Simulation of the propagation of a cylindrical shear wave : non linear and dissipative modelling (original) (raw)
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In this paper the effect of a nonstationary wave on cylindrical bodies with circular and rectangular cross sections is considered. The problem is solved in a flat setting, by a numerical method (FEM). Numerical results were obtained under the influence of the load by the single functions of Heuside. Introduction and statement of the problem. For modern engineering practice, the construction of underground structures is very important and important role is played by research and analysis of wave phenomena occurring in media with various inhomogeneities. The results obtained in this field are decisive for the development of methods for calculating the dynamic effects on structures and structures interacting with various types of ground media. However, in solving the problem posed, it is impossible to achieve significant progress without a deep theoretical analysis of them. The main provisions of the dynamic theory of seismic stability (DTS) are developed in the works [1,2,3] and others. These provisions are as follows [5, 6], a subterranean network of an arbitrary circuit is considered, consisting of elastic rods (pipelines, tunnel trunks) and high rigidity matching structures (observation wells, metro stations, etc.). The movement of the soil surrounding the pipeline during earthquakes is represented as a traveling wave of variable intensity. With this formulation of the problem, only the process associated with the oscillations of the pipeline in the ground is considered, without taking into account the volume of the oscillating soil mass [7]. This takes into account the ground support, friction slipping of the rods in the ground. In this formulation, the problem is solved with the help of a set of differential equations describing the vibrations of the rods, taking into account the dynamic and kinematic conditions for the coupling of the rods. Based on the calculation model described above, the effect of seismic waves on pipelines experiencing longitudinal oscillations has been investigated [8,9]. Among the most commonly used computed methods used in the calculation of underground pipelines, tunnel structures are the finite element method and grids. Variational-difference methods include the Bubnov-Galerkin method, the Ritz method, and the finite element method [10, 11]. Let us dwell on the latter, which has now found wide application for solving practical engineering problems. During the calculations, the calculation area was unevenly divided into rectangular and triangular finite elements. This breakdown was condensed as it approached the ground zone adjacent to the pipe. At present, there are well-developed software complexes for solving planar and spatial problems of linear and nonlinear theory of elasticity according to FEM [7,8,9,10,12,13]. Such problems can be solved on a multiply connected domain of any shape (the exception is the definition of a stress-strain state in a small neighborhood of a singular point). In a rectangular Cartesian coordinate system, a plane region in which a free circular (or square) hole is defined is considered. We consider reinforcement with a ratio of the diameter of the middle contour to the thickness, equal to ten. Before the start of the moment of rotation t = 0, the points of the considered mechanical system are at rest:
On the quasi-analytic treatment of hysteretic nonlinear response in elastic wave propagation
The Journal of the Acoustical Society of America, 1997
Microscopic features and their hysteretic behavior can be used to predict the macroscopic response of materials in dynamic experiments. Preisach modeling of hysteresis provides a refined procedure to obtain the stress-strain relation under arbitrary conditions, depending on the pressure history of the material. For hysteretic materials, the modulus is discontinuous at each stress-strain reversal which leads to difficulties in obtaining an analytic solution to the wave equation. Numerical implementation of the integral Preisach formulation is complicated as well. Under certain conditions an analytic expression of the modulus can be deduced from the Preisach model and an elementary description of elastic wave propagation in the presence of hysteresis can be obtained. This approach results in a second-order partial differential equation with discontinuous coefficients. Classical nonlinear representations used in acoustics can be found as limiting cases. The differential equation is solved in the frequency domain by application of Green's function theory and perturbation methods. Limitations of this quasi-analytic approach are discussed in detail. Model examples are provided illustrating the influence of hysteresis on wave propagation and are compared to simulations derived from classical nonlinear theory. Special attention is given to the role of hysteresis in nonlinear attenuation. In addition guidance is provided for inverting a set of experimental data that fall within the validity region of this theory. This work will lead to a new type of NDT characterization of materials using their nonlinear response.
Wave propagation simulation in a linear viscoelastic medium
Geophysical Journal International, 1988
A new formulation for wave propagation in an anelastic medium is developed. The phenomenological theory of linear viscoelasticity provides the basis for describing the attenuation and dispersion of seismic waves. The concept of a spectrum of relaxation mechanisms represents a convenient description of the constitutive relation of linear viscoelastic solids; however, Boltzmann's superposition principle does not have a straightforward implementation in time-domain wave propagation methods. This problem is avoided by the introduction of memory variables which circumvent the convolutional relation between stress and strain. The formulae governing wave propagation are recast as a first-order differential equation in time, in the vector represented by the displacements and memory variables. The problem is solved numerically and tested against. the solution of wave propagation in a homogeneous viscoelastic medium, obtained by using the correspondence principle.
Geophysical Journal International
In previous studies, the auxiliary differential equation (ADE) method has been applied to 2-D seismic-wave propagation modelling in viscoelastic media. This method is based on the separation of the wave propagation equations derived from the constitutive law defining the stress-strain relation. We make here a 3-D extension of a finite-difference (FD) scheme to solve a system of separated equations consisting in the stress-strain rheological relation, the strain-velocity and the velocity-stress equations. The current 3-D FD scheme consists in the discretization of the second order formulation of a non-linear viscoelastic wave equation with a time actualization of the velocity and displacement fields. Compared to the usual memory variable formalism, the ADE method allows flexible implementation of complex expressions of the desired rheological model such as attenuation/viscoelastic models or even non-linear behaviours, with physical parameters that can be provided from dispersion analysis. The method can also be associated with optimized perfectly matched layers-based boundary conditions that can be seen as additional attenuation (viscoelastic) terms. We present the results obtained for a non-linear viscoelastic model made of a Zener viscoelastic body associated with a non-linear quadratic strain term. Such non-linearity is relevant to define unconsolidated granular model behaviour. Thanks to a simple model, but without loss of generality, we demonstrate the accuracy of the proposed numerical approach.
Modeling Seismic Wave Propagation in 1D/2D/3D Linear and Nonlinear Media
2008
To analyze seismic wave propagation in geological structures, it is possible to consider various numerical approaches: the finite difference method, the spectral element method, the boundary element method, the finite element method, the finite volume method, etc. All these methods have various advantages and drawbacks. The amplification of seismic waves in surface soil layers is mainly due to the velocity contrast between these layers and, possibly, to topographic effects around crests and hills. The influence of the geometry of alluvial basins on the amplification process is also know to be large. Nevertheless, strong heterogeneities and complex geometries are not easy to take into account with all numerical methods. 2D/3D models are needed in many situations and the efficiency/accuracy of the numerical methods in such cases is in question. Furthermore, strong earthquakes implies nonlinear effects in surficial soil layers. To model strong ground motion, it is then necessary to consider the nonlinear dynamic behaviour of soils and simultaneously investigate seismic wave propagation in complex 2D/3D geometries ! Recent advances in numerical formulations and constitutive models in such complex situations are presented and discussed in this paper. Finally, a crucial point concerns the availability of the field/laboratory data to feed such models.
On the rheological models used for time-domain methods of seismic wave propagation
Geophysical Research Letters, 2005
1] After publications by Emmerich and and authors who implemented realistic attenuation in the time-domain methods decided for either of two rheological models -generalized Maxwell body (as defined by Emmerich and Korn) or generalized Zener body. Two parallel sets of papers and mathematical formalisms developed during the years. We have not found any comments on the other rheology. Therefore, we review both models and show that, in fact, they are equivalent. We also derive material-independent anelastic functions. Citation: Moczo, P., and J. Kristek On the rheological models used for time-domain methods of seismic wave propagation, Geophys. Res. Lett., 32, L01306,
Universal Journal of Mechanical Engineering, 2018
A theoretical study of seismic waves propagation in a soil layer with a free surface has a great importance for a prediction in engineering decisions. Wave packets are radiated from an earthquake source and transfer energy. A transformation and a selection of wave packets occur in a process of wave propagating that why waves which arrive in a layer have a length considerably greater than a variation scale of heterogeneity in a medium in a layer near free surface. In the case, when the properties of different layers affect a relatively small degree on a behavior of the waves, an approximation of effective medium gives a fairly good solution. A model of a hypoplastic medium is used for a describing of some effects, which are observed in the time of seismic wave propagation. The model of hypoplastic medium allows describing many effects which are observed in granular soils. We consider a successive application of effective medium and ray methods in order to receive of approximate analytical solutions wishing to describe shear wave propagation in stratified layer, which lies on a half-space.
Modeling Seismic Wave Propagation and Amplification in 1D/2D/3D Linear and Nonlinear Unbounded Media
International Journal of Geomechanics, 2011
To analyze seismic wave propagation in geological structures, it is possible to consider various numerical approaches: the finite difference method, the spectral element method, the boundary element method, the finite element method, the finite volume method, etc. All these methods have various advantages and drawbacks. The amplification of seismic waves in surface soil layers is mainly due to the velocity contrast between these layers and, possibly, to topographic effects around crests and hills. The influence of the geometry of alluvial basins on the amplification process is also know to be large. Nevertheless, strong heterogeneities and complex geometries are not easy to take into account with all numerical methods. 2D/3D models are needed in many situations and the efficiency/accuracy of the numerical methods in such cases is in question. Furthermore, the radiation conditions at infinity are not easy to handle with finite differences or finite/spectral elements whereas it is explicitely accounted in the Boundary Element Method. Various absorbing layer methods (e.g. F-PML, M-PML) were recently proposed to attenuate the spurious wave reflections especially in some difficult cases such as shallow numerical models or grazing incidences. Finally, strong earthquakes involve nonlinear effects in surficial soil layers. To model strong ground motion, it is thus necessary to consider the nonlinear dynamic behaviour of soils and simultaneously investigate seismic wave propagation in complex 2D/3D geological structures! Recent advances in numerical formulations and constitutive models in such complex situations are presented and discussed in this paper. A crucial issue is the availability of the field/laboratory data to feed and validate such models.
A review of the finite-element method in seismic wave modelling
crewes.org
Numerical solutions of the scalar and elastic wave equations have greatly aided geophysicists in both the forward modelling and migration of seismic wavefields in complicated geologic media. In P-and S-wave propagation, the finite-element method is a powerful tool for determining the effect of structural irregularities on wave propagation. Dependence of the wave equation on both spatial and temporal differentials requires solving both spatial and temporal discretization. In the spatial discretization step in 1D and 2D, piecewise linear basis functions and the Galerkin method are the most commonly used tools. After solving spatial discretization with the finite-element method, the wave equation reduces to an ODE (ordinary differential equation). In this regard, different authors used different ODE solver including Runge-Kutta method and finite-difference method. This paper will familiarize the reader with the diverse approaches of solving temporal discretization. An application of finite-element methods to solve seismic wave motion in linear viscoelastic media (where inelastic strains development depends not only on the current state of the stress and strain but on the full history of their development), using memory variable formalism in spatial discretization step, is one of the reviewed sections.