Dislocation layers applied to moving crack problems

In this paper we consider a moving Griffith-type strip crack subjected to three fundamental modes of surface tractions in an infinite, homogeneous, isotropic, elastic solid. Closed form solutions are obtained in detail for the mode II case for the components of the resulting stress fields using a method of dislocation layers and some numerical results are given. Some implications of the results for brittle and ductile crack growth are discussed.     

On some integral equations arising in elastic bimaterial problems

We consider integral equations which arise in anti-plane strain elastic problems where cracks or dislocation pile ups cross an elastic inclusion of circular cross section. An elementary transformation is used to show a connection between these equations and simpler ones which arise when two half-spaces with different elastic constants are welded together. It is then possible to obtain closed form solutions in certain cases.     

Experimental and finite element analyses on stress intensity factors of an elliptical surface crack in a circular shaft under tension and bending

Experimental backtracking technique and finite element analysis have been employed to evaluate the stress intensities along the front of an elliptical surface crack in a cylindrical rod. The finite element solution covers a wide range of crack shapes loaded under end-free and end-constrained axial tension and pure bending. Convenient closed form stress intensity expressions along the whole crack front for each of the loading cases have been given in terms of the crack aspect ratio, crack depth ratio and place ratio.The closed form solutions have been compared against a number of representative solutions collected from the literature. It has been found that different finite element results for the interior points are generally in good mutual agreement, while solutions derived from other methods may sometimes indicate different trends. At the surface interception point agreement is less good because of a complication in the interpretation of stress intensity there.Experimental backtracking results on the end-constrained axial tension case corroborate well with the closed form solution presented. It suggests that the current closed form solution is adequate in describing the stress intensities along the whole crack front of real surface cracks in cylindrical rods.     

Solving variational problems by homotopy–perturbation method

In this paper, we consider some problems in calculus of variations. Homotopy–perturbation method (HPM) is adopted to obtain approximate analytical solutions to variational problems. The solutions are obtained in the form of rapidly convergent infinite series with easily computable terms. Numerical results reveal that HPM is very effective and simple for obtaining approximate solutions to variational problems. Copyright © 2008 John Wiley & Sons, Ltd.     

Strip Yield Model Numerical Application to Different Geometries and Loading Conditions

A new numerical method based on the strip yield analysis approach was developed for calculating the Crack Tip Opening Displacement (CTOD). This approach can be applied to any crack geometry, of either infinite or finite extent, with arbitrary applied loading conditions. The technique is an adaptation of the dislocation-density based boundary element method to obtain crack-face opening displacements at any point on a crack, and succeeds in obtaining the requisite values as a series of definite integrals, the functional parts of each of which are evaluated exactly in closed form. The power of the technique is demonstrated by obtaining solutions to several crack configurations of practical interest.     

Comparison of boundary and finite element methods for moving-boundary problems governed by a potential

Three formulations of the boundary element method (BEM) and one of the Galerkin finite element method (FEM) are compared according to accuracy and efficiency for the spatial discretization of two-dimensional, moving-boundary problems based on Laplace's equation. The same Euler-predictor, trapezoid-corrector scheme for time integration is used for all four methods. The model problems are on either a bounded or a semi-infinite strip and are formulated so that closed-form solutions are known. Infinite elements are used with both the BEM and FEM techniques for the unbounded domain. For problems with the bounded region, the BEM using the free-space Green's function and piecewise quadratic interpolating functions (QBEM) is more accurate and efficient than the BEM with linear interpolation. However, the FEM with biquadratic basis functions is more efficient for a given accuracy requirement than the QBEM, except when very high accuracy is demanded. For the unbounded domain, the preferred method is the BEM based on a Green's function that satisfies the lateral symmetry conditions and which leads to discretization of the potential only along the moving surface. This last formulation is the only one that reliably satisfies the far-field boundary condition.     

Optimum rigid motion with one perspective view

One of the central problems in dynamic computer vision is the determination of motion from correspondence of points on the visible surface of a moving object obtained at two different times. In the problem we consider, the initial position of several feature points on some object is assumed to be known. Then at some later time instant the corresponding perspective projection of those points is given. The moving body may be nonrigid or the measurements may be corrupted with noise. In either case, we determine the multiplicity of possible solution for the best-fitting rigid motion. We show that whenever at least three point correspondences are given, there are at most two possible solutions for the best-fitting rigid motion if the object space is two-dimensional, while there are at most forty possible solutions if the object space is three-dimensional. In the latter case the number of possible solutions drops to four if three point correspondence are known, the body is rigid, and the measurements are noise-free. Several examples are worked out to illustrate our results.©1994 John Wiley & Sons Inc     

Smooth moving punch in an initially stressed transversely isotropic magnetoelastic medium due to shear wave

This study investigates the effect of a semi-infinite smooth moving punch due to shear wave propagation in initially stressed, magnetoelastic, transversely isotropic material. The Wiener–Hopf technique has been employed to determine the closed form expression of dynamic stress concentration due to punch with a load of constant intensity. The substantial effects of magnetoelastic coupling parameters, horizontal and vertical compressive/tensile initial stress, and anisotropy on dynamic stress concentrations has been remarkably traced out. Numerical computations and graphical illustrations, along with comparative study, have been executed for three distinct models: when the strip is comprised of Zinc, Beryl material having hexagonal symmetry, and simply isotropic material.     

Electro-elastic interaction between a moving screw dislocation and collinear interfacial rigid lines

This paper attempts to investigate the problem for the electro-elastic interaction between a piezoelectric moving dislocation and interfacial collinear rigid lines under combined longitudinal shear and in-plane electric field. Using Riemann–Schwarz's symmetry principle integrated with the analysis singularity of complex functions, we present the general elastic solution of this problem and the closed form solution for interface containing single rigid line. The expressions of electro-elastic fields and image force acting on moving dislocation are derived explicitly. The results show that the velocity of moving dislocation has significant effect on the image force. The present solutions contain previously known results as the special cases.     

On the Exact Solution of Boundary Value Problems with Non-separable Geometries,Like a Rough Surface

A Dirichlet problem is considered for the Helmholtz equation and for a class of geometries for which the Helmholts operator does not separate (e.g. a rough surface). It is shown that, contrary to the widely held view, it is possible to obtain the solution of this problem in a closed form which resembles closely the solutions obtained for separable geometries—expansions generated by Sturm-Liouville theory. As with separable geometries, we show in particular that the expansion coefficients can be written explicitly as integrals containing a priori known functions—that matrix inversion is not required for the determination of the expansion coefficients. The problem includes as a special case the scattering of electromagnetic waves from a rough cylindrical surface which is the boundary of a perfect conductor. The method is very general and can be used for much more complicated boundary value problems, such as scattering by dielectric interface.     

Interactions between inclusions and various types of cracks

The problems of a crack inside, outside, penetrating or lying along the interface of an anisotropic elliptical inclusion are considered in this paper. Because the crack may be represented by a distribution of dislocation, integrating the analytical solutions of dislocation problems along the crack and applying the technique of numerical solution on the singular integral equation, we can obtain the general solutions to the problems of interactions between cracks and anisotropic elliptical inclusions. Since there are no analytical solutions existing for the general cases of interactions between cracks and inclusions, the comparison is made with the numerical results obtained by other methods or with the analytical results for the special cases which can be reduced from the present problems. These results show that our solutions are correct and universal     

A dislocation interacts with a finite crack in piezoelectric media

Analytical solutions for the generalized two-dimensional problem of a dislocation interacting with a finite crack in piezoelectric media are formulated via Stroh formalism. Two kinds of electrically boundary conditions along the crack surface, namely, electrically impermeable and electrically permeable are studied in the analysis. Based on the complex variable method and the perturbation technique, the closed form solutions are obtained. The field intensity factors at the crack tip and the image forces on the dislocation due to the crack are computed and discussed.     

Heat Conduction in a Semi-infinite Body with Power-Type Initial and Boundary Conditions

Heat conduction problems in a semi-infinite body with constant thermal properties are investigated. Firstly, two types of basic problems are studied. One type considers the medium with an initial temperature a power function of position subject to zero surface temperature, surface flux, or ambient temperature; the other type considers the medium with zero initial temperature subject to boundary conditions that vary as a power of time. Closed-form analytical solutions for these basic problems are constructed. Using these basic solutions and the supposition principle, analytical solutions for problems with polynomial-type initial and boundary conditions are obtained. The procedure can also be applied to obtain solutions for other types of initial and boundary conditions that can be expanded into power series with an infinite radius of convergence. An illustrative example is presented and discussed. Theoretical proof for the convergence of the series form solution is conducted.     

Explicit solution for a vibrating bar with viscous boundaries and internal damper

We investigate longitudinal vibrations of a bar subjected to viscous boundary conditions at each end and an internal damper at an arbitrary point along the bar’s length. The system is described by four independent parameters and exhibits a variety of behaviors including rigid motion, super stability/instability and zero damping. The solution is obtained by applying the Laplace transform to the equation of motion and computing the Green’s function of the transformed problem. This leads to an unconventional eigenvalue-like problem with the spectral variable in the boundary conditions. The eigenmodes of the problem are necessarily complex-valued and are not orthogonal in the usual inner product. Nonetheless, in generic cases we obtain an explicit eigenmode expansion for the response of the bar to initial conditions and external force. For some special values of parameters the system of eigenmodes may become incomplete, or no non-trivial eigenmodes may exist at all. We thoroughly analyze physical and mathematical reasons for this behavior and explicitly identify the corresponding parameter values. In particular, when no eigenmodes exist, we obtain closed form solutions. Theoretical analysis is complemented by numerical simulations, and analytic solutions are compared to computations using finite elements.     

Free Boundary Determination in Nonlinear Diffusion

Free boundary problems with nonlinear diffusion occur in various applications, such as solidification over a mould with dissimilar nonlinear thermal properties and saturated or unsaturated absorption in the soil beneath a pond. In this article, we consider a novel inverse problem where a free boundary is determined from the mass/energy specification in a well-posed one-dimensional nonlinear diffusion problem, and a stability estimate is established. The problem is recast as a nonlinear least-squares minimisation problem, which is solved numerically using the lsqnonlin routine from the MATLAB toolbox. Accurate and stable numerical solutions are achieved. For noisy data, instability is manifest in the derivative of the moving free surface, but not in the free surface itself nor in the concentration or temperature.     

On the non-linear behaviour of fluid jets

In this paper we consider a one-dimensional theory for the flow of fluid jets, formulated with the help of one-dimensional postulates. The theory is applied to a circular cylindrical jet of incompressible viscous fluid with constant pressure and constant surface tension over its surface. A phase plane analysis is used to obtain some (non-linear) solutions of the governing equations when viscosity is neglected. For steady state problems, including viscosity, earlier work of Matovich and Pearson[2] is obtained as a limiting case of the theory.     

Transient heat conduction analysis in functionally graded materials by the meshless local boundary integral equation method

Advanced computational method for transient heat conduction analysis in continuously nonhomogeneous functionally graded materials (FGM) is proposed. The method is based on the local boundary integral equations with moving least square approximation of the temperature and heat flux. The initial-boundary value problem is solved by the Laplace transform technique. Both Papoulis and Stehfest algorithms are applied for the numerical Laplace inversion to obtain the time-dependent solutions. Numerical results are presented for a finite strip and a hollow cylinder with an exponential spatial variation of material parameters.     

Dissipative heating of a hollow viscoelastic cylinder subjected to the action of normal loads moving along its surface with constant angular velocity

We consider a two-dimensional quasistatic problem of the stress-strain state and dissipative heating of a hollow circular viscoelastic cylinder subjected to the action of two, normal diametrically opposite loads moving along the surface of the cylinder with constant angular velocity. It is assumed that the physicomechanical properties of the material are independent of temperature. In this case, the combined problem of thermoviscoelasticity splits into two linear problems: the problem of the stress-strain state of the cylinder without axial symmetry and the axisymmetric problem of heat conduction with temperature averaged over a cycle and a known heat source. The exact solutions of these problems are obtained. The influence of both the coefficient of heat transfer on the inner boundary of the cylinder and the sizes of the cylinder on the temperature of dissipative heating are investigated. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev, Ukraine. Translated from Problemy Prochnosti, No. 2, pp. 67–73, March–April, 1999.     

STRIP DISTRIBUTED TRANSFER FUNCTION METHOD FOR ANALYSIS OF PLATES

A semi-analytical method, called the strip distributed transfer function method, is developed for analysis of plate structures that are composed of rectangular plates. In the method, a rectangular plate (substructure) is divided into a number of strips; the response of each strip is interpolated in the unknown nodal line displacements, which are functions of the strip longitudinal co-ordinate and time. The nodal line displacements are determined in an exact and closed form by the distributed transfer functions that are defined along the strips. Synthesis of the substructures using the strip distributed transfer functions yields accurate prediction of the static and dynamic response, natural frequencies and buckling loads of the structure. The proposed method is compared with some existing techniques in numerical examples.