A method of time-dependent analysis using elastic solutions for nonlinear materials

The formulation of viscoelastic solutions from elastic equations using the ‘correspondence principle’ and an inverse Laplace transform has been discussed extensively in the literature. Because this method has been developed, many time-dependent solutions can be obtained from closed form elastic solutions and conditions have been delineated in which the ‘quasi-elastic’ approximation of the viscoelastic solution is within acceptable tolerance. This communication shows the feasibility of the application of these methods to formulate approximate nonlinear viscoelastic solutions with nonlinear stress-strain materials, and for want of a specific nonlinear model to demonstrate this, the hyperbolic model was selected. The ‘power law’ is used to model the relaxation modulus of the viscoelastic materials. There are five related development that are discussed here using a simple numerical example to illustrate each of them and they are: (1) a linear elastic solution, (2) a linear viscoelastic solution, (3) a nonlinear elastic solution, (4) a nonlinear viscoelastic solution and finally, (5) a ‘regression’ approximation of the nonlinear viscoelastic solution which is suggested by the series form of the elastic solution. All of these are related to one another and each provides an acceptably accurate solution of the problem it addresses. The latter is of particular practical interest since it can be used to provide answers to problems involving nonlinear viscoelastic materials while requiring only very small calculation times. The problem used as an example is the calculation of the displacement of a circular hole in an infinite plate made of a material with a nonlinear time-dependent stress-strain relationship. The nonlinear elastic form of the solution was developed by matching results from nonlinear finite element analysis.     

Low-frequency shear and structural relaxation in rhyolite melt

The frequency-dependence of the shear viscosity and modulus of rhyolite melt has been determined over a 10–14 log₁₀ Pa s viscosity range and a 0.03–63 rad s^–1 angular frequency range. The frequency-dependent viscosity determined at high frequencies is 5 orders of magnitude lower than the Newtonian viscosity. At lower frequencies, a frequency-independent viscosity identical with the Newtonian viscosity is observed. The measured shear modulus increases from zero to 30.5 ± 2.5 GPa with increasing frequency. The viscoelastic regime consists of a maximum in viscous loss centered on the Max-well relaxation time. The width and height of the loss modulus as a function of frequency is inconsistent with a single relaxation-time. The frequency-dependent shear modulus is best described by a distribution of relaxation-times with a sharp cutoff at times slightly longer than the Maxwell relaxation time, and a long tail at shorter times extending up to 5 orders of magnitude less than the Maxwell relaxation time. This distribution of relaxation-times is in contrast with the single-relaxation-time behavior observed in low viscosity silicate melts.     

Anelasticity and microcreep in polycrystalline MgO at high temperature: an exploratory study

 The frequency dependence of the shear modulus and dissipation in polycrystalline MgO has been determined at high temperature using both microcreep (ɛ = 10⁻⁴) and seismic frequency forced-oscillation (ɛ = 10⁻⁵) measurements. The frequency-dependent and time-dependent data have been described in terms of the elastic, anelastic and viscous components of deformation using the Andrade model. The forced-oscillation measurements show that for temperatures above 700 °C the shear modulus begins to decrease dramatically and the modulus becomes frequency-dependent with increasing temperature. This is accompanied by an increase in dissipation, which also becomes frequency-dependent. The microcreep measurements resolve this frequency-dependent behaviour into an anelastic regime from 700–1050 °C, and a viscoelastic regime from 1100–1300 °C. At 1300 °C, the seismic frequency shear wave speed is ∼60% of the extrapolated low-temperature frequency-independent value, and the dissipation has risen to Q ⁻¹ = 10⁻¹ from 10⁻³ at temperatures below 600 °C. The mechanism by which this frequency-dependent rheology occurs appears to be diffusional creep, which produces viscous slip on the grain boundaries. It is proposed that the anelastic behaviour is due to viscous slip occurring on segments of grain boundaries, with the viscous deformation being accommodated by elastic distortion of adjacent unslipped regions of the grain boundary. At higher temperatures, slippage occurs across the entire grain boundary and viscoelastic behaviour begins to occur. Received: 11 April 2002 / Accepted: 9 January 2003 Acknowledgements Samples were precision-ground by Andrew Wilson, and polished sections prepared by Harri Kokkonen, who also did the SEM work. Uli Faul calculated the volume fraction of grain sizes. The density measurements were done by Lara Weston.     

Dynamic response of an elastic and viscoelastic full-space to a spherical source

The dynamic response due to a spherical source of radius a embedded in an elastic and viscoelastic full-space is investigated at a distance R from the source. Previous solutions to the elastic case are extended to incorporate realistic source pressure functions. The elastic solution is then cast in a scale independent form in order to generalize the application. The results show that the near-field of the spherical source may be defined by R/a < 5. For this region the particle velocity and displacement decrease as R^?2, and the risetime decreases as R^?1. However. in the far-field region (R/a > 5) the particle velocity and displacement decrease as R^?1, and the risetime is independent of R. A non-constant Q model is developed to model viscoelastic attenuation and a complete analytical solution for wave propagation is obtained by cascading the separate mechanisms of geometric attenuation and viscoelastic attenuation. A comparison of our analytical model with the results of dynamic finite element modelling shows excellent agreement. This suggests that the method of cascading the separate transfer functions is a valid approach for wave propagation in viscoelastic media.     

On Preferential Flow, Channeling and Connectivity in Heterogeneous Porous Formations

This paper analyzes the emergence of channeling and preferential flow in heterogeneous porous media. Connectivity is studied through the statistical characterization of the length L of connected, high velocity patterns in both two-dimensional and three-dimensional media. A simple, physically based, fully analytic expression for the probability of L has been derived. It is found that the length L of connected, high velocity channels is flow-related and can be much larger than the conductivity integral scale I. Heterogeneity has a considerable impact on emergence of channeling patterns; connectivity is considerably enhanced in three-dimensional structures as compared to two-dimensional ones. The strong dependence on space dimensionality is a warning against the use of two-dimensional numerical models for assessing connectivity and preferential flow in heterogeneous media. The probability p(L) is employed in order to determine the early arrivals of the breakthrough curve at a given control plane; the simple model can be used for a preliminary assessment of preferential flow. Comparison with numerical simulations confirms that the main connectivity features were adequately captured by the model.     

黏弹性和速度各向异性对薄层反射的影响

煤层具有强波阻抗差、低阻抗的特性,对地震勘探来说是一种典型的薄层。因为其黏弹性和速度各向异性导致薄层反射波场异常,为了定量分析黏弹性和速度各向异性对薄层地震反射的影响,针对强波阻抗差薄层模型,基于波动方程数值模拟方法,正演得到弹性速度各向同性、黏弹性速度各向同性及黏弹性速度各向异性的地震波场,并对比分析了3种模型的弹性波地震反射特征。研究结果表明:黏弹性对薄层的反射产生不可忽略的影响,会明显降低薄层弹性波的反射振幅,但黏弹性速度各向同性与各向异性对薄层的弹性波反射影响差异不大。相比于弹性各向同性情况,黏弹性会引起反射纵波（PP波）瞬时频率最大可达8%的改变量,而对转换横波（PS波）仅有3%的改变量,可以忽略不计。上述研究为利用构造煤品质因子极低的特征来预测构造煤提供了可能。      

基于常Q模型的分数阶粘弹介质数值模拟方法

基于常Q模型的解耦分数阶拉普拉斯算子粘滞波动方程,可以分开模拟振幅衰减和相位错动。但该方程拉普拉斯算子的阶数是随空间变化的,因此数值求解存在一定困难。这里基于截断的泰勒展开,经过一系列近似,推导出拉普拉斯算子的阶数与空间无关的解耦分数阶粘滞弹性波动方程。采用中心差分计算时间导数,使用交错网格伪谱法计算空间导数。数值算例表明,新的方程在处理非均匀介质时具有精度高,计算简便的优点。     

Visco‐elastic load transfer models for axially loaded piles

Viscoelastic or creep behaviour can have a significant influence on the load transfer (t–z) response at the pile–soil interface, and thus on the pile load settlement relationship. Many experimental and theoretical models for pile load transfer behaviour have been presented. However, none of these has led to a closed‐form expression which captures both non‐linearity and viscoelastic behaviour of the soil. In this paper, non‐linear viscoelastic shaft and base load transfer (t–z) models are presented, based on integration of a generalized viscoelastic stress–strain model for the soil. The resulting shaft model is verified through published field and laboratory test data. With these models, the previous closed‐form solutions evolved for a pile in a non‐homogeneous media have been readily extended to account for visco‐elastic response. For 1‐step loading case, the closed‐form predictions have been verified extensively with previous more rigorous numerical analysis, and with the new GASPILE program analysis. Parametric studies on two kinds of commonly encountered loading: step loading, ramp (linear increase followed by sustained) loading have been performed. Two examples of the prediction of the effects of creep on the load settlement relationship by the solutions and the program GASPILE, have been presented. Copyright © 2000 John Wiley & Sons, Ltd.     

Theoretical and numerical analyses of chemical‐dissolution front instability in fluid‐saturated porous rocks

The chemical‐dissolution front propagation problem exists ubiquitously in many scientific and engineering fields. To solve this problem, it is necessary to deal with a coupled system between porosity, pore‐fluid pressure and reactive chemical‐species transport in fluid‐saturated porous media. Because there was confusion between the average linear velocity and the Darcy velocity in the previous study, the governing equations and related solutions of the problem are re‐derived to correct this confusion in this paper. Owing to the morphological instability of a chemical‐dissolution front, a numerical procedure, which is a combination of the finite element and finite difference methods, is also proposed to solve this problem. In order to verify the proposed numerical procedure, a set of analytical solutions has been derived for a benchmark problem under a special condition where the ratio of the equilibrium concentration to the solid molar density of the concerned chemical species is very small. Not only can the derived analytical solutions be used to verify any numerical method before it is used to solve this kind of chemical‐dissolution front propagation problem but they can also be used to understand the fundamental mechanisms behind the morphological instability of a chemical‐dissolution front during its propagation within fluid‐saturated porous media. The related numerical examples have demonstrated the usefulness and applicability of the proposed numerical procedure for dealing with the chemical‐dissolution front instability problem within a fluid‐saturated porous medium. Copyright © 2007 John Wiley & Sons, Ltd.     

Estimation of viscoelastic attenuation of real seismic data by use of ray tracing software: Application to the detection of gas hydrates and free gas

Analysis of P wave velocity profiles and seismic data recorded over the 2002 Hydratech cruise conducted in the Storegga region, North of Norway, has shown the existence of anomalies (a velocity decrease) in some layers of the medium. An elastic propagation model is not sufficient to explain clearly these anomalies, since the viscoelastic attenuation, represented by the quality factor Q_P, is sensitive to physical phenomena of geological media. The combination of the quality factor profile with the velocity profile leads to realistic explanations of these anomalies. In this article, we explain the procedure which we developed for determining the Q_P profile from the P wave velocity profile and the seismic data recorded during Hydratech cruise. Both the Q_P and velocity profiles indicate anomalies in the same layers. Based on previous studies, we interpret that these anomalies are being due to existence of gas hydrates and free gas within these layers.     

