Analytical solutions of long nonlinear internal waves: Part I

The Gardner equation is an extension of the Korteweg–de Vries (KdV) equation. It exhibits basically the same properties as the classical KdV, but extends its range of validity to a wider interval of the parameters of the internal wave motion for a given environment. In this paper, we derive exact solitary wave solutions for the generalized Gardner equation that includes nonlinear terms of any order. Unlike previous studies, the exact solutions are derived without assuming their mathematical form. Illustrative examples for internal solitary waves are also provided. The traveling wave solutions can be used to specify initial data for the incident waves in internal waves numerical models and for the verification and validation of the associated computed solutions.     

A non-hydrostatic model for the numerical study of landslide-generated waves

This paper proposes and demonstrates a two-layer depth-averaged model with non-hydrostatic pressure correction to simulate landslide-generated waves. Landslide (lower layer) and water (upper layer) motions are governed by the general shallow water equations derived from mass and momentum conservation laws. The landslide motion and wave generation/propagation are separately formulated, but they form a coupled system. Our model combines some features of the landslide analysis model DAN3D and the tsunami analysis model COMCOT and adds a non-hydrostatic pressure correction. We use the new model to simulate a 2007 rock avalanche-generated wave event at Chehalis Lake, British Columbia, Canada. The model results match both the observed distribution of the rock avalanche deposit in the lake and the wave run-up trimline along the shoreline. Sensitivity analyses demonstrate the importance of accounting for the non-hydrostatic dynamic pressure at the landslide-water interface, as well as the influence of the internal strength of the landslide on the size of the generated waves. Finally, we compare the numerical results of landslide-generated waves simulated with frictional and Voellmy rheologies. Similar maximum wave run-ups can be obtained using the two different rheologies, but the frictional model better reproduces the known limit of the rock avalanche deposit and is thus considered to yield the best overall results in this particular case.     

Modelling strain localization in granular materials using micropolar theory: mathematical formulations

It has been known that classical continuum mechanics laws fail to describe strain localization in granular materials due to the mathematical ill‐posedness and mesh dependency. Therefore, a non‐local theory with internal length scales is needed to overcome such problems. The micropolar and high‐order gradient theories can be considered as good examples to characterize the strain localization in granular materials. The fact that internal length scales are needed requires micromechanical models or laws; however, the classical constitutive models can be enhanced through the stress invariants to incorporate the Micropolar effects. In this paper, Lade's single hardening model is enhanced to account for the couple stress and Cosserat rotation and the internal length scales are incorporated accordingly. The enhanced Lade's model and its material properties are discussed in detail; then the finite element formulations in the Updated Lagrangian Frame (UL) are used. The finite element formulations were implemented into a user element subroutine for ABAQUS (UEL) and the solution method is discussed in the companion paper. The model was found to predict the strain localization in granular materials with low dependency on the finite element mesh size. The shear band was found to reflect on a certain angle when it hit a rigid boundary. Applications for the model on plane strain specimens tested in the laboratory are discussed in the companion paper. Copyright © 2006 John Wiley & Sons, Ltd.     

基于内变量和张量函数表示定理的本构方程

针对各向同性材料,基于张量函数表示定理,建立了本构关系的张量不变性表示,其中,3个不可约基张量取决于应力的0～2次幂,且相互正交,3个系数由塑性应变增量和应力的不变量表示。基于塑性应变增量的不变量定义内变量,本构关系归结为确定内变量的演化。使用张量函数表示定理,给出了内变量演化方程的一般表达式,它取决于应力不变量的增量,因而与主轴旋转无关。讨论了如何根据试验资料和引入适当的假定,确定具体的演化方程。通过与塑性势理论和多重屈服面理论进行比较,表明所建模型是这些理论的最一般表示,且简捷直观、使用方便。     

二维弹性波动方程F—K正演与偏移

根据Ｈｏｏｋｅ定律，消除应力矢量后得到弹性波动方程位移矢量和波势矢量的非耦合形式递推解，在这个解答中各种类型的波相互分离，但在界面上满足能量守恒定律。正演可变步长延拓；偏移是将震源脉冲和地面记录波场同时下延，然后在每个深度作时间零延迟互相关来完成。由于延拓和成像在频率波数域（Ｆ—Ｋ）中进行，因而算法快速、偏移和正演可方便地扩展到三维空间，能处理各向同性任意非均匀介质中的弹性波。     

基于微形态理论的各向异性散体波动分析

基于散粒体微观力学理论,忽略颗粒转动引起的相对位移,考虑颗粒接触的组构各向异性,根据宏微观能量守恒推导得到了散体材料各向异性微形态本构关系,进而通过单位接触方向积分的递推公式推导出了各向异性本构张量表达式;在此基础上,根据哈密顿原理得到了各向异性散体材料的运动平衡方程和边界条件,从而求得了平面波在各向异性散粒体中的传播规律和频散关系,最后对波的频散关系和频率带隙进行了详细的参数分析。研究表明：该模型预测了散体中包含3类12种位移波：3种纵波、6种横波和3种平面内横向剪切波;横观各向同性条件下,接触各向异性参数a20越大,纵波和横波的频率越大,而平面内横向剪切波的频率越小;正交各向异性条件下,随着接触各向异性参数a22的增大,与2方向运动相关的横波频率增大,而与3方向运动相关的横波频率则减小;但a22的变化对纵波频率影响很小。材料各向异性程度对横波带宽影响不大,但对纵波带宽影响较大：a20的增大使得声?光学波间的带宽减小,而光学波间的带宽增大,当a20>0.84时,声?光学波间的带隙消失;但是a22的增大则使得声?光学波间的带宽增大,而光学波间的带宽减小。退化为各向同性模型后,预测3类波的频散曲线与其他各向同性模型的结果基本一致。     

内波单独作用形成的深水沉积物波

深水沉积物波是一种海底普遍发育、规模较大的波状沉积体,大多数学者将它们解释为等深流沉积或浊流沉积。本文结合内波理论的研究进展，考虑内波沉积作用的水动力学特征，探讨了深水沉积物波的内波成因机制。得出以下几点认识：①海底流动单独作用无法满足沉积物波形成所需的流动层厚度及流动速度，较难解释沉积物波的迁移方向及规则的内部及外部形态。②内波可以引起海底流动，内波比表面波更容易形成更大规模范围内的沉积床形。③内波可以形成大型沉积物波，用内波可以较合理地解释内波的对称波形单元、非对称波形单元及上攀波形单元的成因。波动面离海底距离较大的行进内波及内驻波可以形成对称波形的沉积物波；波动面离海底距离较近的行进内波及内孤立波可以形成非对称波形的沉积物波；内波引起的海底流动进一步增强时，可形成上攀波形沉积物波。④行进内波可以形成向内波传播相反方向迁移的沉积物波，向海盆内部传播的内波可以形成向上坡方向迁移的沉积物波。     

A general basis for yield,failure and potential functions in plasticity

A new concept based on the use of a function expressed as a (complete) polynomial expansion in terms of the three invariants of the stress tensor is proposed for deriving yield, failure and plastic potential functions for use in plasticity based constitutive laws. A mathematical interpretation and physical meaning of the proposed concept are provided by using the idea of the singular nature of constiutive matrices in incremental hypoelastic laws. It is suggested that the proposed function and (polynomial) forms of material moduli can be synonymous. A number of specialized forms of the general function are adopted and their values at failure from advanced three-dimensional tests for a number of (geological) media are evaluated. The results indicate the possibility that there exist invariant numbers associated with the functions(s) that may apply to a wide range of materials. Some ideas on implementation of the proposed concept are also presented.     

深水牵引流形成的床形单元组合

海底上发育深水牵引流形成的各种床形单元,包括等深岩丘及大型沉积物波在内的不同床形单元有规律地组合在一起,对于此类床形组合的确切形成机理,目前仍在探讨之中。本文经详细研究得出以下几点进展：① 运用内波理论可对海底上大型沉积物波各组成单元的成因作出较为合理的解释,向深海方向传播的内波可形成向上坡方向迁移的大型沉积物波;② 在等深流与上覆低密度水体之间的界面上具备产生大规模界面内波的条件;③ 某些底流成因的床形单元组合属于等深流与等深流所引发内波的联合作用的产物,首次提出了等深流－内波沉积组合的概念;④ 在现代海底上及古代地层记录中均发现了等深流－内波沉积组合的实例;⑤ 建立了一个古代地层记录中的深水牵引流沉积组合综合模式。     

