Piezo patch sensor/actuator control of the vibrations of a cantilever under axial load

The effect of axial force in the vibration control of beams is studied by means of an integral equation formulation, which facilitates the numerical solution of the problem of finding the eigenfrequencies and eigenfunctions of a freely vibrating beam controlled by piezo patch sensors and actuators. The basic formulation of the problem is given in terms of a differential equation which is converted to an equivalent integral equation formulation by introducing an explicit Green’s function. After approximating the kernel of the integral equation in terms of Fourier series based on the eigenfrequencies of the host structure, a method of solution is outlined which consists of expressing the integral equation as an infinite series of linear equations and using a finite of number of these equations in the numerical solution. Numerical results are obtained for different values of the gain, the axial load and for three patch sizes. The first three vibration frequencies of the controlled and uncontrolled beam are given. The tip deflections of the beam for different axial loads are also plotted to compare transient behaviour.     

A comparison of localized finite element formulations for two-dimensional wave diffraction and radiation problems

Two of the most promising localized finite element methods are compared: the boundary series element method, in which a series of eigenfunctions is used to represent the far field solution; and the boundary integral element method, in which an integral equation is satisfied at the boundary between localized finite element and outer regions. The methods are applied to water of arbitrary depth. The theory of the two methods is summarized, and typical numerical results are discussed. Consideration is given to the well-known hydrodynamical reciprocal relations, and to the phenomenon of ‘irregular’ frequencies. The relative merits of the two methods are established.     

On the problem of two non-coplanar parallel cracks in a strip

The problem of two non-coplanar parallel Griffith cracks, located symmetrically in a strip is discussed under the assumption of plane strain condition. The crack surfaces are normal to the edges of the strip and are subjected to the same pressure distribution. When the edges are stress free, solution is assumed in the form of a series of complex eigenfunctions. By the use of certain generalized orthonormality relationships and the calculus of residues, two simultaneous Fredholm integral equations are obtained. Similar equations are obtained when the edges of the strip are constrained by smooth rigid planes. Numerical results are given for the case of constant applied pressure.     

On Papkovich-Fadle solutions of crack problems relating to an elastic strip

Papkovich-Fadle eigenfunctions are employed to study a class of crack problems of an elastic strip. A system of series relations is obtained which is reduced to a Fredholm integral equation of the second kind by the use of the generalized orthonormality of the eigenfunctions and the calculus of residues. Four types of crack configuration are considered. Based on the numerical solutions of the integral equations, stress intensity factor and crack energy for two types of edge cracks are reported.Some crack problems of the strip have been attempted by integral transforms which have been widely used in elasticity by Sneddon. It is not clear, however, how they may be used to tackle the problems considered here. The present approach is quite general and straightforward and, may be applied to a wide class of mixed boundary problems.     

Global identities for water-wave scattering potentials

Within the context of the linearised theory of time-harmonic water waves in three dimensions, a number of identities are obtained that are satisfied throughout the fluid domain by the velocity potential in scattering problems. This is done for both incident plane waves and for incident cylindrical waves. The implications of these results for the solution of time-domain scattering problems by the method of expansion in generalised eigenfunctions are discussed. In particular, it is demonstrated explicitly, for both two and three dimensions, that two different formulations of the generalised eigenfunction method are equivalent. Further, a new representation is given for the time-domain solution as an integral over the angles of incidence for particular generalised eigenfunctions.     

The effect of the lattice parameter of functionally graded materials on the dynamic stress field near crack tips

In this paper, the effect of the lattice parameter of functionally graded materials on the dynamic stress fields near crack tips subjected to the harmonic anti-plane shear waves is investigated by means of non-local theory. By use of the Fourier transform, the problem can be solved with the help of a pair of dual integral equations, in which the unknown variable is the displacement on the crack surfaces. To solve the dual integral equations, the displacement on the crack surfaces is expanded in a series of Jacobi polynomials. Unlike the classical elasticity solutions, it is found that no stress singularities are present near crack tips. The non-local elastic solution yields a finite hoop stress at the crack tip, thus allowing us to use the maximum stress as a fracture criterion in functionally graded materials.     

Fracture dynamics analysis of mode III interface moving crack on functionally graded coating

The interface moving crack between the functionally graded coating and infinite substrate structure with free boundary is investigated in this paper. By application of the interface bonding conditions of the two media, all the quantities have been represented by means of a single unknown function. With the help of the exponent model of the shear modulus and density, the dual integral equation of moving crack problem is obtained by Fourier transform. The displacement is expanded into series form using Jacob Polynomial, and then the semi‐analytic solution of dynamic stress intensity factor is derived by Schmidt method. Dynamic stress intensity factor is influenced by those parameters such as crack velocity, graded parameter and coating height.     

Dual boundary element analysis using complex variables for potential problems with or without a degenerate boundary

The dual boundary element method in the real domain proposed by Hong and Chen in 1988 is extended to the complex variable dual boundary element method. This novel method can simplify the calculation for a hypersingular integral, and an exact integration for the influence coefficients is obtained. In addition, the Hadamard integral formula is obtained by taking the derivative of the Cauchy integral formula. The two equations (the Cauchy and Hadamard integral formula) constitute the basis for the complex variable dual boundary integral equations. After discretizing the two equations, the complex variable dual boundary element method is implemented. In determining the influence coefficients, the residue for a single-order pole in the Cauchy formula is extended to one of higher order in the Hadamard formula. In addition, the use of a simple solution and equilibrium condition is employed to check the influence matrices. To extract the finite part in the Hadamard formula, the extended residue theorem is employed. The role of the Hadamard integral formula is examined for the boundary value problems with a degenerate boundary. Finally, some numerical examples, including the potential flow with a sheet pile and the torsion problem for a cracked bar, are considered to verify the validity of the proposed formulation. The results are compared with those of real dual BEM and analytical solutions where available. A good agreement is obtained.     

Crack propagating in a functionally graded strip under the plane loading

In the present paper, a finite crack with constant length (Yoffe type crack) propagating in the functionally graded strip under the plane loading is investigated by means of the Schmidt method. By using the Fourier transform and defining the jumps of displacement components across the crack surface as the unknown functions, two pairs of dual integral equations are derived. To solve the dual integral equations, the jumps of the displacement components across the crack surface are expanded in a series of Jacobi polynomial. Numerical examples are provided to show the effects of the material properties, the thickness of the functionally graded strip, and speed of the crack propagating upon the dynamic fracture behavior.     

Hypersingular integral formulation of elastic wave scattering

A hypersingular boundary integral formulation for calculating two dimensional elastic wave scattering from thin bodies and cracks is described. The boundary integral equation for surface displacement is combined with the hypersingular equation for surface traction. The difficult part in employing the traction equation, the derivation of analytical formulas for the hypersingular integral by means of a limit to the boundary, is easily handled by means of symbolic computation. In addition, the terms containing an integrable logarithmic singularity are treated by a straightforward numerical method, bypassing the use of Taylor series expansions. Example wave scattering calculations for cracks and thin ellipses are presented.     

On the Moving Griffith Crack in a Nonhomogeneous Orthotropic Strip

A finite crack with constant length (Yoffe type crack) propagating in the functionally graded orthotropic strip under the plane loading is investigated by means of the Schmidt method. By using the Fourier transform and defining the jumps of displacement components across the crack surface as the unknown functions, two pairs of dual integral equations are derived. To solve the dual integral equations, the jumps of the displacement components across the crack surface are expanded in a series of Jacobi polynomial. Numerical examples are provided to show the effects of material properties, the thickness of the functionally graded orthotropic strip and the speed of the crack propagating upon the dynamic fracture behavior.     

Diffraction of SH waves by a moving crack

Summary The problem of diffraction of anti-plane shear waves by a running crack of finite length is investigated analytically. Fourier transform method is used to solve the mixed boundary value problem which reduces to two pairs of dual integral equations. These dual integral equations are further reduced to a pair of Fredholm integral equations of the second kind. The iterative solution of the integral equations has been obtained for small wave number. The solution is used to calculate the dynamic stress intensity factor at the edge of the crack.With 2 Figures     

An inverse approach to construct residual stresses existing in axisymmetric structures using BEM

Based on the concept of the inherent strain and BEM, an inverse approach has been proposed for constructing residual stresses existing in axisymmetric structures. Taking into consideration of the efficiency of the algorithm and stability of the solution to the inverse problem, the inherent strains are approximately expressed as a series of smooth basis functions in the form of polynomials. With the presence of the initial stresses in the inverse problem, the boundary integral equation involves a domain integral; in order to preserve the advantage of the BEM, corresponding domain integral is transformed into the boundary integral by means of the dual reciprocity boundary element method. Then, the expression of the sensitivity matrix can be obtained and the distribution of residual stresses constructed efficiently. Numerical examples are given to show the applicability of the presented scheme.     

Effect of a submerged vertical barrier on flexural gravity waves

An explicit solution is provided for the scattering of flexural gravity waves by a rigid vertical barrier submerged in an infinite depth of water. By applying recently developed mode-coupling relation for eigenfunctions, the mixed boundary value problem has been converted to solve dual integral equations with kernel consisting of trigonometric functions. And then complete analytical solutions are derived with an aid of singular integral equations whose solutions are bounded at the end points. The important hydrodynamical scattering quantities such as reflection and transmission coefficients associated with the flexural gravity wave scattering have been obtained analytically in terms of modified Bessel functions and Struve functions. It is observed that these quantities are sensitive to both combined as well as individual effect of plate thickness and barrier depth of submergence. Numerical results are computed and explained graphically for different parameters such as time period and non-dimensional wave length. Further, the effect of compressive force and plate thickness on the flexural gravity waves against a submerged vertical barrier is studied.     

Basic solutions of two parallel mode-I cracks or four parallel mode-I cracks in the piezoelectric materials

In this paper, the stress and the electric intensity factors of two parallel mode-I cracks or four parallel mode-I cracks in the piezoelectric materials were examined by means of the Schmidt method for the permeable electric boundary conditions. The present problem can be solved by using the Fourier transform and the technique of dual integral equation, in which the unknown variables are the jumps of displacements across the crack surfaces, not the dislocation density functions. To solve the dual integral equations, the displacement jumps are directly expanded in a series of Jacobi polynomials. Finally, the effects of the distance between two parallel cracks and the distance between two collinear cracks on the stress and the electric intensity factors in the piezoelectric materials are analyzed. These results can be used for the strength evaluation of the piezoelectric materials with multi-cracks.     

Non-local theory solution of a mode-I crack in a piezoelectric/piezomagnetic composite material plane

The non-local theory solution of a mode-I permeable crack in a piezoelectric/piezomagnetic composite material plane was given by using the generalized Almansi’s theorem and the Schmidt method in this paper. The problem was formulated through Fourier transform into two pairs of dual integral equations, in which the unknown variables are the displacement jumps across the crack surfaces. To solve the dual integral equations, the displacement jumps across the crack surfaces were directly expanded as a series of Jacobi polynomials. Numerical examples were provided to show the effects of the crack length and the lattice parameter on the stress field, the electric displacement field and the magnetic flux field near the crack tips. Unlike the classical elasticity solutions, it is found that no stress, electric displacement and magnetic flux singularities are present at the crack tips in piezoelectric/piezomagnetic composite materials. The non-local elastic solution yields a finite hoop stress at the crack tip, thus allowing us to use the maximum stress as a fracture criterion.     

Dynamic stress intensity factor of a cracked dielectric medium in a uniform electric field

Summary We consider the scattering of normally incident longitudinal waves by a finite crack in an infinite isotropic dielectric body under a uniform electric field. By the use of Fourier transforms, we reduce the problem to that of solving two simultaneous dual integral equations. The solution of the dual integral equations is then expressed in terms of a Fredholm integral equation of the second kind having the kernel that is a finite integral. The dynamic stress intensity factor versus frequency is computed, and the influence of the electric field on the normalized values is displayed graphically.     

Diffraction of transient horizontal shear waves by a finite crack at the interface of two bonded dissimilar elastic solids

Scattering of transient horizontal shear waves by a finite crack located at the interface of two bonded dissimilar elastic solids is investigated in this study. Laplace and Fourier transform technique is used to reduce the problem to a pair of dual integral equations. The solution of the dual integral equation is expressed in terms of the Fredholm integral equation of the second kind having the kernel of a finite integration. Dynamic stress intensity factor is obtained as a function of the material and geometric properties and time.     

Dual boundary integral equations for exterior problems

A dual integral formulation for the interior problem of the Laplace equation with a smooth boundary is extended to the exterior problem. Two regularized versions are proposed and compared with the interior problem. It is found that an additional free term is present in the second regularized version of the exterior problem. An analytical solution for a benchmark example in ISBE is derived by two methods, conformal mapping and the Poisson integral formula using symbolic software. The potential gradient on the boundary is calculated by using the hypersingular integral equation except on the two singular points where the potential is discontinuous instead of failure in ISBE benchmarks. Based on the matrix relations between the interior and exterior problems, the BEPO2D program for the interior problem can be easily reintegrated. This benchmark example was used to check the validity of the dual integral formulation, and the numerical results match the exact solution well.     

A non-axisymmetric cylindrical contact problem

The contact problem under investigation is one whereby a solid circular elastic cylinder of infinite length is rigidly indented by a two piece collar of finite length, each piece being diametrically opposed and extending only partially around one half of the circumference. This case is practically significant in relation to the axisymmetric cylindrical contact problem since in many cases attachment of a component to a cylindrical shaft is achieved by means of a two piece clamp.Shear stresses on the contact interface are taken zero and a radial displacement influence coefficient technique is used to model the integral equation governing this contact problem. Adopting the Papkovich-Neuber solution for the non-axisymmetric cylindrical coordinate case and substituting the appropriate boundary conditions leads to a combined Fourier series, Fourier integral representation for the desired displacements. Convergence of this series—integral is studied and results of interference contact pressure are presented for an illustrative range of the various parameters involved.