Second-order theory for the effective behavior and field fluctuations in viscoplastic polycrystals

A recently developed “second-order” homogenization procedure (Ponte Castañeda (J. Mech. Phys. Solids 50 (2002a, b) 737, 759)) is extended to viscoplastic polycrystals and applied to compute the effective response of a certain special class of isotropic polycrystals. The method itself reduces to a simple expression requiring the computation of the averages of the stress field and the covariances of its fluctuations over the various grain orientations in an optimally selected “linear comparison polycrystal”. Therefore, the method not only allows the determination of the effective behavior of the polycrystal, but as a byproduct also yields information on the heterogeneity of the stress and strain-rate fields within the polycrystal. An application is given for a model 2-dimensional, isotropic polycrystal with power-law behavior for the constituent grains. The resulting predictions for the effective behavior are found to satisfy sharp bounds available from the literature and to be consistent with the results of recent numerical simulations. The associated averages and fluctuations of the stresses and strain rates are found to depend strongly on the strain-rate sensitivity (i.e., nonlinearity) and grain anisotropy. In particular, the stress and strain-rate fluctuations were found to grow and become strongly anisotropic with increasing values of the nonlinearity and grain anisotropy parameters.     

Macroscopic behavior and field fluctuations in viscoplastic composites: Second-order estimates versus full-field simulations

This work presents a combined numerical and theoretical study of the effective behavior and statistics of the local fields in random viscoplastic composites. The full-field numerical simulations are based on the fast Fourier transform (FFT) algorithm [Moulinec, H., Suquet, P., 1994. A fast numerical method for computing the linear and nonlinear properties of composites. C. R. Acad. Sci. Paris II 318, 1417-1423], while the theoretical estimates follow from the so-called “second-order” procedure [Ponte Castañeda, P., 2002a. Second-order homogenization estimates for nonlinear composites incorporating field fluctuations: I—Theory. J. Mech. Phys. Solids 50, 737-757]. Two-phase fiber composites with power-law phases are considered in detail, for two different heterogeneity contrasts corresponding to fiber-reinforced and fiber-weakened composites. Both the FFT simulations and the corresponding “second-order” estimates show that the strain-rate fluctuations in these systems increase significantly, becoming progressively more anisotropic, with increasing nonlinearity. In fact, the strain-rate fluctuations tend to become unbounded in the limiting case of ideally plastic composites. This phenomenon is shown to correspond to the localization of the strain field into bands running through the composite along certain preferred orientations determined by the loading conditions. The bands tend to avoid the fibers when they are stronger than the matrix, and to pass through the fibers when they are weaker than the matrix. In general, the “second-order” estimates are found to be in good agreement with the FFT simulations, even for high nonlinearities, and they improve, often in qualitative terms, on earlier nonlinear homogenization estimates. Thus, it is demonstrated that the “second-order” method can be used to extract accurate information not only for the macroscopic behavior, but also for the anisotropic distribution of the local fields in nonlinear composites.     

Coupled effects of grain size distributions and crystallographic textures on the plastic behaviour of IF steels

Micro-macro scale transition theories were developed to model the inelastic behaviour of polycrystals starting from the local behaviour of the grains. The anisotropy of the plastic behaviour of polycrystalline metals was essentially explained by taking into account the crystallographic textures. Issues like plastic heterogeneities due to grain size dispersion, involving the Hall-Petch mechanism at the grain scale, were often not taken into account, and, only the role of a mean grain size was investigated in the literature. Here, both sources of plastic heterogeneities are studied using: (i) experimental data from EBSD measurements and tensile tests, and, (ii) a self-consistent model devoted to elastic-viscoplastic heterogeneous materials. The results of the model are applied to two different industrial IF steels with similar global orientation distributions functions but different mean grain sizes and grain size distributions. The coupled role of grain size distributions and crystallographic textures on the overall tensile behaviour, local stresses and strains, stored energy and overall plastic anisotropy (Lankford coefficients) is deeply analyzed by considering different other possible correlations between crystallographic orientations and grain sizes from the measured data.     

Effective viscoplastic behavior of polycrystalline aggregates lacking four independent slip systems inferred from homogenization methods; application to olivine

Polycrystalline aggregates lacking four independent systems for the glide of dislocations can deform in a purely viscoplastic regime only if additional deformation mechanisms (such as grain boundary sliding and diffusion) are activated. We introduce an implementation of the self-consistent scheme in which this additional physical mechanism, considered as a stress relaxation mechanism, is represented by a nonlinear isotropic viscoplastic potential. Several nonlinear extensions of the self-consistent scheme, including the second-order method of Ponte-Castañeda, are used to provide an estimate of the effective viscoplastic behavior of such polycrystals. The implementation of the method includes an approximation of the isotropic potential to ensure convergence of the attractive fixed-point numerical algorithm. The method is then applied to olivine polycrystals, the main constituent of the Earth's upper mantle. Due to the extreme local anisotropy of the local constitutive behavior and the subsequent intraphase stress and strain-rate field heterogeneities, the second-order method is the only extension providing qualitative and quantitative accurate results. The effective viscosity is strongly dependent on the strength of the relaxation mechanism. For olivine, a linear viscous relaxation (e.g. diffusion) could be relevant; in that case, the polycrystal stress sensitivity is reduced compared to that of dislocation glide, and the most active slip system is not necessarily the one with the smallest reference stress due to stress concentrations. This study reveals the significant importance of the strength and stress sensitivity of the additional relaxation mechanism for the rheology and lattice preferred orientation in such highly anisotropic polycrystalline aggregates.     

Second-order homogenization estimates for nonlinear composites incorporating field fluctuations: I—theory

This paper is concerned with the development of an improved second-order homogenization method incorporating field fluctuations for nonlinear composite materials. The idea is to combine the desirable features of two different, earlier methods making use of “linear comparison composites”, the properties of which are chosen optimally from suitably designed variational principles. The first method (Ponte Castañeda, J. Mech. Phys. Solids 39 (1991) 45) makes use of the “secant” moduli of the phases, evaluated at the second moments of the strain field over the phases, and delivers bounds, but these bounds are only exact to first-order in the heterogeneity contrast. The second method (Ponte Castañeda, J. Mech. Phys. Solids 44 (1996) 827) makes use of the “tangent” moduli, evaluated at the phase averages (or first moments) of the strain field, and yields estimates that are exact to second-order in the contrast, but that can violate the bounds in some special cases. These special cases turn out to correspond to situations, such as percolation phenomena, where field fluctuations, which are captured less accurately by the second-order method than by the bounds, become important. The new method delivers estimates that are exact to second-order in the contrast, making use of generalized secant moduli incorporating both first- and second-moment information, in such a way that the bounds are never violated. Some simple applications of the new theory are given in Part II of this work.     

Second-order homogenization estimates for nonlinear composites incorporating field fluctuations: II—applications

In Part I of this work, an improved “second-order” homogenization theory was developed. This new theory makes use of generalized secant moduli that are intermediate between the standard secant and tangent moduli of the nonlinear phases, and that depend not only on the averages, or first-moments of the fields in the phases, but also on the second-moments of the field fluctuations, or phase covariance tensors. In this article, the theory, which is known to be exact to second-order in the heterogeneity contrast, is applied to the special cases of rigidly reinforced and porous materials. These are cases corresponding to infinite contrast where fairly explicit analytical expressions of the Hashin-Shtrikman and self-consistent-type may be generated for nonlinear composites. The results show that the new theory improves on the earlier theory (Ponte Castañeda, J. Mech. Phys. Solids 44 (1996) 827) in at least two ways. First, the new theory satisfies rigorous bounds, even near the percolation limit, where field fluctuations become important, and the earlier second-order theory had been found to fail. Second, the new theory predicts fully compressible behavior for porous materials with an incompressible matrix phase, where the earlier theory had also been found to fail. In addition, the new estimates are found to be in better agreement with numerical simulations available from the literature.     

