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Asymptotic expansions of stress tensor for linearly elastic shell
Applied Mathematical Modelling 37 (2013) 7964–7972
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Asymptotic expansions of stress tensor for linearly elastic shell Shen Xiaoqin a,⇑, Li Kaitai b, Ming Yang c a b c
School of Sciences, Xi’an University of Technology, Xi’an 710054, PR China School of Sciences, Xi’an Jiaotong University, Xi’an 710049, PR China Road Traffic Detection and Equipment Engineering Research Center, Chang’an University, Xi’an 710064, PR China
a r t i c l e
i n f o
Article history: Received 1 November 2010 Received in revised form 8 February 2013 Accepted 4 March 2013 Available online 19 March 2013 Keywords: Linearly elastic shell Asymptotic analysis Stress tensor
a b s t r a c t In this paper, the asymptotic expansions of stress tensor for linearly elastic shell have been proposed by new asymptotic analysis method, which is different from the classical asymptotic analysis. The new asymptotic analysis method has two distinguishing features: one is that the displacement is expanded with respect to the thickness variable of the middle surface not to the thickness; another is that the first order term and the second order term of the displacement variable can be algebraically expressed by the leading term. To decompose stress tensor totally into 2-D variable and thickness variable, we have three steps: operator splitting, variables separation and dimension splitting. In the end, a numerical experiment of special hemispherical shell by FEM (finite element method) is provided. We derive the distribution of displacements and stress fields in the middle surface. Crown Copyright Ó 2013 Published by Elsevier Inc. All rights reserved.
1. Introduction The main contents of elastic mechanics is to research the rules how the stress depends on the strain when the forces are applied to the elasticity. The relations between the stress and the strain can be called as constitutive equation. When the relationships between the components of stress and strain are linear or the deformations are ‘‘small’’, the elasticity is called linear elasticity. (seen in [1]). In this paper, we mainly consider linearly elastic shell. Suppose that strain tensor and stress tensor satisfies Hooke’s law in linearly elastic shell. We have proposed the asymptotic expansions of stress tensor for linearly elastic shell by using new asymptotic analysis method, which is different from the classical asymptotic analysis. The new asymptotic analysis method has two distinguishing features: one is that the displacement is expanded with respect to the thickness variable n of the middle surface not to the thickness e; another is that the first order term u1 and the second order term u2 of displacement variable can be algebraically expressed by the leading term u0 . Thus 3-D stress tensor is totally decomposed into 2-D variable and thickness variable, which makes the numerical computation of stress tensor in engineer easier and more convenient than before. To derive the asymptotic expansions of stress tensor, i.e., to decompose stress tensor totally into 2-D variable and thickness variable, we have three steps. We call them ‘‘operator splitting’’, ‘‘variables separation’’ and ‘‘dimension splitting’’. In the end, a numerical experiment of special hemispherical shell by FEM (finite element method) is provided. We derive the distribution of the displacements of the middle surface in new asymptotic method and compare the results of two methods. We derive the distribution of stress fields of the middle surface in new asymptotic method. Numerical results are consistent with the practice and thus prove the validity of the new asymptotic method.
⇑ Corresponding author. E-mail addresses: [email protected], [email protected] (S. Xiaoqin). 0307-904X/$ - see front matter Crown Copyright Ó 2013 Published by Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.apm.2013.03.032
S. Xiaoqin et al. / Applied Mathematical Modelling 37 (2013) 7964–7972
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2. Preliminary ^ e E3 (3D-Euclidean space) An elastic shell with constant thickness 2e is an elastic body whose reference configuration X 3 consists of all points within a distance 6 e from a given surface S E , where the parameter e > 0 is the half thickness of the shell and is thought of as small enough. The surface S is defined as the image by ~ h of the closure of a domain x R2 , where 3 ~ ! E is a smooth injective immersion. Let ~ h:x n denote a continuously varying unit normal vector along S and let Xe ¼ x ðe; eÞ. (seen in [2]). ^ e is given by X ^e ¼ H ~ ðX e Þ, where the mapping H ~:X e R3 ! E3 is defined by Hence the set X
e ; ðx1 ; x2 Þ 2 x: ~ ðx1 ; x2 ; nÞ ¼ ~ H hðx1 ; x2 Þ þ n~ nðx1 ; x2 Þ 8ðx1 ; x2 ; nÞ 2 X ^ e consists of following: upper surface Ct ¼ S fþeg; bottom surface Cb ¼ S feg; lateral surface The boundary of shell X Cl ¼ C0 [ C1 ; C0 ¼ c0 ðe; þeÞ; C1 ¼ c1 ðe; þeÞ, where c ¼ @ x is the boundary of x and c ¼ c0 [ c1 . (seen in [3]) The pair ðx1 ; x2 Þ constitute curvilinear coordinates on S, and ðx1 ; x2 ; nÞ is called semi-geodesic coordinate system (E3 being viewed as a Riemann space and S as a 2D sub-manifold). In elasticity, S is called middle surface of the shell. In the following, Latin indices and exponents: i; j; k; . . ., take their values in the set {1, 2, 3} while Greek indices and exponents: a; b; c; . . ., take their values in the set {1, 2}. In addition, the repeated index summation convention is systematically used. The covariant and contravariant components of the metric tensor of the surface Sare given by
aab ¼ ~ ha ~ hb ;
aab abk ¼ dak ; where ~ ha ¼
@~ h ; @xa
~ ha ~ hb ¼ dba
and the covariant and contravariant components of the curvature tensor on S are defined as follows
bab ¼ ~ n ~ hab ¼ ~ na ~ hb ;
b
ab
¼ aak abr bkr ; where ~ na ¼
@~ n ; @xa
@~ hb ~ hab ¼ a : @x
Similarly the covariant and contravariant components of the third fundamental forms on S are defined as follows
cab ¼ ~ na ~ nb ¼ akr bak bbr ;
cab ¼ aak abr ckr :
Note that the mean curvature H and the Gaussian curvature K is defined by
H¼
1 ab a bab ; 2
K¼
detðbab Þ b ¼ : detðaab Þ a
The covariant derivative on S is defined as (seen in [4])
ra brs ¼ @brs =@ya þ Cr ab bbs Cb as brb
where Cr ab ¼ ~ hr ~ hab denotes the Christoffel symbols on S. In the coordinate system ðxa ; nÞ, the metric tensor of E3 is given by
~i H ~ j; g ij ¼ H
~ ~ i :¼ @ H : g ik g kj ¼ dij ; where H @xi
