Continuum Plasticity Theory Research Articles

A Three‐dimensional Continuum Theory of Dislocation Plasticity ‐ Modelling and Application to a Composite

AbstractThe increasing demand for materials with well defined microstructure, which is accompanied by the advancing miniaturization of devices calls for physically motivated, dislocation‐based continuum theories of plasticity. Only recently rigorous techniques have been developed for performing meaningful averages over systems of moving, curved dislocations, yielding evolution equations for a higher order dislocation density tensor. Our continuum dislocation theory allows for generalizing the planar system towards a three‐dimensional system, where dislocations may have arbitrary orientation and curvature. With the inclusion of curvature, the theory naturally takes into account a deformation‐induced increase in the overall dislocation density without having to invoke ad‐hoc assumptions about dislocation sources. A numerical implementation and some benchmark tests of this continuum theory for dislocation dynamics has already been discussed in the literature. In this paper, we apply this continuum theory to composite materials, where we analyze a plastically deforming matrix with an elastic inclusion. (© 2011 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

On extension of the phenomenological approach in the theory of plasticity

Existing constitutive theories are commonly classified into three large groups: continuum, structural and micro structural. As a rule micro structural models, which are often referred to as physical, are based on the concept of crystallographic slip. One of the interesting exceptions is the theory of plasticity which accounts for micro strains. It postulates generalised deformation laws on mesoscale, and therefore has common properties of micro structural theories. However, it does not account for nanoprocesses, such as behaviour and interaction of separate dislocations, leads to the Von Mises initial yield condition, and accounts for Ilyushin’s postulate of isotropy and other macroscopic phenomenological laws of inelastic deformation. Thus, the theory of micro strains has much in common with continuum theories of plasticity and also establishes connection between continuum and micro structural approaches. In this article previous separate results concerning the theory of microstrains are reviewed and united into a single framework. The resulting generalised variant of the theory of micro strains accounts for both slip and twinning deformation mechanisms and also accurately models polycrystalline grain size distribution. Special cases and some important applications of the theory are discussed.

Dislocation Density-Based Finite Element Method Modeling of Ultrasonic Consolidation

A dislocation density based constitutive model has been developed and implemented into a crystal plasticity quasi‐static finite element framework. This approach captures the statistical evolution of dislocation structures and grain fragmentation at the bonding interface when sufficient boundary conditions pertaining to the Ultrasonic Consolidation process are prescribed. Hardening is incorporated using statistically stored and geometrically necessary dislocation densities (SSDs and GNDs), which are dislocation analogs of isotropic and kinematic hardening respectively. The GND considers strain‐gradient and thus renders the model size‐dependent. The model is calibrated using experimental data from published refereed literature and then validated for the Aluminum 3003 alloy. Introduction As a direct result of ongoing research efforts in ultrasonic consolidation (UC) worldwide [1], it has become apparent that a new approach to modeling of UC bonding is needed. A model which provides a better understanding of the effects of process parameter changes on grain refinement, plastic deformation and bonding during UC will better enable researchers to predict which materials will bond, how the mechanical properties of UC-produced parts can be improved, and how to better design the next generation of UC equipment. The continuum properties of parts made using UC are strongly dependent upon the micromechanics of the bonded interface [1]. Interfacial-scale microstructures can be studied fundamentally using electron microscopy [1] and can be used to correlate atomic and mesoscopic mechanisms of deformation to their continuum counterparts. A dislocation density-based crystal plasticity Finite Element Model (FEM) can capture the statistical distribution of dislocations, partials and various deformation mechanisms at the bonding interface as inputs to predict macroscopic deformation profiles as a function of energy input characteristics. These input characteristics are a function of the process parameters used in a UC machine, namely vibration amplitude, normal force, ultrasonic frequency, welding speed, sonotrode geometry and temperature. Problem Formulation It has been shown that material sheets subjected to UC undergo inhomogeneous plastic deformation through their thickness [1]. Classical continuum plasticity theories do not fully explain this phenomenon [2]. Therefore, a study of strain localization and grain refinement at the material interfaces during UC bonding is required. The following steps lead to the calculation of these localized strains and their effects. Large Deformation Quasi-Static Crystal Plasticity Description The deformation map in space and time is described by the total deformation gradient tensor F (Figure 1). Applying the Kroner-Lee assumption, F is decomposed into elastic Fe and plastic gradient Fp tensors using multiplicative operator theory

