Module 2 WAVE PROPAGATION (Lectures 7 to 9)

Transcription

1 Module 2 WAVE PROPAGATION (Lectures 7 to 9) Lecture 9 Topics 2.4 WAVES IN A LAYERED BODY One-dimensional case: material boundary in an infinite rod Three dimensional case: inclined waves 2.5 ATTENUATION OF STRESS WAVES Material damping Radiation damping 2.4 WAVES IN A LAYERED BODY The model of homogeneous elastic half-space is useful for explaining the existence of body was and Rayleigh waves, and the addition of a softer surficial layer allows Love waves to be described. The earth, however, conditions are much more complicated with many different materials of variable thickness occurring in many areas. To analyze wave propagation under such conditions, and to understand the justification for idealizations of actual conditions when all features cannot be explicitly analyzed, the general problem of wave behavior at interfaces must be investigated One-dimensional case: material boundary in an infinite rod Consider a harmonic stress wave traveling along a constrained rod in the +x direction and approaching an interface between two different materials, as shown in (figure 2.14). Since the wave is traveling toward the interface, it will be referred to as the incident wave. Since it is traveling in material 1, its wavelength will be, and it can therefore be described by (2.66.a) When the incident wave reaches the interface, part of its energy will be transmitted through the interface to continue traveling in the positive x-direction through material 2. This transmitted wave will have a wavelength. The remainder will be reflected at the interface and will travel back through material 1 in the negative x-direction as a reflected wave. The transmitted and reflected waves can Dept. of Civil Engg. Indian Institute of Technology, Kanpur 1

2 be described by (2.66.b) (2.66.c) Figure 2.14: One-dimensional wave propagation at material interface. Incident and reflected waves travel in opposite directions in material 1. The transmitted wave travels through material 2 in the same directions as the incident wave Assuming that the displacements associated with each of these waves are of the same harmonic form as the stresses that cause them; that is, (2.67.a) (2.67.b) (2.67.c) Stress-strain and strain-displacement relationships can be used to relate the stress amplitudes to the displacement amplitudes: (2.68.a) (2.68.b) (2.68.c) From these, the stress amplitudes are related to the displacement amplitudes by (2.69.a) (2.69.b) (2.69.c) At the interface, both compatibility of displacements and continuity of stresses must be satisfied. The former requires that (2.70) Dept. of Civil Engg. Indian Institute of Technology, Kanpur 2

3 And the latter that (2.71) Substituting (equations 2.67 and 2.66 into equations 2.70 and 2.71) respectively indicates that at the interface. (2.72) (2.73) Substituting (equations 2.69 into equation 2.73) and using the relationship gives (2.74) (equation 2.74) can be rearranged to relate the displacement amplitude of the reflected wave to that of the incident wave: (2.75) And knowing, (equation 2.72) can be used to determine as (2.76) Remember that the produce of the density that the wave propagation velocity is the specific impedance of the material. (equations 2.75 and 2.76) indicate that the partitioning of energy at the interface depends only on the ratio of the specific impedances of the materials on either side of the interface. Defining the impedance ratio as, the displacement amplitudes of the reflected and transmitted waves are (2.77) (2.78) After evaluating the effect of the interface on the displacement amplitudes of the reflected and transmitted waves, its effect on stress amplitudes can be investigated from (equations 2.69). (2.79.a) (2.79.b) (2.79.c) Substituting (equations 2.79 into equations 2.77 and 2.78) and rearranging gives (2.80) (2.81) The importance of the impedance ratio in determining the nature of reflection and transmission at interfaces can clearly be seen. Dept. of Civil Engg. Indian Institute of Technology, Kanpur 3

