The role of dislocation pile-up in flow stress determination and strain hardening

Scripta Materialia 116 (2016) 53–56

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Regular Article

The role of dislocation pile-up in flow stress determination and strain hardening Yichao Zhu a,⁎, Yang Xiang a, Katrin Schulz b a b

Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, China Institute for Applied Materials (IAM-CMS), Karlsruhe Institute of Technology, Kaiserstr. 12, 76131 Karlsruhe, Germany

a r t i c l e

i n f o

Article history: Received 30 September 2015 Received in revised form 29 December 2015 Accepted 15 January 2016 Available online xxxx Keywords: Dislocation interaction Dislocation pile-up Flow stress Hardening rate Work hardening

a b s t r a c t Flow stress relation with dislocation densities due to various types of inter-plane interactions between perfect dislocations in face-centred-cubic crystals is formulated. Compared to the widely used Taylor-type hardening relations, the derived formulae resolve more details from the underlying discrete dislocation dynamics. The shear flow stress is found weaker than the critical resolved shear stress to remobilise a singly locked primary dislocation, because dislocations can dynamically pile up against junctions resulting from dislocation interactions. With the derived formulae, the roles played by dislocations belonging to different slip systems in strain hardening are further clarified. © 2016 Elsevier Ltd. All rights reserved.

Predicting the yield stress and the strain hardening behaviour of crystals has been a long-standing challenge. The key to this issue highly relies on an accurate description of dislocation interactions between different slip systems, which demobilise nearby dislocations and induce a higher flow stress in order to remobilise them. Atomistic and discrete dislocation dynamics (DDD) approaches have been adopted to help reveal the role played by dislocations in strain hardening [1–6]. Because of the high computational costs when implementing the abovementioned methods, however, proper formulations of the hardening behaviour at the continuum level where dislocation substructures are represented by density distributions are still necessary. Nowadays the most widely used continuum formulae for predicting the shear flow stress τfl are given by relations of Taylor-type [7,8] pffiffiffiffiffi τfl ¼ αμb ρt ;

ð1Þ

where ρt is the total dislocation density, b is the magnitude of the Burgers vector and μ is the shear modulus. Eq. (1) originates from a dislocation forest model, where τfl is considered as the critical resolved shear stress (CRSS) for a single dislocation to overcome a dislocation forest, which is inversely proportional to the forest intra-spacing. Limitations of using Eq. (1) to formulate hardening still exist for the following two reasons. Firstly, there exit other types of self-locked structures of dislocations whose relation with τfl may not take the form given by Eq. (1). Secondly even for forest hardening, the theoretically assumed value of ⁎ Corresponding author. E-mail address: [email protected] (Y. Zhu).

http://dx.doi.org/10.1016/j.scriptamat.2016.01.025 1359-6462/© 2016 Elsevier Ltd. All rights reserved.

the prefactor α (α ≈ 1 – 1.5 cf. review [9]) is found several times larger than the experimentally measured value (α ≈0.35±0.15 e.g. [10]). Therefore, a continuum formula for the shear flow stress that is more consistent with the underlying discrete dislocation dynamics is still expected. Existing continuum models of dislocations struggle to achieve this goal, because dislocation substructures are described in a locally homogenised manner. As a result, the singularly strong short-range dislocation–dislocation interactions are not well resolved [11–15]. For better resolving the discrete information of dislocations locally, we describe the configuration of dislocations on a primary slip plane by a dislocation continuum embedded by average cells identified by dislocation junctions. Thus formulae for the (local) resolved shear flow stress, which is the threshold stress to maintain dislocation motion within corresponding cells, are derived for cases characterised by various types of dislocation locks. It is found that the commonly recognised gap between theoretical predictions and experimental results in the Taylor-type relations can be properly filled by taking into consideration the role of dislocation pile-up against locks. Being validated through comparisons with existing simulation results, the flow stress formulae quantitatively identify the functions of dislocations during strain hardening that have been discussed in literature: an increasing secondary dislocation density hardens a crystal not only because the CRSS for single dislocations to overcome forests increases (as suggested by the Taylor relation), but also because the mean free paths for primary dislocations get squeezed [6]. In the authors' opinion, the derived formulae also reveal the roles played by primary dislocations for the first time: an increasing primary dislocation density softens a crystal since more dislocations are available for pile-up.

