Theory of Seismic waves I. Elasticity

Theory of elasticity Seismic waves are stress (mechanical) waves that are generated as a response to acting on a material by a force. The force that generates this stress comes from a source of seismic energy such artificial (Vibroseis, dynamite, ... etc) or natural earthquakes. The stress will produce strain (deformation) in the material relating to elasticity theory. Therefore, we need to study a little bit of elasticity theory in order to better understand the theory of seismic waves.

Stress Stress, denoted by , is force per unit area, with units of pressure such as Pascal (N/m2). xx denotes a stress produced by a force that is parallel to the x-axis acting upon a surface (YZ plane) which is perpendicular to the x-axis. xy denotes a stress produced by a force that is parallel to the x-axis acting upon a surface (XZ plane) which is perpendicular to the y-axis.

Stress There should be a maximum of 9 stress components associated with every possible combination of the coordinate system axes (xx, xy, xz, yx, yy, yz, zx, zy, zz). According to equilibrium (body is not moving but only deformed as a result of stress application): ij = ji, meaning that xy = yx, yz = zy, and zx = xz. If the force is perpendicular to the surface, we have a normal stress (xx, yy, zz); while if it’s tangential to the surface, we have a shearing stress (xy, yz, xz).

Stress The stress matrix composed of nine components of the stress:

Strain Strain, denoted by , is the fractional change in a length, area, or volume of a body due to the application of stress. For example, if a rod of length L is stretched by an amount L, the strain is L/L. As a matter of fact, strain is dimensionless.

Strain To extend this analysis to three dimensional case, consider a body with dimensions of X, Y, and Z along the x-, y-, and z-axes respectively. To extend this analysis to three dimensional case, consider a body with dimensions of X, Y, and Z along the x-, y-, and z-axes respectively. If the body is subjected to stress, then generally X will change by an amount of u(x,y,z), Y by an amount of v(x,y,z), and Z by an amount of w(x,y,z)

Strain There are generally 9 strain components corresponding to the 9 stress components (xx, xy, xz, yx, yy, yz, zx, zy, zz) because of equilibrium: ij = ji, meaning that xy = yx, yz = zy, and zx = xz. We can define the following strains: Normal strains ( ) Shear strains ( )

Strain Dilatation () is known as the change in volume (V) per unit volume (V): The strain matrix composed of the nine components of strain:

Components of stress and strain Lectures 04-05 Components of stress and strain If a stretching force is acting in the x-y plane and the corresponding motion is only occurred in the direction of x- axis, we will have the situation depicted in the corresponded figure. The point P moves a distance u to point P’ after stretching while point Q moves a distance ux+ux to point Q’. y x P Q P’ Q’ ux x

Ask students to do similar processing for yy and zz. Lectures 04-05 Normal Strain As we know that normal strain in x- direction is know as the ratio between the change of length of QP to the original length of QP y x P Q P’ Q’ ux x Coordinates P(x,y) Q(x+x,y) P’(x+u,y) Ask students to do similar processing for yy and zz.

Shear Strain If a stretching force is acting in the x-y plane and the corresponding motion is induced either in the direction of x- axis and y-axis, we will have the situation depicted in the corresponded figure. The infinitesimal rectangular PQRS will have displaced and deformed into the diamond P’Q’R’S’. After stretching, points P, Q, S and R move to P’, Q’, S’, and R’ with coordinates. y x P Q P’ Q’ ux x S R S’ R’ uy y

Shear Strain y x P Q P’ Q’ ux x S R S’ R’ uy y Coordinates P(x,y) P’(x+ux,y+uy) Q(x+x,y) S(x,y+y) The deformation in y coordinates in relative to x-axis is given by Ask students to substitute the coordinates of points P, Q, P’, and Q’ to get the shear-strain component in the x-y plane

Hook’s law It states that the strain is directly proportional to the stress producing it. The strains produced by the energy released due to the sudden brittle failure. The energy is released in the form of of seismic waves in earth materials are such that Hooke’s law is always satisfied.

Hook’s law A deformation is the change in size or shape of an object. An elastic object is one that returns to its original size and shape after the act forces have been removed. If the forces acting on the object are too large, the object can be permanently distorted based on its physical properties.

Hook’s law Mathematically, Hooke’s law can be expressed as: where  is the stress matrix,  is the strain matrix, and C is the elastic-constants tensor, which is a fourth-order tensor consisting of 81 elastic constants (Cxxxx to Czzzz).

Hook’s law For example:

Hook’s law In general, the stiffness matrix consists of 81 independent entries

Hook’s law Because σij = σji, there are only 6 independent components in the stress and strain matrices. This means that the elastic-constant tensor decreases to 54 elastic constants.

Hook’s law Because εij = εji, there are only 6 independent components in the stress and strain matrices. This means that the elastic-constant tensor decreases to 36 elastic constants. Moreover, because of the symmetry relations giving Cij = Cji, only 21 independent elastic constants that can exist in the most general elastic material.

