Contribution to the dynamics of a horizontally unstretchable half-space of the westergaard type

Summary The transfer matrix of a layer is derived from the dynamic equations of a continuum reinforced with very thin unstretchable membranes. For the case of a periodically spaced reinforcement, the finite difference method enables to find closed solutions. The elastic wave propagation in a half-space is investigated, and it is shown that the only possible nonattenuating wave has the velocityc ₂ of the shear wave.     

On the stability of an elastic column positioned on an elastic half space

Summary Stability of a heavy elastic column loaded by a concentrated force at the top is analysed. It is assumed that the column is fixed to a rigid circular plate that is positioned on a homogeneous, isotropic, linearly elastic half-space. The constitutive equations for the column are assumed in the form that allows axial compressibility and takes into account the influence of shear stresses. It is shown that eigenvalues of the linearized equations determine the bifurcation points of the full non-linear system of equilibrium equations. Also, the type of bifurcation at the lowest eigenvalue is examined and shown that it could be both super-and sub-critical. The post-critical shape of the column is determined by numerical integration of the equilibrium equations. Received 13 June 1998; accepted for publication 12 November 1998     

Interaction of slowly moving punches with an elastic half-space

The problem of slow dynamic contact interaction of a system of punches remote from each other with an elastic half-space surface in the absence of friction is studied under the assumptions that the diameters of the contact areas are smaller than the minimum distance between the punches and the time required for the shear wave to travel the distance equal to the punch diameter is comparable to the time scale of the process. A first-order asymptotic model is constructed. As an example, the case of steady-state vibrations of a system of two punches is considered. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 6, pp. 198–206, November–December, 2008.     

Propagation of Love waves in non-homogeneous substratum over initially stressed heterogeneous half-space

The paper studies the propagation of Love waves in a non-homogeneous substratum over an initially stressed heterogeneous half-space. The dispersion equation of phase velocity is derived. The velocities of Love waves are calculated numerically as a function of kH and presented in a number of graphs, where k is the wave number, and H is the thickness of the layer. The case of Gibson’s half-space is also considered. It is observed that the speed of Love waves is finite in the vicinity of the surface of the half-space and vanishes as the depth increases for a particular wave number. It is also observed that an increase in compressive initial stresses causes decreases of Love waves velocity for the same frequency, and the tensile initial stress of small magnitude in the half-space causes increase of the velocity.     

Surface waves generated by a line load on a half-space with depth-dependent properties

The reciprocity theorem of elastodynamics is used in this paper to determine the surface waves that are radiated from a time-harmonic line load applied at the surface of a solid body, whose elastic moduli and mass density depend on the distance from the surface. In a high-frequency approximation, the surface wave velocity and expressions for the displacement and stresses of free surface waves are employed in the reciprocity theorem. The general expressions for the surface wave radiated by the oscillating line load, together with a virtual free surface wave, when employed in the reciprocity theorem, yield relatively simple expressions for the amplitude factor of the radiated surface wave. Results show the amplitude factor as a function of the wavenumber.     

STEADY-STATE RESPONSE OF A TIMOSHENKO BEAM ON AN ELASTIC HALF-SPACE UNDER A MOVING LOAD

By introducing the equivalent stiffness of an elastic half-space interacting with a Timoshenko beam, the displacement solution of the beam resting on an elastic half-space subjected to a moving load is presented. Based on the relative relation of wave velocities of the half-space and the beam, four cases with the combination of different parameters of the half-space and the beam, the system of soft beam and hard half-space, the system of sub-soft beam and hard half-space, the system of sub-hard beam and soft half-space, and the system of hard beam and soft half-space are considered. The critical velocities of the moving load are studied using dispersion curves. It is found that critical velocities of the moving load on the Timoshenko beam depend on the relative relation of wave velocities of the half-space and the beam. The Rayleigh wave velocity in the half-space is always a critical velocity and the response of the system will be infinite when the load velocity reaches it. For the system of soft beam and hard half-space, wave velocities of the beam are also critical velocities. Besides the shear wave velocity of the beam, there is an additional minimum critical velocity for the system of sub-soft beam and hard half-space. While for systems of (sub-) hard beams and soft half-space, wave velocities of the beam are no longer critical ones. Comparison with the Euler-Bernoulli beam shows that the critical velocities and response of the two types of beams are much different for the system of (sub-) soft beam and hard half-space but are similar to each other for the system of (sub-) hard beam and soft half space. The largest displacement of the beam is almost at the location of the load and the displacement along the beam is almost symmetrical if the load velocity is smaller than the minimum critical velocity (the shear wave velocity of the beam for the system of soft beam and hard half-space). The largest displacement of the beam shifts behind the load and the asymmetry of the displacement along the beam increases with the increase of the load velocity due to the damping and wave radiation. The displacement of the beam at the front of the load is very small if the load velocity is larger than the largest wave velocity of the beam and the half space. The results of the present study provide attractive theoretical and practical references for the analysis of ground vibration induced by the high-speed train.     

Green’s functions of two-dimensional piezoelectric quasicrystal in half-space and bimaterials

In this paper, we obtain Green’s functions of two-dimensional (2D) piezoelectric quasicrystal (PQC) in half-space and bimaterials. Based on the elastic theory of QCs, the Stroh formalism is used to derive the general solutions of displacements and stresses. Then, we obtain the analytical solutions of half-space and bimaterial Green’s functions. Besides, the interfacial Green’s function for bimaterials is also obtained in the analytical form. Before numerical studies, a comparative study is carried out to validate the present solutions. Typical numerical examples are performed to investigate the effects of multi-physics loadings such as the line force, the line dislocation, the line charge, and the phason line force. As a result, the coupling effect among the phonon field, the phason field, and the electric field is prominent, and the butterfly-shaped contours are characteristic in 2D PQCs. In addition, the changes of material parameters cause variations in physical quantities to a certain degree.     

基于抗损性的钢筋混凝土摇摆柱力学性能研究

抗震韧性将震后建筑的功能维持或快速恢复作为最终目标。构件层次的摇摆机制是实现主体结构功能可恢复的有效途径,但应以构件的抗损性为前提。以钢筋混凝土摇摆柱为对象,比拟弹性半空间局部受载经典拉芒（Flamant）理论,推导摇摆柱轴向应力分布弹性力学解答,以此为基础,假定摇摆柱侧移由柱端接触面转角和柱身弯剪变形引起,建立钢筋混凝土摇摆柱荷载-侧移力学模型,与有限元模拟结果对比验证合理性。开展轴压比和柱端强度分析,结果表明,在柱体抗损的前提下,单纯提高轴压比可提高柱体承载能力,但对侧移能力提升不显著;随着柱端材料强度提升,柱体承载力和侧移能力均显著改善,因此,采用柱端局部增强的方法改善钢筋混凝土摇摆柱工作性能具有可行性。     

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.     

半无限横观各向同性体中共面双裂纹边界元分析

基于双横观各向同性材料基本解的对偶边界元方法,分析了半无限横观各向同性材料中两条共面裂纹的相互作用问题。裂纹面上分别作用着法向和切向两种均布荷载,材料的各向同性面及裂纹面平行于半无限域自由面。数值计算得到了该类裂纹的应力强度因子(SIF)值和两条共面裂纹相互作用的SIF值影响系数。根据数值结果分析了自由面对该类裂纹SIF值的影响,以及裂纹间距与形状、自由面对SIF值影响系数的影响。结果表明,自由面对作用法向力时的该类裂纹SIF值有明显的影响,裂纹间距与形状对SIF值影响系数影响较大,但自由面对SIF值影响系数基本无影响。