Closed form solution for a nonlocal elastic bar in tension

A simple mechanical one-dimensional problem in the context of nonlocal (integral) elasticity is solved analytically. The nonlocal elastic material behaviour is described by the “Eringen model” whose nonlocality features all reside in the constitutive relation. This relation, of integral type, contains an attenuation function (usually assumed symmetric) aimed to capture the diffusion process of the nonlocality effects; it also exhibits a convolution format. The governing equation is a Fredholm integral equation of second kind whose analytical treatment, even for the usual choice of a symmetric kernel, is not easy to develop. In the present paper, assuming a specific shape for the attenuation function, a closed form solution in terms of strains is alternatively obtained by solving a Volterra integral equation of second kind. The latter can be easily solved with standard techniques, at least for the adopted kernel, taking also advantage from the symmetry of the solution. Such a closed form solution is an essential result to validate the effectiveness of numerical procedures aimed to solve more complex mechanical problems in the context of nonlocal elasticity.     

Dynamic response of elastic layer on stiff foundation under time harmonic surface vertical concentrated load

In this paper, the line-load integral equation method proposed in reference [1] is first used for solving the elastodynamic problems. A set of one-dimensional regular integral equation is derived for calculating the dynamic response of elastic layer on stiff foundation under time harmonic surface vertical concentrated load. And the numerical solution of the integral equation is obtained.     

Scattering of elastic waves by elastically transparent obstacles (integral-equation method)

A formulation of elastodynamic diffraction problems for sinusoidally in time varying disturbances in a linearly elastic medium is presented. Starting with the elastodynamic reciprocity relation, an integral representation for the particle displacement is derived. In it, the particle displacement and the traction at the boundary of the obstacle occur. From the integral representation, an associated integral equation is obtained by letting the point of observation approach the boundary of the obstacle. The “obstacle” may be either a rigid body, a void, or a body with elastic properties differing from those of its environment, or a combination of these. The integral equation thus obtained is well-suited for numerical treatment, when obstacles up to a few wavelengths in maximum diameter are considered.     

Contact problems for a circular plate with sliding fixation at the end face

Two mixed elasticity problems of punch indentation into a circular plate placed without clearance in a rigid cylindrical holder with smooth walls are considered. In the first problem, the plate lies without friction on a rigid base, and in the second problem, the plate is rigidly fixed to the base. The problems are solved by a method that was developed for bodies of finite dimensions and is based on the properties of closed systems of orthogonal functions. Each of the problems is reduced to two integral equations, namely, a Volterra integral equation of the first kind for the contact pressure function and a Fredholm integral equation of the first kind for the derivatives of the displacement of the plate upper surface outside the punch. The displacement function is sought as the sum of a trigonometric series and a power function with a root singularity. After truncation, the obtained illposed system of linear algebraic equation has a stable solution. A method for solving Volterra integral equations is given. The contact pressure distribution function and the dimensionless indentation force are determined. Examples of calculation of the plate interaction with the plane punch are given. Contact problems were earlier studied for a rectangle and a circular plate with a stress-free end both without taking account of their fixation [1, 2] and with regard for their fixation [3, 4]. The solution method described here was used to study the interaction of elastic hollow cylinder of finite length with a rigid bandage and a rigid insert [5, 6]. Other papers dealing with contact problems for bodies of finite dimensions, in particular, for a circular plate, should also be mentioned. In these papers, the problems under study were solved by the method of homogeneous solutions [7, 8] and by the method of coupled series-equations [9].     

基于拉氏变换的弹性动力学插值型重构核粒子法

插值型重构核粒子法的形函数具有离散点插值特性和不低于核函数的高阶光滑性，因而不仅可以直接施加本质边界条件，同时也保证了较高的计算精度．本文将弹性动力学方程作拉氏变换后，在变换域内用插值型重构核粒子法求解，最后再借助Durbin数值反演方法求得时间域的解．针对典型的弹性动力学问题，给出了插值型重构核粒子法的数值算例，并验证了本文方法的有效性．     

Analytical solution for the electroelastic dynamics of a nonhomogeneous spherically isotropic piezoelectric hollow sphere

Summary By introduction of a special dependent variable and separation of variables technique, the electroelastic dynamic problem of a nonhomogeneous, spherically isotropic hollow sphere is transformed to a Volterra integral equation of the second kind about a function of time. The equation can be solved by means of the interpolation method, and the solutions for displacements, stresses, electric displacements and electric potential are obtained. The present method is suitable for a piezoelectric hollow sphere with an arbitrary thickness subjected to arbitrary mechanical and electrical loads. Numerical results are presented at the end.The work was supported by the National Natural Science Foundation of China (No. 10172075 and No. 10002016).     

Transient responses of a multilayered spherically isotropic piezoelectric hollow sphere

The dynamic solution of a multilayered spherically isotropic piezoelectric hollow sphere subjected to radial dynamic loads is obtained. By the method of superposition, the solution is divided into two parts: one is quasi-static and the other is dynamic. The quasi-static part is derived by the state-space method, and the dynamic part is obtained by the method of separation of variables coupled with the initial parameter method as well as the orthogonal expansion technique. By using the quasi-static and dynamic parts, the electric boundary conditions as well as the electric continuity conditions, a Volterra integral equation of the second kind with respect to a function of time is derived, which can be solved successfully by means of the interpolation method. The displacements, stresses and electric potentials are finally obtained. The present method is suitable for a multilayered spherically isotropic piezoelectric hollow sphere consisting of arbitrary layers and subjected to arbitrary spherically symmetric dynamic loads. Finally, numerical results are presented and discussed.     

The Analysis of Dynamic Stress Intensity Factor for Semi-Circular Surface Crack Using Time-Domain BEM Formulation

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Dynamic Analysis of a Composite Hollow Sphere Composed of Elastic and Piezoelectric Layers

Dynamic analysis of a two-layered elasto-piezoelectric composite hollow sphere under spherically symmetric deformation is developed. An unknown function of time is first introduced in terms of the charge equation of electrostatics and then the governing equations of piezoelectric layer, in which the unknown function of time is involved, are derived. By the method of superposition, the dynamic solution for elastic and piezoelectric layers is divided into quasi-static and dynamic parts. The quasi-static part is treated independently by the state space method and the dynamic part is obtained by the separation of variables method. By virtue of the obtained quasi-static and dynamic parts, a Volterra integral equation of the second kind with respect to the unknown function of time is derived by using the electric boundary conditions for piezoelectric layer. Interpolation method is employed to solve the integral equation efficiently. The transient responses for elastic and electric fields are finally determined. Numerical results are presented and discussed.     

Thermo-electro-elastic transient responses in piezoelectric hollow structures

The paper presents an analytical method to solve thermo-electro-elastic transient response in piezoelectric hollow structures subjected to arbitrary thermal shock, sudden mechanical load and electric excitation. Volterra integral equation of the second kind caused by interaction between elastic deformation and electric field is solved by using an interpolation method. Thus, the exact expressions for the transient responses of displacement, stresses, electric displacement and electric potential in the piezoelectric hollow structures are obtained by means of Hankel transform, Laplace transform, and their inverse transforms. In Section 2, based on spherical coordinates, the governing equation of thermo-electro-elastic transient responses in a piezoelectric hollow sphere is found and the associated numerical results are carried out. In Section 3, based on cylindrical coordinates, the governing equation of thermo-electro-elastic transient responses in a non-homogeneous piezoelectric hollow cylinder is found and the corresponding numerical results are carried out. The results carried out may be used as a reference to solve other transient coupled problems of thermo-electro-elasticity in piezoelectric structures.     

Exact Electromagnetothermoelastic Solution for a Transversely Isotropic Piezoelectric Hollow Sphere Subjected to Arbitrary Thermal Shock

This paper presents analytical study for electromagnetothermoelastic transient behavior of a transversely isotropic hollow sphere, placed in a uniform magnetic field, subjected to arbitrary thermal shock. Exact solutions for the transient responses of stresses, perturbation of magnetic field vector, electric displacement and electric potential in the transversely isotropic piezoelectric hollow sphere are obtained by means of the Hankel transform, the Laplace transform and their inverse transforms. An interpolation method is used to solve the Volterra integral equation of the second kind caused by interactions among electric, magnetic, thermal and elastic fields. From the sample numerical calculations, it is seen that the present method is suitable for the transversely isotropic hollow sphere, placed in a uniform magnetic field, subjected to arbitrary thermal shock. Finally, the result can be used as a reference to solve other transient coupling problems of electromagnetothermoelasticity.     

