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Product Details
| ISBN-13: | 9780521375993 |
|---|---|
| Publisher: | Cambridge University Press |
| Publication date: | 07/21/2011 |
| Edition description: | Reissue |
| Pages: | 262 |
| Sales rank: | 776,459 |
| Product dimensions: | 7.01(w) x 10.00(h) x 0.55(d) |
About the Author
Eduardo Kausel earned his first professional degree in 1967, graduating as a Civil Engineer from the University of Chile and then worked at Chile's National Electricity Company. In 1969 he carried out post-graduate studies at the Technical University in Darmstadt. He earned his Master of Science (1972) and Doctor of Science (1974) degrees from MIT. Following graduation, Dr Kausel worked at Stone and Webster Engineering Corporation in Boston, and then joined the MIT faculty in 1978, where he has remained since. He is a registered Professional Engineer in the State of Massachusetts, is senior member of various professional organizations (ASCE, SSA, EERI, IACMG), and has extensive experience as consulting engineer. Among the honors he has received are a 1989 Japanese Government Research Award for Foreign Specialists from the Science and Technology Agency; a 1992 Honorary Faculty Membership in Epsilon Chi, the 1994 Konrad Zuse Guest Professor at the University of Hamburg in Germany, the Humboldt Prize from the German Government in 2000, and the 2001 MIT-CEE Award for Conspicuously Effective Teaching. Dr Kausel is best known for his work on Dynamic Soil-Structure Interaction, and for his very successful Green's functions (fundamental solutions) for the dynamic analysis of layered media, which are incorporated in a now widely used program. Dr Kausel is the author of over hundred and fifty technical papers and reports in the areas of structural dynamics, earthquake engineering, and computational mechanics.
Read an Excerpt
Cambridge University Press
0521855705 - Fundamental Solutions in Elastodynamics - A Compendium - by Eduardo Kausel
Excerpt
SECTION I: PRELIMINARIES
1 Fundamentals
1.1 Notation and table of symbols
Except where noted, the following symbols will be used consistently throughout this work:
| a = β∕α | Ratio of S- and P-wave velocities |
| b = bjêj | Body load vector |
| CR | Rayleigh-wave velocity |
| Cn | 3×3 Bessel matrix, cylindrical coordinates and flat layers (see Table 10.2) |
| gj = gijêi | Green's function vector for the frequency domain response due to a unit load in direction j |
| Fn | 3×3 traction matrix, cylindrical coordinates (see Table 10.4) |
| Fn(1), Fn(2) | As Fn above, assembled with first and second Hankel functions |
| F | Fourier transform operator |
| F-1 | Inverse Fourier transform operator |
| gij | Green's function for the frequency domain response in direction i due to a unit load in direction j |
| gij,k = ∂gij∕∂xk | Derivative with respect to the receiver location |
| g′ij,k = ∂gij∕∂x′k | Derivative with respect to the source location |
| Gij | Green's function for the frequency domain response due to a dipole |
| hn(1)(kR), hn(2)(kR) | First and second spherical Hankel functions of order n |
| Hn(1)(kr), Hn(2)(kr) | First and second Hankel functions of order n (Bessel functions of the third kind) |
| Hn | 3×3 displacement matrix, cylindrical coordinates (see Table 10.3) |
| Hn(1), Hn(2) | As Hn above, assembled with first and second Hankel functions |
| Hm | 3×3 spherical Bessel matrix, spherical coordinates (see Table 10.7) |
| 0 t < tα | |
| H (t - tα) = { 1/2 t = tα | Unit step function, or Heaviside function |
| 1 t ≥ tα | |
| i = √-1 | Imaginary unit (non-italicized) |
| i, j, k | Sub-indices for the numbers 1, 2, 3 or coordinates x, y, z |
| î, ĵ k̂ ≡ ê1, ê2, ê3 | Orthogonal unit basis vectors in Cartesian coordinates |
| Jn(kr), Yn(kr) | Bessel functions of the first and second kind |
| J | 3×3 spherical orthogonality condition (Section 9.3, Table 10.6) |
| k | Radial wavenumber |
| kP = ω∕α | P wavenumber |
| kS = ω∕β | S wavenumber |
| kz | Vertical wavenumber |
| kp = √k2 - kP2 | Vertical wavenumber for P waves, flat layers |
| ks = √k2 - kS2 | Vertical wavenumber for S waves, flat layers |
| kp = √kP2 - kz2 | Radial wavenumber for P waves, cylindrical layers |
| kp = √kS2 - kz2 | Radial wavenumber for S waves, cylindrical layers |
| Lmn | 3×3 Spheroidal (co-latitude) matrix (see Tables 10.7, 10.8 ) |
| M | Intensity of moment, torque, or seismic moment |
| Mij | Displacement in direction i due to seismic moment with axis j. |
| p | Load vector |
| p̃ | Load vector in frequency-wavenumber domain |
| p | As above, but vertical component multiplied by -i = -√-1 |
| p | Pressure (positive when compressive) |
| p = √1 - (kP/k)2 | Dimensionless vertical wavenumber for P waves |
| P | Load amplitude |
| Pm | Legendre function (polynomial) of the first kind |
| Pmn | Associated Legendre function of the first kind |
| Qmn | Associated Legendre function of the second kind |
| R | Source-receiver distance in 3-D space |
| r | Source-receiver distance in 2-D space, or range |
| r, θ, z | Cylindrical coordinates (see Fig. 1.2) |
| ȓ, t̑, k̑ | Orthogonal unit basis vectors in cylindrical coordinates |
| ȓ, ŝ, t̑ | Orthogonal unit basis vectors in spherical coordinates |
| R, ɸ, θ | Spherical coordinates (see Fig. 1.3) |
| s = √1 - (kS / k)2 | Dimensionless vertical wavenumber for S waves |
| t | Time |
| tP = r/α | P-wave arrival time |
| tS = r/β | S-wave arrival time |
| tR = r/CR | Rayleigh-wave arrival time |
| T | Torque |
| Tn | Azimuthal matrix |
| Tij | Displacement in direction i due to a unit torque with axis j |
| u | Displacement vector |
| ũ | Displacement vector in frequency-wavenumber domain |
| u | As above, but vertical component multiplied by -i = -√-1 |
| ux, uy, uz ≡ u1, u2, u3 | Displacement components in Cartesian coordinates |
| ur, uθ, uz ≡ u, v, w | Displacement components in cylindrical coordinates |
| uR, uɸ, uθ | Displacement components in spherical coordinates |
| uxz or uij, etc. | Displacement in direction x (or i) due to force in direction z (or j) |
| Uij, uij | Green's funciton for the time domain response in direction i due to a unit load in direction j |
| x, y, z ≡ x1, x2, x3 | Cartesian coordinates (see Fig. 1.1) |
| x′, y′, z′ | Cartesian coordinates of the source |
| α = β√2(1 - ν)/(1 - 2ν) | P-wave velocity |
| β = √μ/ρ | S-wave velocity |
| γi | Direction cosine of R with ith axis (see Cartesian coordinates) |
| 0, t < tS | |
| δ(t - tS) = dH(t - tS) / dt = { ∞, t = tS Dirac-delta singularity function | |
| 0, t ≥ tS | |
| δij = { 1, i = j | Kronecker delta |
| 0, i ≠ j | |
| λ | Lamé constant |
| λ + 2μ = ρ α2 | Constrained modulus |
| μ = ρβ2 | Shear modulus |
| ν | Poisson's ratio |
| τ = tβ∕r = t∕tS | Dimensionless time |
| Ρ | Mass density |
| θ | Azimuth in cylindrical and spherical coordinates |
| θi | Angle between R and the ith axis (γi = cos θi), i = 1, 2, 3 |
| Φ | Dilatational Helmholtz potential |
| Χ = Χ(ω) | Dimensionless component function of Green's functions (in later chapters, a Helmholtz shear potential) |
| X = X(t) | Inverse Fourier transform of Χ(ω) (or the convolution of the latter with an arbitrary time function) |
| ψ = ψ(ω) | Dimensionless component function of Green's functions (in later chapters, a Helmholtz shear potential) |
| Ψ = Ψ(t) | Inverse Fourier transform of ψ(ω) (or the convolution of the latter with an arbitrary time function) |
| Ψ | Helmholtz vector potential (shear) |
| ω | Frequency (rad/s) |
| ΩS = ωr∕β = ωtS | Dimensionless frequency for S (shear) waves |
| ΩP = ωr∕α = ωtP | Dimensionless frequency for P (dilatational) waves |
Figure 1.1: Cartesian coordinates.
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Figure 1.2: Cylindrical coordinates.
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Figure 1.3: Spherical coordinates.
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1.2 Sign convention
Component of vectors, such as displacements and forces, are always defined positive in the positive coordinate directions, and plots of displacements are always shown upright (i.e., never reversed or upside down).
Point and line sources will usually - but not always - be located at the origin of coordinates. When this is not the case, it will be indicated explicitly.
Wave propagation in the frequency-wavenumber domain will assume a dependence of the form exp i(ωt - kx), that is, the underlying Fourier transform pairs from frequency-wavenumber domain to the space-time domain are of the form
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Important consequences of this transformation convention concern the direction of positive wave propagation and decay, and the location of poles for the dynamic system under consideration. These, in turn, relate to the principles of radiation, boundedness at infinity, and causality. Also, this convention calls for the use of second (cylindrical or spherical) Hankel functions when formulating wave propagation problems in infinite media, either in cylindrical or in spherical coordinates, and casting them in the frequency domain.
1.3 Coordinate systems and differential operators
We choose Cartesian coordinates in three-dimensional space forming a right-handed system, and we denote these indifferently as either x, y, z or x1, x2, x3. In most cases, we shall assume that x = x1 and y = y2 lie in a horizontal plane, and that z = x3 is up. For two-dimensional (plane strain) problems, the in-plane (or SV-P) components will be contained in the vertical plane defined by x and z (i.e., x1, x3), and the anti-plane (or SH) components will be in the horizontal direction y (or x2), which is perpendicular to the plane of wave propagation. On the one hand, this convention facilitates the conversion between Cartesian and either cylindrical or spherical coordinates; on the other, it provides a convenient x-y reference system when working in horizontal planes (i.e., in a bird's-eye view). Nonetheless, you may rotate these systems to suit your convenience.
1.3.1 Cartesian coordinates.
a) Three-dimensional space (Fig. 1.1a)
Source-receiver distance
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Direction cosines of R
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First derivatives of direction cosines
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Second derivatives of direction cosines
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Implied summations
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Nabla operator
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Gradient of vector
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where the products of the form î î, etc., are tensor bases, or dyads, that is, ∇u is a tensor. For example, two distinct projections of this tensor are
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Divergence
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Curl
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Curl of curl
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Laplacian
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Wave equation
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b) Two-dimensional space (x-z), (Fig. 1.1b)
Source-receiver distance
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Direction cosines of r
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Implied summations
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1.3.2 Cylindrical coordinates.
Source-receiver distance
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Range
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Azimuth
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Direction cosines
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Basic vectors
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Conversion between cylindrical and Cartesian coordinates
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Nabla operator
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Allowing the symbol ⊗ to stand for the scalar product, the dot product, or the cross product, and considering that
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we can write a generic nabla operation on a vector u in cylindrical coordinates as
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Specializing this expression to the scalar, dot, and cross products, we obtain
Gradient
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where the products of the form ȓȓ, etc., are tensor bases, or dyads.
Divergence
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Curl
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Curl of curl
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Laplacian
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(Note: ∂2/∂θ2 in ∇2 acts both on the components of u and on the basis vectors ȓ, t̑.)
Wave equation
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Expansion of a vector in Fourier series in the azimuth
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in which u ≡ ur , v ≡ uθ , w ≡ uz, and either the lower or the upper element in the parentheses must be used, as may be necessary. Also, un, vn, wn, are the coefficients of the Fourier series, which do not depend on θ, but only on r and z, that is, un = un(r, z), and so forth.
1.3.3 Spherical coordinates.
Source-receiver distance
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Range
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Azimuth
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Polar angle
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Direction cosines
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Basic vectors
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Conversion between cylindrical and Cartesian coordinates
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Nabla operator
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Allowing the symbol ⊗ to stand for the scalar product, the dot product, or the cross product, and considering that
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we can write a generic nabla operation on a vector u in spherical coordinates as
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