Dislocation Climb Influences Grain Size Effect at Elevated Temperature

The Influence of Dislocation Climb on the Mechanical Behaviour of

Polycrystals at Elevated Temperature

Minsheng Huanga,b,c, Zhenhuan Li*a,b, Jie Tongc

a Department of Mechanics, Huazhong University of Science and Technology, Wuhan, Hubei, 430074, China

b Hubei Key Laboratory for Engineering Structural Analysis and Safety Assessment, Luoyu Road 1037, Wuhan, 430074, China

c Mechanical Behaviour of Materials Group, School of Engineering, Anglesea Building, Anglesea Road, University of Portsmouth, Portsmouth, PO1 3DJ, UK

Abstract The tension behaviour of a polycrystalline aluminum with selected grain sizes of submicron/micron was modelled using the climb-assisted discrete dislocation dynamics (DDD) technique. Special focus was given to how dislocation climb influences the flow stress of the polycrystalline aluminum with regard to selected grain sizes at elevated temperature. A periodical representative cell (PRC) consisting of given number of grains was used in the simulations.

Results showed that, at the high temperature considered, dislocation climb plays an important role in defining the mechanical behaviour of the polycrystalline crystal. Specifically, dislocation climb decreases significantly the flow stress and hardening rate while increases the dislocation density by relieving the dislocation pile-ups against the grain boundaries (GBs). In addition, the grain size effect on the yield stress of polycrystalline aluminum is significantly weakened by dislocation climb, especially when the grain size falls within the range of submicron. Another interesting result is that, at high temperature, when both dislocation climb and glide are considered, the grain size effect seems to be insignificant with regard to the applied strain rate, although the strength of material increases with enhanced loading rate.

Keywords: Dislocation climb; Grain size; Discrete dislocation dynamics; Polycrystal; Elevated temperature.

* Corresponding author, [email protected], Fax: 86-27-87543501 1. Introduction

With rapid development of micron/nanometer technology, small sized specimens or structures are extensively used in various NEMSs/MEMSs. The size effect introduced by decreasing specimen size has become a hot topic in both academia and industry. Unlike plastic deformation behaviour of macro-sized specimens, the plastic deformation of micron/submicron sized specimens is usually size-dependent, i.e. the smaller is the stronger. This size effect has been captured by several typical micromechanical tests, such as the thin wire torsion (Fleck et al., 1994; Dunstan et al., 2009; Liu et al., 2013), thin film tension (Espinosa et al., 2004), and micro/nano pillars compression or tension (Uchic et al., 2004; Greer et al., 2005; Kim et al., 2012), where the strength of micro-sized specimens follows the power law s D-n with D being the exterior characterize size (Dunstan and Bushby, 2014). In addition to the specimen size, various interior microstructural sizes, such as the grain size of polycrystals (Hall, 1951; Petch, 1953) , the layer thickness of multilayer (Misra, et al. 2005), the twin thickness of nano-twinned metals(Lu et al. 2009), also strongly influence the strength of materials and follows a similar power law s d -n with d being the interior characterize size. Therefore, the power scale law s l -n seems to be applicable for both specimen size effect and microstructural size effect (Greer and Hosson, 2011). However, there may exist significant differences in the underlying mechanisms in these size effects, which require further investigation. Physically, size effect on the mechanical behaviour of materials stems from exterior and interior constraints (Arzt, 1998). Hall (1951) and Petch (1953) first studied the influence of interior GB constraint on the strength of polycrystals, and

-n used a power law s= s 0 + kd (usually called as H-P relation) to relate the strength s of polycrystalline metals to the grain size d , with s 0 being the friction stress for mobile dislocations or yield stress for materials with sufficiently large grain sizes, n the scaling exponent and k the H-P slope. In this well-known relation, both n and k characterise the intensity of grain size effect, which are closely related to the grain boundary (GB) constraint effect on dislocation motion (Li et al., 2009). By assuming that GB is rigid wall for gliding dislocations and only one single-ended/double-ended pile-up exists in an isolated grain, it can be deduced that the power exponent n equals to 0.5(Hall, 1951; Petch, 1953 and Hirth, 1972). However, a series of experiments and simulations indicated that there are some uncertainties in the Hall- Petch exponent n , i.e. within possible ranges between 0 and 1 (Dunstan and Bushby, 2013), 1/3 and 1 (Baldwin, 1958; Kocks, 1970; Flinn et al., 2001), 0.5 and 1 (Hirth, 1972; Biner and Morris, 2002; Ohno and Okumura, 2007; Li et al. 2009), 0.82 and 1.25 (Evers et al., 2004), 1.19 and 1.50 (Borg, 2007), or 0.5 and 1.5 (Balint et al., 2008). These deviations from the classical Hall–Petch relation with n = 0.5 perhaps originate from striking differences in initial dislocation density (Balint et al., 2008), grain boundary structures (Li et al., 2009), high-order stress/slip gradient (Ohno and Okumura, 2007) and so on. For different GB structures, the dislocation pile-up interact with the GBs by different manners, including dislocation pinning, reflection, absorption, emission and transmission (Shen et al., 1986, 1988). Evidently, the “rigid- GB” assumption made by many researchers may over-estimate the constraint effect of GB on dislocation motion in grains, which also induce both over-estimated grain size effect and strain hardening rate of polycrystals. Recent experimental and computational observations showed that the build-up of long dislocation pile-ups near the compliant GBs is effectively prevented due to absorption or transmission of dislocations at GBs (Li et al. 2009; Danas et al., 2010; Fan et al. 2011; Abuzaid et al. 2012), as a result, strong back-stress on the dislocation sources and follow-up dislocations induced by dislocation pile-ups is releived, rendering the scaling exponent n to deviate from 0.5 as suggested by Hall (1951) and Petch (1953). Besides the dislocation absorption and transmission at the compliant GBs, several other dislocation relaxation mechanisms, including dislocation cross-slip (Brown, 1997; de Sansal et al. 2009) and dislocation climb (Ayas et al., 2014; Danas et al., 2013) can also effectively prevent the build-up of long dislocation pile-ups against GBs (Ayas et al., 2012) and significantly change the extent of the size effect. Different from dislocation glide and cross-slip, climb of edge dislocations requires vacancy diffusion thus significant dislocation climb can only be easily observed at

temperatures above0.3Tm , where Tm is the melting temperature of materials (Ayas et al., 2014). Recent uniaxial compression of Indium micro-pillars at room temperature by Lee et al. (2011) exhibited much weaker size effect than that of Au

(Greer et al., 2005), Ni and Ni3Al (Uchic et al., 2004). A possible explanation is that the occurrence of dislocation climb weakens the size effect as room temperature is

0.3T above m for lower melting-point Indium. In addition, there are also some circumstantial evidences that elevated temperature can weaken the dependence of flow stress on grain size for several polycrystalline metals (Singh et al. 2004; Ono et al. 2003; Atwell et al., 2012; Hagihara et al., 2013). Although these experiments suggested directly or indirectly that the elevated temperature can influence the size effect, there seems few studies on how high temperature favours dislocation climb, which consequently influences the grain size effect (i.e. H-P relation). Besides experiments, DDD simulation, which treats the plastic deformation directly as evolution of a large number of dislocations, has been used in capturing size effect and its underlying mechanisms at micron/submicron scale. To study the H- P effect of polyscrystals and its underlying mechanisms, a series of DDD simulations have been performed, including 2D-DDD (Biner and Morris, 2002; 2003; Balint et al., 2005; 2006a; 2008; Li et al., 2009; Fan et al. 2011; Ahmed and Hartmaier, 2010), 2.5D-DDD (Lefebvre et al., 2005; 2007) and 3D-DDD (Espinosa et al., 2006; Ohashi et al., 2007; de Sansal et al. 2009). Several factors influencing the mechanical behaviour of polycrystals and its grain size effect (H-P relation) have been carefully investigated, including the number of slip systems, the grain arrangement, the dislocation source density and the range of grain sizes (Balint et al., 2005; 2006a; 2008; Biner and Morris, 2002; 2003), the density of grain boundaries (Kumar et al. 2009), dynamic recovery at GBs and GB diffusion (Ahmed and Hartmaier, 2011), dislocation penetrating GB (Li et al., 2009; Kumar et al. 2010; Fan et al. 2011), dislocation curvature (Ohashi et al., 2007) and cross-slip (de Sansal et al. 2009). However, most of these simulations limit the dislocation motion on their fixed slip planes except the modeling of GB-dislocation climb by Ahmed and Hartimaier (2011). More recently, several schemes have been developed to incorporate dislocation climb into the 2D- DDD and 3D-DDD simulations (Keralavarma et al., 2012; Davoudi et al., 2012; Danas and Deshpande, 2013; Ayas et al. 2012; 2014; Ahmed and Hartmaier, 2011; Haghighat 2013; Gao et al. 2011; 2013). By a climb-enabled DDD method, Davioudi et al. (2012) studied the response of an aluminum thin film under uniaxial tension, showing that fewer pile-ups form against the passivation layer and the number of dislocations in formed pile-up also decreases due to dislocation climb. Climb-assisted DDD simulation on smooth and passivated single crystal films by Ayas et al. (2012; 2014) shows that dislocation climb significantly changes the size effect and Bausinger effect of thin film, since climb breaks up the dislocation pile-ups at passivated surfaces. From these simulations it seems that dislocation climb could break up the formation of long dislocation pile-ups against GBs and thus significantly influences the grain size effect on the strength of polycrystalline specimens at a high temperature environment. To our best knowledge, studies of the influence of dislocation climb on the grain size effect or Hall-Petch relation are lacking in published literature. The objective of this work is to carry out a climb-enabled 2D-DDD simulation on the tensile response of a polycrystalline aluminum. The emphasis is on the influence of dislocation climb on the grain size dependent responses and its underlying mechanisms. This paper is organized as follows: Section 2 gives a brief introduction to the climb-enabled 2D-DDD method and the computational model. Section 3 presents the results and discussion. Finally, section 4 ends the paper with some main conclusions.

2. Simulation Methodology and Model

2.1. Climb enabled 2D-DDD method

Climb-enabled discrete dislocation dynamics (DDD) methods, in which both dislocation climb and glide are implemented, have been developed by Davoudi et al. (2012), Keralavarma et al. (2012), Danas and Deshpande (2013) and Ayas et al. (2012; 2014), independently. All of them are based on the 2D (plane strain) small-strain superposition framework by Van der Giessen and Needleman (1995). In this framework, the deformation fields are obtained by superposing the dislocation field

( ) in an infinite domain with a supplementary ( ) field:

u= u + uˆ, e= e + eˆ , s= s + sˆ . (1)

 where the ( ) fields are the sum of contributions from each individual dislocation in an infinite domain as:

n n u  = d u k , s = d s k (2) å k =1 åk =1

k k where nd is the total number of dislocations, and {u ,s } are the displacement and stress fields induced by the kth dislocation, respectively. Obviously, these ( ) fields

inevitably introduce traction T and displacement U on the actual finite boundary considered. To satisfy the applied displacement U and traction T boundary conditions on the Su and St surfaces of finite region considered, respectively, the

supplementary ( ) fields should be added with the corrected boundary conditions as:

ˆ  T= T - T = T -s nt on S t (3) ˆ  U= U - U on Su

n where t is the unit normal of the traction boundary St . Evidently, these

 supplementary fields ( ) can be obtained by linearly elastic finite element (FE) analysis. Different from dislocation glide, dislocation climb is no longer confined to a fixed slip plane. The glide and climb forces (i.e. the Peach–Koehler force), which control the dislocation in-plane and out-of-plane motion, respectively, may be expressed as:

(k ) ( k ) ( k ) fg = m(sˆ + s ) b (4a)

(k ) ( k ) ( k ) fc = - s(sˆ + s ) b (4b)

in which b(k ) is the Burgers vector of the kth dislocation, and {m(k ) , s ( k ) } are the unit vectors normal and tangent to the slip plane, respectively. To consider the influence of Pierels stress, a friction force ffr=t fr b( k ) was introduced and the effective glide

eff( k ) force fc may be written as:

(k ) fr ( k ) fr fc- f if f c > f eff( k ) ( k ) fr ( k ) fr fc= f c + f if f c < - f . (5) (k ) fr 0 if abs ( fc ) < f (k ) The glide velocity vg of the kth dislocation is related to the effective glide force by the drag relation as (Van der Giessen and Needleman, 1995):

(k ) eff ( k ) vg= f g/ B g (6)

where Bg is the glide drag coefficient. The constitutive laws for dislocation nucleation and annihilation are similar to those for glide-only 2D-DDD simulation, as detailed in Van der Giessen and Needleman (1995).

For the dislocation climb induced by vacancy diffusion, the relation between the

(k ) (k ) climb velocity vc and the climb force fc is very complex. Keralavarma et al. (2012) coupled the dislocation climb with the vacancy production and diffusion explicitly to solve the constant-stress creep problem with a large time scale. Danas and Deshpande (2013) incorporated the temperature, the vacancy volume, the equilibrium vacancy concentration, and average dislocation spacing into the climb

(k ) drag coefficient Bc and adopted a simple linear drag-type relation between vc and

(k ) fc similar to Eq.(6). In addition, Davoudi et al. (2012) related the dislocation climb

v(k ) DE f (k ) velocity c to the vacancy self-diffusion energy sd and the climb force c as follows:

骣骣 (k ) * (k ) 2p D0 D Esd 轾 fc DV / b vc = exp琪 -犏 exp 琪 -1 (7) bln( R / b ) kB T臌 kB T where b is the magnitude of the Burgers vector, DV * the vacancy formation volume, kB the Boltzmann constant, T the absolute temperature, D0 the pre-exponential diffusion constant, and the argument R/ b of the logarithm term equal to 2p (Davoudi et al., 2012; Huang et al. 2014). Although the Eq. (7) does not consider directly the coupling of dislocation climb and vacancy diffusion as that by Keralavarma et al. (2012), it can approximately represent the steady-state vacancy diffusion (i.e. the absorption/emission of vacancies by/from the dislocation core), basing on the assumption that the vacancy concentration away from the dislocation core is a constant for equilibrium state in bulk materials. Since Eq. (7) is easy to be incorporated into the DDD framework, it is served as the mobility law of dislocation climb in the present simulations.

Different from the dislocation glide, dislocation climb involves the change of slip plane. If one dislocation of a dislocation dipole climbs out of its glide plane, the traditional dislocation displacement field given in the textbook cannot provide the correct discontinuity in the displacement field (Davoudi et al., 2012). For this, we suggested a simple method to calculate the discontinuous displacement field induced by dislocation climb (Huang et al. 2014). For a typical dislocation climb

th process as shown in Fig.1, the k dislocation located at the point A ( x0, y 0 ) first

climbs to the out-of-plane point B ( x0, y 1 ) and further glides to point C ( x1, y 1 ). To calculate the discontinuous displacement fields correctly, two additional fake dislocations with opposite character are introduced and placed on points A and B.

The fake dislocation A is assumed to have a similar character to the k th dislocation. It should be noted that the fake dislocations are introduced only to correct the discontinuity of displacement field, so they have no stress fields and remain immobile all the time. This is basing on the consideration that, when one dislocation of a dipole (an approximation of infinite prismatic dislocation loop) climbs to a new slip plane, it only leaves two small jogs at two infinite ends for the 2D case considered. Since the stress field of these small jogs can be neglected, the introduced fake dislocations only contribute the displacement fields as (Huang et al. 2014):

b 轾骣y- y 骣 x - x 犏- tan琪c - tan 琪 c (8) 2p 臌桫x - xc 桫 y - y c

where ( xc, y c ) is the coordinate of the fake dislocation. Only the skeleton of the climb-enabled 2D-DDD method is briefly recalled here for the convenience of readers. More details can be found elsewhere (Huang et al. 2014). As mentioned above, both dislocation climb and dislocation penetrating GB can relieve the dislocation pile-ups against GBs and significantly influence the grain size effect of polycrystals. Since the present emphasis is on the influence of dislocation climb, all GBs are assumed to be dislocation-impenetrable (i.e. rigid) in order to eliminate the grain size effect induced by dislocation penetrating GBs. For the same reason, the absorption and reflection of dislocations by GB is also neglected. Theoretically speaking, dislocation climb and glide accompany each other. However, compared with dislocation glide, vacancy diffusion induced dislocation climb is much slower according to Eq.(7) so the time steps for dislocation climb in DDD modeling should be much larger than those for dislocation glide in order to improve the computational efficiency. Following Davoudi et al. (2012), computations for

dislocation climb and glide are uncoupled and the time step for climb Dtc is 100

times larger than that for glide Dtg . Within a given time step Dtc for dislocation

(k ) climb, only when vcD t c b , the kth dislocation can climb successfully and the

int轾v(k )D t / b b climb distance is taken as 臌c c since the minimum space between two potential slip planes equals to b in all calculations.

