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# Equations of motion for a 2-D transverse isotropic discrete system

Figure -a shows the 2-D elastic equivalent of the spring-mass model described in the paper. The medium is constituted by square cells with the mass concentrated in the center of the cell, and the elastic, massless part located at the boundaries. This elastic part is responsible for all the interaction between adjacent cells. As in the 1-D case, only adjacent cells can interact with each other. Therefore, the dynamics of cell [i,j] in Figure -a is controlled only by its own position, and the positions of its eight neighbor cells.

When a cell has a displacement component u, in the x or z direction, relative to its equilibrium position (Figure -b), then one of its neighbor cells will be contracted by u/2 while the other neighbor will be expanded by u/2 if the system is homogeneous. For the case of heterogeneous systems, the contraction would be ,where c1 and c2 are the stiffnesses (C11) of the displaced and contracted cells, respectively.

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Figure 8
(a) A schematic model used to describe a 2-D elastic system. (b) Relations between the displacement of a cell and its neighbors. (c)-(f) The dynamics involved in the four basic displacements that interact with the central cell. The arrows represent some of the acting tractions and are located in the cell that is applying the traction.

It is sufficient to analyze the dynamics associated with the four basic displacements represented in Figures -c to -f since all the other cases can be easily derived from them. The following items refer to each of those figures in turn.

• case c

A displacement ux in cell [i-1,j-1] originates a shear traction Tzx between [i-1,j-1] and [i-1,j] (the traction applied by the first on the second is represented in the figure).

where Lx is the x dimension of an undeformed cell. The rotational equilibrium requires that the total momentum applied to [i-1,j] be zero. Therefore

The same equilibrium condition in [i,j], requires that the same shear traction Txz be applied on it by [i,j-1].

The compression of [i,j-1] by ux/2 creates a compressional traction Tzz (associated with Young's moduli) between [i,j-1] and [i,j), while the asymmetrical compression of [i,j] produces a compressional traction between [i,j] and [i,j+1] given by the equations

The total force applied in [i,j] due to [i-1,j-1]'s displacement then has the following components:

• case d

A displacement ux in cell [i,j-1] creates a shear traction Tzx between [i,j-1] and [i,j], whose magnitude is given by the equation

This traction will be responsible for the only net force applied in [i,j], because the two shear tractions between [i,j] and [i-1,j], and between [i,j] and [i+1,j] compensate each other as well as the two compressional tractions (not represented in the figure) acting in the same interfaces.

The total force applied in [i,j] due to [i,j-1]'s displacement has only an x component, as follows:

• case e

A displacement ux in cell [i-1,j] creates a compressional traction Txx between [i-1,j] and [i,j], and a shear traction Tzx between [i-1,j] and [i-1,j-1] whose magnitudes are given by the equations

Applying the equilibrium conditions in [i-1,j-1] and [i,j-1], it follows that .This traction is responsible for the only net force applied in [i,j], because the two shear tractions between [i,j] and [i-1,j], and [i,j] and [i+1,j] compensate each other as well as the two compressional tractions (not represented in the figure) acting in the same interfaces.

The total force applied in [i,j] caused by [i-1,j]'s displacement also has only an x component:

• case f

A displacement ux in cell [i,j] creates compressional tractions Txx between [i-1,j] and [i,j], and between [i,j] and [i+1,j], and a shear traction Tzx between [i,j-1] and [i,j+1], whose magnitudes are given by the following equations:

The total force applied in [i,j] caused by [i,j]'s displacement also has only an x component, as follows:

For the more general heterogeneous case, some definitions are required to simplify the equations of motion:

• For the stiffness constants, the cells are identified by a number: , , , , , .
• Two equivalent elastic tensors are defined by:

Taking the contributions from the x and z displacements of all nine cells into account leads to the following equations of motion for cell [i,j]:

Next: APPENDIX B Up: Cunha: Modeling a discrete Previous: APPENDIX A
Stanford Exploration Project
12/18/1997