4 (equations 2.77, 2.78, 2.80, and 2.81) indicate that fundamentally different types of behavior occur when the impedance ratio is less than or greater than 1. When the impedance ratio is less than 1, an incident wave can be thought of as approaching a softer material. For this case, the reflected wave will have a smaller stress amplitude than the incident wave and its sign will be reversed (an incident compression pulse will be reflected as a tensile pulse, and vice versa). If the impedance ratio is greater than 1, the incident wave is approaching a stiffer material in which the stress amplitude of the transmitted wave will be greater than that of the incident wave and the stress amplitude of the reflected wave will be less than, but of the same sign, as that of the incident waves. The displacement amplitudes are also affected by the impedance ratio. The relative stress and displacement amplitudes of reflected and transmitted waves at boundaries with several different impedance ratios are illustrated in (table 2.1). The cases of are of particular interest. An impedance ratio of zero implies that the incident wave is approaching a free end across which no stress can be transmitted. To satisfy this zero stress boundary condition the displacement of the boundary (the transmitted displacement) must be twice the displacement amplitude of the incident wave. The reflected wave has the same amplitude as the incident wave but is of the opposite polarity. In other words, a free end will reflect compression waves as a tension wave of identical amplitude and shape and a tension wave as an identical compression wave. An infinite impedance ratio implies that the incident wave is approaching a fixed end at which no displacement can occur. In that case the stress at the boundary is twice that of the incident wave and the reflected wave has the same amplitude and polarity as the incident wave. The case of in which the impedance on each side of the boundary are equal, is also of interest. (Equations 2.77, 2.78, 2.80 and 2.81) indicate the no reflected wave is produced and that the transmitted wave has, as expected, the same amplitude and polarity as the incident wave. In other words, all of the elastic energy of the wave crosses the boundary unchanged and travels away, never to return. Another way of looking at a boundary with an impedance ratio of unity is as a boundary between two identical, semi-infinite rods. A harmonic wave traveling in the positive x-direction (figure 2.15.a) would impose an axial force see (equation 2.5) on the boundary: Dept. of Civil Engg. Indian Institute of Technology, Kanpur 4

5 Figure 2.15 (a) Harmonic wave travelling along two connected semi-infinite rods; (b) semi-infinite rod attached to dashpot. With proper selection of dashpot coefficient, response in semi-infinite rod on left will be identical for both cases. This axial force is identical to that which would exist if the semi-infinite rod on the right side of the boundary were replaced by a dashpot (figure 2.15.b) of coefficient. In other words, the dashpot would absorb all the elastic energy of the incident wave, so the response of the rod on the left would be identical for both cases illustrated in (figure 2.15). This result has important implications for ground response and soil-structure interaction analyses where the replacement of a semi-infinite domain by discrete elements such as dashpots can provide tremendous computational efficiencies. Table2.1: Influence of Impedance Ratio on Displacement and Stress Amplitudes of Reflected and Transmitted Waves Impedanc e ratio Inciden t Displacement Amplitudes Stress Amplitudes Reflected Transmitted Incident Reflected Transmitted Example 4 A vertically propagating shear wave travels upward through a layered soil deposit. Compute the amplitudes of the reflected and transmitted waves that develop when the shear wave reaches the boundary shown in (figure 2.16). Dept. of Civil Engg. Indian Institute of Technology, Kanpur 5

6 Figure 2.16 Solution Although the transmission-reflection behavior in the preceding section was derived for constrained longitudinal waves, extension to the case of shear waves is straightforward. The (shear wave) impedance ratio for an upward-traveling wave is The stress amplitude of the reflected wave is given by (equation 2.79) From (equation 2.80) the stress amplitude of the transmitted wave is The displacement amplitude of the incident wave can be computed from the shear wave equivalent of (equation 2.78.a) The term simply describes the phase angle between stresses and displacements. Then, using (equations 2.76 and 2.77), the displacement amplitude of the reflected and transmitted waves are Dept. of Civil Engg. Indian Institute of Technology, Kanpur 6

7 In this example, the incident wave travels from a material of higher impedance to a material of lower impedance. As a result, the displacement amplitude of the transmitted wave is greater than that of the incident wave, but the stress amplitude is smaller Three dimensional case: inclined waves In general, waves will not approach interfaces at angles as they did in section The orientation of an inclined body wave can strongly influence the manner in which energy is reflected and transmitted across an interface. Fermat s principle defines the propagation time of a seismic pulse between two arbitrary points A and B as the minimum travel time along any continuous path that connects A and B. the path that produces the minimum travel time is called a ray path, and its direction is often represented by a vector called a ray. A wavefront is defined as a surface of equal travel time, consequently, a ray path must (in an isotropic material) be perpendicular to the wavefront as illustrate in (figure 2.17). Snell considered the change of direction of ray paths at interfaces between materials with different wave propagation velocities. Using Fermat s principle, Snell showed that where is the angle between the ray path and the normal to the interface and is the velocity of the wave (p-or s-wave) of interest. This relationship holds for both reflected and transmitted waves. It indicates that the transmitted wave will be reflected (except when ) when the wave propagation velocities are different on each side of the interface. Figure 2.17: Ray path, ray, and wave front for (a) plane wave and (b) curved wave front Consider the case of two half-spaces of different elastic materials in contact with each other. As for the pervious case, the requirements of equilibrium and compatibility and the theory of elasticity can be used to determine the nature of and distribution of energy among the reflected and transmitted waves for the cases of an incident p-wave, an incident SV-wave, and an incident SH-wave. The types of waves produced by incident p-, SV-, and SH-waves are shown in (figure 2.18). Since incident p- and SV-waves involve particle motion perpendicular to the plane of the interface; they will each produce both reflected and refracted p- and SV-waves. An incident SH-wave does not involve particle motion perpendicular to the interface; consequently, only SH-waves are reflected and refracted. The directions and relative amplitudes of the waves produced at the interface depend on both the direction and amplitude of the incident wave. Using Snell s law and the Dept. of Civil Engg. Indian Institute of Technology, Kanpur 7

8 requirements of equilibrium and compatibility, these directions and amplitudes can be determined. Using the notation of Richter (1958): the direction of all waves is easily related to the direction of the incident wave using Snell s law: Figure 2.18: Reflected and refracted rays resulting from incident (a) p-wave, (b) SV-wave, and (c) SH-wave Table2.2 Wave Type Velocity Amplitude Angle with Normal Incident p U A Incident s V B Reflected p U C Reflected s V D Refracted p Y E Refracted s Z F (2.82) Since incident and reflected waves travel through the same material, which shows that the angle of incidence is equal to the angle of angle of reflection for both p- and s-waves. The angle of refraction is uniquely related to the angle of incidence by the ratio of the wave velocities of the material on each side of the interface. Snell s law indicates that waves traveling from higher-velocity materials into lower-velocity materials will be refracted closer to the normal to the interfaces. In other words, wave propagating upward through horizontal layers of successively lower velocity (as is common near the earth s surface) will be refracted closer and closer to a vertical path (figure 2.19). This phenomenon is relied upon heavily by many of the methods of ground response analysis presented. Dept. of Civil Engg. Indian Institute of Technology, Kanpur 8

9 Figure 2.19 Refraction of an SH-wave ray path through series of successively softer (lower ) layers. Note that orientation of ray path becomes closer to vertical as ground surface is approached. Reflected rays are not shown The critical angle of incidence,, is defined as that which produces a refracted wave that travel parallel to the interface ( ). Therefore, (2.84) The concept of critical refraction is used in the interpretation of seismic refraction tests. Assuming that the incident wave is simple harmonic, satisfaction of the requirements of equilibrium and compatibility at the interface give rise to the following systems of simultaneous equations (Richter, 1958), which allows the amplitudes of the reflected and refracted waves (C, D,E and F) to be expressed in terms of the amplitude of the incident p-wave (A). (2.85) Where (the subscripts 1 and 2 refer to materials 1 and 2, respectively). Note that the amplitudes are functions of the angle of incidence, the velocity ratio, and the density ratio. (Figure 2.20) shows the variation of amplitude with angle of p- wave incidence for the following conditions;. The sensitivity of the reflected and refracted wave amplitudes Dept. of Civil Engg. Indian Institute of Technology, Kanpur 9

10 to the angle of incidence is apparent. SV-waves are neither reflected nor refracted at angles of incidence of 0 and, but can carry the majority of the wave energy away from the interface at intermediate angles. Figure 2.20 Ratio of amplitude of (a) reflected p-wave, (b) reflected SV-wave, (c) refracted p-wave and (d) refracted SV-wave to amplitude of incident p-wave versus angle of incidence For an incident SV-wave, both SV- and p-waves are reflected and refracted. The equilibrium/compatibility equations relating the relative amplitudes are which produce the amplitude behavior shown in (figure 2.21). Dept. of Civil Engg. Indian Institute of Technology, Kanpur 10

