Elsevier

Acta Materialia

Volume 54, Issue 12, July 2006, Pages 3177-3190
Acta Materialia

A computational study of the mechanical behavior of nanocrystalline fcc metals

https://doi.org/10.1016/j.actamat.2006.03.007Get rights and content

Abstract

We have conducted a finite-element-based study of the deformation and failure behavior of nanocrystalline face-centered cubic metals. A rate-dependent amorphous plasticity model which accounts for cavitation and related failure phenomena is used to model the grain boundaries, while a crystal plasticity model is used for the grain-interiors. Our numerical simulations using material parameters estimated to represent the macroscopic rate-dependent stress–strain response of nanocrystalline nickel (nc-Ni), show that there is a transition in deformation mechanism from grain-interior shearing to grain-boundary shearing, as the average grain-size decreases from 50 nm to 10 nm, and that the low ductility of nc-Ni is the result of intergranular failure due to grain-boundary shearing and resulting cavitation at triple-junctions and other high stress points in the microstructure. Our numerical simulations also show that the strength of nc-Ni is expected to be slightly higher in compression than in tension, primarily due to the easier operation of cavitation failure of the grain boundaries in tension.

Introduction

Nanocrystalline metals are polycrystalline metallic materials with grain sizes typically less than 100 nm. These materials have been the subject of intense, worldwide research over the past two decades, and due to this research activity the micromechanisms governing their macroscopic mechanical behavior are now beginning to be better understood (e.g. [1], [2]). Recent reviews on the topic, and references to the vast amount of literature, may be found in Kumar et al. [3] and Wolf et al. [4]. Nanocrystalline metals contain a high volume fraction of “grain-boundary”-intercrystalline regions. For example, idealizing a unit cell containing a crystalline grain interior and an intercrystalline grain-boundary region as a sphere of diameter d, with an intercrystalline shell of thickness δ and a crystalline core of diameter d  2δ, the volume fractions of grain-boundary regions for a fixed value of δ  0.5 nm, and grain sizes d of 10 nm and 40 nm are 27.1% and 7.3%, respectively. Thus, a substantial fraction of the atoms in nanocrystalline materials lie in the intercrystalline grain-boundary regions, and these regions play an increasingly significant role as the grain size decreases below the 100 nm level. The nature of the intercrystalline grain-boundary regions depends on how the material has been processed. High-resolution transmission electron microscopy (TEM) studies on nanocrystalline materials show that while many grain boundaries appear sharp and well-defined, others show considerable disorder, with the maximum disordered region (measured perpendicular to grain boundaries) being approximately 2–3 lattice spacings; i.e., less than 1 nm [5].

In this paper we focus our attention on continuum-level modeling of the low temperature mechanical response of nanocrystalline fcc metals. A broad picture of the operative micromechanisms of inelastic deformation in this class of materials in this temperature range is beginning to emerge. The following features of the operative micromechanisms are now reasonably widely-accepted [1], [2], [3], [4], [6], [7], [8]:

  • There is a strong interplay between dislocation-based deformation in the crystalline grain interiors and the inelastic deformation mechanisms operative in the grain-boundary regions.

  • Grain boundaries act as both sources and sinks for dislocations.

  • Let Γ denote the stacking fault energy of the face-centered cubic (fcc) material, G its nominal shear modulus, and b the magnitude of the Burgers vector of a perfect dislocation, then there exists a critical grain size dc, given by dc  (2/3)(Gb2/Γ), above which plastic deformation in nanocrystalline grain interiors occurs by the emission of complete dislocations from grain boundaries, and below which the plastic deformation occurs by emission of partial dislocations. The partial dislocations produce stacking faults as they glide through the grains [9], [10].

  • The number of dislocations or stacking faults that need to traverse a typical grain in the nanocrystalline range to produce an overall strain of the order of 5% is quite small, typically less than 10–15. Thus, dislocation based plasticity in the grain interiors is quite discrete in nature.

  • When the grain size d becomes smaller than dc, dislocation-based slip processes become less effective in producing overall inelastic deformation, and grain-boundary-region-based inelastic deformation mechanisms start to dominate [11], [12], [13].

  • The grain-boundary inelastic deformation mechanisms are loosely called “grain-boundary sliding”, but at low temperatures atomistic simulations show this to be stress-activated shear-shuffling of atoms located in the intercrystalline regions, and the cooperative result of numerous such shear-shuffles leads to substantial overall shearing of the intercrystalline regions; this process is not dominated by thermal diffusion of atoms in the grain-boundary regions. Such a deformation mechanism is reminiscent of that in metallic glasses, where inelastic deformation occurs by local shearing of clusters of atoms – “shear transformation zones” [14].

  • Nanocrystalline fcc materials show a room-temperature strain-rate sensitivity which is almost an order of magnitude higher than their microcrystalline counterparts. The mechanism underlying this enhanced strain-rate sensitivity is not fully understood at present, but is probably due to the enhanced strain rate sensitivity of the intercrystalline grain-boundary regions, relative to the rate sensitivity of the crystalline grain-interiors.