Hook’s law The stress-strain relation of an isotropic elastic material may be described by 2 independent elastic constants, known as Lame constants, and , and: The stress components can be defined as: Note that

Hook’s law In isotropic media, Hooke’s law takes the following form:

Hook’s law Hooke’s law in an isotropic medium is given by the following index equations: These equations are sometimes called the constitutive equations. Students should review Elastic constants in isotropic media (e.g. Young’s modulus, Bulk modulus, Poisson’s ratio, ....., etc.)

One dimensional wave equation To get the wave equation, we will develop Newton’s second law towards our goal of expressing an equation of motion. Newton’s second law simply states: The applicable force have one of two categories: Body Forces: forces such as gravity that work equally well on all particles within the mass- the net force is proportional (essentially) to the volume of the body. Surface Forces: forces that act on the surface of a body-the net force is proportional to the surface area over which the force acts.

Equation of motion Using constitutive equations and Newton’s second law, students are asked to derive the wave equation in one dimension. In order to obtain the equations of motion for an elastic medium we consider the variation in stresses across a small parallelpiped. z y x

Equation of Motion Stresses acting on the surface of a small parallelepiped parallel to the x-axis. Stresses acting on the front face do not balance those acting on the back face. The parallelepiped is not in equilibrium and motion is possible. If we first consider the forces acting in the x-direction, hence the forces will be acting on: Normal to back- and front faces, Tangential to the left- and right-hand faces, and Tangential to the bottom and top faces. z y x

Equation of Motion Normal force acting on the back face force = stress x area Normal force acting on the front face The difference between two forces is given the final normal force acting on the sample in the x-direction

Equation of Motion Tangential force acting on left-hand face Tangential force acting on right-hand face The difference between two forces is given one of tangential forces acting on the sample in the x-direction Ask students to get the other tangential force acting on the sample in the x-direction

Equation of Motion The normal force can be balanced by the mass times the acceleration of the cube, as given by Newton's law: where  • dxdydz is the mass. Cancelling out the volume term on each side, the equation can be written in the following form

Equation of Motion Now we may use Hooke's law to replace stress with displacement: Now, substituting for xx, and remembering that the medium is uniform so that k, m, and r are constants, we have

Equation of Motion The final form of the last equation can be written in the form; This equation equates force per unit volume to mass per unit volume times acceleration. The equation means that Pressure is given by the average of the normal stress components the may cause a change in volume per unit volume.

Ask students to get the wave equation for Shear wave Equation of Motion For an applied pressure P producing a volume change V of a volume V, substituting the k is the modulus of incompressibility (bulk modulus) in the last equation, we will find: Giving P wave equation Ask students to get the wave equation for Shear wave

Equation of Motion in Three dimensions

Equation of Motion (3D) The total force acting on the parallelepiped in the x-direction is given by Making use of the Newton’s second law of motion Mass x Acceleration = resulting force Taking u as the displacement in the x-direction, we will have The equation can be written in the following form Where  is density (1)

Separation of equations of motion in vector form From Hook’s law, the generalized relationship between stress, strain and displacement is given by Substituting stress components in the equation of motion (1), we have for the x-direction Note that for a homogeneous isotropic solid, moduli  and  are constant with respect to x, y and z that do not vary with position. Thus

Separation of equations of motion As we know that represent the divergence operator. If it is applied to: a vector it produces a scalar a tensor it produces a vector It gives the change in volume per unit volume associated with the deformation (  = V/V ). It expresses the local rate of expansion of the vector field.

Separation of equations of motion The gradient operator  is a vector containing three partial derivatives. When applied to a scalar, it produces a vector, a vector, it produces a tensor. The gradient vector of a scalar quantity defines the direction in which it increases fastest; the magnitude equals the rate of change in that direction. Thus, the equation of motion can be written as

Thus, the equation of motion in the x-direction can be written as (2)

Separation of equations of motion Similar to x-direction, the equations of motion in y- and z- directions are given respectively as From equations 2, 3, and 4, we have (3) (4) (5)

Separation of equations of motion Then, equation (5) can be written in the form From the vector analysis, we have Also, the displacement u can be represented in terms of scalar and vector potentials, via Helmholtz’ theorem, then Then equation (6) can be written as (6)

Vectors analysis: Divergence & Curl The divergence is the scalar product of the nabla operator with a vector field V(x). The divergence of a vector field is a scalar! Physically the divergence can be interpreted as the net flow out of a volume (or change in volume). E.g. the divergence of the seismic wavefield corresponds to compressional waves. The curl is the vector product of the nabla operator with a vector field V(x). The curl of a vector field is a vector! The curl of a vector field represents the rotational part of that field (e.g. shear waves in a seismic wavefield)

Vectors analysis Background of mathematics

Separation of equations of motion From the vector analysis, the characteristics of potentials in terms of divergence and curl give that ()=0 and  ()=0. The last equation can be summarized in the following form Using potentials, we can break up the wave equation into two equations

Separation of equations of motion Scalar wave equation (divergence) Vector wave equation (curl)

Separation of equations of motion The scalar potential that satisfies the scalar wave equation gives divergence of a displacement that associated with a change in volume (u  0). This solution produces P waves No shear motion is associated (u = 0) In the vector potential that satisfies the vector wave equation, the displacement is curl (rotation, u  0) that no associated with a change in volume change occurs (u = 0). This solution produces shear motions generating S- waves of probably two independent polarizations No P- wave is associated