11 Figure 2.21 Ratio of amplitude of (a) reflected p-wave, (b) reflected SV-wave, (c) refracted p-wave and (d) refracted SV-wave to amplitude of incident SV-wave versus angle of incidence (2.86) For angles of incidence greater than about 36 in (figure 2.21.a) no p- wave can be reflected, so the incident wave energy must be carried away by the remaining waves. A more detailed discussion of this phenomenon can be found in McCamy et al. (1962). An incident SH-wave involves no particle motion perpendicular to the interface; consequently, it cannot produce p-waves ( ) or SV-waves. The equilibrium/compatibility equations are considerably simplified and easily solved as Dept. of Civil Engg. Indian Institute of Technology, Kanpur 11

12 (2.87) The preceding results show that the interaction of stress waves with boundaries can be quite complicated. As seismic waves travel away from the source of an earthquake, they invariably encounter heterogeneities and discontinuities in the earth s crust. The creation of new waves and the reflection and refraction of ray paths by these heterogeneities cause seismic waves to reach a site by many different paths. Since the paths have different lengths, the motion at the site is spread out in time by this scattering effect. 2.5 ATTENUATION OF STRESS WAVES The preceding sections have considered only the propagation of waves in linear elastic materials. In a homogeneous linear elastic material, stress waves travel indefinitely without change in amplitude. This type of behavior cannot occur, however, in real materials. The amplitudes of stress waves in real materials, such as those that comprise the earth, attenuate with distance. This attenuation can be attributed to two sources, one of which involves the materials through which the waves travel and the other the geometry of the wave propagation problem Material damping In real materials, part of the elastic energy of a traveling wave is always converted to heat. The conversion is accompanied by a decrease in the amplitude of the wave. Viscous damping, by virtue of its mathematical convenience, is often used to represent this dissipation of elastic energy. For the purposes of viscoelastic wave propagation, soils are usually modelled as Kelvin-Voigt solids (i.e., materials whose resistance to shearing deformation is the sum of an elastic part and a viscous part). A thin element of a Kelvin-Voigt solid can be illustrated as in (figure 2.22). Figure 2.22 Thin element of a Kelvin-Voigt solid subjected to horizontal shearing. Total resistance to shearing deformation is given by the sum of an elastic (spring) component and a viscous (dashpot) component Dept. of Civil Engg. Indian Institute of Technology, Kanpur 12

13 The stress-strain relationship for a Kelvin-Voigt solid in shear can be expressed as where is the shear stress, is the shear strain, and is the viscosity of the material. (2.88) Thus the shear stress is the sum of an elastic part (proportional to strain) and a viscous part (proportional to strain rate). For a harmonic shear strain of the form (2.89) The shear stress will be (2.90) Together, (equations 2.89 and 2.90) show that the stress-strain loop of a Kelvin- Voigt solid is elliptical. The elastic energy dissipated in a single cycle is given by the area of the ellipse, or which indicates that the dissipated energy is proportional to the frequency of loading. Real soils, however, dissipate elastic energy hysteretically, by the slippage of grains with respect to each other. As a result, their energy dissipation characteristics are insensitive to frequency. For discrete Kelvin-Voigt systems, the damping ratio,, was shown to be related to the force-displacement (or, equivalently, the stress-strain) loop as shown in (figure 2.23). Since the peak energy stored in the cycle is Figure 2.23 Relationship between hysteresis loop and damping ratio Then (2.91) To eliminate frequency dependence while maintaining the convenience of the voscoelastic formulation, (equation 2.91) is often rearranged to produce an equivalent viscosity that is inversely proportional to frequency. The use of this equivalent viscosity ensures that the damping ratio is independent of frequency: Dept. of Civil Engg. Indian Institute of Technology, Kanpur 13

14 (2.92) A Kelvin-Voigt solid for vertically propagating SH-waves may be represented by a stack of infinitesimal elements of the type shown schematically in figure The one-dimensional equation of motion for vertically propagating SH-waves can be written as (2.93) Substituting (equation 2.88 into 2.93) with and, and differentiating the right side allows the wave equation to be expressed as For harmonic waves, the displacements can be written as (2.94) (2.95) Which, when substituted into the wave (equation 2.94) yields the ordinary differential equation (2.96) Or (2.97) Where is the complex shear modulus? The complex shear modulus is analogous to the complex stiffness. Using (equation 2.92) to eliminate frequency dependence, the complex shear modulus can also be expressed as. This equation of motion has the solution where A and B depend on the boundary conditions and is the complex wave number. (2.98) It can be shown (after Kolsky, 1963) that is given by and only the positive of and the negative root of have physical significance. (2.99) (2.100) Note that for the inviscid case. For a wave Dept. of Civil Engg. Indian Institute of Technology, Kanpur 14

15 propagating in the positive z-direction, the solution can be written as which shows (since is negative) that material damping produces exponential attenuation of wave amplitude with distance. (2.101) Although the Kelvin-Voigt model is by far the most commonly used model for soils, it represents only one of an infinite number of rheological models. By rearranging and adding more springs and dashpots, many different types of behavior can be modeled, although the complexity of the wave equation solution increases dramatically as the number of springs and dashpots increases. Example 5 A harmonic plane wave with a period of 0.3 sec travels through a viscoelastic material ( ). Determine the distance over which the displacement amplitude of the plane wave would be halved. Solution From (equation 2.101) the displacement amplitude at is If represents the location at which the displacement amplitude is halved, then Which leads to Or [ ] For this case From equation (100) ( ) ( ) Dept. of Civil Engg. Indian Institute of Technology, Kanpur 15

16 2.5.2 Radiation damping Since material damping absorbs some of the elastic energy of a stress wave, the specific energy (elastic energy per unit volume) decrease as the wave travels through a material. The reduction of specific energy causes the amplitude of the stress wave to decrease with distance. The specific energy can also decrease by another common mechanism, which can be illustrated by the propagation of stress waves along an undamped conical rod. Consider the unconstrained conical rod of small apex angle shown in (figure 2.24) and assume that it is subjected to stress waves of wavelength considerably larger than the diameter of the rod in the area of interest. If the apex angle is sufficiently small, the normal stress will be uniform across each of two spherical surfaces that bound an element of width, and will act in a direction virtually parallel to the axis of the rod. Letting represent the displacement parallel to the axis of the rod, the equation of motion in that direction can be written, using exactly the same approach used, as Which simplifies to (2.102) (2.103) Figure 2.24 Conical rod of apex angle Substituting the stress-strain and strain-displacement relationships (assuming now that the ends of the element are planar) gives Or (2.104) ( )( ) (2.105) Which is the now-familiar wave equation. Its solution will be of the form [ ] (2.106) Dept. of Civil Engg. Indian Institute of Technology, Kanpur 16

17 Where. (equation 2.106) indicates that the amplitude of the wave will decrease with distance (even though the total elastic energy remains the same). The reduction is of purely geometric origin, resulting from the decrease in specific energy that occurs as the area of the rod increase. Even though elastic energy is conserved (no conversion to other forms of energy takes place), this reduction in amplitude due to spreading of the energy over a greater volume of material is often referred to as radiation damping (also as geometric damping and geometric attenuation). It should be distinguished from material damping in which elastic energy is actually dissipated by viscous, hysteretic, or other mechanisms. When earthquake energy is released from a fault below the ground surface, body waves travel away from the source in all directions. If the rupture zone can be represented as a point source, the wave fronts will be spherical and the preceding analysis can easily be extended to show that geometric attenuation causes the amplitude to decrease at a rate of. It can also be shown (Bullen, 1953) that geometric attenuation of surface waves causes their amplitudes to decrease at a rate of essentially ( ); in other words, surface waves attenuate (geometrically) much more slowly than body waves. This explains the greater proportion of surface wave motion (relative to body wave motion) that is commonly observed at large epicentral distances. This explains the advantages of the surface wave magnitude, relative body wave magnitude, for characterization of distant earthquakes. Dept. of Civil Engg. Indian Institute of Technology, Kanpur 17