Transient analysis of viscoelastic helical rods subject to time-dependent loads

The transient analysis of viscoelastic helical rods subject to time-dependent loads are examined in the Laplace domain. The governing equations for naturally twisted and curved spatial rods obtained using the Timoshenko beam theory are rewritten for cylindrical helical rods. The curvature of the rod axis, effect of rotary inertia and, shear and axial deformations are considered in the formulation. The material of the rod is assumed to be homogeneous, isotropic and linear viscoelastic. The viscoelastic constitutive equations are written in the Boltzmann–Volterra form. Ordinary differential equations in canonical form obtained in the Laplace domain are solved numerically using the complementary functions method to calculate the dynamic stiffness matrix of the problem. The solutions obtained are transformed to the real space using an appropriate numerical inverse Laplace transform method. Numerical results for quasi-static and dynamic response of viscoelastic models are presented in the form of graphics.     

线黏弹性球面发散应力波的频率响应特性

基于线黏弹性球面波Laplace域的理论解, 得到了不同传播距离处粒子速度、粒子位移、应力、应变等力学量的传递函数。以标准线性固体模型为例, 重点讨论了粒子速度频率响应函数的传播特征, 指出随着传播距离的增加, 粒子速度幅频响应函数的高频响应会低于低频响应, 而在理想弹性条件下, 粒子速度幅频响应函数的高频响应一直高于低频响应。以弹性半径为0.025 m的空腔爆炸为例, 采用Laplace数值逆变换方法对粒子速度波形的演化进行了分析, 给出了粒子速度强间断幅值及粒子速度峰值随传播距离变化的衰减规律曲线, 指出黏弹性介质中粒子速度幅值的衰减曲线介于理想弹性介质中粒子速度幅值衰减曲线和黏弹性介质中粒子速度强间断幅值衰减曲线之间。     

Impact of an elastic rod on a deformable barrier: analytical and numerical investigations on models of a valve and a rod-shaped stamping tool

The longitudinal motion of an elastic rod is studied for the case that the rod is suddenly elastically fixed at one end and is hit by a mass at its other end. This configuration represents real settings e.g. as a valve impacts an elastic valve-seat or as a stamping device used in forging is hit by a large mass. The solution of the problem is formulated in the Laplace transformation space. The inverse transformation into the time domain is performed by engaging the so-called Laguerre polynomial technique. This method allows to calculate exact solutions for finite times from a finite number of series elements. Rigorous mathematical proofs not established up to now are given with respect to the convergence of the series encountered and the validity of exchanging the order of inversion of the Laplace transformation and summation of the established series. For comparison also a numerical solution of the problem is presented. An analysis of the energy transfer between rod, impacting mass and elastic barrier elucidates the marked influence of the deformability of the elastic barrier on the stress state in the rod.     

USING LAPLACE TRANSFORM TO SOLVE THE VISCOELASTIC WAVE PROBLEMS IN THE SHPB EXPERIMENTS

对分离式霍普金森压杆(split Hopkinson pressure bar, SHPB) 实验中试件的黏弹性波传播的控制方程组进行Laplace 变换,并结合恰当的初始-边界条件求解,得到变换域的应力、速度、应变等变量的像函数的精确表达式. 采用该方法处理SHPB 实验中涉及黏弹性试件内部应力非均匀性问题,并给出数值反变换解. 作为特例,对于弹性试件分别采用级数展开法和留数定理进行反Laplace 变换,从而给出弹性夹层介质中应力波传播问题的解析解.     

Laplace变换法研究SHPB实验中试件的黏弹性波传播问题

对分离式霍普金森压杆(split Hopkinson pressure bar, SHPB) 实验中试件的黏弹性波传播的控制方程组进行Laplace 变换,并结合恰当的初始-边界条件求解,得到变换域的应力、速度、应变等变量的像函数的精确表达式. 采用该方法处理SHPB 实验中涉及黏弹性试件内部应力非均匀性问题,并给出数值反变换解. 作为特例,对于弹性试件分别采用级数展开法和留数定理进行反Laplace 变换,从而给出弹性夹层介质中应力波传播问题的解析解.      

