Finite Strain Plasticity Theory Research Articles

A micropolar approach to microbuckling problems in unidirectionally reinforced polymer composites

The present contribution considers the application of micropolar continuum theory to predict the microbuckling strength of microbuckling problems in long fiber reinforced composites and to extend simulations into the collapse regime. The approach considers a homogenized description at a coarse length scale where fiber and matrix phases are not individually resolved, but are represented indirectly via an effectively equivalent homogenized solid. The salient characteristic of this approach is its superior numerical efficiency over a fine-scale micromechanical representation where fibers and matrix would need to be discretized separately. A disadvantage of conventionally homogenized models, however, is the circumstance that fiber curvature is not accounted for. Hence, the local fiber bending stiffness cannot be preserved in the homogenization process and the strain localization stage of microbuckling problems is rendered ill-posed. Micropolar homogenization rectifies this deficiency by introducing additional rotational degrees of freedom to account for curvature strain. The present contribution distinguishes itself from earlier micropolar homogenization approaches by adapting a Total-Lagrangian finite strain plasticity theory for micropolar continua to the microbuckling problem under consideration here. From this theoretical basis, a constitutive model and a finite element formulation are derived, and the extra parameters introduced by the micropolar solid are identified.

Physics-Based Deformable Tire–Soil Interaction Model for Off-Road Mobility Simulation and Experimental Validation

A physics-based deformable tire–soil interaction simulation capability that can be fully integrated into the monolithic multibody dynamics computer algorithm is developed by extending a deformable tire model based on the flexible multibody dynamics approach to off-road mobility simulations with a moving soil patch technique and it is validated against test data. A locking-free nine-node brick element is developed for modeling large plastic soil deformation using the multiplicative finite strain plasticity theory along with the capped Drucker–Prager failure criterion. To identify soil parameters including cohesion and friction angle, the triaxial compression test is carried out, and the soil model developed is validated against the test data. In addition to the component level validation for the tire and soil models, the tire–soil interaction simulation capability developed in this study is validated against the soil bin mobility test results. The tire forces and rolling resistance coefficients predicted by the simulation model agree well with the test results. It is shown that effect of the wheel loads and tire inflation pressures is well captured in the simulation model. Furthermore, it is demonstrated that the moving soil patch technique, with which soil behavior only in the vicinity of the rolling tire is solved to reduce the soil model dimensionality, leads to a significant reduction in computational time, thereby enabling use of the high-fidelity physics-based tire–soil interaction model in the large-scale off-road mobility simulation.

On the thermomechanical coupling in dissipative materials: A variational approach for generalized standard materials

This paper deals with the thermomechanical coupling in dissipative materials. The focus lies on finite strain plasticity theory and the temperature increase resulting from plastic deformation. For this type of problem, two fundamentally different modeling approaches can be found in the literature: (a) models based on thermodynamical considerations and (b) models based on the so-called Taylor–Quinney factor. While a naive straightforward implementation of thermodynamically consistent approaches usually leads to an over-prediction of the temperature increase due to plastic deformation, models relying on the Taylor–Quinney factor often violate fundamental physical principles such as the first and the second law of thermodynamics. In this paper, a thermodynamically consistent framework is elaborated which indeed allows the realistic prediction of the temperature evolution. In contrast to previously proposed frameworks, it is based on a fully three-dimensional, finite strain setting and it naturally covers coupled isotropic and kinematic hardening — also based on non-associative evolution equations. Considering a variationally consistent description based on incremental energy minimization, it is shown that the aforementioned problem (thermodynamical consistency and a realistic temperature prediction) is essentially equivalent to correctly defining the decomposition of the total energy into stored and dissipative parts. Interestingly, this decomposition shows strong analogies to the Taylor–Quinney factor. In this respect, the Taylor–Quinney factor can be well motivated from a physical point of view. Furthermore, certain intervals for this factor can be derived in order to guarantee that fundamental physically principles are fulfilled a priori. Representative examples demonstrate the predictive capabilities of the final constitutive modeling framework.

