Incompatibility stresses and lattice rotations due to grain boundary sliding in heterogeneous anisotropic elasticity

Non-uniform grain boundary sliding can induce strain and rotation incompatibilities at perfectly planar interfaces. Explicit analytic expressions of stress and lattice rotation jumps are thus derived at a planar interface in the general framework of heterogeneous anisotropic thermo-elasticity with plasticity and grain boundary sliding. Both elastic fields are directly dependent on in-plane gradients of grain boundary sliding. It is also shown that grain boundary sliding is a mechanism that may relax incompatibility stresses of elastic, plastic and thermal origin although the latter are not resolved on the grain boundary plane. This relaxation may be a driving force for grain boundary sliding in addition to the traditionally considered local shears on the grain boundary plane. Moreover, the obtained analytic expressions are checked by different kinds of bicrystal shearing finite element simulations allowing grain boundary sliding and where a pinned line in the interface plane aims at representing the effect of a triple junction. A very good agreement is found between the analytic solutions and the finite element results. The performed simulations particularly emphasize the role of grain boundary sliding as a possible strong stress generator around the grain boundary close to the triple line because of the presence of pronounced gradients of sliding.


Introduction
Grain boundary sliding (GBS) is a specific deformation mechanism of polycrystals which is especially important during creep [1][2][3][4][5][6]. While GBS can act as a stress relaxation mechanism able to lead to superplasticity, it may also induce cavities at triple points and causes intergranular fracture initiation [3,[6][7][8]. GBS generally needs relatively high homologous temperature and/or relatively small grain sizes [1][2][3][4]. However, it is also sometimes observed with coarse grain sizes at room temperature, particularly in metals with hexagonal close-packed (HCP) structure [5][6][7]9,10]. Due to the low symmetry of the hexagonal lattice, HCP materials are known to be strongly anisotropic and thus to exihbit significant strain incompatibilities between adjacent grains. In those materials, the origin of GBS might hence be the need for an additional mechanism that can accommodate grain incompatibilities [6,7,9]. Some years ago, Mussot et al. [7] derived compatibility conditions at sliding smooth interfaces which showed that GBS modifies strain compatibility conditions as soon as GBS is non-uniform. The relevancy of this result is reinforced by the many experimental evidences indicating that sliding is usually non-uniform along a grain boundary [6,7,9]. Hence, gradients of GBS can induce or relax internal stresses. It must be noticed that the strain incompatibilities of which the study of Mussot et al. [7] referred to are different from the ones generated by uniform gliding of rough interfaces that are believed to control the rate at which GBS occurs [2] or the ones created at triple points that require accommodating processes like migration of the triple edges, diffusion along grain boundaries or dislocation activity in the neighboring grains [3].
The present paper follows the work of Mussot et al. [7] in order to extend strain compatibility conditions at a sliding interface, assumed perfectly planar, to the general context of heterogeneous anisotropic thermo-elasticity with plastic distortions. Moreover, explicit analytic expressions of incompatibility stresses and lattice rotations are looked for. The correctness of the obtained solutions are checked thanks to finite elements (FE) simulations of bicrystals shearing. In the grain boundary plane, a line is pinned in order to mimic the effect of a triple junction. Based on both the obtained theoretical expressions and numerical results, the dual role of grain boundary sliding, as a stress generator and as a stress relaxation mechanism, is then discussed.

Notation and Reduction Conventions
A vector is denoted x ≡ x i , a second-order tensor X ∼ ≡ X ij and a fourth-order tensor X ≈ ≡ X ijkl . The Einstein summation convention over repeated indices is used. The symbol represents tensor multiplication, a : the contracted product between two tensors and ∇, div and curl the gradient, divergence and curl operators, respectively. In rectangular Cartesian coordinates, we have in particular: where ∈ jlm is a component of the third-order alternating Levi-Civita tensor. The jump of a mechanical or material property field f at an interface between two crystals I and I I which is oriented by whereas the average of f across the interface is defined asf = 1 2 f I (x 2 → 0 − ) + f I I (x 2 → 0 + ) . Finally, the contracted Voigt notation [11] (11 → 1, 22 → 2, 33 → 3, 23 → 4, 31 → 5, 12 → 6) is adopted in Section 2.2. The contracted product between tensors A ≈ and B ∼ is then written as (A ≈ : B ∼ ) p = A pq B q where p and q take values ranging from 1 to 6. For consistency, an engineering convention is considered for strain components, i.e., ε 4 = 2ε 23 , ε 5 = 2ε 31 , ε 6 = 2ε 12 , and the Hooke's law in matrix notation is given as: where σ i are stresses, ε e i elastic strains and s ij components of the elastic compliances tensor that include the multiplying factors of 2 and 4 [11].