Modelling of dynamical crack propagation using time-domain boundary integral equations

We study dynamic antiplane cracks in the time domain by the boundary integral equation method (BIEM) based on the integral equation for displacement discontinuity (or crack opening displacement, COD) as a function of stress on the crack. This displacement discontinuity formulation presents the advantage, with respect to methods developed by Das and others in seismology, that it has to be solved only inside the crack. This BIEM is, however, difficult to implement numerically because of the hypersingularity of the kernel of the integral equation. Hence it is rewritten into a weakly singular form using a regularization technique proposed by Bonnet. The first step, following a method due to Sladek and Sladek, consists in converting the hypersingular integral equation for the displacement discontinuity into an integral equation for the displacement discontinuity and its tangential derivatives (dislocation density distribution); the latter involves a Cauchy type singular kernel. The second step is based on the observation that the hypersingularity is related to the static component of the kernel; the static singularity is then isolated and can be expressed in terms of weakly singular integrals using a result due to Bonnet. Although numerical applications discussed in this paper are all for the antiplane problem, the technique can be applied as well to in-plane crack dynamics.

The BIEM is implemented numerically using continuous linear space-time base functions to model the COD on the crack. In the present scheme the COD gradient interpolation is discontinuous at the element nodes while the integral equations are collocated at the element midpoints. This leads to an overdetermined discrete problem which is solved by standard least-squares methods. We use the dynamic BIEM to study a set of problems that appear in earthquake source dynamics, including the spontaneous dynamic crack propagation for a very simple rupture criterion. The numerical results compare favorably with the few exact solutions that are available. Then we demonstrate that difficulties experienced with finite difference simulations of spontaneous crack dynamics can be removed with the use of BIEM. The results are improved by the use of singular crack tip elements.     

Runge–Kutta methods for a semi-analytical prediction of milling stability

On the basis of Runge–Kutta methods, this paper proposes two semi-analytical methods to predict the stability of milling processes taking a regenerative effect into account. The corresponding dynamics model is concluded as a coefficient-varying periodic differential equation with a single time delay. Floquet theory is adopted to predict the stability of machining operations by judging the eigenvalues of the state transition matrix. This paper firstly presents the classical fourth-order Runge–Kutta method (CRKM) to solve the differential equation. Through numerical tests and analysis, the convergence rate and the approximation order of the CRKM is not as high as expected, and only small discrete time step size could ensure high computation accuracy. In order to improve the performance of the CRKM, this paper then presents a generalized form of the Runge–Kutta method (GRKM) based on the Volterra integral equation of the second kind. The GRKM has higher convergence rate and computation accuracy, validated by comparisons with the semi-discretization method, etc. Stability lobes of a single degree of freedom (DOF) milling model and a two DOF milling model with the GRKM are provided in this paper.     

Impact interaction of cylindrical body with a surface of cavity during supercavitation motion in compressible fluid

The paper deals with the early stage of impact of a solid cylindrical body on the surface of a cylindrical cavity for zero and non-zero gap between the cavity surface and the body surface. As a result, the stated mixed non-stationary boundary value problem with the unknown variable in the time boundary is formulated. Its solution is reduced to a joint solution of an infinite system of linear integral Volterra equations of the second kind and the differential equation of the body movement. In the case of simplified formulation, the solution is reduced to the infinite sequence of the linear integral Volterra equations. Hydrodynamic and kinematic characteristics are also obtained.     

Transient responses of a magneto-electro-elastic hollow sphere for fully coupled spherically symmetric problem

By introducing a dependent variable and a special function satisfying the inhomogeneous mechanical boundary conditions, the governing equation for a new variable with homogeneous mechanical boundary conditions is derived. Then by means of the separation of variables technique and the electric and magnetic boundary conditions, the dynamic problem of a magneto-electro-elastic hollow sphere under spherically symmetric deformation is transformed to two Volterra integral equations of the second kind about two functions of time. Cubic Hermite polynomials are adopted to approximate the two undetermined functions at each time subinterval and the recursive formula is obtained to solve the integral equations successfully. The transient responses of displacements, stresses, electric and magnetic potentials are completely determined at the end. Numerical results are presented.     

