Influence of deformation path on the strain hardening behavior and microstructure evolution in low SFE FCC metals

Following our recent studies of the influence of mechanical twinning on the strain hardening of low SFE FCC metals deformed by simple compression, the investigation was extended to two different deformation modes. These were plane strain compression and simple shear carried out on 70/30 brass, which exhibits only strain hardening, and on MP35N, a Co–Ni based alloy that also shows secondary hardening by deformation promoted precipitation. It was found that the magnitude of the primary strain hardening in both alloys, and the secondary hardening in MP35N, was dramatically reduced under simple shear compared to the other deformation paths. This reduced hardening in simple shear appears to be a consequence of the bulk of the deformation twins, and also the secondary hardening precipitates, forming on planes that were parallel to the primary {111} slip planes in this deformation path. These hypotheses are supported by deformation path change tests in which the shear samples that show low flow stress under continued shear, when subjected to simple compression showed a significant increase (jump) in the flow stress, reaching values that are similar to those of the alloy continuously compressed to the same equivalent strain. That is, the reduced strain hardening in shear deformation is due not to reduced twinning, but to the twins produced by shear providing only limited barriers to continued strain by simple shear. Shear banding was found to be more marked in plane strain compression than in simple compression after cold working, and particularly after the additional secondary hardening in MP35N.     

A modified model for simulating latent hardening during the plastic deformation of rate-dependent FCC polycrystals

A strain hardening model for the plastic deformation of rate-dependent FCC crystals is proposed based on experimental observations previously reported for single crystals. This model, which is an extension of that employed by et al. [1983], includes both the self-hardening and latent hardening of the slip systems. The differential hardening of the latent systems is assumed to arise from the interaction between glide dislocations and forests. With this hardening model and a rate-sensitive crystal plasticity theory, the deformation behavior of FCC polycrystals can be predicted from the deformation response of the constituent single crystals. As examples, the uniaxial tensile behaviour of pure aluminum and copper polycrystals is simulated using the extended model, and the results are compared with published experimental data. The effects of latent hardening on polycrystal deformation, especially on flow stress and the formation of tensile textures, are discussed.     

Physics-based plasticity model incorporating microstructure changes for severe plastic deformation

During machining processes, materials undergo severe deformations that lead to different behavior than in the case of slow deformation. The microstructure changes, as a consequence, affect the materials properties and therefore influence the functionality of the component. Developing material models capable of capturing such changes is therefore critical to better understand the interaction process–materials. In this paper, we introduce a new physics model associating Mechanical Threshold Stress (MTS) with Dislocation Density (DD) models. The modeling and the experimental results of a series of large strain experiments on polycrystalline copper (OFHC) involving sequences of shear deformation and strain rate (varying from quasi-static to dynamic) are very similar to those observed in processes such as machining. The Kocks–Mecking model, using the mechanical threshold stress as an internal state variable, correlates well with experimental results and strain rate jump experiments. This model was compared to the well-known Johnson–Cook model that showed some shortcomings in capturing the stain jump. The results show a high effect of rate sensitivity of strain hardening at large strains. Coupling the mechanical threshold stress dislocation density (MTS–DD), material models were implemented in the Abaqus/Explicit FE code. The model shows potentialities in predicting an increase in dislocation density and a reduction in cell size. It could ideally be used in the modeling of machining processes.     

