On dislocation pileups and stress-gradient dependent plastic flow

International Journal of Plasticity 74 (2015) 1e16

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On dislocation pileups and stress-gradient dependent plastic flow Nasrin Taheri-Nassaj*, Hussein M. Zbib School of Mechanical and Materials Engineering, Washington State University, United States

a r t i c l e i n f o

a b s t r a c t

Article history: Received 9 February 2015 Received in revised form 2 June 2015 Available online 18 June 2015

In strain-gradient plasticity, the length scale controlling size effect has been attributed to so-called geometrically necessary dislocations. This size dependency in plasticity can also be attributed to dislocation pileups in source-obstacle configurations. This has led to the development of stress-gradient plasticity models in the presence of stress gradients. In this work, we re-examine this pileup problem by investigating the double pileup of dislocations emitted from two sources in an inhomogeneous state of stress using both discrete dislocation dynamics and a continuum method. We developed a generalized solution for dislocation distribution with higher-order stress gradients, based on a continuum method using the Hilbert transform. We qualitatively verified the analytical solution for the spatial distribution of dislocations using the discrete dislocation dynamic. Based on these results, we developed a dislocation-based stress-gradient plasticity model, leading to an explicit expression for flow stress. Findings show that this expression depends on obstacle spacing, as in the HallePetch effect, as well as higher-order stress gradients. Finally, we compared the model with recently developed models and experimental results in the literature to assess the utility of this method. © 2015 Elsevier Ltd. All rights reserved.

Keywords: A. Dislocations A. Strengthening mechanisms B. Constitutive behaviour A. Yield condition Size effect

1. Introduction Modeling the mechanical behavior of materials presents several challenges due to the complexity of the underlying microstructure. Depending on the objective, a material microstructure can be modeled explicitly or implicitly. Explicit modeling is an effective way to model the microstructure to study material behavior at the grain level, with each grain being a single crystal. By using averaging or homogenization methods, the overall polycrystalline properties of a material can be obtained. It is also possible to model microstructure implicitly by using internal state variables to represent the microstructure. Among the internal state variables, dislocations are the main carriers of plastic deformation, and contribute significantly to mechanical behavior such as strain hardening and ductility. The discrete dislocation dynamics (DDD) method is an explicit approach that considers the motion and interaction of individual dislocation segments to predict stressestrain response by direct simulation of dislocation assemblies. On the other hand, the continuum dislocation dynamics method is an implicit approach based on the concept of dislocation density. In this study, we formulated dislocation density-based models to represent the macroscopic material volume element encompassing mechanisms at the microscopic level. Dislocation

* Corresponding author. Tel.: þ1 509 715 7279. E-mail addresses: [email protected], [email protected] (N. Taheri-Nassaj). http://dx.doi.org/10.1016/j.ijplas.2015.06.001 0749-6419/© 2015 Elsevier Ltd. All rights reserved.

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density-based models can be used to bridge micro-level phenomena and macro-level continuum quantities such as stress and strain (e.g., Domkin, 2005). Dislocation-based crystal plasticity models can be used to describe dislocationeobstacle interactions as well as dislocationedislocation interactions to elucidate underlying dislocation mechanisms and microstructure. In such dislocationeobstacle configurations, dislocations pile up against impenetrable obstacles under applied stresses. For example, in polycrystalline materials, grain boundaries act as natural obstacles to dislocation motion. The classical problem of dislocation pileup goes back to investigations by Hall (1951), Petch (1953), Eshelby (1949) and Bilby and Eshelby (1968). Particularly, Hall (1951) and Petch (1953) investigated the pileup of dislocations against grain boundaries under a constant applied stress, leading pto ffiffiffi the well-known HallePetch relationship. This relationship describes the yield stress of a polycrystal as sy ¼ K= d þ s0 , where K is a material constant, d is the grain size, and s0 is the constant yield stress. The HallePetch relationship reveals the size-dependency of yield strength in polycrystals. It is based on the mechanism of pileup of dislocations against an obstacle (grain boundary) and the critical resolved shear stress required to cause the leading dislocation to penetrate the grain boundary under uniform applied stress. Dislocation pileup problems have been analyzed explicitly using discrete approaches (see, e.g., Eshelby et al., 1951; Chou and Li, 1969; Lardner, 1969; Biby and Eshelby, 1968) or implicitly using continuum approaches (see, e.g., Leibfried, 1951; Le and Stumpf, 1996; Kochmann and Le, 2008; Chou and Louat, 1962; Head and Louat, 1955; Chakravarthy and Curtin, 2011, 2014; Hirth, 2006; Akarapu and Hirth, 2013). In the first approach, the balance of interaction forces among discrete dislocations in a pileup leads to a set of nonlinear algebraic equations for the equilibrium positions. The second approach uses a continuous distribution of infinitesimal dislocations on a slip-plane, as presented by Eshelby (1949) in the PeierlseNabarro description (Nabarro, 1947; Peierls, 1940). The continuum model, along with the transform theory presented by Leibfried (1951), can readily describe the strain fields of pileups. Leibfried (1951) demonstrated that the equilibrium distribution in dislocation pileups could be described in terms of Hilbert Transforms and Tschebyscheff (Chebyshev) Polynomials. Several dislocation pileup analyses have been conducted for homogenous states of stress to formulate models for yield stress. Most notable is the double pileup problem resulting in the well-known HallePetch equation. However, especially on a small scale, the state of stress is generally non-uniform and can affect the assumed constitutive equation for flow strength. Thus, the pileup problem in an inhomogeneous stress state has garnered recent interest. Some studies have examined the pileup problem in the presence of stress gradients. Hirth (2006) proposed the stress-gradient dependent flow strength concept, using a continuum approach to analyze the pileup of dislocations emitted from two sources in the presence of a stress gradient. Hirth applied a continuum method by solving a singular integral equation with a kernel of Cauchy type on a finite interval. Other researchers extended this idea (Akarapu and Hirth, 2013; Chakravarthy and Curtin, 2011, 2014; Liu et al., 2013). For example, Chakravarthy and Curtin (2010, 2011, 2014). used a similar approach for a combined stress gradient for dislocations emitted from one source, with pileups at obstacles on either side of the source so that the dislocation density is zero at the center. These studies resulted in stress-gradient plasticity models that attribute the size effect to the dependence of flow stress on local stress gradients. Studies show the size effect for plastic deformation at a small length scale in torsion experiments of wires with micrometer diameters (see, e.g., Fleck et al., 1994; Dunstan et al., 2009), uniaxial compression or tension experiments of nano and micro€ lken and Evans, 1998; Evans and Hutchinson, pillars (see, e.g., Uchic et al., 2004), bending of thin beams and foils (see, e.g., Sto 2009; Ehrler et al., 2008), etc. The classical theory of plasticity cannot describe size-dependent phenomena since it assumes that material strength is dependent only on local variables. To remedy this problem, several strain-gradient plasticity models have been developed (see, e.g., Zbib and Aifantis, 1988a,b,c, 1989, 1992; Fleck and Hutchinson, 1997). Strain-gradient theories are often based on so-called geometrically necessary dislocations which lead to the dependence of flow stress on strain gradients. One main drawback of these theories is that length scale, as a coefficient that multiplies strain gradient terms in constitutive equations, is phenomenological and must be determined experimentally. Moreover, experimental measurements indicate that length scale depends on stress states and loading conditions, suggesting that it is an evolving internal variable related to the underlying microstructure. Stress-gradient plasticity models, however, arise from the classic analysis of dislocation pileups in source-obstacle configurations (Akarapu and Hirth, 2013; Chakravarthy and Curtin, 2011, 2014; Liu et al., 2013). They consider average obstacle spacing as a material length scale that controls material hardening. In this approach, flow strength relates to the force per unit length on the leading dislocation in the pileup. In terms of the crack problem, this force is based on the concept of a thermodynamic force derived from the rate of change of energy per unit crack length (Hirth and Lothe, 1982). To determine thermodynamic force at both ends of the crack or both tips of the pileup, both are treated equally by assuming symmetry. However, in terms of inhomogeneous stress, the pileup losses its symmetry (Liu and Gao, 1990), and thus the force based on that definition becomes ambiguous. In this work, we identified closed-form analytical solutions for flow stress in the presence of inhomogeneous local stress as the main objective. We also generalized the traditional HallePetch relation to account for spatial stress gradients. The classical HallePetch relationship concerns yield stress vs. grain size in a homogenous state of stress. Our findings show that yield stress is not only dependent on grain size, i.e. obstacle spacing, but also on higher-order stress gradients. Many studies have generalized the traditional HallePetch relationship into two-dimensions (Zhu et al., 2014) or coupled it with a kinetic equation of dislocation evolution in polycrystals (see, e.g., El-Awady, 2015). In this study, we examined an idealized dislocation double pileup problem as a fundamental, simplified case. We derived an analytical solution to inform development of a generalized stress-gradient plasticity model. Although we could consider more complex pileups in anti-planes (see, e.g.,

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Berdichevsky and Le, 2007) or parallel planes and inclined planes, integral equations in these cases cannot be treated analytically and must be solved numerically (see, e.g., Baskaran et al., 2010). Specifically, we developed a stress-gradient dependent flow stress model following the Hirth's method (Hirth, 2006; Akarapu and Hirth, 2013). First, we analyzed the dislocation double pileup problem in the presence of various stress gradients. Then we examined the stress field ahead of the pileup tip. We used both discrete dislocation dynamics analyses (see, e.g., Zbib et al., 1996) and a continuum approach to evaluate pileup density. We used discrete dislocation dynamics (DDD) to demonstrate the physical aspects of the pileup and the need to introduce more than one dislocation source to accommodate a variety of possible stress fields. Furthermore, to avoid ambiguity in determining the force at the tip of the pileup, we avoided the classical definition of thermodynamic force described above. Instead, we assumed that the pileup would collapse when the shear stress near the tip of the pileup reached a critical value, leading to expression for flow stress that depends on both the size of the pileup (as in the classical HallePetch relationship) and the magnitude of the stress gradients. Our findings demonstrate that this approach can easily be extended to higher-order stress gradients by simple superposition of solutions for various stress fields. 2. Discrete dislocation pileup analysis Under an applied stress, dislocations will emit from FrankeRead sources, glide and pile up against obstacles. Emission of dislocations from the sources continues until the pileup reaches equilibrium. To simulate the pileup problem, we first used the discrete dislocation dynamics model developed by Zbib et al. (1996, 1998, 2000, 2002) and Zbib and De la Rubia (2002). In this model, arbitrary three-dimensional curved dislocations are discretized into piecewise, continuous arrays of dislocation segments. Their spatio-temporal dynamics are traced by Newtonian-type equations that include the effect of long-range interactions, surface effects, as well as short-range interactions. To gain a clear view of the pileup mechanism, we restricted the model to a simple box of specific dimensions of size L (¼ 4000 b), where b is the magnitude of the Burgers vector. FrankeRead (FR) sources are placed inside the simulation cell with boundary conditions, as shown in Fig. 1. Under applied shear stress, the dislocations will emit from the FrankeRead sources and move toward the rigid boundaries at x ¼ L=2 and x ¼ þL=2 . The sources are straight, finite dislocation lines parallel to the z-axis (and to each other) in a Cartesian coordinate system. We also considered edge dislocations with Burgers vectors parallel to the x-axis. Furthermore, we assumed that the dislocations could only glide on their glide planes (perpendicular to the y-axis) with no climb. Therefore, all dislocations would stay on their slip planes perpendicular to the obstacles. These obstacles are impenetrable; in this study, we used copper, which has an FCC crystalline structure. Shear stress was applied on the z-plane in the x-direction, and DDD simulations were performed for five stress-field cases: Case Case Case Case Case

0: Constant stress sðxÞ ¼ s0 1: Linear stress sðxÞ ¼ s0 a1 ð2x=LÞ 2: Quadratic stress sðxÞ ¼ s0 a2 ð2x=LÞ2 3: Cubic stress sðxÞ ¼ s0 a3 ð2x=LÞ3 01: Combined sðxÞ ¼ s0  s0 a1 ð2x=LÞ

Fig. 1. Simulated model in discrete dislocation dynamics.

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where a1 …:a3 > 0. For Case 1 (constant stress), only one FR source was placed at the center of the cell (x ¼ 0), and for inhomogeneous cases (Cases 1, 2, 3, and 01) two identical FR sources were placed in cell one at x ¼ L/4, and one at x ¼ L/4, as shown in Fig. 1. Stress distributions are shown in Figs. 2ae6a, and equilibrium configurations obtained from the DDD simulations for the five cases are shown in Figs. 2be6b, respectively. Figs. 2ae6a also show schematics of equilibrium distributions obtained from DDD simulations. 3. Continuum dislocations pileup analysis A discrete dislocation pileup can also be represented by a continuous distribution of infinitesimal dislocations, with the exception of the area very close to the pileup tip. To derive such a continuum representation, we employed the method developed by Hirth (2006) and Akarapu and Hirth (2013) to analyze the dislocation double pileup problem in the presence of

Fig. 2. a) Equilibrium distribution of discrete dislocations in a double ended pileup, b) DDD-simulation result, c) Continuum and discrete dislocation density versus spatial position for the case of constant stress. Green curve is continuum curve and its equivalent DD plot is the red dots. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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Fig. 3. a) Equilibrium distribution of discrete dislocations in a double ended pileup, b) DDD-simulation result, c) Continuum and discrete dislocation density versus spatial position for the case of linear stress. Green curve is continuum curve and its equivalent DD plot is the red dots. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

stress gradients. The governing equation for equilibrium distribution of dislocations is obtained as a balance between applied stress and internal stress resulting from the dislocationedislocation interaction. For any state of stress within the domain L=2  x  þL=2 , equilibrium is given by the following integral equation:

mb2 sðxÞb ¼ 2pð1  nÞ

ZL=2 L=2

nðx0 Þdx0 ; x0  x

L L  xþ : 2 2

(1)

To normalize the length scale, a non-dimensional parameter x ¼ 2x=L is introduced such that nðxÞ ¼ nðxÞ; and x lies in the interval [1, þ1]. This normalization leads to the following integral equation.

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Fig. 4. a) Equilibrium distribution of discrete dislocations in a double ended pileup, b) DDD-simulation result, c) Continuum and discrete dislocation density versus spatial position for the case of quadratic stress. Green curve is continuum curve and its equivalent DD plot is the red dots. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

2ð1  nÞsðxÞ 1 ¼ mb p

Zþ1 1

     n x0 dx0 ¼ Hx n x0 ; x0  x

1  x  þ1;

(2)

where sðxÞ is of power form with integer powers, e.g. sðxÞ ¼ as0 ð2x=LÞm ¼ an s0 xm . The solution to the integral equation (2) can be obtained using the Hilbert Transformation Hx ½nðx0 Þ and Tschebyscheff polynomials (Leibfried, 1951). For the five stress distribution cases analyzed in the previous section using discrete dislocation dynamics, the stress fields and corresponding dislocation density distributions obtained by solving equation (2) are given as follows: Case 0.

Constant stress sðxÞ ¼ s0

As0 ffi x; nðxÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi 1  x2

A¼

2ð1  nÞ mb

(3)

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Fig. 5. a) Equilibrium distribution of discrete dislocations in a double ended pileup, b) DDD-Simulation result, c) Continuum and Discrete Dislocation Density versus spatial position for the case of cubic stress. Green curve is continuum curve and its equivalent DD plot is the red dots. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

sðxÞ ¼ a1 s0 x;

Case 1.

  As0 1 ffi x2  ; n1 ðxÞ ¼ a1 pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1  x2 Case 2.

(4)

sðxÞ ¼ a2 s0 x2

  As0 1 ffi x x2  ; n2 ðxÞ ¼ a2 pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1  x2

(5)

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Fig. 6. a) Equilibrium distribution of discrete dislocations in a double ended pileup, b) DDD-simulation result, c) Continuum and discrete dislocation density versus spatial position for the case of combined stress. In (c) solid curves are continuum dislocation density plots considering different alphas from 0.2 (light blue) to 0.9 (violet). Green one is 0.5 and its equivalent DD plot is the red dots. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Case 3.

sðxÞ ¼ a3 s0 x3

  As0 1 ffi x4  x2  1 ; n3 ðxÞ ¼ a3 pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1  x2

(6)

Case 01. sðxÞ ¼ s0  s0 a1 x



 As0 1 ffi x  a1 x2  : nðxÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1  x2

(7)

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Fig. 7. Size effect in torsion: comparison between the stress-gradient models and experimental data.

Equations (3)e(7) on dislocation density distributions are plotted in Figs. 2c, 3c, 4c, 5c, and 6c, respectively. As can be seen in these figures, dislocations pile up at the rigid boundaries. Note that in Fig. 6c, four continuous plots correspond to different values of a3 . The positive parts of each continuum plot are associated with positive dislocations and the negative parts are negative dislocations. The red dots in Figs. 2ce6c are obtained from DDD results by dividing the spatial domain into finite intervals (¼ 200b in zone II and ¼ 100b in zone I where the dislocation density is more intense), and then presenting the total number of dislocations in each interval as a super-dislocation (the total number of dislocations divided by the size of the interval). As can be deduced from these figures, analytical results from the continuum approach are in qualitative agreement with DDD results, except for singularities at the boundaries. This is because in DDD, the magnitude of the Burgers vector is constant. In developing the above solutions, we did not impose any conditions on nðxÞ at x ¼ 0, although as Chakravarthy and Curtin (2011, 2014) and Liu et al. (2013) imposed the condition nðxÞ ¼ 0 at x ¼ 0. Following the procedure outlined above, one can generate the dislocation density distributions for higher-order states of ffi pffiffiffiffiffiffiffiffiffiffiffiffiffi stress. Table 1 lists solutions up to the 6th order, and the solution is divided into two parts: a singular part as0 A= 1  x2 and a polynomial (regular) part n0i ðxÞ, such that

ai s0 A 0 ffini ðxÞ: ni ðxÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi 1  x2

(8)

For convenience, the regular part can be written in the following compact form:

n0i ðxÞ ¼

iþ1 X

aij xj ;

(9)

j¼0

where the coefficients aij are listed in Table 2.

3.1. Generalized, inhomogeneous states of stress The state of stress is generally non-uniform, resulting in stress gradients. Stress gradients appear as a result of loading or because the materials themselves and their substructures are inhomogeneous. Classical continuum formulation is usually Table 1 The polynomial part of the dislocation density function for various stress states. Case : sðxÞ

s0 A ffi 0 ni ðxÞ ¼ paiffiffiffiffiffiffiffiffi n ðxÞ 2 i

0 : a0 s0 ða0 ¼ 1Þ 1 : a1 s0 x 2 : a2 s0 x2

n00 ðxÞ ¼ x n01 ðxÞ ¼ 12 þ x2

3 : a3 s0 x3

n03 ðxÞ ¼ 18  12x2 þ x4

4 : a4 s0 x4 5 : a5 s0 x5 6 : a6 s0 x6

1x

n02 ðxÞ ¼ 12 x þ x3 n04 ðxÞ ¼ 18 x  12x3 þ x5 1  1x2  1x4 þ x6 n05 ðxÞ ¼ 16 8 2 1 x  1x3  1x5 þ x7 n06 ðxÞ ¼ 16 8 2

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Table 2 P j The coefficients aij in n0i ðxÞ ¼ iþ1 j¼0 aij x . P iþ1 j 0 ni ðxÞ ¼ j¼0 aij x aij j

0

1

2

3

4

5

6

7

8

i 0 1 2 3 4 5 6

0 12 0 18 0 1 16 0

1 0 12 0 18 0 1 16

0 1 0 12 0 18 0

0 0 1 0 12 0 18

0 0 0 1 0 12 0

0 0 0 0 1 0 12

0 0 0 0 0 1 0

0 0 0 0 0 0 1

0 0 0 0 0 0 0

based on the assumption that the state of stress at a material point is the average of the inhomogeneous stress field over a representative volume. Generally, the state of stress at a given material point, say located at x ¼ 0; can be divided into a homogenous part s0 and an inhomogeneous part that express perturbations from this homogenous stress. In one dimension, the state of stress at x ¼ 0 can be expressed by the Taylor series expansion in the form of Equation (10). This equation employs the non-dimensional variable x so that x ¼ 2x=L , where L is the size of the representative volume.

h i sðxÞ ¼ s0 1  a1 x  a2 x2  a3 x3  …  aj xj … :

(10)

The a0i s are the spatial gradients of stress as follow.

  1 vs ; a1 ¼ s0 vx

! 1 v2 s a2 ¼ ; s0 2! vx2

! 1 v3 s a3 ¼ ; ::: s0 3! vx3

(11)

The corresponding dislocation density distribution can be constructed according to the same procedure. The solution is the sum of the solutions obtained for each of the terms in the series. For jth order power series, the solution can be expressed in the general form,

# " j X s0 A 0 0 ffiffiffiffiffiffiffiffiffiffiffiffiffi ffi p ni ðxÞ ¼ ai ni ðxÞ ; no ðxÞ  1  x2 i¼1

(12)

where the regular functions n0i ðxÞ for up to j ¼ 6 are given in Table 2. The minus sign (as opposed to plus sign) in Equation (10) is used for convenience so that a limiting case is obtained, which turns out to be a critical state. The critical state determines macroscopic yield stress that corresponds to a stress state at the right tip of the pileup (x ¼ L/2) with positive coefficients in Equation (10). To elucidate this point, first consider the homogenous part of the stress s0 . This part by itself results in an anti-symmetric dislocation distribution; the magnitude of the resulting shear stress at the right tip of the pileup (x ¼ L/2) is equal to that at the left tip (x ¼ L/2) and, thus, both tips will break simultaneously once the shear stress reaches some critical value. However, when inhomogeneous stress components are added to the homogenous part, the distribution may lose symmetry and the intensities at the left and right tips may no longer be equal. Consider, for example, the special case when s0 and all the coefficients in Equation (10) are positive (a0 s > 0Þ, then the stress and, thus, the pileup on the right side (0 < x < L/2) will be less intense than that at the left side. In this limiting case, it turns out, as discussed in the next section, that the critical stress required to break up the entire pileup is the one needed to overcome the material strength at the right tip (x ¼ L/2). 4. Gradient-dependent flow stress Flow stress is the critical resolved shear stress required to break up the pileup by overcoming the strength of the barrier. Hirth (2006) notes that within a continuum approximation, the pileup becomes equivalent to a shear crack. Therefore, one calculates the total energy per unit depth released in the formation of the pileup (crack), and subsequently determines a thermodynamic force per unit length acting to extend the pileup ðFtip Þ. Alternatively, the force at the tip can be calculated using the method described by Bilby and Eshelby (1968). However, when the applied stress is inhomogeneous, the definition of a thermodynamic force at the tip becomes ambiguous. Rather than resorting to the thermodynamic force concept, we simply analyze the stress field ahead of the tip of the pileup, determine the stress state near the tip, and associate the flow stress with the strength of the barrier. 4.1. Shear stress field ahead of the tip of the pile-up For a given dislocation density distribution, the long-range shear stress field ahead of the tip of the pile-up is given by the relation

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ZL=2 sxy ðy ¼ 0Þ ¼ L=2

11

mb nðx0 Þ dx0 : 2pð1  nÞ ðx  x0 Þ

(13)

Table 3 displays results for various stress distributions with various applied inhomogeneous stress states, up to power 6. Because the solution is for linear elastic media, it is apparent that the stress field for any combinations of applied stresses is simply the superposition of individual solutions. 4.2. Stress near the tip Right Tip: x ¼ L/2 For each of the subsequent cases, we evaluated the shear stress near the right tip of the pileup, asymptotically, for x ¼ L/2 þ R with R << L/2. Results are given in Table 3, which indicate that the near tip stress field for all the cases is given by

rffiffiffi L ; sxy yci ai s0 R

(14)

where i ¼ 0, 1, 2 … 6, and the corresponding constants are ci ¼ 1/2, 1/4, 1/4, 3/16, 2/16, 5/32, 5/32 respectively. Left Tip: x ¼ L/2 The shear stress near the left tip of the pileup, x ¼ L/2R with R << L/2 has the same magnitude of that for the right tip but of opposite sign for the even power cases (cases 0,2,4,6…) and has the same sign for the odd power cases (cases: 1,3,5…). This is because in the even power cases the dislocation distributions and stress fields are anti-symmetric, while in the odd power cases they are symmetric. We assume that the pileup will break up at both ends when the shear stress just ahead of the right tip reaches a critical value, say s, representing the intrinsic strength of the barrier (grain boundary, particle, interface, etc.). Thus, the flow stress  sy can be obtained by letting sxy  ¼ s* in Equation (14) and solving for s0 ¼ sy, leading to

syi ¼

pffiffiffiffiffiffiffiffiffiffi s*2 R 1 pffiffiffi : ci jai j L

(15)

For the homogenous state of stress (Case 0, i ¼ 0), equation (15) reduces to

K sy0 ¼ pffiffiffi; L

K≡

pffiffiffiffiffiffiffiffiffiffiffiffiffi 4s*2 R:

(16)

This equation mirrors the HallePetch relationship, which predicts that the flow stress is inversely proportional to the square root of the size of the pileup.

Table 3 The shear stress fields ahead the tip of the pile-up for various stress states. sxy ðy ¼ 0Þ

Case: sðxÞ 0 : a0 s0

ða0 ¼ 1Þ

Near tip stress x ¼ L/2 þ R, R << L/2 qffiffiffi s0 12 RL

! x ffi1 s0 pffiffiffiffiffiffiffiffi 2 x 1

1 : a1 s0 x 2 : a2 s0 x2 3 : a3 s0 x3 4 : a4 s0 x4 5 : a5 s0 x5 6 : a6 s0 x6

a1 s0 a2 s0

! 2

1þ2x pffiffiffiffiffiffiffiffiffi 2

x 1

2

2

1þ2x pffiffiffiffiffiffiffiffiffi 2

x 1

2

a1 s0 14

x ! !

2

x

5

 x4

4

6

 x5

7

6

a3 s0 a4 s0

x4x þ8x pffiffiffiffiffiffiffiffi ffi

a5 s0

12x p 8x þ16x ffiffiffiffiffiffiffiffi ffi

a6 s0

x2x p 8x þ16x ffiffiffiffiffiffiffiffi ffi

2

x 1 3

8

2

2

16

x 1 5

2

x 1

qffiffiffi

3 a4 s0 16

!

2

3

16

3

!

x 1

L R

3 a3 s0 16

4

14x þ8x pffiffiffiffiffiffiffiffi ffi 8

a2 s0 14

 x2 x

qffiffiffi

! x

5 a5 s0 32 5 a6 s0 32

L R

qffiffiffi

L R

qffiffiffi

L R

qffiffiffi

L R

qffiffiffi

L R

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When the local stress is inhomogeneous and can be expressed as in Equation (10), the stress field ahead of the pileup and near the tip is simply the superposition of solutions listed in Table 3. The corresponding flow stress can be obtained following the same procedure. Table 4 lists various combinations of local stresses with higher-order gradients, expressed in terms of the flow stress sy0 . The absolute values of the gradient coefficients appear in the equations for the reasons discussed below in Section 5. 5. Discussion When local stress state is homogenous, the classical solution for distribution of dislocations in a double-ended pileup against two obstacles is anti-symmetric with n(x ¼ 0) ¼ 0 when one obstacle is located at x ¼ þL/2 and one at x ¼ L/2 (See Equation (3)). This corresponds to the emission of dislocations from a single FrankeRead source located at x ¼ 0, with “positive” dislocations piling up in the range 0  x  L=2 and equally “negative” dislocations in the range L=2  x  0 (Fig. 2). This case leads to the classical HallePetch relationship that predicts increasing strength with decreasing grain size or obstacle spacing. However, when the stress state is inhomogeneous and can be expressed in a power form (i.e. s ¼ axn ), the distribution is symmetric when n is an odd integer (i.e., the stress is antisymmetric around x ¼ 0 in the range L=2  x  L=2) and nðx ¼ 0Þs0. When the stress distribution is symmetric (even integer n), similar to the constant stress case, the dislocation distribution is anti-symmetric and nðx ¼ 0Þ ¼ 0, see Fig. 4 and Equation (5). As suggested by Hirth (2006), in these cases the dislocations must be emitted from multiple sources. DDD analyses show that the distributions can be generated from two FrankeRead sources (see Figs. 3e5). When the stress state is a combination of a constant value and a linear distribution (Case 01), the dislocation density distribution loses symmetry and ðx ¼ 0Þs0 , as can be deduced from Fig. 6c and Equation (7). Fig. 6b shows that using DDD the distribution can be produced from two FrankeRead sources. Liu et al. (2013) and Chakravarthy and Curtin (2011, 2014) developed the dislocation distribution with the condition nðx ¼ 0Þ ¼ 0, which corresponds to only one FranckeRead source in this case, at x ¼ 0. However, this presumes that the stress may not be not be zero at x ¼ 0. With multiple sources, however, there is no need to impose such a condition, and dislocations can be generated for any state of stress. The stress field ahead of the tip of the double-ended pileup is computed for the various stress gradient cases (see Table 3), from which a generalized solution can be constructed with the superposition method. For each stress case, the stress intensity at the tip of the pileup is calculated and given by Equation (14). Since these solutions are based on linear elasticity, the tip stress for any combination of these cases can be obtained by appropriately adding the solutions from each case. The stress intensity at one of the two tips of the double-ended pileup is then used to derive an expression for the macroscopic stress gradient-dependent flow stress s0y . The flow stress is associated with the breakup of the entire pileup when the stress intensity at the two tips of the pileup reaches a critical value equal to the strength of the barrier. When the dislocation distribution under an applied state of stress is anti-symmetric or symmetric the magnitude of the stress intensities at the two tips of the double-ended pileup is identical and both pileups (one on the left and one on the right) will break simultaneously once the stress reaches a critical value. However, when the distribution is neither symmetric nor antisymmetric the magnitudes of the stress intensities at the two tips will not be equal. In this case, the two tips will not break simultaneously at the same stress level. One tip will break at a lower stress than the other. The higher of the two stresses is then defined as the flow stress that is required to break up the entire pileup, leading to the results given in Table 4. To elucidate more on these results, we discuss two stress state scenarios: a) linear, s0 ð1  a1 xÞ (odd power), and b) quadratic, s0 ð1  a2 x2 Þ (even power). We derive the flow stress based on the stress intensities at both tips and for positive and negative coefficients. a) Linear, sðxÞ ¼ s0 ð1  a1 xÞ

Table 4 Flow stress. n

Combined applied stress cases P sðxÞ ¼ s0 ð1  nj¼1 aj xj Þ

Flow stress s0y

0 1

s0 s0 ð1  a1 xÞ

sy0 ¼ pKffiffi

2

sy0 ja1 j

1

s0 ð1  a1 x  a2 x2 Þ 2

L

2

sy0 ja1 j ja2 j  2 2 sy0 ja1 j ja2 j 3ja3 j 1 2  2  8 sy0 ja j ja j 3ja j 3ja j 1 21  22  83  84 sy0 ja j ja j 3ja j 3ja j 5ja j 1 21  22  83  84  165 sy  1

3

3

s0 ð1  a1 x  a2 x  a3 x Þ

4

s0 ð1  a1 x  a2 x2  a3 x3  a4 x4 Þ

5

s0 ð1  a1 x  a2 x2  a3 x3  a4 x4  a5 x5 Þ

6

s0 ð1  a1 x  a2 x2  a3 x3  a4 x4  a5 x5  a6 x6 Þ

1

ja1 j ja2 j 3ja3 j 3ja4 j 5ja5 j 5ja6 j  2  8  8  16  16 2



N. Taheri-Nassaj, H.M. Zbib / International Journal of Plasticity 74 (2015) 1e16

13

The stress distribution is not symmetric for a1 > 0(also Case 01 in Sections pffiffiffiffiffiffiffiffi 2 and 3). The near tip stress at the right tip is the difference between Case 0 and Case 1 (Table 3), yielding s0 ð1  a2 =2Þ L=R and resulting in thep flow ffiffiffiffiffiffiffiffi stress sy0 =ð1  a1 =2Þ. The near tip stress at the left tip is the sum of Case 0 and Case 1 (Table 3), yielding s0 ð1 þ a1 =2Þ L=R and resulting in the flow stress sy0 =ð1 þ a1 =2Þ. We define macroscopic flow stress to be the one needed to break up the entire pileup, and thus the tip that has the lowest stress intensity (weak) determines it. In this case it is the right tip, and so the flow stress is equal to sy0 =ð1  a1 =2Þ for a1 > 0. Following the same arguments as above, for a1 < 0 the left tip is the weak one and so the flow stress in this case is equal to sy0 =ð1 þ a1 =2Þ . Two equations above can be reconciled into one equation for the flow stress given in Table 4, i.e. sy0 =1  ja1 j=1 with both a1 > 0 and a1 < 0. The same conclusion can be drawn for all distributions consisting of a homogenous part plus odd power terms: i.e. a1 x; a3 x3 ; a5 x5 … a) Quadratic, sðxÞ ¼ s0 ð1  a2 x2 Þ Here the stress distribution is anti-symmetric for both a2 > 0 and a2 < 0, so the magnitudes of stress intensity at both the right and the left tips are equal. Following the same procedure as above, and considering the right tip, it turns out that the flow stress in this case is equal to sy0 =ð1  a2 =2Þ for a2 > 0, and sy0 =ð1 þ ja2 j=2Þ for a2 < 0. In this case, we define the flow stress with larger magnitude (upper limit) as being the macroscopic flow stress that will ensure the breakup of any pileup under this state of stress, leading to sy0 =ð1  ja2 j=2Þ for both a2 > 0 and a2 < 0. The same conclusion can be drawn for all distributions consisting of a homogenous part plus even power terms: i.e. a2 x2 ; a4 x4 ; a6 x6 … Table 4 indicates that the flow stress has two contributions, one arising from the size of the pileup (e.g., grain size), and one from the stress gradient terms. This can be written in the general form,

s0y ¼

sy ; 1  f ðja0 sjÞ

(17)

where sy0 is given by Equation (16), corresponds to the constant stress state case. This accounts for the effect of the size of the pileup L (e.g., grain size); f ðja0 sjÞ is a linear function of the stress gradients. Equation (17) illustrates the interaction between the intrinsic size effect captured by sy and the size effect from higherorder stress gradients. Note that for combined linear stress Case 01 (Equation (7)), n ¼ 1 in Table 4, the derived equation for the flow stress, s0y ¼ sy =ð1  a1 =2Þ, is similar to that given in Chakravarthy and Curtin (2011, 2014). This was based on the driving force concept using a method described by Bilby and Eshelby (1968) (i.e. Ftip ¼ ½mbp=4ð1  nÞ limx/L=2 ðx±L=2ÞðnðxÞÞ2 ), with the condition nðx ¼ 0Þ ¼ 0. Using the energy method to derive Ftip , Liu pffiffiffiffiffiffiffiffiffiffiffiffiffiffi et al. (2013) obtained a different expression: s0y ¼ sy = 1  a1 . Equation (17) predicts that flow stress depends on the stress gradient. This dependence becomes strong when dimensions are small. To illustrate this dependence, we consider the phenomenon of size effect in the torsion of micro-wires and the size effect for micro-beam bending. Numerous continuum strain gradient theories and dislocation-density based theories have been used to model size effect at micron and sub-micron scales (see e.g., Fleck and Hutchinson, 1997; Tsagrakis and Aifantis, 2002; Zaiser et al., 2007; Kaluza and Le, 2011; Le and Nguyen, 2013). Gradient plasticity theories were used to interpret size effect for specimens with a micrometer dimension subjected to torsion and bending (e.g., Tsagrakis and Aifantis, 2002), and a density-based continuity model was used to predict size effects in bending of thin films (e.g., Zaiser et al., 2007). We also mention the work by Kaluza and Le (2011) and Le and Nguyen (2013) who used a continuum dislocation theory to investigate torsion of single crystal rods and to model bending of single crystal beams. Below we compare our results with the model of Liu et al. (2013) which was also derived based on a dislocation pileup problem and thus is more suitable for comparing with the results based on the model presented in this paper. 5.1. Size effect in the torsion of micro-wires Consider a metal wire with radius R that is loaded in torsion. Assume that the wire is linear and made of elasticeplastic isotropic material. It can be easily shown that up to plastic yielding, the stress distribution is linear in the radial direction. The first-order stress gradient term becomes a1 ¼ L=2R (Equation (11)), and all other high stress gradient order terms are zero. In this case, the gradient-dependent flow stress is given by:

s0y sy

¼

1 : 1  L=4R

(18)

This equation, in non-dimensional form, predicts that the flow stress in torsion is dependent on the spacing between the internal obstacles spacing L (grain size, etc.) as well as the size of wire R. The equation predicts that flow stress becomes singular when R ¼ L/4. For a very large radius R >> L, the gradient effect diminishes and the flow stress approaches sy . Equation (18) is plotted in Fig. 7 for various values of L. We also plotted data extracted from p experimental results (Fleck ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi et al., 1994; Liu et al., 2012, 2013). The results using the model of Liu et. al. (2013), s0y =sy ¼ 1  L=2R, are also plotted in Fig. 7.

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N. Taheri-Nassaj, H.M. Zbib / International Journal of Plasticity 74 (2015) 1e16

5.2. Size effect in micro-beam bending Consider a thin metal beam of thickness H that is loaded in bending. As before, assume that the beam is linear and composed of elasticeplastic, isotropic material. In this case, and up to plastic yielding, the stress distribution is linear in the thickness direction, resulting in a1 ¼ L=H. Thus, equation (17) yields 0

sy sy

¼

1 : 1  L=2H

(19)

Equation (19) is plotted in Fig. 8 for various values of L. Also in the figure, we plot the experimental data from Moreau et al. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (2005) and the prediction by the model of Liu et. al. (2013), s0y =sy ¼ 1  L=H . 5.3. Comparison with experiments Figs. 7 and 8 shows that experimental data can be captured by the model for a certain range of L, for torsion 9 mm < L < 20 mm, and for bending 3 mm < L < 5 mm. Although this may be attributed to expected statistical variation in the microstructure, the range of variation is large, especially for the torsion case. Furthermore, the models suggest that the strength becomes large as the specimen size becomes comparable to the size of the internal length scale L, as R/L=4 for the torsion case, and as H/L=2 for bending. Such conditions arise solely from the derivation of the flow stress, whether they are based on the stress intensity approach developed in this paper or the energy method. In both approaches, the models assume dislocation pileups in the infinite medial. This assumption, however, breaks down when dealing with dislocations in a small medium, such as micro-wires and micro-beams. In this case, image forces arising from the interaction of dislocations with free surfaces may affect formation of the dislocation pileup, especially when the dimensions are in the sub-micrometer range. This may result in different values for gradient coefficients. Nonetheless, the model describes the physical origin of the size effect, relating it to the local stress state and the resulting equilibrium dislocation configuration, which differs from the mechanism used to motivate strain-gradient theories. In strain-gradient plasticity theories, the size effect in metals is related to the formation of so-called geometrically necessary dislocations, leading to dependence of flow stress on higher-order strain gradients (see, e.g., Zbib and Aifantis, 1988a,b,c, 1989, 1992; Zbib, 1994; Fleck and Hutchinson, 1997). 5.4. Stress-gradient dependent flow stress Following Chakravarthy and Curtin (2011, 2014), within the framework of isotropic plasticity, the effect of the stressqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi gradient on the flow stress can be generalized by using the concept of effective stress. This is defined as s ¼ 3s0ij s0ij =2, where s0ij is the deviatoric stress tensor. Equations (11) and (17), along with results in Table 4, yield the following generalized equation for the stress-gradient dependent flow stress:

Fig. 8. Size effect in bending: comparison between the stress-gradient models and experimental data.

N. Taheri-Nassaj, H.M. Zbib / International Journal of Plasticity 74 (2015) 1e16

s0y ¼

sy ; 1  f ðVsÞ

f ðVsÞ ¼

L L2  2  L3  3  jVsj þ V s þ V s þ … 4s 16s 128s

15

(20)

where Vð:Þ is the first order gradient operator, V2 ð:Þ ¼ V:Vð:Þ, V3 ð:Þ ¼ V:ðV:Vð:Þ, etc. Assuming that the stress gradient terms are small, Equation (20) can be expanded to yield the following linear relationship:

L L2  2  L3  3  jVsj þ s0y ysy 1 þ V s þ V s þ … : 4s 16s 128s

(21)

This linear form is analogous to some strain-gradient models (see, e.g., Zbib and Aifantis, 1988a,b,c, 1989, 1992; Fleck and Hutchinson, 1997), where flow stress appears to depend on the gradient of the effective plastic strain. However, in those theories, the length scale for the strain gradient coefficients is not explicitly defined. It is assumed to be a material parameter that is curve-fitted to experiments. Liu et al. (2013) and Chakravarthy and Curtin (2011, 2014) note that this is not the case in stress-gradient models. In Equation (19), the stress gradient coefficients are given explicitly in terms of obstacle spacing L, and the effective stress as can be deduced from the equation. By applying the linear model given by Equation (21) to the problems of torsion and bending, we obtain the following equations for the flow stress.

s0y sy s0y sy

¼1þ

L 4R

torsion

¼1þ

L 2H

bending

(22)

These linear models are plotted in Figs. 7 and 8, respectively, and predict the experimental data much better than the models given by Equations (18) and (19). In fact, the predictions of the linear model extend to the sub-micrometer region. 6. Summary and conclusions In this work, we re-examined the pileup problem by investigating the double pileup of dislocations in the presence of the inhomogeneous state of stress. We employed both a discrete dislocation dynamics and a continuum approach. It is assumed that multiple dislocation sources can operate simultaneously to accommodate the heterogenous nature of the stress field. Thus, there is no need to impose additional conditions on the anticipated distribution, e.g., n(x) at x ¼ 0. Using a continuum method that incorporates the Hilbert transform, we developed a generalized solution for the dislocation distribution in the presence of higher-order stress gradient, and then verified results using discrete dislocation dynamics simulation. Based on these results, we derived the flow stress using a stress intensity method. We selected this instead of the energy method used by Hirth (2006, 2013) and Liu et al. (2013), which is based on the concept of thermodynamic force. In the stress intensity approach, we determined the stress state near the tip of the pileup and then associated the local stress with the strength of the obstacle. This approach avoids an ambiguous definition of the flow stress when the stress state is inhomogeneous. Consequently, we developed a dislocation-based stress-gradient plasticity model that leads to an explicit expression for the critical stress to overcome the strength of the obstacle. Results show that this stress-gradient plasticity model depends not only on the obstacle spacing of the HallePetch effect, but also on higher-order stress gradients. Assuming that the stress gradient terms are small, this model can be extended to a linear form that is analogous to many strain-gradient models where flow stress appears to depend on the gradient of the effective plastic strain. However, in strain-gradient plasticity theories, the strain gradients coefficients are not defined explicitly, and are assumed to be material parameters. This is not the case for stress-gradient plasticity models in which the stress gradient coefficients are given explicitly. It is noted that while the stress gradient plasticity models can predict the size dependency of material strength at the micro scale, they remain phenomenological and do not provide a direct link to the evolving microstructure. Internal state variables, such as dislocation density and its gradients, can be more physically based. In the method described above along with the dislocation pileup approach, the stress gradients coefficients appear in both the stress field and the dislocation density distributions. Thus, it may be possible to develop relationships between average dislocation density and these coefficients, leading to dependence of flow stress on the dislocation density and its gradients, which is the subject of our next investigation. Finally, we compared our models with the experimental results on micro-wires loaded in torsion and also micro-beams loaded in bending. The experimental data can be captured by the exact model within a certain range of L, which is 9 mm < L < 20 mm for torsion and 3 mm < L < 5 mm for bending. This may be attributed to the assumption regarding derivation of flow stress that considers dislocation pileups in an infinite medium, either based on the stress intensity approach that we used or the energy approach. Therefore, there is ambiguity in regard to pileups in small mediums in the submicrometer range. However, a linear model can effectively predict the experimental data, and can also be used to predict the corresponding flow stress for the sub-micrometer region.

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