Partial contact in two-layered piezoelectric structure with interface occupying periodic profiles

Abstract

The present article establishes a partial contact model to reveal how the contact strength of two piezoelectric materials is influenced by the wavy surface topography. General solutions of governing equations of the piezoelectric materials are presented. Dual series equations are obtained and solved by considering the Werner formulas and Mehler integral. For evaluating various stresses and electric displacements, the related explicit expressions are presented. Numerical tests are conducted to examine in what way the external loading and the piezoelectric coefficient ratio affect the contact performances between two piezoelectric materials. Results delineate that the external loading has a bigger contribution on forming the contact region between two different piezoelectric materials compared with the piezoelectric coefficient ratio.

Appendix

• 1. Expressions of the elements of symmetrical matrix $\mathbf{A}_j =(a_{nk}^{(j)} )_{3\times 3} (j=1,2)$ appearing in Eq. (5)

  $$\begin{aligned} a_{11}^{(j)}= & {} C_{11}^{(j)} \frac{\partial ^{2}}{\partial x^{2}}+C_{44}^{(j)} \frac{\partial ^{2}}{\partial z^{2}},\qquad a_{12}^{(j)} =\left( {C_{13}^{(j)} +C_{44}^{(j)} } \right) \frac{\partial ^{2}}{\partial x\partial z},\nonumber \\ a_{13}^{(j)}= & {} \left( {e_{15}^{(j)} +e_{31}^{(j)} } \right) \frac{\partial ^{2}}{\partial x\partial z}, \qquad a_{22}^{(j)} =C_{44}^{(j)} \frac{\partial ^{2}}{\partial x^{2}}+C_{33}^{(j)} \frac{\partial ^{2}}{\partial z^{2}}, \nonumber \\ a_{23}^{(j)}= & {} e_{15}^{(j)} \frac{\partial ^{2}}{\partial x^{2}}+e_{33}^{(j)} \frac{\partial ^{2}}{\partial z^{2}},\qquad a_{33}^{(j)} =-\in _{11}^{(j)} \frac{\partial ^{2}}{\partial x^{2}}-\in _{33}^{(j)} \frac{\partial ^{2}}{\partial z^{2}}. \end{aligned}$$

  (76)

• 2. Expressions of $c_k^{(j)} (j=1,2,k=0,1,2,3)$ appearing in Eq. (6)

  $$\begin{aligned} c_0^{(j)}= & {} -C_{44}^{(j)} \left[ {\left( {e_{33}^{(j)} } \right) ^{2}+C_{33}^{(j)} \in _{33}^{(j)} } \right] ,\nonumber \\ c_1^{(j)}= & {} -C_{33}^{(j)} \left[ {C_{44}^{(j)} \in _{11}^{(j)} +\left( {e_{15}^{(j)} +e_{31}^{(j)} } \right) ^{2}} \right] -\in _{33}^{(j)} \left[ {C_{11}^{(j)} C_{33}^{(j)} +\left( {C_{44}^{(j)} } \right) ^{2}} \right. \nonumber \\&-\left. {\left( {C_{13}^{(j)} +C_{44}^{(j)} } \right) } \right] -e_{33}^{(j)} \left[ {2C_{44}^{(j)} e_{15}^{(j)} +C_{11}^{(j)} e_{33}^{(j)} -2\left( {C_{13}^{(j)} +C_{44}^{(j)} } \right) \left( {e_{15}^{(j)} +e_{31}^{(j)} } \right) } \right] , \nonumber \\ c_2^{(j)}= & {} -C_{44}^{(j)} \left[ {C_{11}^{(j)} \in _{33}^{(j)} +\left( {e_{15}^{(j)} +e_{31}^{(j)} } \right) ^{2}} \right] -\in _{11}^{(j)} \left[ {C_{11}^{(j)} C_{33}^{(j)} +\left( {C_{44}^{(j)} } \right) ^{2}} \right. \nonumber \\&-\left. {\left( {C_{13}^{(j)} +C_{44}^{(j)} } \right) } \right] -e_{15}^{(j)} \left[ {2C_{11}^{(j)} e_{33}^{(j)} +C_{44}^{(j)} e_{15}^{(j)} -2\left( {C_{13}^{(j)} +C_{44}^{(j)} } \right) \left( {e_{15}^{(j)} +e_{31}^{(j)} } \right) } \right] , \nonumber \\ c_3^{(j)}= & {} -C_{11}^{(j)} \left[ {\left( {e_{15}^{(j)} } \right) ^{2}+C_{44}^{(j)} \in _{11}^{(j)} } \right] . \end{aligned}$$

  (77)

• 3. Expressions of $\alpha _{mn}^{(j)} (j=1,2,m,n=1,2,3)$ appearing in Eq. (7)

  $$\begin{aligned} \alpha _{11}^{(j)}= & {} \left( {C_{13}^{(j)} +C_{44}^{(j)} } \right) \in _{11}^{(j)} +\left( {e_{15}^{(j)} +e_{31}^{(j)} } \right) e_{15}^{(j)}, \nonumber \\ \alpha _{12}^{(j)}= & {} \left( {C_{13}^{(j)} +C_{44}^{(j)} } \right) \in _{33}^{(j)} +\left( {e_{15}^{(j)} +e_{31}^{(j)} } \right) e_{33}^{(j)}, \end{aligned}$$

  (78)
  $$\begin{aligned} \alpha _{21}^{(j)}= & {} -C_{11}^{(j)} \in _{11}^{(j)}, \nonumber \\ \alpha _{22}^{(j)}= & {} -C_{11}^{(j)} \in _{33}^{(j)} -\left( {e_{15}^{(j)} +e_{31}^{(j)} } \right) ^{2}-C_{44}^{(j)} \in _{11}^{(j)}, \nonumber \\ \alpha _{23}^{(j)}= & {} -C_{44}^{(j)} \in _{33}^{(j)}, \end{aligned}$$

  (79)
  $$\begin{aligned} \alpha _{31}^{(j)}= & {} -C_{11}^{(j)} e_{15}^{(j)}, \nonumber \\ \alpha _{32}^{(j)}= & {} -C_{11}^{(j)} e_{33}^{(j)} +C_{13}^{(j)} \left( {e_{15}^{(j)} +e_{31}^{(j)} } \right) ^{2}+C_{44}^{(j)} e_{31}^{(j)}, \nonumber \\ \alpha _{33}^{(j)}= & {} -C_{44}^{(j)} e_{33}^{(j)}. \end{aligned}$$

  (80)

• 4. Expressions of $\eta _{k,xx}^{(j)}, \, \eta _{k,zz}^{(j)},\,\eta _{k,xz}^{(j)},\,\eta _{k,dx}^{(j)}$ and $\eta _{k,dz}^{(j)}\left( {j=1,2,k=1,2,3} \right)$ appearing in Eqs. (36)–(40)

  $$\begin{aligned} \eta _{k,xx}^{(j)}= & {} -C_{11} +C_{13} \beta _{k1}^{(j)} \gamma _k^{(j)} +e_{31} \beta _{k2}^{(j)} \gamma _k^{(j)} \qquad (j=1,2), \end{aligned}$$

  (81)
  $$\begin{aligned} \eta _{k,zz}^{(j)}= & {} -C_{13} +C_{33} \beta _{k1}^{(j)} \gamma _k^{(j)} +e_{33} \beta _{k2}^{(j)} \gamma _k^{(j)} \qquad (j=1,2), \end{aligned}$$

  (82)
  $$\begin{aligned} \eta _{k,xz}^{(j)}= & {} -\left( {C_{44} \gamma _k^{(j)} +C_{44} \beta _{k1}^{(j)} +e_{15} \beta _{k2}^{(j)} } \right) \qquad (j=1,2), \end{aligned}$$

  (83)
  $$\begin{aligned} \eta _{k,dx}^{(j)}= & {} -\left( {e_{15} \gamma _k^{(j)} +e_{15} \beta _{k1}^{(j)} -\in _{11} \beta _{k2}^{(j)} } \right) \qquad (j=1,2), \end{aligned}$$

  (84)
  $$\begin{aligned} \eta _{k,dz}^{(j)}= & {} -e_{31} +e_{33} \beta _{k1}^{(j)} \gamma _k^{(j)} -\in _{33} \beta _{k2}^{(j)} \gamma _k^{(j)} \qquad (j=1,2). \end{aligned}$$

  (85)

• 5. Expressions of the elements of matrices $\mathbf{D}_j =(d_{nk}^{(j)} )_{3\times 3} (j=1,2)$ appearing in Eq. (41)

  $$\begin{aligned} d_{1k}^{(j)} =1, d_{2k}^{(j)} =\eta _{k,zz}^{(j)}, d_{3k}^{(j)} =\eta _{k,dz}^{(j)} \quad \left( {j=1,2,k=1,2,3} \right) . \end{aligned}$$

  (86)