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Y. Zhu et al. / Scripta Materialia 116 (2016) 53–56

Fig. 1. Various types of locking structures in f.c.c. crystals due to the dislocation interactions between different slip systems.

This letter focuses on three types of mostly considered lock structures induced by inter-plane (perfect) dislocation interactions in f.c.c. crystals, and they are summarised in Thompson Tetrahedrons shown in Fig. 1. Here we denote primary and secondary slip systems by α-th and βth slip systems, respectively. For simplicity, we use the term “primary dislocations” to denote dislocations on the primary slip system, and so on for “secondary dislocations”. We consider the situation where all primary dislocations are straight and parallel to secondary slip planes, i.e. all primary dislocations form 60° with their Burgers vector. The structures shown in Fig. 1 can be categorised into two basic groups. In the first group where dislocations from different slip systems are mutually parallel (Fig. 1(a)–(b)), they may react at the intersections of slip planes belonging to different slip systems, provided that the energy of the resultant junctions drops. When the product Burgers vector bint = bα +bβ is of b 110N type, a Lomer lock is formed (Fig. 1(a)); when bint is of b 100N type, a Hirth lock is formed (Fig. 1(b)). In the second group where dislocations from different slip systems are mutually orthogonal, the secondary dislocations become forest dislocations (Fig. 1(c)). The role played by secondary dislocations in the lock structures shown by Fig. 1 can be illustrated as in Fig. 2. At the intersection of two slip planes, a primary dislocation is locked with a secondary dislocation and then becomes an obstacle to primary dislocation motion. Here we assume that locks are always formed at slip plane intersections, because otherwise the strong short-range stress will quickly bind two nearby dislocations from different slip systems to generate one. To remobilise a reactant primary dislocation, the stress at the lock (resolved in the primary slip system) should overtake τ αpin, the maximum pinning stress the secondary dislocation in the lock can exert. For forest hardening mechanisms, the reaction takes place between a primary dislocation and a sheet of forest dislocations. To possess a clearer view over how dislocations dynamically overcome dislocation locks, simple one-dimensional DDD simulations

were conducted. In a region of length 10000b, there are N=40 primary dislocations, and one secondary dislocation that can form a Lomer lock with a primary dislocation is fixed at xj0. The i-th primary dislocation evolves as 0 1 j¼N X dxi μbK αα μbK αβ A α @ ¼ mg b þ τext − dt x −x j xi −x j0 j¼1; j≠i i

ð2Þ

with periodic boundary conditions, where mg is a gliding coefficient (of unit Pa−1 s−1); τ αext is the externally applied stress resolved in the primary slip system; Kαα is a coefficient measuring the elastic interaction between two primary dislocations; K αβ measures the elastic interaction between a primary and a secondary dislocation. A summary for K αα and K αβ associated with various types of lock structures are listed in Table 1, which is derived based on the interaction formulae between two straight dislocations [16]. At the discrete level, the condition for maintaining (primary) dislocation motion at xj0 in the presence of a dislocation lock is formulated by   ταext þ ταint x j0 N ταpin ;

ð3Þ

where τ αint measures the shear stress at xj0 due to all other primary dislocations. In the DDD simulations, τ αpin = 3 × 10−2μ and the Poisson's ratio ν = 1/3. From the simulation results, bifurcation in long-time dislocation behaviour depending on τ αext is observed. When τ αext falls below a critical value τ αcrit ≈ 4 × 10−4μ (which is far less than τ αpin), the system evolves to an equilibrium state; otherwise dislocation motion persists and intermittent bursts in the average speed of all dislocations are observed (see Fig. 3(a)). Here τ αcrit is the threshold shear stress for (local) dislocation motion, that is, the shear flow stress τ αfl resolved in primary slip systems. The reason that τ αcrit is much smaller than τ αpin is because dislocations can pile up near an obstacle (see Fig. 3(b)). Due to these piling-up dislocations, on the one hand, the stresses at the primary dislocations surrounding the dislocation lock decrease, and this may cease the plastic flow. On the other hand, the stress at the lock dramatically increases, and this allows the break-up of a junction under a relatively low external stress. However, if we re-write Eq. (3) on a coarse-grained scale, the (shortrange) shear stress due to dislocation pile-up vanishes, because the continuum resolution is too coarse to distinguish a (locally) non-uniform dislocation structure from a uniform one. In order to overcome this drawback, we describe the discrete system by a dislocation continuum

Table 1 K αα in Eq. (2) is the interaction coefficient between two primary dislocations forming 60° with their Burgers vector; K αβ in Eq. (2) is the interaction coefficient between dislocations from different slip planes; Clock is used in Eq. (9). Fig. 2. A primary dislocation is locked with a secondary dislocation at a slip plane intersection. In order to capture the short-range stress due to (piling-up) primary dislocations, the dislocation substructure is envisaged as a dislocation continuum embedded by averaging cells. Each cell is identified by a dislocation lock at its center. The intra-cell dislocation interactions are formulated discretely. Here all primary dislocations form 60° with the Burgers vector.

Obstacle types

K αα

K αβ

Lomer locks

4−ν 8πð1−νÞ

2−ν 8πð1−νÞ

Hirth locks

4−ν 8πð1−νÞ

ν 8πð1−νÞ

Forest dislocations

4−ν 8πð1−νÞ

0

Clock pffiffi

4 2 3ð4−νÞ pffiffi 4 2ð2−νÞ 3ð4−νÞ pffiffi 2 2 3

Y. Zhu et al. / Scripta Materialia 116 (2016) 53–56

55

Fig. 3. One-dimensional DDD simulation results: (a) Intermittent bursts in the average speed of all dislocations are seen if τ αext N τ αcrit. Comparison with the average dislocation speed computed in the absence of obstacles is made. (b) Dislocations are found piling up against locks. When τ αext b τ αcrit, {xi} are taken from the equilibrium state. When τ αext N τ αcrit, {xi} are taken at one time step before the first speed burst takes place. The density of discrete dislocations (at xi) is defined to be b/(xi −xi−1).

embedded by average cells as illustrated by Fig. 2. Within each cell, a dislocation lock is identified at its centre. This way of representing dislocation substructures is meaningful, because the formation of dislocation locks (driven by short-range stresses) takes place much faster than plastic flow (usually driven by mean-field stresses). Consequently if viewed at a coarser-grained scale, primary dislocations are naturally divided into groups separated by locks. As shown in Fig. 2, the average cell size equals the average horizontal lock spacing, and is calculated by dβsl/ sin θ where θ ≈70.5∘ and dβsl measures the average spacing between neighbouring secondary slip planes. The inter-cell dislocation interactions are formulated in a mean field sense as in conventional continuum models of dislocations [17]. In contrast, the intra-cell dislocation interactions are computed discretely with periodic boundary conditions imposed to the cell, so as to make the stress evaluation (asymptotically) compatible with underlying one-dimensional DDD (for detailed derivations, see [18]). In this way, the shear stress between two primary dislocations in the same cell at xi and xj is formulated by K αατper(xi − xj), where. τper ðxÞ ¼

μbπ sin θ β

dsl

 cot

! πx sinθ β

dsl

ð4Þ

with dβsl/ sin θ being the cell length (Fig. 2). Hence condition (3) becomes ταlong þ K αα

n0 X

  τ per x j0 −xi N τ αpin ;

ð5Þ

i¼1;i≠ j0

where n0 is the number of dislocations in the cell of interest; τ αlong measures the long-range stress consisting of the externally applied stress and the mean-field stress due to dislocations inside other cells. Since the cell size is much smaller than the specimen size, τ αlong is assumed uniform within each cell. Now we consider expressing the discrete sum in Eq. (5) by quantities defined at the continuum level. If all dislocations within a cell stop gliding, the net stress at each dislocation should vanish, yielding K αα 

n0 X

    τper xi −x j þ ταlong −K αβ τper xi −x j0 ¼ 0

ð6Þ

j¼1; j≠i

for all i except when i = j0, where K αβτper(⋅) measures the (periodic) short-range stress due to the secondary dislocation in the cell. It is noted that τper(− x) = − τper(x). Then if we sum Eq. (6) over all i

except for i = j0, all terms in the summation of Eq. (6) get cancelled except for those associated with j0, and we obtain n0   X   τper x j0 −xi ¼ ðn0 −1Þταlong : K αα −K αβ 

ð7Þ

i¼1;i≠ j0

With Eq. (7) one can replace the discrete sum in the flow condition (5), which is then re-written by

ταlong

N

  K αα −K αβ ταpin K αα n0 −K αβ

≈

α K αα −K αβ τpin  ¼ ταfl ; αα n0 K

ð8Þ

where the approximation is made since K αβ is in general much smaller compared to K ααn0. The last identity in Eq. (8) holds due to the definition of the shear flow stress ταfl. At a coarse-grained scale, n0 can be calculated as the ratio of average lock spacing to average primary dislocation spacing. As from which Fig. 2, the average lock spacing equals dβsl/ sin θ,p ffiffiffiffiffiffi can be scaled with the secondary dislocation density by 1=ð ρβ sin θÞ. Similarly, the average primary dislocation spacing dαin is scaledpwith ffiffiffiffiffiffi the primapffiffiffiffiffiffi pffiffiffiffiffiffi ry dislocation density by 1= ρα . Hence n0 ¼ ρα =ð ρβ sin θÞ. Incorporating this into Eq. (8) delivers the expression for the (local) shear flow stress ταfl ¼ C lock ταpin 



ρβ ρα

1=2 ;

ð9Þ

where Clock = (K αα − K αβ)/K αα ⋅ sin θ depends on lock type and is summarised in Table 1. Eq. (9) holds provided its right hand side falls in [τ α0 , τ αpin ]. The upper limit is because the shear flow stress should not overtake the critical shear stress needed to unpin a singly trapped dislocation. If the right side of Eq. (9) exceeds τpin, we let τ αfl = τ αpin. Nevertheless, the right side of Eq. (9) is generally weaker than τ αpin, since primary dislocation densities are normally heavier than secondary dislocation densities, i.e. ρα ≥ ρβ (noted that Clock b 1). To determine the lower limit τ α0 , we consider the case where secondary dislocations are sparsely distributed, i.e. ρβ → 0. In this scenario, the right side of Eq. (9) tends to 0, and the frictional stress to dislocation motion is the CRSS for maintaining single-slip, which is denoted by τ α0 . For Lomer or Hirth locks, their associated τ αpin can be found using atomistic simulations.pIn hardening, τ αpin also depends on forest ffiffiffiffiffiforest ffi density by ταpin ¼ μb ρβ [16]. Incorporating this into Eq. (9) and using

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Y. Zhu et al. / Scripta Materialia 116 (2016) 53–56

Fig. 4. (a) For [123] loading to copper specimens, dislocation densities are associated with total plastic strains through fitting the simulation data in [6]. The primary and the secondary slip systems are ½101ð111Þ and ½101ð111Þ, respectively. (b) A shear-stress-total-strain curve is drawn by using Eq. (10) and the fitted density formulae.

the value for Clock in Table 1 delivers the shear flow stress formula in the case of forest hardening ταfl ¼

pffiffiffi 2 2μb ρβ  pffiffiffiffiffiαffi : 3 ρ

continuum frameworks of dislocation dynamics where other dislocation mechanisms such as cross-slip and Frank-Read sources are also included [19–25].

ð10Þ

Eqs. (9) and (10) relate the shear flow stresses to local dislocation density distributions of different slip planes. Compared to the widely used Taylor-type relations, the derived flow stress formulae resolve more details from the underlying discrete dynamics. From the derivation of Eqs. (9) and (10), one can better perceive the roles played by dislocations belonging to different slip systems in strain hardening. A rising ρα softens a specimen since more primary dislocations can pile up against junctions. In contrast, an increasing ρβ strengthens a crystal, because of a decrease in secondary slip plane spacing which squeezes the mean free paths for primary dislocations. In particular for forest hardening, an increasing ρβ also strengthens crystals by bringing up the CRSS to push a single primary dislocation to glide over the forest (as the Taylor relation (1) suggests). To validate the derived formulae, we refer to averaged DDD simulation results of [123] loading to copper specimens [6]. Here dislocation densities are associated with the total strain γ t by fitting with the data in [6]: log10[ρα(m−2)] = 2.1tan− 1(8.3γt − 0.75) + 11.8 and log10[ρβ(m−2)] = 14.4 − 0.135(γt)−3/2, and the fitting results can be seen in Fig. 4(a). Then we use Eq. (10) to plot the shear flow stress τ αfl against the total strain γt as shown in Fig. 4(b). The obtained result agrees roughly well with that from [6] with three hardening stages identified. For generating Fig. 4(b), following parameters are used: b = 0.25 nm; μ = 44 GPa; the CRSS for single slip τ α0 = 3 MPa. Here the reason that theoretically assumed value of the prefactor α in Eq. (1) is several times larger than the observed value can be understood using Eq. (8). Due to dislocation pile-up against a lock, the shear stress at the lock can be n0 times larger. As a result, a shear stress that is smaller than τpin is able to maintain (local) plastic flow. Summarising, we derive in Eqs. (9) and (10) formulae for the shear flow stress with a dislocation piling-up model that incorporates various types of dislocation interactions from different slip systems. Being validated through comparisons with existing simulation results, the derived formulae quantitatively highlight the distinct roles played by dislocations from different slip systems. Compared to the Taylor-type hardening relations, the formulae resolve more details from the underlying dynamics. Hence they can be integrated into the existing

