<figure><img src="image_55.png" /></figure>

FIGURE 3.36: Representation of the arrangements of atoms and dislocations in the semicoherent Cu/Nb interface, which yields hexagonal-shaped meshes of dislocations with the lowest strain energies. (a) The pre-computing elastic energy landscape for the Cu/Nb system in the NW orientation shows a minimum-energy path (i.e. the horizontal chain in black) from the initial lozenge-shaped Frank-Bilby solution (i.e. a pattern with two sets of dislocations, represented by the red and blue lines) in (c) to the final (fully-relaxed) dislocation structures of the lowest strain energy in (d). The corresponding variation of energy along the minimum-energy path is plotted with respect to the dimensionless coordinate $s_η$ in (b). The initial Frank-Bilby solution and the final hexagonal dislocation structure for the Cu/Nb system are shown in terms of dislocation structure (left-hand sides) by forming the third new set of dislocation junctions (in black) and of the normal stress $\sigma_{33}^{\text{int}}$ in GPa (right-hand sides).

Furthermore, an elliptical shear dislocation loop that lies on the (111)$_{\text{fcc}}$ $\parallel n$ glide plane, with $A b^{\text{loop}} \parallel [\bar{1}0]_{\text{fcc}} \parallel x_1$, is embedded in the Cu material.

In the lower bcc Nb material, a pile-up system with $N = 5$ pure edge dislocations is introduced with line directions $\xi \parallel [01\bar{1}]_{\text{bcc}} \parallel x_2$, on the glide plane normal to $v_{\text{climb}} \parallel [\bar{2}11]$ and Burgers vectors $b^{\text{lat}} \parallel [111]_{\text{bcc}}$. Furthermore, a circular shear dislocation loop resides in the $(011)_{\text{bcc}} \parallel n$ glide plane in Nb, such that the pile-up of edge dislocations is comprised between the circular shear dislocation loop and the semicoherent Cu/Nb interface. The complete dislocated microstructure in the present Cu/Nb bimaterial is displayed in Fig. (3.37a).

According to these dislocation geometries and orientations, both interacting shear dislocation loops in Cu and Nb are not periodically replicated. Moreover, the extrinsic lattice dislocations are fixed in Cu, while the piled-up dislocations can glide only, without bowing around the loops.

## Interfacial dislocation structures

The proper reference state under the condition of vanishing far-field stresses in the NW Cu/Nb bicrystal has been determined in section 3.4, where the magnitude of correct Burgers vectors are given by $b_1 = b_2 = 0.28301$ nm. With respect to the Burgers vectors, the quantized Frank-Bilby eq. (3.121) gives rise to different solutions, with a particular lozenge-shaped dislocation structure that is specifically comprised of two arrays of parallel dislocations (with no local reactions at nodes) with mixed characters $\phi_1^{\text{un}} = \phi_2^{\text{un}} = 37.51^{\circ}$, and the angle between these two unrelaxed sets
<table><caption>TABLE 3.9: Equilibrium positions (in nm) of pure edge piled-up dislocations in Nb measured obliquely along the piled-up direction from the interface in Cu/Nb. Values in brackets indicate the algebraic relative error (in %) due the isotropic approximation of elasticity (with respect to the true anisotropic case as reference).</caption><thead><tr><th># Edge dislocation</th><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th><th>ℓ<sub>pile-up</sub></th><th>f<sub>glide</sub><sup>PK, 1st</sup></th><th>f<sub>climb</sub><sup>PK, 1st</sup></th></tr></thead><tbody><tr><td>Isotropic</td><td>0</td><td>0.28</td><td>4.1</td><td>1.20</td><td>4.0</td><td>2.33</td><td>-3.6</td><td>3.62</td><td>-7.1</td><td>7.44</td><td>-3.9</td><td>14.01</td><td>18.3</td><td>7.99</td><td>-19.5</td></tr><tr><td>Anisotropic</td><td>0</td><td>0.27</td><td>ref</td><td>1.16</td><td>ref</td><td>2.42</td><td>ref</td><td>3.90</td><td>ref</td><td>7.74</td><td>ref</td><td>11.85</td><td>ref</td><td>9.92</td><td>ref</td></tr></tbody></table>

of dislocations is $\phi^{\text{un}} = 15.03^{\circ}$. In addition, $p_1^0 = p_2^0 = 4.33249$ nm, so that the inter-dislocation spacings are given by $d_1^{\text{un}} = d_2^{\text{un}} = 1.12341$ nm. This specific dislocation structure is considered as an initial non-equilibrium state, where local reactions of crossing dislocations to form dislocation segments with a third Burgers vector in hexagonal-shaped networks can be energetically favorable.

As described in section 3.7.1, the intrinsic dislocation pattern in the NW orientation is obtained by pre-computing the elastic strain energy landscape that corresponds to the lozenge-shaped solution predicted by the quantized Frank-Bilby equation. Figure (3.36a) displays the specific landscape for the Cu/Nb system that is related to the initial lozenge-shaped dislocation structure with two sets of dislocations, as depicted by the red and blue lines in Figs. (3.36a) and (c). The minimum-energy dislocation configuration exhibits also a hexagonal-shaped structure with three sets of dislocations, as designated by the fully strain-relaxed pattern. The corresponding minimum-energy path is plotted in Fig. (3.36b). It is therefore shown that the elastic relaxation is accompanied by a decrease in strain energy of ~ 4%, yielding the formation of large set of dislocation junctions with pure edge characters (black lines in Fig. (3.36d)). Changes in the associated normal stresses $\sigma_{33}^{\text{int}}$ between the initial and the final configurations in the Cu/Nb system are computed at $x_3 = 1$ nm, and illustrated on the right-hand side. It is observed that the third new set of dislocation junctions exhibits almost zero normal stresses, while the maximum compressive stress values are reached at the parent dislocation sets.

## Internal forces on lattice dislocations

Table (3.9) summarizes the results (positions to the interface and forces) computed at $x_2 = 0$ nm for the dislocation pile-up in anisotropic and isotropic Cu/Nb dislocated bimaterial at equilibrium, for which the explicit positions are depicted in Fig. (3.37a) for the anisotropic case. Values in brackets indicate the algebraic relative error (in %) due to the isotropic approximation of elasticity (with respect to the full anisotropic case as reference) with significant errors of ~ ±20% in both glide and climb force magnitudes.

Nucleation and multiplication of dislocations in microstructure are traditionally described by applying dislocation criteria, e.g., resolved shear stresses, Hertzian principal shear stresses, von Mises strains or stresses. For illustration, Fig. (3.37b) plots the von Mises stress distribution cut from the mid-section of the microstructure, where the local stress is mainly concentrated around the upper dislocation loop. The Peach-Koelher forces along the leading piled-up dislocation and lattice dislocations are computed at equi-spaced positions along the $x_2$-axis, and are displayed in Fig. (3.37c), including the full-space and the complementary image parts. For clarity, the reference vector that scales the magnitude of the Peach-Koehler force is 10 N/m for the leading dislocation, and 5 N/m for the lattice dislocations. According to the large magnitude in von Mises stress field, the dislocation features may serve as dislocation emission in the bimaterial or/and the semi-coherent interface. Despite the present idealized situations in dislocation features, the complex and heterogeneously distributed force profiles on straight dislocations provide insights into the behavior of dislocations, e.g. the largest forces in magnitude are experienced on the leading piled-up dislocation (L), which would also bow-out away from the interface since the heterogeneous
<figure><img src="image_56.png" /><figcaption>FIGURE 3.37: (a) Geometries and orientations among various dislocations in the anisotropic (111)Cu/(011)Nb bimaterial with interface at $x_3 = 0$ nm. The upper material Cu contains eight infinitely long straight and uniformly spaced parallel dislocation arrays along the $x_2$-axis, with different characters (in dark gray (almost edge) and in red (30° mixed)) and an elliptical shear dislocation loop, while the lower material Nb is comprised of a pile-up system with 5 pure edge dislocations and a circular shear dislocation loop. (b) The corresponding von Mises stress field, associated with the equilibrium piled-up dislocations and all other dislocations, including the hexagonal-shaped dislocation structures at the semi-coherent Cu/Nb interface. (c) Force distribution along all lattice dislocations ($e_i$) and ($m_j$) in Cu as well as along the leading dislocation (L). The reference vector 10 N/m (5 N/m) represents the magnitude scale of the Peach-Koehler force exerted on the leading (lattice) dislocation(s). All lattice and piled-up dislocations are defined along the $x_2$-axis.</figcaption></figure>

step-like force profile is mainly due to the presence of the intrinsic hexagonal-shaped dislocation structure at the semicoherent interface.

From these distributions, the resolved shear stress forces on specific glide planes could therefore be projected to extend the computational procedure of motion for all glide dislocations in bimaterials.

## Self- and Peach-Koehler forces on shear dislocation loops

In order to compute the complete self-forces $f^I$ associated with the dislocation loops, the pre-logarithmic energy factor $E$ in eq. (3.156) is determined by asymptotically reducing the parametric energy-based framework for one set of dislocations [249]. For a single set of Volterra-type dislocations, the corresponding energy per unit length of dislocation is viewed as the work done in forming the dislocation network by cutting and displacing the habit plane at $x_2 = 0$ between $x_1 = r_0$ and $x_1 = d_1 - r_0$, as follows

$$E = d_1 \gamma_e = \frac{1}{2} \int_{r_0}^{d_1 - r_0} t_j(x_1, x_2 = 0) u_j^p(x_1, x_2 = 0) dx_1, \quad (3.161)$$

according to eq. (3.30), where the prescribed displacement jump is $u^p(x_1, x_2 = 0) = -b_1$ for Volterra-type dislocations [260]. However, the inter-distance spacings must be sufficiently large to represent the equivalent energy state for one infinite straight dislocation, as requested by the line tension formulation in section 3.7.2. Here, the inter-distance spacings $d_1$ for one single set
<figure><img src="image_57.png" /><figcaption>FIGURE 3.38: Determination of the self-forces, given by eq. (3.156), on the planar elliptical and circular shear dislocation loops that reside in upper material Cu (red curves) and lower material Nb (blue curves), respectively, as function of the angles $\phi$ and $\alpha$ in $^\circ$. (a) $E + E''$, (b) $H + H''$, (c) $E - H'' - E''$, and (d) the complete algebraic self-forces $f^\Gamma$ that are continuously distributed around both dislocation loops. The symbol ' stands for differentiation with respect to the proper angles that are depicted in the inset of (b). The fcc (bcc) case exhibits symmetric (asymmetric) behavior with respect to the median axes (i.e. the vertical dotted lines), and $f^\Gamma$ can also be positive in Nb (depicted by the shaded regions in blue), which means that the self-forces tend locally to expand the corresponding dislocation loop by the near-edge dislocation elements, i.e. $\alpha = 90 \pm 17^\circ$ and $\alpha = 270 \pm 17^\circ$, while $f^\Gamma$ is always negative for the elliptical shear loop in Cu.</figcaption></figure>

of dislocations is chosen such that the corresponding stress field is equivalent to the stress state produced by one single dislocation. As discussed in Ref. [265], when $d_1$ is fictitiously increased by a multiplicative factor $10^3$, the discrepancy in stress state between such dislocation array with large spacings and the single dislocation case is almost zero. Thus, substituting $d_1$ with $10^3 d_1$ in eq. (3.161), $E$ and $E''$, where ' stands for differentiation with respect to $\phi$, can be numerically be evaluated for infinite character-dependent dislocations in the present anisotropic Cu/Nb material.

As a measure of the stiffness of the dislocations, the term $E + E''$ for Cu and Nb is plotted in Fig. (3.38a) as function of $\phi$, such that pure screw (edge) character is characterized by $\phi = 0^\circ$ ($\phi = 90^\circ$) for infinite dislocations, respectively. These plots are in agreement with the distinguishing classification of the anisotropic curves in Ref. [13], e.g., the appearance of maxima and minima for values of $\phi$ between $\phi = 0^\circ$ and $180^\circ$ as well as the asymmetric (symmetric) case in bcc Nb (fcc Cu) materials about $\phi = 90^\circ$. Furthermore, the term $H + H''$ that arises in eq. (3.156) for the tube self-force contribution, where $H$ is defined in eq. (3.154), is displayed in Fig. (3.38b), while the superposition $E - H - H''$ is plotted in Fig. (3.38c). Here, the symbol ' deals with differentiation with respect to $\alpha$.

Using the geometrical features in terms of curvatures $\kappa$ and relations between $\phi$ and $\alpha$ for both elliptical and circular dislocation loops in Cu and Nb, which can be easily parametrized, the
complete algebraic self-forces $f^\Gamma$ are shown in Fig. (3.38d). Interestingly, as the self-force $f^\Gamma$ is positive close to the edge orientations in Nb, i.e. $90^\circ \le \alpha \le 107^\circ$ and $253^\circ \le \alpha \le 270^\circ$ (which corresponds to $\alpha = 90 \pm 17^\circ$ and $\alpha = 270 \pm 17^\circ$ by symmetry properties), as shown by the shaded blue regions, the line tension provides an expansion reaction in the near-edge orientations that acts along the $-m$ directions, i.e. pointing outward from the centers of the circular dislocation loop, while a global striking behavior is observed for all other non-edge characters, especially for the local near-screw character elements that usually have lower elastic energy [7, 153]. This result is in good qualitative agreement with predictions in Ref. [12] with similar characteristics (i.e. Burgers vector and habit plane) for shear dislocations in highly anisotropic $\alpha$-iron. In Cu, however, the elliptical dislocation loop tends to shrink under the action of the heterogeneously distributed and centripetal line tension self-forces.

Figures (3.39a) and (b) illustrate the self-force profiles (black arrows) that are larger in magnitude in the $x_2 = [112]_{\text{fcc}} \parallel [01\bar{1}]_{\text{bcc}}$ direction in Cu and Nb, respectively, while the self-force contribution tends to locally expand the lower dislocation loop in Nb in the $x_1 \parallel [100]_{\text{bcc}}$ direction (as displayed by the dotted circles in gray). Furthermore, the blue arrows illustrate the interaction force contribution between the two dislocation loops. For instance, the interaction force acting on the upper dislocation loop in Cu is determined by superposing its complementary image force and the full-space part produced by the lower dislocation loop in Nb. It is shown that this force component pulls the elliptical dislocation loop toward the semicoherent interface, i.e. toward the softer material Nb, with the largest magnitude on the minor axis region with screw character elements. On the other hand, the dislocation loop force contribution is almost in-plane for the lower dislocation loop in Nb. Finally, the total Peach-Koehler forces, which include the dislocation loop force contribution and all other contributions from the lattice dislocation arrays (including the piled-up dislocations), are shown in Fig. (3.39) with orange arrows. It can therefore be observed that the Peach-Koehler force tends to rotate out of the $(111)_{\text{fcc}}$ glide plane in the upper dislocation loop around the $[\bar{1}10]_{\text{fcc}}$, and also to shear it by climb-assisted dislocation-glide process, while the same force in Nb tends to expand preferentially the lower circular dislocation loop in the specific $[01\bar{1}]_{\text{bcc}}$ direction onto the $(011)_{\text{bcc}}$ glide plane.

### 3.7.4 Limitations

Based on the previous sections 3.4 and 3.6, and specially in section 3.7, where it should be recognized that the procedure for determining the driving forces is not easily tractable for arbitrarily-shaped dislocation loops in large-scale dislocation dynamics simulations, the inherent assumption that follows from the linear elasticity theory is related to the introduction of a core cutoff radius to eliminate the divergence of the dislocation field solutions. Furthermore, the comparison between the elasticity theory and atomistic predictions leads to discrepancies in the interfacial stored energies, mainly due to the singular consideration of the dislocation cores, as quantified in sections 3.4.6 and 3.6.6. Thus, the remedy to the difficulties encountered and the discrepancies made, lies in the derivation of non-singular solutions for extrinsic and intrinsic dislocation structures. In the context of classical elasticity, singularity-free fields are obtained by convoluting the prescribed displacement jumps with isotropic Gaussian distributions. Conceptually similar to the original Peierls-Nabarro approach [204, 190], this procedure overcomes the long-standing dislocation problems of singular elastic fields in the core regions, which has been applied to interfacial dislocations [257] and more recently to extrinsic dislocation loops [263]. In addition, a second emphasis has been placed on the extension to multilayered magneto-electro-elastic plates with multiple semicoherent interfaces, such that the single semicoherent homo- and hetero-phase interface in pure elastic bimaterials becomes a particular case of the general approach, as described in the following section 3.8.
<figure><img src="image_58.png" /><figcaption>(b) Cu/Nb: Circular loop in lower material Nb</figcaption></figure>

<figure><img src="image_59.png" /></figure>

FIGURE 3.39: Discrete distribution of the local forces (black arrows), of the dislocation loop forces (blue) that include the corresponding full-space solutions and complementary (image) contributions of the loops, and of the complete Peach-Koehler forces (orange), which act along both shear dislocation loops. These force distributions are exerted on the elliptical dislocation loop in Cu (a) and on the circular loop in Nb (b). For both shear dislocation loops, the associated fcc and bcc Burgers vectors lie along the $x_1$-axis. The dotted two small circles in (b) illustrates the local self-stress expansion of the loop in the lower material Nb by the near-edge (since the local character between the Burgers vector and the local tangent is characterized by $\alpha$ in Fig. (3.38)) character elements.

## 3.8 Extension to non-singular fields in multilayered magneto-electro-elastic plates

Multiphysics analyses in man-made (piezoelectric, piezomagnetic, and magneto-electro-elastic (MEE)) multiferroics have attracted tremendous interest of many researchers because of their widespread and advanced applications involving intelligent topological structures, energy harvesting and green energy production, optoelectronics, and self-powered biomedical devices. External surfaces and internal interfaces between alternating dissimilar materials play special roles in magnetism [79, 175, 268], electrical transport [175, 128, 268], mechanical properties [299, 186], and also in multiple coupled magnetic, electric, and mechanical fields, which become crucial to design novel nanostructured composites with outstanding functional and enhanced MEE properties [194, 212, 62, 286]. One significant technological problem during the growth of nanoscale multilayers is related to the lattice mismatches between different layers [58, 298, 83, 293], which induce spurious MEE field concentrations that can markedly enhance or degrade the materials properties, and in the latter case, causing crack initiation and growth, dielectric breakdown and magnetic failure. The present section focuses on atomistically informed conditions for crystalline interfaces with lattice-mismatched dislocation structures in multilayered MEE materials made of $CoFe_2O_4$ (magnetostrictive cobalt ferrite, CFO) and $BaTiO_3$ (piezoelectric barium titanate, BTO).
<figure><img src="image_60.png" /></figure>

FIGURE 3.40: Superposition principle for the semicoherent interfaces in miscible MEE multilayers subjected to external loads. (a) Representative linear and anisotropic free-standing multilayered system that consists of *w* rectangular layers with a couple of semicoherent interfaces at *z* = *z*<sub>j</sub> and *z* = *z*<sub>k</sub>, while the others are perfectly bonded between adjacent layers. The heterophase interfaces possess different internal structures comprised of two planar arrays of infinitely long straight, and periodically-spaced dislocations. The open and filled symbols represent the atomic structure of the lattice-mismatched semicoherent interface, while the solid segments indicate the corresponding misfit dislocations. The imperfect interface at *z*<sub>j</sub> is bonded between layers *j* and *j* + 1, with discontinuity quantities between the upper and lower sides indicated by + and –, and is therefore coplanar to two flat free surfaces at *z* = *z*<sub>0</sub> = 0 (bottom) and *z* = *z*<sub>w</sub> (top). (b) Using the superposition principle, general mechanical, electric, and magnetic boundary conditions are externally and vertically applied on both the top and bottom surfaces of MEE solid with perfectly bonded interfacial boundary conditions (i.e., coherent internal interfaces).

**3.8.1 Boundary-value problem and singularity-free field solutions**

The classical six-dimensional Stroh formalism is extended to a ten-dimensional formalism com-
bined with a Fourier series-based solution procedure to determine the displacement and traction
fields in anisotropic multilayered MEE solids under external loads. Such multilayers are com-
posed of semicoherent interfaces, for which each pure misfit interface consists of two different sets
of infinitely long straight, uniformly spaced, and parallel core-spreading dislocations. Practical re-
cursive operations are explicitly derived with respect to specific internal and external boundary
conditions for multilayered solids with one and two semicoherent heterophase interfaces.

**Basic equations**

Figure (3.40a) shows the representative multilayered system that consists of *w* dissimilar, linear and anisotropic MEE layers with individual finite thickness *h*<sub>*k*</sub> = *z*<sub>*k*</sub> − *z*<sub>*k*−1</sub> for the *k*<sup>th</sup> layer, with *k* = 1, ..., *w*. A global and fixed Cartesian coordinate system with basis vectors (*x*<sub>1</sub>, *x*<sub>2</sub>, *x*<sub>3</sub>) = (*x*, *y*, *z*) is conveniently attached to the multilayers (and alternatively used for clarity in the further notations), where the unit vector normal to the interfaces is *n* || *x*<sub>3</sub> || *z*, while all layered rectangular plates are located in the positive *z*-region. Thus, the flat bottom and top surfaces are located at *z* = *z*<sub>0</sub> = 0 and at *z* = *z*<sub>w</sub> = *H* = Σ<sub>*k*=1</sub><sup>*w*</sup> *h*<sub>*k*</sub>, respectively, within which (mechanical, electric, and magnetic) loads can therefore be applied on these external surfaces, as illustrated in Fig. (3.40b). Furthermore, *s* (≤ *w* − 1) semicoherent interfaces of given crystallographic characters (misorientation and interface plane orientation) containing each up to two different sets of infinitely periodic dislocation patterns are explicitly described by solving the quantized Frank-Bilby equation, as detailed in section 3.2. In the absence of body forces, thermal effects, electric
current densities, electric and magnetic charge densities, the unified formulation of the governing equations of mechanical equilibrium with the Maxwell equations is represented by a single set of partial differential equation [166, 165, 196, 197, 200] as follows

$$ \sigma_{ij,i} = 0, \qquad (3.162) $$

where the index runs from 1 to 3 (from 1 to 5) over repeated lowercase (uppercase) subscripts, unless stipulated otherwise. The extended stress field $\sigma_{ij}$ in eq. (3.162) is defined by

$$ \sigma_{ij} = \begin{cases} \sigma_{ij} & J = j = 1,2,3 \\ D_i & J = 4 \\ B_i & J = 5, \end{cases} \qquad (3.163) $$

with $\sigma_{ij}$ the components of the mechanical stress (in N/m²), $D_i$ the electric displacement (in C/m²), and $B_i$ the magnetic induction (in N/A.m), which satisfy the static constitutive relations for each linear and anisotropic layer of the fully coupled MEE materials, i.e.

$$ \left\{ \begin{aligned} \sigma_{ij} &= c_{ijlm}\gamma_{lm} - e_{kij}E_k - q_{kij}H_k \\ D_i &= e_{ijk}\gamma_{jk} + \epsilon_{ij}E_j + \alpha_{ij}H_j \\ B_i &= q_{ijk}\gamma_{jk} + \alpha_{ji}E_j + \mu_{ij}H_j, \end{aligned} \right. \qquad (3.164) $$

where all materials properties are position-dependent in the multilayers, but homogeneously defined in each layer. In particular, $\gamma_{lm}$ is the elastic strain (dimensionless), $E_k$ is the electric field (in V/m), $H_k$ is the magnetic field (in A/m), and $c_{ijlm}$, $e_{kij}$, $q_{kij}$ and $\alpha_{ij}$ are the elastic moduli (in N/m²), piezoelectric (in C/m²), piezomagnetic (in N/A.m), and magnetoelectric (in C/A.m) coefficients, respectively. Furthermore, $\epsilon_{ij}$ and $\mu_{ij}$ are the dielectric permittivity (in C²/N.m²) and magnetic permeability (in N.s²/C²) coefficients, respectively, for which all repeated indexes are ranged in {1, 2, 3}. Various particular and uncoupled cases (e.g., pure elastic and piezoelectric) can evidently be reduced from eq. (3.164) by setting the appropriate coupling coefficients to zero. Using the shorthand notation, the constitutive relations can be recast as follows

$$ \sigma_{iJ} = c_{iJKl}\gamma_{Kl} = c_{iJKl}u_{K,l}, \qquad (3.165) $$

where the materials constants are defined by

$$ c_{iJKl} = \begin{cases} c_{ijkl} & J, K = j, k = 1,2,3 \\ e_{lij} & J = j = 1,2,3, K = 4 \\ e_{ikl} & J = 4, K = k = 1,2,3 \\ q_{lij} & J = j = 1,2,3, K = 5 \\ q_{ikl} & J = 5, K = k = 1,2,3 \\ -\alpha_{il} & J = 4, K = 5, \text{ or, } K = 4, J = 5 \\ -\epsilon_{il} & J, K = 4 \\ -\mu_{il} & J, K = 5, \end{cases} \qquad (3.166) $$

which satisfy the following symmetries: $c_{ijlm} = c_{jilm} = c_{ijml} = c_{lmij}$, $e_{kij} = e_{kji}$, $q_{kij} = q_{kji}$, $\alpha_{ij} = \alpha_{ji}$, $\epsilon_{ij} = \epsilon_{ji}$, and $\mu_{ij} = \mu_{ji}$. Both extended strain $\gamma_{kl}$ and displacement $u_K$ fields in eq. (3.165) are given
by

$$
\gamma_{kl} = \begin{cases} \gamma_{kl} = \frac{1}{2} (u_{k,l} + u_{l,k}) & K = k = 1,2,3 \\ -E_l = \phi_{,l} & K = 4, \text{ and, } \\ -H_l = \psi_{,l} & K = 5 \end{cases} \qquad u_K = \begin{cases} u_k & K = k = 1,2,3 \\ \phi & K = 4 \\ \psi & K = 5, \end{cases} \tag{3.167}
$$

with $u_k$, $\phi$, and $\psi$, being the elastic displacement (in m), the electrostatic potential (in V), and the magnetostatic potential (in A), respectively. From eq. (3.163), the extended traction $t_J$ with normal is $n_i$ is therefore given by

$$
t_J = \sigma_{ij} n_i = \begin{cases} \sigma_{ij} n_i & J = j = 1,2,3 \\ D_i n_i & J = 4 \\ B_i n_i & J = 5. \end{cases} \qquad (3.168)
$$

The dual variable and position procedure in multilayered systems

For in-plane multilayered MEE plates in presence of periodically-spaced interfacial dislocations,
the extended displacement vector $u_J$ in the physical domain is written in terms of a biperiodic
Fourier series expansion, as follows

$$
u_J (x_1, x_2, x_3 = z) = \operatorname{Re} \sum_{\eta} e^{-i2\pi\eta x_\alpha} \tilde{u}_J (\eta, z), \quad (3.169)
$$

with $\alpha = \{1, 2\}$, as already defined in eq. (3.73) with a negative sign of the exponential in the Fourier transforms. For clarity in the notation, the wavevectors $k$ in eq. (3.73) have been changed with $\eta$ in eq. (3.169), as well as the superscript $k$ with the superimposed tilde for all fields expressed in the frequency domain. Substitution of eq. (3.169) to eq. (3.165) and then to eq. (3.162) results in a system that consists of five homogeneous second-order differential equations in the Fourier-transformed domain, i.e.

$$
4\pi^2 c_{I\alpha J\beta} \eta_\alpha \eta_\beta \tilde{u}_J + i2\pi (c_{I\alpha J3} + c_{I3J\alpha}) \eta_\alpha \tilde{u}_{J,3} - c_{I3J3} \tilde{u}_{J,33} = 0, \quad (3.170)
$$

with $\beta = \{1, 2\}$, and differentiation of the extended displacement is operated with respect to the coordinate variable $x_3$. Furthermore, the derivatives of the Fourier-transformed displacements with the aid of the constitutive equations yield ten convenient relations between the complex transformed-Fourier displacement $\tilde{u}$ and traction $\tilde{t}$ expansion coefficients that can straightforwardly be converted into a linear system of first-order differential equations, i.e.

$$
\frac{d}{dz} \begin{bmatrix} \tilde{u}(\eta, z) \\ \tilde{t}(\eta, z) \end{bmatrix} = \begin{bmatrix} i2\pi\eta \mathbf{T}^{-1}\mathbf{R}^t & \mathbf{T}^{-1} \\ -4\pi^2\eta^2 (-\mathbf{Q} + \mathbf{R}\mathbf{T}^{-1}\mathbf{R}^t) & i2\pi\eta \mathbf{R}\mathbf{T}^{-t} \end{bmatrix} \begin{bmatrix} \tilde{u}(\eta, z) \\ \tilde{t}(\eta, z) \end{bmatrix}, \quad (3.171)
$$

which is satisfied for each homogeneous layer, individually. In eq. (3.171), the five-dimensional
real matrices **Q**, **R**, and **T** are given by

$$
Q_{IK} = c_{jIKs} m_j m_s, \quad R_{IK} = c_{jIKs} m_j n_s, \quad \text{and,} \quad T_{IK} = c_{jIKs} n_j n_s, \tag{3.172}
$$

with $c_{jIK_s}$ being the elastic, electric, and magnetic coefficients defined in eq. (3.166), and

$$
\eta = \begin{bmatrix} \eta_1 \\ \eta_2 \\ 0 \end{bmatrix} = \eta \mathbf{m} \equiv \sqrt{\eta_1^2 + \eta_2^2} \mathbf{m}, \quad \mathbf{m} = \begin{bmatrix} m_1 \\ m_2 \\ 0 \end{bmatrix} = \begin{bmatrix} \cos \theta \\ \sin \theta \\ 0 \end{bmatrix} \equiv \begin{bmatrix} \eta_1 / \eta \\ \eta_2 / \eta \\ 0 \end{bmatrix}, \quad \text{and,} \quad \mathbf{n} = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}, \quad (3.173)
$$
such that the extended matrices in eq. (3.172) are characterized in the oblique plane basis spanned by $(\mathbf{m}(\theta), \mathbf{n})$. For non-zero wavevectors $\eta$ of magnitude $\eta$, the standard and general solution of eq. (3.171) can be presented as follows

$$ 
\begin{aligned}
\tilde{u}_j (\eta, z) &= e^{-i2\pi p \eta z} a_j \\
\tilde{t}_j (\eta, z) &= -i2\pi \eta e^{-i2\pi p \eta z} b_j,
\end{aligned}
\qquad (3.174) $$

where $p$ is the eigenvalue, and $\mathbf{a} = \{a_1, a_2, a_3, a_4, a_5\}^\mathrm{t}$ and $\mathbf{b} = \{b_1, b_2, b_3, b_4, b_5\}^\mathrm{t}$ are the corresponding complex eigenvectors of the following linear ten-dimensional eigensystem, i.e.

$$ \begin{bmatrix} -\mathbf{T}^{-1}\mathbf{R}^{\mathrm{t}} & \mathbf{T}^{-1} \\ -\mathbf{Q} + \mathbf{R}\mathbf{T}^{-1}\mathbf{R}^{\mathrm{t}} & -\mathbf{R}\mathbf{T}^{-1} \end{bmatrix} \begin{bmatrix} \mathbf{a} \\ \mathbf{b} \end{bmatrix} = p \begin{bmatrix} \mathbf{a} \\ \mathbf{b} \end{bmatrix}, \qquad (3.175) $$

after substituting eq. (3.174) into eq. (3.171). The Stroh eigenvalues of eq. (3.175) and the corresponding eigenvectors are conveniently arranged such that $Im p_j > 0$, and $p_{j+5} = \bar{p}_j$, because the remaining five solutions have negative imaginary parts due to the positive definiteness of the magnetic, electric, and elastic strain energy densities. Here and in the following, the overbar denotes the complex conjugate. By superposing the ten eigensolutions, general expressions of the extended displacements and tractions in the Fourier-transformed domain can therefore be expressed in terms of the Stroh formalism in any given layer $j$ bonded by interfaces $z_{j-1}$ and $z_j$, as follows

$$ \begin{bmatrix} -i2\pi\eta \tilde{u}(\eta, z) \\ \tilde{t}(\eta, z) \end{bmatrix} = \begin{bmatrix} \mathbf{A} & \bar{\mathbf{A}} \\ \mathbf{B} & \bar{\mathbf{B}} \end{bmatrix} \begin{bmatrix} \langle e^{-i2\pi p_+ \eta (z-z_j)} \rangle & \mathbf{0}_{5 \times 5} \\ \mathbf{0}_{5 \times 5} & \langle e^{-i2\pi \bar{p}_+ \eta (z-z_{j-1})} \rangle \end{bmatrix} \begin{bmatrix} \mathbf{K}_1 \\ \mathbf{K}_2 \end{bmatrix}, \qquad (3.176) $$

with $z_{j-1} < z < z_j$, while $\mathbf{A}$ and $\mathbf{B}$ the $5 \times 5$ eigenvector matrices defined by

$$ 
\begin{aligned}
\mathbf{A} &= [a_1, a_2, a_3, a_4, a_5] \\
\mathbf{B} &= [b_1, b_2, b_3, b_4, b_5] = \mathbf{R}^{\mathrm{t}}\mathbf{A} + \mathbf{T}\mathbf{A} \langle e^{i2\pi p_+ \eta (z-z_j)} \rangle,
\end{aligned}
\qquad (3.177) $$

where the z-dependent diagonal and exponential matrix in eq. (3.177) is represented by

$$ \langle e^{i2\pi p_+ \eta (z-z_j)} \rangle = \mathrm{diag} \left[ e^{i2\pi p_1 \eta (z-z_j)}, e^{i2\pi p_2 \eta (z-z_j)}, e^{i2\pi p_3 \eta (z-z_j)}, e^{i2\pi p_4 \eta (z-z_j)}, e^{i2\pi p_5 \eta (z-z_j)} \right], \qquad (3.178) $$

and $\mathbf{K}_1$ and $\mathbf{K}_2$ in eq. (3.176) are two $5 \times 1$ complex (and constant) column matrices to be determined by specific boundary conditions in dislocated MEE multilayers. Once the extended displacement $\tilde{u}$ and traction $\tilde{t}$ vectors in the transformed domain are obtained by solving eq. (3.176), the remaining $7 \times 1$ extended in-plane stresses $\tilde{\sigma}^s$ in the transformed domain, i.e., consisting of the in-plane elastic stresses, electric, and magnetic displacements, can be found by using the following relation, i.e.

$$ \tilde{\sigma}_{ij}^s (\eta, z) = -i2\pi\eta c_{ijKl} m_l \tilde{u}_K (\eta, z) + c_{ijKl} m_l \tilde{u}_{K,3} (\eta, z), \qquad (3.179) $$

with $i = \{1, 2\}$, $J = \{1, 2, 4, 5\}$, and $i \le J$. The derivative term on the right-hand side of eq. (3.179) is given in terms of the extended displacements and tractions in the transformed domain by

$$ \tilde{u}_{K,3} (\eta, z) = [c_{ijKl} n_l n_i]^{-1} (\tilde{t}_J (\eta, z) + i2\pi\eta c_{ijKl} m_l n_i \tilde{u}_K (\eta, z)), \qquad (3.180) $$

for which eqs. (3.179) and (3.180) read in vector-tensor form as

$$ 
\begin{aligned}
\tilde{\sigma}^s (\eta, z) &= -i2\pi\eta \mathbf{M}_1 \tilde{u} (\eta, z) + \mathbf{M}_2 \tilde{u}_{,3} (\eta, z) \\
&\equiv \{\tilde{\sigma}_{11}, \tilde{\sigma}_{12}, \tilde{\sigma}_{22}, \tilde{\sigma}_{14} = \tilde{D}_1, \tilde{\sigma}_{24} = \tilde{D}_2, \tilde{\sigma}_{15} = \tilde{B}_1, \tilde{\sigma}_{25} = \tilde{B}_2\}
\end{aligned}
\qquad (3.181) $$
respectively. The two 7 × 5 matrices M₁ and M₂ in eq. (3.181) are explicitly given by

$$
\mathbf{M}_1 = \begin{bmatrix} c_{11K1}m_1 + c_{11K2}m_2 \\ c_{12K1}m_1 + c_{12K2}m_2 \\ c_{22K1}m_1 + c_{22K2}m_2 \\ c_{14K1}m_1 + c_{14K2}m_2 \\ c_{24K1}m_1 + c_{24K2}m_2 \\ c_{15K1}m_1 + c_{15K2}m_2 \\ c_{25K1}m_1 + c_{25K2}m_2 \end{bmatrix} = \begin{bmatrix} c_{11}m_1 + c_{16}m_2 & c_{16}m_1 + c_{12}m_2 & c_{15}m_1 + c_{14}m_2 & e_{11}m_1 + e_{21}m_2 & q_{11}m_1 + q_{21}m_2 \\ c_{61}m_1 + c_{66}m_2 & c_{66}m_1 + c_{62}m_2 & c_{65}m_1 + c_{64}m_2 & e_{16}m_1 + e_{26}m_2 & q_{16}m_1 + q_{26}m_2 \\ c_{21}m_1 + c_{26}m_2 & c_{26}m_1 + c_{22}m_2 & c_{25}m_1 + c_{24}m_2 & e_{12}m_1 + e_{22}m_2 & q_{12}m_1 + q_{22}m_2 \\ e_{11}m_1 + e_{16}m_2 & e_{16}m_1 + e_{12}m_2 & e_{15}m_1 + e_{14}m_2 & -\epsilon_{11}m_1 - \epsilon_{12}m_2 & -\alpha_{11}m_1 - \alpha_{12}m_2 \\ e_{21}m_1 + e_{26}m_2 & e_{26}m_1 + e_{22}m_2 & e_{25}m_1 + e_{24}m_2 & -\epsilon_{21}m_1 - \epsilon_{22}m_2 & -\alpha_{21}m_1 - \alpha_{22}m_2 \\ q_{11}m_1 + q_{16}m_2 & q_{16}m_1 + q_{12}m_2 & q_{15}m_1 + q_{14}m_2 & -\alpha_{11}m_1 - \alpha_{12}m_2 & -\mu_{11}m_1 - \mu_{12}m_2 \\ q_{21}m_1 + q_{26}m_2 & q_{26}m_1 + q_{22}m_2 & q_{25}m_1 + q_{24}m_2 & -\alpha_{21}m_1 - \alpha_{22}m_2 & -\mu_{21}m_1 - \mu_{22}m_2 \end{bmatrix},
\qquad (3.182)
$$

indexed in Voigt notation, in any homogeneous layer.

The present dual variable and position procedure for multilayered structures aims at express-
ing recursive relations for the Fourier expansion coefficients between coherent and semicoherent
interfaces, instead of conventionally solving the entire 10 * 10 system that is composed of the
full-field solutions from all layers, as recently adopted for surface loads in transversely isotropic
layered solids [169] and thermoelasticity of multilayered plates [266, 263]. Substituting z by z<sub>j-1</sub>
and z<sub>j</sub> into the linear system in eq. (3.176), it gives rise to

$$
\begin{bmatrix} -i2\pi\eta \tilde{u}(\eta, z_{j-1}) \\ \tilde{t}(\eta, z_{j-1}) \end{bmatrix} = \begin{bmatrix} \mathbf{A} & \bar{\mathbf{A}} \\ \mathbf{B} & \bar{\mathbf{B}} \end{bmatrix} \begin{bmatrix} \langle e^{i2\pi p_+ \eta h_j} \rangle & \mathbf{0}_{5 \times 5} \\ \mathbf{0}_{5 \times 5} & \mathbf{I}_{5 \times 5} \end{bmatrix} \begin{bmatrix} \mathbf{K}_1 \\ \mathbf{K}_2 \end{bmatrix} \\
\begin{bmatrix} -i2\pi\eta \tilde{u}(\eta, z_j) \\ \tilde{t}(\eta, z_j) \end{bmatrix} = \begin{bmatrix} \mathbf{A} & \bar{\mathbf{A}} \\ \mathbf{B} & \bar{\mathbf{B}} \end{bmatrix} \begin{bmatrix} \mathbf{I}_{5 \times 5} & \mathbf{0}_{5 \times 5} \\ \mathbf{0}_{5 \times 5} & \langle e^{-i2\pi \bar{p}_+ \eta h_j} \rangle \end{bmatrix} \begin{bmatrix} \mathbf{K}_1 \\ \mathbf{K}_2 \end{bmatrix},
\quad (3.183)
$$

respectively. Both unknown complex vectors **K**₁ and **K**₂ in eq. (3.183) can then be eliminated to establish the relation between the expansion coefficients on both interfaces at z<sub>j-1</sub> and z<sub>j</sub> of the layer *j* of interest, i.e.

$$
\begin{bmatrix} -i2\pi\eta \tilde{u}(\eta, z_{j-1}) \\ \tilde{t}(\eta, z_j) \end{bmatrix} = [\mathbf{S}_{10 \times 10}^j] \begin{bmatrix} -i2\pi\eta \tilde{u}(\eta, z_j) \\ \tilde{t}(\eta, z_{j-1}) \end{bmatrix} = \begin{bmatrix} \mathbf{S}_{11}^j & \mathbf{S}_{12}^j \\ \mathbf{S}_{21}^j & \mathbf{S}_{22}^j \end{bmatrix} \begin{bmatrix} -i2\pi\eta \tilde{u}(\eta, z_j) \\ \tilde{t}(\eta, z_{j-1}) \end{bmatrix}, \quad (3.184)
$$

within which the ten-dimensional matrix $\mathbf{S}_{10 \times 10}^j$ is formulated as follows

$$
[\mathbf{S}_{10 \times 10}^j] = \begin{bmatrix} \mathbf{A} \langle e^{i2\pi p_+ \eta h_j} \rangle & \bar{\mathbf{A}} \\ \mathbf{B} & \bar{\mathbf{B}} \end{bmatrix} \begin{bmatrix} \mathbf{A} & \bar{\mathbf{A}} \\ \mathbf{B} & \langle e^{-i2\pi \bar{p}_+ \eta h_j} \rangle \end{bmatrix} \begin{bmatrix} \mathbf{A} \\ \bar{\mathbf{B}} \end{bmatrix} \langle e^{-i2\pi \bar{p}_+ \eta h_j} \rangle^{-1} . \quad (3.185)
$$

For the adjacent layer *j* + 1, the corresponding propagation of the expansion coefficient solu-
tions at both interfaces *z*<sub>j</sub> and *z*<sub>j+1</sub> yields therefore similar relations as eq. (3.184), i.e.

$$
\begin{bmatrix} -i2\pi\eta \tilde{u}(\eta, z_j) \\ \tilde{t}(\eta, z_{j+1}) \end{bmatrix} = [\mathbf{S}_{10 \times 10}^{j+1}] \begin{bmatrix} -i2\pi\eta \tilde{u}(\eta, z_{j+1}) \\ \tilde{t}(\eta, z_j) \end{bmatrix} = \begin{bmatrix} \mathbf{S}_{11}^{j+1} & \mathbf{S}_{12}^{j+1} \\ \mathbf{S}_{21}^{j+1} & \mathbf{S}_{22}^{j+1} \end{bmatrix} \begin{bmatrix} -i2\pi\eta \tilde{u}(\eta, z_{j+1}) \\ \tilde{t}(\eta, z_j) \end{bmatrix}, \quad (3.186)
$$

which can be combined with eq. (3.184) by properly assuming that the interface at z<sub>j</sub> between
the two layers is perfectly bonded, i.e., the transformed displacement and traction vectors are
continuous at $z = z_j$, as specified by

$$ C: \left\{ \begin{array}{l} \left[ \tilde{u}(\eta, z = z_j) \right]_{-}^{+} = \tilde{u}(\eta, z_{j+}) - \tilde{u}(\eta, z_{j-}) = \mathbf{0}_{5 \times 1} \\ \left[ \tilde{t}(\eta, z = z_j) \right]_{-}^{+} = \tilde{t}(\eta, z_{j+}) - \tilde{t}(\eta, z_{j-}) = \mathbf{0}_{5 \times 1} \end{array} \right. \qquad (3.187) $$

Thus, the following recursive relations between interfaces $z_{j-1}$ and $z_{j+1}$ can be derived as

$$ \begin{bmatrix} -i2\pi\eta \tilde{u}(\eta, z_{j-1}) \\ \tilde{t}(\eta, z_{j+1}) \end{bmatrix} = \begin{bmatrix} \mathbf{S}_{10 \times 10}^{j:j+1} \\ \tilde{t}(\eta, z_{j-1}) \end{bmatrix} = \begin{bmatrix} \mathbf{S}_{11}^{j:j+1} & \mathbf{S}_{12}^{j:j+1} \\ \mathbf{S}_{21}^{j:j+1} & \mathbf{S}_{22}^{j:j+1} \end{bmatrix} \begin{bmatrix} -i2\pi\eta \tilde{u}(\eta, z_{j+1}) \\ \tilde{t}(\eta, z_{j-1}) \end{bmatrix}, \quad (3.188) $$

where the superscripts $j:j+1$ means the resulting propagation matrix from layer $j$ to layer $j+1$, with the submatrices $\mathbf{S}_{10 \times 10}^{j:j+1}$ being expressed as

$$ \left\{ \begin{aligned} [\mathbf{S}_{11}^{j:j+1}] &= [\mathbf{S}_{11}^j \mathbf{S}_{11}^{j+1}] + [\mathbf{S}_{11}^j \mathbf{S}_{12}^{j+1}] [\mathbf{I}_{5 \times 5} - \mathbf{S}_{21}^j \mathbf{S}_{12}^{j+1}]^{-1} [\mathbf{S}_{21}^j \mathbf{S}_{11}^{j+1}] \\ [\mathbf{S}_{12}^{j:j+1}] &= [\mathbf{S}_{12}^j] + [\mathbf{S}_{11}^j \mathbf{S}_{12}^{j+1}] [\mathbf{I}_{5 \times 5} - \mathbf{S}_{21}^j \mathbf{S}_{12}^{j+1}]^{-1} [\mathbf{S}_{22}^j] \\ [\mathbf{S}_{21}^{j:j+1}] &= [\mathbf{S}_{21}^{j+1}] + [\mathbf{S}_{22}^{j+1}] [\mathbf{I}_{5 \times 5} - \mathbf{S}_{21}^j \mathbf{S}_{12}^{j+1}]^{-1} [\mathbf{S}_{21}^j \mathbf{S}_{11}^{j+1}] \\ [\mathbf{S}_{22}^{j:j+1}] &= [\mathbf{S}_{22}^{j+1}] [\mathbf{I}_{5 \times 5} - \mathbf{S}_{21}^j \mathbf{S}_{12}^{j+1}]^{-1} [\mathbf{S}_{22}^j]. \end{aligned} \right. \qquad (3.189) $$

For multilayers with a single semicoherent interface, the recursive relations in eq. (3.188) with eqs. (3.189) can be propagated from the bottom surface to the semicoherent interface and then from the semicoherent interface to the top surface, without causing numerical instability issues as obtained by the traditional propagation matrix method [266]. Using the specific displacement discontinuity conditions at the semicoherent interfaces and the traction-free boundary conditions at bottom and top surfaces, the involved unknown expansion coefficients can be numerically solved and propagated to any z-level to determine all z-dependent expansion coefficients of both the Fourier-transformed displacement and traction vectors. This procedure is explicitly derived for two practical traction-free multilayered structures with one and two semicoherent interfaces in the next section. By use of the superposition principle, external uniform loads acting on the bottom and/or top surfaces in the associated MEE solids can consistently be applied using similar recursive relations with perfectly bonded interfacial conditions and subsequently be superposed to the previous dislocation-induced field solutions. When these coefficients are solved by imposing internal and external boundary conditions, the ultimate operations are related to the summation of all the Fourier components altogether to obtain the general and complete full-field solutions in the physical domains by inverse Fourier transforms.

## Disregistry at semicoherent interfaces with core-spreading dislocation structures

Due to the two-dimensional periodicity of the interface dislocation structures for a given neighboring atomic plane, the relative displacement discontinuity condition at semicoherent interfaces at $z = z_j$ between layers $j$ and $j+1$, is defined in both physical and Fourier-transformed domains using a similar biperiodic Fourier series expansion to eq. (3.169), as follows

$$ \left\{ \begin{aligned} \left[ u_J(x_1, x_2, z = z_j) \right]_{-}^{+} &= u_J^p(x_1, x_2, z_j) \\ &= u_J(x_1, x_2, z = z_{j+}) - u_J(x_1, x_2, z = z_{j-}) = \text{Re} \sum_{\eta} e^{-i2\pi\eta_a x_a} \tilde{u}_J^p(n, m, z_j) \\ \left[ \tilde{u}_J(\eta, z = z_j) \right]_{-}^{+} &= \tilde{u}_J(\eta, z = z_{j+}) - \tilde{u}_J(\eta, z = z_{j-}) = \tilde{u}_J^p(n, m, z_j), \end{aligned} \right. \qquad (3.190) $$
with $u^p$ and $\tilde{u}^p$ being the prescribed relative displacement vectors (magnitudes and directions), expressed in the physical and Fourier-transformed domains [249, 250], respectively. The components of the wavevectors $\eta$ parallel to the interface in eq. (3.190) must fulfill the following relation, i.e.

$$ \eta_{\alpha}x_{\alpha} = \eta_{1}(n)x_{1} + \eta_{2}(m)x_{2} = \frac{n}{|\mathbf{p}_{1}|}x_{1} + \frac{m}{|\mathbf{p}_{2}|}x_{2}, \qquad (3.191) $$

by virtue of eq. (3.6), with $\phi = \pi/2$. For pure misfit heterophase interfaces that consist of orthogonal edge dislocation networks with zero interaction energy, the complete displacement jump is described by the superposition of two distinct one-dimensional sawtooth-shaped functions $u_1^p$ and $u_2^p$ with Fourier sine series in the physical domain, as defined in eq. (3.123). The corresponding Fourier-transformed displacement jumps $\tilde{u}_1^p$ and $\tilde{u}_2^p$ for each set of dislocations are therefore related to the total disregistry in eq. (3.190) by

$$ \tilde{u}^p(n, m, z_j) = \tilde{u}_1^p(n, z_j) + \tilde{u}_2^p(m, z_j) = -i \frac{(-1)^{n+1}}{\pi n} \mathbf{b}_1(z_j) - i \frac{(-1)^{m+1}}{\pi m} \mathbf{b}_2(z_j), \qquad (3.192) $$

where the z-dependent Burgers vectors are discretely localized at the interfaces. However, the cores of the misfit dislocations can spread at dissimilar boundaries for interfaces with low shear resistances [188, 124, 171]. Such compact dislocation cores can therefore be spread out by convoluting the discontinuity displacement conditions with specific spreading function on the interface plane to form a continuous distribution of the Burgers vectors. In the context of linear elasticity theory, two isotropic weighted Burgers vector density functions $\omega_{\gamma}(x_{\gamma})$, with $\gamma = \{1, 2\}$, are also introduced as follows

$$ *\mathbf{b}_{\gamma} = \int_{-\infty}^{\infty} \omega_{\gamma}(x_{\gamma}) \, dx_{\gamma} = \mathbf{b}_{\gamma} \int_{-\infty}^{\infty} \omega_{\gamma}(x_{\gamma}) \, dx_{\gamma}, \qquad (3.193) $$

which ensures that both the magnitude and the direction of the Burgers vectors remain unchanged, and $^*b_{\gamma} = b_{\gamma}$ when the density function is reduced to the delta function, i.e., $\omega_{\gamma}(x_{\gamma}) = \delta(x_{\gamma})$. In eq. (3.193) the pre-superscript * indicates the quantities that have been distributed (also, convoluted) by the weighted core-spreading functions. One-dimensional Gaussian distributions of dislocation cores are conveniently prescribed to represent the core-spreading dislocations for each independent set of interfacial dislocations, i.e.

$$ \omega_{\gamma}(x_{\gamma}) = \frac{e^{-x_{\gamma}^{2}/r_{\gamma}^{2}}}{r_{\gamma}\sqrt{\pi}}, \qquad (3.194) $$

where the standard deviation is $\sigma_{\gamma} = r_{\gamma}/\sqrt{2}$, with $r_{\gamma} > 0$ being the dislocation core radius parameters that regularize the classical compact dislocation cores. In practice, the same weighted core-spreading functions are applied to both sets of interfacial dislocations, so that $\omega_1 = \omega_2$, with $r_1 = r_2 = r_0$. Using the advantages offered by the convolution properties of Fourier series expansions, the weighted displacement jump for set 1 from eqs. (3.123) and (3.192) is defined as

$$ \begin{aligned} ^*\mathbf{u}_1^p(x_1, z_j) &= \mathbf{u}_1^p(x_1, z_j) * \omega_1(x_1) \equiv \int_{-\infty}^{\infty} \mathbf{u}_1^p(x_1 - x'_1, z_j) \omega_\gamma(x'_1) \, dx'_1 \\ &= \sum_{\substack{n \ge 1 \\ m=0}} \frac{(-1)^{n+1} e^{-(\pi n r_0 / p_1)^2}}{\pi n} \sin\left(\frac{2\pi n x_1}{p_1}\right) \mathbf{b}_1(z_j) \end{aligned} \qquad (3.195) $$

$$ ^*\tilde{u}_1^p(n, z_j) = -i \frac{(-1)^{n+1} e^{-(\pi n r_0 / p_1)^2}}{\pi n} \mathbf{b}_1(z_j), $$
respectively, while the corresponding displacement jumps for set 2 are analogously given by

$$ *\mathbf{u}_2^p(x_2, z_j) = \sum_{\substack{n=0 \\ m \ge 1}} \frac{(-1)^{m+1} e^{-(\pi m r_0 / p_2)^2}}{\pi m} \sin\left(\frac{2\pi m x_2}{p_2}\right) \mathbf{b}_2(z_j) \quad (3.196) $$

$$ *\tilde{\mathbf{u}}_2^p(m, z_j) = -i \frac{(-1)^{m+1} e^{-(\pi m r_0 / p_2)^2}}{\pi m} \mathbf{b}_2(z_j), $$

in both the physical and Fourier-transformed domains. The non-regularized discontinuous displacement vectors (given by eqs. (3.123) and (3.192)) are also obtained for $r_0 = 0$ in eqs. (3.195) and (3.196). Thus, the interface conditions S on the semicoherent interface associated with core-spreading dislocations in the Fourier-transformed domain are imposed by

$$ S: \left\{ \begin{array}{l} \left[ \left[ \tilde{\mathbf{u}}(\eta, z = z_j) \right]_- \right]^+ = \tilde{\mathbf{u}}(\eta, z_{j+}) - \tilde{\mathbf{u}}(\eta, z_{j-}) \\ = -i \frac{(-1)^{m+1} e^{-(\pi n r_0 / p_1)^2}}{\pi n} \mathbf{b}_1(z_j) - i \frac{(-1)^{m+1} e^{-(\pi m r_0 / p_2)^2}}{\pi m} \mathbf{b}_2(z_j) \quad (3.197) \\ \left[ \left[ \tilde{\mathbf{t}}(\eta, z = z_j) \right]_- \right]^+ = \tilde{\mathbf{t}}(\eta, z_{j+}) - \tilde{\mathbf{t}}(\eta, z_{j-}) = \mathbf{0}_{5 \times 1}, \end{array} \right. $$

for any {$n, m \ge 1$}, which are evidently reduced to eqs. (3.187) for coherent interfaces with zero Burgers vector content.

Figure (3.41a) shows the Gaussian density distribution $b_1 \omega_1(x_1)$ of the single discrete Burgers vector $b_1$, with arbitrarily given values for $b_1 = 0.32$ nm and $r_0 = 2.5 b_1$, while Fig. (3.41b) illustrates the corresponding displacement jumps across the interface with core-spreading dislocations: $*u_1^p$ (red curve, given by eq. (3.195)), and without: $u_1^p$ (black, with $r_0 = 0$). These curves are plotted with 20 harmonics only, with also arbitrarily dislocation spacings $p_1 = 7$ nm, exhibiting that the Fourier series expansion with the core-spreading treatment for interface dislocations converges conditionally and numerically faster than the original expansions without treatment. The relative displacement profile becomes therefore continuously smooth close to the regularized dislocation cores unlike the jump occurring in the original description with compact dislocation cores.

Once the specific interface conditions S in eqs. (3.197) dedicated to interface dislocation patterns are defined, the recursive relations in the layered sub-structures between the semicoherent interfaces and external surfaces can be propagated to obtain the field solutions in the Fourier domain. The following two practical examples give rise to the explicit recursive relations between the transformed displacement and traction vectors that are used for numerical application examples in multilayers with (i) one semicoherent interface, and (ii) two semicoherent interfaces. The multilayered cases of interest with three and more interfaces consist of a straightforward continuation of both subsequent situations with three and more additional recursive sequences.

(i) For a single semicoherent interface in multilayers, the transformed displacement and traction vectors are propagated from the bottom surface at $z = z_0$ to the lower side where the semicoherent is located, i.e., at $z = z_{j-}$, so that eq. (3.188) leads to

$$ \begin{bmatrix} -i2\pi\eta \tilde{\mathbf{u}}(\eta, z_0) \\ \tilde{\mathbf{t}}(\eta, z_{j-}) \end{bmatrix} = \begin{bmatrix} \mathbf{S}_{11}^{1:j} & \mathbf{S}_{12}^{1:j} \\ \mathbf{S}_{21}^{1:j} & \mathbf{S}_{22}^{1:j} \end{bmatrix} \begin{bmatrix} -i2\pi\eta \tilde{\mathbf{u}}(\eta, z_{j-}) \\ \tilde{\mathbf{t}}(\eta, z_0) \end{bmatrix}, \quad (3.198) $$

and also, from the upper side of the interface at $z = z_{j+}$ to the top surface at $z = z_w$, i.e.

$$ \begin{bmatrix} -i2\pi\eta \tilde{\mathbf{u}}(\eta, z_{j+}) \\ \tilde{\mathbf{t}}(\eta, z_w) \end{bmatrix} = \begin{bmatrix} \mathbf{S}_{11}^{j+1:w} & \mathbf{S}_{12}^{j+1:w} \\ \mathbf{S}_{21}^{j+1:w} & \mathbf{S}_{22}^{j+1:w} \end{bmatrix} \begin{bmatrix} -i2\pi\eta \tilde{\mathbf{u}}(\eta, z_w) \\ \tilde{\mathbf{t}}(\eta, z_{j+}) \end{bmatrix}, \quad (3.199) $$
where $S_{10 \times 10}^{1:j}$ and $S_{10 \times 10}^{j+1:w}$ are individually defined by eqs. (3.189). Equations (3.198) and (3.199) together with the given boundary/interface conditions S in eqs. (3.197) solve all the involved transformed-Fourier unknowns. An example for the bilayered system is provided in eq. (3.213) where the system of equations is reordered by arranging all the given quantities to the right-hand side and all the unknowns to be solved to the left-hand side. The field solutions can therefore be propagated to any z-level to determine the transformed propagating values of interest, e.g., using a relation similar to eq. (3.198) if the field point is above the semicoherent interface, or using a relation similar to eq. (3.199) if the field point is below the interface. Finally, operating the summation of the transformed solutions in the Fourier series expansions, the full-field solutions in the physical domain are obtained.

(ii) For two semicoherent interfaces in multilayers, located at $z = z_j$ (corresponding to the previous case (i)) and $z = z_k$ with $z_j < z_k$, the prescribed relative displacement is, in general, different than the interface at $z = z_j$ in terms of dislocation structures, so that

$$ \left[ \left[ \tilde{\mathbf{u}}(\eta, z = z_k) \right]_{-}^{+} = \tilde{\mathbf{u}}(\eta, z = z_{k+}) - \tilde{\mathbf{u}}(\eta, z = z_{k-}) = \right. \\ \left. \left( \neq \left[ \left[ \tilde{\mathbf{u}}(\eta, z = z_j) \right]_{-}^{+} \right) \right), \right] \quad (3.200) $$

with, for instance, different dislocation spacings and magnitudes of both Burgers vectors (but, with similar directions in the present pure misfit interface cases). Thus, while eq. (3.198) is unchanged, eq. (3.199) is split into two propagation relations, from the upper side $z_{j+}$ of the first semicoherent interface at $z_j$ to the lower side of the second semicoherent interface at $z = z_{k-}$, i.e.

$$ \left[ \begin{matrix} -i2\pi\eta \tilde{\mathbf{u}}(\eta, z_{j+}) \\ \tilde{\mathbf{t}}(\eta, z_{k-}) \end{matrix} \right] = \left[ \begin{matrix} \mathbf{S}_{11}^{j+1:k} & \mathbf{S}_{12}^{j+1:k} \\ \mathbf{S}_{21}^{j+1:k} & \mathbf{S}_{22}^{j+1:k} \end{matrix} \right] \left[ \begin{matrix} -i2\pi\eta \tilde{\mathbf{u}}(\eta, z_{k-}) \\ \tilde{\mathbf{t}}(\eta, z_{j+}) \end{matrix} \right], \quad (3.201) $$

and then, from the upper side of the second semicoherent interface at $z_{k+}$ to the top surface at $z = z_w$, i.e.

$$ \left[ \begin{matrix} -i2\pi\eta \tilde{\mathbf{u}}(\eta, z_{k+}) \\ \tilde{\mathbf{t}}(\eta, z_w) \end{matrix} \right] = \left[ \begin{matrix} \mathbf{S}_{11}^{k+1:w} & \mathbf{S}_{12}^{k+1:w} \\ \mathbf{S}_{21}^{k+1:w} & \mathbf{S}_{22}^{k+1:w} \end{matrix} \right] \left[ \begin{matrix} -i2\pi\eta \tilde{\mathbf{u}}(\eta, z_w) \\ \tilde{\mathbf{t}}(\eta, z_{k+}) \end{matrix} \right]. \quad (3.202) $$

Again, eqs. (3.198) and (3.201–3.202) combining with the prescribed boundary conditions in eqs. (3.197) and (3.200) are applied for solving the involved unknowns for the given boundary and interface conditions. After determining the involved boundary and interface values, the transformed displacement and traction vectors at any z-level are obtained by merely propagating the suitable recursive relation (depending upon the relative location of the field point with respect to the two semicoherent interface locations), while the physical-domain solutions are finally deduced by taking the summation of all the Fourier series components.

## Traction boundary conditions at external surfaces

The extended traction boundary conditions are vertically and uniformly applied on the top and bottom surfaces, at $z = z_w$ and $z = z_0$, respectively, as referred to Fig. (3.40b). The mechanical normal traction is described by imposing $t_3$, while the electric and magnetic components are characterized by $t_4$ and $t_5$, respectively. For simplicity, the normal traction components are homogeneously distributed along the $x_2$-axis, and uniformly imposed along the $x_1$-axis only, so that the present case is treated as a two-dimensional plane-strain deformation problem in the $(x_1, x_3)$-plane. In terms of the Cartesian coordinates attached to the present multilayered systems, the
<figure><img src="image_61.png" /><figcaption>FIGURE 3.41: The core-spreading operation for the internal dislocation networks at semicoherent interfaces. (a) The weighted Burgers vector distribution function *b₁, with r₀ = 2.5 b₁ and b₁ = 0.32 nm, as a function of x₁. (b) Disregistries in terms of the original relative displacement u₁p with compact dislocation cores (in black) and the convoluted displacement *u₁p (red) by the core-spreading dislocation function. Both illustrations are carried out with 20 harmonics and arbitrary dislocation spacings p₁ = 7 nm. (c) The prescribed traction boundary condition on the upper surface with 300 harmonics, l = 5 nm, L = 5l, and Γ = 1 (in N/m², C/m², or N/A.m).</figcaption></figure>

prescribed traction t_j^p at the top surface are expressed as

$$t_j^p(x_1, z = z_w) = \begin{cases} -\Gamma & \frac{L-l}{2} \le x_1 \le \frac{L+l}{2} \\ 0 & \text{otherwise,} \end{cases} \qquad (3.203)$$

for J = 3, 4, 5, only, and at the bottom surface as

$$t_j^p(x_1, z = z_0) = t_j^p(x_1, z = z_w). \qquad (3.204)$$

Thus, the same distribution is applied on the bottom surface to ensure the equilibrium condition of zero in vertical direction. In eqs. (3.203) and (3.204) the uniform mechanical, electric, or magnetic fields Γ are applied over the interval with total length l, while L is a reference size to translate the center of the loading area from the global coordinate center to avoid the singularity in the series expansion as can be observed below. Furthermore, the present numerical calculation indicates that L = 5l leads to rapid convergent series. Using the similar discrete Fourier series representation with the previous derivation for the semicoherent interfaces, the surface traction relation at z = z_w is consistently given by

$$t_j^p(x_1, z = z_w) = \operatorname{Re} i \sum_{n \ge 1} e^{-i\pi n x_1 / L} \tilde{t}_j^p(n, z = z_w), \qquad (3.205)$$

where the expansion coefficients $\tilde{t}_j^p$ can be obtained by multiplying both sides of eq. (3.205) by $\sin(\pi m x_1 / L)$, with m integer (i.e., making use of the periodicity of the sine function over the interval $[0, L]$) and integrating along $x_1$ from 0 to L at z = z_w, i.e.

$$\int_0^L \left[ \frac{t_j^p(x_1, z = z_w)}{-\Gamma} \right] \sin\left(\frac{\pi m x_1}{L}\right) dx_1 = \operatorname{Re} \int_0^L i \sum_{n \ge 1} e^{-i\pi n x_1 / L} \sin\left(\frac{\pi m x_1}{L}\right) \tilde{t}_j^p(n, z = z_w) dx_1, \qquad (3.206)$$
which also gives rise to

$$ \Gamma \left[ \frac{L}{\pi m} \cos\left(\frac{\pi m x_1}{L}\right) \right]_{L/2-l/2}^{L/2+l/2} = \frac{L}{2} \tilde{t}_j^p (m, z = z_w) \Leftrightarrow \tilde{t}_j^p (m, z = z_w) = -\frac{4\Gamma}{\pi m} \sin\left(\frac{\pi m}{2}\right) \sin\left(\frac{\pi m l}{2L}\right). \quad (3.207) $$

Thus, the traction boundary condition on the top surface can be expressed as

$$ t_j^p (x_1, z = z_w) = -\mathrm{Re} \sum_{n=1,3,5,...} i \frac{4\Gamma}{\pi n} \sin\left(\frac{\pi n}{2}\right) \sin\left(\frac{\pi n l}{2L}\right) e^{-i\pi n x_1/L}, \quad (3.208) $$

exhibiting a sum over positive odd integers, only.

Figure (3.41c) illustrates the prescribed traction $t_j^p$ from eq. (3.208) with 300 harmonics, and arbitrary values for $l = 5$ nm, $L = 5l$, and $\Gamma = 1$ (in N/m² if $J = 3$, C/m² if $J = 4$, or N/A.m if $J = 5$). It is shown that the external traction boundary condition that acts on the top surface is well-represented in terms of the Fourier series expansion, so that the external loads can consistently and similarly be described with respect to the boundary-value problem as for the semicoherent interface case. Therefore, for $J = 3, 4, 5$, only, the external load conditions L on both the top and the bottom surfaces in the Fourier-transformed domain are finally given by

$$ L: \begin{cases} \tilde{t}_j (n, z = z_w) = -\frac{4\Gamma}{\pi n} \sin\left(\frac{\pi n}{2}\right) \sin\left(\frac{\pi n l}{2L}\right) \\ \tilde{t}_j (n, z = z_0) = \tilde{t}_j (z = z_w), \end{cases} \quad (3.209) $$

where identical expansion coefficients under uniform pressure are applied on the bottom surface, at $z = z_0$. The particular boundary conditions F for free surfaces can also be taken into account by imposing $\Gamma = 0$ in eqs. (3.209), i.e.

$$ F: \begin{cases} \tilde{t}_j (n, z = z_w) = 0 \\ \tilde{t}_j (n, z = z_0) = 0. \end{cases} \quad (3.210) $$

Similar procedure as the internal semicoherent interfaces in section 3.8.1 can be derived for the present external load case to explicitly determine the corresponding displacement and traction field solutions at any z-level in all layers with perfectly bonded (i.e., coherent) interface conditions. Thus, the solutions in the Fourier-transformed domain at $z_f$ in layer j can be obtained from the following set of equations

$$ \begin{bmatrix} -i2\pi\eta \tilde{u}(n, z_0) \\ \tilde{t}(n, z_f) \end{bmatrix} = \begin{bmatrix} \mathbf{S}_{11}^{1:j} & \mathbf{S}_{12}^{1:j} \\ \mathbf{S}_{21}^{1:j} & \mathbf{S}_{22}^{1:j} \end{bmatrix} \begin{bmatrix} -i2\pi\eta \tilde{u}(n, z_f) \\ \tilde{t}(n, z_0) \end{bmatrix} \\ \begin{bmatrix} -i2\pi\eta \tilde{u}(n, z_f) \\ \tilde{t}(n, z_w) \end{bmatrix} = \begin{bmatrix} \mathbf{S}_{11}^{j:w} & \mathbf{S}_{12}^{j:w} \\ \mathbf{S}_{21}^{j:w} & \mathbf{S}_{22}^{j:w} \end{bmatrix} \begin{bmatrix} -i2\pi\eta \tilde{u}(n, z_w) \\ \tilde{t}(n, z_f) \end{bmatrix}, \quad (3.211) $$

which can be recast into the following linear system to be analytically solved for any $n \ge 1$, i.e.

$$ \begin{bmatrix} \mathbf{S}_{11}^{1:j} & \mathbf{0}_{5 \times 5} & -\mathbf{I}_{5 \times 5} & \mathbf{0}_{5 \times 5} \\ \mathbf{S}_{21}^{1:j} & -\mathbf{I}_{5 \times 5} & \mathbf{0}_{5 \times 5} & \mathbf{0}_{5 \times 5} \\ -\mathbf{I}_{5 \times 5} & \mathbf{S}_{12}^{j:w} & \mathbf{0}_{5 \times 5} & \mathbf{S}_{11}^{j:w} \\ \mathbf{0}_{5 \times 5} & \mathbf{S}_{22}^{j:w} & \mathbf{0}_{5 \times 5} & \mathbf{S}_{21}^{j:w} \end{bmatrix} \begin{bmatrix} -i2\pi\eta \tilde{u}(n, z_f) \\ \tilde{t}(n, z_w) \\ -i2\pi\eta \tilde{u}(n, z_0) \\ -i2\pi\eta \tilde{u}(n, z_w) \end{bmatrix} = \begin{bmatrix} -\mathbf{S}_{12}^{1:j} \tilde{t}(n, z_0) \\ -\mathbf{S}_{22}^{1:j} \tilde{t}(n, z_0) \\ \mathbf{0}_{5 \times 5} \\ \tilde{t}(n, z_w) \end{bmatrix}, \quad (3.212) $$

where the submatrices of $\mathbf{S}_{10 \times 10}^{1:j}$ and $\mathbf{S}_{10 \times 10}^{j:w}$ are defined in eqs. (3.189) and the associated boundary
<table><caption>TABLE 3.10: Properties of MEE materials [200] used in the application examples, with $LiNbO_3$ (piezoelectric lithium niobate, LNO), $BaTiO_3$ (pure piezoelectric barium titanate, BTO), and $CoFe_2O_4$ (pure magnetostric- tive cobalt ferrite, CFO). Three MEE material compositions made of BTO and CFO are indicated by $x$ BTO, where $x$ is the volume fraction ratio of BTO.</caption><thead><tr><th colspan="2">Properties</th><th colspan="6">Materials</th></tr><tr><th>Symbol</th><th>Unit</th><th>LNO</th><th>BTO</th><th>0.75 BTO</th><th>0.50 BTO</th><th>0.25 BTO</th><th>CFO</th></tr></thead><tbody><tr><td>c<sub>11</sub></td><td>GPa</td><td>203</td><td>166</td><td>196</td><td>225</td><td>256</td><td>286</td></tr><tr><td>c<sub>12</sub></td><td>GPa</td><td>53</td><td>77</td><td>101</td><td>125</td><td>149</td><td>173</td></tr><tr><td>c<sub>13</sub></td><td>GPa</td><td>75</td><td>78</td><td>101</td><td>124</td><td>147</td><td>170</td></tr><tr><td>c<sub>33</sub></td><td>GPa</td><td>243</td><td>162</td><td>189</td><td>216</td><td>243</td><td>269</td></tr><tr><td>c<sub>44</sub></td><td>GPa</td><td>60</td><td>43</td><td>44</td><td>44</td><td>48</td><td>45</td></tr><tr><td>ε<sub>31</sub></td><td>C/m<sup>2</sup></td><td>0.2</td><td>-4.4</td><td>-3.3</td><td>-2.2</td><td>-1.1</td><td>0</td></tr><tr><td>ε<sub>33</sub></td><td>C/m<sup>2</sup></td><td>1.3</td><td>18.6</td><td>14.0</td><td>9.3</td><td>4.6</td><td>0</td></tr><tr><td>ε<sub>15</sub></td><td>C/m<sup>2</sup></td><td>3.7</td><td>11.6</td><td>8.7</td><td>5.8</td><td>2.9</td><td>0</td></tr><tr><td>ε<sub>11</sub></td><td>10<sup>-9</sup> C<sup>2</sup>/N.m<sup>2</sup></td><td>0.39</td><td>11.2</td><td>8.4</td><td>5.6</td><td>2.9</td><td>0.1</td></tr><tr><td>ε<sub>33</sub></td><td>10<sup>-9</sup> C<sup>2</sup>/N.m<sup>2</sup></td><td>0.26</td><td>12.6</td><td>9.5</td><td>6.3</td><td>3.2</td><td>0.1</td></tr><tr><td>μ<sub>11</sub></td><td>10<sup>-4</sup> N.s<sup>2</sup>/C<sup>2</sup></td><td>0.05</td><td>0.05</td><td>1.51</td><td>2.97</td><td>4.44</td><td>5.90</td></tr><tr><td>μ<sub>33</sub></td><td>10<sup>-4</sup> N.s<sup>2</sup>/C<sup>2</sup></td><td>0.10</td><td>0.10</td><td>0.46</td><td>0.83</td><td>1.20</td><td>1.57</td></tr><tr><td>q<sub>31</sub></td><td>N/A.m</td><td>0</td><td>0</td><td>145</td><td>290</td><td>435</td><td>580</td></tr><tr><td>q<sub>33</sub></td><td>N/A.m</td><td>0</td><td>0</td><td>175</td><td>350</td><td>525</td><td>700</td></tr><tr><td>q<sub>15</sub></td><td>N/A.m</td><td>0</td><td>0</td><td>137</td><td>275</td><td>412</td><td>550</td></tr><tr><td>α<sub>11</sub></td><td>C/A.m</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td></tr><tr><td>α<sub>33</sub></td><td>C/A.m</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td></tr></tbody></table>

conditions are given by eq. (3.209). Once the solutions are found at any field point by considering the interval $[(L - l)/2, (L + l)/2]$ from the prescribed boundary condition in eq. (3.203), the corre- sponding solutions for the displacement and traction fields can simply be translated to the central loading case by replacing $x_1$ with $x_1 - L/2$.

Three analyses on the influence of misfit dislocations on the elastic, electric, and magnetic field solutions are discussed for finite-thickness bi-, tri-, and multi-layered MEE composite mate- rials, respectively, with and without applied mechanical loads. The first two-dimensional illus- trative case deals with one semicoherent interface only, while the subsequent three-dimensional systems are made of two semicoherent interfaces, for which each one contains two sets of interfa- cial dislocations. In the following, all terminal planes between two adjacent crystals are defined by $n \parallel (001)$ in the cube-on-cube orientation, where both $x_1 = [100]$ and $x_2 = [010]$ directions are parallel in the interface planes. The corresponding materials properties used in these examples are listed in Table 3.10.

### 3.8.2 A primary case: 2D bilayered composites

In the present two-dimensional bilayered structure without external loads, the lower layer con- sists of the ferromagnetic (spinel, layer 1) CFO and the upper layer of the ferroelectric (perovskite, layer 2) BTO, with $h_{CFO} = h_{BTO} = 10$ nm. Due to the moderately large 5% lattice mismatch in the CFO/BTO system, with lattice parameters $a_{CFO} = 0.838$ nm and $a_{BTO} = 0.399$ nm for CFO and BTO, respectively [298], a specific semicoherent interface with discrete edge dislocations is located between these two adjacent crystals. The straight parallel dislocations are defined along the $x_1$-axis, for which the infinitely long, straight, parallel dislocations are uniformly spaced by $p_1 = p_{FB} = 8.378$ nm, as predicted by the quantized Frank-Bilby equation. Here, the Burgers vectors are given by $b_1 = a_{||}[100]^t$ along the $x_1$-axis, where the reference in-plane lattice param- eter $a_{||}$ is determined using the procedure proposed in section 3.2.4 for purely elastic CFO/BTO bilayers, i.e., with electric and magnetic constants equal to zero. For this simple case, eqs. (3.201) and (3.202), combining with the specific interface conditions S in eq. (3.197) for a single set of
<figure><img src="image_62.png" /><figcaption>FIGURE 3.42: Illustration of some (elastic, electric, and magnetic) field solutions in CFO/BTO bilayers. (a) Cross-sectional contours of the elastic displacement component $u_3$ (in nm), the electrostatic $\phi$ (in V) and magnetostatic $\psi$ (in mC/s) potentials, with the regularized dislocation core parameter $r_0 = 0.3$ nm. Minimum (maximum) values are linearly displayed in blue (red), while the field solution values are equal to zero in gray. (b) The corresponding distribution of $u_3$, $\phi$, and $\psi$ with respect to $x_3$ at $x_1 = -p_1/2$, i.e., along the vertical z-axis depicted by the white lines in (a). The calculations are performed for the compact dislocation core case, i.e., $r_0 = 0$ (black curves), and the core-spreading case, with $r_0 = 0.3$ (red), and $r_0 = 2.5$ nm (blue). (c) Similar distribution of the stress component $\sigma_{13}$ (in GPa), the electric displacement component $D_3$ (in C/m$^2$), and the magnetic induction component $B_3$ (in Wb/m$^3$).</figcaption></figure>

dislocations, can be recast into the following global linear system, as

$$ \begin{bmatrix} \mathbf{0}_{5 \times 5} & -\mathbf{I}_{5 \times 5} & \mathbf{S}_{11}^{\text{BTO}} & \mathbf{S}_{12}^{\text{BTO}} \\ \mathbf{0}_{5 \times 5} & \mathbf{0}_{5 \times 5} & \mathbf{S}_{21}^{\text{BTO}} & \mathbf{S}_{22}^{\text{BTO}} \\ -\mathbf{I}_{5 \times 5} & \mathbf{S}_{11}^{\text{CFO}} & \mathbf{0}_{5 \times 5} & \mathbf{0}_{5 \times 5} \\ \mathbf{0}_{5 \times 5} & \mathbf{S}_{21}^{\text{CFO}} & \mathbf{0}_{5 \times 5} & -\mathbf{I}_{5 \times 5} \end{bmatrix} \begin{bmatrix} -i2\pi\eta \tilde{u}(n, z_0) \\ -i2\pi\eta \tilde{u}(n, z_{1-}) \\ -i2\pi\eta \tilde{u}(n, z_2) \\ \tilde{t}(n, z_1) \end{bmatrix} = \begin{bmatrix} -i2\pi\eta \tilde{u}(n, z_1) \\ \mathbf{0}_{5 \times 1} \\ \mathbf{0}_{5 \times 1} \\ \mathbf{0}_{5 \times 1} \end{bmatrix}, \quad (3.213) $$

which solves the Fourier-transformed unknowns, i.e., $\{\tilde{u}(n, z_0), \tilde{u}(n, z_{1-}), \tilde{u}(n, z_2), \tilde{t}(n, z_1)\}$, on both external boundaries as well as on the internal interface for all $n \ge 1$, with respect to the corresponding submatrices $S_{\alpha\beta}^{\text{CFO}}$ and $S_{\alpha\beta}^{\text{BTO}}$ for both individual materials, with also $z_0 = 0$, $z_1 = h_{\text{CFO}}$, and $z_2 = h_{\text{CFO}} + h_{\text{BTO}}$.

Figure (3.42) shows the dislocation-induced fields with three different core-spreading parameters, i.e., $r_0 = 0$, $r_0 = 0.3$ and $r_0 = 2.5$ nm, with 64 harmonics that are sufficient to accurately compute the elastic, electric, and magnetic field solutions. For illustration, Fig. (3.42a) displays the periodical contours of the elastic displacement component $u_3$ (in nm), the electrostatic $\phi$ (in
V) and magnetostatic $\psi$ (in mC/s) potentials, within the area of $(x_1, x_3) \in [-10 \text{ nm}, 10 \text{ nm}]^2$, for the intermediate value $r_0 = 0.3$ nm. It is also depicted that the presence of misfit dislocations generates strong short-range electrostatic and magnetostatic potentials in the neighborhood of the semicoherent interface. In particular, these profile should dramatically affect the magnetoelectric effect (induction of magnetization (polarization) by an electric (magnetic) field) and, in general, the coupling between the electric and magnetic fields in laminated piezoelectric/piezomagnetic layers, e.g., the influence of the interfacial dislocations on the effective magnetoelectric coupling coefficients $a_{11}$ and $a_{33}$ in such two-phase systems.

Figure (3.42b) exhibits the distribution of $u_3$, $\phi$ and $\psi$ for the three core-spreading parameters at $x_1 = -p_1/2$ nm, i.e., along the vertical z-axis, as depicted by the white lines in Fig. (3.42a). It is clearly demonstrated that the core widths reduce the intensity of all internal elastic, electric, and magnetic solution fields. For $r_0 = 2.5$ nm, the derivatives of the normal displacement close to the interfaces change in sign compared to the dislocation networks with compact cores, i.e., $r_0 = 0$ nm. All depicted solutions are continuous across the CFO/BTO interface with $r_0 \neq 0$, which is also the case for the elastic stress component $\sigma_{33}$ in Fig. (3.42c) that originally diverges for $r_0 = 0$ using the classical theory of dislocations. However, not all the field quantities are continuous across the semicoherent interfaces, due to the discrete definition of the materials properties along the z-direction, as illustrated in the next section 3.8.3. For $r_0 = 2.5$ nm, the elastic stress $\sigma_{33}$, electric displacement $D_3$ and magnetic induction $B_3$ concentrations are dramatically decreased close to the dislocations as well, as displayed in Fig. (3.42c), which reveals the main role of the spreading cores to release field concentrations produced by the topological interface defects.

### 3.8.3 Energy-based criterion for interlayers in A/B/A trilayers

This section aims at deriving an energy-based criterion of zero net work that computes the critical dislocation spacings and thicknesses of interlayers in heterogeneous MEE materials. Such nanoscale inhomogeneities are typical in strain-induced martensitic transformation domains in ferroelectric systems [127, 167, 96]. More generally, the misfit stabilization of these interlayers with intrinsic dislocation networks can play a decisive role in the self-assembled structural (e.g., precipitation hardening) and functional (e.g., conversion of energies stored in electric and magnetic fields) properties of nanoscale MEE heterostructures. The heterogeneous mechanical problem is also to estimate the complete strain energy stored in the interlayers in the presence of the lattice and stiffness mismatches as well as the interfacial dislocations.

#### From coherent to semicoherent state in trilayers

As shown in Fig. (3.43a) a trilayered A/B/A composite is considered, where the adjacent layers A (interlayer B) are characterized by a finite thickness $h_A$ ($h_B$) and the corresponding anisotropic MEE constants. In the present analysis, $h_A \gg h_B$, so that the thin rectangular-shaped interlayer is assumed to be associated with a large lattice mismatch with flat interfaces and small surface tension. Depending on the lattice constants, the layers A and B are under biaxial tension or compression, such that the coherency strain field in the interlayer B is defined by $\epsilon_B(a_{||}) = (a_{||} - a_B)/a_B$, with $a_{||}$ the reference in-plane lattice parameter for both interfaces in trilayers, which should be different from the previous reference lattice parameter $a_{||}$ in bilayered systems.

In the present analysis, the "reference state" is conceptually created by separating the three layers and by applying uniform distortions to each individual material, as depicted in Fig. (3.43b). After structurally (not chemically) bonding these three distorted layers, the ideal commensurate trilayer is formed, within which forces are needed on both fictitious interface planes to maintain the uniform coherent state, and also the corresponding one-to-one correspondence between lattice planes on the two sides of each interface. In this reference state, which has the interface structure of a single perfect crystal, the interface is also coherent, so that the three layers are in
<figure><img src="image_63.png" /></figure>

FIGURE 3.43: Schematic illustration of the decomposition of the total strain energy $E_t$ in a representative finite-thickness A/B/A trilayers. (a) A strained three-layered structure with specific anisotropic MEE properties is composed of two different types of (virtual and misfit) dislocation networks at the upper and lower semicoherent interfaces. The region R represents the unit three-layered cell, within which the total strain energy is decomposed and computed. (b) The three materials are separated, rotated, and strained, such that the common reference configuration (depicted in green) with the same in-plane lattice is described by uniform displacement gradients applied to A (blue arrows) and B (red arrows). (c) The ideal commensurate trilayer is formed after bonding the three individual solids with the presence of continuous infinitesimal dislocations (i.e., virtual dislocations) to maintain the uniform coherent state, i.e., the three materials are in perfect registry with each other across both interface planes. These two continuous distributions of fictitious infinitesimal dislocations with the same magnitude but opposite signs generate uniform distortions that are non-zero in the interlayer B (orange arrows) and are compensated (also, zero) in both materials A. (d) The atomic structures of both semicoherent interfaces lead to formation of networks of discrete misfit dislocations separated by the regions of coherency that decrease the stored strain energy. The corresponding superposition of the three operations gives rise to non-zero stresses that are short-ranged and heterogeneously distributed in the three layers. The relaxed mismatch strain energy that consists of separating (b) and bonding (c) the three layers is denoted by $E_m$, while $E_d$ is associated with the work done in forming the discrete dislocation networks. The white symbols in (d) correspond to the shifted upper dislocation network, i.e., to the specific cases 5 and 6 in Fig. (3.44).

perfect registry with each other across the interface planes. It is illustrated in Fig. (3.43c) that the continuity of the reference lattice is virtually maintained across the interfaces by the presence of continuous distributions of infinitesimal extrinsic dislocations. Because these two continuous distributions of fictitious infinitesimal dislocations are defined by the same magnitude but opposite signs, the non-zero distortions are added and uniformly distributed in the interlayer B only, and are compensated (also canceled) in both adjacent layers A. Finally, the discrete intrinsic dislocation arrays with short-range elastic fields only (i.e., free of far-field stresses) are superposed to reproduce the "natural state" that defines the semicoherent interfaces with non-uniform internal structures comprised of misfit dislocations, as depicted in Fig. (3.43d) with opposite signs compared to continuous distributions of infinitesimal dislocations. By deviating locally the continuity in the reference configuration, these discrete dislocations emerge to release the elastic stored energy in the heterostructures by alleviating the residual lattice-misfit strains from the ideal commensurable trilayers.

