Gradient Nanomechanics: Applications to Deformation, Fracture, and Diffusion in Nanopolycrystals
Abstract
The term “gradient nanomechanics” is used here to designate a generalized continuum mechanics framework accounting for “bulk-surface” interactions in the form of extra gradient terms that enter in the balance laws or the evolution equations of the relevant constitutive variables that govern behavior at the nanoscale. In the case of nanopolycrystals, the grain boundaries may be viewed either as sources/sinks of “effective” mass and internal force or as a separate phase, interacting with the bulk phase that it surrounds, and supporting its own fields, balance laws, and constitutive equations reflecting this interaction. In either view, a further common assumption introduced is that the constitutive interaction between bulk and “interface” phases enters in the form of higher order gradient terms, independently of the details of the underlying physical mechanisms that bring these terms about. The effectiveness of the approach is shown by considering certain benchmark problems for nanoelasticity, nanoplasticity, and nanodiffusion for which standard continuum mechanics theory fails to model the observed behavior. Its implications to interpreting size-dependent stress-strain curves for nanopolycrystals with varying grain size are also discussed.
1 Introduction
Continuum mechanics was used as an effective tool to model material behavior and processes across a variety of scales ranging from macroscopic (construction/manufacturing industries) and microscopic (optoelectronics/micro-electro-mechanical systems or MEMS technologies) to planetary (earthquakes/tsunamis) and cosmological (star formation/galaxy clustering) ones. Its applicability to the nanoscale has not been explored systematically since this field has emerged only recently and is usually dominated by computer simulations. Moreover, the key submicroscopic mechanisms that such a continuum description should be based upon are not clear. A first goal of this article is to illustrate how the basic structure of an earlier proposal of the author for media with microstructure can be extended to describe deformation and transport processes at the nanoscale. Continuum models for nanoelasticity, nanoplasticity, and nanodiffusion are derived within such a generalized framework. A second objective is to show the effectiveness of these models in describing nanoscale phenomena observed in benchmark configurations that may not be captured by classical continuum theory. They are concerned with the dependence of the effective elastic modulus on the grain size in elastic bicrystals, the inverse Hall–Petch (H-P) effect in nanopolycrystals, and the curvature of concentration-depth profiles observed in nanophase materials. A third objective is to illustrate with two characteristic examples how dislocation theory and fracture mechanics can be revisited with gradient elasticity to dispense with classical singularities and provide information on dislocation core and crack tip effects, which become important at the nanoscale. A final objective is to show how a simplified approximate treatment of the gradient terms can efficiently capture size-dependent stress-strain curves of deforming nanopolycrystals with varying grain size under different temperature and strain rate conditions.
The basic premise that the proposed nanomechanics framework builds upon is the explicit recognition of the critical role of the “surface to volume” ratio in revising the standard continuum mechanics model. This is done by introducing extra terms in the usual balance laws (mass, momentum) or the evolution equations of the basic variables (plastic strain, defect densities) and, then, by making appropriate constitutive assumptions for these extra terms. The constitutive assumptions for the extra terms are motivated directly by the theory of gradient elasticity for the case of elastic deformation (nanoelasticity), gradient plasticity for the case of plastic flow (nanoplasticity), and the theory of double diffusivity for the case of diffusion (nanodiffusion). It turns out that incorporation of gradient-dependent constitutive equations into the aforementioned extended “bulk-surface” continuum framework results in an effective and robust methodology for interpreting deformation, flow, and diffusion problems at the nanoscale. In this connection, it is emphasized that a large number of articles addressing material behavior at the nanoscale are usually based either on a direct and straightforward adoption of corresponding models at macro- and micro-scales or on elaborate but also straightforward computer simulations involving ab-initio and molecular dynamics (MD) procedures based on empirical potentials. The direct adaptation of macro- and micro-scale models to consider material behavior at the nanoscale is questionable, since there is ample experimental evidence of strong size effects observed as a characteristic material or specimen dimension crosses over from the micron to the nano regime; and these effects cannot be captured by such standard models without proper modification. On the other hand, numerical simulations are always restricted by the large computer times required and the unrealistic (as opposed to experimentally imposed) strain rates assumed for such computations. The point of view advanced in the present approach seems to be a reasonable compromise. It maintains the basic continuum mechanics methodology and structure but endows it with an internal length parameter quantifying the effect of higher order gradients on the form of the extra terms modeling the bulk-surface interaction, independently of the details of the underlying submicroscopic mechanisms that bring these terms about (i.e., the terms modeling the exchange of “effective mass” and “effective momentum” between “bulk” and internal or external “surface” points). This point of view is motivated by earlier proposals of the author for a continuum with microstructure,[1, 2, 3] sketched in an effort to model self-diffusion and plastic instabilities in solids. The carriers of effective “mass” and “momentum” in the case of self-diffusion are the vacancies, and the associated surface irregularities resulting by their internal motion are “smoothed out” by appropriate effective boundary conditions for their concentration and flux. In the case of plasticity, the internal carriers of mass and momentum are dislocations and related structural defects and, therefore, dislocation mechanics should be used to motivate the appropriate constitutive equations.