Bounds and self-consistent estimates for elastic constants of random polycrystals with hexagonal, trigonal, and tetragonal symmetries

Peselnick, Meister, and Watt have developed rigorous methods for bounding elastic constants of random polycrystals based on the Hashin-Shtrikman variational principles. In particular, a fairly complex set of equations that amounts to an algorithm has been presented previously for finding the bounds on effective elastic moduli for polycrystals having hexagonal, trigonal, and tetragonal symmetries. A more analytical approach developed here, although based on the same ideas, results in a new set of compact formulas for all the cases considered. Once these formulas have been established, it is then straightforward to perform what could be considered an analytic continuation of the formulas (into the region of parameter space between the bounds) that can subsequently be used to provide self-consistent estimates for the elastic constants in all cases. This approach is very similar in spirit but differs in its details from earlier work of Willis, showing how Hashin-Shtrikman bounds and certain classes of self-consistent estimates may be related. These self-consistent estimates always lie within the bounds for physical choices of the crystal elastic constants and for all the choices of crystal symmetry considered. For cubic symmetry, the present method reproduces the self-consistent estimates obtained earlier by various authors, but the formulas for both bounds and estimates are generated in a more symmetric form. Numerical values of the estimates obtained this way are also very comparable to those found by the Gubernatis and Krumhansl coherent potential approximation (or CPA), but do not require computations of scattering coefficients.     

A selfconsistent formulation for the prediction of the anisotropic behavior of viscoplastic polycrystals with voids

In this work we consider the presence of ellipsoidal voids inside polycrystals subjected to large strain deformation. For this purpose, the originally incompressible viscoplastic selfconsistent (VPSC) formulation of Lebensohn and Tomé (Acta Metall. Mater. 41 (1993) 2611) has been extended to deal with compressible polycrystals. In doing this, both the deviatoric and the spherical components of strain-rate and stress are accounted for. Such an extended model allows us to account for the void and for porosity evolution, while preserving the anisotropy and crystallographic capabilities of the VPSC model. The formulation can be adjusted to match the Gurson model, in the limit of rate-independent isotropic media and spherical voids. We present several applications of this extended VPSC model, which address the coupling between texture, plastic anisotropy, void shape, triaxiality, and porosity evolution.     

Ultrasonic attenuation in polycrystals using a self-consistent approach

The self-consistent method of averaging elastic moduli to define the effective medium of a polycrystal is used to investigate the dynamic problem of wave propagation. An alternative covariance tensor describing the elastic moduli fluctuations of the polycrystal containing self-consistent elastic properties is derived and found to be significantly smaller than the covariance tensor formed through traditional Voigt averaging. Attenuation curves are generated using the self-consistent elastic moduli and covariance tensors and these results are compared with previous Voigt-averaged estimates. The second-order polycrystalline dispersion relation for the self-consistent scheme is found for cases of low and high crystallite anisotropy. The attenuation coefficients and dispersion relations derived through the self-consistent scheme are considerably different than previous estimates. Experimentally measured longitudinal attenuation coefficients support the use of the self-consistent scheme for estimation of attenuation.     

A self-consistent plasticity theory for modeling the thermo-mechanical properties of irradiated FCC metallic polycrystals

A self-consistent theoretical framework is developed to model the thermo-mechanical behaviors of irradiated face-centered cubic (FCC) polycrystalline metals at low to intermediate homologous temperatures. In this model, both irradiation and temperature effects are considered at the grain level with the assist of a tensorial plasticity crystal model, and the elastic-visocoplastic self-consistent method is applied for the scale transition from individual grains to macroscopic polycrystals. The proposed theory is applied to analyze the mechanical behaviors of irradiated FCC copper. It is found that: (1) the numerical results match well with experimental data, which includes the comparison of results for single crystals under the load in different directions, and for polycrystals with the influences of irradiation and temperature. Therefore, the feasibility and accuracy of the present model are well demonstrated. (2) The main irradiation effects including irradiation hardening, post-yield softening, strain-hardening coefficient (SHC) dropping and the non-zero stress offset are all captured by the proposed model. (3) The increase of temperature results in the decrease of yield strength and SHC. The former is attributed to the weakened dislocation–defect interaction, while the latter is due to the temperature-strengthened dynamic recovery of dislocations through the thermally activated mechanism. The present model may provide a theoretical guide to predict the thermo-mechanical behaviors of irradiated FCC metals for the selection of structural materials in nuclear equipment.     

