Multiscale unified prediction of size/scale and Hall-Petch effects in the mechanics of polycrystalline materials

1 Introduction

The study of size effects such as improved strength with decreasing size traces back to the works of Leonardo da Vinci and Galileo Galilei, mostly addressing macro scales; Bazant and Chen [1] cite 377 works on the subject mostly at scales from centimeters to several meters. “Simple” reasoning claims that larger samples show a reduction in strength for probabilistic reasons: a larger sample simply has more flaws to initiate cracks. As size approaches atomic dimensions, the theoretical material strength should be approached. This has important applications within the context of nanoscience and nanotechnology, e.g., in developing nanostructural materials.

Any robust study of size effects must incorporate the structure of the material. However, a large number of studies consider mainly or only the defects in the material as they are, presumably, the main reason for size effects. However, at scales near the ones of significant material structure, and in particular near the average grain size of polycrystalline materials for the present work, it seems natural that size effects should be tied to Hall-Petch effects, which experimentally and theoretically account for defects in the material as well. To our surprise, there are no works present in the literature linking size and Hall-Petch effects, and this is, perhaps, due to difficulties in establishing a link. Another reason may be that theoretical frameworks for the study of size effects, e.g., fracture mechanics, do not have an inherent mechanism for incorporating the material structure at any scale. This will be addressed next as a “preface” to the present work.

2 Size effects in fracture mechanics

For engineering design, size effects is one of the most compelling reasons for adopting fracture mechanics, which predicts a power law dependence of strength on size, with the exponent equal to as opposed to the classic strength of materials where the exponent is equal to zero (no size effect). For material design, size effects have been realized and have been partly the reason for many successes, e.g., high-strength (and toughness, for reasons other than size effects) fiber reinforced composites developed initially in the 1970s where fiber diameters are typically in the micrometer to submillimeter range. Recent technological advances have resulted in materials at the nanoscale with surprising properties, e.g., carbon nanotubes and their potential use in ultralight high-strength composites.

Let us consider a membrane/film or a small piece of a tube containing a thumbnail crack (Figure 1A). On the basis of the Griffith criterion, the failure strength of the film, σ_f, under stress σ is expressed as

Figure 1

(A) A small piece of a film or of a tube of thickness h containing a thumbnail crack-like defect subjected to stress σ. (B) Strength vs. thickness plot, based on the Griffith criterion and schematic of h_cr. Figure based on ref. [1] for the fracture problem and on refs. [2, 3] for the Hall-Petch relation, h, indicating mean grain size in the latter case. The critical mean grain size based on the Hall-Petch relation is unrelated to h_cr. For non-crystalline materials, Hall-Petch effects do not exist.

where γ denotes the surface energy, E is the Young modulus, h is the film thickness (Figure 1A), and a depends on the crack geometry and is approximately equal to for a thumbnail crack extending through half of h and crack length equal to h. Figure 1B shows schematically the plot of Eq. (1), where σ_th denotes the theoretical strength of the material [4]. As σ_f cannot exceed σ_th, a critical thickness, h_cr, is conjectured (Figure 1B). Considering rough estimates for a material (γ=1 J/m², E=100 GPa, σ_th=E/30 [4], and ), Eq. (1) yields h_cr≈30 nm [5]. Of course, the specialization of this argument for metals, where plastic deformation in the vicinity of the defect complicates the situation, needs to be studied in detail.

Fracture mechanics does not include any information on the structure of the material, and as a consequence there is no inherent length scale present in it. However, the Hall-Patch effects have an inherent length scale present, namely the average grain size of a polycrystalline material. As will be shown next, that size is crucial for size effects at the scales corresponding to its vicinity.

3 Hall-Petch effects

In the spirit of the Hall-Petch relation [6, 7] where the strength of a polycrystalline material varies as the inverse square root of the mean grain size, a plot as that of Figure 1B holds, where the horizontal axis now denotes the mean grain size. Further, a critical mean grain size has been identified as the cutoff [2, 3, 8]. Large-scale simulations of a specific metal [9] place the critical mean grain size in the range of 10–15 nm. The critical mean grain size based on the Hall-Petch relation is not related to the critical thickness h_cr. The inverse Hall-Petch effects, i.e., strength as a function of grain size for grain sizes smaller than the critical mean grain size, have received extensive attention. Besides the relevant experimental work and the simulation-based study of these effects [8, 9], continuum-based models, involving higher-order gradients and/or mixtures, offer attractive explanations on these effects, as well as tools for design scientists and engineers [10–14].

It is noted that if the grain size of a thin film scales with the film thickness, the film thickness can be used as the scaling parameter [15]. For a fixed mean grain size, one should be ascertained of size effects, especially at the scale of the mean grain size, and these are not Hall-Petch based; this clearly suggests that h_cr is unrelated to the critical mean grain size.

