2.1. Compatibility Relations
In a continuous medium of infinite volume
V, an infinite planar grain boundary (GB) is considered which separates two crystals
I and
(
Figure 1). The unit vector
designates the normal to the interface and is assumed to be oriented from crystal
I towards crystal
along the direction
(
Figure 1). In what follows, superscripts
I and
correspond to crystals
I (
) and
(
), respectively. This bicrystal is supposed to have been deformed under the action of homogenous macroscopic mechanical loading applied on the external boundary
of
V and of homogeneous temperature variation
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
and the symmetric thermal expansion tensor
are isotherm and uniform in each crystal and that the plastic distortion tensor
depends only on
:
where
is the Heaviside unit step function. GBS is described through the distortion tensor
. Similarly to the expression of the plastic distortion tensor induced by the formation of a dislocation loop [
12,
13],
is singular on the interface and, for a GB with unit normal
, may be expressed as:
where the vector
represents the jump of displacement
at the interface due to sliding:
Moreover, considering the absence of interface decohesion implies
[
7]. The only non-zero components of
are thus
and
. As a consequence of all the preceding assumptions, the total distortion
may be written as follows:
with
the elastic distortion tensor. According to the orthogonal Stokes-Helmholtz decomposition [
14,
15,
16], it is possible to decompose
into compatible and incompatible parts:
such as:
In order to ensure the unicity of this decomposition, the following condition is considered on
[
14,
15,
16]:
where
is the unit normal of
. As a consequence of this decomposition, the Nye tensor
[
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),
is thus given as the unique solution of the following Poisson equation:
Then, by considering the compatibility of the total distortion (
), Equations (
12) and (
16) lead to:
By making the additional hypothesis that the gradients
are uniform,
depends only on
. The same is true for
since it is given as the unique solution of the Poisson Equation (
17). Hence, it is possible to write the elastic distortion as:
where
is a continuous vector such as
[
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
:
where
are spatially uniform constants. By definition, we have
, thus
and accordingly :
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
are indeed directly related to the gradients of
and
according to the relations:
2.2. Stress Jumps
Using the contracted notation of Voigt [
11], relation (
24) makes it possible to write the following system:
where
is the symmetrized gradient of
. The same engineering convention as for strain components is considered, i.e.,
. Considering the continuity of the traction vector at the boundary,
, and using the relation
, the system (
26) becomes:
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
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
the shear modulus and
the Poisson’s ratio, the expressions of the non-zero stress jump components are indeed written in this case as: