## Abstract

A unified physically based microstructural representation of f.c.c. crystalline materials has been developed and implemented to investigate the microstructural behaviour of f.c.c. crystalline aggregates under inelastic deformations. The proposed framework is based on coupling a multiple-slip crystal plasticity formulation to three distinct dislocation densities, which pertain to statistically stored dislocations (SSDs), geometrically necessary dislocations (GNDs) and grain boundary dislocations. This interrelated dislocation density formulation is then coupled to a specialized finite element framework to study the evolving heterogeneous microstructure and the localized phenomena that can contribute to failure initiation as a function of inelastic crystalline deformation. The GND densities are used to understand where crystallographic, non-crystallographic and cellular microstructures form and the nature of their dislocation composition. The SSD densities are formulated to represent dislocation cell microstructures to obtain predictions related to the inhomogeneous distribution of SSDs. The effects of the lattice misorientations at the grain boundaries (GBs) have been included by accounting for the densities of the misfit dislocations at the GBs that accommodate these misorientations. By directly accounting for the misfit dislocations, the strength of the boundary regions can be more accurately represented to account for phenomena associated with the effects of the GB strength on intergranular deformation heterogeneities, stress localization and the nucleation of failure surfaces at critical regions, such as triple junctions.

## 1. Introduction

Statistically stored dislocations (SSDs) in crystalline aggregates subjected to inelastic deformations are generally organized into mosaic patterns consisting of regions of low dislocation density separated by regions of high dislocation density, which are also denoted as dislocation boundaries. In general, two types of dislocation boundaries, which subdivide the grains on two size scales, may result from the inelastic deformation of crystalline materials (e.g. Kuhlmann-Wilsdorf & Hansen 1991; Bay *et al.* 1992). At the smallest scale of approximately a few micrometres, a heterogeneous distribution of SSDs subdivides the grains into a cell-type microstructure of approximately equiaxed and low-density cells, separated by high-density walls also known as incidental dislocation boundaries (IDBs). At a larger scale, geometrically necessary boundaries (GNBs) form due to different active slip systems or different magnitudes of plastic slip among neighbouring regions of individual grains, which can lead to the division of grains into cell blocks (Hughes 2001). Dislocations stored in these boundaries are the geometrically necessary dislocations (GNDs) needed to accommodate lattice misorientations across the GNBs (e.g. Bay *et al.* 1992; Hansen *et al.* 2001; Hughes 2001). SSDs accumulate by the statistical trapping of dislocations during plastic slip (Kocks 1966). Therefore, they are randomly distributed and have no geometrical consequence. On the other hand, gradients of plastic deformation, due to material texture and loading conditions, result in the presence of GNDs (Ashby 1970), which are needed to preserve the lattice continuity through accommodating lattice misorientations.

Experimental analyses of the correlation between the slip pattern and the microstructure (Liu *et al.* 1998; Hansen *et al.* 2001) have indicated that some grain orientations (figure 1, region II) develop GNBs that contain Burgers vectors, which belong to one slip plane. These GNBs are commonly denoted as crystallographic boundaries (Hansen & Jensen 1999) and form when two coplanar active slip systems account for a large fraction of the plastic slip. Some other grain orientations (figure 1, region III) result in the formation of GNBs with Burgers vectors belonging to two active slip planes. These GNBs are denoted as non-crystallographic boundaries. It has been further observed (Liu *et al.* 1998) that crystallographic boundaries have a mixed tilt and twist characteristic, while the non-crystallographic boundaries have a predominantly tilt characteristic. Hence, crystallographic boundaries consist of both types of GNDs, namely edge and screw types, while non-crystallographic boundaries mostly comprise GNDs of edge type. Different microstructures that can develop from these different grain orientations are also shown in figure 1.

Crystal plasticity models have been incorporated within large-scale finite element models to obtain a more detailed understanding of material behaviour at the grain level (e.g. Zikry & Nemat-Nasser 1990; Becker *et al.* 1991; Becker & Panchanadeeswaran 1995; Zikry & Kao 1996; Dai 1997; Zikry & Kameda 1998; Bate 1999; Mika & Dawson 1999; Acharya & Beaudoin 2000; Arsenlis & Parks 2000; Beaudoin *et al.* 2000; Kumar & Dawson 2000; Meissonnier *et al.* 2001; Dawson *et al.* 2002; Ashmawi & Zikry 2003; Evers *et al.* 2004; Rezvanian *et al.* 2006). Most of these approaches have accounted for the effects of texture, geometrical softening and strain hardening.

Some of these crystalline formulations (Ashmawi & Zikry 2003; Evers *et al.* 2004) have been coupled to the evolution of dislocation densities. However, what has generally been lacking is an integrated approach that can account for the different interrelated dislocation densities within a unified framework, such that relevant microstructurally induced phenomena and mechanisms can be accurately predicted and understood for crystalline aggregates subjected to large inelastic strains.

In this study we present a physically based unified formulation of dislocation densities associated with SSDs, GNDs and grain boundary dislocations (GBDs), coupled to a multiple-slip crystalline plasticity formulation for f.c.c. materials. A specialized finite element analysis is then used with this formulation to track intergranular and intragranular microstructural evolution in a crystalline aggregate with Voronoi tessellated grain shapes, and to investigate how GB interfaces, grain interiors and collective grain interactions can affect the evolution of cellular, crystallographic or non-crystallographic microstructural types, grain subdivision and failure modes at critical locations, such as triple junctions.

This paper is organized as follows: in §2 multiple-slip crystal plasticity formulation and the microstructural evolution equations for the SSD, GND and GBD densities are presented; in §3 the computational approach is outlined; in §4 the representation of the polycrystalline aggregate model is introduced; the results are presented in §5; and the summary is given in §6.

## 2. Multiple-slip crystal plasticity formulation

The microstructural formulation for inelastic deformation and subdivision of crystalline grains is based on coupling multiple-slip crystal plasticity to three distinct dislocation density formulations, which pertain to SSDs with a heterogeneous distribution, GNDs and GBDs.

The crystal plasticity kinematics used here is based on that developed by Zikry & Kao (1996) and Kameda & Zikry (1998). The velocity gradient tensor *L* can be obtained from the deformation gradient tensor *F* and its rate of change as(2.1)

The velocity gradient can then be decomposed into its symmetric and skew-symmetric parts; the symmetric part being the deformation rate tensor *D* and the skew-symmetric part being the spin tensor *W*. The total deformation rate tensor and the total spin tensor can then be additively decomposed into elastic and plastic components as(2.2a)(2.2b)The superscript ‘*’ denotes the elastic component and superscript ‘p’ denotes the plastic component. The plastic components can be defined in terms of the crystallographic slip rates as(2.3a)(2.3b)where *α* is summed over all slip systems, and *P*^{(α)} and *ω*^{(α)} are defined as(2.4a)(2.4b) is the unit vector normal to the slip plane of slip system *α* and is the unit vector in the slip direction.

For rate-dependent materials, a general power-law relation for crystallographic slip rate can be stated as(2.5)where *m* is the strain-rate sensitivity parameter; is the reference strain rate; *k* is the Boltzmann constant; *T* is the absolute temperature; Δ*F* is the activation energy; *τ*^{(α)} is the resolved shear stress and is the threshold resistance to slip on system *α* that can be stated as(2.6)where *τ*_{y} is the static yield stress; *c* can range from 0.3 to 0.5; *G* is the shear modulus; and *b* is the magnitude of the Burgers vector. The obstacle density depends on how stored dislocations on every slip system interact with mobile dislocations on slip system *α* and can be quantified through a set of interaction coefficients *i*^{(αs)} (Franciosi & Zaoui 1982) as(2.7) is added to the obstacle density only for the GB elements.

Finally, the co-rotational stress rate tensor can be obtained as(2.8)where the elastic moduli *L*_{ijkl} can be defined in terms of Lame elastic constants.

