Mechanical response and modeling of fully compacted nanocrystalline iron and copper

A comprehensive study on the response of nanocrystalline iron and copper to quasi-static and dynamic loading is reported. Bulk solid nanocrystalline iron and copper specimens used in static and dynamic loading experiments were made by compaction and hot sintering of the nanocrystalline powders. The powders, with grain size 16–96 nm, were obtained by using high energy ball milling. The stress/strain response of dense nanocrystalline iron is found to be grain size and strain rate dependent. The KHL model is modified by incorporating Hall–Petch relation (i.e. yield stress dependence on grain size) and is used to represent the behavior of fully compacted nanocrystalline material. A good correlation with the experimental results is demonstrated.     

Nanocrystalline aluminum and iron: Mechanical behavior at quasi-static and high strain rates,and constitutive modeling

The responses of nanocrystalline aluminum powder of different grain sizes, prepared by ball milling and consolidated into bulk specimens by hot pressing, were determined under quasi-static and dynamic compression. The experiments demonstrated that the reduction in grain size resulted in several-fold increase in hardness and strength; the responses of nanocrystalline aluminum was found to be strain-rate-dependent. Using these measurements, Khan, Huang and Liang (KHL) viscoplastic model was modified by including a bi-linear Hall–Petch type relation to correlate with the response of nanocrystalline aluminum, including the variation of work hardening with grain size. The modified constitutive equation gives results very close to the experimental observations on nanocrystalline aluminum. In addition, the response of nanocrystalline iron, previously published, was also correlated with proposed model. Both correlations of strain-rate-dependent responses for different grain sizes were in good agreement with the experimental results over a wide range of grain sizes (micrometer to nanometer) and strain rates.     

Discrete dislocation plasticity analysis of the grain size dependence of the flow strength of polycrystals

The grain size dependence of the flow strength of polycrystals is analyzed using plane strain, discrete dislocation plasticity. Dislocations are modeled as line singularities in a linear elastic solid and plasticity occurs through the collective motion of large numbers of dislocations. Constitutive rules are used to model lattice resistance to dislocation motion, as well as dislocation nucleation, dislocation annihilation and the interaction with obstacles. The materials analyzed consist of micron scale grains having either one or three slip systems and two types of grain arrangements: either a checker-board pattern or randomly dispersed with a specified volume fraction. Calculations are carried out for materials with either a high density of dislocation sources or a low density of dislocation sources. In all cases, the grain boundaries are taken to be impenetrable to dislocations. A Hall–Petch type relation is predicted with Hall–Petch exponents ranging from ≈0.3 to ≈1.6 depending on the number of slip systems, the grain arrangement, the dislocation source density and the range of grain sizes to which a Hall–Petch expression is fit. The grain size dependence of the flow strength is obtained even when no slip incompatibility exists between grains suggesting that slip blocking/transmission governs the Hall–Petch effect in the simulations.     

Mechanically alloyed nanocrystalline iron and copper mixture: behavior and constitutive modeling over a wide range of strain rates

A comprehensive study on the response of a nanocrystalline iron and copper mixture (80% Fe and 20% Cu) to quasi-static and dynamic loading is performed. The constitutive model developed earlier by Khan, Huang & Liang (KHL) is extended to include the responses of nanocrystalline metallic materials. The strain rate and grain size dependent behaviors of porous nanocrystalline iron-copper mixture were determined experimentally for both static and dynamic loading. A viscoplastic model is formulated by associating the modified KHL model (representing the fully dense matrix behavior), and Gurson's plastic potential which provides the yield criteria for porous material. Simulations of uniaxial compressive deformations of iron-copper mixture with different initial porosity, grain size and at a wide range of strain rate (10⁻⁴ to 10³ s⁻¹) are made. The numerical results correlate well with the experimental observations.     

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.     

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.     

The scale effect on the yield strength of nanocrystalline materials

The microstructure of the nanocrystalline can be divided generally into two parts: grain and interface. When the grain size is about or less than 10 nm, the interface can be divided into grain boundary and triple junctions. The mechanical performance of nanocrystalline materials with complicated microstructures is greatly different from that of the coarse grain materials. In this paper, the nanocrystalline material is considered as a composite with three phases: the grain core, the grain boundaries, and the triple junction. The model analysis for nanocrystalline material deformation is established and the relationship between yield strength and grain size is obtained. The obtained result explains the inverse Hall–Petch relation.     

