Free vibration analysis of rotating pretwisted composite sandwich conical shells with multiple debonding in hygrothermal environment

Engineering Structures 204 (2020) 110058

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Free vibration analysis of rotating pretwisted composite sandwich conical shells with multiple debonding in hygrothermal environment

T

Tripuresh Deb Singhaa, , Mrutyunjay Routb, Tanmoy Bandyopadhyayc, Amit Karmakarc ⁎

a

Mechanical Engineering Department, Govt. College of Engineering and Textile Technology, Serampore, Hooghly 712201, India Mechanical Engineering Department, Govt. College of Engineering, Bhawanipatna, India c Mechanical Engineering Department, Jadavpur University, Kolkata 700032, India b

ARTICLE INFO

ABSTRACT

Keywords: Finite element Free vibration Sandwich Conical shell Debonding

This paper presents a finite element based method to study the influence of elevated temperature and moisture absorption on the free vibration behavior of rotating pretwisted sandwich conical shells consisting of two composite face-sheets and a synthetic foam core. The sandwich structure is assumed to contain single or multiple debonding present either in the face-sheets or at the interface between the face-sheets and core. The finite element formulation consists of an eight-noded isoparametric shell element having five degrees of freedom at each node. Lagrange’s equation is employed to derive the governing equation for free vibration analysis at moderate rotational speeds while the multi-point constraint algorithm is used to model the debonding region present in the composite sandwich shells. Numerical results are presented to depict the effects twist angle, thickness ratio of core to face-sheets, rotational speed and single or multiple debonding on the natural frequency of the composite sandwich conical shells at elevated temperature and moisture concentration.

1. Introduction Sandwich structures comprising of two composite face-sheets and isotropic core are being increasingly used in modern engineering applications like aerospace, marine and automobile industries owing to their low weight combined with high strength and preferred vibration properties [1]. The face-sheets at the top and bottom of the sandwich structures are comparatively thin but have high stiffness while the thick core is made of comparatively low density homogenous or non-homogenous [2] soft material. However, these structures are mostly made to operate in harsh environmental conditions, especially involving high temperature applications and elevated moisture environments. In general, there is a general reduction in the material properties of both composite face-sheets and isotropic core in addition to the induced thermal and moisture strains owing to the differences in thermal and moisture coefficients of matrix and fibers comprising the laminates. This may sometimes lead to a drastic reduction in strength and stiffness of the sandwich structures, which may in turn influence the dynamic behavior of such structures in the hygrothermal environment. Debonding between two layers in composite face-sheets as well as between the face-sheets and core also degrades stiffness of the sandwich construction components [3]. The interaction between the multiple debonding may also severely deteriorate the service life of sandwich ⁎

shells in adverse working environments. Therefore, it becomes inevitable to study the vibration characteristics of pretwisted rotating sandwich structures containing multiple debonding when made to operate in hygrothermal environments. To predict the response of the composite sandwich panels, various theories along with the methods have been proposed in the published literature [4]. The first order shear deformation theory (FSDT) is simple and capable of predicting good results particularly, for thin sandwich plates. Unlike classical plate theory (CPT), it considers the transverse shear deformation effects in the analysis. Besides, the higher order shear deformation theories (HSDT) are generally suggested for accurate predictions of mechanical behavior of thick sandwich plates, but these theories involve more complicated mathematical formulations than FSDT. Static and dynamic analysis of composite sandwich plates was carried out by Lee and Fan [5] wherein face plates of the composite sandwich structures were modeled based on Mindlin's plate theory and displacement fields of the sandwich core were linearly interpolated in terms of thickness of the face plates. Ahmed [6] and Whitney [7] demonstrated a method for accurate determination of the mechanical behavior of thick composite and sandwich plates, which is an extension of the laminated plate theory. Hwu et al. [8] presented a modified FSDT model for obtaining closed-form solutions of natural frequencies for certain particular problems of sandwich plates and shells. Kim [9,10]

Corresponding author. E-mail address: [email protected] (T.D. Singha).

https://doi.org/10.1016/j.engstruct.2019.110058 Received 18 August 2019; Received in revised form 7 November 2019; Accepted 6 December 2019 0141-0296/ © 2019 Elsevier Ltd. All rights reserved.

Engineering Structures 204 (2020) 110058

T.D. Singha, et al.

developed an efficient FSDT incorporating displacement and stress recovery, and enhanced plate theory to study the structural behavior of layered composite and sandwich panels. Zhai et al. [11] obtained an exact solution for the evaluation of dynamic properties of composite cylindrical shells using the discrete layer model based on FSDT. Meunier and Shenoi [12] studied the effects of the material properties and plate geometries on the natural frequencies of sandwich plates employing Reddy’s first and higher order shear deformation theories. Kant and Swaminathan [13] reported an analytical investigation of the free vibration behavior of sandwich plates using refined HSDT. Nayak et at. [14] developed finite element (FE) formulations based on Reddy’s higher-order theory to study the natural frequencies of sandwich plates considering glass/polyester composite face-sheets and HEREX C70 PVC (polyvinyl chloride) foam core. Garg et al. [15] presented the closedform solutions based on HSDT to study the free vibration characteristics of doubly curved sandwich shells. Rahmani et al. [16] proposed a sandwich plate theory that employs FSDT for composite face-sheets and HSDT for flexible core to study the free vibration characteristics of sandwich cylindrical shells. Sayyad and Ghugal [17] studied the natural frequencies of composite sandwich plates using trigonometric shear deformation theory, which includes the trigonometric function to consider shear deformation effect. Kumar et al. [18] employed an efficient FE model based on higher order zig-zag theory to solve the free vibration problem of multi-layered composites and sandwich shallow shells. Conical shells and panels are found in various essential structural parts of aircraft and naval ships due to their special geometrical aspects. The notable works regarding the study of free vibration characteristics of isotropic and multi-layered composite shallow conical shells with pretwist have been carried out by Liew et al. [19], Lim et al. [20] and Lee et al. [21] using FSDT. Sofiyev [22] investigated the free vibration characteristics of laminated orthotropic conical shells using modified FSDT. Recently, an exhaustive review work on the vibration and buckling of functionally graded material (FGM) conical shells using various theories has been reported by Sofiyev [23]. However, comparatively lesser numbers of studies have been reported on the free vibration analysis of composite sandwich conical shells. Wilkins et al. [24] investigated the symmetrical and unsymmetrical vibration modes of sandwich conical shells with various boundary conditions using Love's first-approximation shell theory wherein the transverse shear effect is incorporated. Bardell et al. [25] developed an FE model to study the vibration characteristics of sandwich conical panels and carried out an experiment to verify the model. Nasihatgozar and Khalili [26] analyzed the vibration and buckling of sandwich truncated conical shells using differential quadrature method. Sofiyev and Osmancelebioglu [27] studied the free vibration behavior of sandwich truncated conical shells containing FGM coatings using FSDT. One of the commonly found damages in the sandwich structures is debonding between the face-sheets and core. Besides, laminated composites, which may be used as face-sheets of the sandwich structures, are prone to damages such as matrix cracking [28–30] and delamination between two plies [31–32]. Tracy and Pardoen [33], Hu and Hwu [34], Schwarts-Givli et al. [35,36], and Burlayenko and Sadowski [37] presented the free vibration analysis of sandwich panels having debonding between core and face-sheet. The refined element in FE formulation can be adopted to investigate the vibration and crippling of delaminated sandwich panels as reported by Chakrabarti and Sheikh [38]. Saraswathy et al. [39] studied analytically and experimentally the vibration response of sandwich beams with multiple debonds at the interface between composite face-sheets and honeycomb core. The analytical study of the influence of strains induced due to hygrothermal environment on the bending, buckling and vibration of composite laminates employing the classical laminated plate theory was carried out by Whitney and Ashton [40]. Sai Ram and Sinha [41,42] developed the FE model on the basis of FSDT for laminated composite plates subjected to hygrothermal load and studied the

dependence of natural frequency on temperature and moisture concentration. Panda et al. [43] studied numerically and experimentally the free vibration characteristics of delaminated composite plates exposed in hygrothermal environments. Patel et al. [44] reported the hygrothermal effects on the static and dynamic behavior of layered composite plates using HSDT. Lo et al. [45], and Jin and Yao [46] constructed the mathematical models on the basis of global-local higher order theory for investigating the response of composite laminates subjected to hygrothermal loads. Li and Yu [47] investigated the effects of temperature on vibration, structural and acoustic characteristics of sandwich structures using piecewise low order shear deformation theory (LSDT). Han et al. [48] studied the free vibration and buckling behavior of foam-filled composite corrugated sandwich plates under thermal loading. Matsunaga [49,50] studied the free vibration and stability of laminated composites and sandwich plates subjected to thermal loading and also compared the 2-D single-layer and 3-D layerwise theories for computation of the stresses in these structures. The influence of temperature and moisture on free vibration and transient response of composite plates and shells with multiple delaminations was investigated by Parhi et al. [51] using the finite element method (FEM). Later, Bandyopadhyay and Karmakar [52], and Bandyopadhyay et al. [53] studied hygrothermal effects on bending and free vibration characteristics of delaminated composite rotating pretwisted conical shells. The authors employed FEM for the analysis of pretwisted conical shells and found a good agreement between the results of FEM analysis and the published analytical results. The recent work by Wang and Yuan [54] establishes efficient use of the quadrature element method (QEM) for vibration analysis of curved and prewisted beam with irregular shaped cross-sections. It is evident from the literature survey that a very few studies on the free vibration characteristics of debonded sandwich shells in hygrothermal environment have been carried out. Moreover, there is no available literature, which deals with the free vibration analysis of rotating composite sandwich conical shells with single or multiple debonding in hygrothermal environment. The present work is intended to study the influence of elevated temperature and moisture absorption on the free vibration behavior of rotating pretwisted composite sandwich conical shells with multiple debonded regions present either in the facesheets or at the interface between the face-sheets and the synthetic foam core. An eight-noded isoparametric element having five degrees of freedom at each node is adopted in the FE formulation. Multi-point constraint algorithm is used to model both the single and multiple debonded regions present among the layers of the composite face-sheets as well at the interface between the face-sheets and core. The solution of the standard eigenvalue problem is obtained by using the QR iteration algorithm. The effects of twist angle, thickness ratio, rotational speed, thickness ratio of face-sheet to core and multiple debonding on the fundamental frequency of the composite sandwich conical shells at varying temperatures and moisture concentrations are studied in detail. A close agreement between FSDT data from this work and HSDT data from the literature suggests that FSDT analysis can be conveniently used for thin sandwich plates and shells without any significant loss of accuracy. Thus present work focuses on FSDT analysis, which involves less computation time and mathematical complexity by assuming a shear correction factor of 5/6. The vibration analysis corresponding to the present model could be useful for the designers of turbomachinery applications as the present work could be regarded as a damage identification problem for predicting the reliability of the operation. The contact effect between the debonded layers has not been taken into account in the present study. 2. Theoretical formulation A sandwich shallow conical shell considered in this analysis is composed of two composite face-sheets and an isotropic core as shown in Fig. 1(a)–(c) wherein, s, L, b0 , p0 ,h, hf , hc , v , o denote the cone 2

Engineering Structures 204 (2020) 110058

T.D. Singha, et al.

Fig. 1. Geometry of sandwich shallow conical shell (a) untwisted and (b) twisted, and (c) cross-sectional view at the fixed end.

length, span length; base width, major and minor radii at the fixed end of the conical shell; total thickness, face-sheets thickness, core thickness, vertex angle and base subtended angle, respectively. The neutral plane and the twist angle ( ) of a sandwich conical shell may be expressed in terms of the curvature radius in y-direction(ry ) and the twisted radius (rxy ) as [55]

isoparametric quadratic shell element having five degrees of freedom (u, v , w, x and y ) per node. The shape functions (Ni ) derived from interpolation polynomial are given as

Ni = (1 +

y2

1 2xy + 2 rxy ry

z=

(1)

L rxy

tan =

y p

+

(2)

4s 2tan2

q = (s

Ni = (1

2 )(1

+

i )/2

(for i = 5, 7)

Ni = (1

2)(1

+

i )/2

(for i = 6, 8)

v

x )tan

|ry| = =

tan2

0

2

{ } = { L} + {

{ L} =

(8)

NL }

(

( ) u x

v y

+

w ry

)(

u y

+

v x

+

2w rxy

)(

u z

+

w x

)

(

v z

+

w y

v ry

)

T

(9) (4)

2

The non-linear strain components { T x

0

2 1.5

{

(q 2

T

The linear strain components { L} are given by,

b02

v

+

(7)

where { } = [u0 v0 w0 x y ] and { e} = [u0i v0i w0i xi yi ] The total strain vector { } for the conical shell element may be defined in terms of the Green-Lagrange relations is given below [56]

(3)

d2z dy 2

1 [p4 p4 q

(6)

[Ni]{ e}

The curvature radius ry (x , y ) can be expressed as d2z 1+ dy 2

(for i = 1, 2, 3, 4)

T

=1

2

1)

i

i=1

x )tan 2v tan 20

b0 ( s

+

i

8

where p (x ) and q (x ) are minor and major radii of the ellipse at that particular cross section, respectively and are expressed as

p=

i )(

{ }=

2

z+q q

+

where and are the local natural coordinates of the element. The mid-surface displacement vector { } of the element can be written in terms of nodal displacement vector { e} and shape function matrix [Ni] of the ith node as

Any cross-section of the conical shell at a distance x- from the fixed end along its span will be an ellipse in the (y-z) plane represented by [19], 2

i )(1

p2 ) y 2 ]1.5

NL}

=

1 2

T y T z

0

(5) in which,

In the present study, the FE formulation uses an eight-noded 3

0

0

T y

0

T x

0

0 T z

NL} are expressed as,

T x T y

x y z

=

1 [ ]{ } 2 (10)

Engineering Structures 204 (2020) 110058

T.D. Singha, et al.

x

v x

u x

=

y

=

u y

z

=

u z

w rxy

+ w rxy

+

v y

w x

+

Now,

T

w ry

v T ry

w y

Now, (11)

{ } = [G]{ e}

y HT 1

(C

m2 n2 n2 m2 2mn 2mn

C0) +

2

{ } (T 1

2

as

where m = cos k and n = sin k , k being fiber orientation angle of the k th lamina; C0 and T0 stand for the reference moisture concentration and temperature which have been considered as 0.00% and 300 K, respectively; C and T are the elevated moisture concentration and temperature, respectively. The total strain energy (UT ) of the sandwich conical shell is given by

{

TKe =

1

1

1

[BL ]T [D][BL ]|J|d d

{ vol

T NL } { 0 }Rot d (vol )

+

{ vol

1 2

{ e }T [G]T [ ]T { 0 }Rot d (vol) + vol

1 2

1

[G]T [MRot ][G]|J|d d [G]T [MHT ][G]|J |d d

(22)

u [u v w ] v d (vol) + w

vol

u [ x y z ][A1 ] v d (vol) w

+

2 z

x

y

x

z

2 x

x

y

x

z

+

2 z

y

z

y

z

2 x

+

2 y

(24)

(25)

1 1

[N ]T [mI ][N ]|J |d d

x [N ]T [A1 ] y d (vol) z

(26)

(27)

where [N ] and [mI ] denote the shape function matrix and inertia matrix, respectively. The dynamic equilibrium equation derived from Lagrange’s equation of motion for the conical shell element considering moderate rotational speed and neglecting the Corioli’s effect in hygrothermal environment is expressed as [58]

[Me ]{ ¨e} + ([K e ]L + [K e ]Rot + [K e ]HT ){ e} = {Fe }Rot + {Fe }HT

(28)

The dynamic equilibrium equation for the conical shell is written in global form as

[M ]{ ¨} + ([K ]L + [K ]Rot + [K ]HT ){ } = {FRot } + {FHT }

(18)

(29)

where [M ], [K ]L and [K ] are the global mass, elastic stiffness and geometric stiffness matrices, respectively. { } and { ¨} are the global displacement and acceleration vector, respectively. Assuming the hygrothermal loading is independent of the centrifugal forces, the initial deflections resulting from hygrothermal loading { }HT is initially computed by solving Eq. (30) iteratively,

where { 0 }Rot and { 0 }HT are the initial stresses due to rotational and hygrothermal effects, respectively. The strain energy resulting from initial stresses given by Eq. (18) can be expressed using Eqs. (10) and (11) as

UIS =

1

1

1

vol

(17)

T NL } { 0 }HT d (vol )

1

1

{Fce} =

where [J ] is the Jacobian matrix. The strain energy arising out from the initial stresses (UIS ) due to rotational and hygrothermal effects is given below,

UIS =

1

1 { e }T [Me ]{ e} + { e }T {Fce} 2

[Me] =

(16)

1

1

1

in which element mass matrix [Me] and element centrifugal force vector {Fce} are expressed as

where [T ], { e}, [K e ]L and [BL] are the thickness coordinate matrix, nodal displacement vector, linear stiffness matrix and strain-displacement matrix, respectively. [T ] and [BL] matrices are given in Appendix B. The matrix { L0} consists of the mid-plane strains and curvatures of the conical shell laminate. The linear stiffness matrix of the conical shell element is expressed as

[K e ]L =

1

The expression of the kinetic energy for the shell element is given as

(15)

= [BL ]{ e}

vol

[A1 ] =

(14)

{ L} = [T ]{ L0}

1 2

2 y

{ L }T { L } d (vol) vol 1

(21)

where is the mass density and [A1 ] is the angular velocity component matrix which is expressed as [57,58]

and strain vectors are expressed as

0 L}

1 1 { e }T [K e ]Rot { e} + { e }T [K e ]HT { e} 2 2

(23)

where UL is the elastic strain energy while UIS is the strain energy arising out of the initial stresses resulting from rotation and hygrothermal strains. The elastic strain energy of the conical shell element is expressed as

= 2 { e }T [K e ]L { e}

(20)

The kinetic energy (TK ) of the conical shell due to rotation is given

TK =

(13)

UT = UL + UIS

1 2

= [MHT ]{ } = [MHT ][G]{ e}

[K e ]HT =

T0 ) (12)

UL =

0 }HT

[K e ]Rot =

m2 n2 n2 m2 2mn 2mn

=

[

where [K e ]Rot and [K e ]HT stand for the elemental geometric stiffness matrices arising out of the initial stresses resulting from rotation and hygrothermal loading, respectively and are expressed as,

x xy

= [MRot ]{ } = [MRot ][G]{ e}

UIS =

where [G] consisting of derivatives of the shape functions is given in Appendix B. The hygrothermal strain components { }HT are expressed as [51]

{ }HT =

0 }Rot

]T {

where [MRot ] and [MHT ] consist of the stress resultants arising out of centrifugal and hygrothermal loading, respectively. Subsequently, using Eq. (20), Eq. (19) can further be modified in the following form

w T z

v z

[

]T {

{ e }T [G]T [ ]T { 0 }HT d (vol)

([KL] + [K ]HT ){ }HT = {FHT }

vol

(19) 4

(30)

Engineering Structures 204 (2020) 110058

T.D. Singha, et al.

Fig. 2. Debonded sandwich conical shells.

The nodal equivalent centrifugal force vector is given by

hsub 0 + z sub 2

[Bij ]sub =

(31)

([KL] + [K ]Rot + [K ]HT ){ }HT = {FRot }

The natural frequencies are determined from the standard eigenvalue problem given as below [59] which is solved using the QR iteration algorithm

hsub 0 + z sub 2

{

j

={

0}

1

hsub 0 + z sub 2

[Dij ]sub =

hsub 0 + z sub 2

=

{FeN } (j = 2, 3)

{M }j = [B]j { 0}1 + (zj [B ]j + [D]j ){ }

{FeN } (j = 2, 3)

0 [D]sub = [Bij ]sub z sub [Bij ]sub + [Dij ]sub

0

0

[Aij ]sub =

[Q¯ ij ]sub dz ,

0 2z sub

[Q¯ ij ]sub dz,

hsub 0 + z sub 2

i , j = 1, 2, 6

hsub 0 + z sub 2

[Q¯ ij ]sub zdz

0 2 (z sub ) [Aij ]sub

i, j = 4, 5 (36)

in which transformed reduced stiffness matrix Q¯ ij is given in Appendix C and ks is the shear correction factor in order to account for the parabolic variation of the shear strains. In this study, shear correction factor (ks ) is considered as 5/6. Fig. 3 shows the details of the discretization of the entire planform of the sandwich conical shell. The entire trapezoidal planform of the conical shell is modeled with 64 number of eight noded isoparametric elements with total 225 number of nodes. 3. Results and discussions

(34)

Finite element codes developed on the basis of the above formulation are employed to study the hygrothermal effects on the free vibration characteristics of multiple debonded composite sandwich conical shells. The parameters , 0 , , L, a, nd, h, hc , hf , v and 0 denote the actual angular speed of rotation, fundamental frequency of a non-rotating shell, non-dimensional rotational speed, span-length, debonding length, number of debonding, shell thickness, core thickness, face-sheet thickness, vertex angle and base subtended angle of the cone comprising the sandwich shell, respectively. The fundamental frequencies of composite sandwich conical shells are obtained for various combinations of twist angle ( = 0°, 15° and 30°), non-dimensional rotational speed ( = / 0 = 0.0, 0.5 and 1.0), thickness ratio of core to facesheets (hc / hf =10, 20, 30, 40, 50, 60, 70, 80, 90 and 100), relative size of debonding (a/ L=0.00, 0.25 and 0.50) and multiple debonding (nd=2, 4 and 6) under gradually increasing thermal and moisture strains. In this study, synthetic foam core† (Syntac 350) is considered and its material

0 (35)

where hsub 0 + z sub 2

0 2 z sub ) dz ,

hsub 0 + z sub 2

[Sij ]sub = ks

0 [Sij ]sub

[Q¯ ij ]sub z 2dz

hsub 0 + z sub 2

where {FeN } is the hygrothermal stress resultants. The elasticity matrix for any such sub-laminate enveloping the debonded zone can be expressed as 0 [Aij ]sub z sub [Aij ]sub + [Bij ]sub

[Q¯ ij ]sub (z

hsub 0 + z sub 2

where { 0}j and { } represents strain vector and the curvature vector being identical at the crack tip for elements 1, 2 and 3, wherein z1 is equal to zero for the element 1. The in-plane stress resultants {N }j and the moment resultants {M }j of elements 2 and 3 considering hygrothermal load are given as,

{N }j = [A]j { 0}1 + (zj [A]j + [B ]j ){ }

0 z sub [Aij ]sub

hsub 0 + z sub 2

(33)

+ zj { } (j = 2, 3)

i, j = 1, 2, 6

hsub 0 + z sub 2

where [ ] = ([KL] + [K ]Rot + [K ]HT ) 1 [M ], and = 1/ 2 . A debonding of length a is assumed to be present at a particular distance d from the fixed end along the span and extends entirely along the cordwise-length of the sandwich conical shell as shown in Fig. 2(a). Fig. 2(b) shows a debonded element at the tip of the crack in which the intact lamina of thickness h is modeled as a single element 1 while the debonded laminates are modeled by two distinct sub-laminates 2 and 3 of thicknesses (hsub )2 and (hsub )3, respectively bonded together at the debonded zone [31]. The mid-plane strains between elements 1, 2 and 3 are related as [31] 0}

0 z sub ) dz,

[Q¯ ij ]sub zdz

=

(32)

[ ]{ } = { }

[Q¯ ij ]sub (z

hsub 0 + z sub 2

i, j = 1, 2, 6

hsub 0 + z sub 2

5

Engineering Structures 204 (2020) 110058

T.D. Singha, et al.

Fig. 3. Discretized planform of the sandwich conical shell.

The fundamental frequencies are calculated assuming the original material properties (i.e., T =300 K and C =0.0%), while the reduced material properties of the composite face-sheets are taken into consideration only for debonding and mode shape calculations. The relevant geometrical parameters of the cantilevered composite sandwich conical shell considered are, s=0.4 m, L /s=0.7, v = 0=20°. The boundary conditions considered for the sandwich conical shells are expressed as

Table 1a Elastic moduli of graphite/epoxy composites (face-sheets†) at different temperatures: G13f = G12f , G23f = 0.5G12f , 0.3 × 10-6/K, 12f =0.3, 1f = -6 3 2f =28.1 × 10 /K, f =1600 kg/m [42]. Elastic moduli (GPa)

E1f

300

325

350

375

400

425

130

130

130

130

130

130

6.0

6.0

5.5

5.0

4.75

4.5

9.5

E2f

G12f †

Temperature, T (K)

8.5

8.0

7.5

7.0

At x = 0,

6.75

E1f E2f

kg/m3 [42].

G12f

Moisture concentration, C (%) 0.00

0.25

0.50

0.75

1.00

1.25

1.50

130

130

130

130

130

130

130

6.0

6.0

6.0

6.0

6.0

6.0

6.0

9.5

9.25

9.0

8.75

8.5

8.5

=

y

(37)

=0

The finite element codes developed are validated with those published results to predict its capability in accurate calculations. Table 2 presents the non-dimensional frequencies of simply supported laminated plates (0°/90°/90°/0°) which are compared with those results obtained by Sai Ram and Sinha [42] forC = 0.1% andT = 325 K. The non-dimensional fundamental frequencies of pretwisted shallow conical

Table 1b Elastic moduli of graphite/epoxy composites (face-sheets) at different moisture concentrations: G13f = G12f , G23f = 0.5G12f , 12= 0.3, 1f =0, 2f =0.44, Elastic moduli (GPa)

x

3.1. Validation of results

The subscripts ‘f’ and ‘c’ denote face-sheets and core, respectively.

f =1600

u=v=w=

Table 2 Non-dimensional frequencies( = n L2 / E2 h2 ) , L / b =1, L / h ==100, for (0°/ 90°/90°/0°) simply-supported graphite-epoxy laminated plates.

8.5

properties are [60]: E1c = E2c=2.25 GPa, G12c = G13c = G23c=1.02 GPa, -6 3 12c=0.31, 1c = 2c=31.6 × 10 /°C, 1c = 2c=0.4, c =600 kg/m . The reduced material properties of the composite face-sheets at different temperatures and moisture concentrations are given in Table 1(a) and 1(b), respectively [42]. 6

Mode Number

C = 0.1%

T = 325 K

Sai Ram and Sinha [42]

Present FEM (8 × 8)

Sai Ram and Sinha [42]

Present FEM (8 × 8)

1 2 3

9.429 20.679 40.068

9.395 19.944 39.439

8.088 19.196 39.324

8.049 18.409 38.686

Engineering Structures 204 (2020) 110058

T.D. Singha, et al.

Table 3 Non-dimensional fundamental frequencies ( = n b02 h/ D ) , D = Eh3/ 12(1 − ν2) of pretwisted shallow conical shell with = 0.3, s / h = 1000, v=15°, 0 =30°. Twist angle ( )

Aspect ratio (L/ s )

Liew et al. [19]

Present FEM (8 × 8)

0°

0.6 0.7 0.8

0.35997 0.30608 0.27832

0.34241 0.29412 0.26915

0.6 0.7 0.8

0.28828 0.25752 0.24179

0.27855 0.25077 0.23642

30°

Table 5 Dimensionless fundamental frequencies,

Mode

1 2 3 4

and core: Ec =103.63 MPa, Gc =50 MPa,

c = 130

kg/m3,

G12c =G13c =G23c = 3.45 MPa, Shell/ Plate

Present FEM

FSDT

HSDT

(4 × 4)

(6 × 6)

(8 × 8)

(10 × 10)

(12 × 12)

15.22 27.75 30.36 38.43

15.28 28.69 30.01 38.86

15.218 27.809 31.021 38.769

15.217 27.807 30.420 38.516

15.216 27.778 30.385 38.465

15.216 27.769 30.375 38.452

15.216 27.768 30.371 38.448

( / E2 )f of simply sup-

shells are presented in Table 3 and validated against Liew et al. [19] for different twist angles ( ) and aspect ratios (L/ s ) of the conical shells. A convergence study of the natural frequencies of square sandwich plates (0°/90°/0°/core/0°/90°/0°) with mesh size of 4 × 4, 6 × 6, 8 × 8, 10 × 10 and 12 × 12 is furnished in Table 4(a) and validated with those results (both FSDT and HSDT) of Meunier and Shenoi [12]. A similar convergence study of the fundamental frequencies of cantilever sandwich pretwisted conical shells with mesh size of 4 × 4, 6 × 6, 8 × 8, 10 × 10 and 12 × 12 at T =325 K and C =1.0% is also presented in Table 4(b), and the results are found to be converged with a mesh size of 8 × 8. Table 5 furnishes the conformity of the present results with Garg et al. [15], Kant and Swaminathan [13], and Sayyad and Ghugal [17], which predicts the dimensionless fundamental frequencies of composite sandwich plates and doubly-curved shells. The debonding occurring between two layers in composite face-sheets as well at the interface between the composite face-sheet and core are modeled by incorporating proper continuity conditions [31] at the crack front and a related validation is presented with that of Zak et al. [32] in Fig. 4. A comparison of the non-dimensional fundamental frequencies of rotating cantilever isotropic plates for different rotational speeds with those results obtained by Sreenivasamurthy and Ramamurthi [57] is presented in Table 6. The comparisons of the present FEM results

R /L

Present FEM (8 × 8)

12c

= 0.0,

c =97

kg/m3].

Garg et al. [15]

Kant and Swaminathan [13]

HSDT 9

FSDT

Model-1

Sayyad and Ghugal [17]

Cylindrical Shell

5 10 20

19.60864 16.37386 15.55116

20.30291 16.86094 15.88107

20.86146 17.53292 16.59373

– – –

– – –

Spherical Shell

5 10 20

29.25550 19.52856 16.37817

30.44870 20.33837 16.86979

30.82233 20.89829 17.54220

– – –

– – –

15.27614

15.54053

16.26848

15.5093

15.4647

Plate

c=0.32].

Meunier and Shenoi [12]

2 n (L / h)

ported antisymmetric cross-ply laminated sandwich shells and plates (0°/90°/ core/0°/90°)]. [L / h = 100; hc / hf =10, Material properties for face-sheets: E1f =131 GPa, E2f = E3f = 10.34 GPa, G12f =G13f = 6.895 GPa, G23f =6.205 GPa, 12f = 0.22, f =1627 kg/m3 and for core: E1c =E2c = E3c = 6.89 MPa,

Table 4a Dimensionless natural frequencies [ = n (L2 / h) ( / E2 )c ] for simply supported square sandwich plate (0°/90°/0°/core/0°/90°/0°) [L / h = 10, hc / h=0.88 and ks=0.19; Material properties for face-sheets: E1f =24.51 GPa, E2f =7.77 GPa, G12f = G13f = 3.34 GPa, G13f =1.34 GPa, f =1800 kg/m3, f =0.078

=

Fig. 4. The influence of delamination length on the relative frequencies of a single side clamped eight-layer [( ± 45°)4] graphite-epoxy composite plate [32]. [L=0.25 m, b= 0.09 m, h = 0.008 m; Matrix (epoxy resin): E =3.43 GPa, =0.35, =1250 kg/m3; fibers (graphite): E =275.6 GPa, =0.2, =1900 kg/m3; fiber volume fraction = 0.3].

demonstrate excellent agreement with those results of the published literature. 3.2. Effect of twist angle and thickness ratio (s /h ) The variations of fundamental frequencies of the stationary composite sandwich conical shells [0°/90°/0°/core/0°/90°/0°] with temperature and moisture concentration corresponding to twist angle

Table 4b Convergence study for fundamental frequencies (Hz) of stationary sandwich conical shells [0°/90°/0°/core/0°/90°/0°] with twist angle = 30° [s=0.4 m, s / h=100, L / s=0.7, 0 = v=20°] at T =325 K and C =1%. Mode

1 2 3 4

T =325 K

C =1%

4×4

6×6

8×8

10 × 10

12 × 12

4×4

6×6

8×8

10 × 10

12 × 12

40.897 149.103 208.729 430.538

40.307 148.266 206.488 427.228

40.305 148.243 206.431 426.783

40.305 148.238 206.415 426.576

40.305 148.235 206.409 426.411

36.868 144.351 207.406 429.991

36.329 143.434 205.047 427.827

36.317 143.407 204.976 426.395

36.317 143.401 204.951 426.186

36.317 143.399 204.443 426.079

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fundamental frequency with both temperature and moisture concentration increases with an increase in the twist angle. The data so obtained also corroborates the results furnished in the earlier published paper [55]. This reveals that for untwisted sandwich conical shells, the rate of reduction in structural stiffness with respect to both temperature and moisture concentration is minimum and gradually increases with an increase in twist angle. Similarly, the fundamental frequency decreases monotonically with an increase in thickness ratio (s /h ) irrespective of twist angle, temperature and moisture concentration as shown in Fig. 6. The lower values of frequencies in the case of thin composite sandwich conical shells in hygrothermal environment may be attributed to the lower values of the hygrothermal moments and corresponding geometric stiffness compared to the thick panels at all temperatures and moisture concentrations.

Table 6 Non-dimensional fundamental frequencies [ ¯ = n L2 h2/ D ]of an isotropic 2 ) , =0.3] rotating cantilever plate [L / b=1, h/ L=0.12, D = Eh3/12(1 Non-dimensional speed ( )

Sreenivasamurthy and Ramamurthi [57]

Present FEM 8×8

0.0 0.2 0.4 0.6 0.8 1.0

3.43685 3.51858 3.75280 4.12875 4.56786 5.09167

3.41749 3.49765 3.72638 4.07524 4.51211 5.00991

=0°,15° and 30° are illustrated in Fig. 5. The thickness ratio (s /h ) and thickness ratio of core to face-sheets (hc / hf ) are considered as 100 and 10, respectively. It is evident that at any temperature and moisture concentration, the fundamental frequency of the untwisted shell is the maximum and decreases with an increase in twist angle. The general diminishing trends in fundamental frequency with temperature and moisture concentration are affected by the twist angle of the shell as shown in Fig. 5(a) and (b), respectively. The diminishing rate of the

3.3. Effects of thickness ratio of core to face-sheet (hc / hf ) and rotational speed of sandwich shell Tables 7–10 furnish the numerical values of fundamental frequencies of composite sandwich conical shells [0°/90°/0°/core/0°/90°/ 0°] corresponding to non-dimensional rotational speeds ( )=0.0, 0.5

Fig. 5. Effect of twist angle on fundamental frequency of the non-rotating composite sandwich conical shell corresponding to thickness ratio (s / h ) = 100 at varying (a) temperature and (b) moisture concentration. [(0°/90°/0°/core/0°/90°/0°), s=0.4 m, L / s=0.7, 0 = v=20°, hc / hf =10, =0.0].

Fig. 6. Effect of thickness ratio (s / h ) on fundamental frequency of the non-rotating composite sandwich conical shell corresponding to twist angle =15° at varying (a) temperature and (b) moisture concentration. [(0°/90°/0°/core/0°/90°/0°), s=0.4 m, L / s=0.7, 0 = v=20°, hc / hf =10, =0.0].

