Screw dislocation in nonlocal anisotropic elasticity

doi:10.1016/j.ijengsci.2011.02.011

International Journal of Engineering Science

Volume 49, Issue 12, December 2011, Pages 1404-1414

https://doi.org/10.1016/j.ijengsci.2011.02.011 Get rights and content

Abstract

Based on Eringen’s model of nonlocal anisotropic elasticity, new solutions for the stress fields of screw dislocations in anisotropic materials are derived. In the theory of nonlocal anisotropic elasticity the anisotropy is twofold. The anisotropic material behavior is not only included in the anisotropy of the elastic stiffness properties, but also in the anisotropy of the nonlocality which is expressed by the anisotropy of the length scale parameters, which is incorporated in the anisotropy of the nonlocal kernel function. Particularly, a new two-dimensional anisotropic kernel which is the Green function of a linear differential operator with three length scale parameters is derived analytically. New solutions for the stresses of straight screw dislocations in anisotropic (monoclinic and hexagonal) materials are found. The stresses do not have singularities and possess interesting features of anisotropy, which are presented and discussed.

Introduction

Classical continuum theories like the linear theory of elasticity are intrinsically size independent. The elastic strain and stress fields of defects (dislocations, disclinations) are singular at the defect line. Such singularities are unphysical and their reason is based on the fact that classical elasticity is not valid in the defect core region. An improvement is obtained by using nonlocal elasticity (Eringen, 1983, Eringen, 1987, Eringen, 1992, Eringen, 2002, Kröner and Datta, 1966, Kunin, 1983) instead of classical elasticity. The theory of nonlocal elasticity includes the effect of long range interatomic forces so that it can be used as a continuum model of the atomic lattice dynamics. In the theory of nonlocal elasticity the stress at a reference point r depends on the elastic strain at all other points r′ of the body. From the mathematical point of view, this nonlocal interaction is represented by the so-called nonlocal kernel.

Solutions for straight screw and edge dislocations within nonlocal isotropic elasticity based on Gaussian kernels have been given by Eringen, 1977, Eringen, 1977. The main feature of these solutions is the elimination of the stress field singularities at the dislocation line. Encouraged by the obtained results, Eringen extended nonlocal elasticity to anisotropic materials. Eringen and Balta, 1978, Eringen and Balta, 1979 have examined straight screw and edge dislocations in an infinite nonlocal hexagonal medium. Using a special Gaussian kernel having two length scales, Eringen and Balta, 1978, Eringen and Balta, 1979 found complicated integral expressions for the stresses in nonlocal hexagonal elasticity. Using the same ‘anisotropic’ kernel, Pan (1995) calculated the stress field of a straight screw dislocation in a semi-infinite nonlocal hexagonal medium. Nonlocal anisotropic elasticity was used by Sun and Zhou, 2004, Zhou and Wang, 2005 for the calculation of the stress fields near the crack tip. However, an isotropic kernel function is used in these works (Sun and Zhou, 2004, Zhou and Wang, 2005).

On the other hand, for a class of kernels, which are the Green functions of the Helmholtz equation, nonlocal isotropic elasticity was studied by Eringen, 1983, Eringen, 1992, Eringen, 2002. This nonlocal elasticity of Helmholtz-type was used for the calculation of the stress field and strain energy of a screw dislocation. Fortunately, the singularities of the stress disappeared. This stress field of a screw dislocation coincides with the stress field calculated by Gutkin and Aifantis, 1999, Lazar and Maugin, 2005 within the gradient elasticity framework. The stress field of an edge dislocation was calculated by Lazar (2005) in the framework of nonlocal isotropic elasticity of Helmholtz-type. This result is in agreement with the stress obtained by Gutkin and Aifantis, 1999, Lazar and Maugin, 2005 within gradient elasticity (see also (Lazar, Maugin, & Aifantis, 2005)). The nonlocal kernel and the solutions of screw and edge dislocations based on the so-called nonlocal isotropic elasticity of bi-Helmholtz type were given by Lazar, Maugin, and Aifantis (2006). However, solutions of dislocations in nonlocal anisotropic elasticity based on nonlocal anisotropic kernels which are Green functions are still lacking in the literature. The questions come up which is the nonlocal kernel as a Green function of an ‘anisotropic’ Helmholtz equation in nonlocal anisotropic elasticity and if the stresses can have a simpler form than the complicated integral expressions found by Eringen and Balta, 1978, Eringen and Balta, 1979.

