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NOTATION FOR ELASTIC ANALYSYS

In tensor notation, the relationship between components of stress $\sigma_{ij}$and strain uk,l is given by
   \begin{eqnarray}
\sigma_{ij} = C_{ijkl}u_{k,l},
 \end{eqnarray} (1)
where Cijkl is the stiffness tensor, and repeated indices on the right hand side of (3) are summed. In (1), uk is the kth Cartesian component of the displacement vector ${\bf u}$, and $u_{k,l} = {\partial u_k}/{\partial x_l}$.Whereas for an isotropic elastic medium  the stiffness tensor has the form
   \begin{eqnarray}
C_{ijkl} = \lambda\delta_{ij}\delta_{kl} + \mu\left(\delta_{ik}\delta_{jl} 
+ \delta_{il}\delta_{jk}\right),
 \end{eqnarray} (2)
depending on only two parameters (the Lamé constants, $\lambda$ and $\mu$), this tensor can have up to 21 independent constants for general anisotropic elastic media. The stiffness tensor has pairwise symmetry in its indices such that Cijkl = Cjikl and Cijkl = Cijlk, which will be used later to simplify the resulting equations.

The general equation of motion for elastic wave propagation through an anisotropic medium is given by
   \begin{eqnarray}
\rho\ddot{u}_i = \sigma_{ij,j} = C_{ijkl}u_{k,lj},
 \end{eqnarray} (3)
where $\ddot{u}_i$ is the second time derivative of the ith Cartesian omponent of the displacement vector ${\bf u}$ and $\rho$ is the density (assumed constant). Equation (3) is a statement that the product of mass times acceleration of a particle is determined by the internal stress force $\sigma_{ij,j}$. For the present purposes, we are more interested in the quasistatic limit of this equation, in which case the left-hand side of (3) vanishes and the equation to be satisfied is just the force equilibrium equation
   \begin{eqnarray}
\sigma_{ij,j} = 0.
 \end{eqnarray} (4)

A commonly used simplification of the notation for elastic analysis is given by introducing the strain tensor, where
   \begin{eqnarray}
e_{ij}
= {\textstyle {1\over2}}(u_{i,j} + u_{j,i})
 = {\textsty...
 ...er{\partial x_j}} 
+ {{\partial u_j}\over{\partial x_i}}\right).
 \end{eqnarray} (5)
Then, using one version of the Voigt convention, in which the pairwise symmetries of the stiffness tensor indices are used to reduce the number of indices from 4 to 2 using the rules $11 \to 1$, $22 \to 2$, $33 \to 3$, $23\hbox{ or }32 \to 4$,$13\hbox{ or }31 \to 5$, and $12\hbox{ or }21 \to 6$,we have
   \begin{eqnarray}
\left(
\begin{array}
{c} \sigma_{11} \sigma_{22} \sigma_{33} \
...
 ... e_{11} e_{22} e_{33} e_{23} e_{31} e_{12} \\ end{array}\right).
 \end{eqnarray} (6)
Although the Voigt convention  introduces no restrictions on the stiffness tensor, we have chosen to limit discussion to the form in (6), which is not completely general. Of the 36 coefficients (of which 21 are generally independent), we choose to treat only those cases for which the 12 coefficients shown (of which nine are generally independent) are nonzero. This form includes all orthorhombic, cubic, hexagonal, and isotropic systems, while excluding triclinic, monoclinic, trigonal, and some tetragonal systems, since each of the latter contains additional off-diagonal constants that may be nonzero. Nevertheless, we will restrict our discussion to (6) or to the still simpler case of transversely isotropic (TI) materials.

For TI materials whose symmetry axis is in the x3 direction, another common choice of notation is $C_{11} = C_{22} \equiv a$, $C_{12} \equiv b$,$C_{13} = C_{23} \equiv f$,$C_{33} \equiv c$, $C_{44} = C_{55} \equiv l$, and $C_{66} \equiv m$.There is also one further constraint on the constants that a = b + 2m, following from rotational symmetry in the x1x2-plane. In such materials, (6) may be replaced by
   \begin{eqnarray}
\left(
\begin{array}
{cccccc} \sigma_{11} \sigma_{22} \sigma_{3...
 ...} e_{11} e_{22} e_{33} e_{23} e_{31}
 e_{12} \end{array}\right),
 \end{eqnarray} (7)
in which the matrix has the same symmetry as hexagonal systems and of which isotropic symmetry is a special case (having $a=c=\lambda+ 2\mu$, $b=f=\lambda$, and $l=m=\mu$).


next up previous print clean
Next: BACKUS AVERAGING OF FINE Up: Berryman: Poroelastic shear modulus Previous: INTRODUCTION
Stanford Exploration Project
7/8/2003