冲击波作用下悬臂梁瞬态响应的传递函数方法

利用传递函数方法对爆炸冲击波作用下的悬臂梁进行分析，得到其瞬态响应的封闭形式的解析解。首先，针对给定的爆炸冲击波载荷和初始条件，对梁的控制方程和边界条件进行Laplace变换，然后通过引入状态向量将其改写成状态空间形式，并利用传递函数方法求得其在频域内的解析解，最后利用Crump方法并结合ε算法进行Laplace逆变换，求得悬臂梁在时域内的瞬态响应。给出数值算例，通过与有限元的比较验证了方法的正确性。     

Generalized thermoelastic diffusion problem for an infinitely long hollow cylinder for short times

In this work, we consider the problem of a thermoelastic infinitely long hollow cylinder in the context of the theory of generalized thermoelastic diffusion with one relaxation time. The outer surface of the cylinder is taken traction free and subjected to a thermal shock, while the inner surface is taken to be in contact with a rigid surface and is thermally insulated. Laplace transform techniques are used. The solution is obtained in the Laplace transform domain by using a direct approach. The solution of the problem in the physical domain is obtained numerically using a numerical method for the inversion of the Laplace transform based on Fourier expansion techniques. The temperature, displacement, stress and concentration as well as the chemical potential are obtained. Numerical computations are carried out and represented graphically.     

Application of the Laplace transform dual reciprocity boundary element method in the modelling of laser heat treatments

The Laplace Transform Dual Reciprocity Boundary Element Method (LTDRM or LT-DRBEM) provides with an alternative numerical technique to finite difference (FDM) or finite element methods (FEM) for solving transient diffusion problems. With this method, solutions are calculated directly at any specific time thus avoiding the use of time-stepping schemes. Besides, domain integrals are removed from the problem formulation.In this work we study the applicability of the LT-DRBEM method for laser heat treatment modelling purposes. A simple model was developed based on a two dimensional transient heat conduction equation, in which the laser beam is included as a heat flux boundary condition of gaussian shape. Results corresponding to a stationary and a moving beam are presented and discussed. Non-linear formulations of the problem as those given by temperature dependent material properties are also considered. Good accuracy results were obtained for the stationary beam approach, whereas severe limitations were found for the moving beam case.     

A Thick Plate Problem in the Theory of Generalized Thermoelastic Diffusion

In this work, the problem of a thermoelastic thick plate with a permeating substance in contact with one of the bounding planes is considered in the context of the theory of generalized thermoelastic diffusion with one relaxation time. The bounding surface of the half-space is taken to be traction free and is subjected to a time-dependent thermal shock. The chemical potential is also assumed to be a known function of time on the bounding plane. Laplace transform techniques are used. The solution is obtained in the Laplace transform domain by using a direct approach. The solution of the problem in the physical domain is obtained numerically using a numerical method for the inversion of the Laplace transform based on Fourier expansion techniques. The temperature, displacement, stress, and concentration as well as the chemical potential are obtained. Numerical computations are carried out and represented graphically.     

Generalized Thermoelastic Diffusion Problem for an Infinite Medium with a Spherical Cavity

In this study, the problem of a thermoelastic infinite medium with a spherical cavity is considered in the context of the theory of generalized thermoelastic diffusion with one relaxation time. The surface of the spherical cavity is taken to be traction free and subjected to heating. Laplace transform techniques are used. The solution is obtained in the Laplace transform domain by using a direct approach. The solution of the problem in the physical domain in obtained numerically using a method based on Fourier expansion techniques. The temperature, displacement, stress, and concentration as well as the chemical potential are obtained. Numerical computations are carried out and represented graphically.     

