Papers by Keyword: Integral Equation

Abstract: The research purpose is to develop an approach for determining the stress concentration near the holes in composite structure elements reinforced with carbon fibres. The research is performed on the basis of a numerical-analytic approach using the method of singular integral equations. The paper studies the stress concentration near the holes in composite plate elements of the structures, which are reinforced with carbon fibres. The stresses are determined based on the singular integral equations. The integral equations are solved numerically using the mechanical quadrature method. The stress in the strip is studied at: longitudinal tension; pure bending; three-point bending; with periodically spaced holes. An approach to calculating the stresses in composite strips weakened by holes of different shapes, based on the method of integral equations, has been developed. The equation kernels are formulated on the basis of Green's functions, under which the boundary conditions on straight-line boundaries are satisfied identically. A methodology for calculating the stress concentration near the holes of arbitrary shape in plate elements of the structures has been developed. The results obtained can be used when calculating the strength of composite materials reinforced with carbon fibres.

Abstract: The article shows the process of analyzing and constructing the stiffness matrices for four-node elements in parallelogram and rhombus shapes using the method of potential deformation energy.

Abstract: In this paper the integral equation approach is used to describe the propagation of continuumdamage in three dimensional solids. The governing equation is of integral type and contains bothboundary and domain integrals. Such integrals are computed with the aid of the NURBS functions.The subvolume involved by the damage is modelled by a special mapping procedure that avoids theuse of the internal cells. The implementation is verified on a test case for which an analytical solutionis available.

Abstract: The problem of fracture mechanics concerning contact interaction between elastic infinite plate and elastic compound semi–space is investigated. Plate and semi–space are weakened by finite through cracks, which are perpendicular to surface of heterogeneity in the same plane. Assuming that structure is deformed in antiplane deformation state it is required to determine the contact stress distribution and fracture stress intensity factors dependence of structure heterogeneity and geometrical parameters. Using the Fourier integral transform the problem is reduced to find the solutions of system of two singular integral equations. System solutions behavior at integration domain endpoints is investigated for all cases. In some special cases of cracks location, equations kernels can also contain fixed singularities. An efficient numerical method to solve such equations is suggested. Numerical calculations are done and results are shown in tables and graphs, which express contact stresses and stress intensity factors dependence on problem parameters and simultaneously reveal dangerous cases of fracture of the structure.

Abstract: In statement of the steady-state filtration theory and within the framework of the Darcy`s law plane boundary value problems for the strip-like and wedge-shaped porous ground base are considered when through some system of segments on one face of the base the fluid with a certain vertical velocity or with a certain pressure is injected inside the base. These solutions are reduced to integral equations by means of integral transforms.

Abstract: In this paper, we discuss a class of linear integral equation with piecewise continuous function. Firstly, we change the integral equation to a differential equation with the initial condition. Secondly, the differential equation is solved by the constant variation formula and integration by parts. Explicit solution of the integral equation is given clearly.

Abstract: The purpose of the present work is to investigate the dynamic interaction between plate and structure in a poroelastic half-space produced by moving load using a semi-analytical method. Biots dynamic equations are solved by the Fourier transform technique. The forces and displacements of the superstructure, plate and poroelastic medium at any time are obtained by using the numerical inverse transform technique. Based on the derivation and numerical examples presented above, the following conclusions are drawn: the horizontal displacements and vertical displacements of each floor of the superstructure increase with increasing the velocities of moving load. The vertical displacements in each floor have small change.

Abstract: The quasi-Green function method (QGFM) is applied to solve the free vibration of clamped orthotropic thin plates with parallelogram boundary shape on Winkler foundation. Firstly the model governing differential equation of the problem is reduced to the boundary value problem of the biharmonic operator, and then it is reduced to the Fredholm integral equation of the second kind by Greens formula. A quasi-Green function is established by using the fundamental solution and the boundary equation of the problem. This function satisfies the homogeneous boundary condition of the problem. The singularity of the kernel of the integral equation is overcome by choosing a suitable form of the normalized boundary equation. The comparison with ANSYS finite element solution shows a good agreement. The proposed method is a novel and effective mathematical one.

Abstract: The quasi-Greens function method (QGFM) is applied to solve the bending problem of simply supported polygonal shallow spherical shells on Pasternak foundation. A quasi-Greens function is established by using the fundamental solution and the boundary equation of the problem. And the function satisfies the homogeneous boundary condition of the problem. Then the differential equation of the problem is reduced to two simultaneous Fredholm integral equations of the second kind by the Greens formula. The singularity of the kernel of the integral equation is overcome by choosing a suitable form of the normalized boundary equation. The comparison with the ANSYS finite element solution shows a good agreement, and it demonstrates the feasibility and efficiency of the proposed method.

Abstract: Based on Biot’s theory and the minimum potential energy principle as well as the thin plate theory, the superstructure, raft and soil are assumed to be a whole system according to the substructure method. The system must satisfy the continuity conditions at the interface between the superstructure, raft and soil surface. Considering the compatibility condition that the vertical displacement of the interface between the raft and the saturated soil should be equal, the integral equation accounting for the vertical coupling of the superstructure-raft system with the saturated soil subjected to a moving load is constructed. Using the numerical inverse transform technique, the forces and displacements of the superstructure, plate and saturated soil at any time are obtained. Some numerical results are presented to demonstrate the capacity of the proposed model. Also, the influence of load velocity on the superstructure will be investigated in this paper.