Screw dislocation in nonlocal anisotropic elasticity

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.     

Dislocations in the field theory of elastoplasticity

By means of linear theory of elastoplasticity, solutions are given for screw and edge dislocations situated in an isotropic solid. The force stresses, strain fields, displacements, distortions, dislocation densities and moment stresses are calculated. The force stresses, strain fields, displacements and distortions are devoid of singularities predicted by the classical elasticity. Using the so-called stress function method we found modified stress functions of screw and edge dislocations.     

Volume defects in nonlocal micropolar elasticity

Using the concept of prescribed arrays of point forces and point couples, the long range displacement, the stress and the interaction energy fields associated with an isolated volume defect in nonlocal micropolar continuum elasticity are calculated. It is shown that all the physically possible volume defects can be described by a distribution of body forces and body couples. Interestingly these forces and couples are not singular extending therefore the validity of the classical continuum treatment of defects. Line defects such as dislocations and disclinations can be analyzed in a similar way.     

Stress and couple-stress fields produced by Frank disclinations in an isotropic elastic micropolar continuum

The stress and couple-stress fields due to Frank disclinations in an infinitely extended isotropic, elastic and micropolar continuum are estimated. The author starts with a brief review on the expressions for the elastic fields due to distributed dislocations and disclinations. Those expressions are converted into line-integral expressions for the elastic fields due to Frank disclinations. They are specialized to the case of infinite straight disclinations. The stress and couple-stress fields are calculated for both twist and wedge disclinations.     

Enhanced modeling of laminated and sandwich plates via strain energy transformation

An enhanced first-order shear deformation theory has been developed for the deformation and stress recovery of laminated and sandwich plates. Based on the definition of Reissner–Mindlin’s plate theory, the relationships between three-dimensional and first-order shear deformation theories have been derived. It is assumed that the displacements, in-plane strains and stresses of Reissner–Mindlin’s plate theory can approximate those of three-dimensional theory, in the least-square sense. Their relationships have been systematically established and verified through the strain energy transformation. These relationships provide the closed-form recovering relations for three-dimensional variables expressed in terms of the variables of Reissner–Mindlin’s plate theory. An efficient higher-order plate theory is utilized to obtain the in-plane warping functions. Comparisons of deflection, stresses and shear correction factors of both laminated and sandwich plates using the present theory are made with the original first-order shear deformation theory and three-dimensional exact solutions.     

Non-local theory solution for the anti-plane shear of two collinear permeable cracks in functionally graded piezoelectric materials

In this paper, the non-local theory of elasticity is firstly applied to obtain the behavior of two collinear cracks in functionally graded piezoelectric materials under anti-plane shear loading for permeable electric boundary conditions. To make the analysis tractable, it is assumed that the material properties vary exponentially with coordinate vertical to the crack. By means of the Fourier transform, the problem can be solved with the help of a pair of triple integral equations that the unknown variable is the jump of the displacement across the crack surfaces. These equations are solved by use of the Schmidt method. Numerical examples are provided. Unlike the classical elasticity solutions, it is found that no stress and electric displacement singularities are present near the crack tips. The non-local elastic solutions yield finite stresses at the crack tips, thus allows us to use the maximum stress as a fracture criterion. The finite stresses at the crack tips depend on the distance between two collinear cracks, the functionally graded parameter and the lattice parameter of the materials, respectively.     

Three-dimensional linear elastic distributions of stress and strain energy density ahead of V-shaped notches in plates of arbitrary thickness

This paper presents an analytical solution, substantiated by extensive finite element calculations, for the stress field at a notch root in a plate of arbitrary thickness. The present approach builds on two recently developed analysis methods for the in-plane stresses at notch root under plane-stress or plane strain conditions, and the out-of-plane stresses at a three-dimensional notch root. The former solution (Filippi et al., 2002) considered the plane problem and gave the in-plane stress distributions in the vicinity of a V-shaped notch with a circular tip. The latter solution by Kotousov and Wang (2002a), which extended the generalized plane-strain theory by Kane and Mindlin to notches, provided an expression for the out-of-plane constraint factor based on some modified Bessel functions. By combining these two solutions, both valid under linear elastic conditions, closed form expressions are obtained for stresses and strain energy density in the neighborhood of the V-notch tip. To demonstrate the accuracy of the newly developed solutions, a significant number of fully three-dimensional finite element analyses have been performed to determine the influences of plate thickness, notch tip radius, and opening angle on the variability of stress distributions, out-of-plane stress constraint factor and strain energy density. The results of the comprehensive finite element calculations confirmed that the in-plane stress concentration factor has only a very weak variability with plate thickness, and that the present analytical solutions provide very satisfactory correlation for the out-of-plane stress concentration factor and the strain constraint factor.     

Modeling fracture in the context of a strain-limiting theory of elasticity: a single anti-plane shear crack

This paper is the first part of an extended program to develop a theory of fracture in the context of strain-limiting theories of elasticity. This program exploits a novel approach to modeling the mechanical response of elastic, that is non-dissipative, materials through implicit constitutive relations. The particular class of models studied here can also be viewed as arising from an explicit theory in which the displacement gradient is specified to be a nonlinear function of stress. This modeling construct generalizes the classical Cauchy and Green theories of elasticity which are included as special cases. It was conjectured that special forms of these implicit theories that limit strains to physically realistic maximum levels even for arbitrarily large stresses would be ideal for modeling fracture by offering a modeling paradigm that avoids the crack-tip strain singularities characteristic of classical fracture theories. The simplest fracture setting in which to explore this conjecture is anti-plane shear. It is demonstrated herein that for a specific choice of strain-limiting elasticity theory, crack-tip strains do indeed remain bounded. Moreover, the theory predicts a bounded stress field in the neighborhood of a crack-tip and a cusp-shaped opening displacement. The results confirm the conjecture that use of a strain limiting explicit theory in which the displacement gradient is given as a function of stress for modeling the bulk constitutive behavior obviates the necessity of introducing ad hoc modeling constructs such as crack-tip cohesive or process zones in order to correct the unphysical stress and strain singularities predicted by classical linear elastic fracture mechanics.     

One dimensional nonlocal integro-differential model & gradient elasticity model : Approximate solutions and size effects

In the present work, the close similarity that exists between Mindlin’s strain gradient elasticity and Eringens nonlocal integro-differential model is explored. A relation between length scales of nonlocal-differential model and gradient elasticity model has been arrived. Further, a relation has also been arrived between the standard and nonstandard boundary conditions in both the cases. C⁰-based finite element methods (FEMs) are extensively used for the implementation of integro-differential equations. This results in standard diagonally dominant global stiffness matrix with off diagonal elements occupied largely by the kernel values evaluated at various locations. The global stiffness matrix is enriched in this process by nonzero off diagonal terms and helps in incorporation of the nonlocal effect, there by accounting the long-range interactions. In this case, the diagonally dominant stiffness matrix has a band width equal to influence domain of basis function. In such cases, a very fine discretization with larger number of degrees of freedom is required to predict nonlocal effect, thereby making it computationally expensive. In the numerical examples, both nonlocal-differential and gradient elasticity model are considered to predict the size effect of tensile bar example. The solutions to integro-differential equations obtained by using various higher-order approximations are compared. Lagrangian, Bèzier and B-Spline approximations are considered for the analysis. It has been shown that such higher-order approximations have higher inter-element continuity there by increasing the band width and the nonlocal character of the stiffness matrix. The effect of considering the higher-order and higher-continuous approximation on computational effort is made. In conclusion, both the models predict size effect for one-dimensional example. Further, the higher-continuous approximation results in less computational effort for nonlocal-differential model.     

Analytical Full-field Solutions of a Piezoelectric Layered Half-plane Subjected to Generalized Loadings

The two-dimensional problem of a planar transversely isotropic piezoelectric layered half-plane subjected to generalized line forces and edge dislocations in the layer is analyzed by using the Fourier-transform method and the series expansion technique. The full-field solutions for displacements, stresses, electrical displacements and electric fields are expressed in explicit closed forms. The complete solutions consist only of the simplest solutions for an infinite piezoelectric medium with applied loadings. It is shown in this study that the physical meaning of this solution is the image method. The explicit solutions include Green's function for originally applied loadings in an infinite piezoelectric medium and the remaining terms are image singularities which are induced to satisfy free surface and interface continuity conditions. The mathematical method used in this study provides an automatic determination for the locations and magnitudes of all image singularities. The locations and magnitudes of image singularities are dependent on the piezoelectric material constants of the layered half-plane and the location of the applied loading. With the aid of the generalized Peach-Koehler formula, the image forces acting on dislocations are derived from the full-field solutions of the generalized stresses. Numerical results for the full-field distributions of stresses and electric fields in the piezoelectric layered half-plane and image forces for edge dislocations are presented based on the available analytical solutions.     

Nonlocal nonlinear bending and free vibration analysis of a rotating laminated nano cantilever beam

In this article, we present the nonlocal, nonlinear finite element formulations for the case of nonuniform rotating laminated nano cantilever beams using the Timoshenko beam theory. The surface stress effects are also taken into consideration. Nonlocal stress resultants are obtained by employing Eringen’s nonlocal differential model. Geometric nonlinearity is taken into account by using the Green Lagrange strain tensor. Numerical solutions of nonlinear bending and free vibration are presented. Parametric studies have been carried out to understand the effect of nonlocal parameter and surface stresses on bending and vibration behavior of cantilever beams. Also, the effects of angular velocity and hub radius on the vibration behavior of the cantilever beam are studied.     

Effective stress in the gauge theory of dislocations

Standard practices of the calculus of variations are used to modify the Lagrangian function of elasticity so that boundary tractions and initial linear momentum densities can be specified. Taken in conjunction with the Yang-Mills type minimal coupling theory of dislocations, these practices lead directly to a demonstration that the effective stress and linear momentum are what drive the dislocation fields in finite material bodies without disclinations.     

Stress and strain energy of a periodic array of interfacial wedge disclination dipoles in a transversely isotropic bicrystal

Analytical closed-form solutions are obtained for the elastic stress and strain energy density fields of a periodic array of interfacial wedge disclination dipoles in a bicrystal. The adjoining crystals are transversely isotropic with maximum dissimilar in-plane crystallographic orientations (0 and π/2). The solutions are obtained by the method of image dislocations. The strain energy per unit area of the bicrystal interface is also obtained numerically. The results show that significant discrepancies can exist between the bicrystal and the isotropic homogeneous solutions. The rates of decrease/increase of the strain energy density and stresses from the interface are smaller in the crystal whose larger stiffness direction is perpendicular to the interface. Also, the strain energy of the bicrystal boundary is a function of the dipole arm length (2a) and period (L). The maximum strain energy occurs at a/L=0.25 and is estimated to be ∼8.9 J/m² if the dipole period is 10 nm and the disclination strength is π/2.     

Investigation of the interaction of two collinear cracks in anisotropic elasticity materials by means of the nonlocal theory

In this paper, the interaction of two collinear cracks in anisotropic elasticity materials subjected to an anti-plane shear loading is investigated by means of the nonlocal theory. By use of the Fourier transform, the problem can be solved with the help of a pair of triple integral equations, in which the unknown variable is the displacement on the crack surface. To solve the triple integral equations, the displacement on the crack surface is expanded in a series of Jacobi polynomials. Unlike the classical elasticity solutions, it is found that no stress singularity is present at the crack tip. The nonlocal elastic solutions yield a finite hoop stress at the crack tip, thus allowing us to use the maximum stress hypothesis as a fracture criterion. The magnitude of the finite stress field depends on the crack length, the distance between two cracks and the lattice parameter of materials.     

Investigation of anti-plane shear behavior of a Griffith permeable crack in functionally graded piezoelectric materials by use of the non-local theory

In this paper, the non-local theory of elasticity is applied to obtain the behavior of a Griffith crack in functionally graded piezoelectric materials under the anti-plane shear loading for the permeable electric boundary conditions. To make the analysis tractable, it is assumed that the material properties vary exponentially with coordinate vertical to the crack. By means of the Fourier transform, the problem can be solved with the help of a pair of dual-integral equations that the unknown variable is the jump of the displacement across the crack surfaces. These equations are solved by use of the Schmidt method. Numerical examples are provided. Unlike the classical elasticity solutions, it is found that no stress and electric displacement singularities are present near the crack tips. The non-local elastic solutions yield a finite hoop stress at the crack tips, thus allows us to using the maximum stress as a fracture criterion. The finite hoop stresses at the crack tips depend on the crack length, the functionally graded parameter and the lattice parameter of the materials, respectively.     

Crack-tip fields for matrix cracks between dissimilar elastic and creeping materials

The stress fields near the tip of a matrix crack terminating at and perpendicular to a planar interface under symmetric in-plane loading in plane strain are investigated. The bimaterial interface is formed by a linearly elastic material and an elastic power-law creeping material in which the crack is located. Using generalized expansions at the crack tip in each region and matching the stresses and displacements across the interface in an asymptotic sense, a series asymptotic solution is constructed for the stresses and strain rates near the crack tip. It is found that the stress singularities, to the leading order, are the same in each material; the stress exponent is real. The oscillatory higher-order terms exist in both regions and stress higher-order term with the order of O(r°) appears in the elastic material. The stress exponents and the angular distributions for singular terms and higher order terms are obtained for different creep exponents and material properties in each region. A full agreement between asymptotic solutions and the full-field finite element results for a set of test examples with different times has been obtained.     

Dualistic theory of non-Riemannian material manifolds

The state of a material body can be represented by two different types of material non-Reimannian spaces. One is the strain space whose geometrical structures are specified by the strains and plastic imperfections (dislocations and disclinations). The other is the stress space whose geometrical structures are specified by the generalized stress functions and stresses. It is shown that there is an interesting duality between these two spaces, where the stored energy plays a fundamental role. The dualistic theory is presented. The potentials for dislocations and disclinations, as well as the forces acting on dislocations and disclinations, are derived by this approach.