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 c_{1} and c_{2} are the stiffnesses (C_{11}) of the displaced and contracted cells, respectively.
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.
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).
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 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 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:
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
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:
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
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:
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:
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:
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]: