Introduction to Seismology Geology 5640/6640 Introduction to Seismology 23 Jan 2015 Last time: Complex Numbers; Tensors; Stress • A complex number z = x + iy = r ei has both real and imaginary parts. Its conjugate z* = x – iy = r e–i Tensors are generalizations of scalars and vectors that can be used to describe spatially-varying properties. Tensorial order n determines the number of components of the tensor (= 3n for 3D) • Stress is a second-order tensor describing force per unit area acting in three directions acting on each of three planes… A stress traction is a vector that acts on one particular plane: Read for Mon 26 Jan: S&W 29-52 (§2.1-2.3) © A.R. Lowry 2015

We denote the stress components acting on the plane with normal (i.e. a plane on which x = constant) as:

ij More generally we can denote: index i denotes index j denotes direction of normal to the plane acted on by the force index j denotes direction of the force Hence σxx is a normal stress and σxy and σxz are stresses in the plane (called “shear stresses”). We can do same with the y direction: σyx, σyy, σyz act on the plane parallel to xz-plane. For the z-direction, σzx, σzy, σzz act on the plane parallel to the xy-plane. Thus we can use all nine σij to represent the internal force distribution at point P for any plane passing through P.

We combine our 9 components of stress defined in these coordinate planes to represent the stress on any surface through the medium. The 9 σij define the state of stress at point P. The full stress tensor is given by : Recall that strain is the deformation that results from a given stress. A common PhD exam question is to derive stress and strain & elucidate their difference, particularly in relation to your specific discipline…

Note that our text introduces these concepts with slightly different diagrams & notation:

(Note that many texts also use  to denote stress instead of ). In order for the body to remain in static equilibrium, we require that the stresses balance. This effectively requires that ij = ji

The stress tensor is symmetric, with only six independent elements. The diagonal terms ij for i = j are the normal stresses. These are defined (in our usage at least) positive outward, implying a volumetric expansion for strain, and tension at the point. Hence compression has negative sign. The off-diagonal terms, ij; i ≠ j, are shear stresses.

Units of stress are force per unit area, or kg m-1 s-2. The mks (SI) unit is Pa; but also commonly expressed in bars (= 106 dyn cm-2). 1 bar = 105 Pa. For any state of stress, there exists a set of orthogonal coordinate axes for which stress is entirely normal (i.e., shear stresses are zero). These normal stresses are called principal stresses, and the coordinate axis directions are principal axes. These correspond to the eigenvalues and eigenvectors of the tensor, and can be found using linear algebra. Eigenvalues are the set of scalar values  for which for some choice of eigenvector . Eigenvectors define angles for transformation between the axis system of and the principal axis system.

can be rearranged as , where is the identity matrix (e.g., for 3D): This system of equations has a nontrivial (nonzero) solution only when the determinant is equal to zero: which turns out to be a (3D = 3rd-order) polynomial equation in . Once the roots of the polynomial are found, the eigenvectors can be solved for each  by substituting and solving.

Example: Let be and . Then This has polynomial roots  = 1, 2, 3.

For our particular application (solving for principal stress and axes), we can write the problem in indicial notation as: where ij denotes the Kronecker delta: Within the Earth, stress is dominated by pressure due to the weight of overlying material. We commonly remove that effect by removing the mean stress M = Tr[ij]/3 = ii/3 to get a deviatoric stress :