Stresses in inhomogeneous elastic–viscoelastic–elastic sandwich plates via hyperbolic shear deformation theory

Abstract

A hyperbolic shear deformation theory is proposed to investigate the bending analysis of inhomogeneous elastic/viscoelastic/elastic (EVE) sandwich plates. The sandwich plate is consisting of two elastic material faces and viscoelastic material core. Three kinds of symmetric sandwich plates that are classified depending on thickness of each layer are presented. A fourth type of sandwich plate is considered without viscoelastic core. The equilibrium equations have been solved by using Illyushin’s approximation method as well as the effective moduli method. The deflection and stresses of simply supported EVE sandwich plates have been presented due to a hyperbolic theory and compared with those due to other familiar plate theories. The effect of different parameters on bending analysis of sandwich plates is discussed.

Appendices

Appendix 1

$$C_{11} = - \alpha^{2} A_{11} - \beta^{2} A_{66} ,$$

(42)

$$C_{12} = - \alpha \beta \left( {A_{12} + A_{66} } \right),$$

(43)

$$C_{13} = \alpha \left[ {\alpha^{2} B_{11} + \beta^{2} \left( {B_{12} + 2B_{66} } \right)} \right],$$

(44)

$$C_{14} = - \alpha^{2} D_{11} - \beta^{2} D_{66} + \cosh \left( {\frac{1}{2}} \right)\left( {\alpha^{2} B_{11} + \beta^{2} B_{66} } \right),$$

(45)

$$C_{15} = C_{24} = - \alpha \beta \left( {D_{12} + D_{66} } \right) + \alpha \beta \cosh \left( {\frac{1}{2}} \right)\left( {B_{12} + B_{66} } \right),$$

(46)

$$C_{22} = - \alpha^{2} A_{66} - \beta^{2} A_{11} ,$$

(47)

$$C_{23} = \beta \left[ {\alpha^{2} \left( {B_{12} + 2B_{66} } \right) + \beta^{2} B_{11} } \right],$$

(48)

$$C_{25} = - \alpha^{2} D_{66} - \beta^{2} D_{11} + \cosh \left( {\frac{1}{2}} \right)\left[ {\alpha^{2} B_{66} + \beta^{2} B_{11} } \right],$$

(49)

$$C_{33} = - \alpha^{4} F_{11} - 2\alpha^{2} \beta^{2} \left( {F_{12} + 2F_{66} } \right) - \beta^{4} F_{11} ,$$

(50)

$$C_{34} = \alpha \left[ {\alpha^{2} H_{11} + \beta^{2} \left( {H_{12} + 2H_{66} } \right)} \right] - \alpha \cosh \left( {\frac{1}{2}} \right)\left[ {\alpha^{2} F_{11} + \beta^{2} \left( {F_{12} + 2F_{66} } \right)} \right],$$

(51)

$$C_{35} = \beta \left[ {\beta^{2} H_{11} + \alpha^{2} \left( {H_{12} + 2H_{66} } \right)} \right] - \beta \cosh \left( {\frac{1}{2}} \right)\left[ {\beta^{2} F_{11} + \alpha^{2} \left( {F_{12} + 2F_{66} } \right)} \right],$$

(52)

$$\begin{aligned} C_{44} & = - \alpha^{2} S_{11} - \beta^{2} S_{66} + L_{44} + \cosh \left( {\frac{1}{2}} \right)\left[ {2\alpha^{2} H_{11} + 2\beta^{2} H_{66} - 2G_{44} } \right. \\ & \quad \left. { + \cosh \left( {\frac{1}{2}} \right)\left( {J_{44} - \alpha^{2} F_{11} - \beta^{2} F_{66} } \right)} \right], \\ \end{aligned}$$

(53)

$$C_{45} = - \alpha \beta \left( {S_{12} + S_{66} } \right) + \alpha \beta \cosh \left( {\frac{1}{2}} \right)\left[ {H_{12} + 2H_{66} - \cosh \left( {\frac{1}{2}} \right)\left( {F_{12} + F_{66} } \right)} \right],$$

(54)

$$\begin{aligned} C_{55} & = - \alpha^{2} E_{66} - \beta^{2} E_{22} + L_{44} \\ & \quad + \cosh \left( {\frac{1}{2}} \right)\left[ {2\alpha^{2} H_{66} + 2\beta^{2} H_{11} - 2G_{44} + \cosh \left( {\frac{1}{2}} \right)\left( {J_{44} - \alpha^{2} F_{66} - \beta^{2} F_{11} } \right)} \right] \\ \end{aligned}$$

(55)

