On uniqueness and continuous dependence of solutions in viscoelastic mixtures

This note deals with the isothermal linear theory of porous viscoelastic mixtures. Questions of uniqueness and continuous dependence for solutions of various classes of initial boundary value problems in mixtures consisting of two constituents: a porous elastic solid and a porous Kelvin–Voigt material are studied. The Lagrange identity and Logarithmic convexity methods are used to establish uniqueness and continuous dependence results, with no definiteness assumptions upon the internal energy.     

粘弹性力学的对应原理及其数值反演方法

积分变换是处理粘弹性混合边值问题的重要数学工具,积分变换的应用使粘弹性混合边值问题在象空间与相应弹性混合边值问题对应起来,从而使粘弹性混合边值问题的求解可以继承和借鉴弹性问题的求解方法,再利用积分反演方法就可求得时间域粘弹性边值问题的解．本文结合国内外的研究成果,就粘弹性力学中存在的各种对应原理及数值反演方法进行了归类和总结．结合在求解粘弹性边值问题中的应用,对各类方法的特点进行了评述,并指出存在的问题及发展新的数值方法的研究重点．      

Flow of viscoelastic fluids through a porous channel—I

The flow of viscoelastic fluids through a porous channel with one impermeable wall is computed. The flow is characterized by a boundary value problem in which the order of the differential equation exceeds the number of boundary conditions. Three solutions are developed: (i) an exact numerical solution, (ii) a perturbation solution for small R, the cross-flow Reynold's number and (iii) an asymptotic solution for large R. The results from exact numerical integration reveal that the solutions for a non-Newtonian fluid are possible only up to a critical value of the viscoelastic fluid parameter, which decreases with an increase in R. It is further demonstrated that the perturbation solution gives acceptable results only if the viscoelastic fluid parameter is also small. Two more related problems are considered: fluid dynamics of a long porous slider, and injection of fluid through one side of a long vertical porous channel. For both the problems, exact numerical and other solutions are derived and appropriate conclusions drawn.     

Microscopic Roles of “Viscoelasticity” in HPMA polymer flooding for EOR

A polymer solution with a transient network structure due to the entanglement of long chain molecules exhibits a viscoelastic behavior when it flows through a tortuous and diverging/converging channel in porous media. A constitutive equation is first developed to represent the viscoelastic behavior of polymer solutions in this article. Then a 3D viscoelastic polymer flooding model is established to examine the effect of elasticity of polymers on EOR (enhanced oil recovery). The model is validated in comparison with laboratorial coring data. The simulated results show that the oil recovery of viscoelastic polymer flooding can be enhanced by larger displacement efficiency due to its microscopic roles. In the meanwhile, the injection pressure required increases correspondingly if the elastic effect is significant. Relaxation time as a major characteristic parameter of viscoelastic polymer plays a decisive role, and therefore the HPAM (partially hydrolyzed polyacrylamide) with evident elastic property is recommended in chemical flooding.     

半封闭分数导数粘弹性衬砌隧道的频域响应

混凝土衬砌具有粘弹性性质,以往的经典Kelvin模型、弹性理论和壳体理论都不能刻画其蠕变的全过程。本文基于饱和多孔介质理论,在频率域研究了轴对称荷载和流体压力作用下饱和粘弹性土中半封闭分数导数型衬砌隧洞的稳态动力响应。在引入隧洞部分透水边界条件的基础上,通过分数阶导数粘弹性模型描述衬砌的应力—位移本构关系,并利用衬砌内边界以及接触面的连续性条件,得到了饱和土和衬砌的应力、位移和孔压解答。考察了分数导数阶数、材料参数以及衬砌和土体相对渗透系数的影响。研究表明：分数导数阶数对系统响应影响较大,且依赖于衬砌的材料参数。另外,相对渗透系数对系统响应的影响很大。     

Axisymmetrical analytical solution for vertical vibration of end-bearing pile in saturated viscoelastic soil layer

Based on elasticity and the theory of saturated porous media, and regarding the pile and the soil as a single phase elastic and a saturated viscoelastic media, respectively, the dynamical behavior of vertical vibration of an end-bearing pile in a saturated viscoelastic soil layer is investigated in the frequency domain using the Helmholtz decomposition and variable separation method. The axisymmetrical analytical solutions for vertical vibrations of the pile in a saturated viscoelastic soil layer are obtained, and the analytical expression of the dynamical complex stiffness of the pile top is presented. Responses of dynamic stiffness factor and equivalent damping of pile top with respect to the frequency are shown in figures using a numerical method. Effects of the saturated soil parameters, modulus ratio of the pile to soil, slenderness ratio of pile and pile＇s Poisson ratio, etc. on the stiffness factor and damping are examined. It is shown that, due to the effect of the transversal deformation of the pile and the radial force of the saturated viscoelastic soil acting on the pile, the dynamic stiffness factor and the damping derived from the axisymmetrical solution are greatly different from those derived from the classical Euler-Bernoulli rod model, especially at some specific excitation frequencies. Therefore, there are limitations on applicability of the Euler-Bernoulli rod model in analyzing verticai vibration of the pile. More accurate analysis should be based on a three-dimensional model.     

