Stress distribution of a sector of cylindrical sandwich shell with FG-CNT core and piezoelectric face sheets subjected to blast pressure

ABSTRACT

A novel layerwise theory along with higher-order shear deformation theory is studied to determine the stress distribution in a three-layer simply supported sector of cylindrical sandwich shell with piezoelectric face sheets and functionally graded carbon nanotube (FG-CNT) core. The out-of-shell displacement of the sandwich shell at each layer is assumed to be a quadratic polynomial function of the radial component in addition to a function of the coordinate components within the shell. The sandwich shell is subjected to the internal blast pressure so that the positive and negative phases of the pressure are considered. The 19 equations of motion have been derived by Hamilton’s principle and Maxwell’s static equations. The results indicate that the thickness, length and opening angle of the cylindrical shell have more effect on the stress distribution. In addition, the type of FG-CNT core has an effect on stress components in terms of time.

KEYWORDS:

• layerwise theory
• blast pressure
• cylindrical sandwich shell

Nomenclature

b	=	The coefficient describing the rate of decay of the pressure-time curve.
$C_{ij}^{(k)}$	=	The effective elastic constant in kth piezoelectric layer (k=1, 3, and i, j= 1, …, 6).
$D_i$	=	The electric displacement components (i= 1, 2, 3).
dmax	=	The maximum contact depth.
E0	=	The elastic moduli of the spherical ball.
Eeff	=	The effective elastic modulus.
$E_i^{\left(k\right)}$	=	The electric field density (i= 1, 2, 3) in kth layer.
$e_i$	=	The piezoelectric constant (i= 1, 2, 3, 4, 5).
$E_{ii}^{CNT}$	=	The elastic modulus of the FG-CNT (i, j= 1, 2, 3).
Em	=	The elastic modulus of the matrix.
$G_{ij}^{CNT}$	=	The shear modulus of the FG-CNT (i, j= 1, 2, 3).
Gm	=	The shear modulus of the matrix.
h	=	The thickness of the sandwich shell.
h1	=	The thickness of the bottom face sheet.
h2	=	The thickness of core.
h3	=	The thickness of the top face sheet.
Pmax	=	The peak over-pressure.
Pmin	=	The peak under-pressure.
$Q_{ij}^{(2)}$	=	The core stiffness tensor (i, j= 1, …, 6).
R	=	The radius of the middle layer of the core.
td	=	The positive phase duration.
tn	=	The negative phase duration.
u0	=	The displacement of the core’s mid-surface along ξ1.
$u_1^{\left(k\right)}$	=	The displacement components along ξ1.
$u_2^{\left(k\right)}$	=	The displacement components along ξ2.
$u_3^{\left(k\right)}$	=	The displacement components along ξ.
v0	=	The displacement of the core’s mid-surface along ξ2.
V0	=	The initial velocity of the spherical ball.
Vm	=	The volume fraction of the matrix.
$V_{CNT}$	=	The volume fraction of the FG-CNT.
w0	=	The displacement of the core’s mid-surface along ξ.
${\alpha }_i^{\left(k\right)}$	=	The mass fraction of the FG-CNT.
${\beta }_i^{\left(k\right)}$	=	The coefficients should be calculated in kth layer (i= 1, 2, 3).
${\gamma }_{ij}^{\left(k\right)}$	=	The coefficients should be calculated in kth layer (i= 1, 2).
${\gamma }_{ij}^{\left(k\right)}$	=	The shear strain in kth layer (i= 1, 2, 3).
δK	=	The virtual kinetic energy.
δU	=	The virtual strain energy.
δW	=	The virtual work.
${\epsilon }_i^{\left(k\right)}$	=	The normal strain in kth layer.
${\eta }_1$	=	The efficiency CNT/matrix parameter number 1.
${\eta }_2$	=	The efficiency CNT/matrix parameter number 2.
${\eta }_3$	=	The efficiency CNT/matrix parameter number 3.
${\lambda }_i^{\left(k\right)}$	=	The coefficients should be calculated in kth layer (i= 1, 2).
${\mu }_i$	=	The dielectric constant (i= 1, 2, 3).
ν0	=	The Poisson’s ratio of the spherical ball.
νm	=	The Poisson’s ratio of the matrix.
${\nu }_{ij}^{CNT}$	=	The Poisson’s ratio of the FG-CNT (i, j= 1, 2, 3).
${\rho }^{CNT}$	=	The density of the FG-CNT.
Φ(k)	=	The electric potential (k=1, 3).
${\psi }_i^{\left(k\right)}$	=	The coefficients should be calculated in kth layer (i= 1, 2, 3).

Additional information

Funding

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.