Other relationships

It is useful to think of equations (uA) and (uB) as equations for the changes in the the constituent porosities $\delta\phi_A$ and $\delta\phi_B$. To relate these values to the expressions above, we need another pair of equations. First, note that from (weldedpor)

= v_A_A + v_B_B + v_A (_A-_B),   so we need an expression for the change in v_A. For welded contact, we obtain such an expression by noting that by definition

v_A + v_A = V_A(1+B<>u_A)V_A(1+B<>u_A) +V_B(1+B<>u_B),   which upon expansion and neglect of second order terms yields

v_A = v_Av_B(B<>u_A - B<>u_B),   while for welded contact $\delta v_B = - \delta v_A$.Note that, if A and B expand or contract at the same rate so $\nabla\cdot\tilde{\bf u}_A = \nabla\cdot\tilde{\bf u}_B$,then $\delta v_A = 0$ as expected.

We also want to view the combined solid volume $V_s = V_A(1-\phi_A) + V_B(1-\phi_B)$as a whole in order to recover Biot's macroscopic equations for the inhomogeneous material. Then, it is important to recognize that the solid dilatations must satisfy

B<>u_s = v_AB<>u_A + v_BB<>u_B,   and, similarly, the solid pressures must satisfy

(1-)p_s = v_Ap_A + v_Bp_B.   Relation (avesoliddil) may be easily derived by considering the denominator of the right hand side of (volumefraction), whereas (avesolidpressure) is just a statement of force conservation across the material boundary.