It is useful to think of equations (uA) and (uB) as equations for the changes in the the constituent porosities and . 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 vA. 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 .Note that, if A and B expand or contract at the same rate so ,then as expected.
We also want to view the combined solid volume 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.