algorithm for the solution of this variational inequality that yields the energy reduction and, up to higher order terms, the R‐linear convergence of the stresses with respect to the number of loops. Applications include several plasticity models: linear isotropic‐kinematic hardening, linear kinematic hardening, and multisurface plasticity as model for nonlinear hardening laws. For perfect plasticity, the adaptive algorithm yields strong convergence of the stresses. Numerical examples confirm an improved linear convergence rate and study the performance of the algorithm in comparison with the more frequently applied maximum refinement rule. Copyright © 2006 John Wiley & Sons, Ltd.

Elastoplasticity beyond small deformations

This article is concerned with some essential aspects of representative ideas and viewpoints as well as developments in formulations of rate-independent elastoplasticity beyond the small deformation range. Kinematical and physical foundations underlying the basic variables, and the essential constitutive structures implied by the work postulate are examined and discussed, pro and con, from an integrated viewpoint based on certain coherent threads running through various formulations. Their constitutive and computational implications are elucidated and contrasted by analysing the main structural features based on their basic variables. Emphasis is placed on new issues and new understanding, and on physically pertinent variables and formulations.

Computational analysis of deformation and fracture in a composite material on the mesoscale level

Presented in this paper are the computational results on deformation and fracture in an aluminum-matrix composite. 2D numerical simulations were carried out using the finite-difference method. To describe the mechanical behavior of elasto-plastic matrix and brittle inclusions use was made of different models—elasto-plastic formulation with the strain hardening and a cracking model with a fracture criterion of Huber type, respectively. The fracture criterion takes into account a difference in critical values for different local stress–strain states: tension and compression. The initial structure of the mesovolume, elastic and strength material constants, as well as the strain hardening function were chosen from the experiments. It has been shown that the composite mesovolume exhibits complex mechanical behavior controlled by both shear band formation in the matrix and cracking of inclusions. When applied to simulation of deformation and fracture in a heterogeneous medium, the criterion proposed allows one to describe adequately direction of crack propagation under different types of loading (tension and compression). The computational results have been analyzed in details, taking into account analytical solution for inclusion problem.

Lower semicontinuity and existence of minimizers in incremental finite‐strain elastoplasticity

AbstractWe study incremental problems in geometrically nonlinear elastoplasticity. Using the multiplicative decomposition Dφ= FelFpl we consider general energy functionals of the formℐ(φ, Fpl) = ∫ΩU(x, DφF‐1pl, Fpl, 𝒢(Fpl)) dx‐<ℓ, φ>,which occur as the sum of the stored energy and the dissipation in one time step. Here 𝒢(Fpl) is the dislocation tensor which takes the form ${1\over {\rm det} F_{\rm pl}}$ curl3(Fpl),Fpl T in dimension d = 3.Imposing the usual constraint det Fpl ≡ 1 and suitable growth and polyconvexity conditions on U we show that the minimum of ℐ is attained in the natural Sobolev spaces. Moreover, we are able to treat multiple time steps by controlling the stored and dissipated energies. We also address the relation of the incremental problem to the time‐continuous energetic formulation of elastoplasticity.

Thermo-visco-plastic behaviour of fibre-reinforced polymer composites

The effect of temperature and strain rate on the tensile behaviour on a series of polymeric matrix-unidirectional glass–fibre composites was studied. Dynamic mechanical analysis (DMA) experiments, as well as tensile tests at three different strain rates and three different temperatures below T g were performed on off-axis specimens of three different orientations. The strong temperature and strain rate dependence, exhibited by the materials examined, was further described theoretically by applying a formulation of finite elastoplasticity. Constitutive laws based on the material anisotropy, were applied, in combination with constitutive equations of hypoelasticity, written in their objective form. Moreover, empirical equations for the hardening coefficients, arising from the thermal activation theory, were proposed to formulate the temperature and strain rate effect.

Numerical study of localization in soil systems

A numerical study of the mechanical behavior of heterogeneous soil systems, consisting of a bulk of sand with embedded stiff gravel inclusions or soft clay inclusions, is performed. A solution scheme using parallel computing is employed when analyzing two different categories of problems. First, a homogenization problem is studied, where use of a single representative volume element subjected to plane strain compression offers the possibility to investigate the coupling between the response at a local scale and at a global scale. Second, a plane strain footing problem with different heterogeneous soil systems is analyzed using a traditional finite element formulation. The material model utilized for the soil is a large deformation formulation of non-associated elasto-plasticity with an isotropic hardening law, able to represent dilation. It was found that the shape of the gravel or clay inclusions in the systems had no significant effect on the global responses, whereas the strain localizations in the two different soil systems, sand–gravel and sand–clay, were found to have different character. The effect of the initial density on the response was clearly observed in the localization patterns.