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Table 7 Fundamental frequencies (Hz) of rotating sandwich conical shells [0°/90°/0°/core/0°/90°/0°] with twist angle = 0° [s=0.4 m, s / h=100, L / s=0.7, different temperatures (K). h c /hf

10 20 30 40 50 60 70 80 90 100

= 0.0

= 0.5

10 20 30 40 50 60 70 80 90 100

at

350 K

400 K

425 K

300 K

350 K

400 K

425 K

300 K

350 K

400 K

425 K

68.29 54.47 47.56 43.33 40.43 38.31 36.69 35.40 34.35 33.48

66.47 53.20 46.64 43.10 40.23 38.13 36.52 35.24 34.20 33.34

64.59 51.89 45.69 42.87 40.02 37.93 36.34 35.08 34.05 33.20

63.63 51.21 45.21 42.75 39.90 37.84 36.25 35.00 33.98 33.13

87.86 71.00 62.31 56.91 53.20 50.46 48.36 46.69 45.33 44.19

85.55 69.37 61.13 56.62 52.93 50.22 48.14 46.49 45.14 44.01

83.17 67.70 59.91 56.32 52.66 49.98 47.91 46.28 44.95 43.83

81.95 66.84 59.29 56.17 52.52 49.85 47.80 46.17 44.89 43.74

129.08 105.40 92.86 85.00 79.54 75.52 72.41 69.94 67.93 66.24

125.76 103.03 91.13 84.56 79.15 75.16 72.09 69.64 67.65 65.98

122.32 100.59 89.35 84.13 78.75 74.80 71.76 69.34 67.36 65.71

120.60 99.33 88.44 83.91 78.55 74.62 71.59 69.18 67.22 65.57

thickness ratio of core to face-sheets (hc / hf ) at any rotational speed, temperature and moisture concentration. The rate of reduction (i.e., slope of the curve) of the fundamental frequency gradually decreases with an increase in thickness ratio of core to face-sheets (hc / hf ) . This reduction with an increase in the core thickness (hc ) relative to facesheet thickness (hf ) may be attributed to the lower stiffness values of the isotropic foam core compared to the composite face-sheets in the sandwich conical shell. It may also be noted that the deviation in fun-

= 0.0

= 0.5

0

=

v=20°]

at

= 1.0

300 K

350 K

400 K

425 K

300 K

350 K

400 K

425 K

300 K

350 K

400 K

425 K

64.66 51.59 45.06 41.06 38.32 36.32 34.78 33.56 32.57 31.74

57.57 47.00 42.21 39.49 36.90 35.02 33.59 32.47 31.55 30.73

50.14 41.97 38.55 37.59 35.22 33.51 32.22 31.22 30.40 29.57

46.57 39.75 36.78 35.48 34.25 32.65 31.45 30.52 29.76 28.93

83.99 67.93 58.06 54.48 50.93 48.32 46.32 44.72 43.42 42.33

76.92 61.97 55.93 52.43 49.07 46.63 44.76 43.28 42.08 40.99

69.85 55.21 52.36 49.94 47.76 44.64 42.95 41.63 40.56 39.46

66.55 52.49 50.57 48.39 45.58 43.51 41.93 40.71 39.72 38.61

124.39 101.63 89.59 82.00 76.75 72.88 69.89 67.51 65.57 63.94

111.24 92.92 84.13 78.97 74.00 70.37 67.58 65.38 63.59 61.95

97.47 85.40 78.05 75.31 70.74 67.43 64.91 62.93 61.33 59.69

91.64 81.42 75.47 73.15 68.85 65.75 63.40 61.57 60.08 58.43

Table 9 Fundamental frequencies (Hz) of rotating sandwich conical shell [0°/90°/0°/Core/0°/90°/0°] with twist angle =0° [s=0.4 m, s / h=100, L / s=0.7, different moisture concentrations (%). h c /hf

v=20°]

300 K

Table 8 Fundamental frequencies (Hz) of rotating sandwich conical shells [0°/90°/0°/core/0°/90°/0°] with twist angle =30°[s=0.4 m, s / h=100, L / s=0.7, different temperatures (K).

10 20 30 40 50 60 70 80 90 100

=

= 1.0

and 1.0 and thickness ratios of core to face-sheets (hc / hf )=10, 20, 30, 40, 50, 60, 70, 80, 90 and 100 at various temperatures and moisture concentrations. Some of these numerical results are presented graphically in Figs. 7 and 8 to show the variation of fundamental frequency with thickness ratio of core to face-sheets and non-dimensional speed of the sandwich conical shell, respectively. It is evident from Tables 7–10 and Fig. 7 that the fundamental frequency of both twisted and untwisted sandwich shell decreases monotonically with an increase in

h c /hf

0

= 0.0

= 0.5

0

=

v=20°]

at

= 1.0

0.25%

0.75%

1.0%

1.5%

0.25%

0.75%

1.0%

1.5%

0.25%

0.75%

1.0%

1.5%

67.15 53.42 46.98 43.18 40.30 38.18 36.57 35.29 34.25 33.38

64.79 52.60 45.79 42.89 39.90 37.92 36.33 35.07 34.04 33.19

63.57 50.92 45.17 42.74 39.76 37.79 36.21 34.96 33.94 33.09

61.04 49.17 43.92 42.43 39.63 37.54 35.98 34.74 33.74 32.90

86.42 69.68 61.63 56.79 53.10 50.30 48.23 46.57 45.27 44.09

83.49 67.59 60.05 56.39 52.45 49.93 47.87 46.23 44.94 43.77

81.92 66.45 59.25 56.11 52.35 49.83 47.78 46.15 44.86 43.70

78.25 64.11 57.62 55.71 52.07 49.47 47.44 45.82 44.54 43.39

126.99 103.43 91.77 84.72 79.29 75.27 72.19 69.74 67.73 66.06

122.68 100.34 89.52 84.16 78.53 74.77 71.73 69.31 67.34 65.69

120.44 98.74 88.36 83.88 78.27 74.51 71.49 69.10 67.14 65.50

112.65 95.45 85.99 83.31 78.01 74.04 71.06 68.69 66.77 65.15

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Table 10 Fundamental frequencies (Hz) of rotating sandwich conical shell [0°/90°/0°/core/0°/90°/0°] with twist angle =30° [s=0.4 m, s / h=100, L / s=0.7, different moisture concentrations (%). h c /hf

10 20 30 40 50 60 70 80 90 100

= 0.0

= 0.5

0

=

v=20°]

at

= 1.0

0.25%

0.75%

1.0%

1.5%

0.25%

0.75%

1.0%

1.5%

0.25%

0.75%

1.0%

1.5%

60.58 48.60 43.30 40.08 37.44 35.51 34.03 32.87 31.93 31.11

54.67 43.31 38.97 37.74 35.36 33.65 32.32 31.31 30.49 29.67

50.12 41.25 36.95 36.32 34.12 32.55 31.33 30.41 29.67 28.85

40.89 35.88 33.27 32.65 31.02 29.88 28.98 28.32 27.78 26.97

78.72 64.10 57.35 53.22 49.85 47.35 45.35 43.76 42.52 41.49

71.05 58.44 51.62 50.22 47.03 44.87 43.16 41.74 40.68 39.59

67.81 55.17 50.50 48.26 45.38 43.32 41.77 40.56 39.59 38.50

59.66 48.92 47.72 43.54 42.38 39.86 38.66 37.81 37.11 36.02

116.83 95.94 86.23 80.11 75.04 71.32 68.44 66.17 64.32 62.69

105.25 88.10 77.88 75.60 71.02 67.69 65.10 63.11 61.50 59.88

100.59 82.14 74.68 71.59 65.82 63.12 61.00 59.42 58.14 56.52

91.37 73.63 69.82 65.71 62.56 60.33 58.55 57.25 56.20 54.57

damental frequencies at two temperatures (T =300 K and 425 K) or moisture concentrations (C =0.25% and 1.50%) of the sandwich conical shell for any twist angle and rotational speed gradually decreases with an increase in thickness ratio of core to face sheets (hc / hf ) especially at higher twist angles. This is due to the fact that the composite face-sheets are more prone to the hygrothermal effects compared to the isotropic foam core at a particular twist angle. The thermal and moisture effects

decrease with an increase in the proportion of the core (i.e., an increase in hc / hf ), thereby depicting a decrease in the deviation. The fundamental frequency increases monotonically with an increase in non-dimensional rotational speed in both untwisted and twisted sandwich conical shells for all values of the thickness ratios of core to face-sheets at any temperature and moisture concentration as shown in Tables 7–10 and Fig. 8. This behavior may be explained due to the fact that an