Appendix 2

The functions $\pi \left( t \right)$ and $g_{{\mu_{m} }} \left( t \right)$ may be derived by deducing Laplace–Carson transform of these functions from known Laplace–Carson transform of function $\omega \left( t \right)$ as

$$\omega^{*} \left( s \right) = s\mathop \int \limits_{0}^{\infty } \omega \left( t \right){\text{e}}^{ - st} {\text{d}}t,$$

(56)

using Eq. (28), one gets

$$\omega^{*} \left( s \right) = s\mathop \int \limits_{0}^{\infty } \left( {c_{1} + c_{2} {\text{e}}^{{ - t/t_{s} }} } \right){\text{e}}^{ - st} {\text{d}}t,$$

(57)

then by integrating the above function, we get

$$\omega^{*} \left( s \right) = - s\left[ {\left. {\frac{{c_{1} }}{s}{\text{e}}^{ - st} } \right|_{0}^{\infty } + \left. {\frac{{c_{2} }}{\alpha + s}{\text{e}}^{{ - \left( {\frac{1}{{t_{s} }} + s} \right)t}} } \right|_{0}^{\infty } } \right] = c_{1} + \frac{{c_{2} s}}{\alpha + s},$$

(58)

but we have

$$\pi^{*} \left( s \right) = \frac{1}{{\omega^{*} \left( s \right)}} = \frac{1}{{c_{1} + \frac{{c_{2} s}}{\alpha + s}}},$$

(59)

then

$$\pi^{*} \left( s \right) = \frac{\alpha + s}{{c_{1} \alpha + \left( {c_{1} + c_{2} } \right)s}} = \frac{1}{{c_{1} }}\left( {\frac{{c_{1} \alpha + \left( {c_{1} + c_{2} } \right)s - c_{2} s}}{{c_{1} \alpha + \left( {c_{1} + c_{2} } \right)s}}} \right),$$

(60)

or

$$\pi^{*} \left( s \right) = \frac{1}{{c_{1} }}\left( {1 - \frac{{c_{2} st_{s} }}{{c_{1} + \left( {c_{1} + c_{2} } \right)st_{s} }}} \right) = \frac{1}{{c_{1} }}\left( {1 - c_{3} \frac{{st_{s} }}{{c_{4} + st_{s} }}} \right),$$

(61)

where

$$c_{3} = \frac{{c_{2} }}{{c_{1} + c_{2} }}, c_{4} = \frac{{c_{1} }}{{c_{1} + c_{2} }}.$$

(62)

So, $\pi \left( t \right)$ can be obtained by using inverse Laplace–Carson as

$$\pi \left( t \right) = \frac{1}{{c_{1} }}\left( {1 - c_{3} {\text{e}}^{{ - c_{4} \tau }} } \right), \tau = t/t_{s} .$$

(63)

Similarly,

$$g_{{\mu_{m} }}^{*} \left( s \right) = \frac{1}{{1 + \mu_{m} \omega^{*} \left( s \right)}} = \frac{1}{{1 + \mu_{m} \left( {c_{1} + \frac{{c_{2} s}}{\alpha + s}} \right)}}, m = 1,2,$$

(64)

or

$$g_{{\mu_{m} }}^{*} \left( s \right) = \frac{1}{{1 + c_{1} \mu_{m} }}\left( {1 - \frac{{c_{2} \mu_{m} s}}{{\alpha \left( {1 + c_{1} \mu_{m} } \right) + s\left[ {1 + \left( {c_{1} + c_{2} } \right)\mu_{m} } \right]}}} \right),$$

(65)

where $\mu_{1} = \frac{1}{2}$ and $\mu_{2} = 2$. So,

$$g_{{\mu_{m} }}^{*} \left( s \right) = \frac{1}{{1 + c_{1} \mu_{m} }}\left( {1 - c_{5}^{m} \frac{{st_{s} }}{{c_{6}^{m} + st_{s} }}} \right),$$

(66)

in which

$$c_{5}^{m} = \frac{{c_{2} \mu_{m} }}{{1 + \left( {c_{1} + c_{2} } \right)\mu_{m} }}, c_{6}^{m} = \frac{{1 + c_{1} \mu_{m} }}{{1 + \left( {c_{1} + c_{2} } \right)\mu_{m} }},$$

(67)

and then $g_{{\mu_{m} }} \left( t \right)$ may be obtained in its final form as

$$g_{{\mu_{m} }} \left( t \right) = \frac{1}{{1 + c_{1} \mu_{m} }}\left( {1 - c_{5}^{m} {\text{e}}^{{ - c_{6}^{m} \tau }} } \right).$$

(68)