A class of two-dimensional dual integral equations and its application

Because exact analytic solution is not available, we use double expansion and boundary collocation to construct an approximate solution for a class of two-dimensional dual integral equations in mathematical physics. The integral equations by this procedure are reduced to infinite algebraic equations. The accuracy of the solution lies in the boundary collocation technique. The application of which for some complicated initialboundary value problems in solid mechanics indicates the method is powerful.

Modified multiplying-factor integration method for solving exponential function dual integrals in crack problems

Crack problems are often reduced to dual integral equations,which can be solved by expanding the displacement integral equation as a series in the form of Chebyshev-like or Jacobi polynomials.Schmidt’s multiplying-factor integration method has been one of the most favorable techniques for determining the expansion coefficients by constructing a well-posed system of linear algebraic equations.However,Schmidt’s method is less efficient for numerical computation because the matrix elements of the linear equations are evaluated from dual integrals.In this study,we propose a modified method to construct linear equations to efficiently determine the expansion coefficients.The modified technique is developed upon the application of certain multiplying factors to the traction integral equation and then integrating the resulting equation over“source”regions.Such manipulations simplify the matrix elements as single integrals.By carrying out numerical examples,we demonstrate that the technique is not only accurate but also very efficient.In particular,the method only needs approximately 1/5 of the computation time of Schmidt’s method.Therefore,this method can be used to replace Schmidt’s method and is expected to be very useful in solving crack problems.

Dual-resource integrated scheduling method of AGV and machine in intelligent manufacturing job shop

In view of the fact that traditional job shop scheduling only considers a single factor, which affects the effect of resource allocation, the dual-resource integrated scheduling problem between AGV and machine in intelligent manufacturing job shop environment was studied. The dual-resource integrated scheduling model of AGV and machine was established by comprehensively considering constraints of machines, workpieces and AGVs. The bidirectional single path fixed guidance system based on topological map was determined, and the AGV transportation task model was defined. The improved A* path optimization algorithm was used to determine the optimal path, and the path conflict elimination mechanism was described. The improved NSGA-Ⅱ algorithm was used to determine the machining workpiece sequence, and the competition mechanism was introduced to allocate AGV transportation tasks. The proposed model and method were verified by a workshop production example, the results showed that the dual resource integrated scheduling strategy of AGV and machine is effective.

DYNAMIC BEHAVIOR OF TWO PARALLEL SYMMETRY CRACKS IN MAGNETO-ELECTRO-ELASTIC COMPOSITES UNDER HARMONIC ANTI-PLANE WAVES

The dynamic behavior of two parallel symmetry cracks in magneto-electro-elastic composites under harmonic anti-plane shear waves is studied by Schmidt method. By using the Fourier transform, the problem can be solved with a pair of dual integral equations in which the unknown variable is the jumps of the displacements across the crack surfaces. To solve the dual integral equations, the jumps of the displacements across the crack surface were expanded in a series of Jacobi polynomials. The relations among the electric filed, the magnetic flux and the stress field were obtained. From the results, it can be obtained that the singular stresses in piezoelectric/piezomagnetic materials carry the same forms as those in a general elastic material for the dynamic anti-plane shear fracture problem. The shielding effect of two parallel cracks was also discussed.

TORSIONAL OSCILLATION OF RIGID DISK IN INFINITE TRANSVERSELY ISOTROPIC ELASTIC CYLINDER

In the present analysis torsional oscillation of a rigid disk in an infinite transversely isotropic elastic cylinder is considered. The effects of anisotropy in the stress intensity factor are shown graphically.

SCATTERING OF HARMONIC ANTI-PLANE SHEAR WAVES BY AN INTERFACE CRACK IN MAGNETO-ELECTRO-ELASTIC COMPOSITES

The dynamic behavior of an interface crack in magneto-electro-elastic composites under harmonic elastic anti-plane shear waves is investigated for the permeable electric boundary conditions. By using the Fourier transform, the problem can be solved with a pair of dual integral equations in which the unknown variable was the jump of the displacements across the crack surfaces. To solve the dual integral equations, the jump of the displacements across the crack surface was expanded in a series of Jacobi polynomials. Numerical examples were provided to show the effect of the length of the crack, the wave velocity and the circular frequency of the incident wave on the stress, the electric displacement and the magnetic flux intensity factors of the crack. From the results, it can be obtained that the singular stresses in piezoelectric/piezomagnetic materials carry the same forms as those in a general elastic material for anti-plane shear problem.

DYNAMIC BEHAVIOR OF TWO PARALLEL SYMMETRIC PERMEABLE CRACKS IN A PIEZOELECTRIC STRIP

The dynamic behavior of two parallel symmetric cracks in a piezoelectricstrip under harmonic anti-plane shear waves is studied using the Schmidt method for permeable cracksuface conditions. The cracks are parallel to the edge of the strip. By means of the Fouriertransform, the problem can be solved with the help of two pairs of dual integral equations. Theseequations are solved using the schmidt method. The results show that the stress and the electricdisplacement intensity factors depend on the geometry of the cracks, the frequency of incidentwaves, the distance between cracks and the thickness of the strip. It is also found that theelectric displacement intensity factors for the permeable crack surface conditions are much smallerthan those for the impermeable crack surface conditions.

Vertical vibrations of elastic foundation resting on saturated half-space

This paper is mainly concerned with the dynamic response of an elastic foundation of finite height bounded to the surface of a saturated half-space. The foundation is subjected to time-harmonic vertical loadings. First, the transform solutions for the governing equations of the saturated media are obtained. Then, based on the assumption that the contact between the foundation and the half-space is fully relaxed and the halfspace is completely pervious or impervious, this dynamic mixed boundary-value problem can lead to dual integral equations, which can be further reduced to the Predhohn integral equations of the second kind and solved by numerical procedures. In the numerical extortples, the dynamic colnpliances, displacements and pore pressure are developed for a wide range of frequencies and material/geometrical properties of the saturated soil-foundation system. In most of the cases, the dynamic behavior of an elastic foundation resting on the saturated media significantly differs from that of a rigid disc on the saturated half-space. The solutions obtained can be used to study a variety of wave propagation problems and dynamic soil-structure interactions.

INVESTIGATION OF THE BEHAVIOR OF A GRIFFITH CRACK AT THE INTERFACE BETWEEN TWO DISSIMILAR ORTHOTROPIC ELASTIC HALF-PLANES FOR THE OPENING CRACK MODE

The behaviors of an interface crack between dissimilar orthotropic elastic half-planes subjected to uniform tension was reworked by use of the Schmidt method.By use of the Fourier transform,the problem can be solved with the help of two pairs of dual integral equations,of which the unknown variables are the jumps of the displacements across the crack surfaces.Numerical examples are provided for the stress intensity factors of the cracks.Contrary to the previous solution of the interface crack,it is found that the stress singularity of the present interface crack solution is of the same nature as that for the ordinary crack in homogeneous materials.When the materials from the two half planes are the same,an exact solution can be otained.

THE SCATTERING OF HARMONIC ELASTIC ANTI-PLANE SHEAR WAVES BY A FINITE CRACK IN INFINITELY LONG STRIP USING THE NON-LOCAL THEORY

In this paper, the scattering of harmonic anti-plane shear wavesby a finite crack in infinitely long strip is studied using thenon-local theory. The Fourier transform is applied and a mixedboundary value problem is formulated. Then a set of dual integralequations is solved using the Schmidt method instead of the first orthe second integral equation method. A one-dimensional non-localkernel is used instead of a two-di- mensional one for the anti-planedynamic problem to obtain the stress occurring at the crack tips.Contrary to the classical elasticity solution, it is found that nostress singularity is present at the crack tip. The non-local dynamicelastic solutions yield a finite hoop stress at the crack tip, thusallowing for a fracture criterion based on the maximum dynamic stresshypothesis. The finite hoop stress at the crack tip depends on thecrack length, the width of the strip and the lattice parameters.

Behavior of two parallel symmetry permeable cracks in functionally graded piezoelectric materials subjected to an anti-plane shear loading

The behavior of two parallel symmetry permeable cracks in functionally graded piezoelectric materials subjected to an anti-plane shear loading was investigated. To make the analysis tractable, it was assumed that the material properties varied exponentially with coordinate vertical to the crack. By using the Fourier transform, the problem could be solved with the help of two pairs of dual integral equations, in which the unknown variables were the jumps of the displacements across the crack surfaces. To solve the dual integral equations, the displacement on the crack surfaces was expanded in a series of Jacobi polynomials. The normalized stress and electrical displacement intensity factors were determined for different geometric and property parameters for permeable electric boundary conditions. Numerical examples were provided to show the effect of the geometry of the interacting cracks and the functionally graded material parameter upon the stress intensity factors of cracks.

Interior-stress fields produced by a general axisymmetric punch

This work is a supplement to the work of Sneddon on axisymmetric Boussinesq problem in 1965 in which the distributions of interior-stress fields are derived here for a punch with general profile.A novel set of mathematical procedures is introduced to process the basic elastic solutions(obtained by the method of Hankel transform,which was pioneered by Sneddon)and the solution of the dual integral equations.These processes then enable us to not only derive the general relationship of indentation depth D and total load P that acts on the punch but also explicitly obtain the general analytical expressions of the stress fields beneath the surface of an isotropic elastic half-space.The usually known cases of punch profiles are reconsidered according to the general formulas derived in this study,and the deduced results are verified by comparing them with the classical results.Finally,these general formulas are also applied to evaluate the von Mises stresses for several new punch profiles.