A new nonlinear higher-order shear deformation theory with thickness variation for large-amplitude vibrations of laminated doubly curved shells

Journal of Sound and Vibration ] (]]]]) ]]]–]]]

Contents lists available at SciVerse ScienceDirect

Journal of Sound and Vibration journal homepage: www.elsevier.com/locate/jsvi

A new nonlinear higher-order shear deformation theory with thickness variation for large-amplitude vibrations of laminated doubly curved shells M. Amabili n Canada Research Chair (Tier 1) Department of Mechanical Engineering, McGill University, Macdonald Engineering Building, 817 Sherbrooke Street West, Montreal, Quebec, Canada H3A 0C3

a r t i c l e i n f o

abstract

Article history: Received 7 November 2012 Received in revised form 21 February 2013 Accepted 27 March 2013 Handling Editor: M.P. Cartmell

A consistent higher-order shear deformation nonlinear theory is developed for shells of generic shape allowing for thickness variation by using six variables; geometric imperfections are also taken into account. The geometrically nonlinear strain–displacement relationships are derived retaining full nonlinear terms in the in-plane displacements. They are presented in curvilinear coordinates in a formulation that can be readily implemented in computer codes. This new theory is applied to laminated circular cylindrical shells complete around the circumference and simply supported at the ends. Linear (natural frequencies) and geometrically nonlinear (large-amplitude forced response) vibrations are studied by using the present theory and results are compared to those obtained by using the refined Amabili–Reddy higher-order shear deformation nonlinear shell theory, which neglects thickness variations. & 2013 Elsevier Ltd. All rights reserved.

1. Introduction Thickness variation can be important for shells in case of (i) high-frequency vibrations and (ii) for soft materials, e.g. biological tissues. In fact, (i) modes with dominant thickness oscillation can appear in the high-frequency range and (ii) biological tissues are often subjected to very large strains associated with very large thickness variation. Classical shell theories, which neglect shear deformation and rotary inertia, give inaccurate natural frequencies for moderately thick or laminated anisotropic shells and plates (e.g. [1]). In order to overcome this limitation, shear deformation theories have been introduced, but they still neglect thickness variation. These theories can be classified as first-order and higher-order shear deformation theories [2]; in the first category, a shear correction factor is required for the equilibrium since a uniform shear strain is assumed through the shell thickness. Higher-order shear deformation theories overcome this limitation since a realistic shear stress distribution through the shell thickness is assumed, which also satisfies the condition of zero shear stresses at both top and bottom shell surfaces. Several higher-order shear deformation shell theories have been proposed. Librescu [3] developed a nonlinear shell theory by expanding the shell displacements with cubic terms in the transverse coordinate. A linear higher-order shear deformation theory of shells has been introduced by Reddy [4] and Reddy and Liu [5]. Arciniega and Reddy [6] have improved the theory developed in [5]. A review of shell theories has been presented by Reddy and Arciniega [7].

n

Tel.: +1 514 398 3068; fax: +1 514 398 7365. E-mail address: [email protected]

0022-460X/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jsv.2013.03.024

Please cite this article as: M. Amabili. A new nonlinear higher-order shear deformation theory with thickness variation for large-amplitude vibrations of laminated doubly curved shells, Journal of Sound and Vibration (2013), http://dx.doi.org/ 10.1016/j.jsv.2013.03.024i

2

M. Amabili / Journal of Sound and Vibration ] (]]]]) ]]]–]]]

Reddy [8] developed the nonlinear higher-order shear deformation theory of plates, taking into account von Kármán type nonlinear terms. Dennis and Palazotto [9] and Palazotto and Dennis [10] have extended Reddy shell theory [4] to nonlinear deformation by introducing the von Kármán type nonlinear terms. These theories have been also discussed in the books of Amabili [2] and Reddy [11]. In the existing higher-order shear deformation geometrically nonlinear shell theories, the von Kármán type nonlinear terms (i.e. those involving the normal displacement only) have been added to the linear equations, so that consistent derivation has not been performed. Moreover, the von Kármán type nonlinear terms are known for being accurate for classical shell theories only for small displacements and moderately small rotations. Therefore, it is important to derive a consistent higher-order shear deformation theory that keeps all the nonlinear terms in the normal and in-plane displacements. For this reason, a new nonlinear higher-order shear deformation theory that retains in-plane nonlinear terms has been recently derived by Amabili and Reddy [12] by using a consistent approach. This theory belongs to the class of the equivalent single layer (ESL) theories and has the novelty to retain nonlinear terms with in-plane displacements, neglected in other formulations. It must be observed that another class of theories, the layer wise model (LWM) has also been developed, but not retaining geometrically nonlinear terms, see e.g. Carrera [13]. Amabili [14] and Alijani and Amabili [15] have applied the theory developed in Ref. [12] to laminated closed and open circular cylindrical shells and have shown that it gives an important accuracy improvement for thick laminated deep shells, in particular for vibration modes with a low number of circumferential waves, with respect to commonly used nonlinear higher-order shear deformation theories. The theory developed by Amabili and Reddy [12] introduces the artificial constraint of no thickness deformation, reducing the problem to five variables (3 displacements and two rotations). However, they discussed that their new theory could be extended to take into account thickness variation. An accurate linear shell theory that takes into account thickness variation has been developed by Carrera et al. [16,17]. In particular, in Ref. [17] natural frequencies of laminated spherical and cylindrical panels are investigated by using this theory. The effect of transverse normal stress in linear vibrations of laminated shells and plates has been investigated by Carrera [13] by using a model taking into account continuity of interlaminar transverse and shear stresses and zigzag form of the displacement distribution in the shell thickness. Nayak et al. [18] and introduced a new shell finite element containing three translations, two rotations of the normals about the shell mid-surface, and one drilling rotational degree of freedom per node. This 4-node element has been used to study the transient dynamic response of composite shells. Awrejcewicz et al. [19] studied nonlinear vibrations of shells of complex shape and variable thickness. Many other studies on vibrations and nonlinear vibrations of laminated shells and plates are available in the literature, e.g. [20,21]. In the present study, a consistent higher-order shear deformation nonlinear theory is developed for shells of generic shape allowing for thickness variation by using six variables; geometric imperfections are also taken into account. The geometrically nonlinear strain–displacement relationships are derived retaining full nonlinear terms in the in-plane displacements. They are presented in curvilinear coordinates in a formulation that can be readily implemented in computer codes. This new theory is applied to laminated circular cylindrical shells complete around the circumference and simply supported at the ends. Linear (natural frequencies) and geometrically nonlinear (large-amplitude forced response) vibrations are studied by using the present theory and results are compared to those obtained by using the refined Amabili–Reddy higher-order shear deformation nonlinear shell theory [12], which neglects thickness variations. 2. Nonlinear higher-order shear deformation theory with thickness variation A laminated shell of arbitrary shape, made of a finite number of orthotropic layers, oriented arbitrarily with respect to the shell principal curvilinear coordinates (α1, α2), is considered, as shown in Fig. 1. The development of the theory remains the same for shells made of isotropic, orthotropic or functionally graded materials. The displacements of an arbitrary point (α1, α2) on the middle surface of the shell are denoted by u, v and w, in the α1, α2 and z directions, respectively; w is taken positive outward from the center of the smallest radius of curvature. Initial geometric imperfections of the shell associated with zero initial tension are denoted by displacement w0 in normal direction, also taken positive outward. The thickness h of the shell is assumed to be small compared to the principal radii of curvature of the shell, so that only moderately thick shells can be considered. The displacements (u1, u2, u3) of a generic point (see Figs. 1 and 2) are related to the middle surface displacements by u1 ¼ ð1 þ z=R1 Þu þ zϕ1 þ z2 ψ 1 þ z3 γ 1 þ z4 θ1 ;

(1a)

u2 ¼ ð1 þ z=R2 Þv þ zϕ2 þ z2 ψ 2 þ z3 γ 2 þ z4 θ2 ;

(1b)

u3 ¼ w þ zχ þ w0 ;

(1c)

where ϕ1 and ϕ2 are the rotations of the transverse normals at z¼ 0 about the α2 and α1 axes, respectively, and χ is the thickness variation per unit thickness. Then ψ1, ψ2, γ1, γ2, θ1 and θ2 are functions to be determined in terms of u, v, w, ϕ1 and ϕ2. Thus, the six variables describing the shell deformation are u, v, w, ϕ1, ϕ2 and χ. In Eqs. (1) the in-plane displacements have been expanded up to the 4th order in z while the normal displacement has been assumed to be linear in z. Obviously, a more refined expression of the variation of the normal displacement can be introduced, see e.g. Carrera [13], including zigzag functions, which can be piece-wise linear functions with slope discontinuity at the interlaminar interface, and higher Please cite this article as: M. Amabili. A new nonlinear higher-order shear deformation theory with thickness variation for large-amplitude vibrations of laminated doubly curved shells, Journal of Sound and Vibration (2013), http://dx.doi.org/ 10.1016/j.jsv.2013.03.024i

M. Amabili / Journal of Sound and Vibration ] (]]]]) ]]]–]]]

3

Fig. 1. Doubly curved shell and coordinate system.

Fig. 2. Displacement of a point on the middle surface of the shell and of a generic point at distance z from the middle surface.

order functions in z. Eq. (1a) and 1(b) give a high-order distribution of shear effects through the thickness, while Eq. (1c) gives a uniform strain ε33 through the shell thickness. The thickness variation is given by hχ, being h the shell thickness. A positive χ gives a thickness increase. The shear and normal Green's strains for three-dimensional elasticity are [2] 

 ∂u1 1 1 ∂u3 u1 1 ∂u1 1 ∂A1 u3 ∂u1 γ 13 ¼ þ − þ u2 þ þ 1 þ z=R1 A1 ∂α1 R1 A1 ∂α1 A1 A2 ∂α2 ∂z R1 ∂z 

 1 ∂u2 1 ∂A1 ∂u2 1 ∂u3 u1 ∂u3 þ þ ; (2a) − u1 − A1 ∂α1 A1 A2 ∂α2 A1 ∂α1 R1 ∂z ∂z γ 23 ¼



 ∂u2 1 1 ∂u3 u2 1 ∂u2 1 ∂A2 u3 ∂u2 þ − þ u1 þ þ 1 þ z=R2 A2 ∂α2 R2 A2 ∂α2 A1 A2 ∂α1 ∂z R2 ∂z 

 1 ∂u1 1 ∂A2 ∂u1 1 ∂u3 u2 ∂u3 þ þ ; − u2 − A2 ∂α2 A1 A2 ∂α1 A2 ∂α2 R2 ∂z ∂z ε33 ¼

∂u3 1 þ 2 ∂z

"


 # ∂u1 2 ∂u2 2 ∂u3 2 þ þ : ∂z ∂z ∂z

(2b)

(2c)

