Elastic Green’s Function in Anisotropic Bimaterials Considering Interfacial Elasticity

Abstract

The two-dimensional elastic Green’s function is calculated for a general anisotropic elastic bimaterial containing a line dislocation and a concentrated force while accounting for the interfacial structure by means of a generalized interfacial elasticity paradigm. The introduction of the interface elasticity model gives rise to boundary conditions that are effectively equivalent to those of a weakly bounded interface. The equations of elastic equilibrium are solved by complex variable techniques and the method of analytical continuation. The solution is decomposed into the sum of the Green’s function corresponding to the perfectly bonded interface and a perturbation term corresponding to the complex coupling nature between the interface structure and a line dislocation/concentrated force. Such construct can be implemented into the boundary integral equations and the boundary element method for analysis of nano-layered structures and epitaxial systems where the interface structure plays an important role.

Appendices

Appendix A: Complex Variable Methods in Elasticity

In this appendix we outline several concepts of complex variable methods that are commonly used in linear elasticity problems and also in this manuscript. The goal of this appendix is to provide the reader with enough background to be able to interpret solutions derived in the present manuscript. A full description of complex variable techniques is beyond the scope of this appendix and manuscript, but can be found in many linear elasticity textbooks [28].

Let $S$ be an arbitrary point set in the complex plane, a complex function ${\boldsymbol{\theta}} ( z )$ can be defined as

$$ {\boldsymbol{\theta}} ( z ) = \boldsymbol{u} ( x,y ) + {\mathrm{i}}\boldsymbol{v} ( x,y ) , $$

(64)

at a point $z = x + {\mathrm{i}}y$, where $u$ and $v$ are real valued functions of $x$ and $y$.

The complex derivative ${\boldsymbol{\theta}}' ( z )$ is defined as

$$ {\boldsymbol{\theta}}' ( z ) = \frac{\partial {\boldsymbol{\theta}}}{ \partial z}, $$

(65)

with

$$\begin{aligned} 2\frac{\partial{\boldsymbol{\theta}}}{\partial z} =& \biggl( \frac{ \partial}{\partial x} - {\mathrm{i}} \frac{\partial}{\partial y} \biggr) ( \boldsymbol{u} + {\mathrm{i}}\boldsymbol{v} ) = \boldsymbol{u}_{,x} + \boldsymbol{v}_{,y} -{\mathrm {i}} ( \boldsymbol{u} _{,y} - \boldsymbol{v}_{,x} ) , \end{aligned}$$

(66)

$$\begin{aligned} 2\frac{\partial{\boldsymbol{\theta}}}{\partial\overline{z}} =& \biggl( \frac{\partial}{\partial x} + {\mathrm{i}} \frac{\partial}{ \partial y} \biggr) ( \boldsymbol{u} + {\mathrm{i}}\boldsymbol{v} ) = \boldsymbol{u}_{,x} - \boldsymbol{v}_{,y} + {\mathrm{i}} ( \boldsymbol{u}_{,y} + \boldsymbol{v}_{,x} ) . \end{aligned}$$

(67)

Holomorphic function: A function ${\boldsymbol{\theta}} ( z )$ is said to be holomorphic in a region $S$ if it is single valued in $S$ and its complex derivative ${\boldsymbol{\theta}}' ( z )$ exists at each point of $S$.

Conjugate of an holomorphic function: If ${\boldsymbol {\theta}} ( z )$ is an arbitrary holomorphic function defined in $S$, then for certain regions of $S$ it is possible to use this function to define an associated complex function which is also holomorphic in the region which is the image of $S$ in its boundary. In other words, the complex function ${\overline{\boldsymbol{\theta} ( \overline{z} ) }}$ is holomorphic in the image region of $S$.

Continuation theorem: Suppose two holomorphic complex functions ${\boldsymbol{\theta}}_{+} ( z )$ and ${\boldsymbol {\theta}} _{-} ( z )$ defined in regions $S_{+}$ and $S_{-}$ respectively. Suppose $S_{+}$ and $S_{-}$ intersect in a domain $S$ for which $\forall z_{s} \in S$

$$ {\boldsymbol{\theta}}_{+} ( z_{s} ) = {\boldsymbol{ \theta}} _{-} ( z_{s} ) . $$

(68)

Then, the complex function ${\boldsymbol{\theta}} ( z )$,

$$ \boldsymbol{\theta} ( z ) = \left \{ \textstyle\begin{array}{rcr} {\boldsymbol{\theta}}_{+} ( z ) ,\quad z \in S_{+} \\ {\boldsymbol{\theta}}_{-} ( z ) ,\quad z \in S_{-}, \\ \end{array}\displaystyle \right . $$

(69)

is holomorphic in the union of $S_{+}$ and $S_{-}$, with ${\boldsymbol {\theta}} _{+} ( z )$ being the analytical continuation of ${\boldsymbol{\theta}} _{-} ( z )$ into $S_{+}$ and ${\boldsymbol{\theta}}_{-} ( z )$ being the analytical continuation of ${\boldsymbol{\theta}}_{+} ( z )$ into $S_{-}$.

Liouville’s theorem: If the complex function ${\boldsymbol {\theta}} ( z )$ is holomorphic and single valued in the whole plane including points at infinity, then ${\boldsymbol{\theta}} ( z )$ is a constant.

Cauchy’s theorem: If the complex function ${\boldsymbol {\theta}} ( z )$ is holomorphic in the simply connected region enclosed by a contour $C$ and is continuous on $C$, then

$$ \int_{C} \boldsymbol{\theta} ( z ) dz= 0. $$

(70)

Translating technique: In linear anisotropic elasticity, for bimaterials problems like the one treated in this manuscript, the general solution is generally expressed as follows:

$$ \left. \begin{aligned} {\mathbf{u}}_{+} &= \boldsymbol{A}_{+}\boldsymbol{f}_{+} ( z ) + \overline{ \boldsymbol{A}}_{+}\overline{\boldsymbol{f}_{+} ( z ) } \\ {\boldsymbol{\varPhi}}_{+}&=\boldsymbol{B}_{+} \boldsymbol{f}_{+} ( z ) + \overline{\boldsymbol{B}}_{+} \overline{\boldsymbol{f}_{+} ( z ) } \end{aligned} \right\}\quad z\in S_{+}, $$

(71)

and

$$ \left. \begin{aligned} {\mathbf{u}}_{-} &= \boldsymbol{A}_{-}\boldsymbol{f}_{-} ( z ) + \overline{ \boldsymbol{A}}_{-}\overline{\boldsymbol{f}_{-} ( z ) } \\ {\boldsymbol{\varPhi}}_{-}&=\boldsymbol{B}_{-} \boldsymbol{f}_{-} ( z ) + \overline{\boldsymbol{B}}_{-} \overline{\boldsymbol{f}_{-} ( z ) } \end{aligned} \right\}\quad z\in S_{-}. $$

(72)

When invoking the method of analytical continuation to solve Eq. (71) and Eq. (72), an implicit solution is expressed as a function of a new complex analytical potential (which is based the boundary conditions relationships) generally defined in the form

$$ {\boldsymbol{\varPsi}} ( z ) = \bigl[{\boldsymbol{\varPsi}}_{1} (z ), { \boldsymbol{\varPsi}}_{2} (z ), {\boldsymbol{\varPsi}}_{3} (z ) \bigr]^{T} = {\boldsymbol{C}}\bigl\langle g_{\alpha} ( z ) \bigr\rangle {\boldsymbol{q}}, $$

(73)

where ${\boldsymbol{C}}$ and ${\boldsymbol{q}}$ are a complex matrix and complex vector respectively. The function $g_{\alpha} ( z )$ is a complex function, and the complex argument $z$ has the form $z_{\alpha}=x_{1}+\mu x_{2}$ without indicating the subscript of $\mu$. The angular bracket $\langle\cdot\rangle$ denotes the diagonal matrix in which each component is varied according to its subscript, e.g., $\langle g_{\alpha}\rangle= \text{diag} [ g _{1}, g_{2}, g_{3} ]$. Within the translating technique [36], the explicit full domain solution can be obtained based upon a mathematical operation such that

$$ {\boldsymbol{\varPsi}} ( z ) = \sum_{k=1}^{3} \bigl\langle g_{k} ( z_{\alpha} ) \bigr\rangle {\boldsymbol{C}} {\boldsymbol{I}}_{k} {\boldsymbol{q}}, $$

(74)

where

$$ {\boldsymbol{I}}_{1} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} ,\qquad {\boldsymbol{I}}_{2} = \begin{bmatrix} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{bmatrix} ,\qquad {\boldsymbol{I}}_{3} = \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix} . $$

(75)

After finding the implicit solution and the explicit solution by using the translating technique, the explicit full field solutions for the displacements and stresses can then be obtained by the general solution shown in Eq. (71) and Eq. (72).

Appendix B: Case Study: Transverse Interfacial Compliance

As an illustration of the general formulation presented in the subsection above, we consider in this subsection a special case in which the interface Poisson’s behavior is neglected (i.e., $\mathbb{H} ^{S}_{II}=\boldsymbol{0}$) and the transverse interfacial compliance is the only interfacial property considered (i.e., $\varLambda^{\bot} \neq\boldsymbol{0}$), such that the interfacial coupling tensor $\mathbb{M}_{+}$ reduces to

$$ \mathbb{M}_{+} = {\mathrm{i}} \bigl[ \boldsymbol{\varLambda}^{\bot} \bigr] ^{-1} \boldsymbol{H}. $$

(76)

Thus, Eqs. (41)–(42) can be rewritten as

$$\begin{aligned} \boldsymbol{H} \boldsymbol{\delta}\boldsymbol{\xi}_{+} ( z ) - { \mathrm{i}}\boldsymbol{\varLambda}^{\bot} \boldsymbol{\delta} \boldsymbol{ \xi}'_{+} ( z ) = & {\mathrm{i}} \boldsymbol{ \varLambda}^{\bot} \bigl[ \boldsymbol{I} + \boldsymbol{H} ^{-1} ( \overline{\boldsymbol{M}_{+}} - \overline{ \boldsymbol{M}_{-}} ) \bigr] \overline{\boldsymbol{B}_{+}} \overline{ \boldsymbol{f}''_{0} ( \overline{z} ) },\quad z \in S_{+}, \end{aligned}$$

(77)

$$\begin{aligned} \overline{\boldsymbol{H}} \boldsymbol{\delta} {\boldsymbol{\xi}}_{-} ( z ) + {\mathrm{i}}\boldsymbol{\varLambda}^{\bot} \boldsymbol{\delta} { \boldsymbol{\xi}}'_{-} ( z ) = & - {\mathrm{i}} \boldsymbol{\varLambda}^{\bot} \overline{\boldsymbol{H} ^{-1}} ( \boldsymbol{M}_{+} + \overline{\boldsymbol {M}_{+}} ) { \boldsymbol{B}_{+}} {\boldsymbol{f}''_{0}} ( {z} ) ,\quad z \in S_{-}. \end{aligned}$$

(78)

In this specific case, the general solution associated with the homogeneous problem can be expressed as $\boldsymbol{\delta} \boldsymbol{\xi}_{\pm,gen} ( z ) =\boldsymbol{v}e^{\mp {\mathrm{i}}\nu^{\pm}z}$ where $\boldsymbol{v}$ and $\nu^{\pm}$ are complex vectors and a complex scalars respectively. Consequently, solving Eqs. (77)–(78) is equivalent to find the complex scalars $\nu$ that satisfy the following condition

$$ \begin{aligned} \det \bigl( \boldsymbol{H} - \nu^{+} \boldsymbol{\varLambda}^{\bot} \bigr) = {0}~ \quad \text{for}~S_{+}, \\ \det \bigl( \overline{\boldsymbol{H}} - \nu^{-} \boldsymbol{ \varLambda}^{\bot} \bigr) = {0}\quad \text{for}~S_{-}, \end{aligned} $$

(79)

with $\nu^{+}=\overline{\nu^{-}}$. If we assume that the diagonal terms of the real diagonal matrix $\boldsymbol{\varLambda}^{\bot}$ are all strictly positive [30, 37], the eigenvalue problems represented in Eq. (79) is equivalent to

$$ \begin{aligned} \bigl[ \bigl( \boldsymbol{ \varLambda}^{\bot} \bigr) ^{-1/2} \boldsymbol{H} \bigl( \boldsymbol{\varLambda}^{\bot} \bigr) ^{-1/2} - \nu^{+} \boldsymbol{I} \bigr] \bigl( \boldsymbol{\varLambda}^{\bot } \bigr) ^{1/2} \boldsymbol{v} = \boldsymbol{0}, \\ \bigl[ \bigl( \boldsymbol{\varLambda}^{\bot} \bigr) ^{-1/2} \overline{ \boldsymbol{H}} \bigl( \boldsymbol{\varLambda}^{\bot} \bigr) ^{-1/2} - \nu^{-} \boldsymbol{I} \bigr] \bigl( \boldsymbol{ \varLambda}^{\bot } \bigr) ^{1/2} {\boldsymbol{v}} = \boldsymbol{0}. \end{aligned} $$

(80)

Since the second-order tensor $\boldsymbol{H}$ is Hermitian and positive-definite, one can show [13] that the $3\times3$ matrix $( \boldsymbol{\varLambda}^{\bot} ) ^{-1/2} \boldsymbol{H} ( \boldsymbol{\varLambda}^{\bot} ) ^{-1/2}$ is also Hermitian and positive-definite. As such, a possible eigen-decomposition of this matrix (through a unitary matrix denoted $\boldsymbol{\varOmega}$) provides a simple expression for the eigenvalues of Eq. (79) such that

$$ \langle\nu_{*} \rangle= \bigl[ \bigl(\boldsymbol{\varLambda}^{\bot} \bigr)^{-1/2} \overline {\boldsymbol{\varOmega}} \bigr]^{T} \boldsymbol{H} \bigl[ \bigl( \boldsymbol{\varLambda}^{\bot} \bigr) ^{-1/2} \boldsymbol{\varOmega} \bigr] . $$

(81)

In the special case where the two media are both isotropic and assuming the interface also isotropic, the bimaterial tensor $\boldsymbol{H}$ and interfacial tensor $\boldsymbol{\varLambda}^{\bot}$ can be simply expressed [1, 37] as

$$ \boldsymbol{H}= \begin{bmatrix} \frac{\kappa_{+}+1}{4\mu_{+}}+\frac{\kappa_{-}+1}{4\mu _{-}}&-{\mathrm{i}} ( \frac{\kappa_{-}+1}{4\mu_{-}}-\frac{\kappa_{+}+1}{4\mu _{+}} ) &0 \\ {\mathrm{i}} ( \frac{\kappa_{-}+1}{4\mu_{-}}-\frac{\kappa_{+}+1}{4 \mu_{+}} ) &\frac{\kappa_{+}+1}{4\mu_{+}}+\frac{\kappa _{-}+1}{4\mu _{-}}&0 \\ 0&0&\frac{1}{\mu_{+}}+\frac{1}{\mu_{-}} \\ \end{bmatrix} \quad \text{and}\quad \boldsymbol{\varLambda}^{\bot}= \begin{bmatrix} \alpha^{\bot}&0&0 \\ 0&\alpha^{\bot}&0 \\ 0&0&\gamma^{\bot} \end{bmatrix}, $$

(82)

where $\alpha^{\bot} > 0$ and $\gamma^{\bot} > 0$ are interfacial properties and $\kappa$ and $\mu$ are the bulk and shear moduli, respectively. In this case, explicit expressions [13] of the eigenroots are given by

$$\begin{aligned} \nu_{1} = & \frac{1}{4 \alpha^{\bot}} \biggl[ \frac{\kappa_{+}+1}{ \mu_{+}}+ \frac{\kappa_{-}+1}{\mu_{-}} + \biggl\vert \frac{\kappa _{-}-1}{\mu _{-}}-\frac{\kappa_{+}-1}{\mu_{+}} \biggr\vert \biggr] , \end{aligned}$$

(83)

$$\begin{aligned} \nu_{2} = & \frac{1}{4 \alpha^{\bot}} \biggl[ \frac{\kappa_{+}+1}{ \mu_{+}}+ \frac{\kappa_{-}+1}{\mu_{-}} - \biggl\vert \frac{\kappa _{-}-1}{\mu _{-}}-\frac{\kappa_{+}-1}{\mu_{+}} \biggr\vert \biggr] , \end{aligned}$$

(84)

$$\begin{aligned} \nu_{3} = & \frac{1}{\gamma^{\bot}} \biggl[ \frac{1}{\mu_{+}}+ \frac{1}{ \mu_{-}} \biggr] . \end{aligned}$$

(85)

Note that these expressions are consistent with those given by Sudak and Wang [13] in the case of an imperfect interface (spring-like model).

Using Eq. (81) and introducing the complex vector function $\boldsymbol{\phi}_{\pm}$, with $\boldsymbol{\delta}{\boldsymbol{\xi}}_{+} ( z ) = ( \boldsymbol{\varLambda}^{\bot} ) ^{-1/2} \boldsymbol{\varOmega}\boldsymbol{\phi}_{+} ( z )$, and $\boldsymbol{\delta}{\boldsymbol{\xi}}_{-} ( z ) = ( \boldsymbol{\varLambda} ^{\bot} ) ^{-1/2} \overline{\boldsymbol{\varOmega}} \boldsymbol{\phi}_{-} ( z )$, Eq. (77) can be completely decoupled (i.e., the differential equations in $\boldsymbol{\delta }{\boldsymbol{\xi}} _{i}$ are independent) such that

$$\begin{aligned} \boldsymbol{\phi}'_{+} ( z ) + {\mathrm{i}}\bigl\langle \nu^{+} _{*} \bigr\rangle \boldsymbol{\phi}_{+} ( z ) = & - \overline{\boldsymbol{\varOmega}}^{T} \bigl( \boldsymbol{ \varLambda} ^{\bot} \bigr) ^{1/2} \bigl[ \boldsymbol{I} + \boldsymbol {H}^{-1} ( \overline{\boldsymbol{M}_{+}} - \overline{\boldsymbol{M}_{-}} ) \bigr] \overline{\boldsymbol{B}_{+}} \overline{\boldsymbol{f}''_{0} ( \overline{z} ) },\quad z \in S_{+}, \end{aligned}$$

(86)

$$\begin{aligned} \boldsymbol{\phi}'_{-} ( z ) - {\mathrm{i}}\bigl\langle \nu^{-} _{*} \bigr\rangle \boldsymbol{\phi}_{-} ( z ) = & - \boldsymbol{\varOmega}^{T} \bigl( \boldsymbol{ \varLambda}^{\bot} \bigr) ^{1/2} \boldsymbol{H}^{-1} ( \overline{\boldsymbol{M}_{+}} + {\boldsymbol{M}_{+}} ) { \boldsymbol{B}_{+}} {\boldsymbol{f}''_{0} ( {z} ) },\quad z \in S_{-}. \end{aligned}$$

(87)

Following the methodology outlined in Sect. 3.5.1, solutions to Eqs. (86)–(87) are obtained from the particular solutions and be expressed as

$$\begin{aligned} \boldsymbol{\phi}_{+} ( z ) = & \bigl\langle e^{- {\mathrm{i}} \nu^{+}_{*} z } \bigr\rangle \int^{z}_{z^{+}_{ref}} \bigl\langle e^{{\mathrm{i}} \nu^{+}_{*} \omega}\bigr\rangle \overline{\boldsymbol{\varOmega}}^{T} \bigl( \boldsymbol{ \varLambda}^{\bot} \bigr) ^{1/2} \bigl[ \boldsymbol{I} + \boldsymbol{H}^{-1} ( \overline{\boldsymbol{M}_{+}} - \overline{ \boldsymbol{M}_{-}} ) \bigr] \overline{\boldsymbol{B}_{+}} \biggl\langle \frac{1}{ ( \omega- \overline{z}^{0}_{*} ) ^{2}} \biggr\rangle d\omega\overline{\boldsymbol{q}} , \end{aligned}$$

(88)

$$\begin{aligned} \boldsymbol{\phi}_{-} ( z ) = & \bigl\langle e^{ {\mathrm{i}} \nu^{-}_{*} z } \bigr\rangle \int^{z}_{z^{-}_{ref}} \bigl\langle e^{-{\mathrm{i}} \nu^{-}_{*} \omega}\bigr\rangle \boldsymbol{\varOmega}^{T} \bigl( \boldsymbol{\varLambda} ^{\bot} \bigr) ^{1/2} \boldsymbol{H}^{-1} ( \overline{ \boldsymbol{M}_{+}} + {\boldsymbol{M}_{+}} ) { \boldsymbol{B}_{+}} \biggl\langle \frac{1}{ ( \omega- {z}^{0}_{*} ) ^{2}} \biggr\rangle d \omega{\boldsymbol{q}} . \end{aligned}$$

(89)

The integration bounds $z^{+}_{ref}$ and $z^{-}_{ref}$ have to be chosen so that stresses at infinity vanish, e.g., $z^{+}_{ref}={\mathrm{i}} \infty$ and $z^{-}_{ref}=-{\mathrm{i}}\infty$. Following the methodology defined for the general solution, explicit expression of the perturbation terms are given by

$$\begin{aligned} \boldsymbol{\delta}\boldsymbol{f}'_{+} ( z ) = & \sum _{i=1}^{3} \sum _{j=1}^{3} \sum_{k=1}^{3} \bigl\langle W_{+}^{ij} ( z _{*} ) e^{- {\mathrm{i}}\nu^{+}_{k} z_{*}} \bigr\rangle \boldsymbol{B}^{-1}_{+} \bigl( \boldsymbol{\varLambda}^{\bot} \bigr) ^{-1/2} \boldsymbol{ \varOmega} \boldsymbol{I_{k}} \boldsymbol{I_{ij}} \overline{ \boldsymbol{q}},\quad z\in S_{+}, \end{aligned}$$

(90)

$$\begin{aligned} \boldsymbol{\delta}\boldsymbol{f}'_{-} ( z ) = & \sum _{i=1}^{3} \sum _{j=1}^{3} \sum_{k=1}^{3} \bigl\langle W_{-}^{ij} ( z _{*} ) e^{ {\mathrm{i}}\nu^{-}_{k} z_{*}} \bigr\rangle \boldsymbol{B} ^{-1}_{-} \bigl( \boldsymbol{\varLambda}^{\bot} \bigr) ^{-1/2} \overline{ \boldsymbol{\varOmega}} \boldsymbol{I_{k}} \boldsymbol{I_{ij}} {\boldsymbol{q}} ,\quad z\in S_{-} , \end{aligned}$$

(91)

with

$$\begin{aligned} {W}^{ij}_{+} ( z_{*} ) = & \int^{z_{*}}_{{\mathrm{i}} \infty} e^{{\mathrm{i}}\nu^{+}_{i} \omega} \overline{ \varOmega}_{ki} \bigl( \varLambda^{\bot}_{kk} \bigr) ^{1/2} \bigl[ \delta_{km} + {H}_{km} ^{-1} ( \overline{M_{+}}_{mn} - \overline{M_{-}}_{mn} ) \bigr] \overline{B_{+}}_{nj} \bigl( \omega- \overline{z}^{0}_{j} \bigr) ^{-2} d \omega, \end{aligned}$$

(92)

$$\begin{aligned} {W}^{ij}_{-} ( z_{*} ) = & \int^{z_{*}}_{-{\mathrm{i}} \infty} e^{-{\mathrm{i}}\nu^{+}_{i} \omega} { \varOmega}_{ki} \bigl( \varLambda ^{\bot}_{kk} \bigr) ^{1/2} {H}_{km}^{-1} ( \overline{M_{+}}_{mn} + {M_{+}}_{mn} ) {{B_{+_{nj}}}} \bigl( \omega- {z}^{0}_{j} \bigr) ^{-2} d\omega. \end{aligned}$$

(93)

It should be noted that in this so-called simplified case, integration bounds can be readily chosen and uniquely defined. The diagonal terms of the matrices ${W}^{ii}_{\pm}$ are directly related to the generalized exponential function $E_{2} ( z )$.

Appendix C: Examples of Interfacial Coupling Tensor for Selected Tilt Grain Boundaries

In this appendix, we explicitly illustrate that the assumption which considers the interfacial coupling tensor $\mathbb{M}_{+}$ as diagonalizable is realistic. It should be noted that these examples do not provide a guarantee/proof that any given interface may satisfy this condition as it is material and interface specific.

As such, let us consider a tilt boundary about a given misorientation tilt axis. Figure 2 schematically shows such a boundary, with the misorientation about the tilt axis is defined as $2\theta$.

Fig. 2

Schematic representation of a symmetric tilt grain boundary

Considering the new reference coordinate systems in crystal “+” and crystal “−” which are obtained by rotating the reference coordinate system about the $x_{3}$-axis by an angle $\theta$ and $-\theta$ respectively, yields the rotation matrix $\boldsymbol{\varXi}$ given by

$$ {\boldsymbol{\varXi}} ( \theta ) = \begin{bmatrix} \cos\theta& \sin\theta& 0 \\ -\sin\theta& \cos\theta& 0 \\ 0 & 0 & 1 \end{bmatrix} . $$

(94)

The components of the rotated stiffness tensor $\mathbb{C}_{ijkl}^{ ( m ) }$, where $m=\{+,-\}$ are related to the stiffness tensor $\mathbb{C}_{ijkl}$ in the crystal lattice coordinate by

$$ \mathbb{C}_{ijkl}^{ ( m ) } = \varXi_{ip} \varXi_{jq}\varXi _{kr}\varXi_{ls} \mathbb{C}_{pqrs}. $$

(95)

The matrices $\boldsymbol{Q}$, $\boldsymbol{R}$ and $\boldsymbol{T}$ defined in Eq. (3) in each crystal are then defined as

$$\begin{aligned} {\boldsymbol{Q}}_{m} = \mathbb{C}_{i1k1}^{ ( m ) }={ \boldsymbol{\varXi}} \bigl[ {\boldsymbol{Q}}\cos^{2}\theta+ { \boldsymbol{T}}\sin ^{2}\theta + \bigl( {\boldsymbol{R}} + { \boldsymbol{R}}^{T} \bigr) \cos \theta\sin\theta \bigr] \boldsymbol{ \varXi}^{T}, \end{aligned}$$

(96)

$$\begin{aligned} {\boldsymbol{R}}_{m}= \mathbb{C}_{i1k2}^{ ( m ) }={ \boldsymbol{\varXi}} \bigl[ {\boldsymbol{R}}\cos^{2}\theta- \boldsymbol{R}^{T}\sin ^{2}\theta+ ( {\boldsymbol{T}} - { \boldsymbol{Q}} ) \cos \theta\sin\theta \bigr] \boldsymbol{\varXi}^{T}, \end{aligned}$$

(97)

$$\begin{aligned} {\boldsymbol{T}}_{m}=\mathbb{C}_{i2k2}^{ ( m ) }={ \boldsymbol{\varXi}} \bigl[ {\boldsymbol{T}}\cos^{2}\theta+ { \boldsymbol{Q}}\sin ^{2}\theta - \bigl( {\boldsymbol{R}} + \boldsymbol{R}^{T} \bigr) \cos \theta\sin\theta \bigr] \boldsymbol{ \varXi}^{T}. \end{aligned}$$

(98)