Controlling Plastic Flow across Grain Boundaries in a Continuum Model

A framework for modeling controlled plastic flow through grain boundaries using a continuum plasticity theory, phenomenological mesoscopic field dislocation mechanics (PMFDM), is presented in this article. The developed tool is used to analyze the effect of different classes of constraints to plastic flow through grain boundaries, as it relates to dislocation microstructure development and mechanical response of a bicrystal. It is found that in the case of low misorientation angle between adjacent grains, impenetrable grain boundaries cause significant work hardening as compared to penetrable grain boundaries due to the accumulation of excess dislocations along them. However, a penetrable grain boundary with a high misorientation angle effectively behaves as an impenetrable boundary, with respect to the stress-strain response.

Application of a 3D-Continuum Theory of Dislocations to a Problems of Constrained Plastic Flow: Microbending of a Thin Film

AbstractThe advancing miniaturisation of e.g. microelectronic devices leads to an increasing interest in physically motivated continuum theories of plasticity in small volumes. Such theories need to be based on the averaged dynamics of dislocations. Preserving the line-like character of these defects, however, posed serious problems for the development of dislocation-based continuum theories, while continuum theories based on scalar dislocation densities necessarily stay on a phenomenological level. Within this work we apply a dislocation-based continuum theory, which is based on a physically meaningful averaging of dislocation lines, to the benchmark problem of bending of a free-standing thin film.

A Plastic-Damage Model for Lightweight Concrete and Normal Weight Concrete

A new plastic-damage constitutive model applicable to lightweight concrete (LWC) and normal weight concrete (NWC) is proposed in this paper based on both continuum damage mechanics and plasticity theories. Two damage variables are used to represent tensile and compressive damage independently. The effective stress is computed in the Drucker-Prager multi-surface plasticity framework. The stress is then computed by multiplication of the damaged part and the effective part. The proposed model is coded as a user material subroutine and incorporated in a finite element analysis software. The constitutive integration algorithm is implemented by adopting the operator split involving elastic predictor, plastic corrector and damage corrector. The numerical study shows that the algorithm is efficient and robust in the finite element analysis. Experimental investigation is conducted to verify the proposed model involving both static and dynamic tests. The very good agreement between the numerical results and experimental results demonstrates the capability of the proposed model to capture the behaviors of LWC and NWC structures for static and impact loading.

An elastoplatic framework for coupling hydraulic and mechanical behavior of unsaturated soils

Combining theory of mixture with interfaces (TMI) and the continuum theory of plasticity, a theoretical framework for modeling the elastoplastic constitutive behavior of unsaturated soils is presented. We show that capillary hysteresis, i.e., soil water characteristic curves (SWCCs), is associated with a dissipation mechanism due to the irrecoverable change in volume fraction of water. Within this context, plastic deformation and capillary hysteresis can be consistently simulated in a hierarchical way. Plastic deformation is described by using an intergranular stress tensor that does not require any empirical coefficients and the capillary hysteresis is simulated within the framework of cyclic plasticity. This framework preserves all the advantages of unsaturated soil models based on the effective stress, two stress state variables, and the theory of mixtures. As a first step in developing a comprehensive constitutive model within this framework, a detailed model to simulate SWCCs is presented and compared with experimental data and a good comparison is obtained. A brief description of a novel experimental setup used to rapidly measure the SWCCs is also presented. Finally an isotropic coupled hydraulic–mechanical model is presented and the coupling effects between plastic deformation and capillary hysteresis are investigated.

Modeling of the Size Effects on the Behavior of Metals in Microscale Deformation Processes

For the accurate analysis and design of microforming process, proper modeling of material behavior at the micro/mesoscale is necessary by considering the size effects. Two size effects are known to exist in metallic materials. One is the “grain size” effect, and the other is the “feature/specimen size” effect. This study investigated the feature/specimen size effect and introduced a scaling model which combined both feature/specimen and grain size effects. Predicted size effects were compared with three separate experiments obtained from previous research: a simple compression with a round specimen, a simple tension with a round specimen, and a simple tension in sheet metal. The predicted results had a very good agreement with the experiments. Quantification of the miniaturization effect has been achieved by introducing two parameters, α and β, which can be determined by the scaling parameter n, to the Hall–Petch equation. The scaling model offers a simple way to model the size effect down to length scales of a couple of grains and to extend the use of continuum plasticity theories to micro/mesolength scales.

Size effects and idealized dislocation microstructure at small scales: Predictions of a Phenomenological model of Mesoscopic Field Dislocation Mechanics: Part I

A Phenomenological Mesoscopic Field Dislocation Mechanics (PMFDM) model is developed, extending continuum plasticity theory for studying initial-boundary value problems of small-scale plasticity. PMFDM results from an elementary space-time averaging of the equations of Field Dislocation Mechanics (FDM), followed by a closure assumption from any strain-gradient plasticity model that attempts to account for effects of geometrically necessary dislocations (GNDs) only in work hardening. The specific lower-order gradient plasticity model chosen to substantiate this work requires one additional material parameter compared to its conventional continuum plasticity counterpart. The further addition of dislocation mechanics requires no additional material parameters. The model (a) retains the constitutive dependence of the free-energy only on elastic strain as in conventional continuum plasticity with no explicit dependence on dislocation density, (b) does not require higher-order stresses, and (c) does not require a constitutive specification of a ‘back-stress’ in the expression for average dislocation velocity/plastic strain rate. However, long-range stress effects of average dislocation distributions are predicted by the model in a mechanistically rigorous sense. Plausible boundary conditions (with obvious implication for corresponding interface conditions) are discussed in some detail from a physical point of view. Energetic and dissipative aspects of the model are also discussed. The developed framework is a continuous-time model of averaged dislocation plasticity, without having to rely on the notion of incremental work functions, their convexity properties, or their minimization. The tangent modulus relating stress rate and total strain rate in the model is the positive-definite tensor of linear elasticity, and this is not an impediment to the development of idealized microstructure in the theory and computations, even when such a convexity property is preserved in a computational scheme. A model of finite deformation, mesoscopic single crystal plasticity is also presented, motivated by the above considerations. Lower-order gradient plasticity appears as a constitutive limit of PMFDM, and the development suggests a plausible boundary condition on the plastic strain rate for this limit that is appropriate for the modeling of constrained plastic flow in three-dimensional situations.

Elastoplastic modeling of progressive interfacial debonding for particle-reinforced metal-matrix composites

The purpose of this study is to model the elastoplastic behavior of particle-reinforced metal-matrix composites with particle-matrix interfacial debonding. The partially debonding process at the interface is represented by the debonding angles. The equivalent orthotropic elastic moduli are constructed for the debonded yet isotropic particles to characterize the reduction of the load-transfer ability in the debonded directions. To simulate the debonding evolution and the transition between various debonding modes, the volume fractions of various particles are expressed in terms of the Weibull's statistical functions. Micromechanical homogenization procedures are utilized to estimate the effective moduli and the overall yield function of the resultant multi-phase composites. The associative plastic flow rule and isotropic hardening law are postulated based on the continuum plasticity theory. The effects of partially interfacial debonding on the overall yield surfaces and stress-strain relations of the composites are investigated and illustrated via numerical examples as well.

