Size-dependent electro-static analysis of smart micro-sandwich panels with functionally graded core
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O R I G I NA L PA P E R
M. Shaban
· H. Mazaheri
Size-dependent electro-static analysis of smart micro-sandwich panels with functionally graded core
Received: 25 February 2020 / Revised: 7 July 2020 © Springer-Verlag GmbH Austria, part of Springer Nature 2020
Abstract This paper investigates the electro-static response of smart micro-sandwich panels with functionally graded core, nano-composite facing, and piezoelectric layers. To consider the size-dependent effect, nonlocal theory is adopted to governing equations and boundary conditions. Three-dimensional approach is implemented that takes into account shear effect of thick core and drastic changes of sandwich panel and free from spurious constraints of equivalent plate theories. Closed-form solution is obtained in systematic step-by-step procedure for smart micro-panels to solve the coupled nonlocal differential equations analytically. The obtained results are compared with finite element results for the case of isotropic thick plate and excellent agreement between result is revealed. A detailed parametric study is conducted to bring out the effect of size-dependent parameter, functionally graded material, length-to-thickness ratio, and aspect ratio on the bending behavior of smart micro-sandwich panels.
List of symbols A ai j B [c] D [e] E
Length Displacement coefficients Width Stiffness matrix Electrical displacement vector Piezoelectric constant matrix Electric field vector
E 1CNT E MATRIX Em Ec hi n, m p, q q0 si
Elastic modulus of CNT Elastic modulus of matrix Young modulus of metal phase Young modulus of metal phase Layer thickness Half wave numbers in the x1 and x2 direction Constant of wave number Load amplitude Real root of characteristic equation
M. Shaban (B) · H. Mazaheri Mechanical Engineering Department, Bu-Ali Sina University, Hamadan, Iran E-mail: [email protected] H. Mazaheri E-mail: [email protected]
M. Shaban, H. Mazaheri
ui Ui V VCNT Vm U¯ , V¯ , W¯ xi
Displacement components Displacement component of x3 Voltage difference in actuator Volume fractions of CNTs Volume fractions of matrix Dimensionless displacement components Global Cartesian coordinate
Greek symbols α δ {ε} γi [η] ηi μ, λ ν { σ} σ¯ i j ςi j τ¯i j ξi j
ψ
FG gradient material Nonlocal parameter Strain vector Complex root of characteristic equation Dielectric constant matrix Efficiency parameter for volume fraction of nanotube Lame constants Poisson’s ration Stress vector Dimensionless normal stress component Stress coefficients Dimensionless shear stress component Displacement constant coefficient Dimensionless electric potential Electric potential
Subscripts 1,2,3 a b c s t
Cartesian coordinate Actuator Bottom facing Core Sensor Top facing
1 Introduction Micro-sandwich panels can be considered an important feature of many micro-electro-mechanical systems (MEMS) and devices. They have attracted a great deal of interest in recent years especially in the field of mechanical engineering. These structures consist of two facings and a thick core
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