A stochastic thermoelastic diffusion interaction in an infinitely long annular cylinder

In this work, we study the effect of a stochastic heating on an elastic solid in the form of an infinitely long circular cylinder under the existence of a permeating substance in contact with the outer surface. The stochastic heating is driven by an additive Gaussian white noise. Random elastic, thermal and diffusive properties are obtained in terms of the induced noise. The model is considered in the context of the generalized thermoelastic diffusion theory with one relaxation time. The Laplace transform technique is used to obtain the analytical solution in the transformed domain, while a numerical inversion method for Laplace transform is used to obtain an approximate solution for the temperature, displacement, stress, concentration of the diffusive material and chemical potential. Statistically, first and second moments of the physical properties are derived analytically and discussed.     

Fractional Order Thermoelastic Wave Assessment in a Nanoscale Beam Using the Eigenvalue Technique

Strength of Materials - This paper presents an analytical approach associated with Laplace transforms and a sequential concept over time to obtain the increment of temperature in nanoscale beam...     

Modelling and analysis of coupled nonlinear oscillations of a floating body in two degrees of freedom

Summary We describe a mathematical model to investigate the effect of coupled nonlinear oscillations of a floating body in time domain under the influence of sinusoidal waves. To account for hydrodynamic forces, a mathematical formulation for added mass moments of inertia, damping and restoring moments is presented for roll and yaw. Using perturbation technique, we obtain order wise solutions in the normalized domain wherein the assumption on small distortion holds. On applying Laplace transform, a zeroth-order solution is obtained in closed form whereas for higher order solutions we resort to the Runge-Kutta method with adaptive step size algorithm. For analyzing the model result we perform numerical experiments for a vessel of mass 19190 tons under the action of a beam wave of frequency 0.74 rad/sec and 1.0 m wave height. The validity of the numerical scheme is checked by comparing with the analytical solution for uncoupled zeroth-order roll and then we proceed to examine the effect of coupled behavior of roll and yaw for higher-order approximations. The inter-dependence between wave frequency (ω), system frequency (β) and damping factor (ζ) is obtained to reveal system stability. Model results indicate an artificial increase in amplitude for uncoupled roll, and also emphasize the contribution of viscous damping in roll in contrast to added mass in yaw.     

The transient response of a functionally graded piezoelectric rod subjected to a moving heat source under fractional order theory of thermoelasticity

The transient thermo-piezoelectric response of a functionally graded piezoelectric rod subjected to a moving heat source is investigated in the context of fractional order theory of thermoelasticity proposed by Sherief. The material properties of the functionally graded piezoelectric rod are assumed to vary exponentially along the length, except for the thermal relaxation time and the specific heat, which are taken to be constant. To solve the governing equations of the problem, Laplace transform is applied, eliminating the time effect; the analytical solutions of the displacement, stress, temperature, and electric field in Laplace domain are obtained. Subsequently, the solutions of the considered variables in time domain are obtained by numerical Laplace inversion and illustrated graphically. In calculation, the effect of the fractional order parameter on the variations of the considered variables is presented.     

传递函数法在非局部弹性梁动力学分析中的应用

采用传递函数方法进行了非局部弹性梁的动力学分析。非局部弹性梁内一点的应力与梁某一区域内任意一点的应变均有关系。本文基于Eringen的非局部弹性积分型本构关系,采用幂指数型核函数,利用Laplace变换导出梁的四阶偏微分形式振动方程,通过定义状态向量,将控制方程化为一阶微分方程组,并采用传递函数方法进行了求解,针对两种边界条件给出了非局部弹性梁的固有频率和固有振型。结果表明,同阶频率下,非局部弹性梁的频率比局部梁的频率低,振型基本一致。     

Impulsive effect on an elastic solid with generalized thermodiffusion

The theory of generalized thermoelastic diffusion with one relaxation time is employed to study the distribution of temperature, displacement components, stresses, concentration and chemical potential in a semi-infinite medium having an impulsive mechanical load at the origin. Using the joint Laplace and Fourier transforms, the governing equations are transformed into a vector–matrix differential equation which is then solved by the eigenvalue approach. The solution of the problem in the physical domain is obtained numerically using a numerical method for the inversion of the Laplace and Fourier transforms. Results of this work are presented graphically and are compared with the results of generalized thermoelasticity and classical elasticity deduced as special cases.     