Calculation of Relaxation of Residual Stress and Change of Yield Strength in Shot Peened Layer

Residual stresses distribution after shot peening and its relaxation during uni-axial loading were investigated experimentally and theoretically on stainless steel of AISI304. An analytical model was proposed based upon continuum plasticity theory and elasto-plasticity finite element (FE) analysis, in which both relaxation of shot peening induced residual stress and variation of yield strength in the surface layer were taken into consideration. The results show that the value of yield strength in surface layer is altered after shot peening, and the relaxation of residual stress takes place at the point where the combination of applied strain and residual strain exceeds the true value of yield strength. A fundamental agreement was proved between measurements and predictions in simple tension and compression.

On failure of continuum plasticity theories on small scales

Abstract We argue that continuum plasticity theories cannot describe plastic behavior if the characteristic length of the problem becomes of the order of the decorrelation length of the dislocation structure.

Finite element approximation of field dislocation mechanics

A tool for studying links between continuum plasticity and dislocation theory within a field framework is presented. A finite element implementation of the geometrically linear version of a recently proposed theory of field dislocation mechanics (J. Mech. Phys. Solids 49 (2001) 761; Proc. Roy. Soc. 459 (2003) 1343; J. Mech. Phys. Solids 52 (2004) 301) represents the main idea behind the tool. The constitutive ingredients of the theory under consideration are simply elasticity and a specification of dislocation velocity and nucleation. The set of equations to be approximated are non-standard in the context of solid mechanics applications. It comprises the standard second-order equilibrium equations, a first-order div–curl system for the elastic incompatibility, and a first-order, wave-propagative system for the evolution of dislocation density. The latter two sets of equations require special treatment as the standard Galerkin method is not adequate, and are solved utilizing a least-squares finite element strategy. The implementation is validated against analytical results of the classical elastic theory of dislocations and analytical results of the theory itself. Elastic stress fields of dislocation distributions in generally anisotropic media of finite extent, deviation from elastic response, yield-drop, and back-stress are shown to be natural consequences of the model. The development of inhomogeneity, from homogeneous initial conditions and boundary conditions corresponding to homogeneous deformation in conventional plasticity, is also demonstrated. To our knowledge, this work represents the first computational implementation of a theory of dislocation mechanics where no analytical results, singular solutions in particular, are required to formulate the implementation. In particular, a part of the work is the first finite element implementation of Kröner's linear elastic theory of continuously distributed dislocations in its full generality.

Some Issues of Rough Surface Contact Plasticity at Micro- and Nano-Scales

ABSTRACTRough surface contact plasticity at microscale and nanoscale is of crucial importance in many new applications and technologies, such as nano-imprinting and nano-welding. This paper summarizes our recent progress in understanding contact plasticity from a multiscale point of view, and also presents our perspectives. We first discuss a contact model based on fractal roughness and continuum plasticity theory. Interestingly, our simple, elastic-plastic contact model of the Weierstrass-Archard type gives rise to many practical scaling relations of contact pressure, contact compliance etc. The usefulness of those predictions is discussed for experimental measurements of the thermal/electrical contact resistance. A material length scale can be introduced by a nonlocal plasticity theory, or implicitly by dislocation mechanics modeling. The recent work on micro-plasticity of surface steps gives a variety of surface yielding and hardening behaviors, depending on interface adhesion, roughness features and slip systems. As a consequence, a rough surface contact at mesoscale can lead to the formation of a boundary layer with sub-layer dislocation structures, which cannot be predicted by existing strain gradient plasticity theories. The micromechanical analysis of surface plasticity could serve as the connection between microscale bulk dislocation plasticity and nanoscale atomistic simulations.

A method for linking thermally activated dislocation mechanisms of yielding with continuum plasticity theory

This work combines classical continuum mechanics, the continuously dislocated continuum theory as developed by Kröner and Bilby with discrete dislocation theory to develop quantities that permit models involving interactions between individual dislocations to be incorporated into a description of multiaxial yielding of a material. Two quantities distinguish this approach from earlier efforts: firstly, a dislocation mobility tensor relating the velocity of a dislocation configuration to the net Peach–Koehler force on the configuration and, secondly, a vector quantity representing the dislocation content of the materials. The theory of thermally activated motion of dislocations past obstacles is employed to relate the dislocation velocity to stress by a stress-dependent mobility tensor whose components are determined by the nature of the interaction of the moving dislocation with the obstacle. An example is presented in which the obstacle is a forest dislocation that affects a gliding dislocation through mutual interaction of their stress fields. The development leads to a quantity that can be used as a plastic potential for the construction of an associated flow law.

Effect of particle cracking on elastoplastic behaviour of metal matrix composites

AbstractA micromechanics‐based elastoplastic and damage constitutive model is proposed to predict the overall mechanical behaviour of particle‐reinforced metal matrix composites. Unidirectionally aligned spheroidal elastic particles, some of which contain penny‐shaped cracks, are randomly distributed in the elastoplastic metal matrix. These imperfect particles, attributed to progressive particle cracking, are modelled by using the double‐inclusion concept. An ensemble‐volume averaged homogenization procedure is employed to estimate the effective yield function of the damaged composites. The associative plastic flow rule and the hardening law are postulated based on the continuum plasticity theory. The underlying damage evolution of particles is considered in accordance with the Weibull's statistical function to characterize the varying probability of reinforcement cracking. The elastoplastic mechanical behaviour of particulate composites under uniaxial loading conditions is simulated and compared with available experimental results. Copyright © 2003 John Wiley & Sons, Ltd.

Dislocation Plasticity Effects on Interfacial Fracture

Analyses are reviewed where plastic flow in the vicinity of an interfacial crack is represented in terms of the nucleation and glide of discrete dislocations. Attention is confined to cracks along a metal-ceramic interface, with the ceramic idealized as being rigid. Both monotonic and fatigue loading are considered. The main focus is on the stress and deformation fields near the crack tip predicted by discrete dislocation plasticity, in comparison with those obtained from conventional continuum plasticity theory. The role that discrete dislocation plasticity can play in interpreting interface fracture properties in the presence of plastic flow is discussed.

Fracture and cavitation in a constrained thin metal layer under a scale effect in layered materials

Dislocation activities are confined within a thin metal layer. Therefore instead of continuum plasticity theory, individual dislocation activities are considered in order to analyse their effects on fracture, especially interface fracture. Three failure modes may occur in the thin ductile layer: interface fracture, metal fracture and metal cavitation. These failure modes are studied and the competition between them is examined. It seems that interface fracture occurs prior to metal fracture provided that the cohesive strengths of the interface and the metal are similar. In general, the fracture toughness of the thin layer will increase with increasing layer thickness. However, at a layer thickness of about 10 mm, the layer is more likely to fail by interface debonding, prior to any failure by ductile cavitation. Finally, using material and geometric parameters, a relation is given which determines the competition between crack fracture and cavity instability.

Elastoplastic modeling of metal matrix composites with evolutionary particle debonding

A microstructural-overall two-level elastoplastic and damage model is proposed to predict the overall mechanical behavior of particle-reinforced metal matrix composites. Unidirectionally aligned spheroidal elastic particles, some of which are partially debonded from the matrix, are randomly distributed in the ductile medium. These imperfect particles are modeled by fictitious orthotropic inclusions without debonding. An ensemble-volume averaged homogenization procedure is employed to estimate the effective yield function of the said composites. The associative plastic flow rule and the hardening law are postulated based on the continuum plasticity theory. The evolution of volume fraction of debonding particles is considered in accordance with Weibull’s statistical function to characterize the varying probability of reinforcement debonding. The uniaxial elastoplastic stress–strain behavior of particle composites is investigated as the first application. In particular, comparisons between the proposed uniaxial stress–strain predictions and experimental data are performed to illustrate the capability of the proposed method. Furthermore, the effect of stress triaxiality is discussed under either the purely hydrostatic or axisymmetric loading on the overall elastoplastic behavior of composites.

Geometrically necessary dislocation and size-dependent plasticity

There has recently been a strong interest in modeling size-dependent plasticity in metals based on the concept of geometrically necessary dislocations. This article presents a brief summary of our viewpoints on geometrically necessary dislocations and their role in the development of continuum plasticity theories with an intrinsic material length scale.