Reflection of plane wave at an initially stressed rotating piezo-electro-magnetic-fiber-reinforced Composite half-space

Abstract

The mechanism of Nondestructive testing, especially for metal flaw detection, intelligent monitoring of infrastructure involves the imaging of Primary wave reflection in piezo composites using a transmitter and receiver circuit, transducer tool, and display devices. The signal received in the transducer carries information about crack location, flaw size, orientation, and other characteristics. Keeping these facts in consideration and with an aim to explore the reflection of a plane wave in a highly anisotropic material the present study is modeled. In the present study, the reflection profile of plan waves at the stress-free surface of a rotating Piezo-Electro-Magnetic-Fiber-Reinforce Composite (PEMFRC) half-space subjected to normal and shearing initial stresses is discussed. The micro-mechanics of PEMFRC are established and the effective material properties are derived. Incidence of a qP/qSV wave generates four reflected waves via. quasi-longitudinal/transverse (qP/qSV) waves, electro-acoustic and magneto-acoustic waves. The propagation directions of all reflected waves are graphically demonstrated. The amplitude ratios are derived in closed-form expressions, using secular equations and relevant boundary conditions by the help of Cramer’s rule. Furthermore, using these expressions and the energy flux relation, the energy ratios reflected bulk/surface waves and interaction energies are derived, which exhibit the influence of the existing parameters. Special cases of normal/grazing incidences of qP/qSV waves are discussed. Using the numerical data of BaTiO$_3$ as matrix and CoFe$_2$O$_4$ as fiber constituent, the energy ratios of reflected and interaction waves as well as total energy are illustrated graphically, and the energy conservation law is established.

Appendix

$c_{11}=\frac{c_{11}^f c_{11}^m}{v_f c_{11}^m + (1-v_f) c_{11}^f}, \quad c_{12}=c_{11} \left(\frac{v_f c_{12}^f}{c_{11}^f}+\frac{(1-v_f) c_{12}^m}{c_{11}^m}\right),\quad c_{13}=c_{11}\left(\frac{v_f c_{13}^f}{c_{11}^f}+\frac{(1-v_f) c_{13}^m}{c_{11}^m}\right),$

$c_{22}=v_f c_{22}^f + (1-v_f) c_{22}^m + c_{12}^2/c_{11}-v_f (c_{12}^f)^2/c_{11}^f - (1-v_f)(c_{12}^m)^2/c_{11}^m,$

$c_{23}=v_f c_{23}^f + (1-v_f) c_{23}^m + c_{12} c_{13}/c_{11}-v_f c_{12}^f c_{13}^f/c_{11}^f - (1-v_f)c_{12}^m c_{13}^m/c_{11}^m,$

$c_{33}=v_f c_{33}^f + (1-v_f) c_{33}^m + c_{13}^2/c_{11}-v_f (c_{13}^f)^2/c_{11}^f - (1-v_f)(c_{13}^m)^2/c_{11}^m,$

$e_{31}=c_{11} \left( v_f e_{31}^f/c_{11}^f + (1-v_f)e_{31}^m/c_{11}^m \right) , \quad q_{31}=c_{11} \left( v_f q_{31}^f/c_{11}^f + (1-v_f)q_{31}^m/c_{11}^m \right) ,$

$e_{32}= v_f e_{32}^f + (1-v_f) e_{32}^m + c_{12} e_{31}/c_{11}-v_f c_{12}^f e_{31}^f/c_{11}^f - (1-v_f)c_{12}^m e_{31}^m/c_{11}^m,$

$e_{33}=v_f e_{33}^f + (1-v_f) e_{33}^m + c_{13} e_{31}/c_{11}-v_f c_{13}^f e_{31}^f/c_{11}^f - (1-v_f)c_{13}^m e_{31}^m/c_{11}^m,$

$q_{33}=v_f q_{33}^f + (1-v_f) q_{33}^m + c_{13} q_{31}/c_{11}-v_f c_{13}^f q_{31}^f/c_{11}^f - (1-v_f)c_{13}^m q_{31}^m/c_{11}^m,$

$\epsilon _{33}=v_f \epsilon _{33}^f + (1-v_f) \epsilon _{33}^m + e_{31}^2/c_{11}-v_f (e_{31}^f)^2/c_{11}^f - (1-v_f)(e_{31}^m)^2/c_{11}^m,$

$\mu _{33}=v_f \mu _{33}^f + (1-v_f) \mu _{33}^m + q_{31}^2/c_{11}-v_f (q_{31}^f)^2/c_{11}^f - (1-v_f)(q_{31}^m)^2/c_{11}^m,$

$\alpha _{33}=v_f \alpha _{33}^f + (1-v_f) \alpha _{33}^m + e_{31} q_{31}/c_{11}-v_f e_{31}^f q_{31}^f/c_{11}^f - (1-v_f)e_{31}^m q_{31}^m/c_{11}^m,$

$c_{55}^c = c_{55}^f(1 - v_f) + c_{55}^m v_f, \quad \epsilon _{11}^c = \epsilon _{11}^f (1 - v_f) + \epsilon _{11}^m v_f, \quad \mu _{11}^c = \mu _{11}^f (1 - v_f) + \mu _{11}^m v_f,$

$\mu _{11}=\frac{q_{15}^m \mu _{11}^m}{\mu _{11}^c} \left( \frac{\mu _{11}^m}{q_{15}^m} -\frac{q_{15}^m}{c_{55}^c+\frac{(1-v_f)^2 (e_{15}^f)^2}{\epsilon _{11}^c}} \right) , \quad q_{15}=\frac{q_{15}^m \mu _{11}^m}{\mu _{11}^c} \left( \frac{1}{v_f} -\frac{c_{55}^m}{c_{55}^c+\frac{(1-v_f)^2 (e_{15}^f)^2}{\epsilon _{11}^c}} \right) ,$

$c_{55}= \left( c_{55}^c + \frac{(1-v_f)(e_{15}^f)^2}{\epsilon _{11}^c} \right) \left( \frac{1}{v_f} - \frac{q_{15} \mu _{11}^c}{q_{15}^m \mu _{11}^f} \right) , \quad \alpha _{11}= \left( e_{15}^f - \frac{(1-v_f) e_{15}^f \epsilon _{11}^f}{\epsilon _{11}^c} \right) \left( \frac{\mu _{11}^m}{q_{15}^m}- \frac{\mu _{11}\mu _{11}^c}{q_{15}^m \mu _{11}^f} \right) ,$

$\epsilon _{11}= \left( e_{15}^f - \frac{(1-v_f) e_{15}^f \epsilon _{11}^f}{\epsilon _{11}^c} \right) \left( \frac{\alpha _{11} \mu _{11}^c}{q_{15}^m \mu _{11}^f} \right) +\frac{(\epsilon _{11}^f)^2}{\epsilon _{11}^c}, \quad e_{15}= \left( e_{15}^f - \frac{(1-v_f) e_{15}^f \epsilon _{11}^f}{\epsilon _{11}^c} \right) \left( \frac{1}{v_f} - \frac{q_{15} \mu _{11}^c}{q_{15}^m \mu _{11}^f} \right) ,$

$S_{1m}=ik(c_{55} q_m +c_{55} W +e_{15} \Phi + q_{15} \Psi ), \quad S_{2m}= ik(c_{13} + c_{33} q_m W +e_{33} q_m \Phi + q_{33} q_m \Psi ),$

$S_{3m}=ik (e_{31} + e_{33} q_m W - \epsilon _{33} q_m \Phi - \alpha _{33} q_m \Psi ), \quad S_{4m}= ik (q_{31} + q_{33} q_m W -\alpha _{33} q_m \Phi - \mu _{33} q_m \Psi ),$

$a_{11}=c_{11}+\tau _{11}^0+(c_{55}+\tau _{33}^0)q^2 +2\tau _{31}^0 q-\rho ( c^2+(\frac{\Omega }{k})), \quad a_{12}=(c_{13}+c_{55})q,\quad a_{13}=(e_{13}+e_{15})q,$

$a_{14}= (q_{13}+q_{15})q,\quad a_{21}=(c_{13}+c_{55})q,\quad a_{22}=c_{55}+\tau _{11}^0 + (c_{33} +\tau _{33}^0)q^2 -2\tau _{31}^0 q -\rho c^2,$

$a_{23}=(e_{15}+e_{33})q,\quad a_{24}=(q_{15}+q_{33})q,\quad a_{31}=(e_{13}+e_{15})q, \quad a_{32}=(e_{15}+e_{33})q, \quad a_{33}=-(\epsilon _{11}+\epsilon _{33}q^2),$

$a_{34}= -(\alpha _{11}+\alpha _{33}q^2), \quad a_{41}=(q_{13}+q_{15})q, \quad a_{42}=(q_{15}+q_{33})q, \quad a_{43}=-(\alpha _{11}+\alpha _{33}q^2),$

$a_{44}=-(\mu _{11}+\mu _{33}q^2),$

$P= \begin{bmatrix} S_{11} +\tau _{13}^0+ \tau _{33}^0 q_1 &{} S_{13}+\tau _{13}^0+ \tau _{33}^0 q_3 &{} S_{15}+\tau _{13}^0+ \tau _{33}^0 q_5 &{} S_{17}+\tau _{13}^0+ \tau _{33}^0 q_7\\ S_{21} +(\tau _{13}^0+ \tau _{33}^0 q_1)W_1 &{} S_{23}+(\tau _{13}^0+ \tau _{33}^0 q_3)W_3 &{} S_{25}+(\tau _{13}^0+ \tau _{33}^0 q_5)W_5 &{} S_{27}+(\tau _{13}^0+ \tau _{33}^0 q_7)W_7\\ S_{31} &{} S_{33} &{} S_{35} &{} S_{37}\\ S_{41} &{} S_{43} &{} S_{45} &{} S_{47} \end{bmatrix},$

$Q=[-s_{1r}+\tau _{13}^0+ \tau _{33}^0 q_r, -s_{2r}+(\tau _{13}^0+ \tau _{33}^0 q_r)W_r, -s_{3r}, -s_{4r}], \text{ where } \quad r=2,4, X=[U_1,\,U_3,\,U_5,\,U_7]$.