Deformation due to expanding ring load in modified couple stress thermoelastic diffusion in time and frequency domain

The two-dimensional problem of expanding ring load in a modified couple stress theory of thermoelastic diffusion with heat sources in time and frequency domains is investigated. The mathematical formulation prepared for thermoelastic diffusion solids with one and two relaxation times using Laplace and Hankel transforms. The displacements, stress components, temperature change, and chemical potential are obtained in a transformed domain. Numerical computation is performed for these quantities and the resulting quantities are shown graphically for the time and frequency domains. Comparisons are made with the results predicted by the two theories and different values of time and frequency. Particular cases of interest are also deduced.     

Numerical operational methods for time-dependent linear problems

A general and systematic discussion on the use of the operational method of Laplace transform for numerically solving complex time-dependent linear problems is presented. Application of Laplace transform with respect to time on the governing differential equations as well as the boundary and initial conditions of the problem reduces it to one independent of time, which is solved in the transform domain by any convenient numerical technique, such as the finite element method, the finite difference method or the boundary integral equation method. Finally, the time domain solution is obtained by a numerical inversion of the transformed solution. Eight existing methods of numerical inversion of the Laplace transform are systematically discussed with respect to their use, range of applicability, accuracy and computational efficiency on the basis of some framework vibration problems. Other applications of the Laplace transform method in conjunction with the finite element method or the boundary integral equation method in the areas of earthquake dynamic response of frameworks, thermaliy induced beam vibrations, forced vibrations of cylindrical shells, dynamic stress concentrations around holes in plates and viscoelastic stress analysis are also briefly described to demonstrate the generality and advantages of the method against other known methods.     

A two-dimensional generalized thermoelastic diffusion problem for a half-space subjected to harmonically varying heating

A two-dimensional problem for a thermoelastic half-space is considered within the context of the theory of generalized thermoelastic diffusion with one relaxation time. The upper surface of the half-space is taken to be traction free and subjected to harmonically varying heating with constant angular frequency of thermal vibration. Laplace and Fourier transform techniques are used. The solution in the transformed domain is obtained by a direct approach. Numerical inversion techniques are used to obtain the inverse double transforms. Numerical results are discussed and represented graphically.     

Impact behavior of an inclined edge crack in a layered medium with a graded nonhomogeneous interfacial zone: antiplane deformation

Summary The impact response of an inclined edge crack in a layered medium with a functionally graded interfacial zone is investigated under the state of antiplane deformation. The interfacial zone is modeled by a nonhomogeneous interlayer having the power-law variations of shear modulus and mass density between the coating and the substrate of dissimilar homogeneous properties. Based on the Laplace and Fourier integral transform technique and the coordinate transformations of basic field variables, the transient crack problem is reduced to the solution of a singular integral equation with a generalized Cauchy kernel in the Laplace transform domain. The crack-tip response in the physical domain is recovered through the inverse Laplace transform to evaluate the dynamic mode III stress intensity factors as functions of time. The peak values of the dynamic stress intensity factors are further obtained versus the crack orientation angle, addressing the effects of crack obliquity on the overshoot characteristics of the transient crack-tip behavior for various combinations of material and geometric parameters of the layered medium.     

Solute transport modelling with the variable temporally dependent boundary

In this present study, analytical and numerical solutions are obtained for solute transport modelling in homogeneous semi-infinite porous medium. The dispersion coefficient is assumed to be initial dispersion and velocity is assumed to be temporally dependent with initial seepage velocity. Also, the concept of dispersion is directly proportional to the square of the seepage velocity used for finding the solution. Initially, the domain is not solute free. At one end of the domain, source concentration with the effect of different temporally dependent functions taken into account. The concentration gradient assumed to be zero due to no mass flux at other end of the domain. Laplace Transform Technique is used to obtain the exact solution, whereas Explicit Finite Difference method is used for approximate solution. The different types of temporally dependent velocity are used for the graphical representation of the solution. The accuracy of the solution explored by the Relative error analysis.     

Two-dimensional dynamic analysis of thermal stresses in a finite-length FG thick hollow cylinder subjected to thermal shock loading using an analytical method

This work is aiming to present an analytical method to study the dynamic behavior of thermoelastic stresses in a finite-length functionally graded (FG) thick hollow cylinder under thermal shock loading. The thermo-mechanical properties are assumed to vary continuously through the radial direction as a nonlinear power function. Using Laplace transform and series solution, the thermoelastic Navier equations in displacement form are solved analytically. The solution of the displacement field in the FG cylinder is obtained in the Laplace domain. Also, the fast Laplace inverse transform method (FLIT) is employed to transfer the results from Laplace domain to time domain. The effects of thermal shock loading on the dynamic characteristics of the FG thick hollow cylinder are studied in various points across the thickness of cylinder for various grading patterns of FGMs. A good agreement can be seen in the comparison of the obtained results based on the presented analytical method with published data.     

The effect of viscoelasticity on the stress distribution of adhesively single-lap joint with an internal break in the composite adherends

The aim of this research is to study the effect of a break in the laminated composite adherends on stress distribution in the adhesively single-lap joint with viscoelastic adhesive and matrix. The proposed model involves two adherends with E-glass fibers and poly-methyl-methacrylate matrix that have been adhered to each other by phenolic-epoxy resin. The equilibrium equations that are based on shear-lag theory have been derived in the Laplace domain, and the governing differential equations of the model have been derived analytically in the Laplace domain. A numerical inverse Laplace transform, which is called Gaver–Stehfest method, has been used to extract desired results in the time domain. The results obtained at the initial time completely matched with the results of elastic solution. Also, a comparison between results obtained from the analytical and finite element models show a relatively good match. The results show that viscoelastic behavior decreases the peak of stress near the break. Finally, the effect of size and location of the break, as well as volume fraction of fibers, on the stress distribution in the adhesive layer is fully investigated.     

Nanoscale electron beam induced etching: a continuum model that correlates the etch profile to the experimental parameters

In this paper, we relate experimental electron beam induced etching profiles to various electron limited and mass transport limited regimes via a continuum model. In particular, we develop a series of models with increasing complexity and demonstrate the effects and interactions that the precursor gas adsorption kinetics, the electron flux distribution, and the etch product desorption kinetics have on the resultant nanoscale etching profile. Unlike analogous electron beam induced deposition models, it is shown that one must consider the diffusion, desorption, and possible re-dissociation of the resultant etch product to understand the observed etching profiles. To confirm the explanation of the etch results, a defocus experiment was performed showing transitions from the electron flux limited to the mass transport limited to the etch product dissociation limited regimes.     

Transient Heat Conduction in a Functionally Graded Cylindrical Panel Based on the Dual Phase Lag Theory

The transient heat conduction in a functionally graded cylindrical panel is investigated based on the dual phase lag (DPL) theory in this article. Except for the phase lags which are assumed to be constant, all the other material properties of the panel are assumed to change continuously along the radial direction according to a power-law formulation with different non-homogeneity indices. The heat conduction equations based on the DPL theory in the cylindrical coordinate system are written in a general form which are then used for the analyses of four different geometries: (1) a hollow cylinder of an infinite length; (2) a hollow cylinder of a finite length; (3) a cylindrical panel of an infinite length; and (4) a cylindrical panel of a finite length. Using the Laplace transform, the analytical solutions for temperature and heat flux are obtained in the Laplace domain. The solutions are then converted into the time domain by employing the fast Laplace inversion technique. The exact expressions for the radial thermal wave speed are obtained for the four different geometries. The numerical results are displayed to reveal the effect of different approximations of the DPL theory on the temperature distribution for various non-homogeneity indices. The results are verified with those reported in the literature.