### (a) Microstructural evolution for SSDs

In this section the flow kinetics and microstructural evolution equations are formulated for a cell-type microstructure that is representative of the heterogeneous distribution of SSDs, based on a modification of the microstructural model originally proposed by Nix *et al.* (1985), and the composite model of Mughrabi (1987) and Estrin *et al.* (1998). In a heterogeneous distribution of SSDs, a microstructure consists of high-density regions of cell walls with a local density of *ρ*_{w} separated by low-density cell interiors with a local density of *ρ*_{c}.

#### (i) Microstructural evolution in the cell interiors and cell walls

The net rate of change of the SSD density in the cell interiors for slip system *α* can be stated in terms of the sum of a storage rate that represents the statistical storage of mobile dislocations and a recovery rate that represents the cross-slip of screw dislocations of opposite signs stored in the cell interiors (Rezvanian *et al.* 2006) as(2.9)where is the crystallographic slip rate of cell interiors on slip system *α*. is the mean free path for the mobile dislocations on slip system *α* and can be stated as(2.10)where *C* is a material constant and *h*^{αs} quantifies the mutual immobilization effects of the slip systems. *A* in equation (2.9) depends on the frequency of the cross-slip attempts and the length of a potential site for such an attempt, *w*_{f} is the forward activation energy for cross-slip attempts and *w*_{b} is the backward activation energy, which is assumed to be equal to *w*_{cs}, that is the activation enthalpy of cross-slip. The forward activation energy can be stated as(2.11)where is a portion of *τ*^{(α)} that acts over the cell interiors on slip system *α*.

Assuming that edge dislocations become stored in the cell walls as mobile screw dislocations glide in the cell interiors, and that the recovery within the cell walls occurs by diffusion-controlled climb and the annihilation of oppositely signed edge dislocations, the resultant rate of change of density of the SSDs in the cell walls becomes (Rezvanian *et al.* 2006)(2.12)where *δ*_{w} is the cell wall thickness and *D*_{L} is the lattice diffusivity. Thick fuzzy cell walls are produced by low-temperature deformations, and thin orderly walls by high-temperature deformations (Nix *et al.* 1985). In this study the loading is quasi-static at room temperature. Therefore, it is reasonable to assume that the cell wall thickness remains constant. The crystallographic slip rates of cell interiors , and the resolved shear stresses acting on cell interiors , need to be determined before the dislocation rates can be obtained.

#### (ii) Heterogeneous microstructure

The mechanics of a heterogeneous microstructure can be used to derive relations between the local microstructural variables (, , , ), and the microstructural variables at the slip system level (, , , ). In addition to the information that is obtained in this scheme in regard to the local microstructural variables, it is also shown that the formulation can be extended for each slip system that represents the average distance along the slip direction between the high-density cell walls.

Assuming that the total strain rate is the same in the cell walls and cell interiors, and that it is equal to the slip system average strain rate , a strain rate equilibrium can be written as(2.13)where is the slip rate corresponding to the cell walls and is the slip rate corresponding to the cell interiors on slip system *α*. /*G* and /*G* are the corresponding local elastic strain rates. Equilibriums of equation (2.13) will be solved for and , because is known from equation (2.5) and since the local crystallographic slip rates will have the same form as equation (2.5); therefore, and will be known as well. It should be noted that and are initially equal to *τ*^{(α)}, which means that the SSDs are initially uniformly distributed. The local SSD densities, and , are also initially equal. However, they grow with different rates, with growing at a higher rate as indicated by equation (2.12), and therefore, the local components of slip resistance, and , will also increase with different rates. and have the general form of equation (2.6) as(2.14)The average density of SSDs used in equation (2.7) can be related to and as(2.15)where is the average fraction of the cell interiors on slip system *α*. Since the local fractions must sum to unity, the fraction of the cell walls on slip system *α* is .

Stress equilibrium can then be used to obtain , where there must be a balance between the overall slip system resolved shear stress *τ*^{(α)} and its local components as(2.16) can then be stated in terms of the cell wall thickness , which is assumed to remain constant at room temperature, and the average distance between the cell walls can be regarded as the average dislocation cell size as(2.17)With the cell wall thickness being constant, the change in the fraction of the high SSD density regions on a slip plane is due to the formation of new cell walls in the cell interiors, which would also cause the dislocation cells to become smaller as the deformation evolves.

### (b) Representation of the GNDs

In the plastic deformation of crystalline materials, the mismatch of slip at the boundaries of the grains or intragranular regions can result in gradients of plastic strain. This subsequently results in the formation of GNDs, which are required for the compatible deformations of different parts of the grains (Hughes & Hansen 1997). Investigations of the relationship between plastic strain gradients and the density of dislocations required to accommodate the geometrical consequences of such a gradient originated with the classical work of Nye (1953). Nye tensor provides a direct measure of the number of GNDs from the plastic strain gradients. Plastic strain gradients appear either as the result of the geometry of the loading or owing to the material's anisotropic plastic properties. The GND densities can be determined from the gradients of the crystallographic slips. Furthermore, the type and the sign of the GNDs can also be known.

Gradients of the plastic strain along the slip direction of slip system *α* can result in GNDs of edge type with density of *ρ*_{G,e} to form on the slip system *α* as(2.18)

The plastic strain gradients along the transverse direction defined as , where is slip system's normal direction, can result in GNDs of screw type with density of *ρ*_{G,s} to form on the slip system *α* as(2.19)where *α*′ is the collinear system of slip system *α*. Since the screw dislocations are assumed to be free to cross-slip between collinear systems, their density on each slip system is determined from both of the planes they can glide on. The plastic strain gradients in the normal direction do not contribute to GND formation (Ashby 1970).

Since GND polarity has no effect on their role as short-range barriers, the total GND density used in equation (2.7) can be stated as(2.20)

### (c) Representation of the GBDs

Low-angle GBs (misorientations of less than 10–15°) can normally be considered as a collection of dislocations. As misorientations across the GBs exceed the limit below which boundaries are considered to be low angle, the spacing between misfit dislocations becomes small enough that individual misfit dislocations can no longer be physically characterized. Low-angle boundaries contain large areas of good fit separated by misfit dislocations. Hence, the density of the GB misfit dislocations can be approximated by accounting for geometrical considerations to maintain lattice continuity across the boundaries (Evers *et al.* 2004).

As shown in figure 2, the magnitude of the resolved Burgers vector *b*^{GB} of a slip system at the GB interface is *b*/cos(*α*_{2}), where *α*_{2} is supplementary for the angle between the unit vector normal to the GB plane and the normal to the slip plane (figure 2; *α*_{1}). Therefore, can be obtained as(2.21)

As shown in figure 3, the misfit length *h* between a slip system of grain 1 and its corresponding system in grain 2 is related to the magnitude of their resolved Burgers vectors as(2.22)where the subscript A1 indicates slip system A in grain 1 and A2 indicates its match in grain 2. With *h* representing the spacing between the misfit dislocations at the GB for slip systems A1 and A2, the density of the misfit dislocations for these two systems is(2.23)Using equation (2.21), equation (2.23) can be restated as(2.24)

Using the symmetries of an f.c.c. lattice, there are 12 possible configurations for slip systems of one grain to pair with slip systems of an adjacent grain at the GB. In low-angle GBs, the density of the misfit dislocations is proportional to the boundary energy, and hence the stable configuration would be the one with the smallest total GBD density.

## 3. Finite element computational approach

The finite element method used here is based on the modification of the method developed by Zikry (1994) for multiple-slip dislocation density-based crystal plasticity. The deformation rate tensor and the plastic part of it are needed to update the material stress state. An implicit finite element method is used to obtain the total deformation rate tensor. Nodal displacements are obtained by the quasi-Newton BFGS method. In this investigation, planar deformations will be investigated, and hence four-node quadrilateral elements with bilinear shape functions will be used.

By knowing the nodal displacements, the deformation gradient tensor is computed, and the velocity gradient tensor can then be obtained. Once the velocity gradient tensor is known, the total deformation rate tensor and the total spin tensor can be updated. To obtain the inelastic components of these two tensors, the density of the dislocations pertaining to the microstructure has to be updated.

The density of the SSDs is obtained by solving the governing ordinary differential equations of evolution using a fifth-order Runge–Kutta method. The GND densities are determined from the gradient of the crystallographic slip. For each slip system, the nodal values of shear slip are obtained by interpolation from the integration points. These nodal values of shear slip are then used to calculate the shear slip gradients at the integration points. It should also be noted that the updated resolved shear stresses are the effective resolved shear stress that is obtained by adding the back stress tensor to the applied stress tensor. The gradients of the GND densities, which are needed to update the resolved back stresses, are calculated in the same manner that was used to obtain the plastic slip gradients. To determine the GBD densities at the boundary elements, the set of the 12 slip systems in every GB element is matched with the slip systems of its neighbouring element in 12 different ways. The desired configuration is the one that results in the least amount of dislocation density. For every boundary element, at each increment, all the possible configurations are analysed and the densities are calculated, from which the smallest one is taken as the density at that GB element (Rezvanian 2006).

Voronoi tessellation is used to generate two-dimensional grains. Starting with a set of initial points that are randomly distributed over the domain of the sample, Voronoi tessellation (e.g. Aboav 1970; Weyer *et al.* 2002) is used to assign a polygon to each initial point. A highly refined mesh is needed in the GB regions to accurately represent the GBD densities and the local strains and stresses. Therefore, a boundary region is generated on each side of every boundary line with a specified width by projecting the vertices of each polygon by a distance along the bisectors of their corresponding angles. An inner polygon is then created from these projected vertices. The area between the inner and outer polygons is the boundary region, meshed with the desired refinement for numerical accuracy (Rezvanian 2006).

## 4. Polycrystalline aggregate model

A 50-grain polycrystal sample of 500 by 500 μm was generated by Voronoi tessellation with random initiation points. Based on a convergence analysis for the grain interiors and the GB regions, 1800 quadrilateral elements were used in this analysis. The geometry and loading and boundary conditions are shown in figure 4. The bottom boundary is constrained in the *z*-direction, and the left boundary is constrained in the *y*-direction. A quasi-static displacement loading with a strain rate of 0.0005 s^{−1} is applied on the top boundary along the *z*-direction. The material properties of the grains, which are representative of polycrystalline pure copper, and the material properties used in the crystal plasticity formulation are given in table 1. The grain orientations are selected in such a way that 20% of the grains correspond to region I of the inverse pole figure, 56% correspond to region II and 24% to region III (figure 5). This texture distribution results in random low-angle GBs for the polycrystalline aggregate. The slip systems and their corresponding slip planes and slip directions are given in table 2.

## 5. Results and discussion

As noted earlier, polycrystalline aggregates can subdivide into smaller microstructural constituents when subjected to inelastic deformations. At the smallest scale, there are dislocation cells, and at the larger scale, GNBs can form cell blocks that surround the dislocation cells. This subdivision is mainly due to crystallographic texture that leads to intercrystalline inhomogeneities. Since deformations are assumed to be compatible at the GBs, at least for athermal deformations where there is no GB sliding, intragranular deformation inhomogeneities can arise that eventually result in gradients of the plastic slip. This, in turn, results in intragranular texture that leads to intragranular lattice misorientations. GNBs are then needed to preserve lattice continuity, if there are no pre-existing failure surfaces. The type of the GNBs then determines the type and the composition of the local microstructures that develop throughout the aggregate.

In this study the inelastic deformations of an f.c.c. crystalline aggregate with random low-angle grain boundaries (misorientations of less than 15°) are investigated to illustrate how and why different types of heterogeneous microstructures form, and how and where microstructurally induced inelastic failure modes can initiate and propagate.

Under an applied quasi-static tensile loading, the deformation is dominated by three slip systems: 9 ; 10 ; and 12 . The magnitude of the shear slip corresponding to each of these slip systems is shown in figure 6 at a nominal strain of 20%. It should also be noted that slip systems 10 and 12 are coplanar.

For these three active slip systems, the contours of density of the GNDs of edge type are shown in figure 7, and those of GNDs of screw type are shown in figure 8 at a nominal tensile strain of 20%. The analysis of these two sets of contours indicates that where non-coplanar systems, which are systems 9 and 10 or systems 9 and 12, have considerable GND densities of edge type (figure 7, dashed circle), the GND densities of screw type are significantly lower (figure 8, dashed circle). This is consistent with the experimental observations of Liu *et al.* (1998) and Hansen *et al.* (2001) that indicate where Burgers vectors of the GNBs belong to two slip planes (figure 1, non-crystallographic GNBs), the corresponding GNDs are dominantly edge dislocations. Hence, it can be shown from the model where the non-crystallographic microstructures develop and also specify the type of dislocations that comprise them (figure 7, dashed circle).

Furthermore, it can also be seen from figures 7 and 8 that at regions where the two coplanar systems account for a significant portion of the GND accumulation of edge type (figure 7, full circle), a GND density of screw type of approximately the same magnitude is present (figure 8, full circle). This is also consistent with the slip system analysis of the experimental results of Liu*et al.* (1998) and Hansen *et al.* (2001) that indicate where Burgers vectors of the GNBs belong to two coplanar systems (figure 1, crystallographic GNBs), the corresponding GNDs have a mix character of both edge and screw dislocations. Hence, based on these predictions, it can be where the crystallographic microstructures develop. The type of dislocations associated with these microstructures can also be predicted (figure 7, full circle).

The dotted circles in figures 7 and 8 exemplify regions with low GND density that implies a microstructure consisting of mainly dislocation cells (as shown in figure 1).

Contours of the stored dislocation density in dislocation cell interiors and cell walls for the three active slip systems are shown in figures9 and 10, respectively. A comparison of the distribution of the SSDs in the cell interiors and cell walls with the distribution of the accumulated plastic slips shown in figure 6 exhibits a similar pattern, which indicates that the formation and evolution of dislocation cells and walls is strongly dependent on the plastic slip. At 20% nominal strain, densities of stored dislocations in cell interiors range from about as low as their initial density of 1.0×10^{10} m^{−2} to above 1.0×10^{13} m^{−2}, and they attain local maximum values of up to 6.7×10^{14} m^{−2}. As expected, the stored dislocations in the cell walls have higher densities, but their distribution across the aggregate is similar to that of the stored dislocations in the cell interiors. Their local maximum densities on the three active slip systems are as high as 3.95×10^{15} m^{−2}. It should be noted that the SSDs are initially uniformly distributed and develop heterogeneity as the inelastic deformations increase. The dislocation cell size contours in figure 11 also show a similar dependence on the distribution of the accumulated plastic slips. As the density of stored dislocations in the cell interiors increases, new cell walls start to form, and therefore, the average size of the cells decreases. At 20% nominal strain, the average distance between dislocation walls on slip system 10 has decreased to less than 1 μm, while on slip systems 9 and 12 the average cell wall spacings are dominated by regions of larger dislocation cells with dimensions of greater than 2 μm.

The lattice misfit at the GBs results in the misfit dislocations at the boundary planes. The high density of misfit dislocations makes the GBs more resistant to slip than the grain interiors and helps a heterogeneous pattern of stress distribution and texture to develop across the aggregate. The densities of the misfit dislocations at the GB elements are shown in figure 12. The initial GBD densities for the initial distribution of the grains are shown in figure 12*a*, and the GBD densities at 20% nominal strain are shown in figure 12*b*. As seen from these contours, the GBD densities are approximately four to six orders of magnitude greater than the densities of the other types of stored dislocations.

Lattice rotations throughout the deformation cause lattice misfits at the GBs to change. At a nominal strain of 20%, these lattice rotations decrease the lattice mismatch at some GBs (see the GBs marked by dotted ovals in figure 12*a*,*b*), while some other GBs have an increase in lattice mismatch (see the GBs marked by dashed ovals in figure 12*a*,*b*), and some GBs remain unchanged. Quantitatively, a decrease of approximately 5% is seen in the total misfit dislocations (figure 12*c*). Since the energy of low-angle GBs is known to be directly related to its dislocation content, this decrease can be indicative of a collective grain deformation and lattice rotation behaviour that results in more stable configurations for GBs.