The paper is organized as follows: We start with the basic framework of nonlocal anisotropic elasticity in Section 2. In the next section we determine a new two-dimensional nonlocal kernel which is the Green function of a partial differential equation of second order with three length scale parameters. The main result of this work is presented in Section 4, where a new analytical solution for the stress fields for a straight screw dislocation in nonlocal anisotropic elasticity is obtained. In order to gain more insight to the significance of the results, we consider monoclinic materials as example. The stress fields of a straight screw dislocation for hexagonal materials are given in Section 5. The solutions are highly influenced by orientational effects due to the anisotropy. Conclusions and discussion are given in the last section.

Section snippets

Basic equations

The basic equations of linear, nonlocal anisotropic elasticity, in the static case with vanishing body forces, are (Eringen, 1983, Eringen, 1987, Eringen, 2002): $\partial_{j} t_{ij} = 0,$ $t_{ij} (r) = \int_{V} c_{ijkl} (r, r^{'}) e_{kl} (r^{'}) d v (r^{'}), t_{ij} = t_{ji}$ $e_{ij} = \frac{1}{2} (β_{ij} + β_{ji}),$ where e_ij is the ‘classical’ elastic strain tensor and β_ij is the elastic distortion tensor. In presence of dislocations both tensors are incompatible. t_ij is the stress tensor of nonlocal elasticity given by the integral constitutive relation (2) which states that the

Determination of the nonlocal kernel of nonlocal anisotropic elasticity

In this section we specify the linear differential operator L and the nonlocal kernel function α for nonlocal anisotropic elasticity. We choose an elliptic differential operator of generalized Helmholtz-type. A suitable differential operator to describe the effects of anisotropic nonlocality can be written as follows $L = 1 - L_{ij} \partial_{i} \partial_{j} .$ The second rank tensor L_ij incorporates anisotropic length scale effects and has the dimension: (length)². Thus, L_ij is a symmetric, second rank, length scale tensor and

Screw dislocation in anisotropic medium

We consider a screw dislocation in an anisotropic medium, where the xy-plane is one of the crystal symmetry. For example, a monoclinic material with 2-fold rotation axis (diad axis) parallel to the z-axis is such an anisotropic medium. We consider as example for the numerical analysis the crystal caesium dihydrogen phosphate (CDP) CsH₂PO₄ with the elastic constants c₄₄ = 8.10 × 10⁹ N/m², c₅₅ = 5.20 × 10⁹ N/m² and c₄₅ = −2.25 × 10⁹ N/m² (Prawer, Smith, & Finlayson, 1985).

Screw dislocation in hexagonal medium

Because hexagonal crystals represent technically an important class of materials with stable dislocation structures, we apply the solution of a screw dislocation, found in Section 4.2, to the case of hexagonal symmetry. We consider the xz-plane as the basal plane and the Burgers vector as well as the dislocation line at the z-direction in the basal plane. In general, a hexagonal material has five nonzero independent elastic constants (Ting, 1996). Evidently, for the anti-plane strain problem

Conclusions and discussion

In this paper, we have examined screw dislocations in the framework of nonlocal anisotropic elasticity. We have calculated a new two-dimensional kernel as a Green function of the anisotropic Helmholtz-type equation in nonlocal anisotropic elasticity. This kernel contains three parameters for length scale effects. We applied the nonlocal anisotropic elasticity to screw dislocations. We have found new mathematical solutions for the stress fields which are simpler than those of Eringen and Balta

Acknowledgement

This work was supported by an Emmy-Noether grant of the Deutsche Forschungsgemeinschaft (Grant No. La1974/1-3).

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Screw dislocation in nonlocal anisotropic elasticity

Abstract

Introduction

Section snippets

Basic equations

Determination of the nonlocal kernel of nonlocal anisotropic elasticity

Screw dislocation in anisotropic medium

Screw dislocation in hexagonal medium

Conclusions and discussion

Acknowledgement

Mechanics of Materials

International Journal of Engineering Science

International Journal of Engineering Science

International Journal of Solids and Structures

Scripta Materialia

International Journal of Engineering Science

International Journal of Solids and Structures

European Journal of Mechanics A/Solids

International Journal of Engineering Science

Nonlocal stress at Griffith crack

Crystal Lattice Defects Amorph. Mat.

Screw dislocation in nonlocal elasticity

Journal of Physica D: Applied Physics

Screw dislocation in nonlocal hexagonal elastic crystals

Crystal Lattice Defects

Edge dislocation in nonlocal hexagonal elastic crystals

Crystal Lattice Defects

On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves

Journal of Applied Physics