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 ${c_2 \over (c_1+c_2)} u$,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 u_x in cell [i-1,j-1] originates a shear traction T_zx between [i-1,j-1] and [i-1,j] (the traction applied by the first on the second is represented in the figure).

  $$T_{zx} = C_{55} {u_x \over 2 L_x}$$

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

  $$T_{xz} = { T_{zx} \over 2}.$$

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

  The compression of [i,j-1] by u_x/2 creates a compressional traction T_zz (associated with Young's moduli) between [i,j-1] and [i,j), while the asymmetrical compression of [i,j] produces a compressional traction $T^{\prime}_{zz}$ between [i,j] and [i,j+1] given by the equations

  $$T_{zz} = C_{13} {u_x \over 2 L_x} \mbox{\hspace{0.5cm}}\mbox... ...x{\hspace{0.5cm}} T^{\prime}_{zz} = -C_{13} {u_x \over 4 L_x}.$$

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

  $$F_x = C_{55} {u_x \over 4} L_y \mbox{\hspace{0.5cm}}\mbox{\h... ...{13} {u_x \over 4} + C_{55}{u_x \over 4}{L_z \over L_x}\ \}L_y.$$

• case d

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

  $$T_{zx} = C_{55} {u_x \over 2 L_x}.$$

  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:

  $$F_x = C_{55} {u_x \over 2} L_y.$$

• case e

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

  $$T_{xx} = C_{11} {u_x \over L_x} \mbox{\hspace{0.5cm}}\mbox{\... ...5cm}}\mbox{\hspace{0.5cm}} T_{zx} = - C_{55} {u_x \over 2 L_x}.$$

  Applying the equilibrium conditions in [i-1,j-1] and [i,j-1], it follows that $T_{xz} = T^{\prime}_{xz} = T_{zx}/2$.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:

  $$F_x = \{C_{11} u_x {L_z \over L_x}\ - C_{55} {u_x \over 2} \}L_y.$$

• case f

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

  $$T_{xx} = - C_{11} {u_x \over L_x} \mbox{\hspace{0.5cm}}\mbox... ...5cm}}\mbox{\hspace{0.5cm}} T_{zx} = - C_{55} {u_x \over 2 L_x}.$$

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

  $$F_x = - \{ 2 C_{11} u_x {L_z \over L_x}\ + C_{55} {u_x \over 2} \} L_y$$

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: $[i-1,j-1] \equiv 1$, $[i,j-1] \equiv 2$, , $[i,j] \equiv 5$, , $[i+1,j+1] \equiv 9$.
• Two equivalent elastic tensors are defined by:

  $$W^{k,l}_{mn} = {C^k_{mn} \over C^k_{mn} + C^l_{mn}} \mbox{\h... ...C}^{k,l}_{mn} = 2 {C^k_{mn} C^l_{mn} \over C^k_{mn} + C^l_{mn}}$$

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]:

$$\begin{eqnarray} \rho^{i,j} L_x L_z {d^2 u^{i,j}_x(t) \over dt^2} & = & C^1_{55... ... {C^9_{55} \over 4} {L_z \over L_x}\ \} {u^{i+1,j+1}_z}, \nonumber\end{eqnarray}$$

$$\begin{eqnarray} \rho^{i,j} L_x L_z {d^2 u^{i,j}_z(t) \over dt^2} & = & C^1_{55... ... {C^9_{55} \over 4} {L_x \over L_z}\ \} {u^{i+1,j+1}_x}. \nonumber\end{eqnarray}$$