Eqs. (2a)–(2c) are nonlinear; in Eqs. (2a) and (2b), R1 and R2 (functions of the coordinates α1 and α2) are the principal radii of curvature in α1 and α2 directions, respectively, and A1 and A2 are the Lamé parameters. The shear deformation, see Eqs. (2a) and (2b), is neglected in classical shell theories, which is a very good approximation for thin and moderately thick isotropic shells and for very thin laminated shells [2,22]. For laminated shells that cannot be considered very thin, shear deformation should be retained in order to obtain accurate results. However, shear deformation plays a smaller role than inplane strains and bending (since it is even neglected in classical theories), so that it is reasonable to keep only linear displacement terms in Eqs. (2a) and (2b). This assumption leads to the possibility of expressing the displacement field in Eq. (1a)–(1c) by using only linear expressions in u, v, w, ϕ1, ϕ2 and χ. Therefore, the following relationships are obtained for Please cite this article as: M. Amabili. A new nonlinear higher-order shear deformation theory with thickness variation for large-amplitude vibrations of laminated doubly curved shells, Journal of Sound and Vibration (2013), http://dx.doi.org/ 10.1016/j.jsv.2013.03.024i

M. Amabili / Journal of Sound and Vibration ] (]]]]) ]]]–]]]

4

the transverse shear strains: γ 13 ¼


 ∂u1 1 ∂u3 u1 þ − ; 1 þ z=R1 A1 ∂α1 R1 ∂z

(3a)

γ 23 ¼


 ∂u2 1 ∂u3 u2 þ − : 1 þ z=R2 A2 ∂α2 R2 ∂z

(3b)

The expressions for the transverse shear strains are obtained by substituting Eqs. (1a)–(1c) into (3a) and (3b); the vanishing of the shear strains τ13 and τ23 at the top and the bottom surfaces of the shell requires γ 13 jz ¼

7 h=2

¼ 0;

γ 23 jz ¼

7 h=2

¼ 0;

(4a,b)

since τ13 ¼ G13 γ 13 and τ23 ¼ G23 γ 23 , where G13 and G23 are the shear moduli in 1–3 and 2–3 directions, respectively. The following approximation for moderately thick shell is introduced:
 h h h ≃ 1∓ : (5) 2R7 h 2R 2R Collecting first-order and third-order terms in z from Eq. (4a) into two separate equations, the following expressions are obtained since these expressions change sign if evaluated at z ¼ 7 h=2and must be set equal to zero: 1 4R21 −

ψ 1−

1 γ þ θ1 ¼ 0; 4R1 1

(6a)

1 1 ∂χ 1 ∂w ϕ þ ψ1 þ − ¼ 0: 2R1 1 2 A1 ∂α1 2R1 A1 ∂α1

(6b)

Eqs. (6a) and (6b) give ψ1 ¼

1 1 ∂w 1 ∂χ ϕ þ − ; 2R1 1 2R1 A1 ∂α1 2 A1 ∂α1

(7a)

1 1 γ − ψ : 4R1 1 4R21 1

(7b)

θ1 ¼ Then, Eq. (4a) gives γ1 ¼

−4

" ϕ1 þ

2

3h

# 2
∂w h ∂w ∂χ þ 2 ϕ1 − −R1 : A1 ∂α1 8R1 A1 ∂α1 A1 ∂α1

(7c)

Similarly, from Eq. (4b), the following expressions are obtained: ψ2 ¼

1 1 ∂w 1 ∂χ ϕ þ − ; 2R2 2 2R2 A2 ∂α2 2 A2 ∂α2

(8a)

1 1 γ − ψ ; 4R2 2 4R22 2

(8b)

θ2 ¼

γ2 ¼

−4

"

2

3h

ϕ2 þ

# 2
∂w h ∂w ∂χ þ 2 ϕ2 − −R2 : A2 ∂α2 8R2 A2 ∂α2 A2 ∂α2

(8c)

By substituting Eqs. (1a)–(1c) into (3a) and (3b) and using the approximations (5) and Eqs. (7) and (8), the following strain–displacement relationships are obtained for the shear strains keeping terms up to z3: ð0Þ

ð1Þ

ð2Þ

ð0Þ

ð1Þ

ð2Þ

γ 13 ¼ γ 13;0 þ zðk13 þ z k13 þ z2 k13 Þ;

(9a)

γ 23 ¼ γ 23;0 þ zðk23 þ z k23 þ z2 k23 Þ;

(9b)

where

ð0Þ

k13 ¼ 0;

ð1Þ

γ 13;0 ¼ ϕ1 þ

∂w ; A1 ∂α1

(10a)

γ 23;0 ¼ ϕ2 þ

∂w ; A2 ∂α2

(10b)

k13 ¼ −

  ∂w u ϕ þ þ 3; 1 2 A1 ∂α1 R1 h 4

ð2Þ

k13 ¼ 0;

(11a–c)

Please cite this article as: M. Amabili. A new nonlinear higher-order shear deformation theory with thickness variation for large-amplitude vibrations of laminated doubly curved shells, Journal of Sound and Vibration (2013), http://dx.doi.org/ 10.1016/j.jsv.2013.03.024i

M. Amabili / Journal of Sound and Vibration ] (]]]]) ]]]–]]]

ð0Þ

k23 ¼ 0;

ð1Þ

k23 ¼ −

  ∂w v ϕ2 þ þ 3; A2 ∂α2 R2 h 4

2

ð2Þ

k23 ¼ 0:

5

(11d–f)

Eqs. (10) and (11) show that the strain and stress distribution through the thickness is parabolic and therefore the shear correction factor is no longer required.From Eq. (2c) it is obtained ε33 ¼ ε3;0 ¼ χ þ

u2 2R21

þ

v2 2R22

;

(12)

where in Eq. (12) only nonlinear terms in the middle surface displacement have been retained and ε33 has been assumed to be constant along z, which is consistent with Eq. (1c). The expressions of the other Green's strain tensor εij in the curvilinear coordinate system are obtained from the theory of surfaces and the shell deformation (e.g. see [2]) 2

ε11 ¼ ε1 þ 12ðε21 þ ω21 þ Θ Þ;

(13a)

2

ε22 ¼ ε2 þ 12ðω22 þ ε22 þ Ψ Þ;

(13b)

γ 12 ¼ ω1 þ ω2 þ ðε1 ω2 þ ε2 ω1 þ Θ Ψ Þ;

(13c)

where ε1 ¼


 1 1 ∂u1 1 ∂A1 u3 þ u2 þ ; 1 þ z=R1 A1 ∂α1 A1 A2 ∂α2 R1

(14a)

ε2 ¼


 1 1 ∂u2 1 ∂A2 u3 þ u1 þ ; 1 þ z=R2 A2 ∂α2 A1 A2 ∂α1 R2

(14b)

ω1 ¼


 1 1 ∂u2 1 ∂A1 − u1 ; 1 þ z=R1 A1 ∂α1 A1 A2 ∂α2

(14c)

ω2 ¼


 1 1 ∂u1 1 ∂A2 − u2 ; 1 þ z=R2 A2 ∂α2 A1 A2 ∂α1

(14d)

Θ¼


 1 1 ∂u3 u1 − þ ; 1 þ z=R1 A1 ∂α1 R1

(14e)

Ψ¼


 1 1 ∂u3 u2 − þ : 1 þ z=R2 A2 ∂α2 R2

(14f)

The strain–displacement equations for the higher-order shear deformation theory to be added to Eqs. (9a), (9b) and (12), keeping terms up to z3 and using approximation (5), are obtained by substituting Eqs. (1), (6) and (7) into Eqs. (13) and (14) ð0Þ

ð1Þ

ð2Þ

ð0Þ

ð1Þ

ð2Þ

ε11 ¼ ε1;0 þ zðk1 þ z k1 þ z2 k1 Þ; ε22 ¼ ε2;0 þ zðk2 þ z k2 þ z2 k2 Þ; ð0Þ

ð1Þ

ð2Þ

γ 12 ¼ γ 12;0 þ zðk12 þ z k12 þ z2 k12 Þ; where ε1;0 ¼

ε2;0 ¼

"

2 1 ∂u 1 ∂A1 w 1 ∂u 1 ∂A1 w 2 ∂v 1 ∂A1 þ vþ þ þ vþ þ − u A1 ∂α1 A1 A2 ∂α2 R1 2 A1 ∂α1 A1 A2 ∂α2 R1 A1 ∂α1 A1 A2 ∂α2
2 #

 ∂w u w0 ∂u 1 ∂A1 w ∂w0 ∂w u þ ; − þ vþ − þ þ A1 ∂α1 R1 R1 R1 A1 ∂α1 A1 A2 ∂α2 A1 ∂α1 A1 ∂α1 R1 "
2
 1 ∂v 1 ∂A2 w 1 1 ∂u 1 ∂A2 1 ∂v 1 ∂A2 w 2 þ uþ þ − v þ þ uþ A2 ∂α2 A1 A2 ∂α1 R2 2 A2 ∂α2 A1 A2 ∂α1 A2 ∂α2 A1 A2 ∂α1 R2
 #

 1 ∂w v 2 w0 1 ∂v 1 ∂A2 w ∂w0 ∂w v − þ uþ − þ ; þ þ A2 ∂α2 R2 R2 R2 A2 ∂α2 A1 A2 ∂α1 A2 ∂α2 A2 ∂α2 R2

(15a) (15b) (15c)

(16a)

(16b)


 1 ∂v 1 ∂u 1 ∂A1 1 ∂A2 1 ∂u 1 ∂A1 w þ − u− vþ þ vþ A1 ∂α1 A2 ∂α2 A1 A2 ∂α2 A1 A2 ∂α1 A1 ∂α1 A1 A2 ∂α2 R1

 
1 ∂u 1 ∂A2 1 ∂v 1 ∂A2 w 1 ∂v 1 ∂A1 − v þ þ uþ − u  A2 ∂α2 A1 A2 ∂α1 A2 ∂α2 A1 A2 ∂α1 R2 A1 ∂α1 A1 A2 ∂α2

γ 12;0 ¼

Please cite this article as: M. Amabili. A new nonlinear higher-order shear deformation theory with thickness variation for large-amplitude vibrations of laminated doubly curved shells, Journal of Sound and Vibration (2013), http://dx.doi.org/ 10.1016/j.jsv.2013.03.024i

M. Amabili / Journal of Sound and Vibration ] (]]]]) ]]]–]]]

6




 1 ∂w u 1 ∂w v w0 1 ∂u 1 ∂A2 þ − − − v þ A1 ∂α1 R1 A2 ∂α2 R2 R1 A2 ∂α2 A1 A2 ∂α1


 w0 1 ∂v 1 ∂A1 1 ∂w0 1 ∂w v 1 ∂w0 1 ∂w u þ − u þ − − þ ; A1 ∂α1 A2 ∂α2 R2 A2 ∂α2 A1 ∂α1 R1 R2 A1 ∂α1 A1 A2 ∂α2 ð0Þ

k1 ¼


 ∂ϕ1 w v ∂A1 1 1 ϕ ∂A1 χ − 2þ − þ þ ; þ 2 R1 R2 A1 ∂α1 R1 A1 A2 ∂α2 A1 A2 ∂α2 R1

ð1Þ

∂ϕ1 ∂u ∂A1 ∂w ∂2 w þ þ − 2 ∂α1 R1 ∂α1 2A21 ∂α1 ∂α1 2A1 ∂α21 
  ∂A1 1 1 v ∂w − ϕ2 − − þ R1 2R2 R1 R2 2R2 A2 ∂α2 A1 A2 ∂α2

k1 ¼ −

−

ð2Þ

ð0Þ

1 R2 A2

1 2A1 A22

∂A1 ∂χ ∂A1 ∂χ ∂2 χ þ 3 − 2 ; ∂α2 ∂α2 2A1 ∂α1 ∂α1 2A1 ∂α2

(17b)

1

!