In practice, both semicoherent interfaces in this natural state have the same internal structures with two sets of dislocations, i.e., in terms of the dislocation spacings $p_1 = p_2$ and the magnitude of both Burgers vectors $b_1 = b_2 = b$, except that the directions are defined by $b_1^l = a_{|||}[100]^t$ and $b_2^l = a_{|||}[010]^t$ at the lower interface, and with opposite signs, $b_1^u = -a_{|||}[100]^t$ and $b_2^u = -a_{|||}[010]^t$ at the upper interface, such that the interlayers are formed by periodic arrays of dislocation dipoles in MEE trilayers.

## Coherency and dislocation-induced energies

In accordance with the aforementioned three-step strategy to characterize the heterophase interlayer B, the total energy $E_t$ per unit area that is contained in the elementary region R in Fig. (3.43a)
is conveniently expressed as

$$E_t = 2E_d + E_m, \qquad (3.214)$$

where $E_d$ is the stored dislocation-induced energy due to the heterogeneous short-range stresses generated by a single set of intrinsic dislocation dipoles at the upper and lower interfaces. On the other hand, $E_m$ in eq. (3.214) is the relaxed mismatch strain energy due to the differences in lattice parameter between layers A and B by introducing the continuous distributions of fictitious infinitesimal dislocations. The factor 2 in front of $E_d$ is associated with the second set of dislocation dipoles that is orthogonal to the first set with zero interaction energy, so that the following calculations can conveniently be described in two dimensions.

By taking the advantages of the translational periodicity for one set of interfacial dislocations and using the divergence theorem, the dislocation-induced energy contribution per unit area $E_d$ in eq. (3.214) reads

$$E_d = \frac{1}{2p_1} \int_R (\sigma_{ij}u_{i,j} - D_i\phi_{i,j} - B_i\psi_{i,j}) dx_1 dx_3 = \frac{1}{2p_1} \int_{\partial R} \sigma_{ij}\hat{n}_i u_j d\ell = -\frac{1}{2p_1} \int_{\rho(r)} \sigma_{ij}\hat{n}_i b_j d\ell, \quad (3.215)$$

without electric and magnetic charge densities. The complete dislocation-induced stress field in eq. (3.215) has been derived in the previous section 3.8.1, while $\partial R$ corresponds to the boundary of the periodic region $R$ and $\rho$ to the cut along the line between two discrete dislocations from the lower and upper interfaces, as depicted in Fig. (3.43a). The proper cut $\rho$ excludes the regions of compact dislocation cores by introducing an out-of-plane cutoff parameter $r$, so that the stress divergence near the dislocation cores is removed, with in practice: $r = r_0/4$. In the following calculations with core-spreading dislocations, however, this exclusion is not necessary to compute the line integrals, so that $r = 0$. Due to the periodicity of the traction and displacement on the external boundary $\partial R$ and the zero-traction conditions at the free surfaces, the specific traction is also reduced to the limiting stress $\sigma_{ij}$ acting on $\rho$, where $\hat{n}_j$ denotes the unit vector normal to $\rho$, as displayed in Fig. (3.43a). Evaluation of the integral in eq. (3.215) can also be performed using the appropriate dislocation-induced stresses, which intrinsically depend on the coupled elastic/electric/magnetic field solutions by virtue of eq. (3.164) and on the thicknesses of the three layers as well as the internal dislocation spacings.

Similarly to the work done by Willis and co-workers [283, 284], the relaxed mismatch energy $E_m$ in eq. (3.214) is considered as a result of the elastic superposition of the lattice-mismatched strain and the strain-annihilator fields generated by the continuous distribution of infinitesimal dislocations. On the one hand, the determination of the coherent reference state in nano-trilayers (in general, nano-multilayers) would necessitate atomistics simulations because of the complexity of inhomogeneous anisotropic MEE trilayered systems with finite thicknesses. For large (but finite) thicknesses of layers A, however, the lattice parameter of material A can reasonably be selected as the reference state, so that $a_{||} = a_A$, and also the coherency strain field in A is $\epsilon_A(a_A) = 0$, yielding to zero uniform distortions applied in both layers A in Fig. (3.43b), while the corresponding field in B is $\epsilon_B(a_A) = (a_A - a_B)/a_B = f_m$. As discussed in section 3.2, the continuous distribution of fictitious dislocations with infinitesimal Burgers vectors and spacings can be represented by a linear (macroscopic) displacement field in $x_1$, as $b_i x_1 / p_1$, which generates a corresponding uniform distortion, i.e., $(b_i \hat{n}_j + b_j \hat{n}_i) / 2p_1$. Hence, the genuine mismatch energy $E_m$ is given by

$$E_m = \frac{1}{2} B c_{ijkl} B \epsilon_{kl}^m B \epsilon_{ij}^m h_B, \qquad (3.216)$$

with $B c_{ijkl}$ the elastic constants of the interlayer B, and $B \epsilon_{ij}^m$ the relaxed mismatch strain field defined by

$$B \epsilon_{ij}^m = f_m \delta_{ij} + \frac{b_i \hat{n}_j + b_j \hat{n}_i}{p_1}, \qquad (3.217)$$
<table><caption>TABLE 3.11: Different configurations in trilayered A/B/A composites, where the specific characteristics are schematically illustrated in Fig. (3.43a) with $h_B$ being the middle layer thickness, $p_1$ the inter-dislocation spacing, and $r_0$ the core-spreading parameter. These configurations are applied to both BTO/CFO/BTO and CFO/BTO/CFO stacking sequences. In addition, cases 5 and 6 correspond to case 4 in the BTO/CFO/BTO and CFO/BTO/CFO trilayers, respectively, within which the upper dislocation array is shifted by half dislocation spacing with respect to the lower dislocation array, as depicted in Fig. (3.43d).</caption><thead><tr><th>case 1*</th><th>case 1**</th><th>case 2*</th><th>case 2**</th><th>case 3*</th><th>case 3**</th><th>case 4*</th><th>case 4**</th></tr></thead><tbody><tr><td>h<sub>B</sub> = 2 nm</td><td>h<sub>B</sub> = 2 nm</td><td>h<sub>B</sub> = 12 nm</td><td>h<sub>B</sub> = 12 nm</td><td>p<sub>1</sub> = 8.378 nm</td><td>p<sub>1</sub> = 8.378 nm</td><td>p<sub>1</sub> = 12 nm</td><td>p<sub>1</sub> = 12 nm</td></tr><tr><td>r<sub>0</sub> = 0.3 nm</td><td>r<sub>0</sub> = 2.5 nm</td><td>r<sub>0</sub> = 0.3 nm</td><td>r<sub>0</sub> = 2.5 nm</td><td>r<sub>0</sub> = 0.3 nm</td><td>r<sub>0</sub> = 2.5 nm</td><td>r<sub>0</sub> = 0.3 nm</td><td>r<sub>0</sub> = 2.5 nm</td></tr><tr><td>varying p<sub>1</sub></td><td>varying p<sub>1</sub></td><td>varying p<sub>1</sub></td><td>varying p<sub>1</sub></td><td>varying h<sub>B</sub></td><td>varying h<sub>B</sub></td><td>varying h<sub>B</sub></td><td>varying h<sub>B</sub></td></tr></tbody></table>

<table><caption>TABLE 3.12: Critical values (in nm) for the six different cases (see text for details), i.e., the critical dislocation spacings for cases 1 and 2, while the others deal with the critical thicknesses of the interlayer. Numerical calculations are performed for layered MEE structure made of three layers, with both BTO and CFO materials.</caption><thead><tr><th rowspan="2">Trilayers</th><th colspan="10">Cases</th></tr><tr><th>1*</th><th>1**</th><th>2*</th><th>2**</th><th>3*</th><th>3**</th><th>4*</th><th>4**</th><th>5*</th><th>5**</th><th>6*</th><th>6**</th></tr></thead><tbody><tr><td>BTO/CFO/BTO</td><td>22.28</td><td>14.95</td><td>12.20</td><td>10.01</td><td>~ ∞</td><td>~ ∞</td><td>12.92</td><td>4.44</td><td>17.26</td><td>4.45</td><td></td><td></td></tr><tr><td>CFO/BTO/CFO</td><td>28.53</td><td>17.75</td><td>28.53</td><td>10.73</td><td>~ ∞</td><td>~ ∞</td><td>23.54</td><td>9.26</td><td></td><td></td><td>26.99</td><td>12.06</td></tr></tbody></table>

which is homogeneously distributed in the interlayer B for two sets of continuous distributions of dislocation dipoles.

## Critical dislocation spacings and interlayer thicknesses

In the following calculations, the layered MEE structure is made of three layers, with A = BTO and B = CFO, for which two different stacking sequences are discussed, i.e., the BTO/CFO/BTO and CFO/BTO/CFO trilayers. The thicknesses of both adjacent layers A are fixed and sufficiently large compared to the dislocation spacings predicted by the Frank-Bilby equation in bilayered systems, i.e., $h_A = 40$ nm. In extremely thin (but stable) multiferroics and miniaturized magnetoelectric memory devices, the relations and size effects between the interfacial dislocation spacings and the interlayer thickness $h_B$ become desirable for novel technological paradigms by dislocation engineering. In particular, the estimate of the critical quantities $κ_c$, e.g., dislocation spacings and interlayer thicknesses, are obtained by finding the values $κ_c$ such that eq. (3.214) yields to

$$E_t(\kappa_c) = 2E_d(\kappa_c) + E_m(\kappa_c) = 0, \qquad (3.218)$$

exhibiting an energy balance criterion between the dislocation-induced energy contribution and the relaxed mismatch strain energy from the perfectly coherent trilayered state. Based on a comparison of energy states in eq. (3.218), the critical values for the inter-dislocation distances and thicknesses correspond to the situation where the background relaxed mismatch stress is completely balanced by the stresses generated by the misfit dislocations. Thus, the zero total energy criterion leads to the structural characteristics in trilayers for which the coherently strained interlayer is stabilized by the presence of discrete misfit dislocations at both interfaces of the interlayers.

Figures (3.44a) and (b) illustrate the determination of the critical dislocation spacings and interlayer thicknesses, respectively, by plotting the dislocation stored energy $2E_d$ (black curves), the relaxed mismatch strain energy $E_m$, and the total energy $E_t = 2E_d + E_m$, for six different cases. All energy contributions are expressed in J/m². For the given i'th case, the solid (dotted) curves are associated with the BTO/CFO/BTO (CFO/BTO/CFO) trilayers, while the sub-case i* (i***) is related to field solutions with a core parameter $r_0 = 0.3$ nm ($r_0 = 2.5$ nm), also called here: "condensed dislocation cores" ("core-spreading dislocations"). Thus, the present results exclude the
<figure><img src="image_64.png" /></figure>

FIGURE 3.44: Estimate of critical quantities in MEE trilayers, i.e., dislocation spacings in (a) and thicknesses in (b), for different cases (see text for details of these cases) with condensed dislocations and core-spreading cores. The coherency energy $E_m$ (red curves) can be recovered (except for case 3) by the work done in forming the discrete dislocation networks $E_d$ (black curves), such that the critical quantities are obtained when the total strain energy $E_t$ (blue curves) is zero, as depicted by the vertical solid and dotted arrows. The results for the BTO/CFO/CFO (CFO/BTO/CFO) systems are indicated with solid (dotted) lines. The specific case 5 (6) corresponds to case 4, within which the upper dislocation array is shifted by half the dislocation spacings in the BTO/CFO/CFO (CFO/BTO/CFO) system, as displayed in Fig. (3.43d). For comparison, the thin solid and dotted lines in cases 5 and 6 indicate the results from cases 4* and 4**, respectively.

calculations with the unrealistic compact dislocation cores, i.e., calculations with $r_0 = 0$ nm. The considered cases are:

- Cases 1 and 2 exhibit the effect of the dislocation spacings $p_1$ on the energy profiles, with fixed finite thickness for the interlayers B (= CFO or BTO, depending on the stacking sequence), i.e., $h_B = 2$ nm and $h_B = 12$ nm, respectively. Thus, the specific calculations of case 1** are performed with $r_0 = 2.5$ nm and $h_B = 12$ nm.

- Cases 3 and 4 illustrate the influence of the intermediate thicknesses $h_B$ on the energy profiles, with fixed dislocation spacings, i.e., $p_1 = p_{FB} = 8.378$ nm and $p_1 = 12$ nm, respectively.

- Case 5 (case 6) corresponds to the previous case 4, within which the upper dislocation array is shifted by half the dislocation spacings $p_1$ with respect to the unchanged lower dislocation network in the BTO/CFO/BTO (CFO/BTO/CFO) trilayer, as displayed by dislocations in white at the upper interface in Fig. (3.43d).

Table 3.11 summarizes the aforementioned configurations, while Table 3.12 reports the predictions of the critical quantities for the different cases, obtained from Fig. (3.44) when $E_t(\kappa_c) = 0$.
<figure><img src="image_65.png" /><figcaption>FIGURE 3.45: Comparison of elastic, electric, and magnetic field quantities induced by dislocation networks with condensed dislocation cores (contours in left-hand sides, with $r_0 = 0.3$ nm) and with core-spreading regions (right-hand sides, with $r_0 = 2.5$ nm) in A/B/A trilayers, with A = BTO and B = CFO, i.e., the elastic displacement components $u_1$ and $u_3$ (from -0.01 to 0.01 nm), the electrostatic $\phi$ (from -0.01 to 0.01 V) and magnetostatic $\psi$ (-0.01 to 0.01 mC/s) potentials, the electric displacement component $D_3$ (-0.2 to 0.2 C/m$^2$), the magnetic induction component $B_3$ (-8 to 8 Wb/m$^3$), and the stress components $\sigma_{11}$, $\sigma_{33}$, $\sigma_{12}$, and $\sigma_{13}$ (in GPa). Minimum (maximum) values are linearly displayed in blue (red), while the field solution values are equal to zero in gray.</figcaption></figure>
<figure><img src="image_66.png" /></figure>

FIGURE 3.46: Variation of elastic displacement component $u_3$ (in nm) and stress components $\sigma_{11}$, $\sigma_{12}$, and $\sigma_{33}$ (in GPa) with respect to $x_3$ at $x_1 = 0$, i.e., along the vertical z-axis, midway between two interfacial dislocation dipoles in Fig. (3.45). Both misfit dislocation arrays have Burgers vectors with the same magnitudes, but with opposite signs. Calculations are performed for interfacial dislocations with condensed dislocation cores (plots in left-hand sides) and spread cores (right-hand sides) in trilayered BTO/CFO/CFO (blue/red/blue) and CFO/BTO/CFO (red/blue/red) systems. The results are illustrated for cases 3\* and 3\*\*, with specific thicknesses, i.e., $h_B = 5$ nm (black curves) and $h_B = 12$ nm (red curves), as depicted by the black \* and red \* asterisks in Fig. (3.44).

Comparing rows 2 and 3 in Table 3.12, it is concluded that the largest critical values are always associated with the CFO/BTO/CFO sequences, as displayed by the vertical dotted versus solid arrows in Fig. (3.44). Here, BTO is elastically softer than CFO, which therefore reduces the magnitude of $E_m$ for the CFO/BTO/CFO trilayers, compared to the BTO/CFO/BTO ones. On the other hand, the stacking MEE sequence has less influence on dislocation-induced energy $E_d$ than the coherency energy $E_m$, especially for case 2, even though the elastic constants for these two materials are considerably different, as listed in Table 3.10. In contrast to purely elastic calculations, it is worth remembering that the present predictions result from the elastic/electric/magnetic coupling phenomenon, which also resorts to the coupled constitutive relation in eq. (3.164) with three distinct (elastic, electric, and magnetic) contributions. Whereas the first elastic part gives rise to different stress distributions from both stacking sequences, the piezoelectric and piezomagnetic terms are therefore able to counterbalance the stress difference that is generated using the purely elastic constitutive relations alone. For both cases 1 and 2 with fixed thicknesses, the positive coherency energy decreases (increases) when $p_1 < \hat{p}_1$ ($> \hat{p}_1$), and is equal to zero when $p_1 = \hat{p}_1$, i.e., $E_m(\hat{p}_1) = 0$. The latter corresponds to the fully relaxed mismatch strain case, i.e., $B_e \epsilon_{ij}^m = 0$ in eq. (3.217), for which the interlayers are entirely accommodated by the continuous distribution of virtual dislocations.

Because the core-spreading regions affect the short-range stress concentration close to the interfaces (not the coherency energy $E_m$) the dislocation-induced energy $E_d$ is reduced in magnitude when the core-spreading parameter $r_0$ increases, so that $E_m$ becomes more dominant than $E_d$ for large values of the regularized dislocation cores. Furthermore, the energy variations show that $E_d$ decreases monotonically in magnitude when increasing $p_1$ for the condensed cores (e.g.,
<figure><img src="image_67.png" /><figcaption>FIGURE 3.47: Similar illustration as in Fig. (3.46) for electrostatic potential $\phi$ (in V), electric displacement $D_3$ (in C/s$^2$), magnetostatic potential $\psi$ (in mC/s), and magnetic induction $B_3$ (in Wb/m$^3$).</figcaption></figure>

cases 1\* and 2\*), while $E_d$ becomes fairly constant with respect to $p_1$ for core-spreading dislocations (cases 1\*\* and 2\*\*). Significant differences between these two profiles are observed for very small dislocation spacings (equivalently, for high interfacial dislocation densities). Case 1 versus 2 illustrates that the critical inter-dislocation distance (density) decreases (increases) with increasing the thickness of the interlayers for both sequences, which is qualitatively in accordance with experimental investigations in bilayers [183]. Case 3 shows that $E_m \approx 0$ when $p_1 = p_{FB}$, while $E_d$ decreases slowly in magnitude with increasing $h_B$, so that no critical thicknesses are reached. This theoretical result suggests that the equilibrium inter-dislocation distances are larger in finite-thickness trilayers than in the semi-infinite bicrystals. Case 3 versus 4 demonstrates that the critical interlayer thicknesses decrease when increasing the dislocation spacings for both MEE sequences and both condensed and core-spreading dislocations, which is due to stronger elastic interactions for high dislocation spacings. Again, case 4 illustrates that the core-spreading regions have a great influence on the determination of the critical interlayer thicknesses. For the shifted case 5 versus 6, the critical values for thicknesses are larger than the previous unshifted cases, except for case 4\*\* that has the same value as case 5\*\* for large core-spreading dislocation widths. Furthermore, $E_d$ is close to zero for small thickness values, as qualitatively expected using the classical theory of dislocation dipoles.

## Size effects on the coupled MEE field solutions in trilayers

Figure (3.45) illustrates the influence of the spreading dislocation cores on various elastic, electric, and magnetic solution field components for both particular cases 3\* and 3\*\* (i.e., with $r_0 = 0.3$ nm and $r_0 = 2.5$ nm, respectively), where $h_B = h_{CFO} = 12$ nm are comparable with the internal inter-dislocation spacings. This MEE system is identified by the red asterisk in Fig. (3.44). Similarly to the previous primary bilayered case, the general tendency is that the spreading-core regions release significantly the aforementioned solution fields in magnitude, such that the larger the spreading widths are, the lower the complex distribution and concentration of these elastic, electric, and magnetic quantities become (especially close to the interfaces). Thus, the present core-spreading treatment can therefore be regarded as flattening and stretching operations in the $z$- and
$x$- directions, respectively, of the released elastic, electric, and magnetic concentration originated from the compact cores. Interestingly, whereas the electrostatic and magnetostatic potentials are non-zero in both materials, the electric displacement $D_3$ and the magnetic induction $B_3$ are strictly equal to zero in the magnetostrictive CFO and piezoelectric BTO layers, respectively. The theoretical coexistence of these highly localized electric and magnetic characteristics that emerge from the interfacial dislocations should unambiguously produce remarkable effects on the electric and magnetic properties in MEE heterostructures, as substantial energy electron fluxes in laminated structures. This suggests also that interfacial dislocation networks cannot always be considered as detrimental, but can present an opportunity to enhance the material performance, and to produce exceptional/exotic performances through dislocation technological concepts.

All red curves in Fig. (3.46) illustrate the variation of the elastic displacement $u_3$ and stress components $\sigma_{11}$, $\sigma_{12}$, and $\sigma_{33}$ at $x_1 = 0$, i.e., along the vertical $z$-axis between two interfacial dislocations in Fig. (3.45), for which the interlayer thickness in both stacking sequences is $h_B = 12$ nm. For comparison, the black curves that correspond to the similar MEE system with $h_B = 5$ nm, identified by the black asterisk in case 3 from Fig. (3.44), are plotted as well (converted to the same interface locations for easy comparison, with a similar treatment in Fig. (3.47)). Calculations are performed for interfacial dislocations with condensed dislocation cores (plots in left-hand sides, with $r_0 = 0.3$ nm) and spreading-core dislocations (right-hand sides, with $r_0 = 2.5$ nm) in both trilayered BTO/CFO/BTO (blue/red/blue) and CFO/BTO/CFO (red/blue/red) systems. It can quantitatively be shown that the magnitudes in the normal displacement and stress field components are dramatically released by the core-spreading operations. The normal displacement between two misfit dislocations is continuous across both interfaces with similar characteristic double-well shaped profiles close to the internal boundaries as in Fig. (3.42b), with positive (negative) values at the lower (upper) interfaces. The elastic stress component $\sigma_{33}$ is continuous across the interfaces as well, as expected by the required boundary conditions in the MEE trilayers. The corresponding profiles of $\sigma_{33}$ are different in the interlayers for $h_B = 5$ nm (black curve) and $h_B = 12$ nm (red curve), for which the former (latter) exhibit parabolic (double-well) profiles in both CFO and BTO interlayers, with large differences in magnitude (that are reduced by spreading the dislocation cores). On the other hand, $\sigma_{11}$ and $\sigma_{12}$ in Fig. (3.46) are discontinuous across the interfaces, which is intrinsically ascribed by the heterogeneous elastic properties of the adjacent layers that differ from the interlayers. It can also be observed that the shear component in the middle layers is very sensible to the associated thicknesses, which increases with decreasing thicknesses due to the strong elastic interactions (and also, the superposition of $\sigma_{12}$) between both adjacent dislocation networks with opposite signs. The $\sigma_{11}$ component, however, results from the superposition of positive and negative regions produced by these adjacent networks, which yields to weaken size effects in the interlayers. It is reasonable to point out that such size effects in the interlayer thicknesses would considerably affect the glide and climb components of the Peach-Kohler force acting on lattice dislocations in the interlayers, and also the corresponding microscopic plastic deformation mechanisms and related macroscopic mechanical properties in MEE multilayers.

Figure (3.47) illustrates similar plots as in Fig. (3.46), but for electric quantities (electrostatic potential $\phi$, and electric displacement $D_3$) and for magnetic quantities (magnetostatic potential $\psi$ and magnetic induction $B_3$). All quantities are continuous across both semicoherent interfaces, and differences in profiles are more discernible between both stacking sequences for the electric and magnetic measures than the relatively small variations in the elastic fields, as excepted. Interestingly, the electric displacement $D_3$ and the magnetic induction $B_3$ have alternatively analogous profiles in CFO and BTO interlayers depending on the stacking sequence. Similar features as in the elastic variations in the parabolic versus double-well distributions are emphasized with respect to the interlayer thicknesses, which are extremely reduced by the spreading-core operations. Important size effects on the electric displacement $D_3$ (the magnetic induction $B_3$) are observed in the intermediate layer BTO (CFO) in the CFO/BTO/CFO (BTO/CFO/BTO) trilayers.
<figure><img src="image_68.png" /><figcaption>FIGURE 3.48: Distribution of superposed field quantities in the six-layered MEE materials along two lines in the $x_3$-direction, i.e., at $x_1 = 0$ (solid lines) and $x_1 = (p_1 + p_2)/4$ (dotted lines), as displayed in (a). The total field solutions (red curves) result from the superposition of the external load (blue curves) and the dislocation-induced (black curves) fields. Elastic (b) displacement $u_3$ (in nm) and (c) stress $\sigma_{33}$ (in GPa) components. Electric (d) potential $\phi$ (in V), (e) displacement $D_3$ (in C/s$^2$) and (f) field $E_3$ (in V/µm) components. Magnetic (g) potential $\psi$ (in mC/s), (h) induction $B_3$ (in Wb/m$^3$), and (i) field $H_1$ (in A/µm) components.</figcaption></figure>

### 3.8.4 Dislocation-induced response under applied external loading

Two three-dimensional MEE systems are investigated and compared, i.e., the tri- LNO/BTO/CFO (green/orange/maroon) and the six- LNO/BTO/0.75 BTO/0.50 BTO/0.25 BTO/CFO layered systems in the same cube-on-cube orientations as previously discussed, for which the lead-free ferroelectric LiNbO$_3$ (LNO) is a piezoelectric material as well as the widely used BTO material. Here, 0.25 BTO means 25% of BTO in the MEE composite made of BTO and CFO, so that the intermediate 0.75 BTO/0.50 BTO/0.25 BTO trilayer can be regarded as a buffer sequence to progressively accommodate the lattice mismatch between BTO and CFO. For these two cases, the same mechanical load is applied to both external surfaces and two semicoherent interfaces are considered, so that both lower and upper interfaces in the six-layered system are located between LNO and BTO, and, between 0.25 BTO and CFO, respectively. The following calculations aim at introducing the capabilities of the present framework to investigate the distribution of the elastic, electric, and magnetic field solutions in complex MEE multilayers under externally applied loads, with buffer sequences in presence of topological defects at two semicoherent interfaces.
<figure><img src="image_69.png" /><figcaption>FIGURE 3.49: Three-dimensional spatial distribution in the six-layered MEE composite of the (a) von Mises stress (in GPa), (b) electric displacement $D_1$ (in C/s²), and (c) magnetic induction $B_3$ (in Wb/m³) components.</figcaption></figure>

## Interaction between internal dislocation fields and externally mechanical loads

Both interfaces have different internal structures in terms of dislocation spacings and Burgers vectors. Here, the internal dislocation structure at the lower LNO/BTO interface is described by the same dislocation spacings as in the former studies in trilayers, i.e., $p_1^l = p_2^l = 8.378$ nm, with $b_1^l = b_2^l = a_{BTO} = 0.399$ nm, while the upper 0.25 BTO/CFO interface is characterized by a higher dislocation density, where $p_1^u = p_2^u = 5.911$ nm, and $b_1^u = b_2^u = a_{CFO}/2 = 0.419$ nm. The cross-sectional illustration in Fig. (3.48a) illustrates both internal structures in the tri- and six-layered systems of interest. Importantly, all dislocations have Burgers vectors with the same sign, and all three-dimensional calculations are performed with $r_0 = 0.3$ nm. Because of the miniaturized dimensions of ultrathin multiferroics in the experimental literature [89, 24], nominal nanoscale thicknesses are arbitrarily chosen: $h_{LNO} = h_{CFO} = 3$ nm, $h_{BTO} = 1.5$ nm, and the thickness of each intermediate buffer layer (i.e., for the layers of 0.25 BTO, 0.50 BTO, and 0.75 BTO) is equal to 1 nm. The mechanical load that is applied to both external surfaces is $\Gamma = 1$ GPa, over $l = 10$ nm, while the corresponding responses are computed using totally 1024 harmonics in both directions.

To complete the present results, external electric and magnetic loadings could also be applied and compared to the mechanical loads in the six-layered heterostructure.

## Distribution of the MEE field solutions in the six-layered heterostructure

Figure (3.48) focuses on the variations of some elastic, electric, and magnetic field components in the six-layered heterostructure, resulting also from the superposition (red curves) of the external load (blue) and the dislocation-induced (black) solutions, along two vertical (solid and dotted) lines in the z-direction, as depicted in Fig. (3.48a). The solid lines are located at $x_1 = 0$, while the dotted line are located midway between two adjacent dislocations, at $x_1 = (p_1 + p_2)/4$.

All considered elastic, electric, and magnetic quantities are continuous across the five internal interfaces, except the vertical electric $E_3$ and horizontal magnetic $H_1$ fields that reveal strong and sharp discontinuities at the interfaces. The later are computed by inverting eq. (3.164) and also solving for the extended strains, as

$$ \begin{bmatrix} \gamma \\ E \\ H \end{bmatrix} = \begin{bmatrix} c & -e^t & -q^t \\ e & \epsilon & \alpha \\ q & \alpha & \mu \end{bmatrix}^{-1} \begin{bmatrix} \sigma \\ D \\ B \end{bmatrix}, \qquad (3.219) $$

reading in the vector-tensor form.

All non-homogeneous solutions are more disturbed and disrupted close to the dislocation cores (dotted lines) with pronounced changes at the interfaces, revealing that the long-range interactions between adjacent dislocation networks have important effects on the distribution of the
MEE field components. Significant differences between solid and dotted lines show that the solu-
tion fields are not uniformly distributed with dramatically changes in sign (e.g., the vertical elec-
tric displacement D₃ and the horizontal magnetic induction H₁, which are both discontinuously
distributed), so that the atomic-scale measure of the layered magnetoelectric effects in dislocated
composites by semicoherent interfaces should be interpreted with considerable precautions.

As a conclusive illustration, three-dimensional visualization of three elastic (von Mises stress,
electric (positive horizontal displacement D₁), and magnetic (positive vertical magnetic induction
B₃)) quantities in the six-layered multilayer are exhibited in Fig. (3.49). These figures illustrate
the highly localized nature of these field components, which are dramatically located at both,
lower, and upper interfaces, respectively. For instance, the in-plane von Mises stress concentra-
tion along the misfit dislocations indicate the possible nucleation sources for plastic deformation
mechanisms and cracks, preferentially at upper interface with the highest dislocation density. The
horizontal displacement D₁ is asymmetrically concentrated at the lower interface and the lower
LNO layer, while the misfit dislocation intersections introduce preferential sites with maximum
magnetic induction D₃ that is also diffused in the upper CFO layer.
# Chapter 4

# Conclusion and future works

## 4.1 Concluding remarks

During my first years at the French Alternative Energies and Atomic Energy Commission, a three-dimensional continuum thermodynamically consistent formalism for combining elastoplasticity and phase-field theories has been developed for displacive phase transformations in finite strains. In accordance with the Clausius-Duhem inequality, explicit expressions for the Helmholtz free energy and constitutive relations have been used to determine the displacive driving forces for pressure-induced martensitic phase transitions. Inelastic forces are obtained by a representation of the energy landscape using the concept of reaction pathways for multivariants with respect to the point group symmetry properties of crystalline lattices. In particular, the Mao-Bassett-Takahashi transition path is used to characterize the transformational distortion along the reaction pathways for iron. On the other hand, the elastic forces are formulated for the general case that accounts for large strains and rotations, nonlinear and anisotropic elasticity with different pressure-dependent properties of stable and intermediate phases.

Implemented in a fully Lagrangian code, the nonlinear formalism is applied to analyze the forward and reverse polymorphic phase transformations under high pressure compression in single-crystal iron, within which the multiple lattice-related variants for (low-pressure) cubic and and (high-pressure) hexagonal structures are distinctly generated. Two loading conditions are investigated, i.e. the quasi-static and shock-wave regimes in Refs. [252] and [253], respectively. The first application shows that a forward bcc → hcp transformation of the initial single-crystal bcc phase into a polycrystal of hcp variants is energetically unfavorable due to the large amplitude of the stored elastic energy interactions between phases, and also remains incomplete without plasticity. However, the polymorphism bcc → hcp → bcc martensitic transformations occurs when plasticity is active. This simulation result is due to the effect of the plastic dissipation that releases considerably the elastic strain energy in the formation of a polycrystalline iron with an unexpected selection of variants.

On the other hand, the second dynamics simulations with plasticity accurately reproduce important observable characteristics reported by the experimental literature. For instance, the free-surface velocity exhibits that the shock wave is unstable, which breaks up into elastic, plastic and phase-transition waves for which the bcc-to-hcp phase transformation pressure is in agreement with experiments. The present split three-wave structure is characterized by the dynamical evolution of the strain from one- to three-dimensional compression with a local stress state that also relaxes to a nearly hydrostatic state. Similar plastic relaxations, however without structural phase transformation, have already been revealed in shock-compressed copper by diffraction experiments and large-scale molecular dynamics simulations. Furthermore, the microstructural stress-informed analyzes complement the extensive studies hitherto examined by molecular dynamics simulations with multi-million-atoms in the last two decades. The heterogeneous plastic deformation is quantitatively found to play a significant role in nucleating and selecting the shock-induced
variants at high pressure, which significantly differs from samples loaded under hydrostatic ex-
ternal compression. The Lagrangian time-position diagrams reveal that the prompt plastic relax-
ation to a nearly hydrostatic local state from uniaxial shock-compression is responsible for the
peculiar multiphase microstructure with a gradient selection of high-pressure variants behind the
phase-transition wave front. The existence of two sets of variants, so-called "release" and "reload"
variants appearing in separated zones, results from a nucleation instability that leads to a spe-
cific fingerprint of the nonlinear dynamics of unstable shock waves induced by structural phase
transformations.

The continuum formalism for phase transitions is, however, incomplete. In particular, the for-
mation of homo-phase grain boundaries and the heterophase interfaces between low- and high-
pressure phases during the coexistence of the solid-solid phases should be accompanied by a
loss the lattice coherence. This lattice mismatch by rotations and strains is ignored in the afore-
mentioned simulations, while experimental observations have revealed that coherent interfaces
break down the perfectly-matching interfaces through the presence of misfit dislocation structures
at such (semicoherent) interfaces in a variety of conditions. A lattice-based approach has there-
fore been developed to overcome this significant limitation, first dedicated to materials that are
mapped to a common reference state using displacement gradients alone. The ad-hoc strategy has
been conveniently applied to interface between fcc and bcc crystals, which could be formed during
the temperature-driven polymorphic bcc-fcc phase transition in iron as well as the pressure-driven
bcc-fcc-hcp transitions, with the fcc phases as intermediate phases.