2.2. Models and boundary conditions

A two-dimensional (2D) representative cell (RC) model, as shown in Fig.2, was built to perform 2D-DDD simulations for the monotonic tension of polycrystalline

x y h h aluminums. The edge lengths of the RC in the and directions are x and y , respectively. All the considered RCs contain 36 randomly oriented grains for comparison. The ratio of the edge length to the grain size d is fixed at

hx d= h y d = 6 . To investigate the grain size effect (i.e. H-P relation), a range of grain size d= {0.3125, 0.625, 1.25, 2.5,5.0}m m was considered. As indicated in

Fig.2, a strain-controlled monotonic load was applied to the RC in y-direction at different strain rates e˙ = {25,50,100} s to study the loading-rate effect induced by dislocation climb. These selected strain rates seem too high to study the effect of dislocation climb, since the diffusion time seems too limited for these loading conditions. However, due to the limitation in the time scale of the DDD method, e˙ = {25,50,100} s are the lowest loading rates feasible for computational purposes.

In fact, our loading rates are still much smaller than those employed by other climb- assisted DDD simulations, such as 1000/ s by Ayas et al. (2012; 2014) and 4000/ s by Davoudi et al. (2012). To make the influence of dislocation climb more remarkable, high temperature must be adopted to achieve a high climb velocity. This is why T= 800K and T= 823K were selected by Davoudi et al. (2012) and Ayas et T 800K 0.3T al. (2014), respectively. For the same reason, = much larger than m is also chosen here. In both x and y directions, the following periodic boundary conditions were applied:

A B C D Ux- U x = U x - U x A C B D (9) Uy- U y = U y - U y = e˙ hdt

A B C D A B C D where {Ux, U x , U x , U x } and {Uy, U y , U y , U y } are the displacements at representative points A, B, C and D in x- and y-directions, respectively.

Since the present emphasis is on the influence of dislocation climb, in order to ensure significant dislocation climbs to occur at the strain rates considered, all simulations were run at an elevated temperature T= 800 K similar to Davoudi et al. (2012). For aluminum considered here, the initial dislocation density falls into the range of (30 ~ 300)mm-2 (Keralavarma and Benzerga, 2007; Segurado and

-2 LLorca, 2010; Davoudi et al., 2012) and (30 ~ 60)mm is the typical value at room temperature (Cleveringa et al., 2000; Keralavarma and Benzerga, 2007). Since a high temperature T= 800 K is considered in the present paper, a high density of dislocation sources randomly placed on slip planes was selected as 120mm-2 . The

source strength t nuc was assumed to follow a Gaussian distribution with a mean

value t nuc =100MPa and a standard deviation of dt =15MPa , which is basically

consistent to the value of t nuc =100MPa and dt = 20MPa by Davoudi et al. (2012) also for aluminum at T= 800 K . For simplicity, no dislocation obstacle has been included in the present simulations. The gliding drag coefficient Bg was taken as

32m Pa s at 800K (Olmsted et al. 2005). The material constants and parameters in

Eq.(7) for climb are given in Table 1 in representative of aluminum (Davoudi et al.,

2012) . It should be pointed out that, although the elastic constants m = 26GPa and n = 0.35 also been employed in the climb-enabled DDD simulations around

T= 800 K by Davoudi et al.(2012), Keralavarma et al. (2012) and Ayas et al. (2014), they are corresponding values for aluminum at room temperature actually. Since the elastic constants only influence the dislocation stress field, this approximation should not change the effect of dislocation climb on H-P relation significantly.

Table 1 Material parameters of polycrystalline aluminum

Shear Poisson's Burgers Vacancy self Pre-exponential modulus m ratio v vector b diffusion energy diffusion

DEsd constant D0

26GPa 0.35 1.28eV 0.1185cm2 / s 0 2.86A

For the FCC aluminum, three active slip systems with an intersection angle of

600 between any two slip systems were considered in each grain, as illustrated in

Fig. 2. This is an approximation of the three slip systems in plane strain FCC crystals with relative in-plane angle {00 , 54.736 0 } (Rice, 1987). As mentioned above, all the grain boundaries (GBs) in Fig. 2 are assumed to be rigid with no dislocation penetration permitted. Moreover, as all grains are treated as elastically homogeneous, the RCs are discretized into finite elements of a regular shape, as shown in Fig.2, to improve the calculation efficiency.

3. Results and Discussion

Since the load was applied by a given strain rate e˙ , the overall strain e in the y direction of the RC can be readily obtained by e= e˙t , with t being the total simulation time. Unless otherwise specified, the loading rate e˙ is fixed at e˙ = 50 s in the following text. In addition, the average tension stress s of the RC in the y direction can be calculated as follows:

1 s=[ sˆ(x , y ) + s ( x , y )] dA (10) A in which sˆ(x , y ) and s(x , y ) are the stresses on the Gaussian point (x , y ) obtained by discrete dislocation and FE computations, respectively, and A is the area of the RC simulated. In this section, the influences of dislocation climb on the overall stress- strain s~ e of the RC, as well as the dislocation distribution, dislocation density, microscopic deformation fields and Hall-Petch relation, were investigated carefully.

3.1 Effect of Dislocation Climb on Stress-Strain Response

To investigate the influence of dislocation climb on stress-strain behaviour of polycrystalline aluminum, the computationally obtained s~ e curves with and without consideration of dislocation climb are plotted in Fig.3 for the selected grain sizes d= {0.3125, 1.25, 5.0}m m . The same dislocation-source density and spatial distribution were assigned for the cases with or without dislocation climb. It can be clearly seen from Fig.3 that, when dislocation climb is taken into account, the plastic flow stress and hardening rate are significantly decreased for all the grain sizes considered. Specially, for the polycrystal with a grain size d= 5.0m m , a perfect flow

(maybe slight softening) is captured when dislocation climb is taken into account, as opposite to a strong hardening behaviour when dislocation climb is not considered.

Moreover, if the stress at plastic strain e p = 0.2% is defined as the nominal yield

stress s 0.2 , corresponding to the value of the intersection point between the

e p = 0.2% offset line (the green dashed line in Fig.3) and the stress-strain curve, it

can be found that s 0.2 decreases by 221MPa (from 485MPa to 264MPa) for d= 0.3125m m and only by 37MPa (from 169MPa to 132MPa) for d= 5.0m m when the dislocation climb is included. This means that, even though the Hall-Petch relation is still valid for the cases with the consideration of dislocation climb, the two important parameters k and n which measure the extent of grain size effect may be significantly changed by dislocation climb, which will be further analyzed in Section 3.3.

To further explain the influence of dislocation climb on the mechanical behaviour of polycrystals, the evolutions of dislocation density r with applied strain e are plotted in Fig.4 (a) for both cases with and without dislocation climb. It can be clearly seen that, at a given strain, the dislocation density r in polycrystals with smaller grains is higher than that in polycrystals with larger grains whether dislocation climb is considered or not. However, for a given grain size d , the dislocation density r with dislocation climb is generally larger than that without it at any applied strain e as shown in Fig.4 (a). This is not surprising, since the plastic strain e p with dislocation climb is larger at any given total strain e as indicated by

e Fig.3. Larger plastic strain p generally needs more dislocations to accommodate,

r which induce a higher dislocation density with dislocation climb enabled. To make

this more clearly, the dislocation density evolution with the plastic strain e p is plotted in Fig.4 (b). It shows that, only small dislocation density is needed to produce

e the same plastic strain p when the dislocation climb is permitted, especially for smaller grain size d= 0.3125m m . In other words, one dislocation can produce larger

e p with dislocation climb considered. These can be explained as follows. When the dislocation climb is taken into account, the edge dislocations can move away from their original slip planes. On one hand, this process can relieve the pile-up by reducing the number of its constituent dislocations against the GB and then decrease its back stress on dislocation sources. As a result, more dislocations can be generated from the sources. On the other hand, the climbing dislocation can continue to glide on a new slip plane but not be hindered by the pile-up, which can spread higher plastic deformation over a wider region. Overall, at a given applied strain e , the dislocation climb can induce more dislocations (larger N ) generated from the

i sources, and each nucleated dislocation can produce high plastic strain e p . According

N i to s=E( e - e p ) , a lower flow stress s can be inevitably induced by dislocation i=1 climb as shown in Fig.3. It seems that, when dislocation climb becomes active at elevated temperature, the hardening by pile-up and back-stress on dislocation sources turns less dominant, while the Taylor hardening by long-range interactions of parallel dislocations might become more and more important. To study the influence of dislocation climb on the effect of loading rate, the stress-strain responses of ploycrystals at three different loading strain rates e˙ = {25,50,100} s are plotted in Fig.5, with and without dislocation climb considered. The dislocation source density and other parameters were kept the same as those used before. To save space, only the result for grain size d= 5.0m m is provided here. It can be clearly seen from Fig.5 that, if dislocation climb is neglected, all curves at the three loading rates are approximately overlapped with each other and thus no evident effect of strain loading rate can be captured. This really has to be this case, because the glide only DDD simulations have no time scale apart from the

t Frank-Read source nucleation time nuc . As long as the strain loading rate is not too large, which ensures the Frank-Read sources have sufficient time to nucleate, the simulated response by glide only DDD should be quasi-statically irrespective of e˙ . However, when the dislocation climb is taken into account, the flow stress s at any given strain significantly decreases with decreasing strain loading rate. Dislocation climb makes the loading rate effect on the stress-strain response appear. This can be expected since the diffusion time scale is introduced in the dislocation climb-assisted DDD. This is also consistent with the experimentally observed remarkable effect of deformation rate on the compressive stress of indium micropillars, where dislocation climb was shown to be important even at room temperature (Lee et al., 2011). In order to investigate further the influence of dislocation climb on the effect of loading rate, the variation of dislocation density with strain is plotted in Fig.6 for the three loading rates. It can be seen that, when dislocation climb is ignored, the dislocation densities at different loading rates are basically the same. This is evident since there should be no rate effect in the glide only DDD simulations as mentioned above. However, when dislocation climb is considered, the dislocation density increases significantly with decreasing loading rate. At lower loading rates, more dislocations may climb when more time is available, pile-ups and their shielding effect on dislocation sources may be relieved and thus more dislocations can be multiplied from the dislocation sources. As a result, for lower loading rates, larger plastic strain and smaller flow stress can be induced by dislocation climb, as shown in Fig.5.

3.2 Effect of Dislocation Climb on Microscopic Deformation Fields

The influence of dislocation climb on the stress-strain behaviour of polycrystalline materials and its size effect are closely related to the microscopic deformation field, the dislocation pattern and their evolutions at elevated temperature. To study the effect of dislocation climb on the microscopic deformation field of polycrystals, a parameter named as equivalent plastic slip  is defined following Balint et al. (2006b) as:

u+ u  G =g1 + g 2 + g 3 with g i = m i s,t t,s n i ( i =1 ~ 3) (11) s2 t

i i i where g is the slip quantity for the ith slip system, ms and nt are its tangential and normal vectors, respectively, and u the displacement field induced by all dislocations. This plastic slip G in polycrystalline RC is plotted in Fig.7 for both with and without dislocation climb considered, where Fig.7 (a) and (b) for grain size d= 5.0m m at an applied strain e = 0.004 while Fig.7 (c) and (d) for d= 0.3125m m at e = 0.005 . It can be clearly seen that the slip quantity G tends to localize in some narrow regions and form heterogeneous slip bands, especially for the larger grain size d= 5.0m m . Comparing Fig.7 (a) with Fig.7 (b) for d= 5.0m m , it seems that, with dislocation climb considered, not only the number of slip bands but also the intensity of the slip bands are much larger. Similar results can also be found by comparing