Seismic wave propagation in laterally inhomogeneous poroelastic media via BIEM

This work addresses in‐plane pressure P and vertically polarized shear SV seismic wave propagation in a finite, laterally inhomogeneous, multilayered poroelastic geological region resting on the homogeneous elastic half‐space. The particular approach followed here is based on a combination of the (i) viscoelastic approximation (isomorphism) to Biot's equations of dynamic poroelasticity and on the (ii) boundary integral equation method (BIEM) using frequency‐dependent fundamental solutions of the governing wave equations. The problem is formulated under plane strain conditions and time‐harmonic motions are assumed. Validation of the viscoelastic isomorphism and verification of the BIEM is done by solution of benchmark examples. These simulation studies reveal that the proposed methodology is able to depict a sensitivity of the seismic signals recovered to the following parameters: (i) poroelastic properties of fluid saturated layers; (ii) lateral geological inhomogeneity; (iii) surface topography and (iv) frequency content and direction of the incident wave. It is concluded that the combination of viscoelastic isomorphism with BIEM software provides an effective numerical tool for evaluating site‐effect phenomena in multilayered, fluid saturated geological regions with complex geometry. The numerical results obtained demonstrate that dynamic poroelasticity interacting with other physical peculiarities of the Earth's surface layers, such as lateral heterogeneity, material properties along the wave path, local geological profile and type of elastic wave, gives rise to complex seismic signals on the free surface at the site of interest. Copyright © 2010 John Wiley & Sons, Ltd.     

基于MPI的面波有限差分正演模拟

近年来,瑞利波波形反演技术因其避开了常规频散曲线计算,直接进行波场计算和反演不再受水平层状介质理论假设的限制,得到广大学者的高度重视。但瑞利波波形反演过程中需要不断进行波场正演和逆推计算。另外,由于浅地表速度较小,模拟计算时需要较小的网格间距才能避免数值频散,这无疑大大增加了正演模拟的计算量。对于这一问题,通常采用并行化设计来提高正演模拟的计算效率。本文基于消息传递接口（MPI）并行有限差分算法,以区域分解思路将模型区间分解成若干子区域,各区域互相通信,共同完成对模型的正演计算。并详细给出了区域分解、坐标转换、区域通信、波场合并等并行方案中的具体实现方法和实现步骤。通过对弹性模型、Kelvin黏弹性模型和标准线弹性固体（SLS）黏弹性模型不同并行方案的计算结果进行分析,验证了本文并行方案的可行性和有效性。并行计算结果表明,与单处理器计算时间相比,增加处理器数目可以明显减少计算时间,但随着处理器数目的增加,不同处理器之间的通信时间也增大;因此,并行时需要选择合适的处理器数目。对于黏弹性介质模型,SLS黏弹性模型的并行计算效率优于Kelvin黏弹性模型。     

Iterative analysis of pore‐dynamic models discretized by finite elements

This work proposes an iterative procedure to analyze dynamic linear/nonlinear fully saturated porous media considering time‐domain finite element discretization. In this iterative approach, each phase of the coupled problem is treated separately, uncoupling the governing equations of the model. Thus, simpler, smaller, and better conditioned systems of equations are obtained, rendering more attractive techniques. A relaxation parameter is introduced in order to improve the efficiency and robustness of the iterative solution, and an expression to compute optimal values for the relaxation parameter is discussed. At the end of the paper, numerical examples are presented, illustrating the effectiveness and potentialities of the proposed methodology. Copyright © 2013 John Wiley & Sons, Ltd.     

Enhanced Velocity Mixed Finite Element Methods for Flow in Multiblock Domains

The paper presents a new approach to discretizing flow in porous media via mixed finite element methods on non-matching multiblock grids. The velocity space along the interfaces is enhanced to give flux-continuous approximation. No additional matching conditions need to be imposed. The computational complexity of the resulting algebraic problem is comparable to the one for the single-block case. A priori error estimates for the pressure and the velocity and numerical experiments confirming the theory are presented.     

