Constitutive modeling of strain rate effects in nanocrystalline and ultrafine grained polycrystals

We present a variational two-phase constitutive model capable of capturing the enhanced rate sensitivity in nanocrystalline (nc) and ultrafine-grained (ufg) fcc metals. The nc/ufg-material consists of a grain interior phase and a grain boundary affected zone (GBAZ). The behavior of the GBAZ is described by a rate-dependent isotropic porous plasticity model, whereas a rate-independent crystal-plasticity model which accounts for the transition from partial dislocation to full dislocation mediated plasticity is employed for the grain interior. The scale bridging from a single grain to a polycrystal is done by a Taylor-type homogenization. It is shown that the enhanced rate sensitivity caused by the grain size refinement is successfully captured by the proposed model.     

The competition of grain size and porosity in the viscoplastic response of nanocrystalline solids

Many important processing techniques for nanocrystalline solids, such as ball milling and compaction, are frequently accompanied by the presence of voids in the end products. These voids can apparently lower the yield strength of the material. In order to address the issue of competition between grain size and porosity, we develop an explicit, analytical composite model that allows us to determine the viscoplastic response of a porous, nanocrystalline solid. The development made use of the concept of a three-phase composite comprising of the plastically harder grain interior, plastically softer grain-boundary affected zone (GBAZ), and porosity. A homogenization theory that accounts for the evolution of porosity during plastic flow is established. This establishment is built upon the extension of a linear viscoelastic composite to a non-linear viscoplastic one, in which the viscoplastic behavior of the constituent phases is represented by a unified constitutive law. Then by means of a field fluctuation method, the local strain rates are linked to the applied total strain rate. Such a linkage in turn provides the secant viscosity of the constituent phases at every stage of deformation. In order to test the applicability of the developed theory, we have applied it to model the viscoplastic response of an iron and an iron–copper mixture tested by Khan et al. [Khan, A.S., Zhang, H., Takacs, L., 2000. Mechanical response and modeling of fully compacted nanocrystalline iron and copper. Int. J. Plasticity 16, 1459–1476] and Khan and Zhang [Khan, A.S., Zhang, H., 2000. Mechanically alloyed nanocrystalline iron and copper mixture: behavior and constitutive modeling over a wide range of strain rates. Int. J. Plasticity 16, 1477–1492]. It is demonstrated that the theory is capable of capturing the major features of the tested results at various grain sizes and porosities. Our calculations further point to the change of yield strength in the Hall–Petch plot from an initial increase to level off, and then to decline, at various porosities under a constant strain-rate loading. This in turn brings about the existence of a critical grain size in the nano-meter range at which the material exhibits maximum yield strength. Moreover, this critical grain size tends to move to the left in the Hall–Petch plot as the GBAZ becomes softer.     

Crystal plasticity model with enhanced hardening by geometrically necessary dislocation accumulation

A strain gradient dependent crystal plasticity approach is used to model the constitutive behaviour of polycrystal FCC metals under large plastic deformation. Material points are considered as aggregates of grains, subdivided into several fictitious grain fractions: a single crystal volume element stands for the grain interior whereas grain boundaries are represented by bi-crystal volume elements, each having the crystallographic lattice orientations of its adjacent crystals. A relaxed Taylor-like interaction law is used for the transition from the local to the global scale. It is relaxed with respect to the bi-crystals, providing compatibility and stress equilibrium at their internal interface. During loading, the bi-crystal boundaries deform dissimilar to the associated grain interior. Arising from this heterogeneity, a geometrically necessary dislocation (GND) density can be computed, which is required to restore compatibility of the crystallographic lattice. This effect provides a physically based method to account for the additional hardening as introduced by the GNDs, the magnitude of which is related to the grain size. Hence, a scale-dependent response is obtained, for which the numerical simulations predict a mechanical behaviour corresponding to the Hall-Petch effect. Compared to a full-scale finite element model reported in the literature, the present polycrystalline crystal plasticity model is of equal quality yet much more efficient from a computational point of view for simulating uniaxial tension experiments with various grain sizes.     

THE RATE-INDEPENDENT CONSTITUTIVE MODELING FOR POROUS AND MULTI-PHASE NANOCRYSTALLINE MATERIALS

To determine the time-independent constitutive modeling for porous and multiphase nanocrystalline materials and understand the effects of grain size and porosity on their mechanical behavior, each phase was treated as a mixture of grain interior and grain boundary, and pores were taken as a single phase, then Budiansky's self-consistent method was used to calculate the Young's modulus of porous, possible multi-phase, nanocrystalline materials, the prediction being in good agreement with the results in the literature. Further, the established method is extended to simulate the constitutive relations of porous and possible multi-phase nanocrystalline materials with small plastic deformation in conjunction with the secant-moduli approach and iso-strain assumption. Comparisons between the experimental grain size and porosity dependent mechanical data and the corresponding predictions using the established model show that it appears to be capable of describing the time-independent mechanical behaviors for porous and multi-phase nanocrystalline materials in a small plastic strain range. Further discussion on the modification factor, the advantages and limitations of the model developed were present.     

THE RATE-INDEPENDENT CONSTITUTIVE MODELING FOR POROUS AND MULTI-PHASE NANOCRYSTALLINE MATERIAL

To determine the time-independent constitutive modeling for porous and multiphase nanocrystalline materials and understand the effects of grain size and porosity on their mechanical behavior, each phase was treated as a mixture of grain interior and grain boundary, and pores were taken as a single phase, then Budiansky's self-consistent method was used to calculate the Young's modulus of porous, possible multi-phase, nanocrystalline materials, the prediction being in good agreement with the results in the literature. Further, the established method is extended tosimulate the constitutive relations of porous and possible multi-phase nanocrystalline materials with small plastic deformation in conjunction with the secant-moduli approach and iso-strain assumption. Comparisons between the experimental grain size and porosity dependent mechanical data and the corresponding predictions using the established model show that it appears to be capable of describing the time-independent mechanical behaviors for porous and multi-phase nanocrystalline materials in a small plastic strain range. Further discussion on the modification factor, the advantages and limitations of the model developed were present.     

On the evolution of crystallographic dislocation density in non-homogeneously deforming crystals

A set of evolution equations for dislocation density is developed incorporating the combined evolution of statistically stored and geometrically necessary densities. The statistical density evolves through Burgers vector-conserving reactions based in dislocation mechanics. The geometric density evolves due to the divergence of dislocation fluxes associated with the inhomogeneous nature of plasticity in crystals. Integration of the density-based model requires additional dislocation density/density-flux boundary conditions to complement the standard traction/displacement boundary conditions. The dislocation density evolution equations and the coupling of the dislocation density flux to the slip deformation in a continuum crystal plasticity model are incorporated into a finite element model. Simulations of an idealized crystal with a simplified slip geometry are conducted to demonstrate the length scale-dependence of the mechanical behavior of the constitutive model. The model formulation and simulation results have direct implications on the ability to explicitly model the interaction of dislocation densities with grain boundaries and on the net effect of grain boundaries on the macroscopic mechanical response of polycrystals.     

The role of partial mediated slip during quasi-static deformation of 3D nanocrystalline metals

We present dislocation simulations involving the collective behavior of partials and extended full dislocations in nanocrystalline materials. While atomistic simulations have shown the importance of including partial dislocations in high strain rate simulations, the behavior of partial dislocations in complex geometries with lower strain rates has not been explored. To account for the dissociation of dislocations into partials we include the full representation of the gamma surface for two materials: Ni and Al. During loading, dislocation loops are emitted from grain boundaries and expand into the grain interiors to carry the strain. In agreement with high strain rate simulations we find that Al has a higher density of extended full dislocations with smaller stacking fault widths than Ni. We also observe that configurations with smaller average grain size have a higher density of partial dislocations, but contrary to simplified analytical models we do not find a critical grain size below which there is only partial dislocation-mediated deformation. Our results show that the density of partial dislocations is stable in agreement with in situ X-ray experiments that show no increase of the stacking fault density in deformed nanocrystalline Ni (Budrovic et al., 2004). Furthermore, the ratio between partial and extended full dislocation contribution to strain varies with the amount of deformation. The contribution of extended full dislocations to strain grows beyond the contribution of partial dislocations as the deformation proceeds, suggesting that there is no well-defined transition from full dislocation- to partial dislocation-mediated plasticity based uniquely on the grain size.     

Brittle versus ductile transition of nanocrystalline metals

The brittle versus ductile transition for conventional metals is dictated by the competition between dislocation emission and cleavage. For nanocrystalline metals with grain size below 25 nm, however, dislocation activities are suppressed and the classic theory fails to apply. In this paper, one of the competing mechanisms that control the brittle versus ductile transition of nanocrystalline metals is found to be the grain boundary dominated creep deformation versus the grain boundary decohesion. A model is proposed to quantify the crack propagation in nanocrystalline metals. The effects of material properties, initial configuration and applied loads on the property of crack propagation are addressed. It is concluded that either the increases in the initial crack length, the applied load and the grain boundary damage, or the deterrence in creep deformation, accelerate the crack propagation, and vice versa.     

Investigating the deformation of nanocrystalline copper with microscale kinematic metrics and molecular dynamics

Atomistic simulations are employed to investigate the deformation of nanocrystalline copper and the associated strain accommodation mechanisms at 10 K as a function of grain size. Volume-averaged kinematic metrics based on continuum mechanics theory are formulated to analyze the results of molecular dynamics simulations. The metrics rely on both reference and current configurations, along with nearest neighbor lists to estimate nanoscale behavior of atomic deformation fields in nanocrystalline copper. Various deformation mechanisms are activated in the structures, and shown to depend on average grain size of the nanocrystalline structure. Furthermore, grain boundaries, along with dislocation glide, become an important source of strain accommodation as grain size is reduced. It is demonstrated that the metrics capture the contributions of various mechanisms, and provide a sense of the history of atomic regions undergoing both elastic and plastic deformation. The significance of this research is that unique kinematic signatures of the mechanisms are uncovered using certain metrics, and we are able to resolve the contributions of the deformation mechanisms to the overall strain of the structure using Green strain.     

Finite element analysis of grain-by-grain deformation by crystal plasticity with couple stress

Rigid–plastic crystal plasticity with the rate-sensitive constitutive behavior of a slip system has been formulated within the framework of a two-dimensional finite element method to predict the grain-by-grain deformation of single- and polycrystalline FCC metals. For that purpose, individual grains are represented by several numbers of finite elements to describe the sub-grain deformation behavior, and couple stress has been introduced into the equilibrium equation to be able to describe the size effect as well as to prevent mesh-dependent predictions. A modified virtual work-rate principle with an approximate interface constraint has been suggested to use a C ⁰-continuous element in the finite element implementation, and the couple stress work-rate has been formulated on the basis of an assumed constitutive behavior. Simulated plane-strain compressions of a single crystal cube show that the shearing and the deformation load are closely related to the imbedded lattice orientation of the crystal grain, and that the sub-grain deformation and the load magnitude can be controlled by the couple stress hardening. It is also confirmed that almost the same predictions are obtained for different mesh systems by considering the couple stress hardening. Simulated plane-strain compressions of a bi-crystal show considerably curved grain-by-grain surface profiles after large reduction for several combinations of the imbedded lattice orientation. The high couple stress hardening predicted around grain boundaries is supposed to be related to the grain size effect. It is also supposed that consideration of couple stress is necessary to predict the sub-grain or the grain-by-grain deformation, and the couple stress hardening may be used to describe the state of microstructures in grain.     