The Flow Analysis of Viscoelastic Fluid with Fractional Order Derivative in Horizontal Well

The fractional calculus approach is introduced into the seepage mechanics. A three-dimensional relaxation model of viscoelastic fluid is built. The models based on four boundary conditions of exact solution in Laplace space for some unsteady flows in an infinite reservoir is obtained by using the Laplace transform and Fourier sine and cosine integral transform. The pressure transient behavior of non-Newtonian viscoelastic fluid is studied by using Stehfest method of the numerical Laplace transform inversion and Gauss–Laguerre numerical integral formulae. The viscoelastic fluid is very sensitive to the order of the fractional derivative. The change rules of pressure are discussed when the parameters of the models change. The plots of type pressure curves are given, and the results can be provided to theoretical basis and well-test method for oil field.     

黏弹性界面断裂分析的增量“加料”有限元法

提出了一种适用于黏弹性界面裂纹问题的增量“加料” 有限元方法. 利用弹性界面裂纹尖端位移场的解答,通过对应原理和拉普拉斯逆变换近似方法,得到了黏弹性界面裂纹的尖端位移场. 用该位移场构造了黏弹性界面裂纹“加料” 单元和过渡单元位移模式,推导了增量“加料” 有限元方程,求解有限元方程可获得应力强度因子和应变能释放率等断裂参量. 建立了典型黏弹性界面裂纹平面问题“加料” 有限元模型,计算结果表明,对于弹性/黏弹性界面裂纹和黏弹性/黏弹性界面裂纹,该方法都能得到相当精确地断裂参量,并能很好地反映蠕变和松弛特性,可推广应用于黏弹性界面断裂问题的计算分析.      

A hybrid technique for the solution of unsteady Maxwell fluid with fractional derivatives due to tangential shear stress

The article describes the unsteady motion of viscoelastic fluid for a Maxwell model with fractional derivatives. The flow is produced by cylinder, considering time dependent quadratic shear stress ft² on Maxwell fluid with fractional derivatives. The fractional calculus approach is used in the constitutive relationship of Maxwell model. By applying Laplace transform with respect to time t and modified Bessel functions, semianalytical solutions for velocity function and tangential shear stress are obtained. The obtained semianalytical results are presented in transform domain, satisfy both initial and boundary conditions. Our solutions particularized to Newtonian and Maxwell fluids having typical derivatives. The inverse Laplace transform has been calculated numerically. The numerical results for velocity function are shown in Table by using MATLAB program and compared them with two other algorithms in order to provide validation of obtained results. The influence of fractional parameters and material constants on the velocity field and tangential stress is analyzed by graphs.     

竖向力作用在粘弹性土体内部时的解答与应用

选择三参量固体模型描述土体的粘弹性本构关系,利用半空间体内部受竖向集中力的Mindlin弹性解,根据粘弹性理论中的准静态弹性-弹粘性对应原理,推导了竖向力作用在粘弹性半无限土体内部的理论解。通过对应力和位移解答进行Laplace逆变换,给出了应力与位移的时域解。作为解答的应用,建立了粘弹性半无限体内部矩形面积上作用有三角形分布、均匀分布荷载时的粘弹性沉降计算公式。为了便于计算与工程应用,根据粘弹性理论解编制了计算程序。结果验证和实例分析表明,文中理论解是正确的,并为工程实际应用提供了理论依据。     

Viscoelastic analysis of von Kármán laminated plates under in-plane compression with initial deflection

The creep responses of simply-supported cross-ply and angle-ply viscoelastic laminates, having various magnitudes of imperfections, under in-plane compression are examined. The non-linear strain-displacement relation is based on the von Kármán assumption. The stress function is obtained by solving the compatibility equation by the use of the Laplace transform, and the deflection is calculated from the moment equation by the Galerkin method and a numerical integration scheme. The numerical results of deflection history and edge shortening for the glass/epoxy laminates are presented for illustrating the effect of imperfections and viscoelastic properties on the creep behavior. The solutions based on the quasi-elastic approach are also presented for comparison.     