  • The ductility of these materials, as measured by elongation in tensile experiments, is significantly reduced from the values observed for their microcrystalline counterparts, and seldom exceeds 5%. Since this occurs even for materials at the high end of grain sizes in the nanocrystalline range (∼100 nm), one can infer that even though there is some accommodation of the overall imposed deformation by dislocation-mediated plasticity in the grain interiors, this accommodation is insufficient. Additional, intergranular accommodation mechanisms, such as cavitation and microcracking, must be operative. It must be noted that while widespread distributed grain-boundary damage prior to final macroscopic fracture has not been experimentally observed, atomistic simulations do show void formation and decohesion leading to intergranular fracture in nanocrystalline materials [6], [7], [8].

As reviewed in [3], [4], much of the understanding of the micromechanisms operative during the inelastic deformation of nanocrystalline materials has been obtained from large-scale molecular dynamics (MD) studies published in the past few years. Although MD methods of studying atomic-level mechanical response of materials are useful for gaining valuable insight, these methods are at present not suitable for carrying out simulations of deformation and failure under conditions similar to those under which physical experiments on nanocrystalline materials are carried out, i.e., macroscopic-sized specimens with complicated boundary conditions, involving realistic strain rates. MD simulations are inherently limited to small, idealized microstructures and extremely high strain rates, typically >107/s, which corresponds to a strain of 1% in 1 ns.

In contrast to MD methods, finite-element methods (FEM) for simulation of micromechanical interactions and prediction of local as well as overall response of materials have been effectively used to study the mechanical response of variety of composite material systems in recent years, and such methods do not possess the major limitations of the MD methods listed above. However, use of continuum-mechanical-based FEM methods is contingent upon the assumptions of continuum mechanics – suitable smoothness of displacement fields, the notion of stress, and balance laws of linear and angular momentum – continuing to hold at the nanoscale. Further, just as the results from MD simulations depend crucially on the reliability of the interatomic potentials used in such studies, the results of FEM simulations depend crucially on the reliability of the continuum-level constitutive equations used in such analyses. While the concepts of continuum elasticity are expected to be approximately applicable at small scales approaching the nano-level, the concepts of classical continuum plasticity being applicable at this scale are highly questionable. Nevertheless, based on a pragmatic engineering approach, and bolstered by the success of (length-scale-independent) crystal-plasticity theories and attendant FEM simulation methodologies to represent grain-scale shear localization phenomena and texture evolution (cf., e.g. [15]), a few investigators have recently carried out continuum-level FEM simulations of the inelastic deformation and failure response of nanocrystalline materials (cf., e.g. [17], [18]). Although in their infancy, such FEM-based simulations also provide valuable insights on the deformation and failure response of nanocrystalline materials; insights which cannot solely be obtained from atomistic simulations or physical experiments.

For example, Wei and Anand [17] coupled a single-crystal plasticity constitutive model for the grain interior, with a cohesive interface constitutive model to account for grain-boundary sliding and separation phenomena. They recognized that a standard crystal plasticity model for the grain-interior deformation, which implicitly assumes enough dislocation nucleation and multiplication to result in a sufficiently smooth macroscopic response, is inadequate to represent the limited amount of inelastic deformation due to emission and eventual absorption of the relatively fewer (partial or complete) dislocations from grain boundaries in nanocrystalline materials. However, since elastic anisotropy and crystallographic texture effects are still important in nanocrystalline materials, and since the few dislocations in these materials are still expected to move on slip systems, the mathematical structure of a continuum crystal plasticity theory is still useful as an indicator of the limited inelasticity due to crystalline slip within the nanocrystalline grains. Their cohesive interface model for grain boundaries accounts for both reversible elastic, as well irreversible inelastic sliding-separation deformations at the grain boundaries prior to failure. The tensile and shear properties of the cohesive grain boundaries were estimated by fitting results of numerical simulations to experimentally-measured stress–strain curves for nanocrystalline electrodeposited Ni. Unfortunately, such a fitting procedure makes it difficult to unambiguously characterize the grain-boundary properties, and they used a value for the shear strength of the grain boundaries which was very similar in magnitude to the tensile strength of the boundaries; as is clear from the recent atomistic simulations of grain-boundary response in Sansoz and Molinari [7], this assumption is clearly unrealistic. Also, Wei and Anand [17] restricted their models for the grain interiors and grain boundaries to be completely rate-independent.

Warner et al. [18], following the methodology of Wei and Anand [17], have also used a continuum FEM method to model the plastic deformation of nanocrystalline copper by using crystal plasticity for the grain interiors and cohesive elements for grain boundaries. For the traction–separation relations for the grain boundaries, they obtained estimates for the shear and normal response of the boundaries from quasi-continuum atomistic calculations. Since their atomistic calculations were carried out at 0 K, they had to use a large ad hoc thermal-correction (cf. their Eq. (8)) to estimate the shear strength for the grain boundaries in Cu at room temperature.

