Defect redistribution within a continuum grain boundary plasticity model

Author’s Accepted Manuscript Defect redistribution within a continuum grain boundary plasticity model P.R.M. van Beers, V.G. Kouznetsova, M.G.D. Geers www.elsevier.com/locate/jmps

PII: DOI: Reference:

S0022-5096(15)00151-9 http://dx.doi.org/10.1016/j.jmps.2015.06.002 MPS2664

To appear in: Journal of the Mechanics and Physics of Solids Received date: 4 November 2014 Revised date: 15 May 2015 Accepted date: 9 June 2015 Cite this article as: P.R.M. van Beers, V.G. Kouznetsova and M.G.D. Geers, Defect redistribution within a continuum grain boundary plasticity model, Journal of the Mechanics and Physics of Solids, http://dx.doi.org/10.1016/j.jmps.2015.06.002 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Defect redistribution within a continuum grain boundary plasticity model P.R.M. van Beers, V.G. Kouznetsova∗, M.G.D. Geers Eindhoven University of Technology, Department of Mechanical Engineering, P.O. Box 513, 5600 MB Eindhoven, The Netherlands

Abstract The mechanical response of polycrystalline metals is significantly affected by the behaviour of grain boundaries, in particular when these interfaces constitute a relatively large fraction of the material volume. One of the current challenges in the modelling of grain boundaries at a continuum (polycrystalline) scale is the incorporation of the many different interaction mechanisms between dislocations and grain boundaries, as identified from fine-scale experiments and simulations. In this paper, the objective is to develop a model that accounts for the redistribution of the defects along the grain boundary in the context of gradient crystal plasticity. The proposed model incorporates the nonlocal relaxation of the grain boundary net defect density. A numerical study on a bicrystal specimen in simple shear is carried out, showing that the spreading of the defect content has a clear influence on the macroscopic response, as well as on the microscopic fields. This work provides a basis that enables a more thorough analysis of the plasticity of polycrystalline metals at the continuum level, where the plasticity at grain boundaries matters. Key words: Defect redistribution, Grain boundary, Crystal plasticity, Interface model

1. Introduction Based on the knowledge acquired in the last few decades on the relation between material properties and the underlying microstructure, it has become possible to process metals such that specific functionality can be obtained. The understanding of this interaction between structure ∗

Corresponding author. Tel.: +31402475885 Email address: [email protected] (V.G. Kouznetsova)

Preprint submitted to Journal of the Mechanics and Physics of Solids

June 10, 2015

and property expands even further as computational and experimental resources increase, enabling studies at different time and length scales. An example is the mechanical behaviour of metallic materials. In particular, topics of interest relevant to this work include (i) the miniaturisation in industrial applications involving metals with dimensions approaching microstructural length scales, such as the grain size, and (ii) the ultra-high strength of fine-grained metals. In both cases the behaviour of grain boundaries (GBs) is essential, since they constitute a relatively large fraction of the material volume. Accordingly, one of the current challenges in modelling plasticity of polycrystalline metals is to account more accurately for the role of grain boundaries (McDowell, 2008, 2010). Different interaction mechanisms between dislocations and grain boundaries have been identified from experiments and simulations, see Forwood and Clarebrough (1991); Priester (2013); Sutton and Balluffi (1995); Valiev et al. (1986) and references therein. In general, the details of the interaction mechanisms that take place depend on the initial structure of the grain boundary, the magnitude of the applied forces, and the degree of thermal activation. Grain boundaries can act as obstacles to the motion of lattice dislocations that accumulate into a dislocation pile-up, or as sources of dislocations generating dislocations in the adjacent grains and at the interface. A lattice dislocation may also be transmitted through the interface from one grain to the other in a direct or indirect manner, in the latter case leaving behind a residual defect in the grain boundary. Another mechanism which has a strong effect on the mechanical properties of materials, is the redistribution of the defect content along the grain boundary, as observed in experiments (Clark and Smith, 1979; Couzini´e et al., 2003; Darby et al., 1978; Dingley and Pond, 1979; Ishida et al., 1968; Johannesson and Th¨ olen, 1972; Lee et al., 1990; Liu and Balluffi, 1985; Medlin et al., 1997; Pond and Smith, 1977; Poulat et al., 1999; Pumphrey and Gleiter, 1974; Varin et al., 1978; Wang and Sui, 2009) and fine-scale simulations (Cheng et al., 2008; de Koning et al., 2003; Dewald and Curtin, 2007; Ishida and Mori, 1985; Jin et al., 2008; Kurtz et al., 1999; Pestman et al., 1991; Shimokawa et al., 2007; Yu and Wang, 2012). If the grain boundary acts as a sink and a lattice dislocation is absorbed into the grain boundary, it can (i) remain localised, or (ii) dissociate and spread along the boundary plane. The spreading observed by transmission electron microscopy 2

(TEM) consists in a widening and subsequent disappearance of the dislocation diffraction contrast. However, high-resolution TEM has revealed that non-negligible decomposition products may exist, which are not resolved by conventional TEM. Three types of models have been proposed to describe these accommodation phenomena (Lojkowski, 1991; Nazarov, 2000; Priester, 1997): (i) delocalisation models, which assume continuous spreading of the dislocation cores; (ii) dissociation models, according to which the dislocation decomposes into two or more discrete dislocations that have smaller Burgers vectors; (iii) incorporation models, which consider the accommodation of the dislocation into the intrinsic grain boundary defect structure. The three models yield similar kinetics of spreading. The individual interactions discussed above cannot be resolved in continuum (crystal) plasticity frameworks. This demands the development of a new model that describes the defect redistribution along the grain boundary in an average sense. In recent years, from the perspective of continuum scale modelling, first steps have been made to incorporate grain boundary interface mechanics into strain gradient plasticity (Dahlberg et al., 2013; Fredriksson and Gudmundson, 2007; Massart and Pardoen, 2010; Polizzotto, 2009; Voyiadjis and Deliktas, 2010; Voyiadjis et al., 2014) and strain gradient crystal plasticity (Borg, 2007; Ekh et al., 2011; Gurtin, 2008; Liu et al., 2011; ¨ Ozdemir and Yal¸cinkaya, 2014; Wulfinghoff et al., 2013) frameworks. The latter has the advantage of naturally incorporating the crystallographic misorientation across a grain boundary, which is of particular importance when considering the interactions between dislocations and grain boundaries. However, in these frameworks, constitutive equations at the grain boundary, e.g. the continuum scale expression of the grain boundary energy, still lack a solid physical background. At the same time, atomistic simulations have indicated that the grain boundary energy is dependent on the grain boundary structure (Hasson et al., 1972; Rittner and Seidman, 1996; Tschopp and McDowell, 2007; Wang et al., 1984; Wolf, 1990). Recently, based on the output of atomistic simulations, van Beers et al. (2015a) developed a multiscale approach leading to a continuum representation of the initial grain boundary structure and energy. In this approach, the grain boundary structure is represented by an intrinsic net defect density, which is related to the atomistically calculated grain boundary energies through a loga3

rithmic expression. It was shown that the grain boundary energies in the complete misorientation range of symmetric tilt grain boundaries in aluminium and copper can be accurately described by this multiscale method. The work of van Beers et al. (2015b) incorporated this atomisticto-continuum approach into the grain boundary crystal plasticity model developed earlier. The resulting framework was numerically implemented in order to examine the particular role and influence of the grain boundary energetic contributions on the macroscopic response. It was thereby observed that the effect of the energetic contribution on the macroscopic response depends strongly on the misorientation between adjacent grains, whereby it was assumed that the grain boundary net defect remains localised. The novel contribution in this paper is the development of a model that incorporates the defect redistribution along the grain boundary into the continuum grain boundary plasticity framework of van Beers et al. (2013, 2015b). A diffusion type evolution law is proposed for the redistribution of the net defect within the grain boundary, which allows for a nonlocal relaxation of the grain boundary net defect. The continuum model consists of a set of equations, including a balance and constitutive equations, for a vectorial grain boundary net defect field along the grain boundary surface. The incorporation of a strongly nonlocal interaction term leads to an integro-differential equation for the net defect evolution, which is further reformulated into a differential form. A numerical implementation of the resulting model framework investigates the effect of the defect redistribution on the macroscopic response in the case of a copper bicrystal loaded in simple shear. The paper is organised as follows. Section 2 summarises the grain boundary crystal plasticity framework and the incorporation of the internal grain boundary structure and energy. In Section 3, the continuum model of the defect redistribution is developed and discussed. Section 4 covers the solution procedure and the finite element implementation of the framework. The model problem and the numerical study of the influence of the defect redistribution on the macroscopic response are described in Section 5. Finally, conclusions are provided in Section 6. Cartesian tensors and tensor products are used throughout the paper: a, A and n A denote a vector, a second-order tensor and an nth-order tensor, respectively, unless indicated otherwise. The following notation for vector and tensor operations is adopted: transpose AT ij = Aji , the gradient 4

operator ∇ =

∂ ∂xi ei ,

the dyadic product ab = ai bj ei ej , the cross product a × b = εijk aj bk ei where

εijk is the Levi-Civita symbol, the derivative

∂a ∂b

=

∂ai ∂bj ei ej ,

the dot product A · B = Aij Bjk ei ek and

the double dot product A : B = Aij Bji , with ei (i = 1, 2, 3) the unit vectors of a Cartesian basis.

2. Bulk and interface frameworks This section summarises the dislocation based strain gradient crystal plasticity framework extended with a grain boundary interface model as developed in van Beers et al. (2013), followed by a brief description of the incorporation of the grain boundary structure and energy as established in van Beers et al. (2015a,b). 2.1. Bulk crystal formulation This work departs from the bulk crystal formulation originally proposed by Evers et al. (2004a,b) and Bayley et al. (2006). Attention is restricted to a small deformation framework. The total distortion is decomposed into its elastic (e) and plastic (p) contributions

∇u = He + Hp ,

(1)

where He represents the stretching and rotation of the lattice, while the plastic distortion Hp describes the local deformation due to slip given by

Hp =

X

γ α sα mα .

(2)

α

Here, γ α is the slip on slip system α having slip plane normal mα and slip direction sα . The symmetric part of the elastic distortion He gives the elastic strain Ee

Ee =

1 2


He + HeT .

(3)

For a given slip system α the resolved shear stress or Schmid stress τ α is given by

τ α = σ : sα mα

with

σ = 4 C : Ee .

(4) 5

The latter equation is the conventional linear elastic relationship, which relates the stress tensor σ to the elastic strain Ee via the fourth order elasticity tensor 4 C. In this work isotropic elasticity is considered. In the crystal plasticity approach adopted, dislocation densities on individual slip systems act as key variables. Gradients in slip induce geometrically necessary dislocation (GND) densities of α(e)

edge (e) and screw (s) character, denoted ρg α(e)

ρg

1 = − ∇γ α · sα b

α(s)

and ρg

α(s)

and ρg

, respectively

1 = ∇γ α · lα , b

(5)

with unit vector lα = sα × mα and b the Burgers vector magnitude. The evolution of slip is accomplished through a visco-plastic power-law given by 

α

γ˙ = γ˙ 0

|τ α − τbα | sα

1

m

sign (τ α − τbα ) .

(6)

Parameters γ˙ 0 and m are positive constants. The slip resistance sα for slip system α reflects the obstruction of dislocation glide through short-range interactions of all dislocations on coplanar and intersecting slip systems. Both the densities of geometrically necessary dislocations (GNDs), ρξg , and statistically stored dislocations (SSDs), ρξs , contribute to the slip resistance (Ashby, 1970) α

s = Gb

rX ξ

X Aαξ ρξs + Aαξ ρξg ,

(7)

ξ

where Aαξ represents the full dislocation interaction matrix as given by Franciosi and Zaoui (1982). The components Aαξ are the coefficients a0 , a1 , a2 and a3 , which represent the strength of six types of interactions: self hardening (a0 ), coplanar (a1 ), cross slip (a1 ), Hirth lock (a1 ), glissile junction (a2 ) and Lomer-Cottrell lock (a3 ). The generalised form of Essmann and Mughrabi (1979) is employed to describe the evolution of the SSD densities ρξs ρ˙ ξs

1 = b



 1 ξ ξ − 2yc ρs γ˙ with ρξs (t = 0) = ρξs0 . ξ L

6

(8)

The dislocation accumulation is governed by the average dislocation segment length K ζ P , ξζ ρ + ξζ ρζ H H s g ζ ζ

Lξ = qP

(9)

where H ξζ are the immobilisation coefficients, structured analogously to the interaction coefficients Aαξ . The dislocation annihilation is controlled by a critical annihilation length yc . In Eq. (6), the back stress τbα is obtained by the projection of the internal stress σ int onto slip system α

τbα = −σ int : sα mα .

(10)

The internal stress can be interpreted as a fine-scale correction accounting for microstructural dislocation interactions that are not resolved in the long-range elastic strain field, e.g. due to dislocation pile-ups. Spatial gradients in the densities of geometrically necessary dislocations induce long- and short-range non-vanishing stress fields. The stress fields of dislocations in an infinite elastically isotropic medium are used in the derivation of this internal stress. Furthermore, a linear spatial distribution of the GND densities within a volume with radius R around a material point is assumed. The length scale parameter R determines the extent of the nonlocal influence. Accounting for the contributions from all slip systems α, the following expression for the internal stress is obtained (see Bayley et al. (2006) for more details)

σ int =

GbR2 X GbR2 X α(e) α(s) ∇ρg · 3 Cα(e) + ∇ρg · 3 Cα(s) , 8 (1 − ν) α 4 α

(11)

where the third order slip system orientation tensors

3

Cα(e) = 3mα sα sα + mα mα mα + 4νmα lα lα − sα sα mα − sα mα sα ,

3

Cα(s) = −mα lα sα − mα sα lα + lα mα sα + lα sα mα ,

(12)

with G the shear modulus and ν the Poisson’s ratio. Since the focus of this paper is on the grain 7

boundary behaviour, a simplified expression is used here, taking into account only the ‘self’ internal stress components (Bayley et al., 2006), i.e. only the last two terms in the third order slip system orientation tensors in Eq. (12). More details on this gradient crystal plasticity framework can be found in Evers et al. (2004a,b) and Bayley et al. (2006). 2.2. Grain boundary interface formulation The interface modelling framework, developed in van Beers et al. (2013), incorporates the possibility to describe grain boundary behaviour other than the conventional higher order boundary conditions of either impenetrable or completely transparent grain boundaries. Although the framework allows for the incorporation of macroscopic sliding and opening of the interface, these mechanisms are not incorporated here, i.e. continuity of displacements across the interface is assumed. The different fine-scale phenomena taking place at the boundary, e.g. absorption, emission and transmission, are not resolved separately at the scale of the model, but the net effect of these interactions is lumped into a continuum scale quantity termed ‘interface normal slip’ q. The interface normal slip components of edge (e) and screw (s) dislocations at the interface Γ between adjacent grains A (slip systems α) and B (slip systems β) are defined as

grain A:

q α(e) = γ α sα · ngb ,

grain B:

q β(e) = −γ β sβ · ngb ,

q α(s) = −γ α lα · ngb , q β(s) = γ β lβ · ngb .

(13)

The grain boundary normal ngb points from grain A to grain B. The evolution of the interface normal slip q α on slip system α at the grain A side of Γ is modelled with a visco-plastic power-law, similar to slip in the bulk crystal (the expressions for grain B are obtained by substituting β for α)

q˙α(e) = q˙0

q˙α(s) = q˙0

α(e) ! 1 Fbk − Feα(e) n Rα(e) α(s) ! 1 Fbk − Feα(s) n Rα(s)

  α(e) α(e) sign Fbk − Fe ,

  α(s) α(s) . sign Fbk − Fe 8

(14)

The parameters n and q˙0 are positive constants (similar to the bulk crystal parameters m and α(e)

γ˙ 0 ). The quantities Fbk

α(s)

and Fbk

are the bulk-induced interface microforces and follow from the

projections of the slip system microstress ξ α α(e)

Fbk = −ξ α · sα ,

α(s)

Fbk = ξ α · lα .

(15)

The microstress ξ α can be determined from the microstructurally resolved internal stress σ int and is given by (see van Beers et al. (2013) for the details and Gurtin (2002) for the original concept of the microstress)  GbR2 X ζ(e) 3 ζ(e) GbR2 X ζ(s) 3 ζ(s) : sα mα , ρg C + ρg C ξ = 8 (1 − ν) 4 α



ζ

(16)

ζ

and similarly for grain B. Quantities Rα(e) and Rα(s) represent the interface normal slip resistances for edge and screw dislocations on slip system α, respectively. The energetic interface microforces α(e)

Fe

α(s)

and Fe α(e)

Fe

=

are determined from the grain boundary energy φe

∂φe , ∂q α(e)

α(s)

Fe

=

∂φe , ∂q α(s)

(17)

and can be interpreted as interface ‘back stresses’. The grain boundary energy φe depends on the grain boundary structure and its evolution, as discussed in the next section. 2.3. Grain boundary structure and energy The structure and energy of a grain boundary depends on its initial state and the subsequent deformation history of the adjacent grains. In general, when lattice dislocations interact with a grain boundary, the structure of the interface and the defect content are altered. In order to obtain a continuum representation of the grain boundary structure, van Beers et al. (2013, 2015b) proposed the following vectorial measure of the grain boundary net defect density at a material point on the interface

Bgb = B0 + B .

(18) 9

Here, B0 represents the initial grain boundary defect and corresponds to the contribution of ‘intrinsic’ grain boundary dislocations (Balluffi et al., 1972; Hirth and Balluffi, 1973). These dislocations form the ‘equilibrium’ structure of the boundary. The vector B represents the defect resulting from the interaction of bulk lattice dislocations with the grain boundary during plastic deformation. These ‘extrinsic’ grain boundary dislocations are not part of the equilibrium structure of the boundary. The following form of the extrinsic net defect density B due to the interaction with lattice dislocations of edge and screw character from both sides of the grain boundary is defined

B=

X

  X cα(e) q α(e) + cα(s) q α(s) sα + cβ(e) q β(e) + cβ(s) q β(s) sβ ,

α

(19)

β

with the coefficients

cα(e) = lα · (ngb × k) = (sα · ngb )(mα · k) − (sα · k)(mα · ngb ) cα(s) = sα · (ngb × k) = (mα · ngb )(lα · k) − (mα · k)(lα · ngb )

(20)

and similar for cβ(e) and cβ(s) by substituting β for α. Here, k = p/|p| is the unit boundary period vector. Definition (19) is a general expression that is consistent with an ‘infinitesimal’ Burgers circuit at the grain boundary (Gurtin, 2008). In van Beers et al. (2013, 2015b) a simplified expression for B was used, with coefficients cα equal to 1, which corresponds to a planar configuration. The grain boundary energy φe is linked to the grain boundary structure and its evolution via the relation

φe = φe Bgb q α(e,s) , q β(e,s)





,

(21)

which depends on the grain boundary net defect density Bgb that evolves through the interface normal slip measures q. In order to move towards a physically based expression for the grain boundary energy in Eq. (21), van Beers et al. (2015a) developed a continuum representation of the initial boundary structure and energy based on the output of atomistic simulations. It incorporates: 10

(i) grain boundary energies and boundary period vectors obtained from atomistic simulations, (ii) structural unit and dislocation descriptions of the grain boundary structure and (iii) the FrankBilby equation to determine the grain boundary dislocation content. In this atomistic-to-continuum approach, the key concept is the attribution of an intrinsic net defect Bp , representing the initial grain boundary structure, to a boundary period p. A corresponding scalar intrinsic net defect density |B0 |, with B0 = Bp /|p|, is related to the initial grain boundary energy φe0 through a logarithmic relationship valid for the whole misorientation range around a certain rotation axis

φe0 = M |B0 | − N |B0 | ln(|B0 |) ,

(22)

where M and N are two parameters to be identified by fitting atomistic simulation results. This logarithmic relationship resembles the Read-Shockley equation for low-angle boundaries (Read, 1953; Read and Shockley, 1950) and can be interpreted as a generalisation to the complete misorientation range, including high-angle grain boundaries. This approach has been applied in van Beers et al. (2015a) to symmetric tilt grain boundaries with the h100i, h110i and h111i rotation axes in aluminium and copper, thereby obtaining fairly accurate descriptions of the continuum scale grain boundary energy. As indicated in van Beers et al. (2015b), a reliable physically based description of the evolution of the structure and energy of a grain boundary upon plastic deformation remains a challenge at present. No systematic (atomistic scale) simulations exist that track the evolution of the boundary structure and energy as a consequence of the interaction with multiple dislocations. Therefore, a straightforward generalisation of the expression of the initial grain boundary energy (Eq. (22)) is here proposed

φe = M |Bgb | − N |Bgb | ln (|Bgb |) + 21 Ce N |Bgb − B0 |2 ,

(23)

where Ce is a positive constant and the net defect density is defined in Eq. (18) with Eq. (19). Based on van Beers et al. (2013) a quadratic term is incorporated to yield convex properties. More details on the continuum representation of the grain boundary structure and energy can 11

be found in van Beers et al. (2015a,b). So far, only local expressions for the grain boundary net defect have been considered. The subsequent section will incorporate nonlocal effects due to the defect redistribution along the grain boundary.

3. Defect redistribution The development of a continuum model that takes into account the redistribution of the defect content along the grain boundary within the context of strain gradient crystal plasticity is elaborated in this section. After detailing the derivation of the model, the resulting model is discussed. 3.1. Derivation of the model The continuum redistribution model derived here consists of a defect balance equation and constitutive equations (cf. Ubachs et al. (2004)). Here, a redistribution of a vectorial quantity at a surface is considered, which differs from the consideration of a scalar field within a volume, as done for scalar diffusion problems. 3.1.1. Balance equation The grain boundary net defect balance equation reads d dt

Z

Bgb dΓ = −

Z L

Γ

nΓ · J dL +

Z S dΓ ,

(24)

Γ

which states that the change of the grain boundary net defect density Bgb defined on the grain boundary surface Γ is equal to (i) what is lost or gained by flow J through the edge L of the surface (having unit normal nΓ pointing outward, which lies in the GB plane (if planar)) and (ii) what is created or consumed by sources and sinks S at the grain boundary surface (see Fig. 1). The source/sink term S of the net defect density is due to the in- and outflow of bulk lattice dislocations in the normal direction ngb on the lateral faces of the grain boundary and reads

S=

X α

  X cα(e) q˙α(e) + cα(s) q˙α(s) sα + cβ(e) q˙β(e) + cβ(s) q˙β(s) sβ , β

12

(25)

n¡ grain B

L

n GB grain A

¡

Figure 1: Schematic of the grain boundary surface Γ having unit normal ngb pointing from grain A to grain B on the lateral face. The edge L of the grain boundary surface has unit normal nΓ pointing outward, which lies in the grain boundary plane (if planar).

which corresponds to the time derivative of the extrinsic net defect density in Eq. (19). The surface divergence theorem will be employed and reads (Brand, 1947; Steinmann, 2008) Z

Z ∇s · a dΓ =

Γ

gb

(∇s · n )n Γ

gb

Z

nΓ · a dL ,

· a dΓ +

(26)

L

where ∇s is the surface gradient ∇s = ∇ · (I − ngb ngb ), a is an arbitrary vector or tensor and ∇s · ngb is a measure of the curvature of the grain boundary surface. Application of the surface divergence theorem, assuming a flat grain boundary, leads to the following local balance equation for the evolution of the grain boundary net defect density Bgb at a material point x at the grain boundary surface Γ dBgb = −∇s · J + S . dt

(27)

13

3.1.2. Constitutive equations As a constitutive assumption, the flow J of the grain boundary net defect density is related to a driving force given by the surface gradient of a ‘chemical’ potential µ

J = − 4 η : ∇s µ .

(28)

Here, 4 η is a fourth order mobility tensor, which is assumed to be isotropic for simplicity, i.e. 4η

= η 4 I, with the fourth order unit tensor 4 I = δil δjk ei ej ek el , and η a positive mobility parameter.

Substituting Eq. (28) into the net defect density evolution equation (27) gives dBgb = η∆s µ + S , dt

(29)

with the surface Laplacian operator ∆s = ∇s · ∇s . Note that no change in the grain boundary net defect density occurs due to redistribution, if the chemical potential is homogeneous along the grain boundary. The chemical potential µ is obtained by taking the variation of the grain boundary energy density ψ e with respect to Bgb at the point x

δψ e =

∂ψ e · δBgb (x) ∂Bgb (x)

with µ =

∂ψ e . ∂Bgb (x)

(30)

The following two contributions to the grain boundary energy density ψ e at a grain boundary material point x as a function of the grain boundary net defect density Bgb are considered

e ψ e (Bgb (x), Bgb (y)) = ψle (Bgb (x)) + ψnl (Bgb (x), Bgb (y)) .