岩石三轴蠕变试验黏弹性解析及参数识别

基于岩石三轴蠕变试验,推导了相应的黏弹性解析解,并对三参量H-K流变模型给出 了黏弹性解析解的显式以及流变参数的识别方法. 在此基础上,针对花岗岩及片岩两种岩石 的室内蠕变分级加载试验曲线进行了流变参数的识别与反演. 分析表明,同一岩样在不同加 载级别下的弹性及黏弹性参数在量值上均有所不同,其中黏弹性模量和黏滞系数的量值随 载荷的增加出现损伤衰减,基于试验反演所获参数的解析蠕变曲线与试验蠕变曲线吻合较好.     

变边界粘弹性轴对称问题的复变函数法

对边界几何形状、位置随时间变化的变边界结构,给出了用复变函数求解粘弹问题的解析方法。文中用拉普拉斯变换结合平面弹性复变方法,对内外边界变化时粘弹性轴对称问题进行求解。引入两个与时间、空间相关的解析函数,给出了变边界情况下应力、位移以及边界条件与解析函数的关系。当解析函数形式部分确定,则可用边界条件求解其中与时间相关的待定函数。求解待定函数的方程一般情况下为一系列积分方程,特殊情况可求得解析解。对轴对称问题中应力边值问题、位移边值问题以及混合边值问题,分别利用边界条件求得相关系数,从而得到了应力与位移的解析表达。当取Boltzmann粘弹模型时,进行不同边值问题的分析。分析显示,应力、位移的形态与大小均与边界变化过程相关,与固定边界粘弹性问题有较大不同。本文解答可用于粘弹性轴对称问题内外边界任意变化及各种边值问题的力学分析。此外,该法可进一步进行荷载非对称、复杂孔型变边界问题的求解。     

Reflection of longitudinal displacement wave at the viscoelastically supported boundary of micropolar half-space

The reflection of elastic waves at the boundary of micropolar half-space with the viscoelastic support is studied in this paper. The spring-dashpot model is used to model the viscoelastic support. The boundary condition includes the force stress, couple stress, the displacements and the micro-rotation and the contribution from the spring and the dashpot on them. The amplitude ratios and phase shifts of reflection waves with respect to the incident wave are obtained from the visco-elastic boundary conditions. Further, the energy flux ratios of the reflection waves to the incident wave are estimated and the energy flux conservation with consideration of the energy dissipation of visco-elastic boundary is used to validate the numerical results. Based on the numerical results, the influences of spring and dashpot are studied respectively. It is found that the elastically supported boundary and the viscously supported boundary have evident different influences on the amplitude ratio and the phase shift. The causes resulting in these deviations are related with the instantaneous elasticity of elastic boundary and the time-delay effects of viscous boundary.     

Stability of buoyancy-driven convection in an Oldroyd-B fluid-saturated anisotropic porous layer

The nonlinear stability of thermal convection in a layer of an Oldroyd-B fluid-saturated Darcy porous medium with anisotropic permeability and thermal diffusivity is investigated with the perturbation method. A modified Darcy-Oldroyd model is used to describe the flow in a layer of an anisotropic porous medium. The results of the linear instability theory are delineated. The thresholds for the stationary and oscillatory convection boundaries are established, and the crossover boundary between them is demarcated by identifying a codimension-two point in the viscoelastic parameter plane. The stability of the stationary and oscillatory bifurcating solutions is analyzed by deriving the cubic Landau equations. It shows that these solutions always bifurcate supercritically. The heat transfer is estimated in terms of the Nusselt number for the stationary and oscillatory modes. The result shows that, when the ratio of the thermal to mechanical anisotropy parameters increases, the heat transfer decreases.     

Irreversible deformation with subsequent unloading of a spherical viscoelastoplastic layer

Analytic solutions are obtained for a sequence of one-dimensional quasistatic problems describing viscoelastic deformation processes in the material of a hollow ball and the plastic flow nucleation and evolution processes occurring in the ball as the pressure on the outer boundary increases. The unloading process under slow removal of the loading pressure is considered as well. The stress fields and the elastic and plastic strain fields in the spherical layer material, the law of motion of the elastoplastic boundary, and the residual stress level and distribution are computed. It is assumed that at the stage preceding the plastic flow the material obeys the viscoelastic Voigt model and the loading surface is determined by the von Mises plastic flow condition.     

Potential method in the linear theory of swelling porous elastic soils

This paper deals with the isothermal linear theory of swelling porous elastic soils in the case of fluid saturation. Internal and external boundary value problems of steady vibrations are investigated using the potential method. The uniqueness and existence theorems of classical solutions of the aforementioned problems are proved.     

Mesoscale bounds in viscoelasticity of random composites

Under consideration is the problem of size and response of the representative volume element (RVE) of spatially random linear viscoelastic materials. The model microstructure adopted here is the random checkerboard with one phase elastic and another viscoelastic, perfectly bonded everywhere. The method relies on the hierarchies of mesoscale bounds of relaxation moduli and creep compliances (Huet, 1995, 1999) obtained via solutions of two stochastic initial boundary value problems, respectively, under uniform kinematic and uniform stress boundary conditions. In general, the microscale viscoelasticity introduces larger discrepancy in the hierarchy of mesoscale bounds compared to elasticity, and this discrepancy grows as the time increases.     

On viscoelastic effects in non-Newtonian steady flows through porous media

This paper presents a mathematical model for describing approximately the viscoelastic effects in non-Newtonian steady flows through a porous medium. The rheological behaviour of power law fluids is considered in the Maxwell model of elastic behaviour of the fluids. The equations governing the steady flow through porous media are derived and an analytical solution of these equations in the case of a simple flow system is obtained. The conditions for which the viscoelastic effects may become observable from the pressure distribution measurements are shown and expressed in terms of some dimensionless groups. These have been found to be relevant in the evaluation of viscoelastic effects in the steady flow through porous media.     

Development of an analytical framework for viscoelastic corrugated-core sandwich plates and validation against FEM

In this study, an analytical procedure for the bending problem of a viscoelastic sandwich plate with a corrugated core is presented. Reissner–Mindlin plate theory and N-termed Prony series are employed to define the elastic and time-dependent contributions of the governing equations, respectively. Three different corrugation patterns, i.e., rectangular, trapezoidal, and triangular, are examined. Moreover, the structure is analyzed under both simply support and clamp boundary conditions. The calibrated material parameters of polymethyl methacrylate (PMMA) for the Generalized Maxwell rheological model are employed to show the viscoelastic response of the structure. A 3D finite element simulation of the problem is also conducted to confirm the accuracy of the analytical formulation. The two well-known creep and stress relaxation phenomena of the viscoelastic materials are examined for the mentioned corrugation cores and both boundary conditions analytically and numerically. The time-dependent dimensionless deflection and resultant von Mises stress distributions are provided. Besides, the variation of the results with various rise-times and applied load are studied in detail. The von Mises stress contours of the upper surface of the structure at the end of the creep test are also presented. The finite element method outcomes verify the analytical results with excellent compatibility. The proposed analytical procedure can be used as an efficient tool to study the effects of various parameters such as material, geometrical constants, and corrugation pattern on bending of viscoelastic sandwich plates with corrugated core problems for design and optimization, which involves a high number of simulations.

     

粘弹性地基上弹性梁的自由振动分析

本文将文克尔弹性地基梁模型中的弹簧用粘弹性元件来替代,建立了三元件文克尔粘弹性地基止粘弹性梁的静力和动力本构方程,求出了粘弹性地基上弹性梁的自由振动的级数解。并且对不同的振动情况进行讨论,最后给出了算例及结论。     

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.     

粘弹性界面裂纹奇异场

对于许多粘弹性问题,通常可以利用对应性原理,即由弹性问题的结果得到对应的粘弹性问题在拉普拉斯变换域内的解,再通过反演变换求得最终时域中的解.但是,由于界面裂纹场存在着振荡奇异性,弹性问题解的形式就已经非常复杂,对应的粘弹性问题要通过反演变换直接求得准确的解析解几乎是不可能的.本文在利用对应性原理时做了更简单的准静态处理,即将弹性结果中的材料参数用粘弹性材料参数做对应替代,得到了粘弹性界面裂纹场近似的经典解,并与有限元分析结果作了比较.同时,利用Comninou接触模型,对粘弹性界面裂纹在远场拉剪混合加载情况下的裂尖应力场和接触区做了考察,并与经典解作了比较.     

Generalized variational principles of the viscoelastic body with voids and their applications

From the Boltzmann‘ s constitutive law of viscoelastic materials and the linear theory of elastic materials with voids, a constitutive model of generalized force fields for viscoelastic solids with voids was given. By using the variational integral method, the convolution-type functional was given and the corresponding generalized variational principles and potential energy principle of viscoelastic solids with voids were presented. It can be shown that the variational principles correspond to the differential equations and theinitial and boundary conditions of viscoelastic body with voids. As an application, a generalized variational principle of viscoelastic Timoshenko beams with damage was obtained which corresponds to the differential equations of generalized motion and the initial and boundary conditions of beams. The variational principles provide a way for solving problems of viscoelastic solids with voids.