Two-dimensional generalized thermoelasticity problem for a half-space subjected to ramp-type heating

In this paper, we will consider a half-space filled with an elastic material, which has constant elastic parameters. The governing equations are taken in a unified system from which the field equations for coupled thermoelasticity as well as for generalized thermoelasticity can be easily obtained as particular cases. A linear temperature ramping function is used to more realistically model thermal loading of the half-space surface. The medium is assumed initially quiescent. Laplace and Fourier transform techniques are used to obtain the general solution for any set of boundary conditions. The general solution obtained is applied to a specific problem of a half-space subjected to ramp-type heating. The inverse Fourier transforms are obtained analytically while the inverse Laplace transforms are computed numerically using a method based on Fourier expansion techniques. Some comparisons have been shown in figures to estimate the effect of the ramping parameter of heating with different theories of thermoelasticity.     

Finite element analysis of two-temperature generalized magneto-thermoelasticity

A general finite element model is proposed to analyze transient phenomena in thermoelastic solids. Youssef model of two-temperature generalized magneto-thermoelasticity is selected for an homogenous, isotropic, conducting and elastic medium, which is subjected to thermal shock, and a magnetic field with constant intensity acts tangent to the bounding plane. The numerical solution of the nondimensional governing partial differential equations of the problem has been shown graphically.     

State-space approach of two-temperature generalized thermoelasticity of one-dimensional problem

In this paper, we will consider a half-space filled with an elastic material, which has constant elastic parameters. The governing equations are taken in the context of the two-temperature generalized thermoelasticity theory [Youssef, H., 2005a. The dependence of the modulus of elasticity and the thermal conductivity on the reference temperature in generalized thermoelasticity for an infinite material with a spherical cavity, J. Appl. Math. Mech., 26(4), 4827; Youssef, H., 2005b. Theory of two-temperature generalized thermoelasticity, IMA J. Appl. Math., 1–8]. The medium is assumed initially quiescent. Laplace transform and state space techniques are used to obtain the general solution for any set of boundary conditions. The general solution obtained is applied to a specific problem of a half-space subjected to thermal shock and traction free. The inverse Laplace transforms are computed numerically using a method based on Fourier expansion techniques. Some comparisons have been shown in figures to estimate the effect of the two-temperature parameter.     

Pull-in analysis of electrically actuated viscoelastic microbeams based on a modified couple stress theory

A new beam model is developed for the viscoelastic microbeam based on a modified couple stress model which contains only one material length scale parameter. The governing equations of equilibrium together with initial conditions and boundary conditions are obtained by a combination of the basic equations of modified couple stress theory and Hamilton’s principle. This new beam model is then used for an electrically actuated microbeam-based MEMS structure. The dynamic and quasi-static governing equations of an electrically actuated viscoelastic microbeam are firstly given where the axial force created by the midplane stretching effect is also considered. Galerkin method is used to solve above equation and this method is also validated by the finite element method (FEM) when our model is reduced into an elastic case. The numerical results show that the instantaneous pull-in voltage, durable pull-in voltage and pull-in delay time predicted by this newly developed model is larger (longer) than that predicted by the classical beam model. A comparison between the quasi-static model results and the dynamic model results is also given.     

State-space approach of two-temperature generalized thermoelasticity of infinite body with a spherical cavity subjected to different types of thermal loading

In this work, we will consider an infinite elastic body with a spherical cavity and constant elastic parameters. The governing equations are taken in the context of the two-temperature generalized thermoelasticity theory (Youssef in J Appl Math Mech 26(4):470–475 2005a, IMA J Appl Math, pp 1–8, 2005). The medium is assumed initially quiescent. Laplace transform and state space techniques are used to obtain the general solution for any set of boundary conditions. The general solution obtained is applied to a specific problem when the bounding plane of the cavity is subjected to thermal loading (thermal shock and ramp-type heating). The inverse Laplace transforms are computed numerically using a method based on Fourier expansion techniques. Some comparisons have been shown in figures to estimate the effect of the two-temperature and the ramping parameters.     

弹性半空间二维出平面自由波场的一维化时域算法

提出了一种计算出平面SH波斜入射时弹性半空间自由波场时域计算的一维化有限元方法。首先利用Snell定律确定平面波沿水平方向的传播规律,在用有限元法对弹性半空间进行离散化时,竖向单元尺寸根据波动有限元模拟精度要求确定,而水平向有限元网格尺寸根据水平向波的传播规律和采用的离散时间步长确定,使得有限元离散模型中任意节点的运动可以用水平向相邻节点的运动表示,从而将二维有限元节点运动方程组化为一维的形式。求解此一维方程组,可得到弹性半空间中一列节点的运动,再根据行波的传播规律,可确定全空间自由波场。理论分析和数值算例表明,该方法具有较高的精度和良好的稳定性。     

Nonlinear finite element analysis of laminated composite spherical shell vibration under uniform thermal loading