Diffraction theory of a reflection grating

The reflection of a monochromatic plane electromagnetic wave by an electrically perfectly conducting grating is investigated. The vectorial electromagnetic problem is reduced to two separate scalar problems: those corresponding to E- and H-polarization respectively. A Green's function formulation of the problem is employed. For both cases an integral equation of the second kind for the remaining unknown function on the surface of the grating is derived. A numerical solution of this integral equation is obtained with the aid of either a (discrete) Fourier transform or a cubic spline approximation. Some numerical results of both the echellette grating and the sinusoidal grating are presented.     

Velocity-based reciprocal theorems in elastodynamics and BIEM implementation issues

Reciprocal theorems in elastodynamics are introduced as extensions of respective theorems from elastostatics. Inasmuch as the latter is a subset of the former, the aim here is to present an elastodynamic reciprocal theorem that also includes elastostatics as a special case when the time variable becomes irrelevant. This is accomplished by introducing a velocity-based reciprocal theorem, whose basic properties are subsequently explored. The next step is to use this theorem and formulate a numerical approach based on boundary integral equation statements and compare them with existing formulations based on conventional reciprocity relations. The applications presented here involve the standard mechanical oscillator and the unidimensional axial element as two simple, yet important problems of structural dynamics. Along with the numerical results, a thorough stability analysis of the corresponding time-stepping algorithms is formulated. In both cases, the superior performance of the methodologies built on velocity-based reciprocal theorems is clearly demonstrated.     

Transient thermal stresses in an orthotropic hollow cylinder for axisymmetric problems

For the thermoelastic dynamic axisymmetric problem of a finite orthotropic hollow cylinder, one comes closer to reality to involve the effect of axial strain than to consider the plane strain case only. However, additional mathematical difficulties should be encountered and a different solution procedure should be developed. By the separation of variables, the thermoelastic axisymmetric dynamic problem of an orthotropic hollow cylinder taking account of the axial strain is transformed to a Volterra integral equation of the second kind for a function of time, which can be solved efficiently and quickly by the interpolation method. The solutions of displacements and stresses are obtained. It is noted that the present method is suitable for an orthotropic hollow cylinder with an arbitrary thickness subjected to arbitrary axisymmetric thermal loads. Numerical comparison is made to show the effect of the axial strain on the displacements and stresses. The project supported by the National Natural Science Foundation of China (10172075) and China Postdoctoral Science Foundation (20040350712)     

Magneto–thermo–electro–elastic transient response in a piezoelectric hollow cylinder subjected to complex loadings

The article presents an analytical solution for magneto–thermo–electro–elastic problems of a piezoelectric hollow cylinder placed in an axial magnetic field subjected to arbitrary thermal shock, mechanical load and transient electric excitation. Using an interpolation method solves the Volterra integral equation of the second kind caused by interaction among magnetic, thermal, electric and mechanical fields, the electric displacement is determined. Thus, the exact expressions for the transient responses of displacement, stresses, electric displacement, electric potential and perturbation of the magnetic field vector in the piezoelectric hollow cylinder are obtained by means of Hankel transforms, Laplace transforms, and inverse Laplace transforms. From sample numerical calculations, it is seen that the present method is suitable for a piezoelectric hollow cylinder subjected to arbitrary thermal shock, mechanical load and transient electric excitation, and the result carried out may be used as a reference to solve other transient coupled problems of magneto–thermo–electro–elasticity.     

Explicit solutions for shearing and radial stresses in curved beams

In this paper the formulae for the shearing and radial stresses in curved beams are derived analytically based on the solution for a Volterra integral equation of the second kind. These formulae satisfy both the equilibrium equations and the static boundary conditions on the surfaces of the beams. As some applications, the resulting solutions are used to calculate the shearing and radial stresses in a cantilevered curved beam subjected to a concentrated force at its free end. The numerical results are compared with other existing approximate solutions as well as the corresponding solutions based on the theory of elasticity. The calculations show a better agreement between the present solution and the one based on the theory of elasticity. The resulting formulae can be applied to more general cases of curved beams with arbitrary shapes of cross-sections.