3. Formal Taylor expansions of elasticity tensor and strain tensor From Hooke’s law for linearly elastic shell, we have
rij ðuÞ ¼ Aijkl ekl ðuÞ ijkl
ð3-1Þ
where r ðuÞ; A ; ekl ðuÞ is stress tensor, elasticity tensor and strain tensor respectively; u is the displacement of any point in the shell. In order to obtain the asymptotic expansions of tress tensor, we firstly deduce the formal Taylor expansions of elasticity tensor and strain tensor. We assume that the elastic material constituting the shell is isotropic and homogenous and that the reference configuration of the shell is a natural state. The contravariant components of the elasticity tensor are given by ij
Aijkl ¼ kg ij g kl þ l g ik g jl þ g il g jk ; where k P 0 and
l P 0 are Lamé constants.
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Let
8 abrs a ¼ kaab ars þ lðaar abs þ aas abr Þ; > > < abrs ab rs ar bs a s br ¼ kb b þ lðb b þ b b Þ; b > > : abrs rs ab bs ar br as ¼ kðaab b þ ars b Þ þ lðaar b þ abs b þ aas b þ abr b Þ: c and
hðnÞ ¼ 1 2Hn þ Kn2 þ : We give the following expression for the contravariant components of the elasticity tensor
" # 8 4 X > a brs a brs k 4 > a b rs > A ¼h a þ Ak n ; > > > > k¼1 > > > > > < Aabr3 ¼ Aab3r ¼ Aa3br ¼ A3abr ¼ 0;
ð3-2Þ
> Aab33 ¼ A33ab ¼ kg ab ; A3333 ¼ k þ 2l; > > > > > a3b3 > > A ¼ A3a3b ¼ Aa33b ¼ A3ab3 ¼ lg ab ; > > > > : a333 A ¼ A3a33 ¼ A33a3 ¼ A333a ¼ 0; where
8 abrs > A > > 1 > > < Aabrs 2 abrs > > A > 3 > > : abrs A4
¼ 2cabrs 8Haabrs ; abrs
¼ 2ð12H2 KÞaabrs 10Hcabrs þ 4b ¼ 8HðK 4H 2
2
Þaabrs
2 abrs
¼ ð4H KÞ a
þ ð8H
2
2KÞcabrs
þ 2HðK 4H
2
Þcabrs
; abrs
8Hb
;
2 abrs
þ 4H b
:
Note that
8 g ðx; nÞ ¼ aab ðxÞ 2nbab ðxÞ þ n2 cab ðxÞ; > > > ab > > < g a3 ðx; nÞ ¼ g 3a ðx; nÞ ¼ 0; g 33 ðx; nÞ ¼ 1;
ð3-3Þ
ab > > g ab ðx; nÞ ¼ h2 ðaab ðxÞ 2Kb ðxÞn þ K 2 n2 cab ðxÞÞ; > > > : 3a g ðx; nÞ ¼ g a3 ðx; nÞ ¼ 0; g 33 ðx; nÞ ¼ 1:
In order to express the 3D strain tensor in the S-coordinate system, we define:
8 r r 1 k > > > eab ðuÞ ¼ 2 ðaak db þ abk da Þrr u ; > > < 1 eab ðuÞ ¼ ðbak drb þ bbk dra Þrr uk ; > > > > >2 : eab ðuÞ ¼ 12 ðcar dkb þ cbr dkr Þrk ur ; Note that:
8 > > cab ðuÞ ¼ eab ðuÞ bab u3 ; > > > > < 1
1
c3a ðuÞ ¼ 12 ra u3 þ aab @u@nb ;
1
b
cab ðuÞ ¼ eab ðuÞ þ cab u3 rk bab uk ; c3a ðuÞ ¼ bab @u@n ; > > > > > >2 2 1 : cab ðuÞ ¼ eab ðuÞ þ 12 rkc ab uk ; c3a ðuÞ ¼ 1=2cab @u@nb and
qab ðuÞ ¼ ra rb u3 þ bak rb uk þ bbk ra uk cab u3 þ rk bab uk : Where cab and qab are respectively the covariant components of the linearized change of metric tensor and the linearized change of curvature tensor (seen in [5]). From the Codazzi formulae
ra bbk ¼ rb bak ; we deduce the following formulae for the linearized 3D strain tensor eij ðuÞ ¼ 12 ð5j ui þ 5i uj Þ:
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8 1 2 > > e ðuÞ ¼ cab ðuÞ þ cab ðuÞn þ cab ðuÞn2 þ ; > < ab 1
2
ð3-4Þ
e3a ðuÞ ¼ ea3 ðuÞ ¼ c3a ðuÞ þ c3a ðuÞn þ c3a ðuÞn2 þ ; > > > 3 : e33 ðuÞ ¼ @u : @n 4. Main results: asymptotic expansions of stress tensor
To derive the asymptotic expansions of stress tensor, i.e., to decompose stress tensor totally into 2-D variable and thickness variable, we have three steps. We call them ‘‘operator splitting’’, ‘‘variables separation’’ and ‘‘dimension splitting’’. First step: operator splitting we can get the expressions of the stress tensor in terms of 2-D variables on S. Theorem 4.1. The contravariant components of the stress tensors associated with the displacement uðx; nÞ are in form of