(31)

Here, the local energy density ψle is given by Eq. (23), repeated here for completeness 1 ψle (Bgb (x)) = M |Bgb (x)| − N |Bgb (x)| ln (|Bgb (x)|) + Ce N |Bgb (x) − B0 (x)|2 . 2

(32)

e , which accounts for the effect of the net defect density Bgb (y) at The nonlocal energy density ψnl

14

grain boundary material points y on the grain boundary energy density at a point x, is inspired by Bates and Han (2005); Giacomin and Lebowitz (1997); Ubachs et al. (2004) for two-phase systems, and given as

e ψnl (Bgb (x), Bgb (y))

C = 4

Z

W (x, y)|Bgb (x) − Bgb (y)|2 dΓ .

(33)

Γ

where W (x, y) is a weight function that weighs the contribution of the grain boundary net defect density Bgb (y) and C is a positive constant. Calculating the chemical potential µ (Eq. (30)) gives

µ=

e (Bgb (x), Bgb (y)) ∂ψle (Bgb (x)) ∂ψnl + , ∂Bgb (x) ∂Bgb (x)

(34)

with the derivatives of the local and nonlocal energy densities reading  Bgb (x) ∂ψle (Bgb (x))  gb = M − N − N ln(|B (x)|) + Ce N (Bgb (x) − B0 (x)) , ∂Bgb (x) |Bgb (x)|

(35)

e (Bgb (x), Bgb (y)) ∂ψnl C = gb ∂B (x) 2

(36)

Z

W (x, y)(Bgb (x) − Bgb (y)) dΓ .

Γ

This latter expression can be rewritten into (see Ubachs et al. (2004) for another nonlocal diffusion problem)   e (Bgb (x), Bgb (y)) ∂ψnl gb gb = λ B (x) − B (x) , ∂Bgb (x)

(37) gb

where the nonlocal grain boundary net defect density B (x), a line interface coefficient λ and a scaled weight function W ∗ (x, y) are defined as Z

C B (x) = W (x, y)B (y) dΓ , λ = 2 Γ gb

∗

gb

Z W (x, y) dΓ Γ

and W ∗ (x, y) =

CW (x, y) . (38) 2λ

The integral formulation, Eq. (38), can be reformulated into a differential form (Engelen et al., 2003; Peerlings et al., 1996; Ubachs et al., 2004) using a Taylor series expansion around material

15

point x at the grain boundary plane truncated after the quadratic term 1 Bgb (y) = Bgb (x) + (y − x) · ∇s Bgb (x) + (y − x)(y − x) : ∇s ∇s Bgb (x) . 2

(39)

Substituting this expansion into the integral definition of the nonlocal grain boundary net defect density (Eq. (38)) gives gb

B (x) =

Z

Z W ∗ (x, y)Bgb (x) dΓ + W ∗ (x, y)(y − x) · ∇s Bgb (x) dΓ+ Γ Γ Z 1 ∗ W (x, y)(y − x)(y − x) : ∇s ∇s Bgb (x) dΓ . 2 Γ

(40)

An isotropic weight function is assumed, i.e. it depends only on the distance ρ = |x − y| between the points y and x, for which a Gaussian function is taken (Baˇzant and Jir´asek, 2002)   1 ρ2 W (ρ) = exp − 2 . 2πL2 2L

(41)

Here, L is the length scale that determines the extent of the nonlocal spatial interactions in the grain boundary. Note that for an infinite boundary plane Z W (ρ) dΓ = 1 .

(42)

Γ

Elaboration of the integral expression (Eq. (40)) for an infinite grain boundary plane gives gb

B (x) = Bgb (x) + c∆s Bgb (x) ,

(43)

where the terms involving odd derivatives vanish due to the isotropy of the weight function and the anti-symmetry of the terms y − x with respect to x. The term with ∇s ∇s condensates to a single term involving the surface Laplacian operator ∆s with the parameter c = 21 L2 . Expression gb

(43) is also called an ‘explicit’ gradient approximation of the nonlocal integral (38), since B (x) is given explicitly in terms of Bgb (x) and its derivatives. At this point, note that substituting the

16

gradient approximation (43) and nonlocal expression (37) into the chemical potential (34) yields

µ=

∂ψle (Bgb (x)) − κ∆s Bgb (x) . ∂Bgb (x)

(44)

where κ = λc is a gradient energy coefficient. Expression (44) resembles the chemical potential derived for two-phase systems (Gurtin, 1996; Penrose and Fife, 1990), and can be considered as a weakly nonlocal formulation. The strongly nonlocal formulation, which will be used in this work, is found by applying the surface Laplacian operator to Eq. (43), multiplying it with c, subtracting the result from Eq. (43) and neglecting the higher-order terms. This leads to the relation (Peerlings et al., 1996; Ubachs et al., 2004) gb

gb

B (x) − c∆s B (x) = Bgb (x) ,

(45)

gb

which is a Helmholtz equation for B . In contrast to Eq. (43), expression (45) is called an ‘implicit’ gradient approximation. The Helmholtz equation of the type (45), here used for the determination of the nonlocal grain boundary net defect density, implies a strongly nonlocal formulation, as shown by Peerlings et al. (2001) in the context of damage mechanics. It retains the truly nonlocal character of the nonlocal integral expression (38), in contrast to the explicit gradient approximation in Eq. (43), and it implies another weight function than the one given in Eq. (41). Although the length scale L governs the intensity of the spatial interactions within the grain boundary, the explicit gradient approximation is only weakly nonlocal, i.e. the interactions are limited to an infinitesimal neighbourhood. In this case, the truly nonlocal character is lost due to the truncation of the Taylor series expansion. The reader is referred to Peerlings et al. (2001) for more details on weakly and strongly nonlocal formulations.

17

3.1.3. Summary continuum redistribution model The continuum model for defect redistribution along the grain boundary consists of the following set of equations. The grain boundary net defect balance equation dBgb = −∇s · J + S dt

with

Bgb (t = 0) = B0 ,

(46)

where the flow J of the net defect is given by  J = −η∇s µ = −η∇s

  ∂ψle gb gb +λ B −B . ∂Bgb

(47)

Here, the nonlocal grain boundary net defect density is obtained from the Helmholtz equation gb

B

gb

− c∆s B

= Bgb ,

(48)

the derivative of the local energy density reads   Bgb ∂ψle gb + Ce N (Bgb − B0 ) , gb = M − N − N ln(|B |) ∂B |Bgb |

(49)

and the source/sink term is given by

S=

X α

  X cα(e) q˙α(e) + cα(s) q˙α(s) sα + cβ(e) q˙β(e) + cβ(s) q˙β(s) sβ .

(50)

β

The continuum defect redistribution model summarised here introduces the need for additional boundary and initial conditions for the solution of Eqs. (46) and (48). The boundary condition for the redistribution equation (46) is the zero flow of net defect density at the edge L, i.e. the homogeneous (natural) Neumann condition nΓ · J = 0, which means that there is no loss or gain of the net defect density through L. More advanced triple junction models could be used here as well. For the Helmholtz equation (48), a homogeneous Neumann boundary condition is used, i.e. gb

nΓ · ∇ s B

= 0 (cf. Peerlings et al. (2001); Ubachs et al. (2004)). With this condition the global

18

net defect density in the grain boundary is preserved in the nonlocal averaging Z

gb

B

Z dΓ =

Bgb dΓ .

(51)

Γ

Γ

3.2. Discussion The continuum model consisting of the set of equations (46)-(50) incorporates the redistribution of the defect content within the grain boundary. It is an extension of the model for the grain boundary net defect as summarised in Section 2.3. Note that the model without redistribution is recovered if the mobility η and the length scale L are set to zero. The former means that the dislocations cannot move along the grain boundary, whereas the latter implies that the nonlocal e is disregarded. Eq. (46) then represents the rate form of Eqs. (18) grain boundary energy density ψnl

and (19). Furthermore, the spreading of the defect content occurs via a diffusion-like equation and shows similarities with phase field modelling (Chen, 2002; Moelans et al., 2008; Provatas and Elder, 2010). In fact, the evolution equation (46) with (47) is of the Cahn-Hilliard type (Cahn, 1961; Cahn and Hilliard, 1958). The grain boundary defect content is conserved if the source/sink term S = 0. The mobility of dislocations in the grain boundary η is proportional to

Dgb b kT

(Grabski, 1988),

where Dgb is the grain boundary self-diffusion, k is the Boltzmann constant, T is the absolute temperature and b is, as before, the magnitude of the Burgers vector. The grain boundary selfdiffusion follows an Arrhenius form (Sutton and Balluffi, 1995)