(17c)

(17d)

! 
  ∂ϕ2 ∂v ∂A2 ∂w ∂2 w ∂A2 1 1 u ∂w þ þ 2 − ϕ − − þ − R1 R2 2R1 A1 ∂α1 2∂α2 R2 ∂α2 2A2 ∂α2 ∂α2 2A2 ∂α22 A1 A2 ∂α1 1 R2 2R1

ð2Þ k2

ð0Þ

−


 ∂ϕ2 w u ∂A2 1 1 ϕ ∂A2 χ − 2þ − þ þ 1 A2 ∂α2 R2 A1 A2 ∂α1 R1 R2 A1 A2 ∂α1 R2

−

k12 ¼

R21

4

k2 ¼

ð1Þ

χ

(17a)

!

∂ϕ1 ∂2 w ϕ ∂A1 ∂w ∂A1 ∂w ∂A1 þ 2 þ 2 − þ 2 A ∂α 1 1 A1 ∂α21 A1 A2 ∂α2 A31 ∂α1 ∂α1 A1 A22 ∂α2 ∂α2 3h ! ! ϕ ∂A1 1 1 ∂w ∂A1 1 1 þ − − 2 þ A1 A2 ∂α2 6R22 2R1 R2 A1 A22 ∂α2 ∂α2 6R22 2R1 R2 ! 1 2∂ϕ1 ∂2 w ∂w ∂A1 þ 2 − 3 − 2 3R1 A1 ∂α1 A1 ∂α21 A1 ∂α1 ∂α1
 1 ∂A1 ∂χ 1 1 2 ∂A1 ∂χ 2 ∂2 χ þ þ ; þ − 2 ∂α ∂α 3 A1 A2 2 2 2R1 6R2 3R1 A1 ∂α1 ∂α1 3R1 A21 ∂α21

k1 ¼ −

k2 ¼ −

1 R1 A1

(16c)

χ R22

−

1 2A21 A2

∂A2 ∂χ ∂A2 ∂χ ∂2 χ þ − ; ∂α1 ∂α1 2A32 ∂α2 ∂α2 2A22 ∂α2

(17e)

2

∂ϕ2 ∂2 w ϕ ∂A2 ∂w ∂A2 ∂w ∂A2 ¼− 2 þ 2 þ 1 − 3 þ 2 2 A ∂α A A ∂α ∂α 2 2 1 2 1 A2 ∂α2 A2 ∂α2 2 A1 A2 ∂α1 ∂α1 3h ! ! ϕ1 ∂A2 1 1 ∂w ∂A2 1 1 þ − − þ 2 A1 A2 ∂α1 6R21 2R1 R2 A1 A2 ∂α1 ∂α1 6R21 2R1 R2 ! 1 2∂ϕ2 ∂2 w ∂w ∂A2 þ 2 − 3 − 2 2 A ∂α 2 2 3R2 A2 ∂α2 A2 ∂α2 ∂α2
 1 ∂A2 ∂χ 1 1 2 ∂A2 ∂χ 2 ∂2 χ þ 2 þ þ ; − 3 A1 A2 ∂α1 ∂α1 2R2 6R1 3R2 A2 ∂α2 ∂α2 3R2 A22 ∂α22 4

!



 ∂ϕ1 ∂ϕ2 ϕ ∂A1 ϕ ∂A2 ∂u 1 1 ∂v 1 1 þ − 1 − 2 þ − − þ þ ; A1 ∂α1 R1 R2 A2 ∂α2 A1 ∂α1 A1 A2 ∂α2 A1 A2 ∂α1 A2 ∂α2 R1 R2 ð1Þ

k12 ¼ −

(17f)

(17g)




 ∂ϕ1 1 1 ∂ϕ2 1 1 1 ∂u ∂v − − þ − − R1 R2 A2 ∂α2 A1 ∂α1 A2 ∂α2 R2 2R1 A1 ∂α1 R1 2R2

∂2 w 1 1 1 ϕ1 ∂A1 ϕ ∂A2 þ þ 2 þ þ 2A1 A2 ∂α1 ∂α2 R1 R2 A1 A2 2R1 ∂α2 2R2 ∂α1 ! u ∂A1 v ∂A2 ∂w ∂A1 ∂w ∂A2 þ 2 − − þ 2 R1 ∂α2 R2 ∂α1 R1 A1 ∂α1 ∂α2 R2 A2 ∂α2 ∂α1 þ

1 ∂A2 ∂χ 1 ∂A1 ∂χ ∂2 χ þ − ; 2 ∂α ∂α 2 ∂α ∂α A A 1 2 ∂α1 ∂α2 A1 A2 1 2 A2 A1 2 1

(17h)

Please cite this article as: M. Amabili. A new nonlinear higher-order shear deformation theory with thickness variation for large-amplitude vibrations of laminated doubly curved shells, Journal of Sound and Vibration (2013), http://dx.doi.org/ 10.1016/j.jsv.2013.03.024i

M. Amabili / Journal of Sound and Vibration ] (]]]]) ]]]–]]]

7

Fig. 3. Laminated shell.

Fig. 4. Material principal coordinates (x, y, z) and shell coordinates (a1, a2, z). The thin lines represent the direction of the fibers.

∂ϕ1 ∂ϕ2 ∂2 w ∂w ∂A1 ∂w ∂A2 þ þ2 −2 2 −2 A1 A2 ∂α1 ∂α2 A1 A2 ∂α1 ∂α2 A1 A22 ∂α2 ∂α1 3h A2 ∂α2 A1 ∂α1 !  ϕ ∂A1 ϕ ∂A2 ϕ ∂A1 2 ϕ2 ∂A2 2 ∂ϕ1 1 1 − 2 þ − þ − 1 þ 1 A1 A2 ∂α2 A1 A2 ∂α1 A1 A2 ∂α2 3R21 A1 A2 ∂α1 3R22 A2 ∂α2 6R21 2R1 R2 ! ! ∂ϕ2 1 1 ∂w ∂A1 1 1 þ 2 − þ þ A1 ∂α1 6R22 2R1 R2 A1 A2 ∂α1 ∂α2 6R21 2R1 R2 ! ! ∂w ∂A2 1 1 ∂2 w 1 1 1 þ þ þ − þ A1 A2 ∂α1 ∂α2 6R21 6R22 R1 R2 A1 A22 ∂α2 ∂α1 6R22 2R1 R2


 1 ∂A2 ∂χ 1 5 1 ∂A1 ∂χ 1 5 2∂2 χ 1 1 − þ : þ þ þ − 3A1 A2 ∂α1 ∂α2 R1 R2 A1 A22 ∂α1 ∂α2 2R1 6R2 A2 A21 ∂α2 ∂α1 2R2 6R1 ð2Þ

k12 ¼ −

4

2

(17i)

Eqs. (17a)–(17i) give the changes in curvature and torsion of the middle surface, and they have been obtained retaining only linear terms; in fact, nonlinear terms in the changes in curvature and torsion play a very small role, at least for moderate vibration amplitudes of the order of the shell thickness [22]. Eqs. (16a)–(16c), giving the middle surface strains, are coincident with those obtained by using Novozhilov nonlinear shell theory [2,23], which neglects shear deformation and ð1Þ ð1Þ ð1Þ rotary inertia. Moreover, it can be observed that k1 ; k2 ; k12 are negligible for shells that are not very thick. 3. Elastic strain and kinetic energies, including rotary inertia, for laminated shells The stress–strain relations for the k-th orthotropic lamina (see Fig. 3) of the shell in the material principal coordinates (x,y,z), as shown in Fig. 4, are given by 9ðkÞ 2 9 8 3 8 εxx > c11 c12 c13 0 0 0 ðkÞ > sx > > > > > > > > > > 6 7 > >s > > > > εyy > 0 0 0 7 > > > > 6 c21 c22 c23 y > > > > > > > > 6 7 > > > > = < sz = 6 c31 c32 c33 7 < εzz > 0 0 0 6 7 ; (18) ¼6 7 γ τ 0 0 0 0 0 G > 6 7 > yz > yz yz > > > > > > > > 6 7 > > > > > > > γ xz > 6 0 τxz > 0 7 0 0 0 Gxz > > > > > > > 4 5 > > > > > > > ; ; : τxy > :γ > 0 0 0 0 0 Gxy xy where Gxy, Gxz and Gyz are the shear moduli in x–y, x–z and y–z directions, respectively, and the coefficients cij are given in Appendix A; τxz and τyz are the shear stresses and the superscript (k) refers to the k-th layer within a laminate. Eq. (18) is obtained under the transverse isotropy assumption with respect to planes orthogonal to the x axis, i.e. assuming fibers in the Please cite this article as: M. Amabili. A new nonlinear higher-order shear deformation theory with thickness variation for large-amplitude vibrations of laminated doubly curved shells, Journal of Sound and Vibration (2013), http://dx.doi.org/ 10.1016/j.jsv.2013.03.024i

M. Amabili / Journal of Sound and Vibration ] (]]]]) ]]]–]]]

8

z k =K

h K-1

k x

h k-1

hK

hk

h1

h

h0

k =1 Fig. 5. Notation for thickness of individual layers of a laminate of global thickness h and number of layers K.

