Effective medium description of the resonant elastic-wave response at the periodically-uneven boundary of a half-space

Effective medium description of the resonant elastic-wave response at the periodically-uneven boundary of a half-space

Armand Wirgin LMA, CNRS, UMR 7031, Aix-Marseille Univ, Centrale Marseille, F-13453 Marseille Cedex 13, France, (wirgin@lma.cnrs-mrs.fr)
September 29, 2019
Abstract

A periodically-uneven (in one horizontal direction) stress-free boundary covering a linear, isotropic, homogeneous, lossless solid half space is submitted to a vertically-propagating shear-horizontal plane, body wave. The rigorous theory of this elastodynamic scattering problem is given and the means by which it can be numerically solved are outlined. At quasi-static frequencies, the solution is obtained from one linear equation in one unknown. At higher, although still low, frequencies, a suitable approximation of the solution is obtained from a system of two linear equations in two unknowns. This solution is shown to be equivalent to that of the problem of a vertically-propagating shear-horizontal plane body wave traveling in the same solid medium as before, but with a linear, homogeneous, isotropic layer replacing the previous uneven boundary. The thickness of this layer is equal to the vertical distance between the extrema of the boundary uneveness and the effective body wave velocity therein is equal to that of the underlying solid, but the effective shear modulus of the layer, whose expression is given in explicit algebraic form, is different from that of the underlying solid, notably by the fact that it is dispersive and lossy. It is shown that this dispersive, lossy effective layer, overriding the nondispersive, lossless solid half space, gives rise to two distinctive features of low-frequency response: a Love mode resonance and a Fixed-base shear wall pseudo-resonance. This model of effective layer with dispersive, lossy properties, enables simple explanations of how the low-frequency resonance and pseudo-resonance vary with the geometric parameters (and over a wide range of the latter) of the uneven boundary.

Keywords: dynamic response, effective medium, boundary uneveness.

Abbreviated title: An effective layer that responds like a grating to a plane wave

Corresponding author: Armand Wirgin,
e-mail: wirgin@lma.cnrs-mrs.fr

Contents

1 Introduction

This study originated in the search (ongoing since the last sixty years) for simple explanations (largely-lacking until now) of the main features (amplitude and spectral properties) of seismic response (vibrations at locations near and/or above the ground) in above-sea level natural geophysical configurations such as individual, or groups of, ridges, hills, mountains, etc. [1, 2, 3, 4, 6, 7, 8, 10, 11, 13, 16, 18, 22, 25, 27, 28, 30, 32, 33, 36, 37, 39, 40, 44, 45, 51, 52]. Most of these studies have been either of experimental or numerical nature. The experimental studies usually concerned in situ configurations, so that considering the extreme diversity of the latter, there is little hope of deriving general laws from them. The numerical studies usually deal with periodic, small-height interface or boundary uneveness idealizations of the geophysical configurations (as well as of other elastic-wave devices [29, 41, 44]), and have not, either, given rise to satisfactory explanations of many (including low-frequency) features of the response.

Since earthquakes also affect man-made, above-ground, structures such as buildings, city blocks and even whole cities, which are, in a sense similar to small-scale hills or mountains, there has been some research on this question too [9, 19, 20, 21, 23, 36, 42, 46, 47, 49, 50]. Contrary to the studies concerning the natural uneven boundary configurations, some publications (e.g., [9, 36]), based on homogenization techniques resulting in the reduction of the uneven boundary to a flat boundary loaded by a periodic set of mass-spring oscillators (analogous to fixed-base shear walls (FBSW) [42]), have enabled to account for some of the main features (including what appear to be FBSW resonances) of long wavelength (with respect to the characteristic dimensions of the representative features as well as their separation) seismic response of the uneven boundary representing a city.

Homogenization of empirical nature is a very old and even contemporary practice in geophysical problems. At present, homogenization has developed into a full branch of applied mathematics [5, 15, 17, 24, 26, 31] which has attracted the attention of physicists (mostly solid-state) interested in developing materials with unusual properties [12, 15, 34, 38, 43, 48]. These so-called ’metamaterials’ are usually composed of periodic, or nearly- periodic assemblies of resonating elements similar to the blocks or buildings (thought of as single-degree-of-freedom oscillators) in [9]. A feature of homogenization, which is very useful as a predictive tool, is that it relies on (e.g., see the discussion of the NRW technique in [34], or results in [38, 43]) the equivalence, as regards low-frequency response to a wave, of the original geometrically-complex (although usually periodic) medium, boundary or interface to a geometrically-simpler, homogeneous medium or flat boundary. This is obtained at the expense of rendering more complex the constitutive properties of the media that are involved, but this complexity is precisely what accounts for the unusual (e.g., anomalous dispersion) response of the configuration.

Herein, we focus our attention on a very simple model of an uneven boundary (e.g., a mountain range, with periodicity along one horizontal coordiante) separating air from a solid underlying medium) submitted to a vertically propagating seismic wave. We show, by an effective medium method that is somewhat similar to the NRW technique [34], that this uneven boundary responds, as concerns its amplitude and spectral features at low (but beyond static) frequencies, in much the same manner as a homogeneous layer whose upper and lower horizontal faces correspond to the highest and lowest planes of the uneven boundary. In particular, we show theoretically, and verify numerically that the low-frequency resonant response of the uneven boundary is dominated by the excitation of what is similar to a Love mode and a fixed-base shear-wall mode. Finally, we give mathematically-explicit expressions for the constitutive parameters of the effective layer.

2 Exact solution of the problem of the response of the uneven boundary to the plane-wave solicitation

2.1 The boundary-value problem

In a cartesian coordinate system , with origin at , the uneven, on the average flat and horizontal, boundary separates two half-spaces, the upper one being occupied by the vacuum and the lower one being occupied by a homogeneous, isotropic, linear elastic solid. The uneveness of is periodic in terms of , oscillating between and with period , and does not depend on .

The elastic wave sources are assumed to be located in and to be infinitely-distant from so that the solicitation takes the form of a body (plane) wave in the neighborhood of . Its polarization is shear-horizontal () so that only one (i.e., the -) component of the incident displacement field is non-nil, i.e., , wherein and the angular frequency, the frequency.

Since neither the incident wavefield nor the geometric and compositional features of the configuration depend on , the total wavefield depends only on and , which means that the to-be-considered problem is 2D and can be examined in the sagittal plane.

Figure 1: Sagittal plane view of the periodically-uneven boundary (’grating’ for short) consisting of rectangular protrusions emerging from the ground plane / The medium below the boundary is a homogeneous, isotropic, elastic solid. The central protrusion domain is (width , height ), its left-hand neighbor is the domain and its right-hand neighbor is the domain , etc. The grating, of period , is solicited by a SH plane body wave whose wavevector (lying in the sagittal plane) makes an angle with the axis.

Fig. 1 depicts the problem in the sagittal plane in which: is the portion of below and the (composite) domain constituted by the remainder of , with the -th subdomain (henceforth termed ’protrusion’) of rectangular cross section (width and height ). ( the sagittal plane trace of .

The medium in is assumed to be non-dispersive over the range of frequencies of interest, and its shear modulus to be real. The shear-wave velocity in this solid is the real quantity , with the mass density.

The wavevector of the plane wave solicitation lies in the sagittal plane and is of the form wherein is the angle of incidence (see fig. 1), and . We shall assume in the last part of this study.

The total wavefield is and in is designated by . The incident wavefield is

(1)

wherein is the spectral amplitude of the solicitation.

The plane wave nature of the solicitation and the -periodicity of entails the quasi-periodicity of the field, whose expression is the Floquet condition

(2)

Consequently, as concerns the response in , it suffices to examine the field in .

The boundary-value problem in the space-frequency domain translates to the following relations (in which the superscripts and refer to the upgoing and downgoing waves respectively) satisfied by the total displacement field in :

(3)
(4)
(5)
(6)
(7)
(8)
(9)

wherein () denotes the first (second) partial derivative of with respect to . Eq. (4) is the space-frequency SH wave equation, (5)-(7) the stress-free boundary conditions, (8) the expression of continuity of displacement across the junction between the and the central block, and (9) the expression of continuity of stress across this junction.

Since is of half-infinite extent, the field therein must obey the radiation condition

(10)

2.2 Field representations via domain decomposition and separation of variables (DD-SOV)

As the preceding descriptions emphasize, it is natural to decompose the domain below the stress-free surface into the central protrusive domain above the ground and the half space domain beneath the ground.

Applying the SOV technique, The Floquet condition, and the radiation condition gives rise, in the lower domain, to the field representation:

(11)

wherein:

(12)
(13)

and, on account of (1),

(14)

with the Kronecker delta symbol.

In the central protrusion, the SOV, together with the free-surface boundary conditions (5), (7), lead to

(15)

in which

(16)
(17)

2.3 Exact solutions for the unknown coefficients

Eqs. (6) and (9) entail

(18)

which, on account of the SOV field representations and the identity

(19)

( is the Kronecker delta) yields

(20)

wherein

(21)

with sinc and sinc(0)=1.

Eq. (8) entails

(22)

which, on account of the SOV field representations, and the identity

(23)

with the Neumann symbol (=1 for and =2 for ), enables us to find

(24)

We thus have at our disposal two coupled expressions (i.e., (20) and (24) which make it possible to determine the two sets of unknowns , . Note that the number of members of each of these sets is infinite which is the fundamental source of complexity of the problem at hand and the principal reason why one should strive to simplify the theoretical analysis. This will be done in a later section.

2.4 Linear system for the set of unknown coefficients

Iinserting (20) into(24) yields, after the summation interchange, the system of linear equations:

(25)

wherein

(26)
(27)

Once the are determined they can be inserted into (20) to determine the , i.e.,

(28)

Until now everything has been rigorous provided the equations in the statement of the boundary-value problem are accepted as the true expression of what is involved in the elastic wave response of our grating and certain summation interchanges are valid. In order to actually solve for the sets and (each of whose populations is considered to be infinite at this stage) we must resort either to numerics or to approximations.

2.5 Numerical issues concerning the system of equations for

We strive to obtain numerically the set from the linear system of equations (25). Once this set is found, it is introduced into (20) to obtain the set . When all these coefficients (we mean those whose values depart significantly from zero) are found, they enable the computation of the elastic wave response (i.e., the displacement field) in all the subdomains of the configuration via (1), (3), (11), (15).

Concerning the resolution of the infinite system of linear equations (25), the procedure is basically to replace it by the finite system of linear equations

(29)

in which signifies that the series in is limited to the terms , and to increase so as to generate the sequence of numerical solutions , ,….until the values of the first few members of of these sets stabilize and the remaining members become very small (this is the so-called ’reduction method’ [35] of resolution of an infinite system of linear equations).

Note that to each is associated via (28), i.e.,

(30)

The so-obtained numerical solutions (it being implicit that and ), which for all practical purposes can be considered as ’exact’ for sufficiently-large (of the order of 25 for the range of frequencies and uneveness parameters considered herein) and which are in agreement with numerical results obtained by a finite element method [19, 21], constitute the reference by which we shall measure the accuracy of the approximate solutions of the next section.

From this point on, we assume that the plane wave is normally-incident (i.e., ) onto the uneven boundary . The consequences of this are

(31)

2.5.1 Dependence of reduction method solutions on for varying frequency and various

Figs. 2-4 tell us how the reduction method solutions evolve with for varying and various . In all these figures, , , , , , and the reference solutions (black) curves are obtained for .

Figure 2: The black curves represent the reference spectra and the red curves the approximate spectra . The upper (lower) panels correspond to the real (imaginary) parts of these functions. The left-hand, middle and right-hand panels are for respectively. Case , ,
Figure 3: Same as fig. 2 except that , .
Figure 4: Same as fig. 2 except that , .

A notable feature of these figures is that the larger is , the better is the agreement with the reference results (obtained for large ) at a given (especially close to resonant and/or high) frequency, or stated otherwise: the closer to a resonant or the higher the frequency, the greater must be for the -th order solution to agree with the reference solution. Note that the required value of is not a linear function of frequency .

2.5.2 Dependence of reduction method solutions on for varying frequency and various

Figs. 5-7 tell us how the reduction method solutions evolve with for varying and various . In all these figures, , , , , , and the reference solutions (black) curves are obtained for .

Figure 5: Same as fig. 2 except that , ,
Figure 6: Same as fig. 2 except that , .
Figure 7: Same as fig. 2 except that , .

A notable feature of these figures is that the larger is , the better is the agreement with the reference results (obtained for large ) at a given (especially close to resonant and/or high) frequency, or stated otherwise: the closer to a resonant or the higher the frequency, the greater must be for the -th order solution to agree with the reference solution. Note that the required value of is not a linear function of frequency .

3 The approximation of the response of the uneven boundary

When , the consequence of (25) is

(32)

whence

(33)

or (taking account of (31))

(34)

whence

(35)

A word is here in order about the possibility of resonance showing up in these response functions. Resonances, typically those associated with the excitation of Love modes [13], occur (i.e., at a discrete set of frequencies) when the denominator in the response functions are equal or very nearly equal to zero, therefore leading to infinite or very large response. For this to occur, while assuming, as we have done in this study that the medium in is lossless, would require that the term in the denominators of (34) and (35) be real and negative in relation to the term, but this is impossible because the factor multiplying is imaginary. It follows that the approximation of the uneven boundary response cannot account for resonant behavior, which fact was already observed in the numerical results presented in sects. 2.5.1-2.5.2. We shall return to this issue in sect. 4.5.

4 Relation of the approximate solution to the exact solution of another problem

The approximate solution (34)-(35) of the uneven boundary problem resembles the exact solution of a problem in which the uneven boundary is replaced by a flat-faced layer occupied by a linear, homogeneous, isotropic solid, the solicitation being the same (as regards the polarization of the plane body wave) as for the uneven boundary. To substantiate this assertion, we first derive the solution of the layer problem.

4.1 Description of the problem of the response to a plane wave of a homogeneous layer above a half space

The bottom flat face of the layer (in firm contact with the underlying solid medium) occupies the entire plane and the upper flat face of the layer occupies the entire plane. The half-space above the layer is occupied by the vacuum and the half-space below the layer by a linear, homogeneous, isotropic, lossless, non-dispersive solid.

The elastic wave solicitation is the same (i.e, as regards the polarization of the plane body wave) as in the uneven boundary problem. The wavefield associated with the elastic wave solicitation is .

Since neither the incident wavefield nor the geometric and compositional features of the configuration depend on , the total wavefield depends only on and , which means that the to-be-considered problem is 2D and can be examined in the sagittal plane.

Figure 8: Sagittal plane view of the configuration comprising a homogeneous layer in firm contact with the underlying homogeneous solid across . The half-space below is , the layer domain is (height ) and the half-space above the layer is . The configuration is solicited by a plane body wave whose wavevector (lying in the sagittal plane) makes an angle with the axis.

Fig. 8 depicts the problem in the sagittal plane in which: is the half-space domain occupied by a linear, homogeneous, isotropic, lossless, non-dispersive solid, the domain of the layer occupied by another linear, homogeneous, isotropic material which might be lossy and/or dispersive, and the half-space above the layer occupied by the vacuum.

The shear moduli of the lower medium and layer are and respectively, with positive real and generally-complex. The shear-wave velocities in the lower medium and layer are and respectively, with positive real and generally-complex.

The wavevector of the plane wave solicitation lies in the sagittal plane and is of the form wherein is the angle of incidence (see fig. 8), and .

The total wavefield in is designated by . The incident wavefield is

(36)

wherein is the spectral amplitude of the solicitation.

4.2 The boundary-value problem of the response of the layer/halfspace to a plane wave

The boundary-value problem in the space-frequency domain translates to the following relations satisfied by the total displacement field in :

(37)
(38)
(39)
(40)
(41)
(42)

4.3 DD-SOV field representations

Applying the DD-SOV technique, and the radiation condition gives rise, in the lower domain, to the field representation:

(43)

wherein:

(44)
(45)

In the layer, the SOV, together with the free-surface boundary condition (39), lead to

(46)

in which

(47)
(48)

4.4 Exact solution for the unknown coefficients

The introduction of the field representations into (40)-(41) yields the two equations

(49)
(50)

the exact solution of which is:

(51)
(52)

4.5 Comparison of the approximate uneven boundary problem solution to the exact layer problem solution

The comparison of (34)-(35) with (51)-(52) shows that the zeroth-order approximate solution of the uneven boundary problem is structurally-similar to the exact solution of the homogeneous layer problem. This suggests that the layer response is ’equivalent’ to the approximate uneven boundary response when

(53)

which, of course, implies

(54)

The translation of this equivalence is a series of relations between the parameters of the layer and their counterparts in the grating. The most obvious of these relations are three in number.

Henceforth, we shall assume which means that the solicitation in both problems is that of a normally-incident plane body wave.

4.5.1 First series of relations between the layer parameters and the grating parameters

Eq. (53) is satisfied provided:

(55)

Note that 1a)-1f) constitute explicit solutions for all six layer problem parameters. Also, note that all the parameters of the layer problem, just as those of the uneven boundary problem, do not depend on the frequency. It turns out (see figs. 9-11 hereafter and [47, 49]) that these effective media account very well for the response of the uneven boundary at very low frequencies, and in any case, before the onset of resonances,

4.5.2 Second series of relations between the layer parameters and the grating parameters

Eq. (53) is satisfied provided:

(56)

Note that now only 2a)-2e) constitute explicit solutions for five of the layer problem parameters whereas the obtention of requires solving a nonlinear equation for each frequency and the solution of this equation is not unique.

4.5.3 Third series of relations between the layer parameters and the grating parameters

Eq. (53) is satisfied provided:

(57)

Note that now only 3a)-3e) constitute explicit solutions for five of the layer problem parameters whereas the obtention of requires solving a nonlinear equation for each frequency and the solution of this equation is not unique.

4.5.4 Further comments on consequences of the approximation of grating response

If, for a reason to be evoked further on, one chooses one of the three solutions as a means of identifying some or all of the effective medium parameters (i.e., those denoted by upper-case letters), then he should be aware of the fact that these choices all derive from a approximation of the uneven boundary response, which, as shown previously in sect. 3, is only valid at very low frequencies and cannot account, by any means, for resonant behavior of the uneven boundary.

4.5.5 Numerical results for the effective medium response derived from the approximation of the uneven boundary response as well as from the effective shear modulus

Figs. 9-11 show how the approximate response spectra evolve with . In all these figures, , , , , , and the reference solutions blue curves are obtained for .

Figure 9: The upper and lower left-hand panels represent the real and imaginary parts respectively of obtained, via , from the approximation of . In the right hand panels: the black curves represent the reference spectra and the red curves the approximate spectra . The upper (lower) panels correspond to the real (imaginary) parts of these functions. Case , .
Figure 10: Same as fig. 9 except that , .
Figure 11: Same as fig. 9 except that , .

We observe in these figures that, as expected, the effective layer approximation of response obtained from does not account for the resonant behavior in the neighborhood of .

5 The approximation of uneven boundary response

Previous graphs (in sects. 2.5.1-2.5.2) show that there exists a similarity between the reference uneven boundary response and the approximation of this response even in the neighborhood of what appear to be resonances. We now try to exploit this numerical similarity by showing how it arises in theoretical manner. This will lead to the notion of an effective medium capable of accounting for a resonance associated with the excitation of a Love mode and to another resonant-like feature which we later qualify ’pseudo-resonance’.

The linear system that must be solved is

(58)

wherein

(59)
(60)

from which it follows that: