A constitutive model of cyclic plasticity

This paper addresses a constitutive model of cyclic plasticity with special emphasis on the yield-point phenomena. In order to point out the deformation characteristics of a mild steel, four types of experiments were conducted, i.e. uniaxial tension at several crosshead speeds, cyclic straining, and stress- and strain-controlled ratchetting. A viscoplastic constitutive model of cyclic plasticity is proposed on the premise that the phenomena of sharp yield point and the subsequent abrupt yield drop result from rapid dislocation multiplication and the stress-dependence of dislocation velocity. Besides, cyclic plasticity behavior, such as the Bauschinger effect, cyclic hardening/softening characteristics and ratchet-strain accumulation, is described by some kinematic and isotropic hardening rules. The cyclic stress–strain responses predicted by this model agree well with the corresponding experimental results.     

A novel constitutive model for semiconductors: The case of silicon

The photovoltaic industry relies heavily on solar-grade silicon multicrystals. Understanding their mechanical behavior requires the development of adequate constitutive models accounting for the effects of both high dislocation densities and complex loading situations in a wide range of temperature, strain rate, and impurity contents. The traditional model of Alexander and Haasen poses several limitations. We introduce in this work a novel constitutive model for covalent single crystals and its implementation into a rate-dependent crystal plasticity framework. It is entirely physically based on the dislocation generation, storage and annihilation processes taking place during plastic flow. The total dislocation density is segmented according to the dislocation mobility potential and their character. A dislocation multiplication law for the yield region more accurate than the one of Alexander and Haasen is proposed. The influence of additional dislocation sources created on forest trees, usually disregarded in models for semiconductors, is assessed. The dislocation velocity law combines three potentially rate-limiting mechanisms: the standard double kink mechanism, jog dragging and the influence of localized obstacles. The model is valid at finite strains, in multiple slip conditions and captures accurately the high temperature- and strain rate sensitivity of semiconductors. The experimental stress overshoot is qualitatively reproduced only when jog dragging is accounted for. Localized obstacles are shown not to have any significant effect on dislocation motion in silicon. The broader case of extrinsic semiconductors is discussed and the influence of dissolved oxygen on the upper yield stress of silicon monocrystals is successfully reproduced.     

Modeling the plastic response of thin metal membranes

Using a dislocations-based model of slip and crystal plasticity, we show by illustrative examples that the experimentally observed increase in the yield stress of very thin metallic membranes most likely is due to the variation of grain orientations through the thickness of the membrane, as well as the surface hardness due to oxidation or contamination, both of which generally are insignificant when there is a sufficient number of interior crystals through the membrane thickness; the overall effect may well be produced by a combination of these two causes. We show that crystal plasticity models can account for such size effects without a need for resorting to phenomenological strain-gradient models. We illustrate this using Nemat-Nasser's dislocations-based slip-induced crystal plasticity model that inherently includes length scales, although other rate-dependent slip models, e.g., the classical power-law slip model, most likely would qualitatively produce similar results. Our numerical results, based on the experimentally supported dislocation-induced slip model and the values of the model parameters given in Nemat-Nasser and Li [1998. Flow stress of F.C.C. polycrystals with application to OFHC Cu. Acta Mater. 46, 565-577], correlate well, both qualitatively and quantitatively, with the experimental results reported by Hommel and Kraft [2001. Deformation behavior of thin copper films on deformable substrates. Acta Mater. 49, 3935-3947] and Espinosa et al. [2004. Plasticity size effect in free-standing submicron polycrystalline FCC films subjected to pure tension. J. Mech. Phys. Solids 52, 667-689] for thin copper membranes, suggesting that, for submicron-sized samples, the classical crystal plasticity with slip models, does qualitatively account well for the small-size effects, and that quantitative predictions are obtained when, in addition, a physics-based dislocation model that includes length scales, is used. It is thus concluded that the length-scale effect and the size effect are two separate issues in metal plasticity, both of which are nicely accounted for by physics-based dislocation models of crystal plasticity without a need to include the plastic strain gradient.     

Study on the rate-dependent cyclic deformation of super-elastic NiTi shape memory alloy based on a new crystal plasticity constitutive model

In this paper, a crystal plasticity based constitutive model (Yu et al., 2013) is extended to describe the rate-dependent cyclic deformation of super-elastic NiTi shape memory alloy by considering the internal heat production. Two sources of internal heat productions are included in the proposed model, i.e., the mechanical dissipations of inelastic deformation and the transformation latent heat in the NiTi shape memory alloy. With an assumption of uniform temperature field in the alloy specimen, a simplified evolution law of temperature field is obtained by the first law of thermodynamics and the heat boundary conditions. An explicit scale-transition rule is adopted to extend the proposed single crystal model to the polycrystalline version. The capability of the extended polycrystalline model to describe the rate-dependent cyclic deformation of super-elastic NiTi shape memory alloy is verified by comparing the predictions with the corresponding experimental ones. The comparison demonstrates that the proposed constitutive model considering the internal heat production predicts the rate-dependent cyclic deformation of super-elastic NiTi shape memory alloy fairly well.     

Gradient plasticity constitutive model reflecting the ultrafine micro-structure scale: the case of severely deformed copper

A further development of the mechanism-based strain gradient plasticity model well established in literature is reported. The major new element is the inclusion of the cell size effect in dislocation cell forming materials. It is based on a ‘phase mixture’ approach in which the dislocation cell interiors and dislocation cell walls are treated as separate ‘phases’. The model was applied to indentation testing of copper severely pre-strained by equal channel angular pressing. The deformation behaviour and the intrinsic length scale parameter of the gradient plasticity model were related to the micro-structural characteristics, notably the dislocation cell size, resulting from the deformation history of the material.     

Strain gradient plasticity modeling of the cyclic behavior of laminate microstructures

Two recently proposed Helmholtz free energy potentials including the full dislocation density tensor as an argument within the framework of strain gradient plasticity are used to predict the cyclic elastoplastic response of periodic laminate microstructures. First, a rank-one defect energy is considered, allowing for a size-effect on the overall yield strength of micro-heterogeneous materials. As a second candidate, a logarithmic defect energy is investigated, which is motivated by the work of Groma et al. (2003). The properties of the back-stress arising from both energies are investigated in the case of a laminate microstructure for which analytical as well as numerical solutions are derived. In this context, a new regularization technique for the numerical treatment of the rank-one potential is presented based on an incremental potential involving Lagrange multipliers. The results illustrate the effect of the two energies on the macroscopic size-dependent stress–strain response in monotonic and cyclic shear loading, as well as the arising pile-up distributions. Under cyclic loading, stress–strain hysteresis loops with inflections are predicted by both models. The logarithmic potential is shown to provide a continuum formulation of Asaro's type III kinematic hardening model. Experimental evidence in the literature of such loops with inflections in two-phased FFC alloys is provided, showing that the proposed strain gradient models reflect the occurrence of reversible plasticity phenomena under reverse loading.     

织构可调塑性本构模型标定及其应用

提出了利用率相关晶体塑性模型标定织相可调本构模型的求解步骤,得出了一组依赖于晶粒间相互作用假设而独立于具体板材织构的本构相关系数.以此为基础再结合板材织构系数所得出的本构模型系数可避免出现屈服面非外凸的情形.利用所提求解步骤对在不同热处理条件下产生不同织构的AL5052铝合金板的深拉成形过程进行了有限元模拟.结果再现了典型织构在板材成形过程中所出现的塑性各向异性,从而表明求解步骤的可行性.     

Forming simulation of aluminum sheets using an anisotropic yield function coupled with crystal plasticity theory

This paper describes the application of a coupled crystal plasticity based microstructural model with an anisotropic yield criterion to compute a 3D yield surface of a textured aluminum sheet (continuous cast AA5754 aluminum sheet). Both the in-plane and out-of-plane deformation characteristics of the sheet material have been generated from the measured initial texture and the uniaxial tensile curve along the rolling direction of the sheet by employing a rate-dependent crystal plasticity model. It is shown that the stress–strain curves and R-value distribution in all orientations of the sheet surface can be modeled accurately by crystal plasticity if a “finite element per grain” unit cell model is used that accounts for non-uniform deformation as well as grain interactions. In particular, the polycrystal calculation using the Bassani and Wu (1991) single crystal hardening law and experimental electron backscatter data as input has been shown to be accurate enough to substitute experimental data by crystal plasticity data for calibration of macroscopic yield functions. The macroscopic anisotropic yield criterion CPB06ex2 (Plunkett et al., 2008) has been calibrated using the results of the polycrystal calculations and the experimental data from mechanical tests. The coupled model is validated by comparing its predictions with the anisotropy in the experimental yield stress ratio and strain ratios at 15% tensile deformation. The biaxial section of the 3D yield surface calculated directly by crystal plasticity model and that predicted by the phenomenological model calibrated with experimental and crystal plasticity data are also compared. The good agreement shows the strength of the approach. Although in this paper, the Plunkett et al. (2008) yield function is used, the proposed methodology is general and can be applied to any yield function. The results presented here represent a robust demonstration of implementing microscale crystal plasticity simulation with measured texture data and hardening laws in macroscale yield criterion simulations in an accurate manner.     

A cyclic plasticity model for single crystals

The Armstrong–Frederick type kinematic hardening rule was invoked to capture the Bauschinger effect of the cyclic plastic deformation of a single crystal. The yield criterion and flow rule were built on individual slip systems. Material memory was introduced to describe strain range dependent cyclic hardening. The experimental results of copper single crystals were used to evaluate the cyclic plasticity model. It was found that the model was able to accurately describe the cyclic plastic deformation and properly reflect the dislocation substructure evolution. The well-known three distinctive regimes in the cyclic stress–strain curve of the copper single crystals oriented for single slip can be reproduced by using the model. The model can predict the enhanced hardening for crystals oriented for multislip, showing the model's ability to describe anisotropic cyclic plasticity. For a given loading history, the model was able to capture not only the saturated stress–strain response but also the detailed transient stress–strain evolution. The model was used to predict the cyclic plasticity under a high–low loading sequence. Both the stress–strain responses and the microstructural evolution can be appropriately described through the slip system activation.     

Effect of source and obstacle strengths on yield stress: A discrete dislocation study

The combined effect of dislocation source strength τ_s, dislocation obstacle strength τ_obs, and obstacle spacing L_obs on the yield stress of single crystal metals is investigated analytically and numerically. A continuum theory of dislocation pileups emanating from a finite-strength source and impinging on asymmetric obstacles gives a closed-form expression for the yield stress. A 2d discrete dislocation model for a single-source/obstacle problem agrees well with the analytic model over a wide range of material parameters. Discrete dislocation simulations for a full tensile bar with statistically distributed sources and obstacles show that the distribution of obstacles plays a significant role in controlling the yield stress. Over a wide range of parameters, the simulations agree well with the analytic model using an effective obstacle spacing L_obs^* chosen to capture the strength-controlling statistically weaker pileup configurations. The analytic model can thus be used to guide the choice of source and obstacle parameters to obtain a desired yield stress. The model also shows how different combinations of internal source and obstacle parameters can generate the same macroscopic yield stress, and points to several internal length scales that could relate to size-dependent plasticity phenomena.     

An endochronic plasticity formulation for filled rubber

The nature of elastomeric material demands the consideration of finite deformations, nonlinear elasticity including damage as well as rate-dependent and rate-independent dissipative properties. While many models accounting for these effects have been refined over time to do better justice to the real behavior of rubber-like materials, the realistic simulation of the elastoplastic characteristics for filled rubber remains challenging.The classical elastic-ideal-plastic formulation exhibits a distinct yield-surface, whereas the elastoplastic material behavior of filled rubber components shows a yield-surface free plasticity. In order to describe this elastoplastic deformation of a material point adequately, a physically based endochronic plasticity model was developed and implemented into a Finite Element code. The formulation of the ground state elastic characteristics is based on Arruda and Boyce (1993) eight-chain model. The evolution of the constitutive equations for the nonlinear endochronic elastoplastic response are derived in analogy to the Bergström–Boyce finite viscoelasticity model discussed by Dal and Kaliske (2009).     

Constitutive modeling of cyclic plasticity deformation of a pure polycrystalline copper

Cyclic plasticity experiments were conducted on a pure polycrystalline copper and the material was found to display significant cyclic hardening and nonproportional hardening. An effort was made to describe the cyclic plasticity behavior of the material. The model is based on the framework using a yield surface together with the Armstrong–Frederick type kinematic hardening rule. No isotropic hardening is considered and the yield stress is assumed to be a constant. The backstress is decomposed into additive parts with each part following the Armstrong–Frederick type hardening rule. A memory surface in the plastic strain space is used to account for the strain range effect. The Tanaka fourth order tensor is used to characterize nonproportional loading. A set of material parameters in the hardening rules are related to the strain memory surface size and they are used to capture the strain range effect and the dependence of cyclic hardening and nonproportional hardening on the loading magnitude. The constitutive model can describe well the transient behavior during cyclic hardening and nonproportional hardening of the polycrystalline copper. Modeling of long-term ratcheting deformation is a difficult task and further investigations are required.     

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.     

An analysis of exhaustion hardening in micron-scale plasticity

The rate-dependent behavior of micron-scale model planar crystals is investigated using the framework of mechanism-based discrete dislocation plasticity. Long-range interactions between dislocations are accounted for through elasticity. Mechanism-based constitutive rules are used to represent the short-range interactions between dislocations, including dislocation multiplication and dislocation escape at free surfaces. Emphasis is laid on circumstances where the deformed samples are not statistically homogeneous. The calculations show that dimensional constraints selectively set the operating dislocation mechanisms, thus giving rise to the phenomenon of exhaustion hardening whereby the applied strain rate is predominantly accommodated by elastic deformation. When conditions are met for this type of hardening to take place, the calculations reproduce some interesting qualitative features of plastic deformation in microcrystals, such as flow intermittency over coarse time-scales and large values of the flow stress with no significant accumulation of dislocation density. In addition, the applied strain rate is varied down to 0.1 s⁻¹ and is found to affect the rate of exhaustion hardening.     

A substructure mixtures model for the cyclic plasticity of single slip oriented nickel single crystal at low plastic strain amplitudes

The cyclic plasticity behavior of nickel single crystals oriented for single slip is characterized by uniaxial, symmetric, tension–compression, strain controlled tests carried out at constant plastic strain amplitudes ranging from 5(10⁻⁵) to 1(10⁻³). Annealed single crystals are cycled in this manner to post-cyclic saturation and microstructural characterizations, including transmission electron microscopy and optical micrographs of specimen surface replicas are used to verify and evaluate dislocation substructures. Stress–strain and microstructure data are used to construct a mixtures model that couples cyclic plasticity models for three substructures as well as a model for reverse magnetostriction (Villari effect) that is a significant component of inelastic strain at the lower plastic strain amplitudes. The model is used to correlate the stress–plastic strain hysteresis loop responses over the range of plastic strain amplitudes and from cumulative plastic strains from 0.3 to post-cyclic saturation. Complex evolution of substructure plastic strain amplitudes toward their so-called intrinsic values upon the formation of persistent slip bands is modeled. Additionally, bulk Young’s modulus is found to vary significantly with plastic strain amplitude and cumulative plastic strain. A correlation of this behavior is included.     

Micromechanical modeling of the elasto-viscoplastic behavior of semi-crystalline polymers

A micromechanically based constitutive model for the elasto-viscoplastic deformation and texture evolution of semi-crystalline polymers is developed. The model idealizes the microstructure to consist of an aggregate of two-phase layered composite inclusions. A new framework for the composite inclusion model is formulated to facilitate the use of finite deformation elasto-viscoplastic constitutive models for each constituent phase. The crystalline lamellae are modeled as anisotropic elastic with plastic flow occurring via crystallographic slip. The amorphous phase is modeled as isotropic elastic with plastic flow being a rate-dependent process with strain hardening resulting from molecular orientation. The volume-averaged deformation and stress within the inclusions are related to the macroscopic fields by a hybrid interaction model. The uniaxial compression of initially isotropic high density polyethylene (HDPE) is taken as a case study. The ability of the model to capture the elasto-plastic stress-strain behavior of HDPE during monotonic and cyclic loading, the evolution of anisotropy, and the effect of crystallinity on initial modulus, yield stress, post-yield behavior and unloading-reloading cycles are presented.     

Plastic spin associated with a non-normality theory of plasticity

A plasticity model using a vertex-type plastic flow rule on a smooth yield surface for an anisotropic solid has been proposed recently. This model is here completed by incorporating the effect of plastic spin. Simple shear with a large shear strain is one of the hardest tests on finite strain anisotropic plasticity models, and it is here shown which plastic spin expression is needed to avoid unrealistic oscillatory behavior of the shear stress under large shear strains. The idea of using non-normality with a smooth yield surface originates from a recent proposal of using an abrupt strain path change to determine the subsequent yield surface shape. For this method both polycrystal plasticity calculations and experiments have shown a vertex-type response on the apparently smooth yield surface.     

Modeling the Bauschinger effect for sheet metals,part I: theory

It is essential to model the Bauschinger effect correctly for sheet metal forming process simulation and subsequent springback prediction when material points are subjected to cyclic loading conditions. The combined nonlinear hardening model for time independent cyclic plasticity, proposed by Chaboche and co-workers, is examined and a simple modification is suggested for the isotropic part of the hardening rule to utilize the conventional tensile test data directly. This modification is useful for the materials whose reverse loading curves saturate to the monotonic loading curve. In addition, an anisotropic nonlinear kinematic hardening model (ANK model) is proposed in an attempt to represent the Bauschinger effect more realistically. Possible offset in flow stress is modeled by treating the back stress evolution during reverse loading differently from the initial loading. This strategy coupled with the modified isotropic hardening rule seems to provide a way to model the Bauschinger effect consistently over multiple cycles. Two types of auto-body alloys are examined in this paper. Associated material parameters are determined by employing available tension-compression test data and multi-cycle bend test data. A developed finite element formulation is applied to analyze simple validation type of problems. The cyclic stress–strain curves generated from the proposed ANK model match remarkably well with measured data.