Z.M. Xiao

Differential quadrature method for Mindlin plates on Winkler foundations

This paper presents an approximate analysis of rectangular plates resting on Winkler foundations based on the Mindlin plate theory. The plates are subject to any combination of free, simply supported and clamped boundary conditions. Solutions to the problem are obtained using the differential quadrature method (DQM) by solving the governing differential equations. Numerical results are compared with existing literature to establish the validity and accuracy of the method. This study shows numerically the effects of shear deformation on the deflections and stress resultants at some selected locations. The distributions of the bending and twisting moments and shear force for several plates are presented graphically by varying the relative thickness ratio h/a to further show the significant effect of shear deformation.

Differential quadrature method for thick symmetric cross-ply laminates with first-order shear flexibility

A global numerical technique, the differential quadrature (DQ) method, is examined here for its suitability to solve the boundary-value problem of symmetric cross-ply laminates using the first-order shear deformation plate theory by Whitney and Pagano [J. Appl. Mech.37, 1031–1036 (1970)]. The bending behaviours of symmetric cross-ply laminates, subject to different boundary constraints, are investigated. In this study, the method is used to transform the sets of governing differential equations and boundary conditions of the laminated plates into sets of linear algebraic equations. Boundary conditions along the edges are implemented through the discrete grid points by constraining the displacements, bending moments and rotations. The theoretical formulations and solution procedures of the method are illustrated through solving several numerical examples. The accuracy and validity of the present formulation, if available, are examined by direct comparison with the known values.

A screw dislocation interacting with a coated fiber

Abstract A close-form analytical solution is obtained for the stress field due to a screw dislocation near a coated fiber inhomogeneity in isotropic material. The forces on dislocations are derived. The equilibrium positions of the dislocation are discussed in detail for various material constants’ combinations. It is found that when the coating layer is thick, the elastic property of the fiber (inclusion) has no significant influence on the force of the dislocation, therefore the equilibrium and stability of the dislocation can be obtained similarly from the two-phase model adopted by Dunders (1967. Recent Advances in Engineering Science 2, 223–233). On the other hand when the coating layer is thin, if both the fiber and the coating layer are “softer” (i.e., have lower modulus) than the matrix, the dislocation is always attracted by the fiber, and if both the fiber and the coating are “harder” (i.e., have higher modulus) than the matrix, the dislocation is always repelled by the fiber. As a result, there are no equilibrium positions under these two conditions. While if the fiber is harder and the coating is softer than the matrix, there is at least one unstable equilibrium position near the coating–matrix interface, if the fiber is softer and the coating is harder than the matrix, there is at least one stable equilibrium position near the coating–matrix interface.

On the interaction between an edge dislocation and a coated inclusion

Abstract An analytical solution is derived for the stress field due to an edge dislocation located near a coated inclusion in a solid by using the Muskhelishvili complex variable method. The force on the dislocation is calculated. Examples for various coating thickness and material constant combinations are given and discussed. It is shown that when a coating layer is thick, the elastic properties of the inclusion have no significant influence on the force on the dislocation. Therefore, the equilibrium position and the stability of the dislocation can be obtained in a manner similar to the two-phase model adopted by Dundurs and Mura (Dundurs, J., Mura, T., 1964. Journal of Mechanics and Physics of Solids 12, 177–189.). If the coating layer is thin, both the shear modulus and Poisson’s ratio of both the inclusion and the coating can affect and change greatly the equilibrium position and the stability of the dislocation.

Stress intensity factor for a Griffith crack interacting with a coated inclusion

Stress investigation for the interaction problem between a coated circular inclusion and a near-by line crack has been carried out. The crack and the coated inclusion (a coated fiber) are embedded in an infinitely extended isotropic matrix, with the crack being along the radial direction of the inclusion. Two loading conditions, namely, the tensile and shear loading ones are considered. During the solution procedure, the crack is treated as a continuous distribution of edge dislocations. By using the solution of an edge dislocation near a coated fiber as the Greens function, the problem is formulated into a set of singular integral equations which are solved by Erdogan and Gupta (1972) method. The expressions for the stress intensity factors of the crack are then obtained in terms of the asymptotic values of the dislocation density functions evaluated from the integral equations. Several numerical examples are given for various material and geometric parameters. The solutions obtained from the integral equations have been checked and confirmed by the finite element analysis results.

