Misfit stresses in a composite core-shell nanowire with an eccentric parallelepipedal core subjected to one-dimensional cross dilatation eigenstrain

The boundary-value problem in the classical theory of elasticity for a core-shell nanowire with an eccentric parallelepipedal core of an arbitrary rectangular cross section is solved. The core is subjected to one-dimensional cross dilatation eigenstrain. The misfit stresses are given in a closed analytical form suitable for theoretical modeling of misfit accommodation in relevant heterostructures.

NW with an eccentric parallelepipedal core of an arbitrary rectangular cross section, which is characterized by a one-dimensional (1D) cross dilatation eigenstrain with respect to the shell material. In the present work, we solve the corresponding boundary-value problem in the theory of elasticity and find its solution in a closed analytical form. Using numerical calculations, we illustrate our solution by the stress maps which prove the satisfaction of boundary conditions of the problem and show some interesting features in the stress distribution over the NW cross section.

Model
Our model is a long elastic cylinder containing an eccentric inclusion (core) in the form of a long parallelepiped with an arbitrary rectangular cross section (figure 1), in contrast with the special case in [11], where a symmetric core with a square cross section was considered. Both the cylinder and core are elastically homogeneous and isotropic and have identical elastic moduli: the shear modulus G and the Poisson ratio ν. The outer radius of the cylinder is R. The cross section of the core is a rectangle given by the coordinates of his vertexes. The core is subjected to a 1D dilatation eigenstrain * yy ε , in contrast with the case of isotropic 3D dilatation eigenstrain in [11]. It allows us to take into account the misfit-strain anisotropy in real composite NWs.
The stress field in such a composite cylinder ) ( cyl y ij σ can be given by the sum of a stress field ) ( y ij σ ∞ caused by the core in an infinite body, and an extra stress field ) ( y ij σ which is required to fulfill the boundary conditions on the cylinder free surface: The boundary conditions read 0 ) ( The non-vanishing stress components ij σ ∞ are determined by the formulas [13]: Any stress function ij Ψ in the cylindrical and Cartesian coordinate system can be calculated through the complex potentials [14] as are the complex potentials that are unknown analytical functions of the complex variable ) , respectively. Subtracting equations (5) from (6), we obtain the following formula which is convenient for the fulfillment of the boundary conditions: With taking into account the stress finiteness at r → 0, we will search the functions ) where А n and В n are complex constants in the general case. Now we could rewrite equation (7) as Let us find now the extra stress field ) ( y ij σ in the same form as that given by equation (3): The boundary conditions (2) with account for equation (7) are Let us introduce the new complex variables and consider the terms in (11) separately: To satisfy the boundary conditions (2), we should represent equations (12) and (13) in terms of the power series. Finally, equation (11) reads Comparing the coefficients at n ζ in both equations (9) and (14) Substitution of equations (15) and (8) to (5), (6) gives the complex form for the stress function of the extra stress tensor: Transforming equations (16), we come to the final components of the extra stress tensor: where ρ and θ are the polar coordinates of the core vertex, and Thus, the problem is solved.

Results
The distribution of stress components  . In metallic NWs, this stress level is quite enough for the onset of plastic relaxation processes even at room temperature, while in semiconductor NWs, it occurs under higher temperatures. The relaxation processes are expected to start either on the core-shell interface or on the shell surface.

Conclusions
In this paper, we have obtained a closed-form analytical solution for misfit stresses in a core-shell NW with an eccentric parallelepipedal core which is subjected to a 1D dilatation eigenstrain. It is shown that the solution satisfies the boundary conditions of the problem and gives a clear insight on the stress distribution in the NW. The stress fields are very inhomogeneous over the cross section of the NW and strongly screened by its free surface. Some stress components are concentrated at the edges of the core, while the others are concentrated at the core faces and on the shell surface. The stress magnitude can be sufficient for the onset of plastic relaxation processes in the NW under suitable temperature either on the core-shell interface or on the shell surface. We expect that our solution will be widely used in the theoretical description of the mechanisms for misfit accommodation in such NWs.