direction parallel to axis x, so that Ey ¼Ez, Gxy ¼Gxz, νxy ¼νxz and νyz ¼νzy. In particular, Eq. (18) gives sz ¼ ðνxy Ey =ð1−νyz −2νxy νyx ÞÞεxx þ ½ðνyz þ νxy νyx ÞEy =ðð1 þ νyz Þð1−νyz −2νxy νyx ÞÞεyy þ ½ð1−νxy νyx ÞEy =ðð1 þ νyz Þð1−νyz −2νxy νyx ÞÞεzz . Here it is observed that sz should vanish at z¼ −h/2 and z¼ h/2, but due to the assumption of constant sz along z this cannot be satisfied. Eq. (18) can be transformed to the shell coordinates (α1, α2, z) by the following equation [2]: 9ðkÞ 9 8 8 ε11 > s1 > > > > > > > > > > > > > >ε > > > > s2 > > > > 22 > > > > > > > > > > > > > = = < < ε s3 33 ; (19) ¼ ½Q ðkÞ γ > τ > > > > > > > > 23 > > 23 > > > > > > > > > > > γ > τ > > > > > > 13 > > 13 > > > ; ; :τ > :γ > 12 12 where ½Q ðkÞ is the 6  6 matrix of the material properties of the k-th layer expressed in the shell principal coordinates and it is given in Appendix A. Eq. (19) can be rewritten as 28 9 93 8 8 9 9 8 ð0Þ ð2Þ > > ε1;0 > k1 > k1 > 0 > > > > > > > > > > > > > > > 6> > >7 > > > > > > > > ð0Þ > ð2Þ > 6> > > > > > > > > ε2;0 > 0 > > >7 > > > > > k k > > > > > > > 6> > >7 > > > 2 > > 2 > > > > > > > > > 6> = =7 = = < < < < ε 0 6 7 3;0 0 0 ðkÞ 2 3 7: þ z (20) þ z þ z fsðkÞ g ¼ ½Q 6 ð1Þ 6> γ 23;0 > > >7 > > 0 > > 0 > > k23 > > > > > 6> 7 > > > > > > > > > > > > > > > 6> 7 > > > > >7 > > > 0 > > 0 > > kð1Þ > > γ 13;0 > 6> > > > > > > 13 > > > > > > > > > 4> > >5 > > > > > > ; ; > > : :γ ð0Þ > ; ; : : kð2Þ > 12;0 k 0 12

ð1Þ k1 ;

ð1Þ k2 ;

12

ð1Þ k12

In Eq. (20), and have been neglected, supposing a moderately thick shell. The elastic strain energy US of the shell is given by US ¼

1 K ∑ 2k¼1

Z

a

Z

0

b

Z

hðkÞ hðk−1Þ

0

ðkÞ ðkÞ ðkÞ ðkÞ ðkÞ ðsðkÞ 1 ε11 þ s2 ε22 þ K z s3 ε33 þ τ 12 γ 12 þ τ13 γ 13 þ τ23 γ 23 Þ

ð1 þ z=R1 Þð1 þ z=R2 ÞA1 A2 dα1 dα2 dz; (k−1)

(21)

(k)

, h ) are the z coordinates of the k-th layer, see Fig. 5, and where K is the total number of layers in the laminated shell, (h Kz is the normal stress correction factor that takes into account that the actual s3 is different from the uniform distribution assumed here. For simplicity a shell of rectangular base of dimensions a and b in α1 and α2 directions, respectively, has been considered in Eq. (21). The kinetic energy TS of the shell, including rotary inertia, is given by TS ¼

1 2

K

∑ ρðkÞ S

k¼1

Z

a 0

Z

b 0

Z

ðkÞ

h

ðk−1Þ

h

ðu_ 21 þ u_ 22 þ u_ 23 Þð1 þ z=R1 Þ ð1 þ z=R2 ÞA1 A2 dα1 dα2 dz:

(22)

The z and z3 terms vanish after integration on z in the case of a laminate with symmetric density with respect to the z axis. In particular, for a laminate with the same density for all the layers and uniform thickness, the following simplified expression is obtained: 

 Z aZ b 1 17 _ 2 _ 2 χ_ 2 41 2 2 41 _ 1 u_ _ 2 v_ _ 2 þ h2 T S ¼ ρS h ðϕ1 þ ϕ2 Þ þ þϕ þ þ u_ 2 þ v_ 2 þ w þϕ 2 315 120R1 15R2 15R1 120R2 12 0 0 Please cite this article as: M. Amabili. A new nonlinear higher-order shear deformation theory with thickness variation for large-amplitude vibrations of laminated doubly curved shells, Journal of Sound and Vibration (2013), http://dx.doi.org/ 10.1016/j.jsv.2013.03.024i

M. Amabili / Journal of Sound and Vibration ] (]]]]) ]]]–]]]

9

L

R θ

x

Fig. 6. Circular cylindrical shell: coordinate system.

" #
!
 _  _ _ _2 1 u_ 2 v_ 2 1 1 1 ∂w ∂w 8ϕ w u_ u_ _χ þ þ þ þ þ − − 1 þ þ w_ 4 R1 R2 6 R1 R2 12R1 R2 A1 ∂α1 252A1 ∂α1 120R1 30R2 315
 _  u_ ∂_χ _ _ ∂w ∂w 8ϕ v_ v_ v_ ∂_χ 5 þ − − 2 − − A1 A2 dα1 dα2 þ Oðh Þ: þ A2 ∂α2 252A2 ∂α2 120R2 30R1 315 12 A1 ∂α1 12 A2 ∂α2

(23)

4. Boundary conditions and discretization of a circular cylindrical shell In order to reduce the system to finite dimensions, the middle surface displacements u, v and w, the two rotations ϕ1 and ϕ2 and the thickness variation per unit thickness χ are expanded by using approximate functions. Circular cylindrical shells with simply supported boundary conditions are analyzed in the following part of the study, as shown in Fig. 6. In particular, R1 ¼ ∞, R2 ¼ R, α1 ¼ x, α2 ¼ θ, A1 ¼ 1, A2 ¼ R, a ¼ L and b ¼ 2π. The following boundary conditions are imposed at the shell ends, x¼0, L: w ¼ 0;

v ¼ 0;

ϕ2 ¼ 0;

N x ¼ 0;

Mx ¼ 0;

χ ¼ 0;

(24a,b,d) (24e,f)

where Nx is the axial stress resultant per unit length and Mx is the axial stress moment resultant per unit length, i.e., ( )   Z hðkÞ K Nx 1 sðkÞ ¼ ∑ ð1 þ z=RÞdz: (25) x ðk−1Þ Mx z k¼1 h Moreover, the four displacements and the two rotations must be continuous in θ. The following base of shell displacements, which satisfy identically the geometric boundary conditions (24a–d), is used to discretize the system: M1

uðx; θ; tÞ ¼ ∑

M2

N

∑ ½um;j;c ðtÞcosðjθÞ þ um;j;s ðtÞsinðjθÞcosðλm xÞ þ ∑ um;0 ðtÞcosðλm xÞ;

m¼1j¼1

(26a)

m¼1

3M 1

vðx; θ; tÞ ¼ ∑

2N

∑ ½vm;j;c ðtÞsinðjθÞ þ vm;j;s ðtÞcosðjθÞsinðλm xÞ;

(26b)

m¼1j¼1

M1

wðx; θ; tÞ ¼ ∑

M2

N

∑ ½wm;j;c ðtÞcosðjθÞ þ wm;j;s ðtÞsinðjθÞsinðλm xÞ þ ∑ wm;0 ðtÞsinðλm xÞ;

m¼1j¼1 M1

ϕ1 ðx; θ; tÞ ¼ ∑

(26c)

m¼1 M2

N

∑ ½ϕ1m;j;c ðtÞcosðjθÞ þ ϕ1m;j;s ðtÞsinðjθÞcosðλm xÞ þ ∑ ϕ1m;0 ðtÞcosðλm xÞ;

m¼1j¼1

(26d)

m¼1

M1

ϕ2 ðx; y; tÞ ¼ ∑

N

∑ ½ϕ2m;j;c ðtÞsinðjθÞ þ ϕ2m;j;s ðtÞcosðjθÞsinðλm xÞ;

(26e)

m¼1j¼1

M1

χðx; θ; tÞ ¼ ∑

N

M2

∑ ½χ m;j;c ðtÞcosðjθÞ þ χ m;j;s ðtÞsinðjθÞsinðλm xÞ þ ∑ χ m;0 ðtÞsinðλm xÞ;

m¼1j¼1

(26f)

m¼1

Please cite this article as: M. Amabili. A new nonlinear higher-order shear deformation theory with thickness variation for large-amplitude vibrations of laminated doubly curved shells, Journal of Sound and Vibration (2013), http://dx.doi.org/ 10.1016/j.jsv.2013.03.024i

M. Amabili / Journal of Sound and Vibration ] (]]]]) ]]]–]]]

10

where j is the number of circumferential waves, m is the number of longitudinal half-waves, λm ¼ mπ=L and t is the time; um,j(t), vm,j(t), wm,j(t), ϕ1m;j , ϕ2m;j and χ m;j are the generalized coordinates that are unknown functions of t; the additional subscript c or s indicates if the generalized coordinate is associated to cosine or sine function in θ except for v, for which the notation is reversed (no additional subscript is used for axisymmetric terms). The integers N, M1 and M2 must be selected with care in order to obtain the required accuracy and acceptable dimension of the nonlinear problem. By using a different number of terms in the expansions, it is possible to study the convergence and the accuracy of the solution. The minimal expansion giving accurate results for excitation in the neighborhood of resonance of mode (m ¼1, n) is 3

uðx; θ; tÞ ¼ ½u1;n;c ðtÞcosðnθÞ þ u1;n;s ðtÞsinðnθÞcosðλ1 xÞ þ ∑ u2m−1;0 ðtÞcosðλ2m−1 xÞ;

(27a)

m¼1

2

vðx; θ; tÞ ¼ ∑ ½v1;kn;c ðtÞsinðknθÞ þ v1;kn;s ðtÞcosðknθÞsinðλ1 xÞ þ ½v3;2n;c ðtÞsinð2nθÞ þ v3;2n;s ðtÞcosð2nθÞsinðλ3 xÞ;

(27b)

k¼1

3

wðx; θ; tÞ ¼ ½w1;n;c ðtÞcosðnθÞ þ w1;n;s ðtÞsinðnθÞsinðλ1 xÞ þ ∑ w2m−1;0 ðtÞsinðλ2m−1 xÞ;

(27c)

m¼1 3

ϕ1 ðx; θ; tÞ ¼ ½ϕ11;n;c ðtÞcosðnθÞ þ ϕ11;n;s ðtÞsinðnθÞcosðλ1 xÞ þ ∑ ϕ12m−1;0 ðtÞcosðλ2m−1 xÞ;

(27d)

ϕ2 ðx; θ; tÞ ¼ ½ϕ21;n;c ðtÞsinðnθÞ þ ϕ21;n;s ðtÞcosðnθÞsinðλ1 xÞ;

(27e)

m¼1

3

χðx; θ; tÞ ¼ ½χ 1;n;c ðtÞcosðnθÞ þ χ 1;n;s ðtÞsinðnθÞsinðλ1 xÞ þ ∑ χ 2m−1;0 ðtÞsinðλ2m−1 xÞ:

(27f)

m¼1

This expansion has 28 generalized coordinates (degrees of freedom) and guarantees good accuracy for the calculation performed in the present work. Initial geometric imperfections of the shell are considered only in radial direction. They are assumed to be associated with zero initial stress. The imperfection w0 is expanded in the same form of w ~1 M

w0 ðx; θÞ ¼ ∑

~2 M

N~

∑ ½Am;n cosðnθÞ þ Bm;n sinðnθÞsinðλm xÞ þ ∑ Am;0 sinðλm xÞ;

m¼1n¼1

(28)

m¼1

~ 1 and M ~ 2 are integers indicating the number of where Am;n , Bm;n and Am;0 are the modal amplitudes of imperfections; N~ , M terms in the expansion. Eq. (24e) is not identically satisfied by the shell displacements given in Eqs. (26) and (27). In particular, after integration with respect to z as shown in Eq. (25), it becomes 9 28 ! ε > ðkÞ2 ðk−1Þ2 = < x;0 > N h −h ðkÞ ðk−1Þ ðkÞ ðkÞ 6 εθ;0 h N x ¼ ∑ fQ ðkÞ ; Q ; Q g −h þ 4 11 12 15 > > 2R ; :γ k¼1 xθ;0 8 ð0Þ 9 8 ð2Þ 9 3 k > k > > ! > ! > > = hðkÞ2 −hðk−1Þ2 hðkÞ3 −hðk−1Þ3 = hðkÞ4 −hðk−1Þ4 hðkÞ5 −hðk−1Þ5 7 < x > < x > ð0Þ ð2Þ 7 ¼ 0; at x ¼ 0; L; þ þ (29) þ kθ þ kθ 5 > > > > 2 3R 4 5R > > > ; : kð0Þ ; : kð2Þ > xθ

xθ

ðkÞ ðkÞ where Q ðkÞ 11 ; Q 12 ; and Q 15 are given by Eqs. (A6)–(A8) in Appendix A and the strain–displacement relations are given in ^ θ; tÞ must be added to Appendix B. In order to satisfy Eqs. (29) up to second-order nonlinear terms, a second-order term uðx; the expansion of u in Eqs. (26a) and (27a). In particular, eliminating in Eq. (29) null terms at x ¼0, L, the following expression is obtained: h 28  2 i ∂w ∂w0 9 ∂u^ 1 ∂v 2 > þ ∂x ∂x > þ ∂w > > ! ∂x þ 2 ∂x ∂x > > ðkÞ2 ðk−1Þ2 = < 6 N h −h 1 ∂u 2 ðkÞ ðk−1Þ ðkÞ ðkÞ ðkÞ 6 1 h −h Nx ¼ ∑ fQ 11 ; Q 12 ; Q 15 g6 þ 2 R ∂θ > 4> 2R > > k¼1 > > ; : ∂v 1 ∂u þ ∂x R ∂θ 9 9 8 8 3 ! > 0 > ! 0 > ðkÞ2 ðk−1Þ2 > ðkÞ3 ðk−1Þ3 = h −h = hðkÞ4 −hðk−1Þ4 hðkÞ5 −hðk−1Þ5 < < h −h 7 0 0 þ þ (30a) þ þ 5 ¼ 0; at x ¼ 0; L; > > 2 3R 4 5R ; ; : kð0Þ > : kð2Þ > xθ xθ

^ θ; tÞ has been neglected in second-order terms. In Eq. (30), all the linear terms ∂ v=∂x þ ð1=RÞð∂ u=∂θÞ, where the term uðx; ð0Þ ð2Þ kxθ and kxθ can be eliminated since they establish a linear relationship which is satisfied by using the minimization of energy in the process of building the Lagrange equations of motion; in fact, this is equivalent to the Rayleigh–Ritz method Please cite this article as: M. Amabili. A new nonlinear higher-order shear deformation theory with thickness variation for large-amplitude vibrations of laminated doubly curved shells, Journal of Sound and Vibration (2013), http://dx.doi.org/ 10.1016/j.jsv.2013.03.024i

M. Amabili / Journal of Sound and Vibration ] (]]]]) ]]]–]]]

11

and therefore it is necessary only to satisfy only geometrical boundary conditions. Therefore Eq. (30a) can be simplified into 28 h  2 i ∂ w ∂ w 9 2 > = < ∂∂xu^ þ 12 ∂∂xv þ ∂∂xw þ ∂x ∂x 0 > N ðkÞ 6 ; Q g N x ¼ ∑ fQ ðkÞ 4  11 12 2 > > 1 1 ∂ u ; : k¼1 2 R ∂θ !# ðkÞ2 ðk−1Þ2 h −h ðkÞ ðk−1Þ  h −h þ ¼ 0; at x ¼ 0; L; (30b) 2R which immediately gives N

Nx ¼ ∑

k¼1

ðkÞ

ðk−1Þ

h −h

ðkÞ2

þ

h

ðk−1Þ2

−h 2R

þQ ðkÞ 12

!" Q ðkÞ 11

∂u^ 1 þ ∂x 2


 # 1 1 ∂u 2 ¼ 0; 2 R ∂θ

"


∂v ∂x

! 2
2 # ∂w ∂w ∂w0 þ þ ∂x ∂x ∂x

at x ¼ 0; L:

By introducing the notation N

ðkÞ

ðk−1Þ

D11 ¼ ∑ Q ðkÞ 11 h −h

ðkÞ2

þ

h

k¼1 N

ðkÞ

ðk−1Þ

D12 ¼ ∑ Q ðkÞ 12 h −h

ðk−1Þ2

−h 2R

ðkÞ2

þ

h

k¼1

(30c)

! ;

ðk−1Þ2

−h 2R

! ;

Eq. (30c) can be expressed in the following form: "

 # !
 ∂u^ 1 ∂v 2 ∂w 2 ∂w ∂w0 1 1 ∂u 2 þ þ ¼ 0; þ þ D12 Nx ¼ D11 ∂x 2 ∂x ∂x ∂x ∂x 2 R ∂θ

at x ¼ 0; L:

(30d)

In Eq. (30d) it is necessary to retain only the resonant mode (m,n). The expression of u^ can be obtained from Eq. (30d) as  1

aðtÞ þ bðtÞcosð2nθÞ þ cðtÞsinð2nθÞ sinð2mπx=LÞ−ðmπ=LÞ 8

 N~ M~ 1  i

Ai;j cosðjθÞ þ Bi;j sinðjθÞ  wm;n;c ðtÞcosðnθÞ þ wm;n;s ðtÞsinðnθÞ ∑ ∑ m þ i j¼0i¼1 ^ ¼− uðtÞ

sin½ðm þ iÞπx=L;

(31)

where aðtÞ ¼ ðmπ=LÞðw2m;n;c þ w2m;n;s þ v2m;n;c þ v2m;n;s Þ þ ðD12 =D11 Þ½Ln2 =ðmπR2 Þðu2m;n;c þ u2m;n;s Þ; bðtÞ ¼ ðmπ=LÞðw2m;n;c −w2m;n;s þ v2m;n;s −v2m;n;c Þ þ ðD12 =D11 Þ½Ln2 =ðmπR2 Þðu2m;n;s −u2m;n;c Þ; cðtÞ ¼ ð2mπ=LÞðwm;n;c wm;n;s þ vm;n;c vm;n;s Þ−2ðD12 =D11 Þ½Ln2 =ðmπR2 Þum;n;c um;n;s : The boundary condition (24f) is identically satisfied for symmetric laminates if the term z=R is neglected in Eq. (25), ð0Þ ð2Þ i.e. for thin shells. This is due to the expressions of kx and kx given in Appendix B, which are zero at x¼ 0, L for the expansions assumed in Eq. (26a)–(26f). Additional terms must be added to the expansion of the in-plane displacement u for asymmetric laminates and moderately thick shells. In fact, bending and stretching are coupled for asymmetric laminates. 5. Lagrange equations of motion The virtual work W done by the external forces is written as Z 2π Z L ðqx u þ qy v þ qz wÞdxRdθ; W¼ 0

(32)

0

where qx, qθ and qz are the distributed forces per unit area acting in x, θ and z directions, respectively, applied at the middle surface. Only a single harmonic force orthogonal to the shell is considered; therefore, qx ¼qθ ¼0. The external distributed load qz applied to the shell, due to the normal concentrated harmonic force f~ and uniform pressure p, is given by ~ ~ qz ¼ f~ δðx−xÞδðθ− θÞcosðωtÞ þ p;

(33)

where ω is the excitation frequency, t is the time, δ is the Dirac delta function, f~ gives the force magnitude positive in z direction and x~ and θ~ give the position of the point of application of the force. Here, the point excitation is located at middle length of shell, that is, x~ ¼ L=2, θ~ ¼ 0. Eq. (32) can be rewritten in the following form: Z 2π Z L W ¼ f~ cosðωtÞðwÞx ¼ L=2;θ ¼ 0 þ pwdxRdθ 0

0

Please cite this article as: M. Amabili. A new nonlinear higher-order shear deformation theory with thickness variation for large-amplitude vibrations of laminated doubly curved shells, Journal of Sound and Vibration (2013), http://dx.doi.org/ 10.1016/j.jsv.2013.03.024i

12

M. Amabili / Journal of Sound and Vibration ] (]]]]) ]]]–]]] 3

¼ f~ cosðωtÞðwÞx ¼ L=2;θ ¼ 0 þ 4LRp ∑ w2m−1;0 ðtÞ=ð2m−1Þ:

(34)

m¼1

The nonconservative damping forces are assumed to be of viscous type and are taken into account by using the Rayleigh dissipation function Z L Z 2π 1 _ 1 2 þ h2 ϕ _ 2 2 ÞdxRdθ; _ 2 þ h2 χ_ 2 þ h2 ϕ ðu_ 2 þ v_ 2 þ w (35) F¼ c 2 0 0 where c has a different value for each term of the mode expansion; in particular " ! !# ^ ^ M N M _ 2m;n u_ 2m;n þ v_ 2m;n w 1 Lπ N 2 2 2 _ _ þ χ _ c þ ϕ þ ϕ þ : F¼ ∑ ∑ cm;n ∑ ∑ m;n 1m;n 2m;n m;n 2 2 2 2 n¼0m¼1 h h n¼0m¼1

(36)

In Eq. (36) displacements are non-dimensinalized dividing by h, while rotations are already non-dimensional. The damping coefficient cm,n is related to the modal damping ratio that can be evaluated from experiments by ζ m;n ¼ cm;n =ð2 μm;n ωm;n Þ, where ωm;n is the natural circular frequency of mode (m, n) and μm,n is the modal mass of this mode. The following notation is introduced for brevity: ^ and n ¼ 1; …; N or N: ^ q ¼ fum;n ; vm;n ; wm;n ; ϕ1m;n ; ϕ2m;n ; χ m;n gT ; m ¼ 1; …; M or M

(37)

The generic element of the time-dependent vector q is referred to as qj; the dimension of q is N, which is the number of degrees of freedom (dofs) used in the mode expansion.The generalized forces Qj are obtained by differentiation of the Rayleigh dissipation function and of the virtual work done by external forces: Qj ¼ − The Lagrange equations of motion are d dt

∂F ∂W þ : ∂ q_ j ∂ qj

! ∂T P ∂T P ∂U P − þ ¼ Q j; ∂q_ j ∂qj ∂qj

(38)

j ¼ 1; …; N;

(39)

where ∂ T P =∂ qj ¼ 0. The complicated term, derived from the maximum potential energy of the plate, giving quadratic and cubic nonlinearities, can be written in the form N N N ∂U ¼ ∑ f j;i qi þ ∑ f j;i;k qi qk þ ∑ f j;i;k;l qi qk ql ; ∂qj i¼1 i;k ¼ 1 i;k;l ¼ 1

j ¼ 1; …; N;

(40)

where the coefficients f have long expressions that include also geometric imperfections. It is interesting to observe that in Eq. (40) there are quadratic and cubic terms. 5.1. Inertial coupling in the equations of motion For shells with rotary inertia, inertial coupling arises in the equations of motion (see Eqs. (22) and (23)) so that they cannot be immediately transformed in the form required for numerical integration. In particular, the equations of motion take the following form: Mq€ þ Cq_ þ ½K þ N2 ðqÞ þ N3 ðq; qÞq ¼ f 0 cosðωtÞ;

(41)

where M is the non-diagonal mass matrix of dimension N  N (N being the number of degrees of freedom), C is the damping matrix, K is the linear stiffness matrix, which does not present terms involving q, N2 is a matrix that involves linear terms in q, therefore giving the quadratic nonlinear stiffness terms, N3 is a matrix that involves quadratic terms in q, therefore giving the cubic nonlinear stiffness terms, f0 is the vector of excitation amplitudes and q is the vector of the N generalized coordinates, defined in Eq. (37). In particular, by using Eq. (40), the generic elements kj,i, n2j; i and n3j; i , of the matrices K, N2 and N3, respectively, are given by kj;i ¼ f j;i ;

N

n2j;i ðqÞ ¼ ∑ f j;i;k qk ; k¼1

N

n3j;i ðq; qÞ ¼ ∑ f j;i;k;l qk ql :

(42a–c)

k;l ¼ 1

Eq. (41) is pre-multiplied by M−1 in order to diagonalize the mass matrix, as a consequence that the matrix M is always invertible; the result is Iq€ þ M−1 Cq_ þ ½M−1 K þ M−1 N2 ðqÞ þ M−1 N3 ðq; qÞq ¼ M−1 f 0 cosðωtÞ;

(43)

which can be rewritten in the following form: ~ þ M−1 N2 ðqÞ þ M−1 N3 ðq; qÞq ¼ f~ 0 cosðωtÞ; Iq€ þ C~ q_ þ ½K

(44)

Please cite this article as: M. Amabili. A new nonlinear higher-order shear deformation theory with thickness variation for large-amplitude vibrations of laminated doubly curved shells, Journal of Sound and Vibration (2013), http://dx.doi.org/ 10.1016/j.jsv.2013.03.024i

M. Amabili / Journal of Sound and Vibration ] (]]]]) ]]]–]]]

13

Table 1 Natural frequencies (in Hz) of the lowest mode of the thick laminated shell for different number of circumferential waves n; for n ¼0 the lowest mode is torsional. Comparison of results obtained by using the present model (with thickness variation) and results obtained by Amabili [14] with higher-order shear deformation theory developed by Amabili and Reddy [12]. Circumferential waves (n)

Present model

Amabili [14]

Novozhilov theory

0 1 2 3 4 5

782.50 569.09 537.80 970.45 1591.40 2276.83

782.49 567.74 535.47 966.69 1585.28 2267.71

786.40 575.18 575.89 1155.97 2115.24 3377.61

Table 2 Natural frequencies (in Hz) of the lowest mode of the thin laminated shell for different number of circumferential waves n; for n ¼0 the lowest mode is torsional. Comparison of results obtained by using the present model (with thickness variation) and results obtained by Amabili [14] with higher-order shear deformation theory developed by Amabili and Reddy [12]. Circumferential waves (n)

Present model

Amabili [14]

Novozhilov theory

0 1 2 3 4 5

785.07 540.91 325.18 237.46 256.16 353.97

785.07 540.89 325.10 237.03 254.86 351.61

785.11 540.89 325.12 237.17 255.57 353.64

where C~ ¼ M−1 C;

~ ¼ M−1 K and f~ 0 ¼ M−1 f 0 : K

(45a–c)

Eq. (44) is in the form suitable for numerical integration.

5.2. Numerical linear results Initially linear results are obtained in order to validate the present numerical approach. Six natural frequencies are obtained for any set of (m, n), where m is the number of axial half-waves of the mode shape and n is the number of circumferential waves. Here it is opportune to observe that three natural frequencies for any set of (m, n) are instead obtained for a classical shell theory and five for shear-deformable shell theories. The additional mode with respect to sheardeformable theories is the peculiar thickness variation mode, which is an high-frequency mode and cannot be predicted with other theories. Calculations have been performed for the graphite/epoxy laminated circular cylindrical shell with simply supported edges previously studied by Amabili [14]. The dimensions and material properties of this imperfectionfree shell are: R¼0.15 m, L ¼0.52 m, h¼0.03 m, E1 ¼50  109 Pa, E2 ¼2  109 Pa, G12 ¼G13 ¼1  109 Pa, G23 ¼ 0.4  109 Pa, ν12 ¼ν13 ¼0.25, ν23 ¼0.22 and ρ¼1500 kg/m3. This is a thick shell, being R/h¼5. The shell is made of four layers 01/901/901/01 of the same thickness (β¼0 for the internal layer, then two layers at β¼π/2 and finally the external layer with β¼0). In the numerical calculations, the normal stress correction factor Kz has been considered equal to one. The natural frequencies of the lowest frequency modes with number n of circumferential waves from 0 to 5 are given in Table 1. The fundamental mode has n ¼2 circumferential waves. Results in Table 1 are compared to those obtained in reference [14] by using the Amabili–Reddy theory, which neglects thickness variation, and to results obtained by using the classical Novozhilov shell theory [2], which neglects shear deformation, rotary inertia and thickness variation. The largest difference in estimated natural frequency between the present theory and neglecting the thickness variation is 9.12 Hz, corresponding to a percentage difference of 0.4 percent, for the mode with n ¼5 circumferential waves among those in Table 1. All the natural frequencies computed by using the present theory are slightly larger than those evaluated in Ref. [14]. This is due to the removal of the hypothesis s3 ¼0, that is used in [14] and in other shear deformable theories, and from the removes the artificial constraint of no thickness variation, which actually considered alone should decrease the natural frequencies. The difference with the classical Novozhilov theory is instead large and increases with the number of circumferential waves. This large difference is observed here since the laminated shell is thick. A similar shell, with just the thickness reduced ten times (h ¼0.003 m) is studied in Table 2. This can be refereed as a thin laminated shell (R/h¼50). The largest difference of the present theory with reference [14] is again for mode n ¼5 and is 0.67 percent. For a thin laminated shell it is also observed that the classical Novozhilov shell theory gives also excellent accuracy. Please cite this article as: M. Amabili. A new nonlinear higher-order shear deformation theory with thickness variation for large-amplitude vibrations of laminated doubly curved shells, Journal of Sound and Vibration (2013), http://dx.doi.org/ 10.1016/j.jsv.2013.03.024i

14

M. Amabili / Journal of Sound and Vibration ] (]]]]) ]]]–]]]

Fig. 7. Deflection of the pressurized thick laminated circular cylindrical shell. (a) Generalized coordinate w1,0 associated to axisymmetric radial deflection; ——, present theory with thickness deformation, 28 dofs; - -, theory presented in [14]; and (b) generalized coordinate χ1,0 associated to thickness variation (only for present theory).

5.3. Numerical nonlinear results The nonlinear equations of motion have been integrated by using the software AUTO [24] for continuation and bifurcation analysis by using the pseudo-arclength continuation method, starting at zero force from the trivial solution. All the variables have been made non-dimensional to improve numerical accuracy. A non-dimensionalization of the normal force excitation has also be introduced. The non-dimensional force f is defined as f ¼ f~ =ðhω21n μÞ where μ is the modal mass of mode (1,n) for the normal displacement w, given by μ ¼ ρhRπL=2. The non-dimensional pressure P is defined as P ¼ 4LRp=ðhω21n 2μÞ. The solution has been initially continued with the uniform internal pressure amplitude as parameter. Then the solution has been continued with the excitation amplitude as parameter and fixed excitation frequency. Once the desired excitation level has been reached, the solution has been continued by using the excitation frequency as continuation parameter. Pitchfork bifurcations have been detected on the first branch (the one with driven mode active), originating the second branch, which is the one with companion mode participation. In all the forced vibration simulations, a modal damping coefficient ς1;n ¼ 0:001 has been used for the mode (1,n), for all its generalized coordinates. For the other modes, a damping proportionally increasing with frequency has been assumed. Numerical calculations have been performed for the thick laminated shell previously investigated in Section 5.2. In fact, for a thick shell the thickness variation is more significant. Fig. 7 presents the effect of the internal pressure on the axialsymmetric deformation of the shell. In particular, the deflection in the shape of the first axial-symmetric generalized coordinate w1,0 is shown in Fig. 7(a), while the thickness variation χ1,0 is shown in Fig. 7(b) and it is important; up to a 25 percent of thickness reduction due to very high pressurization is observed. In Fig. 7(a) the numerical results obtained with the shell theory that neglects thickness variation [14] are shown for comparison, and results are almost perfectly superimposed to the present results. This means that for static load, the theory in Ref. [14] is very accurate, even if it cannot predict the thickness variation. After reaching the maximum pressure P ¼1200 indicated in Fig. 7, the dynamic response is studied. Fig. 8 presents the frequency-response curve for forced vibrations of the pressurized (P ¼1200) shell in the frequency neighborhood of the Please cite this article as: M. Amabili. A new nonlinear higher-order shear deformation theory with thickness variation for large-amplitude vibrations of laminated doubly curved shells, Journal of Sound and Vibration (2013), http://dx.doi.org/ 10.1016/j.jsv.2013.03.024i

M. Amabili / Journal of Sound and Vibration ] (]]]]) ]]]–]]]

15

Fig. 8. Frequency-response curve of the pressurized thick laminated circular cylindrical shell; only the first branch of the solution is shown for the generalized coordinate w1,n,c; P¼1200, f¼0.086; n¼ 2. Thick continuous and broken lines, present theory; continuous thin line, results presented in [14]. ——, Stable periodic solution (present theory); - -, unstable or quasi-periodic solutions (present theory); BP, pitchfork bifurcation; TR, Neimark–Sacker (torus) bifurcation.

resonance of the fundamental mode (n ¼2) computed by using the present theory (thick continuous and broken lines) and the Amabili–Reddy nonlinear theory [14] (thin continuous line). Due to pressurization, the resonance is largely increased, i. e. 13.82 times in the present case, with respect to the resonance frequency ω1,n of the unpressurized shell. A nondimensional harmonic point force excitation of amplitude f ¼0.086 and frequency ω applied at shell mid-length is assumed. Only the response without companion mode participation (w1,n,s(t) ¼0) is presented in Fig. 8. This figure shows that the nonlinear response predicted by the present theory is slightly moved to higher frequencies with respect to the one computed neglecting thickness variation [14]. However, more significant than the frequency change is the change in the shape of the nonlinear response, that with the present theory shows a hardening behavior with a further folding for larger vibration amplitudes. This is much more pronounced than the behavior predicted by neglecting the thickness variation, which is instead more smooth. Therefore, a significant difference in the nonlinear behavior predicted by the two different theories is predicted for the highly pressurized shell, due to the significant thickness reduction introduced by the pressure. The response in Fig. 8 presents two pitchfork bifurcations and two Neimark–Sacker bifurcations, the latter ones being close to the peak. Between these two Neimark–Sacker bifurcations, a quasi-periodic response is obtained. The branch 1 of the nonlinear response in Fig. 9 corresponds to vibration with zero amplitude of the companion mode w1;n;s ðtÞ, as it was shown in Fig. 8. Branch 1 has two pitchfork bifurcations (BP) at ω=ω1;n ¼ 13.8167 and at 13.8246, where branch 2 appears. This new branch corresponds to participation of both w1;n;c ðtÞ and w1;n;s ðtÞ, giving a traveling-wave response. The companion mode presents a node at the location of the excitation force. Therefore, it is not directly excited; its amplitude is different than zero only for large-amplitude vibrations, due to nonlinear coupling through 1:1 internal resonance. In the frequency region where both w1;n;c ðtÞ and w1;n;s ðtÞ are different from zero, they give rise to a traveling wave around the shell; phase shift between the two coordinates is practically equal to π=2 when the two generalized coordinates have almost the same amplitude. This branch appears for sufficiently large excitation. Branch 2 presents two peaks, at ω=ω1;n ¼13.7947 and 13.8246. It also shows two Neimark–Sacker bifurcations; between these two bifurcations, a quasiperiodic response is obtained. Fig. 9(c) shows the contribution of the first axisymmetric mode, with its large static deflection due to pressurization. In Fig. 9((a)–(c)) results computed neglecting the thickness variation are reported for comparison. It is evident that the frequency region with traveling wave response is smaller when computed taking thickness variation in account. Fig. 9(d) presents the maximum of the generalized coordinate χ 1;n;c associated to the thickness variation and shows that a dynamic thickness variation of about 2.5/100 of the shell thickness is observed in this case at the response peak. Instead Fig. 9(e) shows a large uniform static thickness variation, due to pressurization, with a small dynamic contribution at twice the excitation frequency. The time response for the driven mode w1;n;c ðtÞ, companion mode w1;n;s ðtÞ and the first axisymmetric mode w1;0 ðtÞ is given in Fig. 10((b)–(d)), respectively, for excitation frequency ω=ω1;n ¼13.7947 (i.e. at the response peak of branch 2 for the pressurized shell) with companion mode participation; the excitation is reported in Fig. 10(a) so that the phase relationship can be obtained. In particular, the amplitude of the driven and companion modes, Fig. 10((b) and (c)), for this excitation frequency is practically the same and the phase angle is π/2, giving rise to the described traveling wave. Fig. 10(d) shows that the first axisymmetric mode has a double frequency with respect to the excitation. This generalized coordinate is associated to the dynamic axisymmetric contraction during large-amplitude vibration and must be retained in the expansion in order to avoid large non-physical in-plane stretching of the shell. In fact, thin shells bends easily but they are very stiff to in-plane stretching. The time behavior of the axial generalized coordinate u1;n;c is shown in Fig. 10(e). Fig. 10(f) shows the time Please cite this article as: M. Amabili. A new nonlinear higher-order shear deformation theory with thickness variation for large-amplitude vibrations of laminated doubly curved shells, Journal of Sound and Vibration (2013), http://dx.doi.org/ 10.1016/j.jsv.2013.03.024i

16

M. Amabili / Journal of Sound and Vibration ] (]]]]) ]]]–]]]

Fig. 9. Frequency-response curve of the thick laminated circular cylindrical shell obtained by using the proposed nonlinear theory (thick continuous and broken lines) versus the results presented in [14] (thin continuous line); n¼2, f ¼0.086, P¼ 1200. ——, Stable periodic solution (present theory); - -, unstable or quasi-periodic solutions (present theory); BP, pitchfork bifurcation; TR, Neimark–Sacker bifurcation; 1, branch 1; 2, branch 2. (a) Maximum of the generalized coordinate w1;n;c (driven mode) normalized with respect to the shell thickness h; (b) maximum of the generalized coordinate w1;n;s (companion mode) normalized with respect to h; (c) maximum of the generalized coordinate w1;0 (first axisymmetric mode) normalized with respect to h; (d) maximum of the generalized coordinate χ 1;n;c (first coordinate associated to thickness variation); and (e) maximum of the generalized coordinate χ 1;0 (first axisymmetric coordinate associated to thickness variation).

response of the generalized coordinate ϕ1;n;c associated to rotations of the transverse normals at z ¼0 about the α2 axis. Fig. 10(g) and (h) present the coordinate χ 1;n;c and χ 1;n;s , respectively, associated to the thickness variation. It is interesting to observe that the thickness contraction χ 1;n;c presents the same shape of the deflection coordinate w1;n;c . The same is observed for χ 1;n;s and w1;n;s . 6. Conclusions The proposed theory is an innovative higher-order shear deformation theory retaining nonlinearities in the in-plane displacements and thickness variation. It is the extension of the Amabili–Reddy theory to include thickness variation. Please cite this article as: M. Amabili. A new nonlinear higher-order shear deformation theory with thickness variation for large-amplitude vibrations of laminated doubly curved shells, Journal of Sound and Vibration (2013), http://dx.doi.org/ 10.1016/j.jsv.2013.03.024i

M. Amabili / Journal of Sound and Vibration ] (]]]]) ]]]–]]]

17

Fig. 10. Time responses at the peak amplitude of branch 2 of the frequency-response curve with companion mode participation, ω=ω1;n ¼ 13:7947; n¼2, f¼ 0.086, P ¼1200. (a) Force excitation; (b) generalized coordinate w1;n;c ; (c) generalized coordinate w1;n;s ; (d) generalized coordinate w1;0 ; (e) generalized coordinate u1;n;c ; (f) generalized coordinate ϕ1;n;c ; and (g) generalized coordinate χ 1;n;c ; (h) generalized coordinate χ 1;n;s .

Please cite this article as: M. Amabili. A new nonlinear higher-order shear deformation theory with thickness variation for large-amplitude vibrations of laminated doubly curved shells, Journal of Sound and Vibration (2013), http://dx.doi.org/ 10.1016/j.jsv.2013.03.024i

M. Amabili / Journal of Sound and Vibration ] (]]]]) ]]]–]]]

18

Moreover, it has been derived with consistency without introducing nonlinearities of the von Kármán at the end of the derivation of the theory. Numerical results show that, for the studied cases, the thickness variation has a marginal role in both the prediction of the low-frequency flexural modes and the static deflection due to pressurization. However, the nonlinear forced response at low-frequency excitation of a highly pressurized laminated graphite/epoxy circular cylindrical shell presents significant differences if the thickness variation is neglected or taken into account. The present theory has been developed to study soft tissues that are described by an hyperelastic behavior and that are subjected to thickness variation even more significant that the 25 percent thickness reduction observed here for the pressurized shell. In fact, in case of hyperelastic materials is fundamental to obtain the transverse strain ε3;0 with accuracy in order to calculate the strain-energy function. Therefore, all shell theories that neglect the thickness variation are not really suitable to study hyperelastic materials and the present theory seems to present great advantages. The present theory is also suitable to study high-frequency modes of isotropic and laminated shells, for which thickness variation could be not negligible.

Acknowledgments The author acknowledges the financial support of the NSERC Discovery Grant, Canada Research Chair and Canada Foundation for Innovation programs of Canada, the PSR-SIIRI grant of Québec and the SUPERPANEL FP7 project, Marie Curie Actions, of the European Community.

Appendix A. Stress–strain relations for a layer within a laminate The coefficients in Eq. (18) for a lamina are given by c11 ¼

Ex ð1−νyz Þ ; ð1−νyz −2νxy νyx Þ

c22 ¼ c33 ¼

c12 ¼ c21 ¼ c13 ¼ c31 ¼

ð1−νxy νyx ÞEy ; ð1 þ νyz Þð1−νyz −2νxy νyx Þ

νxy Ey νyx Ex ¼ ; ð1−νyz −2νxy νyx Þ ð1−νyz −2νxy νyx Þ

c23 ¼ c32 ¼

ðνyz þ νxy νyx ÞEy ; ð1 þ νyz Þð1−νyz −2νxy νyx Þ

νij Ej ¼ νji Ei :

(A1a,b)

(A1c,d) (A1e)

Usually, the lamina material axes (x,y) do not coincide with the plate reference axes (α1,α2), while the 3 axis is coincident with z. Then, the strains and stresses on material axes can be related to the reference axes by using the following invertible expressions [11,25]: 9 9 9 9 8 8 8 8 εxx > sx > ε11 > s1 > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > εyy > sy > ε22 > s2 > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > = < sz = < s3 = < εzz = < ε33 > ¼ T1 ; ¼ T2 ; (A2a,b) γ τ γ τ > > > > > > > yz yz 23 > 23 > > > > > > > > > > > > > > > > > > > > > > > > > > > > γ xz > τxz > > > > > > > > > γ 13 > τ13 > > > > > > > > > > > > > > > > > > > > > > > > > : : : : γ xy ; τxy ; γ 12 ; τ12 ; where 2 6 6 6 6 6 T1 ¼ 6 6 6 6 4 2 6 6 6 6 6 T2 ¼ 6 6 6 6 4

cos2 β

sin2 β

0

0

0

sin2 β

cos2 β

0

0

0

0

0

1

0

0

0

0

0

cos β

−sin β

0

0

0

sin β

cos β

−sin βcos β

sin βcos β

0

0

0

cos2 β

sin2 β

0

0

0

sin2 β

cos2 β

0

0

0

0 0

0 0

1 0

0 cos β

0 −sin β

0

0

0

sin β

cos β

−2sinβcosβ

2sinβcosβ

0

0

0

2sin βcos β

3

7 −2sin βcos β 7 7 7 0 7 7; 7 0 7 7 0 5 2 cos2 β−sin β sin βcos β

(A3)

3

7 −sin βcos β 7 7 7 0 7 7; 7 0 7 7 0 5 2 2 cos β−sin β

(A4)

β being the angle between the shell principal coordinate α1 and the material axis x as shown in Fig. 4. Please cite this article as: M. Amabili. A new nonlinear higher-order shear deformation theory with thickness variation for large-amplitude vibrations of laminated doubly curved shells, Journal of Sound and Vibration (2013), http://dx.doi.org/ 10.1016/j.jsv.2013.03.024i

M. Amabili / Journal of Sound and Vibration ] (]]]]) ]]]–]]]

19

It can be shown that [11,26] ðT1 −1 ÞT ¼ T2 : ðkÞ

Therefore, the matrix ½Q 

(A5)

in Eqs. (19) and (20) is given by ½Q ðkÞ ¼ ½ðT2 ÞT CT2 ðkÞ ;

(A6)

where C is the 6  6 matrix of cij and Gij coefficients in Eq. (18). As a consequence of the discontinuous variation of the stiffness matrix ½Q ðkÞ from layer to layer, the stresses may be discontinuous layer to layer. In particular, 2 2 4 2 2 ðkÞ Q ðkÞ 11 ¼ ½c11 cos θ þ sin θð2c12 cos θ þ 4Gxy cos θ þ c22 sin θÞ ;

(A7)

4 2 4 2 2 2 ðkÞ Q ðkÞ 12 ¼ ½c12 ðsin θ þ cos θÞ þ sin θðc11 cos θ þ c22 cos θ−4Gxy cos θÞ :

(A8)

Appendix B. Strain–displacement relations for a circular cylindrical shell

εx;0 ¼

εθ;0 ¼

∂u 1 þ ∂x 2

"

2
2 # ∂u 2 ∂v ∂w ∂w0 ∂w ; þ þ þ ∂x ∂x ∂x ∂x ∂x

"


 # 1 ∂v w 1 1 ∂u 2 1 ∂v w 2 1 ∂w v 2 þ þ þ − þ þ R ∂θ R 2 R ∂θ R ∂θ R R ∂θ R

 w0 1 ∂v w 1 ∂w0 1 ∂w v þ þ − ; þ R ∂θ R ∂θ R R R ∂θ R εz;0 ¼ χ þ

γ xθ;0 ¼

v2 2R2



 ∂v 1 ∂u 1 ∂u ∂u 1 ∂v w ∂v ∂w 1 ∂w v þ þ þ þ þ − ∂x R ∂θ R ∂x ∂θ R ∂θ R ∂x ∂x R ∂θ R
 w0 ∂v ∂w0 1 ∂w v 1 ∂w0 ∂w þ þ − þ ; R ∂θ ∂x R ∂x ∂x R ∂θ R

γ θz;0 ¼ ϕ2 þ

(B5)

1 ∂w ; R ∂θ

(B6)

∂2 χ ; 2∂x2

(B8)


 ∂ϕ1 ∂2 w þ 2 ; ∂x ∂x 3h

(B9)

ð1Þ

ð0Þ

kθ ¼ ð1Þ

ð2Þ




4 2

3h

4

2

1 ∂ϕ2 w χ − þ ; R ∂θ R2 R

(B10)

 1 ∂ϕ2 1 ∂v 1 ∂2 w χ ∂2 χ − þ − 2− 2 2; 2 2 ∂θ R ∂θ 2R ∂θ R 2R ∂θ

(B11)


 1 ∂ϕ2 1 ∂2 w 1 2 ∂ϕ2 1 ∂2 w 2∂2 χ þ 2 2 − 2 þ 2 2 þ 3 2; R ∂θ R ∂θ 3R R ∂θ R ∂θ 3R ∂θ

(B12)

kθ ¼ −

1 R

2

ð0Þ




(B4)

(B7)

kx ¼ − ð2Þ

(B3)

∂ϕ1 ; ∂x

ð0Þ

kx ¼

kx ¼ −

(B2)

∂w ; ∂x

γ xz;0 ¼ ϕ1 þ

kθ ¼ −

;

(B1)

kxθ ¼


 1 ∂ϕ1 ∂ϕ2 1 ∂v 1 ∂u − ; þ þ R ∂θ R ∂x R ∂θ ∂x

(B13)

Please cite this article as: M. Amabili. A new nonlinear higher-order shear deformation theory with thickness variation for large-amplitude vibrations of laminated doubly curved shells, Journal of Sound and Vibration (2013), http://dx.doi.org/ 10.1016/j.jsv.2013.03.024i

M. Amabili / Journal of Sound and Vibration ] (]]]]) ]]]–]]]

20

ð1Þ

kxθ ¼ ð2Þ

kxθ ¼ −




4 2

3h


 1 1 ∂ϕ1 1 ∂ϕ2 1 ∂2 w ∂2 χ − − ; þ þ R R ∂θ 2 ∂x 2R ∂x∂θ R∂x∂θ

1 ∂ϕ1 ∂ϕ2 2 ∂2 w þ þ R ∂θ R ∂x ∂θ ∂x

 þ


 ∂ϕ 1 ∂2 w 2∂2 χ þ 2 ; − 2þ R ∂x ∂θ ∂x 6R 3R ∂x ∂θ 1

2

ð0Þ

(B15)

kxz ¼ 0;

(B16)


 4 ∂w ð1Þ ; kxz ¼ − 2 ϕ1 þ ∂x h

(B17)

ð2Þ

kxz ¼ 0; ð0Þ

kθz ¼ 0; ð1Þ

(B14)

kθz ¼ −

4 2

h


ϕ2 þ

 1 ∂w v þ 3; R ∂θ R

ð2Þ

kθz ¼ 0:

(B18) (B19) (B20)

(B21)

Appendix C. Kinetic energy for a circular cylindrical shell In particular, for a laminate with the same density for all the layers and uniform thickness, the following simplified expression is obtained:

  Z 2π Z L  χ_ 1 17 _ 2 _ 2 2 41 _ 1 u_ þ _ 2 þ h2 þ ðϕ1 þ ϕ2 Þ þ ϕ ϕ_ v_ T S ¼ ρS h u_ 2 þ v_ 2 þ w 12 2 315 15R 120R 2 0 0

 _ _ _ _ v_ 1 v_ 2 1 ∂w 1 ∂w 1 u_ 8 _ 1 ∂w 1 ∂w 8 _ _χþ − − w_ − ϕ þ ϕ þ þ þ 1 2 4 R2 6R ∂x 252 ∂x 30 R 315 R ∂θ 252R ∂θ 120R 315  u_ ∂_χ v_ ∂_χ 5 − Rdθdx þ Oðh Þ: − (C1) 12 ∂x 12 R∂θ If the ratio h/R is negligible (i.e. for thin shells), Eq. (C1) can be simplified.

References [1] M. Amabili, S. Farhadi, Shear deformable versus classical theories for nonlinear vibrations of rectangular isotropic and laminated composite plates, Journal of Sound and Vibration 320 (2009) 649–667. [2] M. Amabili, Nonlinear Vibrations and Stability of Shells and Plates, Cambridge University Press, New York, USA, 2008. [3] L. Librescu, Refined geometrically nonlinear theories of anisotropic laminated shells, Quarterly of Applied Mathematics 45 (1987) 1–22. [4] J.N. Reddy, Exact solutions of moderately thick laminated shells, Journal of Engineering Mechanics 110 (1984) 794–809. [5] J.N. Reddy, C.F. Liu, A higher-order shear deformation theory of laminated elastic shells, International Journal of Engineering Science 23 (1985) 319–330. [6] R.A. Arciniega, J.N. Reddy, A consistent third-order shell theory with application to composite circular cylinders, AIAA Journal 43 (2005) 2024–2038. [7] J.N. Reddy, R.A. Arciniega, Shear deformation plate and shell theories: from Stavsky to present, Mechanics of Advanced Materials and Structures 11 (2004) 535–582. [8] J.N. Reddy, A general non-linear third-order theory of plates with moderate thickness, International Journal of Non-Linear Mechanics 25 (1990) 677–686. [9] S.T. Dennis, A.N. Palazotto, Large displacement and rotation formulation for laminated shells including parabolic transverse shear, International Journal of Non-Linear Mechanics 25 (1990) 67–85. [10] A.N. Palazotto, S.T. Dennis, Nonlinear Analysis of Shell Structures, AIAA Educational Series, Washington, DC, USA, 1992. [11] J.N. Reddy, Mechanics of Laminated Composite Plates and Shells: Theory and Analysis, 2nd edition, CRC Press, Boca Raton, FL, USA, 2004. [12] M. Amabili, J.N. Reddy, A new non-linear higher-order shear deformation theory for large-amplitude vibrations of laminated doubly curved shells, International Journal of Non-linear Mechanics 45 (2010) 409–418. [13] E. Carrera, A study of transverse normal stress effects on vibration of multilayered plates and shells, Journal of Sound and Vibration 225 (1999) 803–829. [14] M. Amabili, Nonlinear vibrations of laminated circular cylindrical shells: comparison of different shell theories, Composite Structures 94 (2011) 207–220. [15] F. Alijani, M. Amabili, Nonlinear vibrations of thick laminated circular cylindrical panels, Composite Structures 96 (2013) 643–660. [16] E. Carrera, S. Brischetto, M. Cinefra, M. Soave, Effects of thickness stretching in functionally graded plates and shells, Composites: Part B 42 (2011) 123–133. [17] A.J.M. Ferreira, E. Carrera, M. Cinefra, C.M.C. Roque, Analysis of laminated doubly-curved shells by a layerwise theory and radial basis functions collocation, accounting for through-the-thickness deformations, Computational Mechanics 48 (2011) 13–25. [18] A.K. Nayak, R.A. Shenoi, J.I.R. Blake, A study of transient response of initially stressed composite sandwich folded plates, Composites: Part B 44 (2013) 67–75. [19] J. Awrejcewicz, L. Kurpa, T. Shmatko, Large amplitude free vibration of orthotropic shallow shells of complex shapes with variable thickness, Latin American Journal of Solids and Structures 10 (2013) 149–162.

Please cite this article as: M. Amabili. A new nonlinear higher-order shear deformation theory with thickness variation for large-amplitude vibrations of laminated doubly curved shells, Journal of Sound and Vibration (2013), http://dx.doi.org/ 10.1016/j.jsv.2013.03.024i

M. Amabili / Journal of Sound and Vibration ] (]]]]) ]]]–]]]

21

[20] M. Ganapathi, Large amplitude free flexural vibrations of laminated composite curved panels using shear-flexible shell element, Defence Science Journal 45 (1995) 55–60. [21] T. Kant, Mallikarjuna, Vibrations of unsymmetrically laminated plates analyzed by using a higher-order theory with a Co finite element formulation, Journal of Sound and Vibration 134 (1989) 1–16. [22] M. Amabili, Comparison of shell theories for large-amplitude vibrations of circular cylindrical shells: Lagrangian approach, Journal of Sound and Vibration 264 (2003) 1091–1125. [23] V.V. Novozhilov, Foundations of the Nonlinear Theory of Elasticity, Graylock Press, Rochester, NY, USA, 1953 (now available from Dover, NY, USA). [24] E.J. Doedel, A.R. Champneys, T.F. Fairgrieve, Y.A. Kuznetsov, B. Sandstede, X. Wang, AUTO 97: Continuation and Bifurcation Software for Ordinary Differential Equations (with HomCont), Concordia University, Montreal, Canada, 1998. [25] J.E. Ashton, J.M. Whitney, Theory of Laminated Plates, Technomic, Stamford, CT, USA, 1970. [26] A.K. Nayak, S.S.J. Moy, R.A. Shenoi, Free vibration analysis of composite sandwich plates based on Reddy's higher-order theory, Composites: Part B 33 (2002) 505–519.

Please cite this article as: M. Amabili. A new nonlinear higher-order shear deformation theory with thickness variation for large-amplitude vibrations of laminated doubly curved shells, Journal of Sound and Vibration (2013), http://dx.doi.org/ 10.1016/j.jsv.2013.03.024i