Compatibility Relations
In a continuous medium of infinite volume V, an infinite planar grain boundary (GB) is considered which separates two crystals I and I I (Figure 1). The unit vector n designates the normal to the interface and is assumed to be oriented from crystal I towards crystal I I along the direction x 2 (Figure 1). In what follows, superscripts I and I I correspond to crystals I (x 2 < 0) and I I (x 2 > 0), respectively. This bicrystal is supposed to have been deformed under the action of homogenous macroscopic mechanical loading applied on the external boundary ∂V of V and of homogeneous temperature variation ∆T and in the absence of body forces. Elasticity, plasticity and GBS are considered as deformation mechanisms in a static small strain setting. It is assumed that the elastic compliance tensor s ≈ and the symmetric thermal expansion tensor α T ∼ are isotherm and uniform in each crystal and that the plastic distortion tensor β p ∼ depends only on x 2 : where H(x 2 ) is the Heaviside unit step function. GBS is described through the distortion tensor β S ∼ .
Similarly to the expression of the plastic distortion tensor induced by the formation of a dislocation loop [12,13], β S ∼ is singular on the interface and, for a GB with unit normal n ≡ x 2 , may be expressed as: where the vector g represents the jump of displacement u at the interface due to sliding: Moreover, considering the absence of interface decohesion implies g 2 = 0 [7]. The only non-zero components of β S ∼ are thus β S 12 and β S 32 . As a consequence of all the preceding assumptions, the total distortion β ∼ may be written as follows: with β e ∼ the elastic distortion tensor. According to the orthogonal Stokes-Helmholtz decomposition [14][15][16], it is possible to decompose β e ∼ into compatible and incompatible parts: such as: In order to ensure the unicity of this decomposition, the following condition is considered on ∂V [14][15][16]: where m is the unit normal of ∂V. As a consequence of this decomposition, the Nye tensor α ∼ [17] (or dislocation density tensor), which may be defined as the curl of the elastic distortion [14], can be written solely from the incompatible part of the elastic distortion: As a result of Equations (14)- (16), β e ⊥ is thus given as the unique solution of the following Poisson equation: Then, by considering the compatibility of the total distortion (curl β (12) and (16) lead to: By making the additional hypothesis that the gradients g i,l are uniform, α ∼ depends only on x 2 .
The same is true for β e ⊥ ∼ since it is given as the unique solution of the Poisson Equation (17). Hence, it is possible to write the elastic distortion as: where z is a continuous vector such as ∇z = β e ∼ [14][15][16]. From relation (19), we have then: By considering again the compatibility of the total distortion, we can write: Combining this relation with Equations (12) and (20), we obtain: which becomes by integration with respect to x 2 : These relations are in agreement with the compatibility relations derived by Mussot et al. [7]. They show that GBS can induce strain and rotation incompatibilities as soon as the sliding is non-uniform. The terms g i,j(j =2) are indeed directly related to the gradients of β S 12 and β S 32 according to the relations:

Stress Jumps
Using the contracted notation of Voigt [11], relation (24) makes it possible to write the following system: where ∇ S g is the symmetrized gradient of g. The same engineering convention as for strain components is considered, i.e., ∇ S g 5 = g 1,3 + By introducing the notation: the system (27) becomes similar to the one obtained for the derivation of incompatibility stresses without GBS and thermal expansion in bicrystals or periodic-layered composites [18][19][20]. Therefore, stress jumps solutions can be directly expressed as: where the non-zero components of the tensor G ≈ are given by [19,20]: Equations (28)-(30) show that incompatibility stresses are directly dependent on in-plane gradients of GBS. It is seen that GBS is a mechanism capable of relaxing incompatibility stresses of elastic, plastic and thermal origin although the latter are not resolved on the grain boundary plane. This relaxation may be a driving force for GBS in addition to the traditionally considered local shears on the grain boundary plane.
It is noteworthy than the expressions of the stress jumps are much simpler if ones considers isotropic homogeneous thermo-elasticity properties [18]. With µ the shear modulus and ν the Poisson's ratio, the expressions of the non-zero stress jump components are indeed written in this case as:

Validation by FE Simulations
The correctness of the analytic formulas (28)-(32) are checked thanks to FE simulations performed with the software ABAQUS/standard. Two rectangular grains, α and β, of dimensions 120 × 90 × 60 are defined as two different parts ( Figure 2) and discretized into a regular mesh made of 173972 8-node linear brick elements (C3D8). The initial position of the upper face of the first grain matches exactly the position of the bottom face of the second grain and corresponds to x 2 = 0. The behavior is supposed to be purely elastic (no plasticity, no thermal expansion) with a possibility of GBS. Zn which has a HCP structure is considered as a reference material. The crystallographic orientations of the two grains are indicated in Figure 3. Six different simulations are performed. They have in common that the top (x 2 = 90) and bottom (x 2 = −90) faces are constrained to be planar (u 2 = constant), that the continuity of u 2 is enforced at the GB ([[u 2 ]] = 0) and that the line at x 1 = −60 and x 2 = 0 is pinned (u 1 = u 2 = u 3 = 0) ( Figure 2). This pinned line may represent a fixed GB junction like a triple line and aims to produce GBS gradients. In order to induce an effective contact between the two parts, a very small compression (ε 22 = 10 −6 ) is applied on the bicrystal external boundary in a first time step for all the simulations. Then, the following additional boundary and interface conditions are applied during a second time step of 1 s:  A classical crystal-elasticity UMAT subroutine is used to compute the displacements and the stresses (see for e.g., [18]). In these simulations, when GBS is allowed, it is completely free to develop (no friction, no critical stress), which corresponds to an extreme, not realistic, situation.

Discussion
In the absence of plasticity and thermal expansion, source of incompatibilities arise from heterogeneous elasticity and in-plane gradients of GBS which are zero in simulations 1 and 4. Figures 4 and 5 both show that, as one moves away from the pinned line, GBS increases but its gradient decreases. Depending on the shear loading direction and the relative position of the two grains α and β, significant differences in the GBS distribution are however observed. It must be noticed that [[ω e 13 ]] depends only on GBS gradients (Equation (32)) and that incompatibility stresses depend on two terms (Equations (28) and (29) (Tables 1-6), even though the theoretical derivation of the analytic model assumes uniform gradients of g. It seems that the agreement is much better when the contribution of GBS gradients on the elastic field jump increases. As a matter of fact, a perfect match is obtained for all the values of [[ω e 13 ]]. Besides, it is noteworthy that, within the present FE configuration, even the fulfillment of the traction vector continuity at the grain boundary is not always perfectly respected.
GBS clearly leads to a relaxation of the main shear stress component in the GB plane: σ 12 in simulations 2 and 3 to be compared with simulation 1 (Tables 1-3) and σ 23 in simulations 5 and 6 to be compared with simulation 4 (Tables 4-6). At the same time, as expected from the term −G ≈ : ∇ S g, incompatibility stresses get more and more pronounced when approaching the pinned line due to growing GBS gradients. This effect is retrieved when looking at the profiles of the Von Mises stress through the GB at different distances from the pinned line (Figures 6-11). Indeed, allowing GBS can induce very strong stress concentrations around GB compared to the case without GBS (Figures 6, 9 and 10) because, in these cases close to the pinned line, the amount of GBS is relatively small and its gradient quite strong. These consequent stresses should then be relaxed by other mechanisms like dislocations glide or climb [2]. On the other hand, far enough from the pinned line, GBS may appear as a relaxation mechanism since the Von Mises stress is much lower with than without GBS in Figures 7, 8 and 11. In these cases, the amount of GBS is relatively high and its gradient has become almost insignificant. As a consequence, the present simulations emphasize the dual role of GBS which can act both as a strong stress generator and as a stress relaxation mechanism along a same GB. It is important to notice that these effects are shown in the case of a perfectly planar interface with no GBS at the triple line, which is thus different from the GBS accommodation processes discussed in [2,3].
When inverting the positions of grains α and β, the tensor G  Tables 5 and 6). It is interesting to underline that a good approximation of the Von Mises stress at GB can then be obtained directly from the analytic terms −G ≈ : ∇ S g: For higher values of average stresses across GB, the expression (33) corresponds to a lower bound estimate of the Von Mises stress at GB.