Structural Health Monitoring with Piezoelectric Wafer Active Sensors, 2nd Edition

(93)

Upon solution, Eqs. (92), (93) are written as

$\{\begin{array}{l}{\sigma }_x^a=\frac{E_a}{1-{\nu }_a^2}({\epsilon }_x^a+{\nu }_a{\epsilon }_y^a)-\frac{E_a(1+{\nu }_a)}{1-{\nu }_a^2}{\epsilon }_{ISA}\\ {\sigma }_y^a=\frac{E_a}{1-{\nu }_a^2}({\nu }_a{\epsilon }_x^a+{\epsilon }_y^a)-\frac{E_a(1+{\nu }_a)}{1-{\nu }_a^2}{\epsilon }_{ISA}\end{array}(\mathrm{PWAS})$ (94)

(94)

$\{\begin{array}{l}{\sigma }_x=\frac{E}{1-{\nu }^2}({\epsilon }_x+\nu {\epsilon }_y)\\ {\sigma }_y=\frac{E}{1-{\nu }^2}(\nu {\epsilon }_x+{\epsilon }_y)\end{array}(\mathrm{structure})$ (95)

(95)

Recall the strain-displacement relations

$\{\begin{array}{l}{\epsilon }_x^a=\frac{\partial u_x^a}{\partial x}\\ {\epsilon }_y^a=\frac{\partial u_y^a}{\partial y}\end{array}(\mathrm{PWAS})$ (96)

(96)

$\{\begin{array}{l}{\epsilon }_x=\frac{\partial u_x}{\partial x}\\ {\epsilon }_y=\frac{\partial u_y}{\partial y}\end{array}(\mathrm{structure})$ (97)

(97)

Substitution of ...