Elasticity solutions of functionally graded piezoelectric plates under electric fields in cylindrical bending

Abstract

This paper presents elasticity solutions for functionally graded piezoelectric plates under electric fields in cylindrical bending. Based on the generalized Mian and Spencer plate theory, the assumption of the material parameters which can vary along the thickness direction of the plate in an arbitrary fashion is kept; however, the materials are extended from elastic materials to piezoelectric materials. The electric potential function is constructed following the forms of the displacement functions in Mian and Spencer plate theory. The essential idea of Mian and Spencer plate theory (J Mech Phys Solids 46:2283–2295, 1998) is that the three-dimensional elasticity equations for inhomogeneous materials can be obtained by two-dimensional solution for homogeneous materials by straightforward substitutions. Through rigorous derivation, the corresponding elasticity solutions of cylindrical bending of functionally graded piezoelectric plates under electric fields are obtained. In the numerical examples, the accuracy of the present solutions is verified and the responses of plates subjected to electrical potential difference and electrical displacement are investigated, respectively.

Appendices

Appendix A: Expressions of the introduced functions related to variable z and integral constants in electric potential case

$$\begin{aligned} g_{i}^{n} \left( z \right)= & {} \int \limits _{-h / 2}^z {\left( {z-\xi } \right) ^{n}\xi ^{i}\frac{\lambda _{33} c_{13} +e_{33} e_{31} }{c_{33} \lambda _{33} +e_{33}^{2} }} d\xi ,\qquad \left( {i,n=0,1} \right) \nonumber \\ h_{i}^{n} \left( z \right)= & {} \int \limits _{-h / 2}^z {\left( {z-\xi } \right) ^{n}\xi ^{i}\left( {c_{11} -c_{13} \frac{c_{13} \lambda _{33} +e_{33} e_{31} }{c_{33} \lambda _{33} +e_{33}^{2} }-e_{31} \frac{c_{13} e_{33} -c_{33} e_{31} }{e_{33}^{2} +c_{33} \lambda _{33} }} \right) } d\xi ,\quad \left( {i,n=0,1} \right) ,\nonumber \\ f_{i}^{n} \left( z \right)= & {} \int \limits _{-h / 2}^z {\left( {z-\xi } \right) ^{n}\xi ^{i}\frac{c_{33} e_{31} -c_{13} e_{33} }{c_{33} \lambda _{33} +e_{33}^{2} }d\xi },\quad \left( {i,n=0,1} \right) ,\nonumber \\ a_{i}^{n} \left( z \right)= & {} \int \limits _{-h / 2}^z {\left( {z-\xi } \right) ^{n}\xi ^{i}\frac{e_{33} }{c_{33} \lambda _{33} +e_{33}^{2} }d\xi },\quad \left( {i,n=0,1} \right) ,\nonumber \\ b_{i}^{n} \left( z \right)= & {} \int \limits _{-h / 2}^z {\left( {z-\xi } \right) ^{n}\xi ^{i}\frac{c_{33} }{c_{33} \lambda _{33} +e_{33}^{2} }d\xi },\quad \left( {i,n=0,1} \right) ,\nonumber \\ f_{5} \left( z \right)= & {} \int \limits _{-h / 2}^z {\frac{g_{5} \left( \xi \right) }{c_{55} }} d\xi , \quad f_{6} \left( z \right) =\int \limits _{-h / 2}^z {\frac{e_{15} }{c_{55} }} d\xi ,\nonumber \\ f_{7} \left( z \right)= & {} \int \limits _{-h / 2}^z {\left\{ {\frac{f_{0}^{0} \left( \xi \right) +e_{15} b_{0}^{0} \left( \xi \right) }{c_{55} }-\left[ {a_{0}^{0} \left( \xi \right) -a_{0}^{0} \left( 0 \right) } \right] } \right\} } d\xi ,\nonumber \\ f_{16} \left( z \right)= & {} \int \limits _{-h / 2}^z {\left\{ {\frac{e_{15} }{c_{55} }\left[ {f_{0}^{0} \left( \xi \right) +e_{15} b_{0}^{0} \left( \xi \right) } \right] +b_{0}^{0} \left( \xi \right) \lambda _{11} } \right\} } d\xi ,\nonumber \\ f_{17} \left( z \right)= & {} \int \limits _{-h / 2}^z {\left\{ {\frac{e_{15} }{c_{55} }\left[ {h_{0}^{0} \left( \xi \right) +e_{15} f_{0}^{0} \left( \xi \right) } \right] +f_{0}^{0} \left( \xi \right) \lambda _{11} } \right\} } d\xi ,\nonumber \\ f_{18} \left( z \right)= & {} \int \limits _{-h / 2}^z {\left\{ {\frac{e_{15} }{c_{55} }\left[ {h_{1}^{0} \left( \xi \right) +e_{15} f_{1}^{0} \left( \xi \right) } \right] +f_{1}^{0} \left( \xi \right) \lambda _{11} } \right\} } d\xi ,\nonumber \\ k_{31}^{*} \left( z \right)= & {} c_{13} \left[ {F_{1} \left( z \right) +F_{0} } \right] -h_{0}^{1} \left( z \right) , \quad k_{32}^{*} \left( z \right) =e_{31} \left[ {F_{1} \left( z \right) +F_{0} } \right] -f_{17} \left( z \right) ,\nonumber \\ k_{33}^{*} \left( z \right)= & {} c_{13} f_{7} \left( z \right) +f_{0}^{1} \left( z \right) , \quad k_{41}^{*} \left( z \right) =c_{13} \left[ {B_{1} \left( z \right) +B_{0} } \right] +h_{1}^{1} \left( z \right) ,\nonumber \\ k_{42}^{*} \left( z \right)= & {} e_{31} \left[ {B_{1} \left( z \right) +B_{0} } \right] +f_{18} \left( z \right) , \quad k_{44}^{*} \left( z \right) =e_{31} f_{7} \left( z \right) +f_{16} \left( z \right) ,\nonumber \\ B_{1} \left( z \right)= & {} -\int \limits _{-h / 2}^z {\left\{ {\frac{1}{c_{55} }\left[ {h_{1}^{0} \left( \xi \right) +e_{15} f_{1}^{0} \left( \xi \right) } \right] -\left[ {g_{1}^{0} \left( z \right) -g_{1}^{0} \left( 0 \right) } \right] } \right\} } d\xi ,\nonumber \\ C_{1} \left( z \right)= & {} g_{1}^{0} \left( z \right) \quad D_{00} \left( z \right) =\int \limits _{-h / 2}^z {\left[ {\frac{k_{33}^{*} \left( \xi \right) \lambda _{33} +k_{44}^{*} \left( \xi \right) e_{33} }{c_{33} \lambda _{33} +e_{33}^{2} }} \right] } d\xi ,\nonumber \\ D_{01} \left( z \right)= & {} \int \limits _{-h / 2}^z {\left[ {\frac{k_{31}^{*} \left( \xi \right) \lambda _{33} +k_{32}^{*} \left( \xi \right) e_{33} }{c_{33} \lambda _{33} +e_{33}^{2} }} \right] } d\xi , \quad D_{02} \left( z \right) =\int \limits _{-h / 2}^z {\left[ {\frac{k_{41}^{*} \left( \xi \right) \lambda _{33} +k_{42}^{*} \left( \xi \right) e_{33} }{c_{33} \lambda _{33} +e_{33}^{2} }} \right] } d\xi ,\nonumber \\ F_{1} \left( z \right)= & {} -\int \limits _{-h / 2}^z {\left\{ {\frac{1}{c_{55} }\left[ {h_{0}^{0} \left( \xi \right) +e_{15} f_{0}^{0} \left( \xi \right) } \right] -\left[ {g_{0}^{0} \left( \xi \right) -g_{0}^{0} \left( 0 \right) } \right] } \right\} } d\xi ,\nonumber \\ G_{1} \left( z \right)= & {} -g_{0}^{0} \left( z \right) , \quad \phi _{10} \left( z \right) =f_{0}^{0} \left( z \right) , \quad \phi _{20} \left( z \right) =-f_{1}^{0} \left( z \right) , \nonumber \\ \phi _{00} \left( z \right)= & {} \int \limits _{-h / 2}^z {\left[ {\frac{k_{33}^{*} \left( \xi \right) e_{33} -k_{44}^{*} \left( \xi \right) c_{33} }{c_{33} \lambda _{33} +e_{33}^{2} }} \right] } d\xi ,\nonumber \\ \phi _{01} \left( z \right)= & {} \int \limits _{-h / 2}^z {\left[ {\frac{k_{31}^{*} \left( \xi \right) e_{33} -k_{32}^{*} \left( \xi \right) c_{33} }{c_{33} \lambda _{33} +e_{33}^{2} }} \right] } d\xi ,\nonumber \\ \phi _{02} \left( z \right)= & {} \int \limits _{-h / 2}^z {\left[ {\frac{k_{41}^{*} \left( \xi \right) e_{33} -k_{42}^{*} \left( \xi \right) c_{33} }{c_{33} \lambda _{33} +e_{33}^{2} }} \right] } d\xi ,\nonumber \\ \end{aligned}$$