Fig.7(c) with Fig.7 (d) for d= 0.3125m m . This means that dislocation climb induces more slip bands, thus can accommodate more plastic deformation, which contributes to the decrease of the flow strength of polycrystals as shown in Fig.(3) and Fig.(5). In addition, some elastic regions with G 0 (the blue regions) which are surrounded by the slip bands can be found in both d= 5.0m m and d= 0.3125m m , which may be considered as “hard elastic cores” embedded in “soft plastic materials”. It is worth noting that the size of “hard cores” in Fig.7 (a) (or Fig.7(c)) is significantly smaller than that in Fig.7 (b) (or Fig.7 (d)). This means that dislocation climb can effectively decrease the size of “hard cores”, which also contributes to the decrease of the flow strength of the material, as shown in Fig.(3) and Fig.(5). To examine the influence of dislocation climb on the dislocation pattern and its evolution, the dislocation slip traces defined as the trajectories of dislocation motion in the polyscystal with d= 0.3125m m at e = 0.015 are plotted in Fig.8 (a) and Fig.8

(b) with and without the consideration of dislocation climb, respectively. The insets are dislocation slip traces within the specified window ( x: 0 ~ 0.3m m and

y : 0 ~ 0.3m m ). It is evident that, when dislocation climb is considered, both the number of slip bands and the width of them are significantly enhanced. On the contrary, when dislocation climb is neglected, the dislocations are only active in the limited number of slip bands and even concentrated on only one slip plane. The most likely reason is that the pile-ups against GBs, which cannot be effectively relaxed by dislocation climbs, impose strong shielding effect on dislocation sources. As a result, only few dislocation sources on isolated slip planes may be activated and nucleate dislocations. This can be more clearly demonstrated by comparing the insets of Fig.8 (a) and (b). By means of climbs, the dislocations can spread in a wide range on both sides of those main slip planes, as shown in the inset of Fig.8 (a). With the increase of dislocation climbs, the slip bands become wider and wider, which provide more room for the dislocations to glide freely and thus accommodate more plastic strains. However, as indicated by the inset of Fig.8(b), if no dislocation climb is considered, most dislocation sources are effectively shielded by dislocation pile-ups since mechanisms such as dislocation climb are no longer available to relax these pile-ups to promote new dislocations. Consequentially, although the dislocation density is lower without dislocation climb considered as given in Fig. 4(a), the flow stress is higher as shown in Fig.3 and Fig.5, since less plastic deformation may be produced by the glide of limited dislocations.

The distribution of the normal stress s yy along the tensile axis at an applied loading strain e = 0.005 is presented in Fig.9 (a) and (b) for d= 0.3125m m with and without the consideration of dislocation climbs, respectively. It is evident that for the two cases considered, the microscopic stress s yy field in the RC is not homogenous although a uniform tensile loading is applied. Significant stress concentration can be observed near the grain boundaries parallel to the tensile axis for the GB arrangement considered. Of course, this heterogonous stress distribution is induced by the heterogonous dislocation distribution, as indicated in Fig.8. Comparing Fig.9 (a) with Fig.9 (b), it can be found that although dislocation climb introduces higher dislocation density near the GBs (Fig.8 (a)), it significantly reduces stress concentration near those vertical GBs. This can be explained as follows. More dislocations induced by climb can produce larger plastic strain, which can reduce the flow stress and relax the stress concentration. In addition, the intensity of the stress concentration mainly depends on the number of dislocations in pile-ups against the GBs. Since climbing can make the dislocations move away from the pile-ups, the number of dislocations in pile-ups may decrease with dislocation climbing, which can also relax the stress concentration to a certain extent. In summary, by decreasing the pile-up induced back stress on dislocation sources and increasing the number and width of slip bands, dislocation climb can increase plastic slip deformation and relieve the stress concentration at GBs, which contribute to the decrease of the flow strength.

3.3 Effect of Dislocation Climb on Hall-Petch Relation To investigate the influence of dislocation climb on the grain size effect (i.e. Hall-

Petch relation), the variation of the nominal yield stress s 0.2 with grain size d are presented in Fig. 10 for both with and without dislocation climb. For each grain size, the simulations were repeated at least three times to capture the dispersion of the

yield stress s 0.2 induced by random distributions of dislocation sources. The yield stress of polycrystals is known to increase with decreasing grain size in the form of

-n the classical Hall-Petch relation as s= s 0 + kd , where k is the Hall-Petch slope, n

is the grain-size sensitivity exponent and s 0 the yield stress of the polycrystal with sufficiently large grain size. It can be clearly seen from Fig.10 that the DDD simulated results both with and without dislocation climb compare well with the Hall-Petch

-n relation s= s 0 + kd , i.e. the yield stress increases with decreasing grain size.

However, the parameters {s 0 ,k , n} for these two cases are quite different. As shown in Fig.10, the parameters in Table 1 (where R2 is the correlation coefficient) can fit well the DDD simulated s ~ d curves with and without dislocation climb considered.

Table 1. The fitting Hall-Petch parameters

Parameters k (MPa) n 2 s R 0 (MPa)

Without dislocation climb 145 90 1 0.97

considered With dislocation climb considered 90 95 0.45 0.98

The parameter s 0 equals to 90MPa with dislocation climb, which is much lower than 145MPa without it. This means that, even for the polycrystals with sufficiently large grains, their yield stress s 0 decreases significantly when dislocation climb is taken into account. In the Hall-Petch relation, although both the slope k and exponent n depict the effect of grain size on yield stress, the power exponent n is a more dominant parameter. On one hand, it can be found from Fig.10 and Table 1 that, if dislocation climb is ignored, the value of n is approximate to 1, which is almost reach the upper limit of the range 0.5 ~ 1 observed by a series of experiments (Ohno and Okumura, 2007; Dunstan and Bushby, 2013). Although such high value seems very surprising, it can be explained as follows. For polycrystals with grain sizes from 0.1~5m m , Ohno and Okumura (2007) found that the higher-order stress work-conjugate to slip gradient significantly influences the initial yielding of polycrystals. Based on the self-energy of geometrically necessary dislocations