In this article, nonlinear free vibration behavior of laminated composite shallow shell under uniform temperature load is investigated. The mid-plane kinematics of the laminated shell is evaluated based on higher order shear deformation theory to count the out of plane shear stresses and strains accurately. The nonlinearity in geometry is taken in Green-Lagrange sense due to the thermal load. In addition to that, all the nonlinear higher order terms are taken in the mathematical model to capture the original flexure of laminated panel. A nonlinear finite element model is proposed to discretise the developed model and the governing equations are derived using Hamilton’s principle. The sets of governing equations are solved using a direct iterative method. In order to validate the model, the results are compared with the available published literature and the limitations of the existing models have been discussed. Finally, some numerical experimentation has been done using the developed nonlinear model for different parameters (thickness ratio, curvature ratio, modular ratio, support condition, lamination scheme, amplitude ratio and thermal expansion coefficient) and their effects on the responses are discussed in detail.     

成层半空间出平面自由波场的一维化时域算法

提出了一种计算出平面SH波斜入射时弹性水平成层半空间中自由波场时域计算的一维化有 限元方法. 在进行有限元网格划分时，竖向单元取满足有限元模拟精度的任意尺寸，水平向 网格尺寸由时间离散步长和水平视波速确定，并自动进行虚拟网格划分. 基底设置人工边界， 并将波动输入转化为等效荷载施加在边界节点上. 然后将集中质量有限元法和中心差分法相 结合建立节点运动方程，并将水平方向相邻节点的运动用该节点相邻时刻的运动表示，从而 将求解节点运动的二维方程组转化为一维方程组. 求解此方程组，即得到自由场中竖向一列 节点的运动. 最后根据行波传播的特点，可方便地确定全部自由波场. 理论分析和数值算例 表明，该方法具有较高的精度和良好的稳定性.     

State-space approach to two-temperature generalized thermoelasticity without energy dissipation of medium subjected to moving heat source

In this work,a model of two-temperature generalized thermoelasticity without energy dissipation for an elastic half-space with constant elastic parameters is constructed.The Laplace transform and state-space techniques are used to obtain the general solution for any set of boundary conditions.The general solutions are applied to a specific problem of a half-space subjected to a moving heat source with a constant velocity.The inverse Laplace transforms are computed numerically,and the comparisons are shown in figures to estimate the effects of the heat source velocity and the two-temperature parameter.     

Effects of three-phase-lag on two-temperature generalized thermoelasticity for infinite medium with spherical cavity

The thermoelastic interaction for the three-phase-lag (TPL) heat equation in an isotropic infinite elastic body with a spherical cavity is studied by two-temperature generalized thermoelasticity theory (2TT). The heat conduction equation in the theory of TPL is a hyperbolic partial differential equation with a fourth-order derivative with respect to time. The medium is assumed to be initially quiescent. By the Laplace transformation, the fundamental equations are expressed in the form of a vector-matrix differential equation, which is solved by a state-space approach. The general solution obtained is applied to a specific problem, when the boundary of the cavity is subjected to the thermal loading (the thermal shock and the ramp-type heating) and the mechanical loading. The inversion of the Laplace transform is carried out by the Fourier series expansion techniques. The numerical values of the physical quantity are computed for the copper like material. Significant dissimilarities between two models (the two-temperature Green-Naghdi theory with energy dissipation (2TGN-III) and two-temperature TPL model (2T3phase)) are shown graphically. The effects of two-temperature and ramping parameters are also studied.     

Two-temperature theory in generalized magneto-thermoelasticity with two relaxation times

The model of one-dimensional equations of the two-temperature generalized magneto-thermoelasticity theory with two relaxation times in a perfect electric conducting medium is established. The state space approach developed in Ezzat (Can J. Phys. Rev. 86(11):1241–1250, 2008) is adopted for the solution of one-dimensional problems. The resulting formulation together with the Laplace transform techniques are applied to a specific problem of a half-space subjected to thermal shock and traction-free surface. The inversion of the Laplace transforms is carried out using a numerical approach. Some comparisons have been shown in figures to estimate the effects of the temperature discrepancy and the applied magnetic field.     

A triangular plate element 2343 using second-order absolute-nodal-coordinate slopes: numerical computation of shape functions

In this paper, the process by which geometrical and structural matrices of plate finite elements employing absolute nodal coordinate formulation (ANCF) are constructed is studied. The kinematic and topological properties of an arbitrary plate finite element are described using universal digital code dncm that provides systematic enumeration of finite elements. This code is formed using the element’s dimension d, the number of nodes it possesses n, the number of scalar coordinates per node c, and a multiplier describing the process of transforming a conventional finite element to an ANCF element m. The detailed generation of a new type of triangular plate finite element 2343 using numerical computation of shape functions is also discussed in the paper. The new triangular element employs position vectors and slope vectors up to second-order mixed-derivative slope vector. A detailed derivation of the equations of motion of the element is also provided and examples of its numerical simulation and validation presented.     

A 1D time-domain method for in-plane wave motions in a layered half-space

A 1D finite element method in time domain is developed in this paper and applied to calculate in-plane wave motions of free field exited by SV or P wave oblique incidence in an elastic layered half-space. First, the layered half-space is discretized on the basis of the propagation characteristic of elastic wave according to the Snell law. Then, the finite element method with lumped mass and the central difference method are incorporated to establish 2D wave motion equations, which can be transformed into 1D equations by discretization principle and explicit finite element method. By solving the 1D equations, the displacements of nodes in any vertical line can be obtained, and the wave motions in layered half-space are finally determined based on the characteristic of traveling wave. Both the theoretical analysis and the numerical results demonstrate that the proposed method has high accuracy and good stability. The project supported by the National Natural Science Foundation of China (50478014), the National 973 Program (2007CB714200) and the Beijing Natural Science Foundation (8061003). The English text was polished by Yunming Chen.     

The Effect of Surface Inhomogeneities on the Propagation of Elastic Waves

We investigate the effect of the surface inhomogeneities (defects) on the propagation of the elastic waves in a semi-infinite isotropic solid body (half-space). A perturbation-theoretical scheme is devised for small surface defects (in comparison with the relevant elastic disturbances propagating in the body), and the elastic waves equations are solved in the first-order approximation. It is shown that surface defects generate both scattered waves localized (and propagating only) on the surface (two-dimensional waves) and scattered waves reflected back in the body. Directional effects, wave slowness and attenuation by diffusive scattering, or possible resonance effects are discussed.     

Vertical vibrations of elastic foundation resting on saturated half-space

This paper is mainly concerned with the dynamic response of an elastic foun- dation 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 half- space is completely pervious or impervious,this dynamic mixed boundary-value problem can lead to dual integral equations,which can be further reduced to the Fredholm integral equations of the second kind and solved by numerical procedures.In the numerical exam- ples,the dynamic compliances,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.     

The torsion of a non-homogeneous stratum with a hyperbolic variation of shear modulus

An exact formulation of the governing dual integral equations for the torsion of a non-homogeneous stratum due to a rigid circular body at its free surface is presented. The stratum varies in shear modulus according to the hyperbolic variation in a contemporary work [1]. It is shown that the unknown static stress distribution under the rigid body is governed by modified Bessel function of the first kind. By comparing the governing functions in the dual integral equations for five cases of elastic media: homogeneous half-space, and stratum, linearly non-homogeneous half-space and stratum and, finally, the present non-homogeneous stratum with hyperbolic variation, it is established that the surface shear modulus is the dominant parameter in the assessment of the stress and displacement fields in a non-homogeneous stratum where lateral variation of elastic properties is negligible.     

A Mathematical Model for Short-Time Filtration in Poroelastic Media with Thermal Relaxation and Two Temperatures

In this work, we derive a set of governing equations for a mathematical model of generalized thermoelasticity in poroelastic materials. This model predicts finite speeds of propagation of waves contrary to the model of coupled thermoelasticity where an infinite speed of propagation is inherent. Next, we prove the uniqueness of solution of these equations under suitable conditions. We also obtain a reciprocity theorem for these equations. A thermal shock problem for a half-space composed of a poroelastic material saturated with a liquid is then considered. The surface of the half-space is assumed to be traction free, permeable, and subjected to heating. The Laplace transform technique is used to solve the problem. Numerical results for the temperature in the elastic body and fluid, displacement of the elastic body, velocity of the fluid, and stresses for both components are obtained and represented graphically.     

A theory of finite elastic consolidation

Presented in this paper is a general theory describing the consolidation of a porous elastic soil. The formulation allows for the occurrence of finite geometry changes and finite elastic strains during the consolidation process. The governing equations have been cast in a rate form and the laws which determine deformation and pore fluid flow, i.e. Hooke's law and Darcy's law, are presented in a frame indifferent manner. A numerical technique is described that provides an approximate solution to the governing equations. The theory and the solution technique are illustrated by several examples of practical interest.     

Surface displacements due to elastic wave scattering by buried planar and non-planar cracks

The scattering of time-harmonic plane longitudinal, shear, and Rayleigh waves by a crack in two dimensions embedded in a semi-infinite homogeneous isotropic elastic half-space has been studied in this paper. Two problems have been considered: a straight crack and a Y-shaped crack. A hybrid numerical technique combining a multipolar representation of the scattered field in the half-space with the finite element method has been used to obtain the far-field displacements as well as the stress-intensity factors for the crack tips. Results for vertical displacement on the free surface of the half-space are presented in this paper.     

Global analysis of secondary bifurcation of an elastic bar

In a three dimensional framework of finite deformation configurations, this paper investigates the secondary bifurcation of a uniform, isotropic and linearly elastic bar under compression in a large range of parameters. The governing differential equations and finite dimensional equations of this problem are discussed. It is found that, for a bar with two ends hinged, usually many secondary bifurcation points appear on the primary branches which correspond to the maximum bending stiffness. Results are shown on parameter charts. Secondary modes and branches are also calculated with numerical methods. The project supported in part by the National Natural Science Foundation of China