At 20% nominal strain, lattice rotations of up to 30° have occurred in the aggregate (figure 13). The positive lattice rotations (anticlockwise) are given in figure 13*a*, and the negative lattice rotations (clockwise) are given in figure 13*b*. The intragranular regions, which have rotated in opposite directions or with different magnitudes, are indicative of grain subdivision at a larger scale than that of the dislocation cells (see the regions denoted by dotted lines in figure 13*a*,*b*). The GNDs will form at the newly developed boundaries by the lattice rotations (Hughes 2001). These gradients of lattice rotations result in plastic slip gradients and the accumulation of GND densities.

The normal stresses (along *z*-direction) normalized by the yield stress are shown in figure 14*a* at a nominal strain of 20%. Owing to the contribution of the GB misfit dislocation densities to the flow stress of the boundary elements, the GBs are more resistant to plastic deformation than the grain interiors. As a result, since no separation is allowed to initiate and develop, the stresses increase significantly in these interfacial boundary regions in comparison with the grain interiors. At 20% nominal strain, the normal stresses in the grain interiors have attained maximum values of about four times the yield stress, while in the GB regions the maximum values of the normal stresses are approximately more than 10 times the yield stress. It can also be seen that high stress gradients occur at some triple junctions (figure 14*b*), which is an indication that triple junctions can be potential sites for crack initiation.

## 6. Summary and conclusions

A unified dislocation density-based multiple-slip crystal plasticity formulation that integrates SSDs, GNDs and GBDs in one framework has been developed and coupled to specialized finite element analysis for polycrystalline aggregates with physically realistic grain shapes. The inelastic finite strain deformation of a pure copper polycrystalline aggregate with randomly distributed low-angle GBs was then investigated to provide significantly improved understanding of how crystalline microstructures develop, and in what ways they can generally affect the large strain behaviour of f.c.c. polycrystalline aggregates.

GND densities were obtained from the gradients of the plastic slip. Through comparison with experimental observations, it was shown that a slip system analysis together with the distribution of the different types of the GNDs can be used to predict where crystallographic, non-crystallographic or cellular microstructures form and the nature of their dislocation composition.

The formulation for SSD densities, based on a cell-type dislocation microstructure, was used to predict how different microstructural variables, which are associated with the inhomogeneous distribution of the SSDs, evolve throughout the aggregate. Based on these results, and as it has been experimentally observed, the cell walls attain higher densities than the cell interiors. It was also predicted that the decreases in the size of dislocation cells are inversely related to increases in the accumulated plastic slip, which is also consistent with experimental measurements and observations.

Grain subdivision was tracked at different scales. Predicted cell wall spacings provided quantitative insights on how the cellular microstructure reduces in size as it evolves across the aggregate. While dislocation cells are indicative of grain subdivision at the smallest scale, unequal lattice rotations of grain interiors clearly indicate how crystalline grains subdivide at a larger scale into crystallites or smaller grains.

Lattice mismatches at the GB planes were taken into account by determining the densities of the misfit dislocations that arise due to the lattice misorientations at the low-angle GBs. An accurate representation of the strength of the GB regions could then be obtained by the incorporation of the GBD densities at the GB regions. The densities of the misfit dislocations at the GBs were approximately four to six orders of magnitude higher than other stored dislocations. As the deformation evolved, the total dislocation density at the GB regions decreased by approximately 5% in comparison with the original total dislocation density at the GBs. The decrease was due to the collective grain deformation and lattice rotation behaviour that results in more stable GB configurations.

Physically realistic grain shapes, texture, and GB misorientations and distributions in combination with the developed microstructural framework also provide a detailed understanding of how the heterogeneous stress and strain distributions develop at different scales throughout the aggregate, and at critical GB locations, such as triple junctions, which are highly susceptible to microcrack formation and material failure.

## Acknowledgments

The authors gratefully acknowledge the support from the Army Research Office, grant DAAG55-98-D-003.

## Footnotes

- Received May 6, 2007.
- Accepted July 12, 2007.

- © 2007 The Royal Society