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When the dimension of a structure falls to the micro-/nanoscale, surface effect is significant and plays a key role in affecting the mechanical behavior. This article studies the influence of surface elasticity on the stress intensity factor of an antiplane shear crack embedded in an elastic strip made of functionally graded materials. Surface elasticity is applied on the strip surfaces and crack faces, and classic elasticity is invoked for the strip interior. An antiplane shear crack problem is solved for a symmetric FGM with a crack parallel to the strip surfaces. The associated problem is converted to a hypersingular integro-differential equation for the out-of-plane displacement on the crack faces through the Fourier transform and then to a singular integro-differential equation with Cauchy kernel. The Galerkin method is applied to expand the crack face displacement as a Chebyshev series, and the singular integro-differential equation reduces to a system of algebraic linear equations. Stress intensity factors at the crack tips and the out-of-plane displacement on the crack faces are calculated numerically. It is found that surface elasticity and gradient index strongly alter the bulk stress and its intensity factors near the crack tips. Positive surface shear modulus decreases the mode III stress intensity factors and negative surface shear modulus has an opposite behavior. The influence of the variation of material gradient on the mode III stress intensity factors is expounded in graph.

Scale-dependent stress intensity factors in an anti-plane cracked orthotropic material layer are evaluated using strain gradient theory. Both volumetric and surface strain gradient material characteristic lengths represented as l and $$l^{'}$$ , respectively, are employed to obtain semi-analytical solutions. The surface strain gradient effect is considered for both positive and negative $$l^{'}$$ values. The layer edges are assumed stress-free and oriented parallel to the crack plane. The presence of orthotropy can either increase or decrease the stress intensity factors depending on if it is greater or smaller than unity. The volumetric strain gradient effect reduces the stress intensity factor and it is more pronounced for smaller layer thickness. It was found that the negative surface gradient leads to a more complaint crack, while the positive surface gradient increases crack stiffness. Overall, the surface gradient effect is less significant in comparison with the volumetric gradient effect.

As outlined in the previous paper, an eigenstrain state due to an interstitial atom, acting as misfitting inclusion in the crystal lattice, may interact with a load stress state and/or a stress state due to a defect. Since the interstitial atoms can be assembled in a cylinder, as in the established case of a so-called Cottrell cloud, the according stress state can be investigated by a cylindrical inclusion with a general eigenstrain state. Both eigenstrain components belonging to the antiplane shear, however, were not dealt with in the past. According stress and deformation equations are now offered in analytical form in this addendum.

The circumferential shear stress along the boundary of a traction-free circular hole in an infinite isotropic solid under antiplane shear is twice the circumferential shear stress along the corresponding circle in an infinite solid without a hole, under the same loading conditions. This result, originally derived by Lin et al. (Yielding, damage, and failure of anisotropic solids. EGF Publication, Mechanical Engineering Publications, London, 1990) by invoking the extended circle theorem of Milne-Thomson, is rederived in this paper using the Fourier series analysis, in the spirit of the original Kienzler and Zhuping (J Appl Mech 54:110–114, 1987) derivation of the plane strain result for the hoop stress around the hole. The formula is also deduced by the complex potential method, without explicitly invoking the extended circle theorem. The ratio of the circumferential stresses along the boundary of a noncircular hole and the congruent curve in a solid without a hole depends on the loading. For an elliptical hole in an infinite solid under remote loading $${{\sigma_{xz}^{0}}}$$ , this ratio is constant and equal to 1 + b/a; under $${{\sigma_{yz}^{0}}}$$ it is equal to 1 + a/b, where a and b are the semi-axes of the ellipse. These constant values coincide with the values of the stress concentration factor in the two cases. If $${{\sigma_{xz}^{0}}}$$ and $${{\sigma_{yz}^{0}}}$$ are both applied, the stress ratio varies along the boundary of the ellipse; a closed form expression for this variation is determined. The portions of the elliptical boundary along which the strain energy density is increased or decreased by the creation of the hole are determined for different types of loading and different aspect ratios of the ellipse. An analytical expression for the stress concentration factor is then obtained for any orientation of the ellipse relative to the applied load.

In this paper, the basic presentation in antiplane shear and inplane electric field of piezoelectric materials is refreshed. In order that the functions used in the formulation can be distinguished by their usage, four analytic functions, or four complex potentials, are introduced. A multiple crack problem for piezoelectric materials is studied. After taking the traction or the electric displacement on the crack face as unknown functions, one can naturally obtain a Fredholm integral equation for the multiple crack problem. It is found that the Fredholm integral equation approach is effective for solving the multiple crack problem. Finally, numerical examples are given.

The dynamic behavior of two parallel symmetric cracks in functionally graded piezoelectric/piezomagnetic materials subjected to harmonic antiplane shear waves is investigated using the Schmidt method. The present problem can be solved using the Fourier transform and the technique of dual integral equations, in which the unknown variables are jumps of displacements across the crack surfaces, not dislocation density functions. To solve the dual integral equations, the jumps of displacements across the crack surfaces are directly expanded as a series of Jacobi polynomials. Finally, the relations among the electric, magnetic flux, and dynamic stress fields near crack tips can be obtained. Numerical examples are provided to show the effect of the functionally graded parameter, the distance between the two parallel cracks, and the circular frequency of the incident waves upon the stress, electric displacement, and magnetic flux intensity factors at crack tips.

The antiplane shear deformation problem of two edge-bonded dissimilar isotropic wedges is considered. In the case when the sum of the two apex angles is equal to 2π, the problem reduces to that of two edge-bonded dissimilar materials with an interfacial crack subjected to concentrated antiplane shear tractions on the crack faces. An explicit expression is extracted for the stress intensity factor at the crack tip. In the special cases of different combinations of the apex angles, the obtained expression for the stress intensity factor may be simplified and relations of a simpler form are given for the stress intensity factor. It is shown that the stress intensity factor is dependent on the material properties as well as the geometry and loading. However, in special cases of equal apex angles as well as the case of similar materials the dependency of the stress intensity factor on the material properties disappears.

This paper presents a novel efficient procedure to analyze the two-phase confocally elliptical inclusion embedded in an unbounded matrix under antiplane loadings. The antiplane loadings considered in this paper include a point force and a screw dislocation or a far-field antiplane shear. The analytical continuation method together with an alternating technique is used to derive the general forms of the elastic fields in terms of the corresponding problem subjected to the same loadings in a homogeneous body. This approach could lead to some interesting simplifications in solution procedures, and the derived analytical solution for singularity problems could be employed as a Green's function to investigate matrix cracking in the corresponding crack problems. Several specific solutions are provided in closed form, which are verified by comparison with existing ones. Numerical results are provided to show the effect of the material mismatch, the aspect ratio, and the loading condition on the elastic field due to the presence of inhomogeneities.

A generalized and unified treatment is presented for the antiplane problem of an elastic elliptical inclusion undergoing uniform eigenstrains and subjected to arbitrary loading in the surrounding matrix. The general solution to the problem is obtained through the use of conformal mapping technique and Laurent series expansion of the associated complex potentials. The resulting elastic fields are derived explicitly in both transformed and physical planes for the inclusion and the surrounding matrix. These relations are universal in the sense of being independent of any particular loading as well as the geometry of the matrix. The complete field solutions are provided for an elliptical inclusion under uniform loading at inifinity, and for a screw dislocation interacting with the elastic elliptical inclusion.

Antiplane shear problem of an infinite medium containing a circular inclusion of different material is investigated in this article. Perfectly bonding between the matrix and the inclusion, as well as partial debonding in the form of a circumferential crack occurring at the interface of these two constituents, are considered. Using the complex variable method in conjunction with a semi-inversed technique, exact expressions for the stress, displacement and stress intensity factor in the problem are obtained for various external loading conditions, including that brought about by stress singularities. A number of numerical results of stress intensity factors for a partially damaged matrix-inclusion system are given, and a discussion is made on the general features of the antiplane shear problem of inhomogeneous matrix-inclusion systems.