D

gb

=

D0gb exp

  Qgb − , Rg T

(52)

where D0gb is a pre-exponential factor, Qgb is the grain boundary self-diffusion activation energy and Rg is the gas constant. In this work, the mobility parameter η is therefore taken as   δDgb b δD0gb b Qgb η= = exp − , kT kT Rg T

(53)

with δ the grain boundary width. The quantities δ and Dgb combined can be measured experimentally (Surholt and Herzig, 1997). In general, the grain boundary self-diffusion depends on the 19

grain boundary structure, i.e. the less ordered the grain boundary, the higher the self-diffusion, due to the more ‘open’ structure for diffusional transport. Hence, high-angle grain boundaries typically have larger values of self-diffusion than low-angle grain boundaries. Furthermore, the grain boundary diffusivity can be anisotropic, which leads to different values of self-diffusion in different directions in the boundary plane (Atkinson, 1985). This anisotropy can be incorporated via the mobility tensor 4 η, but is not considered here. It has been reported from experiments (Elkajbaji and Thibault-Desseaux, 1988; Lee et al., 1990; Mori and Tangri, 1979) and fine-scale simulations (Dewald and Curtin, 2007; Ishida and Mori, 1985; Pestman et al., 1991) that an absorbed lattice dislocation may decompose into two smaller dislocations (possibly after reacting with other defects first), one of which glides along the grain boundary plane, whereas the other remains at the point of impingement. Glide is possible when the Burgers vector of the product is parallel to the grain boundary plane, which should be a physically possible slip plane. Movement of a product with a Burgers vector component lying out of the boundary plane requires both climb and glide. Climb is a thermally activated process and is controlled by grain boundary diffusion. In general, due to the rather special conditions for glide, it has been concluded in Grabski (1988) that, from the experimental point of view, independently of the orientation of the Burgers vector with respect to the grain boundary plane, the mobility of dislocations is controlled by grain boundary diffusion. In our work, at the continuum scale, no distinction is made between glissile and sessile dislocations in the redistribution of the net defect. A net defect density is considered and it is assumed that all redistribution processes occur through physically different types of atomic shuffling collectively described by the evolution equation (46). Finally, in general, the (coupled) processes of grain boundary sliding, grain boundary migration and grain rotation are accompanied with the defect redistribution and the change of the grain boundary structure (Grabski, 1988; Priester, 1997; Valiev et al., 1986). It is expected that these phenomena have a significant influence on the macroscopic behaviour. In the case of a polycrystal, the grain boundary is subjected to additional constraints imposed by the microstructure, such as triple junctions. Other stress relaxation mechanisms such as cavitation and decohesion can also occur. The presence of impurities and grain boundary segregation further increases the complexity 20

grain boundary extended crystal plasticity framework equilibrium equation

continuum net defect redistribution interface normal slip rate

redistribution equation

GND balance equations

Helmholtz equation

+ IC, BC and GB model

+ IC and BC

3D/2D mesh

2D/1D mesh along GB

implicit time integration

GB net defect density

explicit time integration at start of increment

incremental-iterative procedure

Figure 2: Schematic of the staggered solution scheme.

of the problem. These phenomena and features are not modelled in this work.

4. Solution procedure and finite element implementation The grain boundary extended strain gradient crystal plasticity framework combined with the continuum defect redistribution model, complemented by appropriate initial (IC) and boundary conditions (BC), comprises a set of highly nonlinear coupled equations. Different solution strategies exist, one of which is to solve the set of equations in a finite element setting using a staggered solution scheme, schematically shown in Fig. 2. The staggered solution procedure consists of the sequential solution of two parts of the model, namely (i) the bulk and grain boundary model and (ii) the grain boundary redistribution model. The grain boundary redistribution model is solved on a domain of one dimension lower than the bulk and grain boundary model. The finite element implementation of the strain gradient crystal plasticity framework enhanced with a grain boundary model follows the procedure outlined in van Beers et al. (2013). It consists of the equilibrium equation and the GND balance equations in the weak form. The displacements and the GND densities act as the degrees of freedom. The grain boundary is modelled by uncoupling the finite elements adjacent to the grain boundary by inserting double nodes. It is assumed that the displacements are continuous across the grain boundary, which is accomplished by tying the corresponding displacement degrees of freedom. Implicit time integration is employed, which results in a standard incremental-iterative procedure for the solution of this part of the framework. For the 21

solution of this part the grain boundary net defect density distribution along the grain boundary needs to be known, which is obtained from the second part of the framework, see Fig. 2. The finite element implementation of the continuum defect redistribution model solves the weak forms of the redistribution equation (46) and the Helmholtz equation (48), with the local and nonlocal grain boundary net defect densities acting as degrees of freedom. Considering the grain boundary net defect balance equation (46), multiplication with a weight function w, integration over the grain boundary, application of the product rule and employing the surface divergence theorem (26), while considering a flat grain boundary and zero flow of the net defect density at the edge L, i.e. nΓ · J = 0, leads to the weak form dBgb dΓ = w· dt Γ

Z

Z

Z

T

w · S dΓ ∀w .

(∇s w) : J dΓ + Γ

(54)

Γ

Substituting the flow J = −η∇s µ and applying explicit time integration for tn+1 = tn + ∆t yields after elaboration Z

Z

gb

w · B (tn+1 ) dΓ = Γ

Z

gb

w · B (tn ) dΓ + Γ

∆t w · S(tn ) dΓ Γ

  ∂µ T − η∆t ∇s w · : ∇s Bgb (tn ) dΓ ∀w , gb ∂B Γ tn Z

(55)

where the derivative of the chemical potential with respect to the local grain boundary net defect density reads ∂µ = ∂Bgb



   M − N − N ln(|Bgb |) M − N ln(|Bgb |) + C N + λ I − Bgb Bgb . e |Bgb | |Bgb |3

(56)

Next consider the Helmholtz equation (48). The weak form is obtained by multiplying with a weight function w, integrating over the grain boundary, applying the product rule and the surface divergence theorem (26) for a flat grain boundary, while using the homogeneous Neumann condition gb

nΓ · ∇ s B Z

= 0 at the edge L gb

Z

w · B (tn ) dΓ + Γ

T

Z

gb

c (∇s w) : ∇s B (tn ) dΓ = Γ

Γ

22

w · Bgb (tn ) dΓ ∀w .

(57)

Applying the standard finite element discretisation leads to two sets of discretised linear equations ¯ gb (tn ) = f ¯ (tn ) K B¯ B¯ B B ˜ ˜

and K BB B gb (tn+1 ) = fB (tn ) , ˜ ˜

(58)

which are solved at the start of each increment in the incremental-iterative solution procedure of the grain boundary crystal plasticity part of the framework. In order to solve this second part of the framework, the distribution of the interface normal slip rate at the grain boundary q(t ˙ n ), as required for Bgb , needs to be transferred from the first part, see Fig. 2. The finite element meshes for the GND density fields and the local and nonlocal grain boundary net defect densities do not necessarily have to be conforming. The grain boundary integration point values of the interface normal slip rates of the GND density mesh are passed to the net defect density mesh through linear interpolation. For the transfer of the grain boundary net defect density nodal values from the net defect density mesh to the grain boundary integration points of the GND density mesh, the same procedure is applied.

5. Numerical study of defect redistribution The strain gradient crystal plasticity framework enhanced with a grain boundary model, as summarised in Section 2, extended with the net defect redistribution along the grain boundary developed in Section 3, are numerically implemented next, using the solution procedure described in the previous section. The influence of the defect redistribution on the macroscopic behaviour is investigated in the case of a bicrystal problem in simple shear. To this end, first the model problem is explained and the material parameters are given. Then the discretisation and the boundary conditions are described, after which the results are presented. 5.1. Model problem In order to examine the effect of the redistribution of the net defect within the grain boundary on the macroscopic response, an academic problem of a bicrystal under plane strain conditions is considered, as shown in Fig. 3. The bicrystal consists of grains A (blue) and B (red), which have 23

U

e2 = [001]

h

grain A

grain B 1 e1 = √2 [110] 1 e3 = √2 [110]

2d Figure 3: Geometry and boundary conditions for simple shear deformation of a bicrystal. Strong inhomogeneities are introduced in the form of slip bands at the grey regions. Two of the three meshes employed in the staggered solution procedure are depicted, which correspond to the bilinear quadrilaterals (solid squares) for the GND densities (the displacement finite elements are biquadratic quadrilaterals) and the linear elements (solid circles) for the local and nonlocal GB net defect densities.