4 Spatial and temporal scales in materials

Most materials problems involve a breadth of temporal and spatial scales. Data relevant to a problem may be collected through experiments or observations or may be generated by means of models. It is desirable to be able to collectively use data for a particular application. Data isolation at specific scales of interest is not a solution to this, as the scales interact with each other, i.e., changes in the underpinning mechanisms/interactions of the system elements create dynamics that propagate to other parts of the systems in space and time. It may be feasible to isolate data at a specific scale of interest, provided the interaction of the data with ones at other scales is captured. It is precisely this point that is advanced in recent work [16] and is herein extended to specific materials problems and specific spatial correlation functions of material strength.

The scale isolation method has applications in many disciplines of science and engineering, and it is herein demonstrated through the strength of a polycrystalline bar or fiber problem in one spatial dimension and through establishing a relation between size and Hall-Petch effects at spatial scales in the vicinity of the average grain size in a polycrystalline material.

5 Strength of materials at particular scales and relation to Hall-Petch effects

We are not aware of models capable of capturing both size and Hall-Petch effects effectively within a single framework. As both effects are statistical in nature, let the material strength of a bar or fiber subjected to uniaxial load be described as a random process in the spatial dimension x, f(x). We consider a uniaxial problem for the sake of simplicity and illustration of the process. The wavelet transform of f(x) [17, 18] is expressed as

where a denotes spatial scale, b denotes the spatial variable in the x direction, and ψ(·) denotes the wavelet function. The wavelet variance, i.e., the variance of T_f(b, a), is dependent on scale a, i.e., it is decomposed on a scale-by-scale basis. As the spectral density function (SDF) decomposes the variance of a process across frequencies, the wavelet variance decomposes the same variance in scales. Thus, the variance of the strength appeals in characterizing the strength of a structure (bar in this case). The variance of T_f(b, a) is expressed as

where ɛ(k) denotes the SDF of f(x) and implies the Fourier transform of (·).

Consider that data collection from different spatial scales suggests that the uniaxial material strength of a polycrystalline material can be described as a random field characterized by its mean value 〈f(x)〉, its variance, and its autocorrelation function featuring a correlation length, l, directly related to the mean grain size of a polycrystalline material [19]. For this example, we consider that the fusion process suggests that strength is a stationary process with fixed variance and known autocorrelation structure. From Eq. (2), i.e., the expression for the variance in terms of scale, data (the variance in this case) can be expressed at the scale of interest. For bar-length L, the variance at specific scales is expressed as

where denotes the variance of the process over length L, and l₀ is a lower cutoff l such that there are enough internal lengths l in l₀ to obtain the statistics of strength properly. By varying l and keeping L and l₀ fixed, the Hall-Petch effects can be modeled, whereas by fixing l and l₀ and varying L, size effects on strength can be modeled.

Two autocorrelation functions for strength f(x), c_f are considered here, i.e., the Gaussian one

where r denotes the lag, having a Fourier transform

and the exponential one

having a Fourier transform

The continuous wavelet used is¹

and its Fourier transform is

With the above equations, Eq. (2) yields

for the Gaussian autocorrelation, and

for the exponential autocorrelation, where Erfc(·) denotes the complementary error function. From Eqs. (10) and (11), it follows that

for the Gaussian, and

for the exponential correlation. From Eq. (12) or (13), the integral in Eq. (3) can be easily evaluated. The autocorrelation function does not alter the trend as long as it is stationary, even though only few (two herein) autocorrelation functions can be examined analytically.

Figure 2 shows representative results based on the Gaussian autocorrelation function. In particular, Figure 2A shows size effects and Figure 2B the Hall-Petch effects. The Weibull distribution for the strength of a material is well documented in the literature, and so is the behavior of the Hall-Petch effects. Figure 3 shows results similar to Figure 2, yet for the exponential autocorrelation function.

Figure 2

(A) Prediction of size effects for l=1, l₀=5, and variable L based on the Gaussian autocorrelation function. Strength is inversely proportional to its variance from the mean value. One of the curves corresponds to Eq. (3) and the other to size effects where strength is proportional to (1/L)^m, m being a constant, resulting from a Weibull strength distribution. (B) Prediction of Hall-Petch effects for l₀=5l, L=20, and variable l. Strength is inversely proportional to its variance from the mean value. One of the curves corresponds to Eq. (3) and the other where strength is proportional to resulting from Hall-Petch effects.

Figure 3

Similar to Figure 2, yet for the exponential autocorrelation function for strength.

In practical terms, the wavelet variance expressed at particular scales represents information at these scales, yet contains information on the interactions from other scales. It is information fissioned at the particular scales from the one available at all scales based on the complete wavelet variance. The integral in Eq. (3) extends from zero to infinity for the non-fissioned information, yet, for the particular scales of interest, it reduces to the relevant integral and is only evaluated from scale l₀ to scale L. The present work concentrated on spatial scales. However, it is also amenable to temporal scales, with potential applications in time-dependent mechanics of materials problems such as fatigue, creep, relaxation, and dynamic loads.