(Gurtin, 2002), they derived a d-1 -dependent (n= 1 ) term for the averaged resolved shear stress. Further, this high value of n= 1 has been validated by their experiments in both Cu and Al with the grain size in the range of 0.1~5m m (Ohno and Okumura,

2007). In addition, Balint et al. (2008) also investigated the grain size dependence of the flow strength by 2D DDD. Their results show that the exponent n is dependent on both the range of grain size and the initial dislocation source density. For a general value of source density, it was found that n= 1 and n= 0.5 for grain sizes d 2.5m m and d 2.5m m , respectively. Further, recent researches (Szajewski et al., 2013;

Akarapu and Hirth, 2013) have shown that the stress gradient (induced by dislocations in surrounding grains) in grains can also enhance the hardening effect and then increase the coefficient n (to a value even larger than 1). Since the grain size considered in present 2D DDD simulations is in the range of d 5m m , the effects of higher-order stress (slip gradient), stress gradient and initial source density become important which induce a high value of n =1. It should be mentioned that, Ohashi et al. (2007) also suggested n= 1 for polycrystals with fine grains basing on the 3D curved Frank-Read source model, but it cannot be used to explain the present n =1 achieved by present 2D DDD simulations. On the other hand, when dislocation climb is taken into account, the value of n decreases significantly to 0.45, which is even slightly lower than 0.5 . Although the slope k increases from 90 to 95, on the whole, the grain size effect on yield stress s at elevated temperature (800K ) is weakened by dislocation climb. As shown in Fig.10, when the grain size d is less than 1.25mm , this weakening on grain size effect by dislocation climb is very significant. However, when the grain size is above 1.25mm , two curves of s ~ d with or without dislocation climb are almost parallel to each other. In other words, the weakening on grain size effect by dislocation climb becomes insubstantial for polycrystals with a large grain size. To sum up, at high temperature where dislocation climb becomes significant, the grain size effect on yield stress can be weakened significantly by dislocation climb when grain size is at sub-micron scales, as opposed to no substantial weakening on grain size effect exists when grain size is several microns or above. The fake dislocations are especially introduced into the present DDD simulations to keep appropriate displacement discontinuity as mentioned above. Since each two fake dislocations correspond to a dislocation climb event, the density of fake dislocations as byproduct can describe the level of dislocation climb to a certain extent. Considering this, the density of fake dislocations r fake is plotted in Fig.11 as a function of the applied strain e for the selected grain sizes d= {0.3125,1.25,2.5,5.0}m m . It can be seen from this figure that r fake decreases with increasing grain size when d is less than 1.25m m . This means that the extent of dislocation climb is remarkably higher in those grains at submicron scale, which leads to significantly reduced grain size effect at submicron scale as shown in Fig.10. The deviation occurs when the grain size is larger than 2.5mm , for example, r fake for d= 5.0m m is only slightly lower at small applied strain but becomes higher at large applied strain than that for d= 2.5m m . It means that, when d>1.25m m , the extent of dislocation climb has no noticeable decrease with increasing grain size. This might be the reason why the grain size effect is not significantly influenced by the dislocation climb in the grain size range of d>1.25m m .

Table 2. The fitting Hall-Petch parameters

Strain rate /s Parameters n R2 s 0 (MPa) k ( MPa)

25 70 95 -0.45 0.985

50 90 95 -0.45 0.98

100 105 95 -0.45 0.96

As dislocation climb is time-dependent, the applied strain rate may influence the grain size effect. The computationally obtained s 0.2 ~ d curves are plotted in Fig.12 for three strain loading rates e˙ = {25,50,100} s . All these three curves were fitted by

{s ,k , n } the Hall-Petch relation with the parameters 0 and the correlation coefficient

R2 provided in Table 2. It can be seen from Fig.12 that, although the yield stress

decreases significantly with decreasing strain rate, the three curves of s 0.2 ~ d at these different strain rates are almost parallel to each other. This is also indicated by the fitting parameters in Table 2. For all the strain rates considered, both the slope parameter k and the exponent n are the same, with approximately equal

2 correlation coefficient R . Only the parameter s 0 decreases with decreasing strain rate. Obviously, for the present dislocation climb parameters, applied strain rate may change the magnitude of yield stress but does not seem to significantly influence the grain size effect ( k and n ). Although the 2D DDD simulations performed in the present paper cannot capture the hardening effect from jogs, junctions, and the interaction between perpendicular dislocations (forest hardening), it can model the climb of edge dislocations effectively. Since the dislocation climb seems not coupling with other three- dimensional mechanisms (such as jogs and junctions) significantly, the present 2D DDD simulations can mainly capture the influence of dislocation climb on H-P relation. Of course, the dislocation lines generally climb piece by piece, leaving a series of dislocation segments on different slip planes connecting by jogs. This climb way is much easier to happen than the simultaneous climb of the whole dislocation considered in present 2D simulations. In addition, recent investigations (Dunstan and Bushby, 2013; 2014) also show that the effect of 3D dislocation curvature according to the space available in grains seems more important for the Hall-Petch effect than the pile-ups. Obviously, a more accurate investigation on the H-P relation at elevated temperature need further 3D DDD with 3D mechanisms (such as the effect of dislocation curvature) and dislocation climb included.