Plastic deformation modeling of AL-6XN stainless steel at low and high strain rates and temperatures using a combination of bcc and fcc mechanisms of metals

Combination of physically based constitutive models for body centered cubic (bcc) and face centered cubic (fcc) metals developed recently by the authors [Voyiadjis, G.Z., Abed, F.H., 2005. Microstructural based models for bcc and fcc metals with temperature and strain rate dependency. Mech. Mater. 37, 355–378] are used in modeling the plastic deformation of AL-6XN stainless steel over a wide range of strain rates between 0.001 and 8300 s⁻¹ at temperatures from 77 to 1000 K. The concept of thermal activation analysis as well as the dislocation interaction mechanism is used in developing the plastic flow model for both the isothermal and adiabatic plastic deformation. In addition, the experimental observations of AL-6XN conducted by Nemat-Nasser et al. [Nemat-Nasser, S., Guo, W., Kihl, D., 2001. Thermomechanical response of AL-6XN stainless steel over a wide range of strain rates and temperatures, J. Mech. Phys. Solids 49, 1823–1846] are utilized in understanding the underlying deformation mechanisms. The plastic flow is considered in the range of temperatures and strain rates where diffusion and creep are not dominant, i.e., the plastic deformation is attributed to the motion of dislocations only. The modeling of the true stress–true strain curves for AL-6XN stainless steel is achieved using the classical secant modulus for the case of unidirectional deformation. The model parameters are obtained using the experimental results of three strain rates (0.001, 0.1, and 3500 s⁻¹). Good agreement is obtained between the experimental results and the model predictions. Moreover, the independency of the present model to the experiments used in the modeling is verified by comparing the theoretical results to an independent set of experimental data at the strain rate of 8300 s⁻¹ and various initial temperatures. Good correlation is observed between the model predictions and the experimental observations.     

Constitutive modeling of textured body-centered-cubic (bcc) polycrystals

A new latent hardening model for body-centered-cubic (bcc) single crystals motivated by the inapplicability of the Schmid law (Critical Resolved Shear Stress Criterion) is presented. This model is based on the asymmetry of shearing resistance of the {112} slip planes depending on the shearing direction in the sense of ‘twin’ and ‘anti-twin’. For the interpretation of deformation of polycrystalline aggregates depending upon initial texture, a constitutive law for bcc single crystals is developed. This law is based on a rigorous constitutive theory for crystallographic slip that accounts for the effects of strain hardening, rate-sensitivity and thermal softening. The deformation response of textured polycrystal is investigated by means of a Taylor type averaging scheme and an established numerical procedure. Results for textured tungsten polycrystals at low and high strain rates for two different textures [001] and [011] are presented and compared with experimental results. The predictions compare well with experimental observations for the [001] texture. In the [011] texture, due to the reduced symmetry of deformation, lateral tensile stresses develop even under uniaxial compression. These lateral tensile stresses are responsible for observed lack of ductility and transgranular failure in the [011] texture.     

A plane stress anisotropic plastic flow theory for orthotropic sheet metals

A phenomenological theory is presented for describing the anisotropic plastic flow of orthotropic polycrystalline aluminum sheet metals under plane stress. The theory uses a stress exponent, a rate-dependent effective flow strength function, and five anisotropic material functions to specify a flow potential, an associated flow rule of plastic strain rates, a flow rule of plastic spin, and an evolution law of isotropic hardening of a sheet metal. Each of the five anisotropic material functions may be represented by a truncated Fourier series based on the orthotropic symmetry of the sheet metal and their Fourier coefficients can be determined using experimental data obtained from uniaxial tension and equal biaxial tension tests. Depending on the number of uniaxial tension tests conducted, three models with various degrees of planar anisotropy are constructed based on the proposed plasticity theory for power-law strain hardening sheet metals. These models are applied successfully to describe the anisotropic plastic flow behavior of 10 commercial aluminum alloy sheet metals reported in the literature.     

Analysis of stress-strain relations by use of an anisotropic hardening plastic potential

A method of analyzing plastic behavior by use of an anisotropic hardening plastic potential is proposed. The plastic potential surface in deviatoric stress space is assumed to be the same as the equi-plastic-strain surface. Stress-strain relations in combined loading and in multi-axial cyclic loading are calculated by use of the anisotropic hardening plastic potential and the normality rule of the plastic strain increment vector to the plastic potential surface, which are experimentally determined or confirmed by subjecting thinwalled tubular test specimens of $\text{60}\text{40}$ brass to combined axial load, internal pressure and torsion. The calculated results agree fairly well with the experimental observations.     