Following these rotations, the $\boldsymbol{A}$ and $\boldsymbol{B}$ matrices can be determined through Eq. (5). As explained earlier, the three eigenvectors associated with the three eigenvalues with positive imaginary parts define ${\mathbf{A}}_{m}$, $m=\{+,- \}$. Note also that, in order to simplify the numerical computation of the tensors $\mathbf{A}_{m}$ and $\mathbf{B}_{m}$, the eigenvalues and eigenvectors can be computed from the $6\times6$ matrix $\mathbf{N}_{m}$ defined by Ting [38] such that

$$ \mathbf{N}_{m} \begin{bmatrix} \mathbf{a} \\ \mathbf{b} \\ \end{bmatrix} = \mu^{m} \begin{bmatrix} \mathbf{a} \\ \mathbf{b} \\ \end{bmatrix} , $$

(99)

with

$$ \mathbf{N}_{m} = \begin{bmatrix} -\mathbf{T}^{-1}_{m} \mathbf{R}_{m}^{T} & \mathbf{T}^{-1}_{m} \\ \mathbf{R}_{m}^{T} \mathbf{T}^{-1}_{m} \mathbf{R}_{m}^{T} - \mathbf{Q}_{m} & -\mathbf{R}_{m} {\mathbf{T}_{m}^{-1}}^{T} \end{bmatrix} . $$

(100)

As a result, the eigenvalues of $\mathbf{N}_{m}$ are identical to the ones solution to Eq. (5). In addition, the first three components of the first three eigenvectors of $\mathbf{N}_{m}$ are the eigenvectors solution to Eq. (5) and correspond to the components of the matrix $\mathbf{A}_{m}$. In addition, both $\mathbf{A}_{m}$ and $\mathbf{B}_{m}$ need to be normalized to satisfy the following identity [38]:

$$ \mathbf{A}_{m}^{T}\mathbf{B}_{m}+ \mathbf{B}_{m}^{T} \mathbf{A}_{m} = \mathbf{I} . $$

(101)

After obtaining the $\boldsymbol{A}$ and $\boldsymbol{B}$ matrices, the interfacial coupling tensor in the upper and lower half-planes $\mathbb{M}_{m}$ (e.g., $\mathbb{M}_{+} = [ {\mathbb{H}}^{S} _{II} \boldsymbol{M}_{+} - {\mathrm{i}}\boldsymbol{\varLambda}^{\bot} ] ^{-1} \boldsymbol{H}$) can be derived following the method described in Sects. 2 and 3. Keeping in mind that $\mathbb{M}_{+}$ and $\mathbb{M}_{-}$ are conjugate, only the upper half-plane can be considered to demonstrate the diagonalizability of $\mathbb{M}_{m}$.

As a specific example, let us consider two $\varSigma5$ symmetric tilt grain boundaries in copper along the $[ 001 ]$ tilt axis. The elastic constants in the lattice coordinates are given by $\mathbb{C} _{1111} = 167.1$ GPa, $\mathbb{C}_{1122} = 124.0$ GPa, $\mathbb{C} _{2323} = 76.4$ GPa [30]. Characteristics of both grain boundaries considered are tabulated in Table 1 while the components of the corresponding transverse compliance tensor $\mathbf{\varLambda}^{\bot}$ and the interfacial Poisson’s effect tensor $\mathbb{H}^{S}$ are given in Table 2.

Table 1 Crystallographic characteristics of $\varSigma5$ symmetric tilt grain boundaries observed in copper [30]

Table 2 Interface elastic properties of $\varSigma5$ symmetric tilt grain boundaries for copper [30] (only non-zero components are shown)

The six eigenvalues and eigenvectors, solution of the eigenvalue problem (Eq. (5)) are given in Table 3, while Table 4 shows the eigenvalues associated with $\mathbb{M}_{+}$. It should be noted that its eigenvalues are all non-null and distinct and that, consequently, $\mathbb{M}_{+}$ is diagonalizable.

Table 3 Eigenvalues and eigenvectors solutions of Eq. (5) in the upper half-plane for both symmetric tilt grain boundaries considered. The annotated $0^{\ast}$ indicates that the numerical values for both the real and imaginary parts of those complex numbers were below $10^{-17}$ and therefore negligible

Table 4 Eigenvalues of the interfacial coupling tensor $\mathbb{M}_{+}$ for both $\varSigma5$ symmetric tilt grain boundaries