The presentation is divided into four parts. In the first part (Section II), the basic structure of the bulk-surface interaction nanomechanics framework is outlined for the case of nanoelasticity, nanoplasticity, and nanodiffusion. In the second part (Section III), we use such nanoelasticity, nanoplasticity, and nanodiffusion models to discuss certain benchmark nanoscale configurations. We derive results for the grain size dependence of effective elastic moduli, H-P type normal and abnormal behavior in nanopolycrystals, and the curvature observed experimentally in concentration-penetration diffusion profiles in nanophase materials. In the third part (Section IV), we present two representative examples from dislocation theory and fracture mechanics where elastic singularities are eliminated and the details of such nonsingular stress and strain distributions near dislocation cores and crack tips can be obtained in conjunction with their use in related nanoscale applications. Finally, in the fourth part (Section V), size-dependent stress-strain curves for nanopolycrystals of varying grain size are obtained and compared with recent experimental data. The effects of strain rate and temperature are also discussed in good agreement with observed behavior. In this discussion, the effect of strain gradients is accounted for by an oversimplified but robust nanoscopic dimensional argument.
2 Gradient Continuum Nanomechanics Framework
An extended generalized continuum mechanics framework is outlined here for addressing the mechanical response of nanocrystalline (NC) and ultrafine grain (UFG) polycrystals. This extension is based on generalizing the standard continuum mechanics structure by introducing extra terms modeling the “interaction” between bulk and (external or internal) surface points, as well as appropriate constitutive equations for these terms. An alternative physical concept that can be used to achieve such a generalization is to view NC and UFG polycrystals as a mixture of “bulk” and “grain boundary” phases. The two phases can interact mechanically by exchanging mass and momentum, but the overall “composite” should obey the standard balance laws of continuum mechanics with each phase obeying its own constitutive equations. We present this discussion, for convenience, separately for elastic and plastic deformations, and separately for diffusion at the nanoscale. The resulting governing differential model equations are proposed to be used in connection with the determination of the mechanical and diffusion response of polycrystals at the submicron and nano regimes. We will refer to these models as nanoelastiticy, nanoplasticity, and nanodiffusion, respectively.
2.1 Nanoelasticity
2.1.1 Bulk/surface approach
i.e., the equations of gradient elasticity.
2.1.2 Mixture approach
2.2 Nanoplasticity
2.2.1 Bulk/surface approach
2.2.2 Mixture approach
In concluding this section, certain comments on the dependence of the flow stress τ on the first gradient of the effective strain γ would be useful. Such dependence was excluded in Eq. [21] on the basis of a Taylor expansion and isotropy. [The condition τ(x) = τ(–x) suggests that the dependence on ∂_{x}γ drops and only the dependence on ∂_{xx}γ and ( ∂_{x}γ)^{2} may be retained up to terms of second degree and order, if the assumption of linearity is relaxed.] An admissible dependence on first gradients is possible by allowing for the absolute value of ∂_{x}γ, i.e., \( \left| {\nabla \gamma } \right|, \) to enter in the formulation. For example, the constitutive assumption for \(\cal {M}\) in Eq. [11]_{2} may be generalized to read \( {{\cal {M}}} = \partial_{x} {\text{S}} - \bar{c}\left| {\nabla\gamma } \right|^{{{n \mathord{\left/ {\vphantom {n 2}} \right.\kern-\nulldelimiterspace} 2}}} , \) where \( \bar{c} \) and n are constants. The sign of the various gradient coefficients depends on whether the material is under strain hardening (or strain softening) and strain gradient hardening (or strain gradient softening). For strain softening (dκ/dγ < 0), Eq. [22] with c > 0 was used[2,3] to obtain the thickness of shear bands during localization of deformation. In the absence of localization, the coefficient \( \bar{c} \) is taken as positive for strain gradient hardening and negative for strain gradient softening. In this connection, it would be instructive for later purposes to establish contact between the preceding phenomenological arguments and dislocation theory in relation to the role of grain size in controlling material behavior and interpreting size-dependent stress-strain curves. In the absence of localization, during the initial stages of plastic deformation, it easily turns out that the flow stress may be written as \( \tau = \tau_{0} + kl^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}} \left| {\nabla \gamma } \right|^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}} \) by taking c = 0, \( \kappa (\gamma ) = \tau_{0} , \) and n = 1; τ_{0} is the homogeneous portion of the flow stress containing frictionlike and other nongradient contributions, k is a stresslike parameter accounting for strain gradient hardening/softening, and l is an internal length related to dislocation spacing or dislocation source distance. Such square root dependence of τ on \( \left| {\nabla \gamma } \right|^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}} \) may be physically justified by recalling Taylor’s hardening relation \( \tau \propto \sqrt \rho , \) where ρ denotes dislocation density. By splitting the dislocation density to ρ_{S} (density of “statistically distributed” dislocations) and ρ_{G} (density of “geometrically necessary” dislocations), we can write \( \tau = A_{S} \sqrt {\rho_{S} } + A_{G} \sqrt {\rho_{G} } = A_{S} \sqrt {\rho_{S} } \left( {1 + A^{*} \left| {\nabla \gamma } \right|^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}} } \right) \), where the well-known proportionality relationship \( \rho_{G} \propto \nabla \gamma \) is used and the A’s are trivially inter-related. By considering slow variations of ρ_{S} as compared to ρ_{G} and properly identifying the various material functions and parameters (τ_{0}, k, l; ρ_{S}, A_{S}, A_{G}, A^{*}), one can easily arrive at the desired result.
2.3 Nanodiffusion
The approach adopted earlier for a deforming NC or UFG material, viewed as a continuum that can interact with its surface or as a medium consisting of two (bulk and grain boundary) interacting phases, can be applied to model transport processes such as diffusion and heat conduction. The discussion here is focused on (substitutional) diffusion in NC materials for which it has been shown that the diffusivity may be more than ten orders of magnitude higher in comparison to regular bulk lattice diffusion. The two phases that contribute to the overall diffusion process, however, are taken here to correspond to grain boundary or intercrystalline (IC) space (as before) and grain boundary triple junction (TJ) (regular lattice diffusion in the intergranular space is thus neglected). The IC space is characterized by a diffusivity comparable to grain boundary diffusivity of conventional polycrystals and the TJs space is characterized by a diffusivity comparable to conventional surface diffusion. This view may be supported by the fact that the values of activation energies for substitutional diffusion in NC materials are more comparable to surface diffusivities rather than to grain boundary diffusivities in conventional polycrystals.[5,6] For example, at room temperature, the activation energies for diffusion of Cu and Ag in n-Cu are 0.64 and 0.39 eV/atom, while the corresponding activation energies for surface diffusion are 0.69 and 0.30 eV/atom, respectively.
2.3.1 Bulk/surface approach
2.3.2 Mixture approach
It is worth noting that the effective diffusion coefficient may be expressed as D ≡ D_{eff} = fD_{1} + (1 − f)D_{2}, with the identification of the volume fractionlike parameter f as f = κ_{2}/(κ_{1} + κ_{2}).
3 Benchmark Configurations
3.1 Size-Dependent Elastic Moduli
As an application of the “nanoelasticity” gradient-dependent stress-strain relation given by Eq. [9], the dependence of the effective (macroscopic) elastic moduli of NC’s on grain size is obtained by solving an elementary boundary value problem.