Discrete element simulation of shock wave propagation in polycrystalline copper

The implementation of the characteristic of compressive plasticity into the Discrete Element Code, DM2, while maintaining its quasi-molecular scheme, is described. The code is used to simulate the shock compression of polycrystalline copper at 3.35 and 11.0 GPa. The model polycrystal has a normal distribution of grain sizes, with mean diameter 14 μm, and three distinct grain orientations are permitted with respect to the shock direction; 〈1 0 0〉, 〈1 1 0〉, and 〈1 1 1〉. Particle velocity dispersion (PVD) is present in the shock-induced flow, attaining its maximum magnitude at the plastic wave rise. PVD normalised to the average particle velocity of and are yielded for the 3.35 and 11.0 GPa shocks, respectively, and are of the same order as those seen in the experiment. Non-planar elastic and plastic wave fronts are present, the distribution in shock front position increasing with propagation distance. The rate of increase of the spread in shock front positions is found to be significantly smaller than that seen in probabilistic calculations on nickel polycrystals, and this difference is attributed, in the main, to grain interaction. Reflections at free surfaces yield a region of tension near to the target free surface. Due to the dispersive nature of the shock particle velocity and the non-planarity of the shock front, the tensile pressure is distributed. This may have implications for the spall strength, which are discussed. Simulations reveal a transient shear stress distribution behind the shock front. Such a distribution agrees with that put forward by Lipkin and Asay to explain the quasi-elastic reloading phenomenon. Simulation of reloading shocks show that the shear stress distribution can give rise to quasi-elastic reloading on the grain scale.     

Multiscale modeling of oriented thermoplastic elastomers with lamellar morphology

Thermoplastic elastomers (TPEs) are block copolymers made up of “hard” (glassy or crystalline) and “soft” (rubbery) blocks that self-organize into “domain” structures at a length scale of a few tens of nanometers. Under typical processing conditions, TPEs also develop a “polydomain” structure at the micron level that is similar to that of metal polycrystals. Therefore, from a continuum point of view, TPEs may be regarded as materials with heterogeneities at two different length scales. In this work, we propose a constitutive model for highly oriented, near-single-crystal TPEs with lamellar domain morphology. Based on small-angle X-ray scattering (SAXS) and transmission electron microscopy (TEM) observations, we consider such materials to have a granular microstructure where the grains are made up of the same, perfect, lamellar structure (single crystal) with slightly different lamination directions (crystal orientations). Having identified the underlying morphology, the overall finite-deformation response of these materials is determined by means of a two-scale homogenization procedure. Interestingly, the model predictions indicate that the evolution of microstructure—especially the rotation of the layers—has a very significant, but subtle effect on the overall properties of near-single-crystal TPEs. In particular, for certain loading conditions—namely, for those with sufficiently large compressive deformations applied in the direction of the lamellae within the individual grains—the model becomes macroscopically unstable (i.e., it loses strong ellipticity). By keeping track of the evolution of the underlying microstructure, we find that such instabilities can be related to the development of “chevron” patterns.     

Study of the antiplane deformation of linear 2-D polycrystals with different microstructures

The effective behavior and the distribution of local mechanical fields of linearly viscous 2-D polycrystals under antiplane shear is investigated. Several microstructures are considered, and a full-field approach based on the Fast Fourier Transform technique is applied. First, the accuracy of this technique is evaluated on a strictly isotropic 2-phase microstructure. Voronoi tessellation is then used to generate artificial microstructures, and a real (fully recrystallized) polycrystalline microstructure is obtained by electron back-scattering diffraction. Ensemble averages over several configurations using eight crystalline orientations (phases) are performed. Although a slight anisotropy is obtained for the effective behavior of each individual configuration, statistically, the results are in very good agreement with the available analytical isotropic solution. At phase level, a marked asymmetry is obtained for the distribution of local stresses. The intraphase first- and second-order moments of the stress field, calculated for both microstructures are compared with corresponding self-consistent predictions.     