width d in the e1 -direction and height h in the e2 -direction, and are separated by a grain boundary. A grain size d = h = 2 µm will be considered and only edge dislocations are taken into account in this planar configuration. The material used in the simulations is copper. Copper has a face-centered cubic (FCC) crystalline structure, in which 12 slip systems exist, denoted by {111}h110i, where {111} corresponds to the family of slip planes and h110i to the family of slip directions. It has been shown (Kysar et al., 2005; Rice, 1987; Shishvan et al., 2011) that plane strain conditions emanate when certain slip systems act cooperatively, for which the 12 slip systems associated with the FCC crystal reduce to three effective slip systems. This is illustrated in Fig. 4 for the state of plane strain in the [¯1¯10]-[001] plane. The (1¯ 11)[110] and (¯ 111)[110] slip systems can be combined to form an effective in-plane slip system, called the complex system (Rice, 1987). In this paper, the effective complex slip system acts in the [¯ 1¯ 10] direction and is referred to as slip system α = 1. The (111)[0¯11] and (111)[¯101] slip systems operate in equal amounts to form an effective plane strain slip system on the (111) plane in the [¯ 1¯ 12] direction and is here referred to as slip system α = 2. Similarly, the slip system α = 3 is the effective plane strain slip system on the (11¯1) plane and in the [112] direction as a combination of (11¯ 1)[101] and (11¯ 1)[011]. Note that the three effective in-plane slip systems 24

(111) and (111) with [110]

[001]

(111) with [101] and [011]

m2

s3

s2

m3

[001]

φ

m1 ω

s1 [110]

[100]

[110]

ω = 54.74° φ = 70.53°

[010]

(111) with [011] and [101]

Figure 4: An FCC crystal with the simultaneous action of slip systems to resemble the plane strain condition (left) and the three effective plane strain slip systems in the [¯ 1¯ 10]-[001] plane (right).

can be obtained by applying counterclockwise rotations of 0◦ , 54.74◦ and 125.26◦ , respectively, relative to the [¯ 1¯ 10] direction. Furthermore, from geometrical considerations it follows that the √ effective lengths of the Burgers vectors for these three slip systems α = 1, 2 and 3 are b, 12 3b and √ 1 2 3b, respectively (Shishvan et al., 2011; Wang et al., 2009). The orientations of the grains in the bicrystal are explained in Fig. 5. Starting from the reference state, corresponding to the fixed coordinate system, the grains are first rotated (vector transformation) 90◦ counterclockwise to the state where no misorientation between the adjacent grains is present, i.e. a perfect crystal. A target grain boundary of misorientation θ is recovered by applying a rotation of θ/2 in opposite directions for grains A and B, respectively. The grain boundaries that are produced by these operations are symmetric tilt boundaries. For the particular case considered in the present work, the rotation axis is parallel to the [1¯10] direction. The structure and energy of symmetric tilt grain boundaries with this rotation axis in aluminium and copper in the entire misorientation range have been investigated in detail in van Beers et al. (2015a). In the current work, a misorientation θ = 70.53◦ between the grains is considered, which corresponds to the Σ3 (112) symmetric tilt boundary. Note that for this misorientation the third effective slip system in grain A and the second effective slip system in grain B are horizontal, see Fig. 5.

25

5.2. Material parameters The bulk material parameters used in the calculations are representative of copper and are similar to those of Evers et al. (2004b) and Bayley et al. (2006). The values are listed in Table 1, where the interaction and immobilisation matrices Aαξ and H ξζ result from the interaction and immobilisation coefficients a0 and h0 on the diagonal, and the off-diagonal entries a3 and h3 . The interface parameters are listed in Table 2. The interface reference slip rate q˙0 and rate sensitivity exponent n are taken to be the same as their bulk counterparts γ˙ 0 and m. The energetic interface parameters M and N have been determined in van Beers et al. (2015a). The lumped interface self-diffusion parameter δD0gb and the grain boundary self-diffusion activation energy Qgb for highpurity copper are taken from Surholt and Herzig (1997). In order to illustrate the effect of the defect redistribution along the grain boundary, strong inhomogeneities are introduced in both crystals, as shown in Fig. 3. The grey regions represent slip bands, in which the slip resistance sα (Eq. (7)) for slip in the bulk (Eq. (6)) is lowered for the third and second effective slip system (oriented horizontally) in grains A and B, respectively. For this purpose, the initial SSD density ραs0 (Eq. (8)) of these slip systems is reduced with a factor 10. Furthermore, the interface normal slip activity (Eq. (14)) of these two slip systems inside the slip bands (at the GB) is increased by lowering the interface normal resistance R by a factor 25. 5.3. Discretisation and boundary conditions The numerical implementation of the framework follows the solution procedure elaborated in Section 4. The specific initial and boundary conditions for the model problem considered in this reference state

s3 m3

m2

perfect state (θ=0°) m 1A

s2

s1A

s2A

m1

m

s1

2 A

m 1B s2B m

s2A m 2A s3A m 3A

s3A m 3A

1 m 1A sA

s1B m 1B

2 B

[001] [110]

misoriented state (θ=70.53°)

s1B

s3B m 3B

s2B m 2B θ θ 2 2

s3B

m 3B

Figure 5: Orientations of the grains in the bicrystal starting from the reference state (left) via the perfect state (middle) to the misoriented state (right).

26

Table 1: Bulk material parameters representative of copper.

Parameter

Description

Magnitude

E G ν b γ˙0 a0 a3 h0 h3 R m K yc ρs0

Young’s modulus Shear modulus Poisson’s ratio Burgers vector length Reference slip rate Interaction coefficient Interaction coefficient Immobilisation coefficient Immobilisation coefficient GND evaluation radius Rate sensitivity exponent Dislocation segment length constant Critical annihilation length Initial SSD density

144 GPa 54.1 GPa 0.33 0.256 nm 0.001 s−1 0.06 1.0 0.2 1.0 1 µm 0.1 26 1.6 nm 7.0 µm−2

Table 2: Interface parameters used in the numerical simulations.

Parameter

Description

Magnitude

q˙0 n Ce R M N k Rg Qgb δD0gb

Interface reference slip rate Interface rate sensitivity exponent Interface energy constant Interface normal slip resistance Interface energy constant Interface energy constant Boltzmann constant Gas constant Interface self-diffusion activation energy Lumped interface self-diffusion parameter

0.001 s−1 0.1 103 0.25 N mm−1 358 mJ m−2 1434 mJ m−2 1.38 · 10−23 J K−1 8.314 J mol−1 K−1 72.47 kJ mol−1 3.98 · 10−16 m3 s−1

work along with the particular discretisation of the unknown variables are described next for the two parts of the staggered solution scheme, see Fig. 2. Considering first the equilibrium equation, the displacements at the top and bottom boundary are fully prescribed, see Fig. 3. Simple shear deformation is simulated by applying a prescribed displacement U = γht ˙ on the top boundary in the e1 -direction, while the bottom boundary is fully fixed. A constant shear rate of γ˙ = 10−3 s−1 is applied, up to a maximum of 1% macroscopic shear. Traction-free boundary conditions are imposed on the left and right boundary. For the GND 27

balance equations, microhard (no slip, homogeneous Neumann) boundary conditions are applied to the top and bottom boundary, where the specimen is clamped. The left and right boundary are subjected to microfree (zero GND density, homogeneous Dirichlet) boundary conditions. The GND density field is initially zero. The constitutive behaviour at the grain boundary is dictated by the grain boundary model. Each grain in the bicrystal is discretised with 4 elements in the e1 -direction and 16 elements in the e2 -direction, see Fig. 3. The displacement field is discretised with 8-node quadrilaterals using 2 × 2 reduced integration, whereas the GND density field is interpolated with 4-node bilinear shape functions employing 2 × 2 full integration. The slip bands are located at the 13th and 14th row of elements in grain A and at the 3rd and 4th element rows in grain B. The boundary condition for the redistribution equation is the zero flow of net defect density out of the clamped specimen where the top and bottom boundary meet the grain boundary, i.e. the homogeneous (natural) Neumann condition nΓ · J = 0. The initial local grain boundary net defect density is uniform and equals the intrinsic defect density B0 , as follows from the atomistic simulations (van Beers et al., 2015a). For the Helmholtz equation, the homogeneous Neumann gb

boundary condition is used, i.e. nΓ · ∇s B

= 0. The grain boundary is discretised with a 1D

mesh of 32 elements in the e2 -direction, see Fig. 3. The local and nonlocal net defect density fields along the grain boundary are discretised with 1D linear elements using 2 integration points. 5.4. Results To assess the effect of the defect redistribution along the grain boundary on the macroscopic response, as well as the microscopic fields, three different mobilities η are considered by selecting the temperatures T = 293, 323 and 353 K. In this set of simulations, the line interface redistribution coefficient λ and the nonlocal interface length parameter L are fixed at λ = 0.6 J/m2 and L = 100 nm. Furthermore, additional simulations that serve as reference cases are included, where (i) the grain boundary acts as an impenetrable boundary, i.e. a microhard grain boundary with Fe = 0 and R → ∞, and (ii) no redistribution occurs within the grain boundary, i.e. η = 0 and L = 0. Note that the parameters in these reference cases are such that the original models in Section 2 are recovered, i.e. (i) the bulk crystal plasticity model with conventional microhard grain boundary conditions and (ii) the grain boundary extended bulk crystal plasticity model excluding 28