rij ðuÞ ¼ rij0 ðuÞ þ rij1 ðuÞn þ rij2 ðuÞn2 þ :
ð4-1Þ
where
8 ab abkr c ðuÞ þ kaab @u3 ; > > kr @n > r0 ðuÞ ¼ a < 1 ab a bkr r1 ðuÞ ¼ 2c ckr ðuÞ þ aabkr ckr ðuÞ þ 2kbab @u@n3 ; > > > : rab ðuÞ ¼ ð4babkr þ 6Hcabkr 6Kaabkr Þc ðuÞ þ 2cabkr c1 ðuÞ þ aabkr c2 ðuÞ þ 3cab @u3 ; kr kr kr 2 @n
ð4-2Þ
and
8 3a r ðuÞ ¼ aab c3b ðuÞ; > > < 0 1 r31a ðuÞ ¼ 2bab c3b ðuÞ þ aab c3b ðuÞ; > > 1 2 : 3a r2 ðuÞ ¼ 3cab c3b ðuÞ þ 2bab c3b ðuÞ þ aab c3b ðuÞ;
ð4-3Þ
and
8 r33 ðuÞ ¼ c0 ðuÞ; > > < 0 2 33 3 a r 1 ðuÞ ¼ ð2K 4H Þu u ra H; > > : 33 r2 ðuÞ ¼ 2HðK 4H2 Þu3 þ ua ra ðK 2H2 Þ:
ð4-4Þ
Proof. From Hooke’s law for linearly elastic shell, we have
rij ðuÞ ¼ Aijkl ekl ðuÞ According to (3-2) and (3-4), it follows that
rab ðuÞ ¼ Aabkl ekl ðuÞ ¼ Aabkr ekr ðuÞ þ 2Aab3r e3r ðuÞ þ Aab33 e33 ðuÞ ¼ Aabkr ekr ðuÞ þ kg ab e33 ðuÞ
h
i @u3 1 2 ab ¼ kaab akr þ lðaak abr þ aar abk Þ ckr ðuÞ þ ckr ðuÞn þ ckr ðuÞn2 þ k h2 aab ðxÞ 2Kb ðxÞn þ K 2 n2 cab ðxÞ @n ¼ ra0b ðuÞ þ r1ab ðuÞn þ r3ab ðuÞn2 þ ;
where
ra0b ðuÞ; ra1b ðuÞ; ra2b ðuÞ are defined in Eq. (4-2). and
r3a ðuÞ ¼ A3akl ekl ðuÞ ¼ A3akr ekr ðuÞ þ 2A3a3r e3r ðuÞ þ A3a33 e33 ðuÞ ¼ 2lg ab e3b ðuÞ
h
i 1 2 ab ¼ 2l h2 aab ðxÞ 2Kb ðxÞn þ K 2 n2 cab ðxÞ c3a ðuÞ þ c3a ðuÞn þ c3a ðuÞn2
¼ r30a ðuÞ þ r31a ðuÞn þ r33a ðuÞn2 þ ; where
r30a ðuÞ; r31a ðuÞ; r32a ðuÞ are defined in Eq. (4-3). and
r33 ðuÞ ¼ A33kl ekl ðuÞ ¼ A33kr ekr ðuÞ þ 2A333r e3r ðuÞ þ A3333 e33 ðuÞ ¼ kg ab eab ðuÞ þ ðk þ 2lÞ
@u3 @n
h
i 1 2 @u3 ab ckr ðuÞ þ ckr ðuÞn þ ckr ðuÞn2 þ ðk þ 2lÞ ¼ k h2 aab ðxÞ 2Kb ðxÞn þ K 2 n2 cab ðxÞ @n 2 33 33 ¼ r33 0 ðuÞ þ r1 ðuÞn þ r3 ðuÞn þ :
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33 33 where r33 0 ðuÞ; r1 ðuÞ; r2 ðuÞ are defined in Eq. (4-4). This completes the proof. h
Note. After this step we don’t completely decompose stress tensor into 2-D variable and thickness variable because uðx; nÞ still depends on thickness variable n. Second step: variables separation We apply the new asymptotic analysis method to the displacement variable uðx; nÞ. As we all know, 3-D boundary value problem (see [3]) in linearly elastic shell is as follows:
8 rj rij ðuÞ ¼ f i > > > < ui ¼ 0 > rij ðuÞnj ¼ 0 > > : ij r ðuÞnj ¼ g i
in Xe ; on C0 ; on C1 ; on Ct [ Cb :
Consequently, the 3-D variational equations:
8 < Find u 2 VðXe Þ :¼ fv 2 ðH1 ðXÞÞ3 ; v jC ¼ 0:g such that 0 R R R i j pffiffiffi : Aijkl ekl ðuÞeij ðv Þpffiffiffi g dx þ Ct [Cb g ij g i v j ds; g dx ¼ g f v X X ij
for all
v 2 VðXe Þ:
ð4-5Þ
Equivalently, the Ritz minimization problem:
8 Find u 2 VðXe Þ such that > > < JðuÞ ¼ inf v 2VðXÞ Jðv Þ where hR i > R R ijkl pffiffiffi pffiffiffi > : Jðv Þ ¼ 1 A ekl ðv Þeij ðv Þ g dx X g ij f i v j g dx þ Ct [Cb g ij g i v j ds : 2 X where ds denotes the area element along the surface:
ds ¼
pffiffiffi 1 2 adx dx ;
on Ct [ Cb :
P i Different from the asymptotic analysis method uðx; nÞ ¼ ui e proposed by Ciarlet (seen in [6–8]),we suppose that the solution uðx; nÞ to (4-2) exists a formal asymptotic expansion as follows:
uðx; nÞ ¼ u0 ðxÞ þ u1 ðxÞn þ u2 ðxÞn2 þ ¼ u0 ðxÞ þ u1 ðxÞn þ u2 ðxÞn2 þ u3 ðx; n Þn3 ; where
ð4-6Þ
q
@ u 2 ½"; "; uq ðxÞ ¼ q!@ q j¼0 . Note that uðx; nÞ is expanded with respected to the thickness variable n instead of thickness e. Since the problem is linear, we may assume, without loss of generality, that the leading term u0 ðxÞ is of order 0 with respect to n. The successive terms u0 ; u1 ; . . . ; etc, are independent of n. The first term u0 is called the leading term, and uq ; q P 0, is called the term of order q. Dots ‘‘. . .’’ account for the fact that the number of successive terms which will be eventually needed is left unspecified (seen in [9]). Theorem 4.2. Assume that the solution u to the variational problem (4-2) satisfies (4-3), and that the body and surface forces satisfy
f ¼
1 X
fi ni ;
g¼
i¼1
1 X
g i ni :
i¼1
Then there exist two mappings
P1 : V K ðxÞ ! V 1 :¼ fv 2 ðH1 ðxÞÞ3 ; v jc ¼ 0g and
P2 : V K ðxÞ ! V 2 :¼ fv 2 ðL2 ðxÞÞ3 g; such that
u1 ðxÞ ¼ P1 u0 ðxÞ;
u2 ðxÞ ¼ P2 u0 ðxÞ;
At the same time u0 is the solution to 2-D variational equations as follows
Find u0 2 V 2 ðxÞ such that
Z
x
pffiffiffi 2K1 ðu0 ; v 0 Þe þ 2=3K3 ðu0 ; v 0 Þe3 adx ¼< P;
v 0 >; 8 v 0 2 V 2 ðxÞ
S. Xiaoqin et al. / Applied Mathematical Modelling 37 (2013) 7964–7972
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where
8 > < P1 u0 ¼ aak rk u30 ðxÞ~ ea k0 c0 ðu0 Þ~ n; > : P2 u0 ¼ 12 rb ðcab ðu0 Þ þ 2k0 aab c0 ðu0 ÞÞ bab rb u30 ~ ea þ k20 ðD u30 m1 ðu0 Þ ð1 þ 3k0 ÞHc0 ðu0 ÞÞ þ k4 b0 ðu0 Þ ~ n;
2l c0 ðu0 Þ; b0 ðu0 Þ; K1 ðu0 ; v 0 Þ; K3 ðu0 ; v 0 Þare defined in [3], and k ¼ 1 k0 ¼ kþ2 l.
Proof. The proof can be seen in [3]. h Note. The solution of 3-D variational equations is as follows:
uðx; nÞ ¼ u0 ðxÞ þ P1 u0 ðxÞn þ P2 u0 ðxÞn2 :
ð4-7Þ
Third step: dimension splitting Thus combining Theorem 4.1 and Theorem 4.2, we can decompose the 3-D stress tensor into 2-D variable and thickness variable: Theorem 4.3. If uðx; nÞ satisfies (4-3), then the contravariant components of the stress tensors have the following asymptotic expansions:
rij ðuÞ ¼ Rij0 ðu0 Þ þ Rij1 ðu0 Þn þ Rij2 ðu0 Þn2 þ ; where
8 ab R ðu0 Þ ¼ aabkr ckr ðu0 Þ; > > > 0ab < R1 ðu0 Þ ¼ 2cabkr ckr ðu0 Þ aabkr qkr ðu0 Þ 2kk0 bab c0 ðu0 Þ;
> 1 2 > ab > : R2 ðu0 Þ ¼ 4babkr þ 6Hcabkr ckr ðu0 Þ 2cabkr qkr ðu0 Þ þ aabkr ckr ðu0 Þ þ ckr ðu0 Þ þ ckr ðu0 Þ 6K ckr ðu0 Þ ;
ð4-8Þ
and
8 3a R ðu0 Þ ¼ 0; > > > 0 < R31a ðu0 Þ ¼ 12 rb ðcab ðu0 Þ þ k0 aab c0 ðu0 ÞÞ; >
> > : R3a ðu Þ ¼ 1 cab r 3 1 a krml cml ðu0 ÞÞ 12 bam abl þ k0 bab aml rb cml ðu0 Þ; 0 b ðu0 Þ þ 2l bk rr ða 2 2
ð4-9Þ
and
8 33 > > > R0 ðu0 Þ ¼ c0 ðu0 Þ; > > > < R33 ðu Þ ¼ D u3 þ 2k Hc ðu Þ uk r H þ ð2K 4H2 Þu3 ; 0 0 0 0 1 0 0 k 0 > 33 1 a ab c ðu ÞÞ r ðbab r u3 Þ þ aab r u3 r H þ ua r ðK 2H2 Þ > > R ðu Þ ¼ r r ð c bðu Þ þ k a 0 a 0 0 0 a a 0 b b b 0 > 2 0 a 0 2 > > : þð4H2 2KÞk0 c0 ðu0 Þ þ 2HðK 4H2 Þu30 :
ð4-10Þ
Proof. From Eqs. (4-2) and (4-7), it’s clear that
R0ab ðu0 Þ ¼ aabkr ckr ðu0 Þ: From Eqs. (4-3) and (4-7), we have
3a X 1 @ua ðu0 Þ ¼ aab c3b ðu0 Þ ¼ aab rb u30 þ aba 0 ¼ 0: 2 @n 0 From Eqs. (4-4) and (4-7), it’s clear that
R33 0 ðu0 Þ ¼ c0 ðu0 Þ: 33 Similarly, we can get expressions of Ra1b ðu0 Þ; Ra2b ðu0 Þ; R31a ðu0 Þ; R32a ðu0 Þ; R33 1 ðu0 Þ; R2 ðu0 Þ by large quantity computations according to 4-1, 4-2, 4-3, 4-4 and 4-7, 4-8, 4-9, 4-10. The proof can be completed. h
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Fig. 1. The shell with vertical applied forces.
(a)
(b)
(c)
Fig. 2. The displacements of the middle surface in new asymptotic method.
(a)
(b)
(c)
Fig. 3. The displacements of the middle surface in classical asymptotic method.
Table 1 Comparisons of two methods. Methods
Classical asymptotic
New asymptotic
True solution
u10 (h ¼ 0)
9.85146e006
9.86917e006
9.87135e006
u20 (h ¼ 0)
1.46135e005
1.46407e005
1.46441e005
u30 (h ¼ 0)
1.01874e005
1.02056e005
1.02079e005
u10 (h ¼ p2 )
1.10631e043
6.64541e043
3.33379e043
u20 (h ¼ p2 )
1.98923e031
9.97023e032
4.98659e032
u30 (h ¼ p2 )
3.96794e032
1.94933e032
9.73378e033
5. Numerical simulations Assume that the middle surface S of shell is a hemispherical surface whose reference equation is given by the mapping ~ r defined by
~ rð/; hÞ ¼ ðr cos / sin h; r sin / sin h; r cos hÞ; where r > 0 is a constant, 0 6 / 6 2p is longitude and 0 6 h 6 p=2 is colatitude. The distance between an arbitrary point of shell and the origin is d ¼ r þ n, where e 6 n 6 e. Let the radius of the middle surface S be r ¼ 1 m , the thickness be 2e ¼ 0:0005 m and Lam e parameter be k ¼ 2 105 MPa, l ¼ 1 106 MPa. Suppose the equator surface of shell is fixed and the applied forces are vertical (seen in Fig. 1). We let p1;0 ¼ p2;0 ¼ 0; p3;0 ¼ 100 MPa. We use the FEM (finite element method) to do the numerical experiments. The integration domain w ¼ ½0; 2p ½0; p=2 is partitioned into triangles. The mesh is 40 5. We use the P1-element in every unit.
S. Xiaoqin et al. / Applied Mathematical Modelling 37 (2013) 7964–7972
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Fig. 4. The distributions of the stress fields.
Fig. 5. The distributions of the norm of the stress in middle surface.
Firstly, we can get the distribution graphs of the displacements in middle surface (see Fig. 2). Fig. 2 includes (a), (b), (c) 3 parts, where (a) shows the displacements of u10 , (b) shows the displacements of u20 , (c) shows the displacements of u30 . According to Fig. 2, we can find that the displacement of the point in the middle surface in the same colatitude and different longitude is the same. It means the distribution of the displacements is axial-symmetric, which only depends on the colatitude. Furthermore, the length of displacements is smaller as the colatitude is bigger.Thus the new asymptotic method is validate in applicant. In order to compare the new asymptotic method with the classical method, we do the numerical experiments for the classical method in the same conditions (see Fig. 3). Some information is included in the Table 1, from which we can see that it’s better to approximate the true solutions for the new asymptotic method than old. Secondly, we derive the distribution graphs of the stress fields in middle surface by linear Hooke laws (see Fig. 4). In order to get the more intuitional results we define a new norm of stress fields krk :¼ detðrij ðuÞÞ. Fig. 5 shows that the norm of stress fields in the middle surface is axis-symmetric, too. At the same time, the norm of stress fields is smaller as the colatitude is bigger, which is consistent with displacements in middle surface. 6. Conclusions In this paper we have proposed the asymptotic expansions of stress tensor for linearly elastic shell by using new asymptotic analysis method, which is different from the classical asymptotic analysis. The new asymptotic analysis method has two distinguishing features: one is that the displacement is expanded with respect to the thickness variable n of the middle surface not to the thickness e; another is that the first order term u1 and the second order term u2 of displacement variable can be algebraically expressed by the leading term u0 . To decompose stress tensor totally into 2-D variable and thickness variable, we have three steps operator splitting, variables separation and dimension splitting. Finally, a numerical experiment of special hemispherical shell by FEM (finite element method) is provided. We derive the distribution of displacements and stress fields of the middle surface in new asymptotic method. Numerical experiment shows that the distribution of displacements and stress fields in the middle surface is axis-symmetric, i.e., the length of displacements and the norm of stress in the same colatitude and different longitude is the same. At the same time, as the colatitude is bigger, the length of displacements and the mode of stress is smaller, which is consistent with the practice and thus proves the validity of our model.
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In the future, we should go on the works and move it forward to solve the real-world engineering problems. It require us to consider multiple aspects. Acknowledgement I’m very grateful for Professor P.G.Ciarlet’s direction during my visiting to City University of Hong Kong. In addition, this paper is supported by National Natural Science Foundation of China (NSFC 11101330, NSFC 11202159, NSFC 61004122) and Natural Science Foundation of Shaanxi Province (2011JQ1007). References [1] [2] [3] [4] [5] [6]
P.G. Ciarlet, Mathematical elasticity, Theory of Shells, vol. III, North-Holland, 2000. P.G. Ciarlet, An Introduction to Differential Geometry, with Applications to Elasticity, Springer-Verlag, Heidelberg, 2005. Kaitai Li, Xiaoqin Shen, A dimensional splitting method for linearly elastic shell, Int. J. Comput. Math. 84 (6) (2007) 807–824. Kaitai Li, Aixiang Huang, Tensor Analysis and Its Applications, Chinese Scientific Press, 2004 (in Chinese). Shen Xiaoqin, Li Kaitai, Metric tensor and curvature tensor in deformed shell middle surface, J. Xi’an Jiaotong Univ. vol. 41 (12) (2007) 1483–1486. P.G. Ciarlet, V. Lods, Asymptotic analysis of linearly elastic shells. I. Justification of membrane shells equations, Arch. Ration. Mech. Anal. 136 (1996) 119–161. [7] P.G. Ciarlet, V. Lods, B. Miara, Asymptotic analysis of linearly elastic shells. II. Justification of flexural shells, Arch. Ration. Mech. Anal. 136 (1996) 163– 190. [8] P.G. Ciarlet, V. Lods, Asymptotic analysis of linearly elastic shells. III. A justification of Koiters shell equations, Arch. Ration. Mech. Anal. 136 (1996) 191– 200. [9] V. Lods, C. Mardare, Asymptotic justification of the Kirchhoff-Love assumptions for a linearly elastic clamped shell, J. Elast. 58 (2) (2000) 105–154.
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Asymptotic expansions of stress tensor for linearly elastic shell Shen Xiaoqin a,⇑, Li Kaitai b, Ming Yang c a b c
School of Sciences, Xi’an University of Technology, Xi’an 710054, PR China School of Sciences, Xi’an Jiaotong University, Xi’an 710049, PR China Road Traffic Detection and Equipment Engineering Research Center, Chang’an University, Xi’an 710064, PR China
a r t i c l e
i n f o
Article history: Received 1 November 2010 Received in revised form 8 February 2013 Accepted 4 March 2013 Available online 19 March 2013 Keywords: Linearly elastic shell Asymptotic analysis Stress tensor
a b s t r a c t In this paper, the asymptotic expansions of stress tensor for linearly elastic shell have been proposed by new asymptotic analysis method, which is different from the classical asymptotic analysis. The new asymptotic analysis method has two distinguishing features: one is that the displacement is expanded with respect to the thickness variable of the middle surface not to the thickness; another is that the first order term and the second order term of the displacement variable can be algebraically expressed by the leading term. To decompose stress tensor totally into 2-D variable and thickness variable, we have three steps: operator splitting, variables separation and dimension splitting. In the end, a numerical experiment of special hemispherical shell by FEM (finite element method) is provided. We derive the distribution of displacements and stress fields in the middle surface. Crown Copyright Ó 2013 Published by Elsevier Inc. All rights reserved.