Mechanical characterization and constitutive modeling of Mg alloy sheets

In this paper, an experimental mechanical characterization of the magnesium alloys ZE10 and AZ31 is performed and a suitable constitutive model is established. The mechanical characterization is based on uniaxial tensile tests. In order to avoid poor formability at room temperature, the tests were conducted at elevated temperature (200 °C). The uniaxial tensile tests reveal sufficient ductility allowing sheet forming processes at this temperature. The differences in yield stresses and plastic strain ratios (r-values) confirm the anisotropic response of the materials under study. The constitutive model is established so that the characteristic mechanical features observed in magnesium alloys such as anisotropy and compression-tension asymmetry can be accommodated. This model is thermodynamically consistent, incorporates rate effect, is formulated based on finite strain plasticity theory and is applicable in sheet forming simulations of magnesium alloys. More precisely, a model originally proposed by Cazacu and Barlat in 2004 and later modified to account for the evolution of the material anisotropy is rewritten in a thermodynamically consistent framework. The calibrated constitutive model is shown to capture the characteristic mechanical features observed in magnesium alloy sheets.

On the thermomechanical coupling in finite strain plasticity theory with non-linear kinematic hardening by means of incremental energy minimization

The thermomechanical coupling in finite strain plasticity theory with non-linear kinematic hardening is analyzed within the present paper. This coupling is of utmost importance in many applications, e.g., in those showing low cycle fatigue (LCF) under large strain amplitudes. Since the by now classical thermomechanical coupling originally proposed by Taylor and Quinney cannot be used directly in case of kinematic hardening, the change in heat as a result of plastic deformation is computed by applying the first law of thermodynamics. Based on this balance law, together with a finite strain plasticity model, a novel variationally consistent method is elaborated. Within this method and following Stainier and Ortiz (2010), all unknown variables are jointly and conveniently computed by minimizing an incrementally defined potential. In sharp contrast to previously published works, the evolution equations are a priori enforced by employing a suitable parameterization of the flow rule and the evolution equations. The advantages of this parameterization are, at least, twofold. First, it leads eventually to an unconstrained stationarity problem which can be directly applied to any yield function being positively homogeneous of degree one, i.e., the approach shows a broad range of application. Secondly, the parameterization provides enough flexibility even for a broad range of non-associative models such as kinematic hardening of Armstrong–Frederick-type. Different to Stainier and Ortiz (2010), the continuous variational problem is approximated by a standard, fully-implicit time integration. The applicability of the resulting numerical implementation is finally demonstrated by analyzing the thermodynamically coupled response for a loading cycle.

On the effect of different parameterizations of the flow rule on the efficiency of variational constitutive updates

AbstractThe numerical efficiency of so‐called variational constitutive updates for finite strain plasticity theory is analyzed. These updates compute the unknowns such as the plastic strains by minimizing an appropriate functional. Within the present paper, different parameterizations of the flow rule are utilized within the variational constitutive update scheme. It is shown that comparing to the return‐mapping algorithm, the variational updates require significantly less iteration steps and thus, is numerically highly efficient. (© 2010 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Variationally consistent modeling of finite strain plasticity theory with non-linear kinematic hardening

Variational constitutive updates provide a physically and mathematically sound framework for the numerical implementation of material models. In contrast to conventional schemes such as the return-mapping algorithm, they are directly and naturally based on the underlying variational principle. Hence, the resulting numerical scheme inherits all properties of that principle. In the present paper, focus is on a certain class of those variational methods which relies on energy minimization. Consequently, the algorithmic formulation is governed by energy minimization as well. Accordingly, standard optimization algorithms can be applied to solve the numerical problem. A further advantage compared to conventional approaches is the existence of a natural distance (semi metric) induced by the minimization principle. Such a distance is the foundation for error estimation and as a result, for adaptive finite elements methods. Though variational constitutive updates are relatively well developed for so-called standard dissipative solids, i.e., solids characterized by the normality rule, the more general case, i.e., generalized standard materials, is far from being understood. More precisely, (Int. J. Sol. Struct. 2009, 46:1676–1684) represents the first step towards this goal. In the present paper, a variational constitutive update suitable for a class of nonlinear kinematic hardening models at finite strains is presented. Two different prototypes of Armstrong–Frederick-type are re-formulated into the aforementioned variationally consistent framework. Numerical tests demonstrate the consistency of the resulting implementation.

Towards variational constitutive updates for finite strain plasticity models based on non‐associative evolution equations

AbstractIn this contribution, first steps towards variational constitutive updates for finite strain plasticity theory based on non‐associative evolution equations are presented. These schemes allow to compute the unknown state variables such as the plastic part of the deformation gradient, together with the deformation mapping, by means of a fully variational minimization principle. Therefore, standard optimization algorithms can be applied to the numerical implementation leading to a very robust and efficient numerical implementation. Particularly, for highly non‐linear, singular or nearly ill‐posed physical models like that corresponding to crystal plasticity showing a large number of possible active slip planes, this is a significant advantage compared to standard constitutive updates such as the by now classical return‐mapping algorithm. While variational constitutive updates have been successfully derived for associative plasticity models, their extension to more complex constitutive laws, particularly to those featuring non‐associative evolution equations, is highly challenging. In the present contribution, a certain class of non‐associative finite strain plasticity models is discussed and recast into a variationally consistent format. (© 2009 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

On the implementation of rate-independent standard dissipative solids at finite strain – Variational constitutive updates

This paper is concerned with an efficient, variationally consistent, implementation for rate-independent dissipative solids at finite strain. More precisely, focus is on finite strain plasticity theory based on a multiplicative decomposition of the deformation gradient. Adopting the formalism of standard dissipative solids which allows to describe constitutive models by means of only two potentials being the Helmholtz energy and the yield function (or equivalently, a dissipation functional), finite strain plasticity is recast into an equivalent minimization problem. In contrast to previous models, the presented framework covers isotropic and kinematic hardening as well as isotropic and anisotropic elasticity and yield functions. Based on this approach a novel numerical implementation representing the main contribution of the paper is given. In sharp contrast to by now classical approaches such as the return-mapping algorithm and analogously to the theoretical part, the numerical formulation is variationally consistent, i.e., all unknown variables follow naturally from minimizing the energy of the considered system. Consequently, several different numerically efficient and robust optimization schemes can be directly employed for solving the resulting minimization problem. Extending previously published works on variational constitutive updates, the advocated model does not rely on any material symmetry and therefore, it can be applied to a broad range of different plasticity theories. As two examples, an anisotropic Hill-type and a Barlat-type model are implemented. Numerical examples demonstrate the applicability and the performance of the proposed implementation.

A variational formulation for finite deformation wrinkling analysis of inelastic membranes

This paper is concerned with a novel, fully variational formulation for finite deformation analysis of inelastic membranes with wrinkling. In contrast to conventional approaches, every aspect of the physical problem derives from minimization of suitable energy functionals. A variational formulation of finite strain plasticity theory, which leads to a minimization problem for the constitutive updates, serves as the starting point for the derivations. In order to take into account the kinematics induced by wrinkles and slacks, a relaxed version of the finite strain functional is postulated. In effect, the local incremental stress–strain relations are established via differentiation of the relaxed energy functional with respect to the strains. Hence, the presented formulation is fully analogous to that of hyperelasticity with the sole exception that the aforementioned functional depends on history variables and, accordingly, it is path dependent. The advantages associated with the developed variational method are manifold. From a practical point of view, the possibility of applying standard optimization algorithms to solve the minimization problem describing inelastic membranes is remunerative. From a mathematical point of view, on the other hand, the energy of the system induces some sort of natural metric representing an essential requirement for error estimation and thus, for adaptive finite element methods. The presented derivation of the model allows to consider possible material symmetries in the elastic as well as plastic response of the material. As a prototype, a von Mises-type model is implemented. The efficiency and performance of the resulting algorithm are demonstrated by means of numerical examples.

A fully variational algorithmic formulation for wrinkling at finite strains

AbstractThis contribution is concerned with an efficient novel algorithmic formulation for wrinkling at finite strains. In contrast to previously published numerical implementations, the advocated method is fully variational. More precisely, the parameters describing wrinkles or slacks, together with the unknown deformation mapping, are computed jointly by minimizing the potential energy of the considered mechanical system. Furthermore, the wrinkling criteria are naturally included within the presented variational framework. The presented approach allows to employ three–dimensional constitutive models directly, i.e., plane stress conditions characterizing membranes are variationally enforced by minimizing the potential energy with respect to the transversal strains. Since the proposed formulation for wrinkling in membranes is fully variational, it can be conveniently combined with other variational methods (based on energy minimization). As an example, a variationally consistent framework for finite strain plasticity theory is considered. More precisely, the minimization principle characterizing wrinkling in elastic membranes and that describing plasticity in inelastic solids are coupled leading to a novel variational approach for inelastic membranes. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A covariance condition in finite plasticity and related constitutive results

This article investigates the role of covariance in a theory of finite strain plasticity based on the multiplicative decomposition of the deformation gradient. A covariance description for the energy function is developed, and the nature of referential covariance is explored. The implications of the covariance condition on the flow rule and the elastic-plastic tangent tensor are discussed.

Numerical analyses of discontinuous material bifurcation: strong and weak discontinuities

In this paper an algorithmic formulation for numerical analyses of material bifurcation is presented. Conditions for the onset of both weak discontinuities (discontinuous strain rates) and strong discontinuities (discontinuous velocity fields) are summarized. Based on a recently proposed plasticity model formulated within the logarithmic strain space, the condition for the formation of strong discontinuities is extended to anisotropic finite strain plasticity theory. The resulting equations associated with the mode of bifurcation are solved numerically. For that purpose, an equivalent optimization problem is considered. The algorithmic formulation is based on Newton’s method using a consistent linearization. To enlarge the radius of convergence, a line search strategy is applied. The applicability of the proposed implementation as well as its performance and numerical robustness is investigated by means of three-dimensional numerical bifurcation analyses of a Drucker–Prager type plasticity model.

Modelling the compaction behaviour of powders: application to pharmaceutical powders

Stress and density changes in axi-symmetric compaction of pharmaceutical powders are analysed numerically. Data measured in a compression cycle are used with a calibration procedure to assess the mechanical behaviour of powders in compaction based on a Drucker–Prager Cap model. This model is based on the elastic–plastic theory and takes into account the macroscopic characteristics of powders such as cohesion and global friction between particles. Moreover, a yield function is used to limit the admissible stresses in a tablet during a compaction cycle. This yield function depends on the first and second invariants of the stress tensor: pressure and stress deviation. To represent the plastic compaction mechanisms, a strain hardening function is used to expand the yield function with increasing volumetric strain. A finite element method coupled to the finite strain plasticity theory is used to calculate stress and strain changes in a tablet during compression and decompression. The die wall friction is estimated from the transmission effort to the lower punch with the modified equation of Shaxby and Evans. This model and the calibration procedure are applied to lactose powder. Mechanical properties calculated are compared to the experimental data measurements with a Jenike shear cell. The relative density distribution at the end of compaction and after the unloading is analysed. The normal pressure on the die is numerically estimated and analysed in terms of load transferred from powder to die during compaction and load restitution to tablet during decompression.

Numerical modelling of the powder compaction of a cup

The porosity and stress distributions in an axisymmetric cup have been analysed numerically. The finite element programme is based on rate independent, finite strain plasticity theory, and two different porous material models have been implemented. The first material model is a combination of the Gurson model, relevant to rather low porosities, and the Fleck–Kuhn–McMeeking (FKM) model, developed for highly porous powder compacts. The second material model accounts for the possibility of reduced inter-particle cohesion. Friction between tool and work-piece as well as the compaction method have both a significant influence on the porosity distribution in the compacted cup. The inter-particle cohesive strength plays only a minor role in the porosity distribution, but is more important in both the distribution and the level of the von Mises and the hydrostatic stresses.

A re-formulation of the exponential algorithm for finite strain plasticity in terms of cauchy stresses

The role of the stress measure to be chosen as the argument in the definition of yield functions is discussed in the context of finite strain plasticity theory. Motivated by physical arguments, the exponential algorithm for multiplicative finite strain plasticity is revisited such that Cauchy stresses are adopted as arguments in the yield function. Using logarithmic strain measures, the return map algorithm is formulated in principal axes. The algorithmic tangent moduli are obtained in a slightly modified, unsymmetric format compared to the standard formulation in terms of Kirchhoff stresses. However, the global structure of the exponential algorithm is unchanged. The algorithm is applied to the re-formulation of the Cam—Clay model in terms of Cauchy stresses. The typical calibration procedure of the Cam—Clay model based on Cauchy stresses is demonstrated. As an alternative, a modification of the Cam—Clay model, which allows re-calibration of Cauchy stress-based test data to be used within the framework of a Kirchhoff-based finite strain model is also discussed. The relevance of the adequate choice of the stress measure is illustrated by means of selected numerical analyses.

Numerical modelling of cold compaction of metal powder

A finite element programme has been developed for the analysis of porosity and stress distributions in a powder compact, based on rate-independent finite strain plasticity theory. The strain hardening versions of the Gurson model ( J. Engng. Mater. Technol., 1977, 99, 2–15), the more recent FKM model ( J. Mech. Phys. Solids, 1992, 40(5), 1139–1162), developed by Fleck, Kuhn and McMeeking, and a combination of the two models are used. The friction between the mould wall and the metal powder is modelled by a combination of Coulomb friction and a constant friction shear stress, since Coulomb friction is not realistic at high normal pressures. The finite element programme has been used to study the effects of friction, compaction method, and material parameters. Analyses for powder compacts of various geometries are presented to illustrate the method.

The decrease of volume during loading to finite plastic strain

From the 1870s to the present, basic tenets of most theories of plasticity in ordered solids have included the assumption of incompressibility as an internal constraint, the tacit belief that rigid body rotations become large as the finite strain increases, the historical misinterpretation of the location and stress configuration of Tresca's observations for his domain of constant stress, and the unquestioned foregone conclusion that simple shear is always a permissible form of deformation. The laboratory finding in the 1970s that an internal constraint very different from incompressibility applies at finite strain in ordered solids and, with respect to all of the above, the general theory of finite strain plasticity that followed from this laboratory discovery in the 1980s, have raised questions that are described in the present paper. As illustrative, the definitive experiment of P. W. Bridgman in 1946 on the two-dimensional compression of steel is compared with its restudy in more comprehensive detail during the 1980s.

Laboratory experiments on thin-walled tubes at large finite strain: Symmetry, coaxiality, rigid body rotation, and the role of invariants, for the applied stress σ = RTRT, the cauchy stress σ∗ = [IIIV]−1FTRT, and the left cauchy-green stretch tensor V = FRT

From laboratory measurements extending back to the mid-1960s, a frame indifferent general continuum theory for large finite plastic strain in ordered solids has been developed. Its foundations are based on a stress σ that replaces the Cauchy stress σ∗ in the applicable constitutive statements. With W the work per unit volume in the underformed reference configuration, a work function W = traceσV, established from experiment, is consistent with an internal constraint on the left Cauchy-Green stretch tensor, trace V = 3, in whose presence simple shear becomes inadmissible. Compatible with this internal constraint, the decrease in volume during deformation is measured. Also shown in the laboratory is the fact that even when determined just prior to catastrophic collapse, the measured rigid body rotation is too small to play other than a negligible role. In this article, laboratory experimental results are presented which reveal that the Cauchy stress σ∗, the stress σ of the present theory, and the left Cauchy-Green stretch tensor V, are all coaxial and symmetric. Moreover, for the approximation R = I, the second invariant T of the deviatoric stress S and the second invariant Γ of the finite strain E = V − I provide a universal parabolic rule that applies for stress paths of arbitrary composition and direction. This general nonlinear theory of finite strain plasticity during loading is shown here to square in detail with a recently published general nonlinear theory of finite strain elasticity derived from the analytical foundations of rational mechanics by Beatty and Hayes [1992].

Large deflection, rotation, and plastic strain in cantilevered beams

Relatively slender cantilever beams, having a ratio of length to height in ess of 50, are subjected to a series of loads at the free end, resulting in end deflections in which the final displacement is greater than halfunderfored length of the bar. As each load is ado the free end, the curve of deflection, angles of rotation, plastic and elastic strain histories, and rigid body displacement of generic material points are accurately recorded. From thesee data, moments in the underfomed and deformed reference configurations are compared. Since the role of rigid body rotation is found to be negligible, having been cancelled by the rigid body displacement, one may introduce moments in the underormed reference configuration that correlate with ehe experiment in the context of a general incremental theory of finite strain plasticity. The theory satisfies the analytical requirement of material objectivity when, as is found by experiment, the role of the rigid body rotation of principal axes is negligible.