0.7

120

Scalar local GB net defect density (−)

Shear stress (MPa)

100

microhard no redistribution T = 293 K T = 323 K T = 353 K

80

60

40

20

0 0

0.002

0.004 0.006 Applied shear (−)

(a)

0.008

0.01

0.69

microhard no redistribution T = 293 K T = 323 K T = 353 K

0.68

0.67

0.66

0.65

0.64 0

0.2

0.4 0.6 0.8 Dimensionless vertical position (−)

1

(b)

Figure 6: The influence of the net defect redistribution along the grain boundary on the macroscopic response (a) and the distribution of the scalar local grain boundary net defect density at the final applied load (b) for three different temperatures. The reference cases of a microhard grain boundary and no redistribution are included.

redistribution. The results of the simulations are shown in Fig. 6. Fig. 6(a) shows the resulting macroscopic stress-strain response, while Fig. 6(b) plots the scalar local grain boundary net defect density, √ defined as |Bgb | = Bgb · Bgb , along the grain boundary. In the case of the microhard grain boundary, from the onset of plasticity in the bulk crystals, dislocations start to pile up at the interface inducing GND densities that contribute to the rate of strain hardening, see Fig. 6(a), through the slip resistance sα and back stress τbα (Eq. (6)). The local grain boundary net defect density is not altered and remains equal to the initial intrinsic net defect density throughout the simulation. For the situation including the grain boundary plasticity model, but excluding the defect redistribution, the behaviour initially follows the microhard result (Fig. 6(a)) up to the point where the interface normal slip resistance is reached and dislocations are absorbed into the grain boundary, where they continue to accumulate locally due to the absence of redistribution along the grain boundary, see Fig. 6(b). The absorption of dislocations causes the rate at the which dislocations pile up at the grain boundary to decrease, whereas the interface back stress grows due to the local build-up of a net defect in the grain boundary at the regions where the slip bands impinge on the grain boundary. The net effect of these two mechanisms yields a decrease 29

in the macroscopic rate of strain hardening, see Fig. 6(a). Incorporating the redistribution of the grain boundary net defect, the initial behaviour is close to the microhard result up to the point of dislocation absorption into the grain boundary, see Fig. 6(a). For increasing temperature, and hence mobility, the extent to which the net defect content is redistributed increases, see Fig. 6(b). Due to the defect redistribution along the grain boundary, the local accumulation of the net defect relaxes, which lowers the interface back stress at the points of impingement. This decreases the rate of pile-up accumulation of the bulk dislocations at these interface spots and hence reduces the rate of strain hardening, see Fig. 6(a), the effect of which is larger for higher temperature. In order to obtain a better understanding of the influence of temperature on the defect redistribution, the evolution of the local grain boundary net defect density is plotted in Fig. 7 for the three temperatures and the case without redistribution of the net defect. The evolution results provide insight in the competition between the local net defect accumulation through the source term and the redistribution of the net defect along the grain boundary via the diffusion term (see Eq. (46)). In the absence of defect redistribution the evolution is completely governed by the local net defect accumulation, see Fig. 7(a). For the lowest temperature, Fig. 7(b), the evolution is still dominated by the local build-up of the grain boundary net defect. A transition occurs for the intermediate temperature, where the net defect accumulation and redistribution are present in approximately equal amounts (Fig. 7(c)). In the case of the highest considered temperature, the redistribution of the net defect controls the evolution, see Fig. 7(d). Finally, for this set of simulations, 2D distributions of an effective slip measure, defined as the Euclidean norm of the three slip systems, at four applied macroscopic shears, i.e. 0.25%, 0.5%, 0.75% and 1%, are shown in Fig. 8 for the cases excluding redistribution and redistribution at temperature T = 353 K. The slip bands in each grain can be clearly observed. Whereas the differences in the slip distribution between the two cases are minor at 0.25% applied shear, they increase for increasing macroscopic shear, in correspondence with the observations from Figs. 6 and 7. Following the explanation described above, due to the reduced pile-up of dislocations at the grain boundaries in the case of redistribution, slip within the bulk is less restricted and hence more slip develops. Related to this is the decreased rate of strain hardening in the macroscopic 30

no redistribution

T = 293 K 0.01

0.7

0.01 0.009

0.008 0.007

0.68

0.006 0.67

0.005 0.004

0.66

0.003 0.002

0.65

Scalar local GB net defect density (−)

0.69

Applied shear (−)

0.69

0.008 0.007

0.68

0.006 0.67

0.005 0.004

0.66

0.003 0.002

0.65

0.001 0.64 0

0.2 0.4 0.6 0.8 Dimensionless vertical position (−)

1

0.001 0.64 0

0

(a)

0.2 0.4 0.6 0.8 Dimensionless vertical position (−)

1

(b)

T = 323 K

T = 353 K

0.7

0.01

0.7

0.01 0.009

0.008 0.007

0.68

0.006 0.67

0.005 0.004

0.66

0.003 0.002

0.65

Scalar local GB net defect density (−)

0.69

Applied shear (−)

Scalar local GB net defect density (−)

0.009 0.69

0.008 0.007

0.68

0.006 0.67

0.005 0.004

0.66

0.003 0.002

0.65

0.001 0.64 0

0.2 0.4 0.6 0.8 Dimensionless vertical position (−)

(c)

0

1

0

Applied shear (−)

Scalar local GB net defect density (−)

0.009

Applied shear (−)

0.7

0.001 0.64 0

0.2 0.4 0.6 0.8 Dimensionless vertical position (−)

1

0

(d)

Figure 7: The evolution of the scalar local net defect density along the grain boundary for (a) the no redistribution case, (b) redistribution at temperature T = 293 K, (c) redistribution at T = 323 K and (d) redistribution at T = 353 K.

stress-strain response, Fig. 6(a). A quantitative comparison and model validation at the continuum length scale remains a challenge. The current continuum scale redistribution model does not resolve individual fine scale dislocation processes. Furthermore, other phenomena that can occur simultaneously, such as grain boundary sliding and migration, are not taken into account here. However, some qualitative agreement on the redistribution with smaller scale experiments (Lee et al., 1990; Mori and Tangri, 1979; Priester, 1997; Pumphrey and Gleiter, 1974) and numerical simulations (de Koning et al., 2003; 31

shear = 0.005

shear = 0.0075

shear = 0.01

0.03

0.02

0.01

Effective slip (-)

T = 353 K No redistribution

shear = 0.0025

0

Figure 8: 2D distributions of the effective slip measure, defined as the Euclidean norm of the three slip systems, at 0.25%, 0.5%, 0.75% and 1% applied macroscopic shear for the no redistribution case (top row) and redistribution at temperature T = 353 K (bottom row).