Computational simulation of the dependence of the austenitic grain size on the deformation behavior of TRIP steels

Due to the strain-induced martensitic transformation which occurs during plastic deformation, a transformation-induced plasticity (TRIP) phenomenon is generated. With the TRIP phenomenon, the TRIP steel possesses favorable mechanical properties such as high strength, ductility and toughness, and is frequently employed as a structural material. In the past, several researchers clarified experimentally that the strain-induced martensitic transformation and the deformation behavior of TRIP steel depend upon the austenitic grain size. In order to obtain the expected mechanical properties of TRIP steel through control of the austenitic grain size, prediction and control of the material characteristics in the deformation processes is essential. Here, the new strain-induced martensitic transformation kinetics model and constitutive equation of TRIP steels are proposed by considering the dependence of the austenitic grain size. Then, the deformation behavior of a type 304 austenitic stainless steel cylinder is simulated under different environmental temperatures with the various austenitic grain sizes by the finite-element method along with newly-proposed constitutive equations. Finally, the validity of proposed constitutive equations and the possibility of the improvement of the mechanical properties through control of the austenitic grain size are discussed.     

A secant-viscosity composite model for the strain-rate sensitivity of nanocrystalline materials

In order to address the strain-rate sensitivity of nanocrystalline solids, a secant-viscosity composite model is developed in this article. The microgeometry of the composite is taken to consist of the grain-interior phase and the grain-boundary affected zone (GBAZ) as suggested by Schwaiger et al. [Schwaiger, R., Moser, B., Dao, M., Chollacoop, N., Suresh, S., 2003. Some critical experiments on the strain-rate sensitivity of nanocrystalline nickel. Acta Mater. 51, 5159–5172], while the constituent properties are modeled by a unified viscoplastic constitutive law. The drag stress of the grain interior is assumed to follow the Hall–Petch relation, but that of the GBAZ is independent of grain size, d. Then in terms of the secant viscosity of the constituent phases, the strain-rate sensitivity of the nanocrystalline solid is determined with the help of a linear viscous comparison composite and a field-fluctuation approach. To test the applicability of the developed model, it is applied to predict the strain-rate effect of a nanocrystalline Ni, and the grain-size dependence of its stress–strain relations. Our theoretical calculations indicate that the tensile strength of a nanocrystalline Ni with d = 40 nm is about five times that of a microcrystalline one with d = 10 μm under the same strain rate of , and that the nanocrystalline Ni exhibits a much stronger strain-rate effect. These predictions are found to be consistent with the experimental data of Schwaiger et al. Possible grain-size softening with further grain-size reduction such as reported in molecular dynamic simulations is also demonstrated.     

Micromechanics modeling of strength for nanocrystalline copper

We present a model in this paper for predicting the inverse Hall–Petch phenomenon in nanocrystalline (NC) materials which are assumed to consist of two phases: grain phase of spherical or spheroidal shapes and grain boundary phase. The deformation of the grain phase has an elasto-viscoplastic behavior, which includes dislocation glide mechanism, Coble creep and Nabarro–Herring creep. However the deformation of grain boundary phase is assumed to be the mechanism of grain boundary diffusion. A Hill self-consistent method is used to describe the behavior of nanocrystalline pure copper subjected to uniaxial tension. Finally, the effects of grain size and its distribution, grain shape and strain rate on the yield strength and stress–strain curve of the pure copper are investigated. The obtained results are compared with relevant experimental data in the literature.     

Characterization of yielding behavior of polycrystalline metals with single crystal plasticity based on representative characteristic length

Single crystal plasticity based on a representative characteristic length is proposed and introduced into a homogenization approach based on finite element analyses, which are applied to characterization of distinctive yielding behaviors of polycrystalline metals, yield-point elongation, and grain size strengthening. The computational manner for an implicit stress update is derived with the framework of a standard multi-surface plasticity at finite strain, where the evolution of the characteristic lengths are numerically converted from the accumulated slips of all of slip systems by exploiting the mathematical feature of the characteristic length as the intermediate function of the plastic internal variables. Furthermore, a constitutive model for a single crystal reproduces the stress–strain curve divided into three parts. Using two-scale finite element analysis, the macroscopic stress–strain response with yield-point elongation under a situation of low dislocation density is reproduced. Finally, the grain size effect on the yield strength is analyzed with modeling of the grain boundary in the context of the proposed constitutive model and is discussed from both macroscopic and microscopic views.     