It should be emphasized that graphs of the type depicted in Figure 1(b) are obtained by assigning specific values for the gradient coefficients (c_{g} = l_{g}^{2}G_{g}, c_{gb} = l_{gb}^{2}G_{gb}) and assuming that the internal lengths l_{g} and l_{gb} are comparable in size (of the order 0.25 to 0.75 nm for l_{gb} and 1 to 2 nm for l_{g}), as well as that the shear modulus of the grain boundary space is a fraction of the shear modulus of the grain (e.g., G_{gb}= 0.3 – 0.75 G_{g}). It may be of interest to point out that the solution for the average strain \( \bar{\gamma } \) in Eq. [36] may be written as the sum of a homogeneous part \( [\tau^{\infty } /(d + \delta /2)][(\delta /2)G_{gb}^{ - 1} + dG_{g}^{ - 1} ] \) and an inhomogeneous gradient-dependent term \( \tau^{\infty } /G_{\text{grad}} , \) where the gradient modulus G_{grad} depends explicitly on the elastic, internal length, and other geometric parameters of the bicrystal listed previously. It follows then that the effective shear modulus can be written as \( G_{\text{eff}}^{ - 1} = fG_{g}^{ - 1} + (1 - f)G_{gb}^{ - 1} + G_{\text{grad}}^{ - 1} \), where f ≡ f_{g} = d/(d + δ/2) and f_{gb} = 1 − f are assumed to approximate reasonably well the volume fractions of the grain and grain boundary space in the present one-dimensional configuration. In other words, the effective shear modulus is expressed as the sum of a classical term (familiar from standard mixture rule arguments) and a gradient term (which corrects mixture rule type relationships and models, in addition, size effects). In fact, it turns out that, for l_{g} of the order of about 10 times larger than l_{gb}, the ratio G_{eff}/G initially decreases, and, after attaining a minimum at a grain size (d) comparable to the internal length (l_{g}), continues to increase, reaching its asymptotic value (1) for large grain sizes. More details on such type of simple gradient arguments to illustrate the interplay of grain size, grain boundary thickness, and internal length parameters in interpreting stiffening or softening of the shear modulus will be given in a future article, where corresponding size effects for the elastic moduli of nanopolycrystals and other nano-objects (nanolayers, nanowires, and nanotubes) will also be reported.
3.2 Inverse H-P Effect
In this section, we discuss normal and abnormal H-P behavior for various material properties as we cross over from the microcrystalline to the nanocrystalline regime. Such H-P effect has traditionally been considered for the yield strength of polycrystals, along with its inverse behavior as the grain size is reduced below a critical value d_{crit} of the order of a few nanometers. However, recent experimental evidence reveals a similar behavior for the activation volume and the pressure-sensitivity parameters. Certain results on this topic are given subsequently (Section III–B–2).
3.2.1 Inverse H-P behavior for strength
Before we proceed with an illustration of how gradient effects could explicitly be incorporated in Eq. [38], we shall point out that the starting averaging relation, as expressed by the standard mixture rule statement given by Eq. [37], may also be interpreted on the basis of a simple gradient argument for the hardness. To this end, we consider in one dimension a grain of size d constrained on its left and right side by two boundaries, each of thickness δ. The corresponding unit cell is then of size d + δ, containing the grain and the two halves of its left and right boundaries. The hardness H of the unit cell is assumed to be the sum of the local hardness H_{G} at the center of the grain plus the gradient contribution due to the heterogeneous distribution of hardness across the unit cell, i.e., H = H_{G} + l∇H_{G}, where l is an internal length. The gradient of the local hardness ∇H_{G} can be written, in a first approximation, as twice (in order to account for the left and right contribution) the quantity (H_{GB} − H_{G})/[(d + δ)/2], where H_{GB} denotes the hardness at the center of the grain boundary and (d + δ)/2 is the distance between the centers of the grain and its grain boundary. By assuming then that l = δ/4, the final expression for the total hardness H becomes H = (dH_{G} + δH_{GB})/(d + δ), which is precisely Eq. [37].
In connection with the preceding discussion, it should be noted that a large number of articles have been devoted to rationalizing the standard and the inverse H-P relation on the basis of various hypotheses and mechanisms[12, 13, 14] and references quoted therein. A rather interesting approach on the basis of gradient theory with surface/interface energy was provided recently in References 15 and 16. The governing relationship for the strength of the NC polycrystal in this model is obtained as \( H = H_{\text{o}} + kd^{ - 1/2} - (\gamma_{gb} /2a)d^{ - 1} , \) where the first two terms designate a normal H-P dependence and the last term designates an abnormal one (γ_{gb} and a are material parameters related to grain boundary interfacial energy and geometry). It then turns out easily that the critical grain size that the normal H-P relation breaks down is given by d_{crit} = (γ_{gb}/2a)^{2}, and a number of experimental graphs for normal and abnormal H-P behavior can be fitted. In concluding, it is pointed out that such a d^{−1} dependence of strength was arrived at by other authors[12,13] through different arguments.