A note on the assumptions made while computing the postseismic lithospheric deformation

The postseismic lithospheric deformation is usually explained as viscoelastic relaxation of the coseismic stresses. In general, for computing the postseismic deformation, the shear modulus (μ) is relaxed, keeping either the bulk modulus (k) or the La’me parameter (A) fixed. It is shown that the two assumptions yield significantly different results. The assumptionk = const. implies that the medium behaves like an elastic body for dilatational changes which can be justified on physical grounds, but such a justification cannot be given in the case of the assumption λ = const.     

地基非线性波动问题中黏-弹性人工边界研究

建立了考虑地基动力非线性效应的波动模拟的二维和三维黏-弹性人工边界条件。引入考虑动力非线性特征的土体等效线性模型中的动模量变化模式,推导了新的平面内法向、平面内切向以及出平面切向的黏-弹性人工边界公式,同时也给出了新的三维法向、切向黏-弹性人工边界公式,并在实际应用中采用等效线性化方法处理,进行了数值算例分析,结果表明,新的黏-弹性人工边界具有更好的精度,可以用于地基非线性波动问题的研究。     

On the interface error analysis for finite difference wave simulation

A common way to construct a finite difference scheme is to satisfy a desired order of approximation, typically as high as possible. For linear wave propagation problem approximation together with stability delivers convergence of the same order as approximation. If a wave propagation proses is considered convergence to a plane wave solution can be derived analytically by means of the dispersion analysis. However, mentioned techniques are applicable only to homogeneous media and provide no knowledge of reflection/transmission coefficients. In this paper we prove that the only way to get second order accuracy of the solution for media with discontinuous parameters is to use a conservative finite difference scheme of the second order, and the only way to do this is to use the arithmetic mean for the density and the harmonic mean for the bulk modulus in the vicinity of the interface.     

粘弹性准饱和土中球空腔的动力响应

从工程实际出发,采用粘弹性两相介质模型,考虑土骨架的粘性以及流体与固体之间的耦合作用,利用Laplace变换求解了粘弹性准饱和土中球空腔的动力响应问题,得到了变换域内的解析解。借助数值Laplace反变换,数值分析了粘弹性准饱和土中球空腔动力响应的位移、应力及孔压的变化规律。为分析地下结构动力响应提供了一种有效的方法,模型符合工程实际,有一定的工程应用价值。     

Parameter identification of viscoelastic materials

Two parameter identification procedures for linear viscoelastic materials are presented. One is the method using the incremental constitutive relation for linear viscoelastic materials, and the other is the method using the elastic–viscoelastic correspondence principle. Part of back analysis in both methods is formulated based on the boundary control concept (Ichikawa Y, Ohkami T. A parameter identification procedure as a dual boundary control problem for linear elastic materials. Soils and Foundations 1992;32(2):35–44). Two numerical examples are presented to compare the efficiency of both methods.     

NON-LINEAR FAULT BEHAVIOUR NEAR UNDERGROUND EXCAVATIONS — A BOUNDARY ELEMENT APPROACH

A boundary element model for stress/stability analysis of underground excavations in the vicinity of faults is presented. The boundary element formulation adopts the fictitious stress method for the simulation of excavation boundaries and the displacement discontinuity method for the representation of faults. The numerical model employs the Barton–Bandis non-linear joint model for the modelling of the fault behaviour and linear elastic behaviour for the rock. An incremental-iterative in situ stress relaxation algorithm is implemented for the non-linear analysis of the faults. Both deformation and peak strength models of Barton–Bandis are incorporated for modelling the mechanical behaviour of the fault. The non-linear deformation of fault considers the effects of coupling between shear and normal stresses and displacement, joint closure, joint separation, hardening followed by post-peak or residual behaviour. The peak strength model employs a mobilized non-linear shear strength envelope. The differences between linear and non-linear simulation of the fault models are discussed. A comparison of model predictions with the classical Mohr–Coulomb peak strength model with constant joint stiffness is presented. The numerical model is used for a case study of Canadian hard rock underground mine. The shear and normal displacements along the fault during four mining sequences with backfill simulation are presented and discussed.