Anisotropic yield criterion for polycrystalline metals using texture and crystal symmetries

An anisotropic yield criterion for polycrystalline metals which uses texture data and takes advantage of crystal symmetries is presented. A linear transformation is developed to map an anisotropic yield surface for a polycrystal to an appropriate isotropic yield surface. The transformation developed reflects the symmetry of the material being modeled. First, the transformation is determined. Then, information regarding the orientation distribution (texture) of the crystals in a polycrystalline aggregate is used to determine, via averaging, the transformation for the polycrystal. The transformation, along with appropriate isotropic yield surface, provides a phenomenological approach to modeling yield, yet accounts for microstructural texture. The approach reduces to the Hill (1950) anisotropic plasticity theory under certain conditions. The yield surfaces and R-values for various face-centered-cubic ( fcc) polycrystalline textures are computed by this method. Results compare favorably with those given by other theories, and with experiment. The method proves to have the computational efficiency of phenomenological approaches to modeling yield, while effectively incorporating the physics of more complex crystallographic approaches.     

A novel approach for anisotropic hardening modeling. Part II: Anisotropic hardening in proportional and non-proportional loadings,application to initially isotropic material

The modeling of anisotropic hardening, in particular for non-proportional loading paths, is a challenging task for advanced macroscopic models. The complex distortion of the yield locus is related to the activation and cross-hardening of different slip systems, depending on crystallographic orientations. These physical mechanisms can be taken into account in polycrystalline models but the computation times are enormous. The novel approach detailed in Part I (Rousselier et al., 2009) consists in: (i) drastically reducing the number of crystallographic orientations to save the computation cost, (ii) applying a parameter calibration procedure to obtain a good agreement with the experimental database. This methodology is first applied here to the anisotropic hardening in the proportional loadings of the strongly anisotropic aluminum alloy of Part I. Very good modeling is achieved with only eight crystallographic orientations. Different levels of additional hardening in biaxial proportional loading as compared to uniaxial loading can be modeled with the same polycrystalline model. For this, only the parameter calibration has to be performed with different databases. The same methodology has also been applied for the modeling of isotropic behavior. The best compromise between model accuracy and numerical cost is obtained with fourteen orientations. The deviations from isotropy are acceptable in all loading directions. Different levels of hardening in orthogonal loading: simple shear followed by simple tension, are achieved without any modification of the model equations. Only the parameter calibration has to be performed with different hardening levels in the database. FE calculations of a deep drawing test have been performed. The CPU time of the polycrystalline model is only five times larger than that with the simple von Mises model. The CPU time with texture evolution is further increased by a factor of two. The effects of texture evolution in rolling of the initially isotropic fcc material have been investigated. The resulting texture and hardening are qualitatively good.     

Kinematic,stress, and hardening analysis in finite double slip

A kinematic, stress, and hardening analysis of finite double slip in fcc crystals under axial loading is presented. The relative amounts of slip in classic 1925 experiments by Taylor and Elam (determined analytically in [Int. J. Plasticity 9 (1993) 159–179] by comparing theoretical and experimental cones of unextended directions), together with load-extension data and other measurements, are used to calculate resolved shear stress vs. slip curves and assess predictions of several finite distortional hardening theories. In particular, a new hardening rule is introduced that gives very close agreement with the anisotropic experimental results in double slip yet is consistent with the axisymmetric deformation, lattice stability, and isotropic hardening that are characteristic of fcc and bcc crystals in high symmetry axial-load orientations involving 6- and 8-fold slip.     