3.2.2 Inverse H-P Behavior for the Activation Volume and Pressure Sensitivity
The parameter values used for the model prediction (solid line) and its comparison with the experimental data (dots) are υ_{g}^{0} = 1000b^{3}, υ_{gb} = 30b^{3}, δ = 2 nm, and \( k_{g} = 0.3{{\sqrt {nm} } \mathord{\left/ {\vphantom {{\sqrt {nm} } {{\mathbf{b}}^{3} }}} \right. \kern-\nulldelimiterspace} {{\mathbf{b}}^{3} }}; \)b denotes, as usual, the magnitude of the Burgers vector.
3.3 Curvature in Concentration-Penetration Depth Diffusion Profiles
4 Revisiting Dislocation and Fracture Theory
In this section, we discuss the possibility of revisiting classical dislocation and fracture mechanics theory based on linear elasticity by considering certain typical problems within the extended gradient elasticity framework summarized by Eqs. [5]. This is done because at the nanoscale an additional size/internal length dependence (other than the Burgers vector magnitude “b” for dislocation problems, and the crack length “a” for fracture problems) may determine mechanical behavior. Elasticity solutions for dislocation and crack problems were used with great success for interpreting mechanical behavior at meso- and micro-scales for a plethora of geometric and loading configurations, as well as a variety of deformation mechanisms. Since gradient elasticity brings in an additional internal length in a general way, independently of the details of the underlying submicroscopic configurations and related deformation mechanisms, it is suggested that the derivation and use of new modified gradient elasticity formulas may provide a new tool for interpreting behavior at the nanoscale.
4.1 Stability of an Intragrain Dislocation
4.2 Elimination of Singularities at the Tips of Nanocracks
5 Size-Dependent Stress-Strain Curves
The parameter values used for the fits of Figure 9(a) are \( \sigma_{f} = 0.5\;{\text{GPa}},\;\sigma_{s}^{0} = 4\;{\text{GPa}},\;k_{s} = - 140\;{\text{kPa}}\sqrt {\text{m}} , \)\( h_{0} = 730\;{\text{GPa}},\;{\text{and}}\;k_{h} = 34\;{\text{MPa}}\sqrt {\text{m}} . \) The fits in Figures 8 and 9 are discussed in detail in a ERC/LMM-AUT report,[31] where various other possibilities on grain size dependence and gradient-type constitutive equations for nanopolycrystals are examined. In particular, the temperature and strain rate dependence of stress-strain relationships discussed subsequently is also elaborated upon in detail in the aforementioned report, and pertinent results, in comparison with related experiments and simulations, will be published elsewhere.
Figure 10 shows σ−ε fits under varying \( \dot{\varepsilon } \) and T for bulk nc-Cu with grain size of 32 nm.[29] The expression for σ_{s} used for fitting the results of Figure 10(a) is \( \sigma_{s} = [890 + 15\ln \;\dot{\varepsilon }]\;{\text{MPa,}} \) while σ_{f} = 383 MPa and h = 96 GPa; also, \( \dot{\varepsilon }_{0} = 1\;\hbox{s}^{ - 1} \) and \( m = 0.017. \) The parameter values used for fitting the results of Figure 10(b) are \( T_{r} = 296 \) K and T_{m} = 1356 K and q = 2.6. The fits are shown in Figures 10(a)* and (b)*, where solid lines correspond to model predictions and dots correspond to experimental data.
σ_{f}^{0} | σ_{s}^{0} | h_{0} | |||||
---|---|---|---|---|---|---|---|
70 MPa | 265 MPa | 3 GPa | |||||
k_{1} | k_{2} | m_{f} | k_{3} | k_{4} | m_{s} | k_{5} | k_{6} |
386 | 1634 | 0.04 | 437 | 1207 | 0.01 | 43,460 | 153,680 |
with the units for k’s expressed in \( {\text{kPa}}\sqrt {\text{m}}. \) The d^{−1/2} and d^{−1} dependence of \( \left( {\sigma_{s} ,\sigma_{f} ,h} \right) \) is in accordance with the discussion provided at the end of Section III–B–1.
Notes
Acknowledgments
Discussions with K.E. Aifantis and X. Zhang are acknowledged, as well as the support of the ERC grant of the former to the latter for carrying out some of the curve fittings. The same holds for my colleague A. Konstantinidis and doctoral student J. Konstantopoulos for going through portions of the manuscript.
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