Normalize material parameters

In this section we show that (for the \(p\)-system) we can consider without loss of generality \(K_A=\rho_A=1\).

Consider the \(p\)-system:

\(\begin{align} \epsilon_t-(u_x+v_y)&=0, \\ \rho u_t-\sigma_x=0, \\ \rho v_t-\sigma_y=0, \end{align} \)

where \(\sigma=f(K\epsilon)\); i.e., \(\sigma\) is a (non)linear function that depends on the product \(K\epsilon\), \(\rho=\begin{cases} \rho_A &\forall x\in\Omega_A \\ \rho_B & \forall x\in\Omega_B \end{cases}\), and similarly \(K\).

Let

\(\begin{align} \tilde{\rho}=\begin{cases} 1 &\forall x\in\Omega_A \\ \rho_B/\rho_A &\forall x\in\Omega_B \end{cases}, \end{align} \)

and similarly for \(\tilde{K}\). Now, multiply the first equation by \(K_A\) and the second and third by \(K_A/\rho_A\). This leads to:

\(\begin{align} (K_A\epsilon)_t-[(K_Au)_x+(K_Av)_y]&=0, \\ \tilde{\rho} (K_Au)_t-\left(\frac{K_A}{\rho_A}\sigma\right)_x=0, \\ \tilde{\rho} (K_Av)_t-\left(\frac{K_A}{\rho_A}\sigma\right)_y=0, \end{align} \)

Finally, let \(\tilde{\epsilon}=K_A\epsilon\), \(\tilde{u}=K_Au\), \(\tilde{v}=K_Av\) and \(\tilde{\sigma}=K_A/\rho_A\sigma\) to obtain:

\(\begin{align} \tilde{\epsilon}_t-(\tilde{u}_x+\tilde{v}_y)&=0, \\ \tilde{\rho} \tilde{u}_t-\tilde{\sigma}_x=0, \\ \tilde{\rho} \tilde{v}_t-\tilde{\sigma}_y=0, \end{align} \)

with \(\tilde{\sigma}=K_A/\rho_Af(\tilde{K}\tilde{\epsilon})\). This system has the same form as the original system, but here \(\tilde{\rho}_A=\tilde{K_A}=1\), and the parameters in the B-material are scaled accordingly. Therefore, given a problem with the original system, we can solve the tilde version and then recover the non-tilde solution.

Hence, without loss of generality, we can consider, whenever is convenient, \(\rho_A=K_A=1\).