Time domain modeling of damping using anelastic displacement fields and fractional calculus

A fractional derivative model of linear viscoelasticity based on the decomposition of the displacement field into an anelastic part and elastic part is developed. The evolution equation for the anelastic part is then a differential equation of fractional order in time. By using a fractional order evolution equation for the anelastic strain the present model becomes very flexible for describing the weak frequency dependence of damping characteristics. To illustrate the modeling capability, the model parameters are fit to available frequency domain data for a high damping polymer. By studying the relaxation modulus and the relaxation spectrum the material parameters of the present viscoelastic model are given physical meaning. The use of this viscoelastic model in structural modeling is discussed and the corresponding finite element equations are outlined, including the treatment of boundary conditions. The anelastic displacement field is mathematically coupled to the total displacement field through a convolution integral with a kernel of Mittag–Leffler function type. Finally a time step algorithm for solving the finite element equations are developed and some numerical examples are presented.     

General Expressions of Constitutive Equations for Isotropic Elastic Damaged Materials

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The use of a virtual configuration in formulating constitutive equations for residually stressed elastic materials

Residual stress is the stress present in the unloaded equilibrium configuration of a body. Because residual stresses can significantly affect the mechanical behavior of a component, the measurement of these stresses and the prediction of their effect on mechanical behavior are important objectives in many engineering problems. Common methods for the measurement of residual stresses include various destructive experiments in which the body is cut to relieve the residual stress. The resulting strain is measured and used to approximate the original residual stress in the intact body. In order to predict the mechanical behavior of a residually stressed body, a constitutive model is required that includes the influence of the residual stress.In this paper we present a method by which the data obtained from standard destructive experiments can be used to derive constitutive equations that describe the mechanical behavior of elastic residually stressed bodies. The derivation is based on the idea that for each infinitesimal neighborhood in a residually stressed body, there exists a corresponding stress free configuration. We refer to this stress free configuration as the virtual configuration of the infinitesimal neighborhood. The derivation requires that the constitutive equation for the stress free material be known and invertible; it is used to relate the residual stress to the deformation of the virtual configuration into the residually stressed configuration. Although the concept of the virtual configuration is central to the derivation, the geometry of this configuration need not be determined explicitly, and it need not be achievable experimentally, in order to construct the constitutive equation for the residually stressed body.The general mathematical forms of constitutive equations valid for residually stressed elastic materials have been derived previously for a number of cases. These general forms contain numerous unknown material-response functions or material constants that must be determined experimentally. In contrast, the method presented here results in a constitutive equation that is an explicit function of residual stress and includes only the material parameters required to describe the stress free material.After presenting the method for the derivation of constitutive equations, we explore the relationship between destructive experiments and the theory used in the derivation. Specifically, we discuss the use of the theory to improve the design of destructive experiments, and the use of destructive experiments to obtain the data required to construct the constitutive equation for a particular material.     

Generalizing the Boussinesq approximation to stratified compressible flow

The simplifications required to apply the Boussinesq approximation to compressible flow are compared with those in an incompressible fluid. The larger degree of approximation required to describe mass conservation in a stratified compressible fluid using the Boussinesq continuity equation has led to the development of several different sets of ‘anelastic’ equations that may be regarded as generalizations of the original Boussinesq approximation. These anelastic systems filter sound waves while allowing a more accurate representation of non-acoustic perturbations in compressible flows than can be obtained using the Boussinesq system. The energy conservation properties of several anelastic systems are compared under the assumption that the perturbations of the thermodynamic variables about a hydrostatically balanced reference state are small. The ‘pseudo-incompressible’ system is shown to conserve total kinetic and anelastic dry static energy without requiring modification to any governing equation except the mass continuity equation. In contrast, other energy conservative anelastic systems also require additional approximations in other governing equations. The pseudo-incompressible system includes the effects of temperature changes on the density in the mass conservation equation, whereas this effect is neglected in other anelastic systems. A generalization of the pseudo-incompressible equation is presented and compared with the diagnostic continuity equation for quasi-hydrostatic flow in a transformed coordinate system in which the vertical coordinate is solely a function of pressure. To cite this article: D.R. Durran, A. Arakawa, C. R. Mecanique 335 (2007).     

Scattering of harmonic elastic waves by a plane interface crack with linear adhesive tips in a layered half space

In this paper, the scattering of elastic waves by an interface crack with linear adhesive tips in a layered half space is considered. By use of integral transform and integral equation methods, the singular integral equations of this problem are derived, which are transformed into a set of algebraic equations by means of contour integration and Chebyshev polynomials expanding technique. The numerical results of the adhesive region and stress amplitudes are given in this paper.     

Non-linear waves in a fluid-filled inhomogeneous elastic tube with variable radius

In the present work, by employing the non-linear equations of motion of an incompressible, inhomogeneous, isotropic and prestressed thin elastic tube with variable radius and the approximate equations of an inviscid fluid, which is assumed to be a model for blood, we studied the propagation of non-linear waves in such a medium, in the longwave approximation. Utilizing the reductive perturbation method we obtained the variable coefficient Korteweg–de Vries (KdV) equation as the evolution equation. By seeking a progressive wave type of solution to this evolution equation, we observed that the wave speed decreases for increasing radius and shear modulus, while it increases for decreasing inner radius and the shear modulus.     

Interaction between crack and elastic inclusion

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Constitutive equation of a gas-filled two-phase medium

A set of equations governing the consolidation of a two-phase medium consisting of a porous elastic skeleton saturated with a highly compressible liquid (gas), is described. The homogenization method was utilized to deduce the equations. For the equivalent macroscopic medium, mass and momentum conservation equations and the flow equation of pore liquid are presented. Sample material constants were calculated using laboratory test results which were carried out at the Institute of Geotechnics, Technical University of Wroclaw.     

尖端具有吸附性接触的界面裂纹对瞬态弹性波作用下的动态响应

建立并研究一类接触型界面裂纹模型对瞬态弹性波作用下的动态响应问题。文中利用积分变换和积分方程法推导了确定这类问题的奇异积分方程组。采用围道积分技术和切比雪夫多项式展开技术，得到了待定系数的非线性代数方程组。最后给出了裂纹尖端接触区大小和接触应力随时间变化的数值结果，揭示了这种接触裂纹的动力学特性及物理上的合理性。     

Asymptotics, structure, and integration of sound-proof atmospheric flow equations

Relative to the full compressible flow equations, sound-proof models filter acoustic waves while maintaining advection and internal waves. Two well-known sound-proof models, an anelastic model by Bannon and Durran’s pseudo-incompressible model, are shown here to be structurally very close to the full compressible flow equations. Essentially, the anelastic model is obtained by suppressing ? _t ρ in the mass continuity equation and slightly modifying the gravity term, whereas the pseudo-incompressible model results from dropping ? _t p from the pressure equation. For length scales small compared to the density and pressure scale heights, the anelastic model reduces to the Boussinesq approximation, while the pseudo-incompressible model approaches the zero Mach number, variable density flow equations. Thus, for small scales, both models are asymptotically consistent with the full compressible flow equations, yet the pseudo-incompressible model is more general in that it remains valid in the presence of large density variations. For the relatively small density variations found in typical atmosphere–ocean flows, both models are found to yield very similar results, with deviations between models much smaller than deviations obtained when using different numerical schemes for the same model. This in agreement with Smolarkiewicz and Dörnbrack (Int J Numer Meth Fluids 56:1513–1519, 2007). Despite these useful properties, neither model can be derived by a low-Mach number asymptotic expansion for length scales comparable to the pressure scale height, i.e., for the regime they were originally designed for. Derivations of these models via scale analysis ignore an asymptotic time scale separation between advection and internal waves. In fact, only the classical Ogura and Phillips model, which assumes weak stratification of the order of the Mach number squared, can be obtained as a leading-order model from systematic low Mach number asymptotic analysis. Issues of formal asymptotics notwithstanding, the close structural similarity of the anelastic and pseudo-incompressible models to the full compressible flow equations makes them useful limit systems in building computational models for atmospheric flows. In the second part of the paper, we propose a second-order finite-volume projection method for the anelastic and pseudo-incompressible models that observes these structural similarities. The method is applied to test problems involving free convection in a neutral atmosphere, the breaking of orographic waves at high altitudes, and the descent of a cold air bubble in the small-scale limit. The scheme is meant to serve as a starting point for the development of a robust compressible atmospheric flow solver in future work.     

一种孔隙介质力学模型及在水泥基材料的应用

基于细观力学和断裂力学的基本理论提出一个新的分析模型, 对孔隙介质的力学性能进行了分析. 依据孔隙介质内部孔隙的几何描述和状态参数,如孔隙率、形状、尺度及分布等,通过等效夹杂理论获得孔隙介质的等效本构方程,其最终变量为应力、应变和孔隙的形态参数. 根据断裂理论中材料承受载荷作用下破坏增长过程中的能量守恒,对孔隙介质变形过程中机械能、弹性应变能和载荷提供的势能进行分析, 根据能量守恒定律建立能量守恒方程,其最终变量也为应力、应变和孔隙的形态参数. 根据等效本构方程和能量守恒方程,获得孔隙介质承受载荷作用下的应力应变关系. 最后将该力学模型应用于水泥基材料,计算水泥基材料的力学性能并与文献中的结果进行对比分析,结果显示模型的计算结果准确有效.      

Nonlinear dynamics of oriented elastic solids

Based on a continuum model for oriented elastic solids the set of nonlinear dispersive equations derived in Part I of this work allows one to investigate the nonlinear wave propagation of the soliton type. The equations govern the coupled rotation-displacement motions in connection with the linear elastic behavior and large-amplitude rotations of the director field. In the one-dimensional version of the equations and for two simple configurations an exhaustive study of solitons is presented. We show that the transverse and/or longitudinal elastic displacements are coupled to the rotational motion so that solitons, jointly in the rotation of the director and the elastic deformations, are exhibited. These solitons are solutions of a system of linear wave equations for the elastic displacements which are nonlinearly coupled to a sine-Gordon equation for the rotational motion. For each configuration, the solutions are numerically illustrated and the energy of the solitions is calculated. Finally, some applications of the continuum model and the related nonlinear dynamics to several physical situations are given and additional more complex problems are also evoked by way of conclusion.     

Bilinear forms and soliton interactions for two generalized KdV equations for nonlinear waves

Korteweg–de Vries (KdV)-type equations can describe the nonlinear waves in fluids, plasmas, etc. In this paper, two generalized KdV equations are under investigation. Bilinear forms of which are constructed with the Bell polynomials and an auxiliary variable. \(N\) -soliton solutions are given through the Hirota direct method. Via the asymptotic analysis, the soliton interactions of the first generalized KdV equation are analyzed, which turn out to be elastic. Singular breather solutions have been derived from the two-soliton solutions. The collision between soliton and singular breather appears to be elastic, and the bound states of soliton and singular breather are exhibited. Unlike the first one, the other generalized KdV equation can only support the bound states of solitons, for the regular and singular solitons alike.     

Weakly non-linear waves in a fluid-filled elastic tube with variable prestretch

In the present work, by utilizing the non-linear equations of motion of an incompressible, isotropic thin elastic tube subjected to a variable prestretch both in the axial and the radial directions and the approximate equations of motion of an incompressible inviscid fluid, which is assumed to be a model for blood, we studied the propagation of weakly non-linear waves in such a medium, in the long wave approximation. Employing the reductive perturbation method we obtained the variable coefficient KdV equation as the evolution equation. By seeking a travelling wave solution to this evolution equation, we observed that the wave speed is variable in the axial coordinate and it decreases for increasing circumferential stretch (or radius). Such a result seems to be plausible from physical considerations.     

The exact solution for the general bending problems of conical shells on the elastic foundation

The general bending problem of conical shells on the elastic foundation (Winkler Medium) is not solved. In this paper, the displacement solution method for this problem is presented. From the governing differential equations in displacement form of conical shell and by introducing a displacement function U(s,θ), the differential equations are changed into an eight-order soluble partial differential equation about the displacement function U(s,θ) in which the coefficients are variable. At the same time, the expressions of the displacement and internal force components of the shell are also given by the displacement function U(s θ). As special cases of this paper, the displacement function introduced by V.S. Vlasov in circular cylindrical shell^[5], the basic equation of the cylindrical shell on the elastic foundation and that of the circular plates on the elastic foundation are directly derived.Under the arbitrary loads and boundary conditions, the general bending problem of the conical shell on the elastic foundation is reduced to find the displacement function U(s,θ).The general solution of the eight-order differential equation is obtained in series form. For the symmetric bending deformation of the conical shell on the elastic foundation, which has been widely usedinpractice,the detailed numerical results and boundary influence coefficients for edge loads have been obtained. These results have important meaning in analysis of conical shell combination construction on the elastic foundation,and provide a valuable judgement for the numerical solution accuracy of some of the same type of the existing problem.     

层状层电介质空间轴对称问题的状态空间解

从横观各向同性压电介质空间轴对称问题的基本方程出发,建立了压电介质空间轴对称问题的状态变量方程,对状态变量方程进行Hankel变换,得到以状态变量表示的单层压电介质在Hankel变换空间中的解,讨论了3种不同特征根的情况,利用提出的解得到了半无限压电体在垂直集中载荷和点电荷作出下的Boussinesq解。利用传递矩阵方法导出了多层压电介质空间轴对称问题解一般解析式。     

EIGEN THEORY OF VISCOELASTIC MECHANICS FOR ANISOTROPIC SOLIDS

Anisotropic viscoelastic mechanics is studied under anisotropic subspace. It is proved that there also exist the eigen properties for viscoelastic medium. The modal Maxwell's equation, modal dynamical equation (or modal equilibrium equation) and modal compatibility equation are obtained. Based on them, a new theory of anisotropic viscoelastic mechanics is presented. The advantages of the theory are as follows: 1) the equations are all scalar, and independent of each other. The number of equations is equal to that of anisotropic subspaces, 2) no matter how complicated the anisotropy of solids may be, the form of the definite equation and the boundary condition are in common and explicit, 3) there is no distinction between the force method and the displacement method for statics, that is, the equilibrium equation and the compatibility equation are indistinguishable under the mechanical space, 4) each model equation has a definite physical meaning, for example, the modal equations of order one and order two express the volume change and shear deformation respectively for isotropic solids, 5) there also exist the potential functions which are similar to the stress functions of elastic mechanics for viscoelastic mechanics, but they are not man-made, 6) the final solution of stress or strain is given in the form of modal superimposition, which is suitable to the proximate calculation in engineering.     

Bi-variable damage model for fatigue life prediction of metal components

Based on the theory of continuum damage mechanics,a bi-variable damage mechanics model is developed,which,according to thermodynamics,is accessible to derivation of damage driving force,damage evolution equation and damage evolution criteria. Furthermore,damage evolution equations of time rate are established by the generalized Drucker’s postulate. The damage evolution equation of cycle rate is obtained by integrating the time damage evolution equations,and the fatigue life prediction method for smooth specimens under repeated loading with constant strain amplitude is constructed. Likewise,for notched specimens under the repeated loading with constant strain amplitude,the fatigue life prediction method is obtained on the ground of the theory of conservative integral in damage mechanics. Thus,the material parameters in the damage evolution equation can be obtained by reference to the fatigue test results of standard specimens with stress concentration factor equal to 1,2 and 3.     

An equation of motion for a thick viscoelastic plate

In this paper an equation of motion is presented for a general thick viscoelastic plate, including the effects of shear deformation, extrusion deformation and rotatory inertia. This equation is the generalization of equations of motion for the corresponding thick elastic plate, and it can be degenerated into several types of equations for various special cases.