The plastic deformation of non-homogeneous polycrystals

To describe the work hardening process of polycrystals processed using various thermomechanical cycles with isochronal annealing from 500 to 900 °C, a dislocation based strain hardening model constructed in the basis of the so-called Kocks–Mecking model is proposed. The time and temperature dependence of flow stress is accounted via grain boundary migration, and the migration is related to annihilation of extrinsic grain boundary dislocations (EGBD’s) by climb via lattice diffusion of vacancies at the triple points. Recovery of yield stress is associated with changes in the total dislocation density term ρ^T. A sequence of deformation and annealing steps generally result in reduction of flow stress via the annihilation of the total dislocation density ρ^T defined as the sum of geometrically necessary dislocations ρ^G and statistically stored dislocations ρ^S. The predicted variation of yield stress with annealing temperature and cold working stages is in agreement with experimental observations. An attempt is made to determine the mathematical expressions which best describe the deformation behaviour of polycrystals in uniaxial deformation.     

Bounds on the Elastic Moduli of Completely Random Two-Dimensional Polycrystals

Explicit bounds on the elastic moduli of completely random planar polycrystals, the shape and crystalline orientations of the constituent grains of which are uncorrelated, are derived and calculated for a number of crystals of general two-dimensional anisotropy. The bounds on the elastic two-dimensional bulk modulus happen to coincide with the simple third order (in anisotropy contrast) bounds for the subclass of idealistic circular cell polycrystals. The bounds on the shear modulus are close to the much simpler bounds for circular cell polycrystals, which approximate aggregates of equiaxed grains.     

On the accuracy of self-consistent elasticity formulations for directionally solidified polycrystal aggregates

In this work, the elastic properties of directionally solidified (DS) polycrystal aggregates are investigated through a combination of analytical and numerical approaches. The effects of crystallographic misorientations and grain aspect ratios of aggregates with ellipsoidal shaped grains are first examined following a self-consistent approach. Finite element techniques are then used to examine the effects of grain size on the elastic properties of the aggregate and to assess the accuracy of the self-consistent predictions. To that purpose, a finite element procedure is presented to generate numerically realistic 3D DS microstructures from electron back-scatter diffraction (EBSD) lattice orientation measurements on an arbitrary cross-section of a DS material. The elastic stiffnesses predicted numerically and analytically are then compared with experimental data on a Ni-base DS alloy tested uniaxially along arbitrary orientations. The general trend predicted analytically was found to be consistent with the numerical and experimental results. Furthermore, an increase in the misorientation between the [0 0 1] axis of each DS grain with respect to the grain growth direction was found to decrease the elastic anisotropy of the DS material.     

Multiscale modeling of plasticity based on embedding the viscoplastic self-consistent formulation in implicit finite elements

This paper is concerned with the multiscale simulation of plastic deformation of metallic specimens using physically-based models that take into account their polycrystalline microstructure and the directionality of deformation mechanisms acting at single-crystal level. A polycrystal model based on self-consistent homogenization of single-crystal viscoplastic behavior is used to provide a texture-sensitive constitutive response of each material point, within a boundary problem solved with finite elements (FE) at the macroscale. The resulting constitutive behavior is that of an elasto-viscoplastic material, implemented in the implicit FE code ABAQUS. The widely-used viscoplastic selfconsistent (VPSC) formulation for polycrystal deformation has been implemented inside a user-defined material (UMAT) subroutine, providing the relationship between stress and plastic strain-rate response. Each integration point of the FE model is considered as a polycrystal with a given initial texture that evolves with deformation. The viscoplastic compliance tensor computed internally in the polycrystal model is in turn used for the minimization of a suitable-designed residual, as well as in the construction of the elasto-viscoplastic tangent stiffness matrix required by the implicit FE scheme.Uniaxial tension and simple shear of an FCC polycrystal have been used to benchmark the accuracy of the proposed implicit scheme and the correct treatment of rotations for prediction of texture evolution. In addition, two applications are presented to illustrate the potential of the multiscale strategy: a simulation of rolling of an FCC plate, in which the model predicts the development of different textures through the thickness of the plate; and the deformation under 4-point bending of textured HCP bars, in which the model captures the dimensional changes associated with different orientations of the dominant texture component with respect to the bending plane.