4. Conclusions

The tensile mechanical behaviour of polycrystalline aluminums at elevated temperature and associated grain size effect (i.e. the Hall-Petch relation) have been studied by using climb-assisted DDD simulations. A special focus was given to how dislocation climb influences some of the material responses at elevated temperature. DDD simulations with and without the consideration of dislocation climb have been performed, from which some interesting results are obtained:  With dislocation climb considered, significant strain rate dependency of the stress-strain curves at elevated temperature is captured. In addition, the plastic flow stress and hardening rate are significantly decreased, while the dislocation density increases remarkably due to the shielding effects or back stress introduced by dislocation pile-ups on dislocation sources being relieved by dislocation climb.  The distributions of the plastic slip quantity and the tensile stress component in the polycrystals are significantly influenced by dislocation climb, which increases the width and the intensity of the localized dislocation slip bands, while releases the intensity of stress concentration near the GBs by decreasing the number of dislocations in the pile-ups, leading to the decrease of the flow strength and hardening rate of polycrystals at elevated temperature.  Dislocation climb significantly weakens the grain size effect of polycrystals with submicron sized grains while does not appear to influence substantially that of polycrystals with micron sized grains. In addition, with dislocation climb considered, the increase of applied strain rate enhances the strength of polycrystals although not so much the grain size effect.

Acknowledgments: The authors would like to express their thanks for the financial supports from NSFC (11272128 and 11072081).

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Fig. 1. A schematic of the numerical treatment of dislocation climbing from A to B and gliding to C. The left pink dislocation is the dipole counterpart of the dislocation C. Two dashed black fake dislocations were added at points A and B, which can ensure correct displacement discontinuity on the slip planes after climb.

Fig. 2. The periodical polycrystalline RC model with 36 grains.

Fig. 3. The stress-strain curves of polycrystals with selected grain sizes d= {0.3125, 1.25, 5.0}m m at e˙ = 50 / s , with and without the dislocation climb being considered for comparison.

Fig. 4. The evolution of dislocation density for selected grain sizes d= {0.3125, 1.25, 5.0}m m at e˙ = 50 / s as a function of (a) the total applied strain and (b) the plastic strain, with and without the dislocation climb being considered for comparison. Fig. 5. The stress-strain curve of the polycrystal with grain size d= 5.0m m at three loading strain rates e˙ = {25,50,100} s , with and without dislocation climb considered.

Fig. 6. The evolution of dislocation density with applied strain at selected loading rates, with and without dislocation climb.

Fig. 7. The contours of plastic slip G in the polycrystal with a grain size d= 5.0m m at e = 0.004 , with (a) and without (b) dislocation climb being considered, and in the polycrystal with grain size d= 0.3125m m at e = 0.005 , with (c) and without (d) dislocation climb being considered.

Fig. 8. The dislocation slip traces in the polycrystal with grain size d= 0.3125m m at the applied strain e = 0.015 , with (a) and without (b) dislocation climb considered.

Fig. 9. The distribution of tensile stress component s yy in polycrystal with grain size d= 0.3125m m at the applied loading strain e = 0.005 , with (a) and without (b) dislocation climb.

Fig. 10. The variation of yield stress s 0.2 with the grain size d for cases with and without the consideration of dislocation climb. The applied strain rate is fixed at 50 / s .

Fig. 11. The evolution of fake dislocation density with the applied strain in the polycrystals with selected grain sizes at a fixed strain rate 50 / s .

Fig. 12. The s 0.2 ~ d curves at different strain loading rates e˙ ={25,50,100}/ s , with dislocation climb considered. ux / b

C( x1 , y 1 ) B( x0 , y 1 )

Fake dislocation 2 y Dipole dislocation of C

x Fake dislocation 1 A( x0 , y 0 )

Fig. 1. A schematic of the numerical treatment of a dislocation climbing from A to B and gliding to C. The left pink dislocation is the dipole counterpart of the dislocation C. The two dashed black fake dislocations were added at point A and point B to ensure the correct displacement discontinuity on the slip planes after the climb. ˙ u p ˙ u p ˙ d o w n ˙ U y w i t h U y -U y = e h ˙ d o w n U y

Grain 0 size 60

Slip systems 600 hy Grain boundary

FE mesh

C D Fig. 2. The periodical polycrystalline RC model with 36 grains. Fig. 3. The stress-strain curves of polycrystals with selected grain sizes d= {0.3125, 1.25, 5.0}m m at e˙ = 50 / s , with and without the dislocation climb being

considered for comparison. (a) (b)

Fig. 4. The evolution of dislocation density for selected grain sizes d= {0.3125, 1.25, 5.0}m m

at e˙ = 50 / s as a function of (a) the total applied strain and (b) the plastic strain, with and without the dislocation climb being considered for comparison. Fig. 5. The stress-strain curve of the polycrystal with grain size d= 5.0m m at three loading

strain rates e˙ = {25,50,100} s , with and without dislocation climb considered. Fig. 6. The evolution of dislocation density with applied strain at selected loading rates, with and without dislocation climb. (a) (b)

(c) (d)

Fig. 7. The contours of plastic slip G in the polycrystal with a grain size d= 5.0m m at

e = 0.004 , with (a) and without (b) dislocation climb being considered, and in the polycrystal with grain size d= 0.3125m m at e = 0.005 , with (c) and without (d) dislocation climb being

considered. (a) (b)

Fig. 8. The dislocation slip traces in the polycrystal with grain size d= 0.3125m m at the applied

straine = 0.015 , with (a) and without (b) dislocation climb considered. Pa

(a) Pa

Fig. 9. The distribution of tensile stress component s yy in polycrystal with grain size d= 0.3125m m at the applied loading strain e = 0.005 , with (a) and without (b) dislocation

climb. Fig. 10. The variation of yield stress s 0.2 with the grain size d for cases with and without the

consideration of dislocation climb. The applied strain rate is fixed at 50 / s . Fig. 11. The evolution of fake dislocation density with the applied strain in the polycrystals with selected grain sizes at a fixed strain rate 50 / s . Fig. 12. The s 0.2 ~ d curves at different strain loading rates e˙ ={25,50,100}/ s , with

dislocation climb considered.