Stress analysis for a Zener–Stroh crack interacting with a coated inclusion

Abstract A microcrack can be initiated by coalescing dislocations piled up near an inhomogeneity. This mechanism was firstly proposed by Zener and later analyzed by Stroh. The microcrack thus is called a Zener–Stroh crack, a counterpart of the well-known Griffith crack in linear elastic fracture mechanics. In the current study we consider a Zener–Stroh crack initiated near a coated circular fiber in a solid material. The interaction between the crack and the coated inclusion (fiber) for both Mode I and II displacement loading is investigated using the solution of an edge dislocation interacting with a coated fiber derived by Xiao and Chen [International Journal of Solids and Structure, in press] as the Green’s function. The problem is formulated as a set of singular integral equations and is solved numerically by Erdogan–Gupta method. The influence of several parameters, such as the distance between the crack and the inclusion, the material constants combination of the fiber, the coating layer and the matrix, on the stress intensity factors of the crack is analyzed. Several numerical examples are given and the results obtained are discussed in detail.

A ZENER-STROH CRACK NEAR AN INTERFACE

Abstract A micro crack can be initiated by coalescing dislocations piled up along a slip plane. This mechanism was firstly proposed by Zener and later on analyzed by Stroh. The micro crack is thus called Zener-Stroh crack, which is a counterpart of the well-known Griffith crack in linear elastic fracture mechanics. In the present paper, we consider a Zener-Stroh crack initiated near a bi-material interface. Due to image force acted on the dislocations, a Griffith crack mechanism is introduced even where the crack is purely loaded by net dislocations. It is seen that the stress intensity factor, which consists of a Zener-Stroh component and a Griffith component, and the critical crack length are strongly affected by the presence of the interface.

On piezoelectric inhomogeneity related problem—part I: a close-form solution for the stress field outside a circular piezoelectric inhomogeneity

Abstract The stress investigation for a circular piezoelectric fiber sensor embedded in a non-piezoelectric elastic material has been carried out. During the formulation procedure, the inhomogeneity problem is partially decoupled into an elastic problem and a dielectric inhomogeneity problem connected via some eigenstrain and eigen-electric field. The stress field inside the piezoelectric fiber sensor (the inhomogeneity) is evaluated based on Eshelby’s equivalent inclusion method. While the explicit close-form solution for the stress distribution outside the inhomogeneity is obtained through Tanak–Mura’s superposition process. The analytical solutions presented in this paper have been checked and confirmed by finite element analysis results.

Electro-elastic stress analysis for a wedge-shaped crack interacting with a screw dislocation in piezoelectric solid

Abstract The electro-elastic stress investigation on the interaction problem of a screw dislocation near the tip of a semi-infinite wedge-shaped crack in piezoelectric material has been carried out. Explicit closed-form analytical solutions are obtained for the stress intensity factor (SIF) and the electric displacement intensity factor (EDIF) of the crack, as well as the force on dislocation. The derivation is based on the conformal mapping method and the perturbation technique. The dislocation has Burgers vector normal to the isotropic basal plane, with a line force and a line charge being applied at the core of the dislocation. The influence of the location and the wedge angle of the crack on the image force of the dislocation has been discussed in detail. At the same time the effect of the dislocation on the crack behavior has been also examined under different configurations. Two types of PZT materials are used to numerically illustrate the influences of the wedge angle and the location of the dislocation on the image force and the crack intensities. Results obtained in the current study can be fully reduced to various special cases available in the literatures.

On the interaction between a semi-infinite anti-crack and a screw dislocation in piezoelectric solid

Abstract Electro-elastic stress investigation on the interaction between a semi-infinite anti-crack and a screw dislocation under anti-plane mechanical and in-plane electrical loading is carried out. The screw dislocation is subjected to a line-charge and a line-force at the core. Explicit closed-form analytical solutions for the stress fields are derived by means of complex variable and conformal mapping methods. The stress and electric intensity factors at the anti-crack tip are found to be classic 1/ r singularity. The force on the dislocation is also calculated and discussed for two typical PZT materials.