Fig. 7. Variation of fundamental frequency (Hz) with thickness ratio of core to face-sheets (hc / hf ) of the composite sandwich conical shells [(0°/90°/0°/core/0°/90°/ 0°), s=0.4 m, s / h=100, L / s=0.7, 0 = v=20°]

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Fig. 8. Effects of rotational speed and thickness ratio of core to face-sheets (hc / hf ) on fundamental frequency (Hz) of the sandwich conical shells corresponding to different twist angles [(0°/90°/0°/core/0°/90°/0°), [s=0.4 m, s / h=100, L / s=0.7, 0 = v=20°].

increase in rotational speed results in centrifugal stiffening of the sandwich panel resulting in higher values of the fundamental frequency. Moreover, the slope of the curves is slightly higher in the panels with higher proportions of the composite face-sheets. This is a consequence of higher mass density of the composite face-sheets

compared to the low density foam core resulting in higher centrifugal stiffening effects at lower hc / hf values. At all rotational speeds, the higher deviation with an increase in temperature and moisture concentration at lower values of hc / hf represents that stiffening due to rotation and hygrothermal effects are independent of each other.

Fig. 9. Variation of fundamental frequency (Hz) of non-rotating sandwich conical shells with temperature corresponding to different relative sizes of debonding present between the layers in the top face-sheet (FS) and at the interface between the top face-sheet and core (FSC) [0°/90°/core/90°/0°), s=0.4 m, s / h=100, L / s=0.7, 0 = v=20°, hc / hf =40].

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Fig. 10. Variation of fundamental frequency (Hz) of non-rotating sandwich conical shells with moisture concentration corresponding to different relative sizes of debonding present between the layers in the top-face sheet (FS) and at the interface between the top face-sheet and core (FSC) [(0°/90°/core/90°/0°), s=0.4 m, s / h=100, L / s=0.7, 0 = v=20°, hc / hf =40, d/ L =0.5].

Fig. 11. Variation of fundamental frequency (Hz) of non-rotating sandwich conical shell with the relative position (h'/ hf )) of the debonding (a/ L=0.25) across the thickness of the top face-sheet from top surface at temperature T =350 K for (a) hc / hf =10 and (b) hc / hf =20. [(0°/90°)2/core/(90°/0°)2], s=0.4 m, s / h=100, L / s=0.7, 0 = v=20°].

Fig. 12. Variation of fundamental frequency (Hz) of non-rotating sandwich conical shell with the relative position (h'/ hf ) of the debonding (a/ L=0.25) across the thickness of the top face-sheet from top surface at moisture concentration C =0.5% for (a) hc / hf =10 and (b) hc / hf =20. [(0°/90°)2/core/(90°/0°)2, s=0.4 m, s / h=100, L / s=0.7, 0 = v=20°].

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in temperature and moisture concentration irrespective of the location (FS or FSC) and relative size of debonding (a/ L ) in both twisted and untwisted sandwich composite conical shells. In addition, there is also a decrease in the fundamental frequencies with an increase in the relative size of the debonding for both cases (FS and FSC) considered. The rates of decrease of the fundamental frequencies with temperature and moisture concentration are found to diminish slightly with an increase in the relative size of the debonding because of the gradual reduction in structural stiffness. It is also noted that the fundamental frequency is slightly lower in case of the debonding present at the interface between the top composite face-sheet and the core (FSC) than debonding present between the layers in the top composite face-sheet (FS). This can be attributed to the fact that the debonding in case of FSC is nearer to the mid-plane of the sandwich shell compared to the FS case. The variation of the fundamental frequency with the relative position of the single debonding across the thickness (h'/ hf ) of the composite face-sheet from the top of the composite laminate [(0°/90°)2 /core/(90°/0°)2] is presented in Figs. 11 and 12 for T =400 K and C =0.5%, respectively. A gradual reduction in fundamental frequency is observed as the debonding approaches to the mid-plane of the sandwich laminate across the thickness of the face-sheet at elevated temperatures and moisture conditions irrespective of the twist angles and thickness ratio of core to face-sheets (hc / hf ).

Fig. 13. Arrangement of the layers of the composite sandwich conical shell with debonding [(0°/90°)2 /core/(90°/0°)2, d/ L= 0.50, a/ L=0.25].

3.4. Effect of single debonding in sandwich conical shell The variation of fundamental frequency of both untwisted ( =0°) and twisted ( =15°) non-rotating composite sandwich conical shell [0°/ 90°/core/90°/0°] with temperature and moisture concentration corresponding to different relative sizes of debonding (a/ L=0.0, 0.25 and 0.50) present between the layers in the top composite face-sheet (FS) and as well as at the interface between the top composite face-sheet and the core (FSC) are shown in Figs. 9 and 10, respectively. There is a general reduction in the fundamental frequency values with an increase

3.5. Effect of multiple debonding in sandwich conical shell The fundamental frequencies of both stationary and rotating sandwich conical shells having symmetrically located multiple central

Fig. 14. Variation of fundamental frequency (Hz) of rotating composite sandwich conical shell with temperature for different twist angles and number of debonding (nd=2,4 and 6) [(0°/90°)2 /core/(90°/0°)2, =0.5, s=0.4 m, s / h=100, L / s=0.7, 0 = v=20°, a/ L= 0.25, d/ L=0.50]. 13

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Fig. 15. Variation of fundamental frequency (Hz) of composite sandwich conical shell with moisture concentration for different twist angles and number of debonding (nd=2, 4 and 6). [(0°/90°)2/core/(90°/0°)2], = 0.5, s=0.4 m, s / h=100, L / s=0.7, 0 = v=20°, a/ L= 0.25.

debonding sequences as illustrated in Fig. 13 are plotted in Figs. 14 and 15, respectively, for varying temperature and moisture concentration. For all temperatures and moisture concentrations, the highest decrease in fundamental frequency values occurs for multiple debonding (nd= 2) located at the face-sheet core interface (FSC) and is lower for debonding away from the core (nd=4 and nd=6) compared to the intact sandwich conical shells. It can be inferred that the fundamental frequency values are highly dependent on the location of the multiple debonding around the core.

amount of deformation are different. Table 12 presents the three-dimensional surface plots of the first four mode shapes of the pretwisted sandwich conical shells without debonding (a/ L = 0.0) and with single debonding (a/ L = 0.5) at different non-dimensional rotational speeds ( =0.0, 0.5 and 1.0) for temperature, T =300 K. The pattern of the mode shapes (spanwise bending for the first two modes and torsion for third and fourth mode) are not affected by the non-dimensional rotational speed. It is noted that the directions of deformation of the first mode for =0.0 and = 0.5 are same wherein for = 1.0 is opposite to those cases. The direction of deformation of the second and fourth mode shapes of the stationary sandwich conical shell ( = 0.0) is opposite to the rotating ( =0.5 and 1.0) sandwich conical shell. In case of third mode, direction first torsional bending for = 0.0 and = 1.0 are opposite to the = 0.5. It is also found that all four mode shapes of the pretwisted sandwich conical shells without debonding and with single debonding are same.