(A-1)

$$\begin{aligned} B_{0}= & {} -k_{2} f_{5} \left( 0 \right) +\Phi _{2} f_{6} \left( 0 \right) -H_{4} f_{7} \left( 0 \right) -B_{1} \left( 0 \right) ,\nonumber \\ C_{0}= & {} -g_{1}^{0} \left( 0 \right) -H_{4} a_{0}^{0} \left( 0 \right) ,\nonumber \\ D_{0}= & {} \kappa _{3} \left[ {D_{00} \left( z \right) H_{2} +D_{01} \left( z \right) } \right] +\kappa _{4} \left[ {D_{00} \left( z \right) H_{4} +D_{02} \left( z \right) } \right] -a_{0}^{0} \left( z \right) H_{7},\nonumber \\ F_{0}= & {} -k_{1} f_{5} \left( 0 \right) +\Phi _{1} f_{6} \left( 0 \right) -H_{2} f_{7} \left( 0 \right) -F_{1} \left( 0 \right) ,\nonumber \\ G_{0}= & {} g_{0}^{0} \left( 0 \right) -H_{2} a_{0}^{0} \left( 0 \right) , \quad \Phi _{0} \text{= }\bar{{\Phi }}_{1} -\left( {\Phi _{2} C_{2} +\Phi _{1} C_{5} } \right) . \end{aligned}$$

(A-2)

Appendix B: Expressions of stress resultants and moments

$$\begin{aligned} N_{x}= & {} \int \limits _{-h/2}^{h/2} {\sigma _{x} dz=} N_{1} \bar{{u}}_{,x} +N_{3} \bar{{w}}_{,xx} +N_{5} \bar{{u}}_{,xxx} +N_{7} \bar{{w}}_{,xxxx} +N_{0} \nonumber \\ M_{x}= & {} \int \limits _{-h/2}^{h/2} {\sigma _{x} zdz} =M_{1} \bar{{u}}_{,x} +M_{3} \bar{{w}}_{,xx} +M_{5} \bar{{u}}_{,xxx} +M_{7} \bar{{w}}_{,xxxx} +M_{0} \nonumber \\ Q_{x}= & {} \int \limits _{-h/2}^{h/2} {\tau _{zx} dz=Q_{1} \bar{{u}}_{,xx} +Q_{2} \bar{{w}}_{,xxx} } \end{aligned}$$

(B-1)

where

$$\begin{aligned} N_{1}= & {} \int \limits _{-h/2}^{h/2} {\left( {c_{11} +c_{13} {G}'+e_{31} \phi _{1}^{\prime }} \right) } dz,N_{3} =\int \limits _{-h/2}^{h/2} {\left( {c_{11} A+c_{13} {C}'+e_{31} \phi _{2}^{\prime }} \right) } dz,\nonumber \\ N_{5}= & {} \int \limits _{-h/2}^{h/2} {c_{11} F} dz,N_{7} =\int \limits _{-h/2}^{h/2} {c_{11} B} dz,N_{0} =\int \limits _{-h/2}^{h/2} {c_{13} {D}'+e_{31} {\phi }'_{0} } dz,\nonumber \\ M_{1}= & {} \int \limits _{-h/2}^{h/2} {z\left( {c_{11} +c_{13} {G}'+e_{31} \phi _{1}^{\prime }} \right) } dz,M_{3} =\int \limits _{-h/2}^{h/2} {z\left( {c_{11} A+c_{13} {C}'+e_{31} \phi _{2}^{\prime }} \right) } dz,\nonumber \\ M_{5}= & {} \int \limits _{-h/2}^{h/2} {c_{11} zF} dz,M_{7} =\int \limits _{-h/2}^{h/2} {c_{11} zB} dz,M_{0} =\int \limits _{-h/2}^{h/2} {\left( {c_{13} {D}'+e_{31} {\phi }'} \right) z} dz,\nonumber \\ Q_{1}= & {} \int \limits _{-h/2}^{h/2} {\left[ {c_{55} \left( {{F}'+G} \right) +e_{15} \phi _{1} } \right] } dz,Q_{2} =\int \limits _{-h/2}^{h/2} {\left[ {c_{55} \left( {{B}'+C} \right) +e_{15} \phi _{2} } \right] } dz, \end{aligned}$$

(B-2)

Appendix C: Expressions of the introduced functions related to variable z and integral constants in electric displacement case

$$\begin{aligned} F_{10} \left( z \right)= & {} \int \limits _{-h/2}^z {\left[ {c_{55} \kappa _{1} -c_{11} -c_{13} {G}'\left( \xi \right) -e_{31} {\phi }'_{1} \left( \xi \right) } \right] } d\xi , \nonumber \\ B_{10} \left( z \right)= & {} \int \limits _{-h/2}^z {\left[ {c_{55} \kappa _{2} -c_{11} A\left( \xi \right) -c_{13} {C}'\left( \xi \right) -e_{31} {\phi }'_{2} \left( \xi \right) } \right] } d\xi ,\nonumber \\ D_{00}= & {} c_{13} \left( {-h / 2} \right) \left[ {\kappa _{3} F\left( {-h / 2} \right) +\kappa _{4} B\left( {-h / 2} \right) } \right] , \nonumber \\ D_{01}= & {} e_{31} \left( {-h / 2} \right) \left[ {\kappa _{3} F\left( {-h / 2} \right) +\kappa _{4} B\left( {-h / 2} \right) } \right] ,\nonumber \\ D_{10} \left( z \right)= & {} \int \limits _{-h/2}^z {\left\{ {\kappa _{3} \left[ {c_{55} \left( {{F}'+G} \right) +\left( {Fc_{13} } \right) ^{\prime }+e_{15} \phi _{1} } \right] +\kappa _{4} \left[ {c_{55} \left( {{B}'+C} \right) +\left( {c_{13} B} \right) ^{\prime }+e_{15} \phi _{2} } \right] } \right\} } d\xi ,\nonumber \\ D_{20} \left( z \right)= & {} \int \limits _{-h/2}^z {\left\{ {\kappa _{3} \left[ {e_{15} \left( {{F}'+G} \right) +\left( {Fe_{31} } \right) ^{\prime }-\lambda _{11} \phi _{1} } \right] +\kappa _{4} \left[ {e_{15} \left( {{B}'+C} \right) +\left( {e_{31} B} \right) ^{\prime }-\lambda _{11} \phi _{2} } \right] } \right\} } d\xi . \end{aligned}$$

(C-1)

$$\begin{aligned} B_{0}= & {} -\int \limits _{-h/2}^0 {\left\{ {\frac{1}{c_{55} }\left[ {B_{10} \left( z \right) -e_{15} \phi _{2} \left( z \right) } \right] -C\left( z \right) } \right\} } dz,\nonumber \\ C_{0}= & {} \int \limits _{-h/2}^0 {\left[ {\frac{\lambda _{33} c_{13} +e_{33} e_{31} }{c_{33} \lambda _{33} +e_{33}^{2} }A\left( z \right) } \right] } dz,\nonumber \\ D_{0}= & {} \int \limits _{-h/2}^0 {\left\{ {\frac{\lambda _{33} \left[ {D_{10} \left( z \right) +D_{00} \left( z \right) } \right] +e_{33} \left[ {D_{20} \left( z \right) +D_{01} \left( z \right) } \right] }{c_{33} \lambda _{33} +e_{33}^{2} }} \right\} } dz,\nonumber \\ F_{0}= & {} -\int \limits _{-h/2}^0 {\left\{ {\frac{1}{c_{55} }\left[ {F_{10} \left( z \right) -e_{15} \phi _{1} } \right] -G\left( z \right) } \right\} } dz, \quad G_{0} =\int \limits _{-h/2}^0 {\left( {\frac{\lambda _{33} c_{13} +e_{33} e_{31} }{c_{33} \lambda _{33} +e_{33}^{2} }} \right) } dz,\nonumber \\ \Phi _{1}= & {} -\int \limits _{-h/2}^0 {\left( {\frac{c_{33} e_{31} -c_{13} e_{33} }{c_{33} \lambda _{33} +e_{33}^{2} }} \right) dz}, \quad \Phi _{2} \text{= }-\int \limits _{-h/2}^0 {\left[ {\frac{c_{33} e_{31} -c_{13} e_{33} }{c_{33} \lambda _{33} +e_{33}^{2} }A\left( z \right) } \right] dz},\nonumber \\ \Phi _{0}= & {} -\int \limits _{-h/2}^0 {\left\{ {\frac{c_{33} \left[ {D_{20} \left( z \right) +D_{01} \left( z \right) } \right] -e_{33} \left[ {D_{10} \left( z \right) +D_{00} \left( z \right) } \right] }{c_{33} \lambda _{33} +e_{33}^{2} }} \right\} } dz. \end{aligned}$$

(C-2)