Constitutive model for plastic deformation of nanocrystalline materials with shear band

A phase mixture model was used to study the plastic deformation behaviors in hardening stage of nanocrystalline materials. The material was considered as a composite of grain interior phase and grain boundary (GB) phase. The constitutive equations of the two phases were determined in term of their main deformation mechanisms. In softening stage, a shear band deformation mechanism was presented and the corresponding constitutive relation was established. Numerical simulations have shown that the predications fit well with experimental data. The investigation using the finite-element method (FEM) provided a direct insight into quantifying shear localization effect in nanocrystalline materials.     

Atomistic based continuum investigation of plastic deformation in nanocrystalline copper

A continuum model of nanocrystalline copper was developed based on results from independent atomistic calculations on 11 bicrystals containing high angle grain boundaries. The relationship between grain boundary structure and its mechanical response was investigated. Based on the atomistic calculations; a constitutive law for grain boundary interfaces was implemented within a finite element calculation that consisted of a microstructure loaded in compression. The yield strength as a function of grain size was compared to experimental data and molecular dynamics results. Calculations were performed to demonstrate the relationship between intragranular plasticity and grain boundary sliding.     

Role of grain boundary energetics on the maximum strength of nanocrystalline Nickel

Numerical simulations are used to investigate the competing grain boundary and dislocation mediated deformation mechanisms in nanocrystalline Ni with grain sizes in the range 4-32 nm. We present a 3D phase field model that tracks the evolution of individual dislocations and grain boundaries. Our model shows that the transition from Hall-Petch to inverse Hall-Petch as the grain size is reduced cannot be characterized only by the grain size, but it is also affected by the grain boundary energetics. We find that the grain size corresponding to the maximum yield stress (the transition from Hall-Petch strengthening with decreasing grain size to inverse Hall-Petch) decreases with increasing grain boundary energy. Interestingly, we find that for grain boundaries with high cohesive energy the Hall-Petch maximum is not observed for grains in the range 4-32 nm.     

A Review of Fatigue Behavior in Nanocrystalline Metals

Nanocrystalline metals have been shown to exhibit unique mechanical behavior, including break-down in Hall-Petch behavior, suppression of dislocation-mediated plasticity, induction of grain boundary sliding, and induction of mechanical grain coarsening. Early research on the fatigue behavior of nanocrystalline metals shows evidence of improved fatigue resistance compared to traditional microcrystalline metals. In this review, experimental and modeling observations are used to evaluate aspects of cyclic plasticity, microstructural stability, crack initiation processes, and crack propagation processes. In cyclic plasticity studies to date, nanocrystalline metals have exhibited strongly rate-dependent cyclic hardening, suggesting the importance of diffusive deformation mechanisms such as grain-boundary sliding. The cyclic deformation processes have also been shown to cause substantial mechanically-induced grain coarsening reminiscent of coarsening observed during large-strain monotonic deformation of nanocrystalline metals. The crack-initiation process in nanocrystalline metals has been associated with both subsurface internal defects and surface extrusions, although it is unclear how these extrusions form when the grain size is below the scale necessary for persistent slip band formation. Finally, as expected, nanocrystalline metals have very little resistance to crack propagation due to limited plasticity and the lack of crack path tortuosity among other factors. Nevertheless, like bulk metallic glasses, nanocrystalline metals exhibit both ductile fatigue striations and metal-like Paris-law behavior. The review provides both a comprehensive critical survey of existing literature and a summary of key areas for further investigation.     

A finite deformation theory of strain gradient plasticity

Plastic deformation exhibits strong size dependence at the micron scale, as observed in micro-torsion, bending, and indentation experiments. Classical plasticity theories, which possess no internal material lengths, cannot explain this size dependence. Based on dislocation mechanics, strain gradient plasticity theories have been developed for micron-scale applications. These theories, however, have been limited to infinitesimal deformation, even though the micro-scale experiments involve rather large strains and rotations. In this paper, we propose a finite deformation theory of strain gradient plasticity. The kinematics relations (including strain gradients), equilibrium equations, and constitutive laws are expressed in the reference configuration. The finite deformation strain gradient theory is used to model micro-indentation with results agreeing very well with the experimental data. We show that the finite deformation effect is not very significant for modeling micro-indentation experiments.     

Mechanics of very fine-grained nanocrystalline materials with contributions from grain interior, GB zone, and grain-boundary sliding

In this paper, we formulated an atomically-equivalent continuum model to study the viscoplastic behavior of nanocrystalline materials with special reference to the low end of grain size that is typically examined by molecular dynamic (MD) simulations. Based on the morphology disclosed in MD simulations, a two-phase composite model is construed, in which three distinct inelastic deformation mechanisms disclosed from MD simulations are incorporated to build a general micromechanics-based homogenization scheme. These three mechanisms include the dislocation-related plastic flow inside the grain interior, the uncorrelated atomic motions inside the grain-boundary region (the GB zone), and the grain-boundary sliding at the interface between the grain and GB zone. The viscoplastic behavior of the grain interior is modeled by a grain-size dependent unified constitutive equation whereas the GB zone is modeled by a size-independent unified law. The GB sliding at the interface is represented by the Newtonian flow. The development of the rate-dependent, work-hardening homogenization scheme is based on a unified approach starting from elasticity to viscoelasticity through the correspondence principle, and then from viscoelasticity to viscoplasticity through replacement of the Maxwell viscosity of the constituent phases by their respective secant viscosity. The developed theory is then applied to examine the grain size- and strain rate-dependent behavior of nanocrystalline Cu over a wide range of grain size. Within the grain-size range from 5.21 to 3.28 nm, and the strain rate range from 2.5 × 10⁸ to 1.0 × 10⁹/s, the calculated results show significant grain-size softening as well as strain-rate hardening that are in quantitative accord with MD simulations [Schiotz, J., Vegge, T., Di Tolla, F.D., Jacobsen, K.W., 1999. Atomic-scale simulations of the mechanical deformation of nanocrystalline metals. Phys. Rev. B 60, 11971–11983]. We have also applied the theory to investigate the flow stress, strain-rate sensitivity, and activation volume over the wider grain size range from 40 nm to as low as 2 nm under these high strain rate loading, and found that the flow stress initially displays a positive slope and then a negative one in the Hall–Petch plot, that the strain-rate sensitivity first increases and then decreases, and that the activation volume first decreases and then increases. This suggests that the maximum strain rate sensitivity and the lowest activation volume do not occur at the smallest grain size but, like the maximum yield strength (or hardness), they occur at a finite grain size. These calculated results also confirm the theoretical prediction of Rodriguez and Armstrong [Rodriguez, P., Armstrong, R.W., 2006. Strength and strain rate sensitivity for hcp and fcc nanopolycrystal metals. Bull. Mater. Sci. 29, 717–720] on the basis of grain boundary weakening and the report of Trelewicz and Schuh [Trelewicz, J.R., Schuh, C.A., 2007. The Hall–Petch breakdown in nanocrystalline metals: a crossover to glass-like deformation. Acta Mater. 55, 5948–5958] on the basis of hardness tests. In general the higher yield strength, higher strain rate sensitivity, and lower activation volume on the positive side of the Hall–Petch plot are associated with the improved yield strength of the grain interior, but the opposite trends displayed on the negative side of the plot are associated with the characteristics of the GB zone which is close to the amorphous state.