3.6. Mode shapes The effect of thickness ratio of core to face-sheets (hc / hf ) on the first four mode shapes of pretwisted stationary composite sandwich conical shell at T =300 K and 425 K is depicted in Table 11. In this Table, the three-dimensional surface plots (including two-dimensional contour plots) of the mode shapes are presented to understand the deformation pattern of the conical shell. It is evident that the first two modes correspond to first spanwise bending mode (1B) and second spanwise bending mode (2B) while the third and fourth modes correspond to first torsional (1T) and second torsional mode (2T), respectively, at both temperatures T =300 K and 425 K. The mode shapes of the stationary sandwich conical shell for both hc / hf =10 and hc / hf =50 are similar although the direction of the spanwise bending in the first mode and

4. Conclusions The free vibration characteristics of the composite sandwich conical shells in hygrothermal environment have been studied using FEM. The results obtained from the theoretical model based on FSDT are validated with those results of published literature. The main conclusions drawn from the theoretical investigation are

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Table 11 Effect of temperature and face-sheet to core ratio on the first four mode shapes of composite sandwich conical shells [(0°/90°/0°/core/0°/90°/0°), s=0.4 m, s / h=100, L / s=0.2, 0 = v=20°, = 30° =0.0].

listed as follows:

sandwich conical shell (0°/90°/core/90°/0°) with single debonding present either between the layers in the top composite face-sheet (FS) or at the interface between the composite face-sheet and core (FSC) at elevated temperatures and moisture concentrations. It is found that debonding present either between two layers in the composite face-sheets (FS) or at the interface between the facesheet and core (FSC) reduces the fundamental frequency of the composite sandwich conical shells since overall stiffness of the shell degrades irrespective of the twist angles, temperatures and moisture concentrations. The frequency values in case of FSC are found to be slightly lower than the FS case. (3) The effects of number and position of the symmetrically located multiple debonding on the fundamental frequency of the composite sandwich conical shells [(0°/90°)2 /core/(90°/0°)2] are also investigated. It is found that the reduction in fundamental frequency is highly dependent on the location of the multiple debonding around the core.

(1) The fundamental frequency of composite sandwich conical shell (0°/90°/0°/core/0°/90°/0°) without debonding is studied considering different twist angle, thickness ratio of the conical shell, rotational speed and thickness ratio of core to face-sheets in hygrothermal environment. A general reduction in fundamental frequency is observed with an increase in twist angle and thickness ratio of non-rotating composite sandwich conical shells subjected to hygrothermal loadings. The fundamental frequency also decreases monotonically with an increase in thickness ratio of core to facesheets irrespective of twist angles, temperatures and moisture concentrations. Besides, due to centrifugal stiffening, the fundamental frequency of the shell increases with an increase in rotational speed. (2) The work is extended to investigate the influence of size of single debonding on the fundamental frequency of the composite

15

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Table 12 Effect of rotational speed and debonding on the first four mode shapes of composite sandwich conical shells [(0°/90°/0°/core/0°/90°/0°), s=0.4 m, s / h=100, L / s=0.2, 0 = v=20°, = 30° , hc / hf =10, T = 300 K].

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(4) The effects of thickness ratio of core to face-sheets, temperature and non-dimensional rotational speed on the first four mode shapes of the pretwisted sandwich conical shell without debonding and with debonding are studied. The patterns of the first four mode shapes are similar irrespective of the thickness ratio of core to face-sheets and temperature. The direction of deformation in the first fourth

modes of vibration is influenced by the non-dimensional rotational speed of the sandwich conical shells. Declaration of Competing Interest The authors declared that there is no conflict of interest.

Appendix A. Nomenclature major and minor radii of conical shell at any distance x from fixed end displacement in x , y and z directions mid-surface displacement in x , y and z directions rotations of the cross sections perpendicular to y- and x- axis

q (x ) , p (x ) u, v, w u 0 , v 0, w 0 x, y u, v, w { ¨e } , { ¨}

velocity in x , y and z directions nodal and global acceleration vectors in-plane strain components

x , y , xy

1,

2

1,

2

thermal coefficients moisture coefficients

E, G, [K ]L , [K ]Rot , [K ]HT [BL], [D], [M ] {Fce }, [mI ] x, y, z

[Aij ], [Bij ], [Dij ] h' bx , bx+

x

Young’s modulus, rigidity modulus, Poisson’s ratio global linear stiffness matrix, geometric stiffness due to rotation and hygrothermal load. Strain-displacement matrix, elasticity matrix, global mass matrix element centrifugal force vector, inertia matrix components of speed of rotation in x , y and z directions extension, bending-extension coupling and bending coefficients

position of debonding across the thickness from top surface of face- sheet. width of the conical shell at x and x + x

Appendix B. Elements of[G], [T ] and [BL] matrices

8

[G] = i=1

1 0 [T ] = 0 0 0

8

[BL] = i=1

Ni, x Ni, y 0 0 0 0 0 0 0 0 0 0

0 1 0 0 0

0 0 1 0 0

0 0 0 0 0 Ni / Rxy 0 0 Ni, x Ni/ Rxy 0 0 Ni, y Ni / Ry 0 0 0 Ni, x 0 0 Ni/Ry Ni, y 0 0 0 0 Ni, x 0 0 0 Ni, y 0 0 0 0 Ni, x 0 0 0 Ni, y 0 0 Ni 0 0 0 0 Ni

z 0 0 0 0

0 z 0 0 0

0 0 z 0 0

0 0 0 1 0

0 0 0 0 1

Ni, x 0 0 0 0 0 Ni, y Ni /Ry 0 0 Ni, y Ni, x 2Ni/ Rxy 0 0 0 0 0 Ni, x 0 0 0 0 0 Ni, y 0 0 0 Ni, y Ni, x 0 0 Ni, x Ni 0 0 0 Ni, y 0 Ni

Appendix C. Coefficients of Q¯ ij matrix

Q¯ 11 = Q11 m4 + 2(Q12 + 2Q66 ) m2n2 + Q22 n4 Q¯ 12 = (Q11 + Q22

4Q66 ) m2n2 + Q12 (m4 + n4 )

Q¯ 22 = Q11 n4 + 2(Q12 + 2Q66 ) m2n2 + Q22 m4 17

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T.D. Singha, et al.

Q¯ 16 = (Q11

Q12

2Q66 ) m3n + (Q12

Q22 + 2Q66 ) mn3

Q¯ 26 = (Q11

Q12

2Q66 ) mn3 + (Q12

Q22 + 2Q66 ) m3n

Q¯66 = (Q11 + Q22

2Q12

4Q66 ) m2n2 + Q66 (m4 + n4 )

Q¯44 = Q44 m2 + Q55 n2

Q¯45 = (Q55

Q44 ) mn

Q¯55 = Q55

+ Q44 n2

m2

where Q11 = E1/(1

12 21) ,

Q12 =

12 E2/(1

12 21) ,

Q22 = E2/(1

12 21) ,

Q66 = G12 , Q44 = G23 , Q55 = G13 and m = cos k , n = sin k .

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