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.     

Homogenization method for strength and inelastic behavior of nanocrystalline materials

Homogenization techniques are used for modeling the so-called “breakdown” of the Hall–Petch law in the case of nanocrystalline (NC) materials. In this paper, the NC material is modeled as a composite material composed of two phases: the grain core (inclusion) and the grain boundaries (matrix). The deformation of the inclusion phase has a viscoplastic component that takes into account the dislocation glide mechanism as well as Coble creep. The boundary phase is modeled as an amorphous material with a perfect elastic–plastic behavior. An application of the model is developed on pure copper under tensile load. The results are compared with various experimental data.     

Internal-state-variable based self-consistent constitutive modeling for hot working of two-phase titanium alloys coupling microstructure evolution

An internal-state-variable based self-consistent constitutive model was proposed for unified prediction of flow stress and microstructure evolution during hot working of wrought two-phase titanium alloys in both single-beta region and two-phase region. For each constituent phase of titanium alloys, a set of constitutive equations incorporating solution strengthening, Hall–Petch effect, dislocation interaction, and dynamic recrystallization were developed using internal state variable method. The effect of second phase on recystallization was modeled by considering particle stimulated nucleation and exerting drag force on boundary migration. The constitutive equations of constituent phases were implemented into a viscoplastic self-consistent scheme to predict the overall response of the aggregate. Predictions of the model are in good agreement with experimental results of the Ti–6Al–4V alloy and IMI834 alloy. The model can reproduce many features of the hot working of two-phase titanium alloys, including the dependence of flow stress on temperature, strain rate and alloying elements; the increase of strain rate sensitivity with temperature; the stress and strain partitionings between alpha and beta phases; the relatively high apparent activation energy in two-phase region, the decrease of recrystallization kinetics with temperature in two-phase region; and the decrease of recrystallized grain size with Zener–Hollomon parameter in beta working.     

Shakedown analysis for a class of strengthening materials within the framework of gradient plasticity

The classical shakedown theory is extended to a class of perfectly plastic materials with strengthening effects (Hall–Petch effects). To this aim, a strain gradient plasticity model previously advanced by Polizzotto (2010) is used, whereby a featuring strengthening law provides the strengthening stress, i.e. the increase of the yield strength produced by plastic deformation, as a degree-zero homogeneous second-order differential form in the accumulated plastic strain with associated higher order boundary conditions. The extended static (Melan) and kinematic (Koiter) shakedown theorems are proved together with the related lower bound and upper bound theorems. The shakedown limit load problem is addressed and discussed in the present context, and its solution uniqueness shown out. A simple micro-scale structural system is considered as an illustrative example. The shakedown limit load is shown to increase with decreasing the structural size, which is a manifestation of the classical Hall–Petch effects in a context of cyclic loading.     

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.     

Activation volume and deviation from Cottrell–Stokes law at small grain size

Dependence of activation volume with flow stress is examined for metals with grain size lower than 0.3 μm and larger than few tens of nanometers, where plastic deformation is most likely to be governed by a combination of grain boundary sliding and dislocations activity. The experimentally observed deviation from the classic linear behavior given by Cottrell–Stokes law [Basinski, Z.S., 1974. Forest hardening in face centered cubic metals. Scripta Metallurgica 8, 1301–1308] is analyzed, thanks to a modified Orowan equation taking into account of the grain boundaries sliding coupled to dislocations activity. These results are compared to experimental measurements of the activation volume, between room temperature and 120 °C, for a copper nanostructure with a grain size of 100 nm. A constant activation volume is observed at low stress (or high temperature) followed by an increase of activation volume with stress (inverse Cottrell–Stokes behavior). This analysis follows our initial experiments on this fine grained metal prepared by powder metallurgy, which exhibits ductility at near constant stress and strain rate [Champion, Y., Langlois, C., Guérin-Mailly, S., Langlois, P., Bonnentien, J.-L., Hÿtch, M.J., 2003. Near-perfect elastoplasticity in pure nanocrystalline copper. Science 300, 310–311].     