Approximate equations for torsional waves in a nonlinear elastic rod with weak viscoelasticity

Approximate equations are derived for nonlinear torsional waves propagating along a thin circular viscoelastic rod. Ignoring the thermal effect, ‘nearly elastic’ compressible viscoelastic solids are considered in which a weak dependence of stresses on a history of strain is assumed. With the assumption that the rod is subjected to a finite angle of torsion, but that the rod is thin, the displacement is sought in a power series of the radial coordinate. The effects of geometrical and material nonlinearity give rise to the normal stress effect, which introduces deformations in the cross sectional and longitudinal dimensions of rod. Taking account of both the effect of nonlinearity and that of viscoelasticity, one dimensional approximate equations are obtained for the angle of torsion coupled with the longitudinal deformation.     

Viscoelastic composite with a cracked layer under anti-plane shear

A viscoelastic cracked sandwich composite under anti-plane load is presented via the application of the Laplace transform and the complex variable formulation. Using a special technique of analytical continuation associated with the method of image, the triple-layered problem is reduced to a singular function by simple algebraic manipulations. The superposition method and the singular integral equation is applied to deal with the concentrated load and crack problem. Finally, some typical viscoelastic models and its corresponding stresses and stress intensity factors are also discussed. The results show that the time dependent stress intensity factor is affected by the relative strength of creep compliances of each layer.     

求解饱和半空间上弹性圆板固结沉降的积分方程

本文采用解析方法分析了弹性圆板在饮和半空间上的固结沉降。考虑弹性圆板与饮和半空间的接触面上无摩擦力,且饱和半空间表面为全部透水的。运用Ｂｉｏｔ固结理论和积分方程技术,在Ｌａｐｌａｃｅ变换域上建立了弹性圆板固结沉降的对偶积分方程,并化此对偶积分方程为第二类Ｆｒｅｄｈｏｌｍ积分方程。通过对其核函数的有效数值发得到第二类Ｆｒｅｄｈｏｌｍ积分方程的解,再利用Ｌａｐａｃｅ反演技术获得弹性板在时间域中的固结沉     

粘弹性大挠度圆板的轴对称弯曲

本文探讨粘弹性大挠度圆板的轴对称弯曲的基本方程和求解方法.用半逆解和摄动法分析挠度与膜力,对标准线性固体进行数例计算,并与小挠度理论相比较.全部方程与解答可退化得相应的弹性大挠度板的结果.     

A fractional differential constitutive model for dynamic stress intensity factors of an anti-plane crack in viscoelastic materials

Fractional differential constitutive relationships are introduced to depict the history of dynamic stress inten- sity factors （DSIFs） for a semi-infinite crack in infinite viscoelastic material subjected to anti-plane shear impact load. The basic equations which govern the anti-plane deformation behavior are converted to a fractional wave-like equation. By utilizing Laplace and Fourier integral transforms, the fractional wave-like equation is cast into an ordinary differential equation （ODE）. The unknown function in the solution of ODE is obtained by applying Fourier transform directly to the boundary conditions of fractional wave-like equation in Laplace domain instead of solving dual integral equations. Analytical solutions of DSIFs in Laplace domain are derived by Wiener-Hopf technique and the numerical solutions of DSIFs in time domain are obtained by Talbot algorithm. The effects of four parameters α, β, b1, b2 of the fractional dif- ferential constitutive model on DSIFs are discussed. The numerical results show that the present fractional differential constitutive model can well describe the behavior of DSIFs of anti-plane fracture in viscoelastic materials, and the model is also compatible with solutions of DSIFs of anti-plane fracture in elastic materials.     

变截面黏弹性直杆纵振时复频率分析的差分解法

从一维黏弹性本构方程出发,导出了黏弹性变截面直杆纵向振动微分方程的一般形式,采用了有限差分法,并以二阶矩阵表示的递推形式,建立了该问题的复特征值方程组。两种Maxwell黏弹性变截面（指数指数、线性函数）直杆的数值计算表明,该方法运算简单,计算精度高,能适用于求解任意变截面黏弹性直属的纵向自由振动问题。     

State space approach to the generalized thermoelastic problem with temperature-dependent elastic moduli and internal heat sources

A two-dimensional equation of generalized thermoelasticity with one relaxation time in an isotropic elastic medium with the elastic modulus dependent on temperature and with an internal heat source is established using a Laplace transform in time and a Fourier transform in the space variable. The problem for the transforms is solved in the space of states. The problem of heating of the upper and the lower surface of a plate of great thickness by an exponential time law is considered. Expressions for displacements, temperature, and stresses are obtained in the transform domain. The inverse transform is obtained using a numerical method. Results of solving the problem are presented in graphical form. Comparisons are made with the results predicted by the coupled theory and with the case of temperature independence of the elastic modulus.     

黏弹性接触界面端附近的奇异应力场

研究蠕变加载条件下线黏弹性材料接触界面端附近的奇异应力场问题.考虑接触界面的摩擦,假设界面端的滑移方向不改变,相对滑移量微小,且其与位移同量级,由此线性化局部边界条件,根据对应原理得到Laplace变换域中的界面端应力场,导出时域中奇异应力场的卷积积分表达式.对卷积积分核函数进行数值反演,考虑接触材料的两类组合,一是持久模量具有量级上的差异,另一是持久模量接近相同.算例结果证实核函数可以用准弹性法求得的解析式较准确地近似.在此基础上,利用积分中值定理,并引入各应力分量的修正系数,得到黏弹性奇异应力场的简化式.结合核函数的数值反演结果分析修正系数表达式的取值范围,得到如下结论,若两相接触材料的持久模量相差很大,可以采用准弹性解的解析式较准确地描述界面端的奇异应力场;一般情况下,应力场不存在统一的奇异值和应力强度系数,当采用类似于准弹性解的表达式近似给出黏弹性应力场时,可以估计此近似描述的误差限.文中最后采用有限元分析黏弹性板端部嵌入部位的应力场,算例包括了黏弹性板与弹性金属支承、黏弹性板与黏弹性垫层所形成的滑移接触界面端,利用黏弹性有限元的数值结果验证理论分析所得结论的有效性.      

黏弹性功能梯度材料裂纹问题的有限元方法

针对组分材料体积含量任意分布的黏弹性功能梯度材料裂纹问题建立有限元分析途径. 通过Laplace变换,将黏弹性问题转化到象空间中求解,基于反映材料非均匀的梯度单元和裂纹尖端奇异特性的奇异单元计算象空间中的位移、应力和应变场,应用虚拟裂纹闭合方法得到应变能释放率,分别由应力和应变能释放率确定应力强度因子. 给出这些断裂参量在物理空间和象空间之间的对应关系,由数值逆变换求出其在物理空间的相应值. 文中分析两端均匀受拉的黏弹性边裂纹板条,首先针对松弛模量表示为空间函数和时间函数乘积的特殊梯度材料进行计算,结合对应原理验证方法的有效性. 然后分析组分材料体积含量具有任意梯度分布的情形,由Mori-Tanaka方法预测象空间中的等效松弛模量. 计算结果表明,蠕变加载条件下,应变能释放率随时间增加,其增大程度与黏弹性组分材料体积含量相关. 由于梯度材料的非均匀黏弹性性质,产生应力重新分布,导致应力强度因子随时间变化,其变化范围与组分材料的体积含量分布方式有关.     

Effective viscoelastic behavior of particulate polymer composites at finite concentration

Polymeric materials usually present some viscoelastic behavior. To improve the mechanical behavior of these materials, ceramics materials are often filled into the polymeric materials in form of fiber or particle. A micromechanical model was proposed to estimate the overall viscoelastic behavior for particulate polymer composites, especially for high volume concentration of filled particles. The method is based on Laplace transform technique and an elastic model including two-particle interaction. The effective creep compliance and the stress and strain relation at a constant loading rate are analyzed. The results show that the proposed method predicts a significant stiffer response than those based on Mori-Tanaka's method at high volume concentration of particles.