Are you up for some nut-cracking challenge?
[MATH]E_{tot}=\frac{2 \pi} {\big(1-\phi\big)R_o^3}\bigg[3 K_s \big(R_{o}^2u_0-R_i^2u_i\big)^2 + \frac{4 \mu }{\phi}\big(R_i^2u_{i} - \phi R_o^2u_0\big)^2\bigg][/MATH];
where, [MATH]\phi = \frac{R_i^3}{R_o^3}[/MATH], and [MATH]u_o[/MATH] & [MATH]u_i[/MATH] both have different expressions for two different cases - drained and undrained.
For the drained case,
[MATH]u_o = \frac{\big(3K_s\phi+4\mu\big)R_o \ \sigma}{12K_s\mu(1-\phi)}[/MATH], and
[MATH]u_i = \frac{\big(3K_s+4\mu\big) R_i \ \sigma}{12K_s\mu(1-\phi)}[/MATH].
For the undrained case,
[MATH]u_o = \frac{\big(3K_s\phi+4\mu\big)R_o \ \sigma - \big(3K_s+4\mu\big)R_o\phi p}{12K_s\mu(1-\phi)}[/MATH], and
[MATH]u_i = \frac{\big(3K_s+4\mu\big) R_i \ \sigma - \big(3K_s+4\mu\phi\big)R_i \ p}{12K_s\mu(1-\phi)}[/MATH].
What I have been trying to do is to reduce [MATH]E_{tot}[/MATH] to the simplest possible solution for the drained [MATH]E_{tot_d}[/MATH] and undrained [MATH]E_{tot_u}[/MATH] cases, using the expressions for the [MATH]u_o[/MATH] and [MATH]u_i[/MATH] above. Can someone help, please?
Thanks,
Somtico
PS: Remember that [MATH]\phi = \frac{R_i^3}{R_o^3}[/MATH].
[MATH]E_{tot}=\frac{2 \pi} {\big(1-\phi\big)R_o^3}\bigg[3 K_s \big(R_{o}^2u_0-R_i^2u_i\big)^2 + \frac{4 \mu }{\phi}\big(R_i^2u_{i} - \phi R_o^2u_0\big)^2\bigg][/MATH];
where, [MATH]\phi = \frac{R_i^3}{R_o^3}[/MATH], and [MATH]u_o[/MATH] & [MATH]u_i[/MATH] both have different expressions for two different cases - drained and undrained.
For the drained case,
[MATH]u_o = \frac{\big(3K_s\phi+4\mu\big)R_o \ \sigma}{12K_s\mu(1-\phi)}[/MATH], and
[MATH]u_i = \frac{\big(3K_s+4\mu\big) R_i \ \sigma}{12K_s\mu(1-\phi)}[/MATH].
For the undrained case,
[MATH]u_o = \frac{\big(3K_s\phi+4\mu\big)R_o \ \sigma - \big(3K_s+4\mu\big)R_o\phi p}{12K_s\mu(1-\phi)}[/MATH], and
[MATH]u_i = \frac{\big(3K_s+4\mu\big) R_i \ \sigma - \big(3K_s+4\mu\phi\big)R_i \ p}{12K_s\mu(1-\phi)}[/MATH].
What I have been trying to do is to reduce [MATH]E_{tot}[/MATH] to the simplest possible solution for the drained [MATH]E_{tot_d}[/MATH] and undrained [MATH]E_{tot_u}[/MATH] cases, using the expressions for the [MATH]u_o[/MATH] and [MATH]u_i[/MATH] above. Can someone help, please?
Thanks,
Somtico
PS: Remember that [MATH]\phi = \frac{R_i^3}{R_o^3}[/MATH].