As is clear from our brief review, continuum-level FEM simulations of the inelastic deformation and failure response of nanocrystalline materials are still in their infancy. Much needs to be done to refine the constitutive models, specially for the traction–separation relations in approaches using cohesive elements to represent grain boundaries.

Here, motivated by the fact that amorphous metals are the ultimate limit for nanocrystalline metals, as the crystal grain size decreases to zero, and that a new strain-rate-dependent continuum-plasticity theory for amorphous metals at low homologous temperatures is now available [19], [20], we have developed a slightly modified version of the continuum theory for amorphous metals to represent the intercrystalline grain-boundary regions. We do not use interface traction–separation relations; instead we use the stress–strain relations from this continuum theory to represent the grain-boundary response. We have coupled this with a single-crystal plasticity model for the crystalline grain interiors in a “composite”-type approach to model the deformation and failure response of nanocrystalline materials.

It is the purpose of this paper to report on our constitutive models and the results from our numerical simulations. Specifically, we have (a) developed a modified version of the rate-dependent amorphous plasticity model of Anand and Su [19], [20] to account for cavitation and related failure phenomena, and used such a model to represent the grain-boundary response of nanocrystalline materials; and (b) accounted for the transition from partial dislocation to complete dislocation mediated plasticity for the grain interiors. This development is based on the recent work of Zhu et al. [10].

The plan of this paper is as follows. The amorphous plasticity model we have adopted to represent the rate-dependent deformation and failure of grain boundaries is described in Section 2. In Section 3, we describe the model that we have used to represent grain interior plasticity due to partial or perfect dislocations. In Section 4 we study the response of a prototypical grain-boundary as modeled by our theory for amorphous materials. Numerical representation of two-dimensional microstructures with different grain sizes is briefly discussed in Section 5. Applications of the model to represent the deformation and failure response of Ni over a range of grain sizes in the nanocrystalline regime are reported in Section 6. We close in Section 7 with some final remarks.

Section snippets

A constitutive model for the elastic–viscoplastic deformation of grain boundaries

Our continuum-mechanical constitutive model for the intercrystalline grain-boundary regions in nanocrystalline materials is based on a modification of the theory of Anand and Su [19], [20] for amorphous metallic materials, to account for cavitation failure. Using standard notation of modern continuum mechanics, the underlying constitutive equations relate the following basic fields: χ, motion; F = χ with J = det F > 0 deformation gradient; Fp with Jp = det Fp > 0, plastic deformation gradient; Fe = FFp−1

Constitutive model for grain-interior response

We employ the standard kinematical and constitutive framework of rate-dependent single-crystal plasticity based on the multiplicative decomposition F = FeFp of the deformation gradient into an elastic and a plastic part (e.g. [21], [22]). Here the stress Te = (det Fe)Fe−1TFe−⊤ is given by Te=C[Ee]. where C is the anisotropic elasticity tensor and Ee = (1/2)(Ce  1) the elastic strain, with Ce = FeFe For cubic materials C has only three independent constants, which are traditionally denoted by C11, C12, C

Simulation of an amorphous grain-boundary region

Before we proceed to the application of our plasticity theories to modeling the response of nanocrystalline materials, in this section we consider the deformation and failure response of a prototypical “grain-boundary” as modeled by our theory for amorphous materials. Fig. 1a shows a finite-element model of an amorphous grain-boundary region “GB” sandwiched between elastic layers “A”. The bottom edge of the sandwiched layer is held fixed, while u denotes the displacement of the top edge.

Finite-element representation of microstructures with different grain sizes

As shown schematically in Fig. 4a and b, for a fixed grain-boundary thickness of 2δ = 1 nm, the volume fraction of grain-boundary regions decreases as the grain size d increases from 10 nm to 20 nm. Hence, by assuming that the physical thickness of a grain-boundary is fixed at ≈1 nm, we can numerically represent microstructures with different average grain size, by simply adjusting the volume fraction of “grain-boundary” elements to “grain-interior” elements in a finite-element mesh.

Two-dimensional

Application of the model to nanocrystalline nickel

The tensile stress–strain response of nanocrystalline nickel (nc-Ni) has been experimentally investigated by numerous groups (e.g. [10], [16], [25]). For example, Fig. 6 from [16] shows room temperature tensile stress–strain curves for nc-Ni with a grain size of ≈35 nm at three different strain-rates. The material exhibits a high strength of approximately 1500 MPa, but a low ductility of less than 5%. As noted earlier, it also exhibits a positive strain-rate sensitive response, with the rate

Concluding remarks

We have developed a finite-element-based two-dimensional plane-strain mesoscale model to numerically study the deformation and failure behavior of nanocrystalline fcc metals. A rate-dependent amorphous plasticity model which accounts for cavitation and related failure phenomena is used to model the grain-boundary, while a crystal-plasticity model which accounts for the transition from partial dislocation to complete dislocation mediated plasticity is used for the limited plasticity of the grain

Acknowledgements

Discussions with Professors Subra Suresh, Sharvan Kumar, and Chris Schuh are gratefully acknowledged. Financial support was provided by the ONR Contract N00014-01-1-0808 with MIT.

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