Dynamic stress intensity factor (Mode I) of a penny-shaped crack in an infinite poroelastic solid

This paper considers the transient stress intensity factor (Mode I) of a penny-shaped crack in an infinite poroelastic solid. The crack surfaces are impermeable. By virtue of the integral transform methods, the poroelastodynamic mixed boundary value problems is formulated as a set of dual integral equations, which, in turn, are reduced to a Fredholm integral equation of the second kind in the Laplace transform domain. Time domain solutions are obtained by inverting Laplace domain solutions using a numerical scheme. A parametric study is presented to illustrate the influence of poroelastic material parameters on the transient stress intensity. The results obtained reveal that the dynamic stress intensity factor of poroelastic medium is smaller than that of elastic medium and the poroelastic medium with a small value of the potential of diffusivity shows higher value of the dynamic stress intensity factor.     

饱和地基上弹性圆板的固结位移

 采用Laplace变换及Hankel变换研究弹性圆板在饱和地基上的固结位移．在Laplace变换域上建立了求解弹性圆板固结位移的第二类Fredholm积分方程，由数值反变换获得时域解，给出了对工程实践有参考价值的计算结果．计算结果表明，弹性板的固结位移随饱和土的排水泊松比的减少而增加，而弹性板的初始位移随饱和土的不排水泊松比的减少而增加．     

The symplectic approach for two-dimensional thermo-viscoelastic analysis

This paper presents an exact symplectic approach for two dimensional isotropic viscoelastic solids subjected to external force and temperature boundary conditions. With the use of the state space method and the Laplace transform, all general solutions of the governing equations are obtained analytically. By applying the inverse integral transform, the time domain adjoint symplectic relationships between the general solutions are established. Therefore, the problems of the particular solution and the boundary conditions can be analysed either in the Laplace domain or directly in the time domain. As its applications, the boundary condition problems are discussed in the numerical calculations. The results show that, due to the displacement constraints and the temperature influence, local effects are distinct near the boundary, and the effects decay rapidly with the distance from the boundary.     

Finite integration method for nonlocal elastic bar under static and dynamic loads

The finite integration method is proposed in this paper to approximate solutions of partial differential equations. The coefficient matrix of this finite integration method is derived and its superior accuracy and efficiency is demonstrated by making comparison with the classical finite difference method. For illustration, the finite integration method is applied to solve a nonlocal elastic straight bar under different loading conditions both for static and dynamic cases in which Laplace transform technique is adopted for the dynamic problems. Several illustrative examples indicate that high accurate numerical solutions are obtained with no extra computational efforts. The method is readily extendable to solve more complicated problems of nonlocal elasticity.     

Analytical solutions describing the consolidation of a multi-layered soil under circular loading

Analytical solutions describing the consolidation of a multi-layered soil under circular loading are presented. From the governing equations of saturated poroelastic soil in a cylindrical coordinate system, the eighth-order state-space equation of consolidation is obtained by eliminating the variation of time t using the Laplace transform together with the technique of Fourier expansions with respect to the coordinate θ and the Hankel transform with respect to coordinate r. The solution of the eighth-order state-space equation is derived directly by using the Laplace transform and its inversion of the z-domain. Based on the continuity between layers and the boundary conditions, the transfer-matrix method is utilized to derive the solutions for the consolidation of a multi-layered soil under circular loading in the transformed domain. By the inversion of the Laplace transform and the Hankel transform, the analytical solutions in the physical domain are obtained. A numerical analysis based on the solutions is carried out by a corresponding program.     

Thermoelastic transient response of an infinitely long annular cylinder composed of two different materials

This paper deals with the transient response of one-dimensional axisymmetric quasistatic coupled thermoelastic problems. The governing equations, taking into account of the thermome-chanical term, are expressed in terms of temperature increment and displacement. Using the Laplace transform with respect to time, the general solutions of the governing equations are obtained in the transform domain. Also presented are the numerically transient distributions of stress and temperature increments in the real domain for the case of an infinitely long annular cylinder composed of two different materials. The inversion to the real domain is obtained by using a Fourier series technique and matrix operations simultaneously; therefore, no thermoelastic potentials are introduced in the solution process. It shows that the coupling effect behaves as a clear lag in both the stress and the temperature distributions.     

Response of gradient-viscoelastic bar to static and dynamic axial load

Summary. The response of a bar to static or dynamic axial load is studied analytically on the basis of a simple linear theory of gradient viscoelasticity. The governing equations of axial equilibrium and motion are first obtained by combining the basic equations. They are also obtained by a variational statement, which provides in addition all possible boundary conditions. A correspondence principle between the gradient elastic and gradient viscoelastic formulation and solution is established. Thus, the Laplace transformed with respect to time viscoelastic solution is obtained from the corresponding elastic one by simply replacing the elastic modulus by its Laplace transform times the Laplace transform parameter. The time domain response is finally obtained by a numerical inversion of the transformed solution. Two boundary value problems, one quasi-static and one dynamic, are studied and the gradient viscoelasticity effect on the solutions is assessed.     

State space approach to one-dimensional thermal shock problem for a semi-infinite piezoelectric rod

The theory of generalized thermoelasticity, based on the theory of Lord and Shulman with one relaxation time, is used to solve a boundary value problem of one-dimensional semi-infinite piezoelectric rod with its left boundary subjected to a sudden heat. The governing partial differential equations are solved in the Laplace transform domain by the state space approach of the modern control theory. Approximate small-time analytical solutions to stress, displacement and temperature are obtained by means of the Laplace transform and inverse transform. It is found that there are two discontinuous points in both stress and temperature solutions. Numerical calculation for stress, displacement and temperature is carried out and displayed graphically.     

Exact solutions for the unsteady rotational flow of non-Newtonian fluid in an annular pipe

This paper deals with some unsteady transient rotational flows of an Oldroyd-B fluid in an annular pipe. The fractional calculus approach in the constitutive relationship model of an Oldroyd-B fluid is introduced. A generalized Jeffreys model with the fractional calculus is built. Exact solutions for some unsteady rotational flows of an Oldroyd-B fluid in an annular pipe are obtained by using Hankel transform and the theory of Laplace transform for fractional calculus. The well known solutions for a Navier–Stokes fluid, as well as those corresponding to a Maxwell fluid and a second grade one, appear as limiting cases of our solutions.     

Vibrations of linear viscoelastic materials for any hereditary property

A new analytic method for solutions of vibration problems of linear viscoelastisity with arbitrary relaxation kernels is proposed. The problem is reduced to the solution of a set of ordinary integro-differential equations which are solved using the Laplace integral transform, the method of contour integration and the convolution of functions. The branch points of integrand can not be obtained for arbitrary relaxation kernels. For this reason, successive approximation method for calculating the poles and the inverse Laplace transformation of solution is offered. The first approach coincides with the well-known solution obtained by both the averaging method and the method of complex module. It is shown that the limits of these approximations are unique and it is proved that the obtained solution is the exact solution of the considering problem.     

Analytical expressions for stress and displacement fields in viscoelastic axisymmetric plane problem involving time-dependent boundary regions

An analytical solution is developed in this paper for viscoelastic axisymmetric plane problems under stress or displacement boundary condition involving time-dependent boundary regions using the Laplace transform. The explicit expressions are given for the radial and circumferential stresses under stress boundary condition and the radial displacement under displacement boundary condition. The results indicate that the two in-plane stress components and the displacement under corresponding boundary conditions have no relation with material constants. The general form of solutions for the remaining displacement or stress field is expressed by the inverse Laplace transform concerning two relaxation moduli. As an application to deep excavation of a circular tunnel or finite void growth, explicit solutions for the analysis of a deforming circular hole in both infinite and finite planes are given taking into account the rheological characteristics of the rock mass characterized by a Boltzmann or Maxwell viscoelastic model. Numerical examples are given to illustrate the displacement and stress response. The method proposed in this paper can be used for analysis of earth excavation and finite void growth.     

A Two-Temperature Photothermal Interaction in a Semiconductor Medium Containing a Cylindrical Hole

Photothermoelastic interactions in an infinite semiconductor medium containing a cylindrical hole with two temperatures are studied using mathematical method under the purview of the coupled theory of thermal, plasma and elastic waves. The internal surface of the hole is constrained and the carrier density is photogenerated by bound heat flux with an exponentially decaying pulse. Based on Laplace transform and the eigenvalue approach methodology, the solutions of all variables have been obtained analytically. The numerical computations for silicon-like semiconductor material have been obtained. The results further show that the analytical scheme can overcome mathematical problems to analyze these problems.     

粘弹性薄板动力响应问题的多重互易法

本文给出了粘弹性薄板动力响应问题的多重互易法（MRM）.首先在Laplace变换区域中得到了由重调和算子基本解序列给出的粘弹性板动力响应问题的MRM方法,再利用改进的Bellman反变换技术,求得原问题的解.讨论了MRM方法中的迭代误差估计.文末给出了数值算例,计算表明该方法具有较高精度和较快收敛性.适用于长时间的动力问题的计算.     

Time-dependent fundamental solutions for homogeneous diffusion problems

This paper describes the applications of the method of fundamental solutions (MFS) for 1-, 2- and 3-D diffusion equations. The time-dependent fundamental solutions for diffusion equations are used directly to obtain the solution as a linear combination of the fundamental solution of the diffusion operator. The proposed scheme is free from the conventionally used Laplace transform or the finite difference scheme to deal with the time derivative of the governing equation. By properly placing the field points and the source points at a given time level, the solution is advanced in time until steady state solutions are reached. Test results obtained for 1-, 2- and 3-D diffusion problems show good comparisons with the analytical solutions and some with the MFS based on the modified Helmholtz fundamental solutions, thus the demonstration present numerical scheme of MFS with the space–time unification has been demonstrated as a promising mesh-free numerical tool to solve homogeneous diffusion problem.     

Transient Analysis of Elastic Wave Propagation in Multilayered Structures

In this article, explicit transient solutions for one-dimensional wave propagation behavior in multi-layered structures are presented. One of the objectives of this study is to develop an effective analytical method for constructing solutions in multilayered media. Numerical calculations are performed by three methods: the generalized ray method, numerical Laplace inversion method (Durbin's formula), and finite element method (FEM). The analytical result of the generalized ray solution for multilayered structures is composed of a matrix-form Bromwich expansion in the transform domain. Every term represents a group of waves, which are transmitted or reflected through the interface. The matrix representation of the solution can be used to calculate the transient response, without tracing the ray path manually. Numerical inversion of the Laplace transform by Durbin's formula is also used to construct transient responses. This numerical Laplace inversion technique has the advantage of calculating long-time transient responses for complicated multilayered structures. FEM results agree well with calculations obtained by the generalized ray method and numerical Laplace inversion.     

Transient analysis of dynamic crack propagation in piezoelectric materials

Abstract

In this paper, the transient analysis of semi‐infinite propagating cracks in piezoelectric materials subjected to dynamic anti‐plane concentrated body force is investigated. The crack surface is assumed to be covered with an infinitesimally thin, perfectly conducting electrode that is grounded. In analyzing this problem, it has characteristic lengths and a direct attempt towards solving this problem by transform and Wiener‐Hopf techniques (Noble, 1958) is not applicable. In order to solve this problem, a new fundamental solution for propagating cracks in piezoelectric materials is first established and the transient response of the propagating crack is obtained by superposition of the fundamental solution in the Laplace transform domain. The fundamental solution to be used is the responses of applying exponentially distributed traction in the Laplace transform domain on the propagating crack surface. Taking into account the quasi‐static approximation, exact analytical transient solutions for the dynamic stress intensity factor and the dynamic electric displacement intensity factor are obtained by using the Cagniard‐de Hoop method (Cagnard, 1939; de Hoop, 1960) of Laplace inversion and are expressed in explicit forms. Numerical calculations of dynamic intensity factors are evaluated and the results are discussed in detail. The transient solutions for stationary cracks have been shown to approach the corresponding static values after the shear wave of the piezoelectric material has passed the crack tip.     

New hybrid Laplace transform/finite element method for three-dimensional transient heat conduction problem

The paper presents results obtained by the implementation of a new hybrid Laplace transform/finite element method developed by the authors. The present method removes the time derivatives from the governing differential equation using the Laplace transform and then solves the associated equation with the finite element method. Previously reported hybrid Laplace transform/finite element methods¹ have been confined to one nodal solution at a time. When applied to many nodes it takes an excessive amount of computer time. By using a similarity transform method on the matrix of the complex number coefficients this restriction is removed and the reported new method provides a more useful tool for the solution of linear transient problems. Test examples are used to show that the basic accuracy is comparable to that obtainable by analytical, finite difference and finite element methods.     

Dynamic analysis of cracks using the SGBEM for elastodynamics in the Laplace-space frequency domain

Dynamic analysis of a system can be carried out either in the time or frequency domains. Time responses/histories of this system may be directly obtained using time-domain formulations. In the frequency domain, analysis can be performed in either the Fourier or Laplace spaces. The symmetric-Galerkin boundary element method (SGBEM) for 2-D elastodynamics in the Fourier-space frequency domain has been previously reported in the literature. In this paper, the SGBEM for elastodynamics in the Laplace-space frequency domain using the standard continuous quadratic element and its application to dynamic analysis of cracks is presented for the first time. The technique developed is employed together with the fast Laplace inverse transform by Durbin to obtain time-dependent results for several typical examples including both crack and non-crack problems. These results are highly accurate when compared to those obtained from other numerical techniques. It is shown in this work that the very same boundary element code can be utilized to perform frequency domain analysis in either the Fourier or Laplace spaces. However, if time responses are required, the accuracy and computational effectiveness of the analysis may depend on the type of space selected as it determines the type of transforms (inverse Fourier/Laplace transforms) needed for converting frequency solutions to the desired time responses.     

Elastodynamic response of a finite strip with two coplanar cracks under impact load

The elastodynamic response of two coplanar Griffith cracks in a finite elastic strip under in-plane compression and anti-plane shear impact is considered in this paper. Laplace and Fourier transforms are used to reduce two mixed boundary value problems to Cauchy-type singular integral equations in Laplace transform plane, which are solved numerically. The elastodynamic stress-intensity factors are obtained as functions of time and geometry parameters.     

State space approach to generalized thermo-viscoelasticity with two relaxation times

The model of the equation of generalized thermo-viscoelasticity with two relaxation times is established. The state space formulation for thermo-viscoelasticity with two relaxation times is introduced. The formulation is valid for problems with or without heat sources. The resulting formulations together with the Laplace transform technique are applied to a variety of problems. The solutions to a thermal shock problem and to a problem of a layer media both without heat sources are obtained. Also a problem with a distribution of heat sources is considered. A numerical method is employed for the inversion of the Laplace-transforms. Numerical results are given and illustrated graphically for the problems considered. Comparisons are made with the results predicted by the coupled theory.