The lattice-based model combines the closely related Frank-Bilby and O-lattice techniques with the Stroh sextic formalism for the anisotropic elasticity theory of interfacial dislocation networks [260]. Starting from my postdoctoral position at the Massachusetts Institute of Technology, the formalism is used by means of a Fourier series-based analysis to determine the reference states of semicoherent interfaces that gives rise to dislocations whose far-field elastic fields meet the condition of vanishing far-field strains and prescribed misorientations. These interface dislocations are viewed as Volterra dislocations that have been inserted into the reference state, subject to the stated constraints at long range. The complete elastic fields of these dislocations are calculated using heterogeneous anisotropic linear elasticity and interface dislocation configurations consistent with the quantized Frank-Bilby equation. The present model resolves the ambiguity arising from the infinite number of reference states available when the Frank-Bilby equation is analyzed based on geometry alone, i.e. without consideration of the elastic fields. The importance of accounting for the reference state has been illustrated in Refs. [259, 261], for which the selection of incorrect reference states leads to non-zero far field stresses, spurious far-field rotations, or both. Overall, all results reflect the importance of considering the anisotropy of elastic constants in the materials joined at the interface, where unequal partitioning of elastic fields is found.

The corresponding energetics have been quantified and used for rapid computational design of interfaces with tailored misfit dislocation patterns [258, 255]. In particular, the coupled approach with an object kinetic Monte Carlo code has revealed that elastic interactions between radiation-induced point defects and semicoherent interfaces lead to significant increases in interface sink strength, compared to the case with no defect-interface interactions [255]. The original version has also been extended to bilayers of finite thickness terminated with free surfaces [247], layered superlattices with differing layer thicknesses [248] as well as multilayered MEE solids [257] for semicoherent interfaces with relaxed dislocation patterns at semicoherent interfaces [249, 250] and core-spreading dislocation networks [263]. For many complicated lattice structures, the elastic full-field solutions have been compared with atomistic calculations [258, 251], which provide an opportunity for rigorous validation of the anisotropic elasticity theory of interfacial dislocations as well as for collaborations with individuals outside the home laboratory. Recently, a unified formalism for intrinsic dislocation arrays and extrinsic dislocation loops has recently been developed in Ref. [264], improving the first investigation on the estimation of elastic interactions between both types of defects from Refs. [265, 256].
Regarding the active research topic on the role played by the dislocations in interface-dominated materials, three inspiring routes are currently emerging, which are focused mainly on the thermoelasticity of imperfect interfaces as well as the interactions between dislocations and cracks by use of theoretical (continuously distributed dislocations based) and numerical (finite-element based) approaches.

# 4.2 Perspectives

## 4.2.1 Thermoelasticity of semicoherent interfaces

[P24] A. Vattré, E. Pan, V. Chiaruttini. *Free vibration of fully coupled thermoelastic multilayered composites with imperfect interfaces*. Composite Structures, 113203, 2021.

[P23] A. Vattré, E. Pan. *Thermoelasticity of multilayered plates with imperfect interfaces*. International Journal of Engineering Science, 158, 103409, 2021.

In this research line, the thermoelasticity response of the most advanced dislocation-based model from the previous chapter 3 is targeted in the near future, including the presence of intrinsic and extrinsic dislocations in multilayered materials subjected to external thermoelastic loads. A first effort has recently been done in Refs. [266, 267], within which the imperfect interfaces are described by phenomenological constitutive relations.

In the former reference, the three-dimensional solutions for time-harmonic temperature and thermoelastic stresses in multilayered anisotropic layers are derived with imperfect boundary conditions at internal interfaces using the extended Stroh formalism combined with nonlocal effects. For illustration, the residual stress fields in graphite fiber-reinforced epoxy matrix composites are investigated in Fig. (4.1). In particular, a unidirectional graphite-epoxy composite with fibers oriented along the $x_1$-direction (material depicted in grey) and a soft core is considered, where the thermoelastic properties and dimensions of both materials are reported in Ref. [266].

Following Savoia and Reddy [216], the steady-state thermoelastic bending of the three-layered sandwich square plates with $L_x/L_y = 1$ are subjected to a sinusoidal temperature that rises at both bottom and top surfaces, with $\bar{T}^B = 1$ K, and $\bar{T}^T = -1$ K, respectively. Figure (4.1) shows the effects of the ratios of $L_x/H$ and $l/H$ on various thermoelastic field solutions, by varying the lateral length $L_x$ as well as the nonlocal Eringen-based parameter $l$, where the entire thickness $H$ is kept fixed. For thinner plates, the temperature profile tends to a linear distribution through each individual layer, as illustrated in Fig. (4.1a), while nonlinear exponential branches appear in the graphite-epoxy composite plates for larger thicknesses. This trend indicates that when the aspect ratio is small, namely $L_x/H < 5$, the standard thin-plate result may be invalid, even though the temperature remains linear (close to zero) in the middle layer. The corresponding curves associated with the heat flux in Fig. (4.1b) are different from the temperature variation along the vertical $z$-direction. In particular, the normal heat flux is continuous across the interfaces and tends also to be steeper for thinner systems, while significant gradient emerges at the external surfaces as $L_x/H$ decreases.

The in-plane normal stress components $\sigma_{11}$ and $\sigma_{22}$ are displayed in Figs. (4.1c-f) for both extreme aspect ratios with further consideration of nonlocal effect. Three ratios for the nonlocal analysis are examined. It is worth noting that with reference to the composite stiff faces, higher in-plane stress levels occur in the direction perpendicular to the fibers. Moreover, due to material property mismatch between the layers, these in-plane normal stresses are discontinuous at both interfaces, with significant discontinuities in $\sigma_{11}$ when the aspect ratio is small, as shown in Fig. (4.1d). The amplitudes of these discontinuities at internal interfaces are therefore less pronounced for thinner plates, with negligible effect by the nonlocal parameter. However, the nonlocal parameter $l$ has a significant influence on the stress field for extremely thick plates subjected
<figure><img src="image_70.png" /></figure>

FIGURE 4.1: Steady-state thermoelastic bending of a three-layered structure with square plates subjected to a sinusoidal temperature rise at the two external faces. The first terms in the temperature expansion are considered, thus *m* = *n* = 1. The light grey regions are the unidirectional graphite-epoxy composites with fibers oriented along the *x*₁-direction, which are bonded by a soft core material. The through-the-thickness distributions for different values of the aspect ratios *L*<sub>*x*</sub>/*H* and of the nonlocal parameters *l*/*H* are depicted for (a) the temperature *T*, (b) the normal heat flux *q*<sub>3</sub>, (c-d) the in-plane stresses *σ*<sub>11</sub>, and (e-f) *σ*<sub>22</sub>. The standard local case corresponds to the field solutions with *l* = 0.

to thermal loads only, where the nonlocal parameter can completely change the variation trend of
the stresses, switching their signs and altering their magnitudes, as depicted by the blue curves in
Figs. (4.1d) and (4.1f).

**4.2.2 Distributed dislocations for periodic networks of cracks**

[P25] **A. Vattré.** *Kinked and forked crack arrays in anisotropic elastic bimaterials*. Journal of the Mechanics and Physics of Solids, 104744, 2022.

In Ref. [262], the fracture problem of multiple branched crack arrays in anisotropic bimateri-
als has recently been formulated by using the linear elasticity theory of lattice dislocations with
compact cores described in section 3.7.1. Yet, the general full-field solutions are obtained from the
standard technique of continuously distributed dislocations along finite-sized cracks of arbitrary
shapes, which are embedded in dissimilar anisotropic half-spaces under far-field stress loading
conditions. The bimaterial boundary-value problem leads to a set of coupled integral equations of
Cauchy-type that is numerically solved by using the Gauss-Chebyshev quadrature scheme with appropriate boundary conditions for kinked and forked crack arrays. The path-independent $J_k$-integrals as crack propagation criterion are therefore evaluated for equally-spaced cracks, while the limiting configuration of individual cracks is theoretically described by means of explicit expressions of the local stress intensity factors K for validation and comparison purposes on several crack geometries. The non-zero, singular and dimensionless stress components resulting from the idealized configurations of infinitely periodic cracks are illustrated in Fig. (4.2), for which the application setups are given in Ref. [262]. Specially, the $\sigma_{22}^{\text{array cracks}}$ exhibits a small compressive zone along the crack pointing to the upper surface. Figure (4.2) shows the large discontinuities of the in-plane stress component $\sigma_{11}^{\text{array cracks}}$ across both crack and interface planes as well as the traction-free conditions for $\sigma_{22}^{\text{array cracks}}$ and $\sigma_{12}^{\text{array cracks}}$ along the main crack plane that are therefore fully satisfied, as required.

The corresponding non-singular elasticity problem (using the core-spreading treatment from section 3.8.1) for interfacial cracks has recently and successfully been addressed in collaboration with Andreas Kleefeld from the University of Applied Sciences Aachen by use of the Tikhonov method for Fredholm integral equations of the first kind. The novel stress field solutions at interfacial crack tip do not exhibit oscillatory singularity induced by mismatching of the dissimilar materials, while the traction-free conditions are completely fulfilled along the discontinuities. In the near future, the influences of anisotropic elasticity, elastic mismatch, applied stress direction, inter-crack spacings and crack length ratios on the predictions from the crack opening displacement, as well as $J_k$- and K-based fracture criteria could therefore be examined in the light of different configurations from the single kinked crack case in homogeneous media to the network of closely-spaced interfacial cracks at bimaterial interfaces.

### 4.2.3 Towards a general treatment for {interfaces, dislocations, cracks}

[P27] A. Vattré, V. Chiaruttini. *Singularity-free theory and adaptive finite element computations of arbitrarily-shaped dislocation loop dynamics in 3D heterogeneous material structures*. Journal of the Mechanics and Physics of Solids, 104954, 2022.

The long-standing problem of arbitrarily-shaped dislocation loops in three-dimensional heterogeneous material structures has been addressed by introducing novel singularity-free elastic field solutions as well as developing adaptive finite element computations for dislocation dynamics simulations in Ref. [263]. The first framework uses the Stroh formalism in combination with the biperiodic Fourier-transform and dual variable and position techniques to determine the finite-valued Peach-Koehler force acting on curved dislocation loops. On the other hand, the second versatile mixed-element method proposes to capture the driving forces through dissipative energy considerations with domain integrals by means of the virtual extension principle of the surfacial discontinuities. Excellent agreement between theoretical and numerical analyses is illustrated from simple circular shear dislocation loops to prismatic dislocations with complicated simply-connected contours in linear homogeneous isotropic solids and anisotropic elastic multi-materials, which also serves as improved benchmarks for dealing with more realistic boundary-value problems with evolving dislocations. For illustration, the singularity-free Peach-Koehler magnitudes for a prismatic dislocation loop with a complex butterfly-shaped front are presented in Fig. (4.3a), using a given core-spreading radius. The theoretical (numerical) solutions are shown as solid lines (with symbols), while the corresponding driving forces are drawn in pink along the contours in Fig. (4.3b), with and without the two-dimensional shear stress $\sigma_{12}(x_1, x_2, z_s)$ maps in the background for further comparison. The signed magnitudes of the Peach-Koehler forces are plotted against the polar angle $\theta$, for which $\theta = 0^\circ$ corresponds to the points M in the schematics. In general, the very good agreement in terms of stresses and forces in sign and magnitude is also demonstrated, although slight deviations in direction are noticeable when the local radius of
<figure><img src="image_71.png" /><figcaption>FIGURE 4.2: Contours of non-zero and dimensionless stress field components produced by a network of equally- and closely- spaced forked cracks in an anisotropic bimaterial under traction using the concept of the continuously distributed dislocations.</figcaption></figure>

curvature changes drastically in sign. These discrepancies are mainly due to the different core-spreading schemes that have been appropriately adopted for mathematical convenience in each of the theoretical and numerical formulations.

Figure (4.4a) illustrates a large-scale three-dimensional finite element computation that cannot, to the knowledge of the authors, be achieved by existing numerical approaches in the broader literature, corresponding to the Orowan dislocation-precipitate bypass mechanism in a compressed micropillar of polycrystalline copper. An anisotropic copper polycrystalline micropillar with 80 grains is automatically generated from the intersection of a cubical Voronoi tessellation with a representative pillar specimen, in which a shear dislocation loop with a Burgers vector glides in the (111) slip plane of a specific host grain. The latter lies outside the microstructure, so that the outer grain boundary corresponds to the free surface of the computational sample. A high compressive strain of 7.1% is applied and maintained constant on one external face of the specimen, while the opposite face is blocked. At the grain scale, the Orowan bypass mechanism is described by the presence of the infinitely stiff, also elastically mismatched precipitate of arbitrary shape, for which the elastic constants are fictitiously multiplied by a factor of ten, with impenetrable boundaries and without consideration of cross-slip events. The internal grain boundaries are also considered as impenetrable barriers to dislocation motion, so that the dislocation loop is strictly confined to the host grain. The initial number of degrees of freedom associated with the full mesh is ~ 193k,
<figure><img src="image_72.png" /><figcaption>FIGURE 4.3: Prismatic dislocation loops with complex simply-connected fronts as the (a) butterfly- and (b) skull-shaped contours. The corresponding magnitude of the singularity-free Peach-Koehler forces are computed at $z_s$ and are displayed on the left-hand side with respect to the polar angle $\theta$, for which $\theta = 0^\circ$ is represented by the point $M$ in the plots of the right-hand side. The direction and amplitude of the driving forces as well as the shear stress component $\sigma_{12}(x_1, x_2, z_s)$ are depicted for both theoretical and numerical finite element solutions. For the sake of clarity, the Peach-Koehler forces along both dislocation contours are also shown in pink without the stress maps in the background.</figcaption></figure>

while the multiscale problem exhibits three orders of magnitude between the polycrystalline sample length and the representative size of the precipitate. The snapshot in Fig. (4.4b) shows the elastic dislocation/precipitate interaction, and especially the dislocation propagation by bowing around the inclusion as well as the self-coalescence of the dislocation loop once the arms pass the particle in the intermediate configuration. Thus, an Orowan-like dislocation loop is left around the infinitely strong inclusion, providing a new route in understanding of the Bauschinger effect in realistic precipitation-strengthened material structures. The planar propagation of a dislocation loop completely cuts the host grain and also leaves a surface step of the Burgers vector magnitude on the free surface of the micropillar sample, while the slip transmission of the dislocation loops across the neighboring grain boundaries is let for promising future development. Figure (4.4b) summarizes the various stages of the dislocation loop propagation bypassing the inclusion in the polycrystalline copper micropillar, for which the final configuration mesh is composed of ~ 1007k degrees of freedom. The corresponding animation of the Orowan precipitate bypass mechanism is referred to as "Orowan bypass mechanism in a micropillar", computed in less than 20 hours with 291 adaptive remeshing events with an average discrete time step of 0.46 ns.

At first glance and in the current form, the finite-element framework should be considered as a computational tool to carry out calculations with several types of discontinuities, such as grain boundaries, free surfaces, dislocation loops and cracks, in multiphase finite material structures. The main interesting feature of the approach is to unify these discontinuities into a single finite-element entity to revisit the fundamental problems concerning the interactions between dislocation loops and cracks, in particular the emission of dislocations from crack fronts in three dimensions, as well as the interactions between dislocations and stress concentrations at grain boundaries and heterophase interfaces, especially the nucleation and emission of dislocation loops from the internal material boundaries. Although the computational approach undoubtedly opens many perspectives, also with close links to experiments, some extensions can be introduced. A current limitation is related to the use of a single regularization rule at the dislocation fronts, whether the dislocation loops are located in the core of the grains or near the internal interfaces. A more physics-based rule could be provided to offer a better description of the short-range elastic
<figure><img src="image_73.png" /><figcaption>FIGURE 4.4: From the initial dislocation loop embedded in a given grain of the polycrystalline copper micropillar with ~ 193k degrees of freedom in (a) to the various propagation steps followed by the shear dislocation loop in (b), thus leaving residual dislocation edges around the bypassed heterophase precipitate. The final computational mesh involves ~ 1007k degrees of freedom.</figcaption></figure>

fields close to the grain boundaries to analyze the transmission of dislocation loops into neighboring grains, thus overcoming the current impermeability conditions. Furthermore, although the current simulations are performed on a workstation, the numerical framework could benefit from the robust iterative and domain decomposition solvers to handle the discretization of several tens of millions of unknowns. By the use of a parallel mesh generation algorithm for robust domain decomposition techniques, high-performance calculations with a hundred dislocation loops are anticipated to characterize standard dislocation microstructures with typical densities of $10^{12}/10^{14}$ m$^{-2}$ in the 1-to-100 micrometer mesoscale range. Finite element calculations with hundreds of millions of degrees of freedom are therefore expected to achieve such numerical experiments for multiple dislocation loops in three-dimensional material structures. These subsequent boundary-value problems should be accompanied by consideration of additional dislocation junctions, such as the Lomer-Cottrell lock, the Hirth lock and the glissile junction as well as the implementation of the dislocation cross-slip mechanism and energetics, which are left for future investigations. In an extrapolation scenario, computations of several thousand dislocation loops on supercomputers could be carried out with the aim of better understanding dislocation-based strain hardening mechanisms in realistic structures at the macroscale.
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