Behaviors of three BCC metal over a wide range of strain rates and temperatures: experiments and modeling

The elastic–plastic behaviors of three body-centered cubic metals, tantalum, tantalum alloy with 2.5% tungsten, and AerMet 100 steel, are presented over a wide range of strains (15%), strain rates (10⁻⁶–10⁴ s⁻¹) and temperatures (77–600°F). Johnson-Cook and Zerilli-Armstrong models were found inadequate to describe the observations. A new viscoplastic model is proposed based on these experimental results. The proposed constitutive model gives good correlations with these experimental results and strain-rate jump experiments. In the next paper (Liang, R., Khan A.S., 2000. Behaviors of three BCC metals during non-proportional multi-axial loadings and predictions using a recently proposed model. International Journal of Plasticity, in press), multi-axial loading results on these materials and comparison with the proposed model will be presented.     

Grain size,strain rate,and temperature dependence of flow stress in ultra-fine grained and nanocrystalline Cu and Al: Synthesis,experiment, and constitutive modeling

For the first time, high quality bulk nanocrystalline (nc) fcc metals, with least amounts of imperfections, exhibiting high strength and ductility at room and different temperatures, under quasi-static and dynamic types of loading, were prepared and a comprehensive study on their post-yield mechanical properties was performed. This investigation included study of the effect of temperature on stress–strain responses of mechanically milled bulk nc Cu and Al. The samples after preparation through mechanical milling and consolidation processes were subjected to uniaxial compressive loading at quasi-static and dynamic strain rates of 10⁻² s⁻¹ and 1840–3105 s⁻¹, respectively, at temperatures ranging from 223 to 523 K. In both materials strong dependency of flow stress to temperature was observed; this dependency was rather more pronounced when the materials were tested at the quasi-static strain rate. Further, a new grain size and temperature dependent viscoplastic phenomenological constitutive equation, Khan–Liang–Farrokh (KLF) model was developed based on the Khan–Huang–Liang (KHL) constitutive equation. The model was featured to correlate different characteristic behaviors of polycrystalline materials in the plastic regime, as the result of grain refinement. In addition, the viscoplastic responses of bulk Cu and Al of different grain sizes (from sub-micron to nanometer range), and those from bulk nc Cu and Al at different strain rates (quasi-static to dynamic), recently published (21 and 22), were simulated using the newly developed equation. The results confirmed reasonable capability of the developed model to correlate a wide spectrum of the viscoplastic responses of these fcc metals.     

Multiaxial and non-proportional loading responses,anisotropy and modeling of Ti–6Al–4V titanium alloy over wide ranges of strain rates and temperatures

Results from a series of multiaxial loading experiments on the Ti–6Al–4V titanium alloy are presented. Different loading conditions are applied in order to get the comprehensive response of the alloy. The strain rates are varied from the quasi-static to dynamic regimes and the corresponding material responses are obtained. The specimen is deformed to large strains in order to study the material behavior under finite deformation at various strain rates. Torsional Kolsky bar is used to achieve shear strain rates up to 1000 s⁻¹. Experiments are performed under non-proportional loading conditions as well as dynamic torsion followed by dynamic compression at various temperatures. The non-proportional loading experiments comprise of an initial uniaxial loading to a certain level of strain followed by biaxial loading, using a channel-type die at various rates of loadings. All the non-proportional experiments are carried out at room temperature. Experiments are also performed to investigate the anisotropic behavior of the alloy. An orthotropic yield criterion [proposed by Cazacu, O., Plunkett, B., Barlat, F., 2005. Orthotropic yield criterion for hexagonal closed packed metals. International Journal of Plasticity 22, 1171–1194.] for anisotropic hexagonal closed packed materials with strength differential is used to generate the yield surface. Based on the definition of the effective stress of this yield criterion, the observed material response for the different loading conditions under large deformation is modeled using the Khan–Huang–Liang (KHL) equation assuming isotropic hardening. The model constants used in the present study, were pre-determined from the extensive uniaxial experiments presented in the earlier paper [Khan, A.S., Suh, Y.S., Kazmi R., 2004. Quasi-static and dynamic loading responses and constitutive modeling of titanium alloys. International Journal of Plasticity 20, 2233–2248]. The model predictions are found to be extremely close to the observed material response.