An investigation of the combined size and rate effects on the mechanical responses of FCC metals

The molecular dynamics (MD) simulations are performed with single-crystal copper blocks under simple shear to investigate the size and strain rate effects on the mechanical responses of face-centered cubic (fcc) metals. It is shown that the yield stress decreases with the specimen size and increases with the strain rate. Based on the theory of dislocation nucleation, a modified power law is proposed to predict the scaled behavior of fcc metals, which agrees well with the numerical and experimental data ranging from nanoscale to macro-scale. In the MD simulations with different strain rates, a critical strain rate exists for each single-crystal copper block of given size, below which the yield stress is nearly insensitive to the strain rate. A hyper-surface is therefore formulated to describe the combined size and strain rate effects on the plastic yield strength of fcc metals. The preliminary results presented in this paper demonstrate the potential of the proposed simple procedure for engineering design at various spatial and temporal scales.     

Evolution of plastic anisotropy in amorphous polymers during finite straining

The large strain deformation response of amorphous polymers results primarily from orientation of the molecular chains within the polymeric material during plastic straining. Molecular network orientation is a highly anisotropic process, thus the observed mechanical response is strongly a function of the anisotropic state of these materials. Through mechanical testing and material characterization, the nature of the evolution of molecular orientation under different conditions of state of strain is developed. The role of developing anisotropy on the mechanical response of these materials is discussed in the context of assessing the capabilities of several models to predict the state of deformation-dependent response. A three-dimensional rubber elasticity spring system that is capable of capturing the state of deformation dependence of strain hardening is used to develop a tensorial internal state variable model of the evolving anisotropic polymer response. This fully three-dimensional constitutive model is shown to be successfully predictive of the true stress vs. true strain data obtained in our isothermal uniaxial compression and plane strain compression experiments on amorphous polycarbonate (PC) and polymethylmethacrylate (PMMA) at moderate strain rates. A basis is established for providing the polymer designer with the ability to predict the flow strengths and deformation patterns of highly anisotropic materials. A companion paper by Arruda, Boyce, and Quintus-Bosz [in press] shows how the model developed herein is used to predict various anisotropic aspects of the large strain mechanical response of preoriented materials. Additional work has been done to extend the model to include the effects of strain rate and temperature in Arruda, Jayachandran, and Boyce [in press].     

A dislocation density based crystal plasticity finite element model: Application to a two-phase polycrystalline HCP/BCC composites

We present a multiscale model for anisotropic, elasto-plastic, rate- and temperature-sensitive deformation of polycrystalline aggregates to large plastic strains. The model accounts for a dislocation-based hardening law for multiple slip modes and links a single-crystal to a polycrystalline response using a crystal plasticity finite element based homogenization. It is capable of predicting local stress and strain fields based on evolving microstructure including the explicit evolution of dislocation density and crystallographic grain reorientation. We apply the model to simulate monotonic mechanical response of a hexagonal close-packed metal, zirconium (Zr), and a body-centered cubic metal, niobium (Nb), and study the texture evolution and deformation mechanisms in a two-phase Zr/Nb layered composite under severe plastic deformation. The model predicts well the texture in both co-deforming phases to very large plastic strains. In addition, it offers insights into the active slip systems underlying texture evolution, indicating that the observed textures develop by a combination of prismatic, pyramidal, and anomalous basal slip in Zr and primarily {110}〈111〉 slip and secondly {112}〈111〉 slip in Nb.     

Modeling transients in the mechanical response of copper due to strain path changes

When copper is deformed to large strains its texture and microstructure change drastically, leading to plastic anisotropy and extended transients when it is reloaded along a different strain path. For predicting these transients, we develop a constitutive model for polycrystalline metals that incorporates texture and grain microstructure. The directional anisotropy in the single crystals is considered to be induced by variable latent hardening associated with cross-slip, cut-through of planar dislocation walls, and dislocation-based reversal mechanisms. These effects are introduced in a crystallographic hardening model which is, in turn, implemented into a polycrystal model. This approach successfully explains the flow response of OFHC Cu pre-loaded in tension (compression) and reloaded in tension (compression), and the response of OFHC Cu severely strained in shear by equal channel angular extrusion and subsequently compressed in each of the three orthogonal directions. This new theoretical framework applies to arbitrary strain path changes, and is fully anisotropic.     

Benchmark experiments and characteristic cyclic plasticity deformation

Key issues in cyclic plasticity modeling are discussed based upon representative experimental observations on several commonly used engineering materials. Cyclic plasticity is characterized by the Bauschinger effect, cyclic hardening/softening, strain range effect, nonproporitonal hardening, and strain ratcheting. Additional hardening is identified to associate with ratcheting rate decay. Proper modeling requires a clear distinction among different types of cyclic plasticity behavior. Cyclic hardening/softening sustains dependent on the loading amplitude and loading history. Strain range effect is common for most engineering metallic materials. Often, nonproportional hardening is accompanied by cyclic hardening, as being observed on stainless steels and pure copper. A clarification of the two types of material behavior can be made through benchmark experiments and modeling technique. Ratcheting rate decay is a common observation on a number of materials and it often follows a power law relationship with the number of loading cycles under the constant amplitude stress controlled condition. Benchmark experiments can be used to explore the different cyclic plasticity properties of the materials. Discussions about proper modeling are based on the typical cyclic plasticity phenomena obtained from testing several engineering materials under various uniaxial and multiaxial cyclic loading conditions. Sufficient experimental evidence points to the unambiguous conclusion that none of the hardening phenomena (cyclic hardening/softening, strain range effect, nonproportional hardening, and strain hardening associated with ratcheting rate decay) is isotropic in nature. None of the hardening behavior can be properly modeled with a change in the yield stress.     

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.     

Modeling lattice strain evolution at finite strains and experimental verification for copper and stainless steel using in situ neutron diffraction

An elasto-plastic self-consistent (EPSC) polycrystal model is extended to account, in an approximate fashion, for the kinematics of large strains, rigid body rotations, texture evolution and grain shape evolution. In situ neutron diffraction measurements of the flow stress, internal strain, texture and diffraction peak intensity evolutions were performed on polycrystalline copper and stainless steel, up to true tensile strains of ε = 0.3. Suitably adjusted slip system hardening model parameters enable the model to quantitatively describe the flow stress of the polycrystalline aggregate. Quantitative predictions of the texture evolution and the internal strain evolution along the stress axis are good, while predictions of transverse internal strains (perpendicular to the tensile loading direction) are less satisfactory. The latter exhibit a large dispersion from grain to grain around a macroscopic average, and the implications of this finding for the interpretation of in situ neutron diffraction method are explored. Finally, as a demonstration of the applicability of the model to problems involving finite rotation, as well as deformation, simulations of simple shear were conducted which predict a texture evolution in agreement with published experimental data, and other modeling approaches as well.     

A variational multiscale constitutive model for nanocrystalline materials

This paper presents a variational multi-scale constitutive model in the finite deformation regime capable of capturing the mechanical behavior of nanocrystalline (nc) fcc metals. The nc-material is modeled as a two-phase material consisting of a grain interior phase and a grain boundary effected zone (GBAZ). A rate-independent isotropic porous plasticity model is employed to describe the GBAZ, whereas a crystal-plasticity model which accounts for the transition from partial dislocation to full dislocation mediated plasticity is employed for the grain interior. The constitutive models of both phases are formulated in a small strain framework and extended to finite deformation by use of logarithmic and exponential mappings. Assuming the rule of mixtures, the overall behavior of a given grain is obtained via volume averaging. The scale transition from a single grain to a polycrystal is achieved by Taylor-type homogenization where a log-normal grain size distribution is assumed. It is shown that the proposed model is able to capture the inverse Hall-Petch effect, i.e., loss of strength with grain size refinement. Finally, the predictive capability of the model is validated against experimental results on nanocrystalline copper and nickel.