Acknowledgements The financial support from the Hong Kong Research Grant Council through General Research Fund 606313 is gratefully acknowledged. The financial support from the German Research Foundation under the contract of number GU367/36-1 and in the context of the Young Investigator Group (K. Schulz) from the Karlsruhe Institute of Technology (KIT) is also gratefully acknowledged. We would like to thank Professor Peter Gumbsch of KIT for many helpful discussions. References [1] V. Bulatov, F.F. Abraham, L. Kubin, B. Devincre, S. Yip, Nature 391 (6668) (1998) 669–672. [2] D. Rodney, R. Phillips, Phys. Rev. Lett. 82 (8) (1999) 1704–1707. [3] R. Madec, B. Devincre, L. Kubin, T. Hoc, D. Rodney, Science 301 (5641) (2003) 1879–1882. [4] V.V. Bulatov, L.L. Hsiung, M. Tang, A. Arsenlis, M.C. Bartelt, W. Cai, J.N. Florando, M. Hiratani, M. Rhee, G. Hommes, T.G. Pierce, T.D. de la Rubia, Nature 440 (7088) (2006) 1174–1178. [5] L. Kubin, B. Devincre, T. Hoc, Acta Mater. 56 (20) (2008) 6040–6049. [6] B. Devincre, T. Hoc, L. Kubin, Science 320 (5884) (2008) 1745–1748. [7] G.I. Taylor, Proc. Roy. Soc. 145 (855) (1934) 362–387. [8] P. Franciosi, M. Berveiller, A. Zaoui, Acta Metall. 28 (3) (1980) 273–283. [9] D. Kuhlmann-Wilsdorf, Philos. Mag. 79 (4) (1999) 955–1008. [10] R. Madec, B. Devincre, L.P. Kubin, Phys. Rev. Lett. 89 (25) (2002) 255508. [11] I. Groma, F.F. Csikor, M. Zaiser, Acta Mater. 51 (5) (2003) 1271–1281. [12] Y. Xiang, J. Mech. Phys. Solids 57 (4) (2009) 728–743. [13] K. Schulz, D. Dickel, S. Schmitt, S. Sandfeld, D. Weygand, P. Gumbsch, Model. Simul. Mater. Sci. Eng. 22 (2) (2014) 025008. [14] D. Dickel, K. Schulz, S. Schmitt, P. Gumbsch, Phys. Rev. B 90 (9) (2014) 094118. [15] J.L. Dequiedt, C. Denoual, R. Madec, J. Mech. Phys. Solids 83 (2015) 301–318. [16] J.P. Hirth, J. Lothe, Theory of Dislocations, second ed. Wiley, New York, 1982. [17] T. Mura, Philos. Mag. 89 (1963) 843–857. [18] S.J. Chapman, Y. Xiang, Y.C. Zhu, Homogenisation of a Row of Dislocation Dipoles from Discrete Dislocation Dynamics to appear, SIAM J. Appl. Math (2016) arxiv. org/abs/1501.07331. [19] A. El-Azab, Phys. Rev. B 61 (18) (2000) 11956–11966. [20] A. Acharya, J. Mech. Phys. Solids 49 (4) (2001) 761–784. [21] A. Arsenlis, D.M. Parks, J. Mech. Phys. Solids 50 (9) (2002) 1979–2009. [22] Y.C. Zhu, H.Q. Wang, X.H. Zhu, Y. Xiang, Int. J. Plast. 60 (0) (2014) 19–39. [23] T. Hochrainer, S. Sandfeld, M. Zaiser, P. Gumbsch, J. Mech. Phys. Solids 63 (2014) 167–178. [24] Y.C. Zhu, Y. Xiang, J. Mech. Phys. Solids 84 (2015) 230–253. [25] S. Schmitt, P. Gumbsch, K. Schulz, J. Mech. Phys. Solids 84 (2015) 528–544.