Dewald and Curtin, 2007; Jin et al., 2008) exists, where dislocations are absorbed into the grain boundary and, possibly after dissociation, move along the boundary plane. Next, the effect of the line interface redistribution coefficient λ and the nonlocal interface length L on the macroscopic and microscopic behaviour is examined. In the first series of simulations, the temperature and the nonlocal interface length are fixed at T = 293 K and L = 100 nm, whereas the interface redistribution coefficient λ is varied. The results are shown in Fig. 9 for λ = 0.6, 6 · 103 , 6 · 104 and 3 · 105 J/m2 . The influence of the line interface redistribution coefficient λ on the macroscopic response is minor, even for the large range of six orders of magnitude, see Fig. 9(a). A more pronounced influence is observed microscopically when considering the evolution of the scalar local grain boundary net defect density in Fig. 7(b) and Fig. 9(b)-(d). For increasing line interface redistribution coefficient λ, the diffusion term becomes stronger with respect to the source term, resulting in a more homogeneous distribution of the local net defect density. In the second series of simulations, the nonlocal interface length L is varied, while the temperature and the line interface redistribution coefficient are fixed at T = 293 K and λ = 3 · 105 J/m2 . The results are shown in Fig. 10 for L = 100, 200, 400 and 800 nm. Again, for the considered parameter range, the macroscopic influence is limited, see Fig. 10(a). Microscopically, considering the evolution of the scalar local grain boundary net defect density in Fig. 9(d) and Fig. 10(b)-(d), the variation of L has a clear effect. The diffusion contribution becomes even more dominating for increasing L, leading to a larger annihilation of the net defect at the center of the grain boundary, where the positive and negative defect content from the slip bands in grains A and B, respectively, 32

2

λ = 6e3 J/m 0.7

λ = 6e−1 J/m2

0.009 Scalar local GB net defect density (−)

λ = 6e3 J/m2 100

λ = 6e4 J/m2 λ = 3e5 J/m2

Shear stress (MPa)

0.01

80

60

40

20

0.69

0.008 0.007

0.68

0.006 0.67

0.005 0.004

0.66

Applied shear (−)

120

0.003 0.002

0.65

0.001

0.002

0.004 0.006 Applied shear (−)

0.008

0.64 0

0.01

(a)

0.2 0.4 0.6 0.8 Dimensionless vertical position (−)

1

(b) 2

2

λ = 6e4 J/m

λ = 3e5 J/m

0.7

0.01

0.7

0.01 0.009

Scalar local GB net defect density (−)

0.69

0.008 0.007

0.68

0.006 0.67

0.005 0.004

0.66

Applied shear (−)

Scalar local GB net defect density (−)

0.009

0.003 0.002

0.65

0.69

0.008 0.007

0.68

0.006 0.67

0.005 0.004

0.66

0.003 0.002

0.65

0.001 0.64 0

0

0.2 0.4 0.6 0.8 Dimensionless vertical position (−)

1

Applied shear (−)

0 0

0.001 0.64 0

0

(c)

0.2 0.4 0.6 0.8 Dimensionless vertical position (−)

1

0

(d)

Figure 9: The influence of the line interface redistribution coefficient λ in the net defect redistribution along the grain boundary on the macroscopic response (a) and the evolution of the scalar local net defect density for the line interface redistribution coefficients (b) λ = 6 · 103 J/m2 , (c) λ = 6 · 104 J/m2 and (d) λ = 3 · 105 J/m2 .

cancel in the redistribution. To summarise, the results indicate that the grain boundary net defect redistribution has a clear macroscopic influence, but is mainly due to the temperature directly affecting the mobility η of the dislocations within the grain boundary. For the current set of parameters, only a limited effect on the macroscopic response was observed for varying the line interface redistribution coefficient λ and the nonlocal interface length L. All three parameters (η, λ and L) have, however, a significant 33

L = 200 nm

120

0.01 0.009

80

60

40

20

0.69

0.008 0.007

0.68

0.006 0.67

0.005 0.004

0.66

Applied shear (−)

Scalar local GB net defect density (−)

Shear stress (MPa)

100

0.7

L = 100 nm L = 200 nm L = 400 nm L = 800 nm

0.003 0.002

0.65

0.001

0.002

0.004 0.006 Applied shear (−)

0.008

0.64 0

0.01

(a)

0.2 0.4 0.6 0.8 Dimensionless vertical position (−)

1

(b)

L = 400 nm

L = 800 nm

0.7

0.01

0.7

0.01 0.009

0.008 0.007

0.68

0.006 0.67

0.005 0.004

0.66

0.003 0.002

0.65

Scalar local GB net defect density (−)

0.69

Applied shear (−)

Scalar local GB net defect density (−)

0.009 0.69

0.008 0.007

0.68

0.006 0.67

0.005 0.004

0.66

0.003 0.002

0.65

0.001 0.64 0

0.2 0.4 0.6 0.8 Dimensionless vertical position (−)

(c)

0

1

0

Applied shear (−)

0 0

0.001 0.64 0

0.2 0.4 0.6 0.8 Dimensionless vertical position (−)

1

0

(d)

Figure 10: The effect of the nonlocal interface length L in the net defect redistribution along the grain boundary on the macroscopic response (a) and the evolution of the scalar local net defect density for the nonlocal interface lengths (b) L = 200 nm, (c) L = 400 nm and (d) L = 800 nm.

microscopic effect. Note that by substituting the Helmholtz equation (48) into the redistribution equation (46), similar as in expression (44), the gradient energy coefficient κ = λc =

1 2 2 λL

is

identified, which characterises the relative contribution of the nonlocal term into the redistribution. The nonlocal interface length L changes κ quadratically, for this reason having a larger effect as compared to the line interface redistribution coefficient λ. For the bicrystal problem considered here, it is expected that further increase of the parameters η, λ and L will increase the influence on the macroscopic and microscopic behaviour, up to the point where the diffusion contribution 34

completely dominates the evolution, whereby absorbed lattice dislocations directly redistribute and annihilate. As a final remark, in general, the nonlocal term containing the interface redistribution coefficient λ and the nonlocal interface length L, does not only incorporate spatial interactions, but also serves as a regularising term in the cases when the local energy density is nonconvex, e.g. in spinodal decomposition (Ubachs et al., 2004) or dislocation patterning (Yalcinkaya et al., 2011), by penalising strong gradients in the quantity of interest. The local grain boundary energy density chosen in this work (Eq. (32)), based on a straightforward generalisation of the initial grain boundary energy determined from atomistic simulations, already has a convex character depending on the value of Ce . In the specific planar configuration examined here, this can be verified analytically by calculating the eigenvalues of the second derivative of the local energy density, given in Eq. (56) without the λ, which are positive, and hence indicate convexity by positive-definiteness of the second derivative. Hence, pattern formation in the redistributed defects along the grain boundary is not expected.

6. Conclusions Many different interaction mechanisms between dislocations and grain boundaries exist that have an important effect on the mechanical behaviour of polycrystalline materials. To date, significant challenges are present in physically based modelling of these grain boundary plasticity features in macroscopic continuum frameworks. The main contribution of this work and conclusions are: 1. A novel model that incorporates the nonlocal relaxation of the grain boundary net defect due to the redistribution of the defect content along the interface, which is developed within the context of strain gradient crystal plasticity. 2. The continuum redistribution model consists of a set of equations, including a balance and constitutive equations, which lead to an evolution equation of the net defect density at the grain boundary and a Helmholtz equation that accounts for nonlocal spatial interactions within the interface.

35

3. The net defect evolution at a grain boundary material point is determined by: (i) flow of the net defect along the grain boundary and (ii) a source/sink term due to the in- and outflow of lattice dislocations at the grain boundary. Furthermore, defect content of opposite sign can annihilate in the redistribution. 4. The flow of the interface defect resulting in the redistribution is driven by the gradient of a chemical potential, which depends on local and nonlocal defect contributions to the grain boundary energy density. 5. The defect redistribution model, based on a vectorial quantity defined at a surface, has a diffusional character and shows similarities with phase field modelling approaches. 6. The complete framework is implemented numerically employing a staggered solution scheme in order to investigate the influence of the redistribution of the grain boundary defect content on the macroscopic response and the microscopic fields in the case of a copper bicrystal subjected to simple shear. 7. The resulting defect redistribution has a clear macroscopic and microscopic effect, which becomes stronger for increasing temperature, as a result of the increased mobility of the defect content, and for increasing nonlocal interface parameters, which enhance the nonlocal contribution to the redistribution. Although a quantitative comparison and validation at the continuum length scale remains a challenge, some qualitative agreement with small scale experiments and numerical simulations exists. Moreover, other processes that accompany the defect redistribution and the change of grain boundary structure, such as grain boundary sliding, grain boundary migration and grain rotation, are not taken into account in the current modelling framework. These processes mainly occur at higher temperatures at which other mechanisms in the bulk, such as dislocation climb, become thermally activated. This would also necessitate adjustments of the crystal plasticity model, since these mechanisms are not incorporated in the current bulk framework. In polycrystals, the presence of triple junctions poses additional constraints. It is expected that these mechanisms and features can change the microscopic and macroscopic behaviour considerably, the investigations of which are topics for future work. 36

Acknowledgements This research is supported by the Dutch Technology Foundation STW, which is the applied science division of NWO, and the Technology Program of the Ministry of Economic Affairs under Project Nr. 10109.

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