Analytical Study of Underground Explosion-Induced Ground Motion

A theoretical approach is presented to study the ground motion induced by an underground tunnel explosion. The ground motion is caused by two coupled stress waves, i.e., the reflected body wave and the secondary surface wave or Raleigh wave. Based on the principle of conservation of momentum at the wavefronts, the reflected body waves along the ground surface are derived. The interaction of the body wavefront and the ground surface induces the secondary surface wave which transfers outwards on the ground. The particle velocity and particle acceleration on the ground surface are subsequently derived. The analytical results are compared with results from numerical simulations and empirical formulae with different material damping ratios. The effects of the loading density and the material damping on the ground motion are investigated. Finally, the limitations of the proposed theoretical approach for ground motion prediction are discussed.     

隧道岩爆的动力学机理及其控制的实验研究

隧道爆破开挖过程对隧道围岩的扰动,可以用弹性动力学的Lamb问题来描述。利用动态光弹性方法,对Lamb模型进行了验证,研究了冲击荷载下模型内应力波的传播、相互作用以及刻槽的拦截效应。结果表明：（1）冲击荷载作用下,模型内传播的有P波、S波和Rayleigh波,与Lamb模型的力学解答是一致的;（2）等差条纹图的分布规律,可以合理解释岩爆高发区段与开挖面的位置关系,认为P波和Rayleigh波的致裂作用和传播规律是岩爆发生的重要因素;（3）在模型表面的刻槽,可以实现对岩爆有突出贡献的Rayleigh波的大量拦截,说明在隧道表面切槽的岩爆控制方法是可行的。     

地下洞室群对地面运动影响问题的级数解答:SH波入射

采用波函数展开法给出平面SH波入射下半空间中洞室群对地面运动影响问题的一个级数解答，并分析了入射波频率和入射角以及两洞室之间距离等参数对地面运动的影响。数值结果表明，地下洞室群对沿线地面运动具有显著的放大作用：两个洞室情况地面运动可以达到单个洞室情况的1．6倍、无洞室情况的3．5倍以上。建议地铁等地下工程在设计和规划时应考虑工程建设后对沿线设计地震动的影响。     

分层密度流体的边界波动在不透水刚性底床的运动行为——Ⅰ造波条件对波型演化的关系

在一座二维内波实验水槽(12 m×0.7 m×0.5 m)、于总水深H等于40 cm的条件下,以淡、盐水依密度分层布置的两层流体,从事孤立内波生成的研究。内波水槽内的上下层水深比H₁/H₂、造波区位能差(η₀)、上下层的密度比皆为实验主要的控制变因。在各种不同实验条件配合之下,试验及比较以上变因对内波振幅能量及相关传递系数的影响,进而产生几种不同的波动特性:上举型孤立内波、下沉形孤立内波、非周期性的重力内波等,且不同形式的重力内波需要配合固定的造波条件使能达成。     

The empirical Melchior laws and parametric excitation of the Chandler wobble

Models of the motion of the terrestrial reference frame with respect to an inertial frame can describe the motions of the Earth-Moon system, which are traditionally separated into precession, nutation, the polar motion, and rotation about the Earth’s axis. Existing theoretical models do not describe variations in the Earth-orientation parameters with the required accuracy, so that the current polar coordinates and duration of the day must be determined from observations. To improve theoretical models for the time dependence of the coordinates of the Earth’s pole, we examine the possible excitation of the Chandler wobble due to internal properties of the Earth-Moon system. A differential equation describing the parametric resonance in the Earth-Moon system is obtained for the first region of the parametric excitation. The solution of this equation analytically describes the finite amplitudes of the nonlinear conservative system. The theoretical results are compared with the empirical laws of Melchior deduced from observational data on the coordinates of the Earth’s pole.     

Assessment of the major hazard potential of interfacial solitary waves moving over a trapezoidal obstacle on a horizontal plateau

Over the last few decades, a lot of attention has been concentrated on the consequences of marine impacts, especially those caused by the tsunami wave train. Internal solitary waves are similar to the surface waves that commonly occur in the waters of the ocean or large lakes and can have significant effects on oceanic mixing, climate change, the movement of submerged plankton, and the weathering of geological structures. This motion can be severe enough to create natural hazards such as submarine tsunamis in the ocean. These could also even occur in large lakes. The present work aims to contribute to this knowledge base by studying internal wave propagation on a shallow continental shelf following a particular marine impact. A series of laboratory experiments were conducted in order to clarify the movement of an interfacial solitary wave across a uniform slope and a horizontal plateau forming a slope-shelf topography. The results obtained from test runs indicate that the wave maintains its strength, having a direct impact on the natural ecology of the local oceanic environment. Comparison with different seabed topographies is also presented to demonstrate the propagation of an internal wave over a trapezoidal barrier. A better fitting and more appropriate model is employed to examine the relationship between the physical parameters for better predicting the evolution of an internal solitary wave as it moves over a trapezoidal obstacle on a horizontal plateau.     

Analysis of Stochastic Seismic Wave Interaction with a Slippery Rock Fault

Stochastic seismic wave interaction with a slippery rock fault is studied, based on the principle of conservation of momentum at the wave fronts along the fault. By using the displacement discontinuity method, the wave propagation equations are derived for incident longitudinal-(P-) and shear-(S-) waves, respectively. This is an extension of the study by Li and Ma (2010) for blast-induced wave propagation across a linear rock joint. Stochastic seismic waves are generated from a frequency spectrum and used to analyze the seismic wave interaction with a rock fault having a Coulomb-slip behavior. Parametric studies are carried out to investigate the effect of the intensity and impinging angle of the incident seismic waves on wave propagation across a slippery rock fault. Results show that the transmission of the incident P-wave is almost not affected by the fault, on the contrary, this is not the case for an incident S-wave, due to the occurrence of a relative slip which is related to the impinging angle of the incident S-wave. A quantitative study is presented which is of help in understanding the propagation and attenuation laws of seismic waves in discontinuous rock masses.     

材料的率相关与梯度耦合模型的二维内尺度律研究

利用率相关与梯度塑性耦合本构模型,采用简谐波的分析方法对材料的应变局部化及材料的稳定性进行了研究,得到了二阶耦合模型在二维情况下的内尺度律的变化及其与材料稳定性的关系,得到了波长变化的下界及材料稳定性的条件;在此基础上,结合二阶耦合模型在一维情况下的内尺度律变化的规律,对其进行了对比研究,得到材料变形局部化带在一维与二维条件下的关系公式,这对材料变形局部化带宽的研究有积极意义。     

The vortex mechanism of suppression of tsunami waves by underwater obstacles

A theoretical model explaining the effect of anomalous suppression of the energy (up to 70%) of tsunami-type waves by thin (compared to the wavelength) underwater obstacles is developed based on the integral laws of conservation of mass and energy fluxes. It is shown that the analytical dependences for the coefficients of reflection and transmission of waves across an underwater obstacle that have been obtained using the theoretical model proposed by the author agree with the results of the experiments and numerical simulation based on the complete Navier–Stokes equations.     

地下圆形衬砌隧道对沿线地震动的影响(I):级数解

利用波函数展开法,给出了地下圆形衬砌隧道对入射平面P波和SV波散射问题的一个级数解答,当衬砌与半空间介质相同时,该解答退化为文献中的无衬砌隧道的解答。级数解答为进一步定量研究隧道对入射平面P波和SV波的放大作用以及入射波长、入射角度、隧道直径和衬砌刚度等参数对隧道沿线地震动的影响奠定了理论基础。     

地下圆形衬砌隧道对沿线地震动的影响(Ⅰ):级数解

利用波函数展开法，给出了地下圆形衬砌隧道对入射平面P波和SV波散射问题的一个级数解答，当衬砌与半空间介质相同时，该解答退化为文献中的无衬砌隧道的解答。级数解答为进一步定量研究隧道对入射平面P波和SV波的放大作用以及入射波长、入射角度、隧道直径和衬砌刚度等参数对隧道沿线地震动的影响奠定了理论基础。