The least‐squares meshfree method for elasto‐plasticity and its application to metal forming analysis

AbstractA new meshfree method for the analysis of elasto‐plastic deformation is presented. The method is based on the proposed first‐order least‐squares formulation for elasto‐plasticity and the moving least‐squares approximation. The least‐squares formulation for classical elasto‐plasticity and its extension to an incrementally objective formulation for finite deformation are proposed. In the formulation, equilibrium equation and flow rule are enforced in least‐squares sense, i.e. their squared residuals are minimized, and hardening law and loading/unloading condition are enforced pointwise at each integration point. The closest point projection method for the integration of rate‐form constitutive equation is inherently involved in the formulation, and thus the radial‐return mapping algorithm is not performed explicitly. The proposed formulation is a mixed‐type method since the residuals are represented in a form of first‐order differential system using displacement and stress components as nodal unknowns. Also the penalty schemes for the enforcement of boundary and frictional contact conditions are devised and the reshaping of nodal supports is introduced to avoid the difficulties due to the severe local deformation near contact interface. The proposed method does not employ structure of extrinsic cells for any purpose. Through some numerical examples of metal forming processes, the validity and effectiveness of the method are discussed. Copyright © 2005 John Wiley & Sons, Ltd.

Hyper-singular boundary element method for elastoplastic fracture mechanics analysis with large deformation

In this paper a two-dimensional hyper-singular boundary element method for elastoplastic fracture mechanics analysis with large deformation is presented. The proposed approach incorporates displacement and the traction boundary integral equations as well as finite deformation stress measures, and general crack problems can be solved with single-region formulations. Efficient regularization techniques are applied to the corresponding singular terms in displacement, displacement derivatives and traction boundary integral equations, according to the degree of singularity of the kernel functions. Within the numerical implementation of the hyper-singular boundary element formulation, crack tip and corners are modelled with discontinuous elements. Fracture measures are evaluated at each load increment, using the J-integral. Several cases studies with different boundary and loading conditions have been analysed. It has been shown that the new singularity removal technique and the non-linear elastoplastic formulation lead to accurate solutions.

탄소성 최소 제곱 수식화와 이를 이용한 무요소법

A new meshfree method for the analysis of elasto-plastic deformations is presented. The method is based on the proposed first-order least-squares formulation, to which the moving least-squares approximation is applied. The least-squares formulation for the classical elasto-plasticity and its extension to an incrementally objective formulation for finite deformations are proposed. In the formulation, the equilibrium equation and flow rule are enforced in least-squares sense, while the hardening law and loading/unloading condition are enforced exactly at each integration point. The closest point projection method for the integration of rate-form constitutive equation is inherently involved in the formulation, and thus the radial-return mapping algorithm is not performed explicitly. Also the penalty schemes for the enforcement of the boundary and frictional contact conditions are devised. The mam benefit of the proposed method is that any structure of cells is not used during the whole process of analysis Through some numerical examples of metal forming processes, the validity and effectiveness of the method are presented.

An assessment of constitutive models for concrete columns confined with fibre composite sheets

Fibre reinforced composite materials are becoming more frequently used in civil engineering structures. One of the most practical applications of these new materials concerns the strengthening of reinforced concrete columns by means of confinement with fibre composite sheets. In the literature, various theoretical models have been proposed to describe the behaviour of confined concrete columns. These models can be grouped into three families: empirical, nonlinear elastic, and elastoplastic. The last two classes are considered in this paper. Comparisons of numerical predictions with experiments show that the nonlinear elastic approach is not very suitable for fibre composite confinement. The elastoplastic formulation gives predictions that agree quite well with experimentally determined axial load–deformation curves.

On a physically stabilized one point finite element formulation for three-dimensional finite elasto-plasticity

In the case of linear elasticity, a direct connection between the concept of reduced integration with hourglass stabilization and a mixed method can usually be established. In the non-linear case, this is in general not possible. To overcome this difficulty we suggest in this paper a new concept based on a Taylor expansion of the constitutively dependent quantities with respect to the centre of the element. The push-forward of the second (linear) term of the Taylor series for the first Piola–Kirchhoff stress tensor to the current configuration determines the so-called hourglass stabilization part of the residual force vector. Due to the fact that the element uses only one Gauss point and the hourglass stabilization part is computed by means of a simple functional evaluation, the present element technology is very efficient from the computational point of view. In contrast to the 2D case the computation of the Jacobi determinant only in the centre of the 3D element does not yield the correct volume, if the element shape deviates from being a parallelipiped. It is shown in the paper that the error becomes negligibly small for a relatively coarse discretization. The formulation is free of volumetric locking and can compete with shell formulations up to an aspect ratio of about hundred. For bending-dominated problems, at least two elements over the thickness are needed in order to compute the onset of plastification correctly. The element behaves very robustly in finite elasticity and inelasticity, also when large element distortions occur.

Elastoplastic deformation and the limit equilibrium of granular media

A theory of the deformation of a granular medium beyond the limit of elasticity is proposed, which takes into account both elastic and plastic deformations and enables the transition of the medium from a purely elastic state to a limit state to be traced. A system of functions is introduced which define the components of the stress and strain tensors, and a resolving system of equations of elastoplastic deformation of the medium is obtained for the case of a plane strained state. By taking the limit one can obtain the equations of the theory of limit equilibrium from these equations. The elastoplastic problem of the loading of a tube (a thick-walled circular cylinder) by internal pressure is solved. It is shown that the load corresponding to the transition of the tube to a purely plastic state over the whole cross section is not always identical with the load obtained using the limit equilibrium theory. The passage to the limit for which these two loads are the same is determined. The elastoplastic problem of the loading of a circular layer having a hole (a well) at the centre is solved in the new formulation. The depth of the layer for which a region of plastic deformation appears in the neighbourhood of the well are determined. The limit value of the depth at which the layer collapses is obtained. The residual stresses in the neighbourhood of the well when there is a repeated increase in the pressure to the value of the rock pressure are determined. The classical problem of the stressed state of a plane slope is solved in the elastoplastic formulation. A relation is established between the mechanical characteristics of the medium and the angle of the slope for which the slope always remains in the elastic state over the whole depth. It is shown that three different stress-strain states are possible in elastoplastic deformation. The transition to the limit state is traced and it is established that, unlike existing representations, plastic flow of the slope from a rigid plastic material is impossible for slope angles less than the angle of internal friction. For slope angles greater than the angle of internal friction plastic yield does not occur over the whole slope but only over its upper layer of a certain thickness.

Formulation of hybrid Trefftz finite element method for elastoplasticity

The present investigation provides a hybrid Trefftz finite element approach for analysing elastoplastic problems. A dual variational functional is constructed and used to derive hybrid Trefftz finite element formulation for elastoplasticity of bulky solids. The formulation is applicable to either strain hardening or elastic-perfectly plastic materials. A solution algorithm based on initial stress formulation is introduced into the new element model. The performance of the proposed element model is assessed by three examples and comparison is made with results obtained by other approaches. The hybrid Trefftz finite element approach is demonstrated to be particularly suited for nonlinear analysis of two-dimensional elastoplastic problems.

Symmetric and asymmetric deformation transition in the regularly cell-structured materials. Part II: Theoretical study

Symmetric and asymmetric deformation behavior is theoretically analyzed for regularly cell-structured materials. On the basis of the elasto-plastic formulation by the finite element method, the computational model is constructed in order to understand the effect of various parameters on the deformation mode transition. Symmetric deformation changes itself to asymmetric deformation when increasing the nominal compressive strain. Effects of the relative density, the contact length, and the adhesive bonding on this mode transition are investigated under quasi-static in-plane compression. Besides the relative density, the contact length to thickness ratio ( r c/ t) plays an important role on the symmetric–asymmetric deformation transition. The intercell stress transfer also has an influence on the stability of the cell-structure during crushing. The theoretical prediction is compared with the experimental results in (Part 1) to quantitatively discuss the compressive deformation of regularly cell-structured materials.

On non–corotational rates of Oldroyd's type and relevant issues in rate constitutive formulations

The lower and upper Oldroyd rates are two wellknown objective noncorotational rates. A remarkable property of these rates is that they establish direct relationships between finite strain measures ...

Biaxial Failure Model for Fiber Reinforced Concrete

Steel fiber reinforced concrete (SFRC) is widely applied in the construction industry. Numerical elastoplastic analysis of the macroscopic behavior is complex. This typically involves a piecewise linear failure curve including corner singularities. This paper presents a single smooth biaxial failure curve for SFRC based on a semianalytical approximation. Convexity of the proposed model is guaranteed so that numerical problems are avoided. The model has sufficient flexibility to closely match experimental results. The failure curve is also suitable for modeling plain concrete under biaxial loading. Since this model is capable of simulating the failure states in all stress regimes with a single envelope, the elastoplastic formulation is very concise and simple. The finite element implementation is developed to demonstrate the conciseness and the effectiveness of the model. The computed results display good agreement with published experimental data.