1. Introduction The main contents of elastic mechanics is to research the rules how the stress depends on the strain when the forces are applied to the elasticity. The relations between the stress and the strain can be called as constitutive equation. When the relationships between the components of stress and strain are linear or the deformations are ‘‘small’’, the elasticity is called linear elasticity. (seen in [1]). In this paper, we mainly consider linearly elastic shell. Suppose that strain tensor and stress tensor satisfies Hooke’s law in linearly elastic shell. We have proposed the asymptotic expansions of stress tensor for linearly elastic shell by using new asymptotic analysis method, which is different from the classical asymptotic analysis. The new asymptotic analysis method has two distinguishing features: one is that the displacement is expanded with respect to the thickness variable n of the middle surface not to the thickness e; another is that the first order term u1 and the second order term u2 of displacement variable can be algebraically expressed by the leading term u0 . Thus 3-D stress tensor is totally decomposed into 2-D variable and thickness variable, which makes the numerical computation of stress tensor in engineer easier and more convenient than before. To derive the asymptotic expansions of stress tensor, i.e., to decompose stress tensor totally into 2-D variable and thickness variable, we have three steps. We call them ‘‘operator splitting’’, ‘‘variables separation’’ and ‘‘dimension splitting’’. In the end, a numerical experiment of special hemispherical shell by FEM (finite element method) is provided. We derive the distribution of the displacements of the middle surface in new asymptotic method and compare the results of two methods. We derive the distribution of stress fields of the middle surface in new asymptotic method. Numerical results are consistent with the practice and thus prove the validity of the new asymptotic method.
⇑ Corresponding author. E-mail addresses: [email protected], [email protected] (S. Xiaoqin). 0307-904X/$ - see front matter Crown Copyright Ó 2013 Published by Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.apm.2013.03.032
S. Xiaoqin et al. / Applied Mathematical Modelling 37 (2013) 7964–7972
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2. Preliminary ^ e E3 (3D-Euclidean space) An elastic shell with constant thickness 2e is an elastic body whose reference configuration X 3 consists of all points within a distance 6 e from a given surface S E , where the parameter e > 0 is the half thickness of the shell and is thought of as small enough. The surface S is defined as the image by ~ h of the closure of a domain x R2 , where 3 ~ ! E is a smooth injective immersion. Let ~ h:x n denote a continuously varying unit normal vector along S and let Xe ¼ x ðe; eÞ. (seen in [2]). ^ e is given by X ^e ¼ H ~ ðX e Þ, where the mapping H ~:X e R3 ! E3 is defined by Hence the set X
e ; ðx1 ; x2 Þ 2 x: ~ ðx1 ; x2 ; nÞ ¼ ~ H hðx1 ; x2 Þ þ n~ nðx1 ; x2 Þ 8ðx1 ; x2 ; nÞ 2 X ^ e consists of following: upper surface Ct ¼ S fþeg; bottom surface Cb ¼ S feg; lateral surface The boundary of shell X Cl ¼ C0 [ C1 ; C0 ¼ c0 ðe; þeÞ; C1 ¼ c1 ðe; þeÞ, where c ¼ @ x is the boundary of x and c ¼ c0 [ c1 . (seen in [3]) The pair ðx1 ; x2 Þ constitute curvilinear coordinates on S, and ðx1 ; x2 ; nÞ is called semi-geodesic coordinate system (E3 being viewed as a Riemann space and S as a 2D sub-manifold). In elasticity, S is called middle surface of the shell. In the following, Latin indices and exponents: i; j; k; . . ., take their values in the set {1, 2, 3} while Greek indices and exponents: a; b; c; . . ., take their values in the set {1, 2}. In addition, the repeated index summation convention is systematically used. The covariant and contravariant components of the metric tensor of the surface Sare given by
aab ¼ ~ ha ~ hb ;
aab abk ¼ dak ; where ~ ha ¼
@~ h ; @xa
~ ha ~ hb ¼ dba
and the covariant and contravariant components of the curvature tensor on S are defined as follows
bab ¼ ~ n ~ hab ¼ ~ na ~ hb ;
b
ab
¼ aak abr bkr ; where ~ na ¼
@~ n ; @xa
@~ hb ~ hab ¼ a : @x
Similarly the covariant and contravariant components of the third fundamental forms on S are defined as follows
cab ¼ ~ na ~ nb ¼ akr bak bbr ;
cab ¼ aak abr ckr :
Note that the mean curvature H and the Gaussian curvature K is defined by
H¼
1 ab a bab ; 2
K¼
detðbab Þ b ¼ : detðaab Þ a
The covariant derivative on S is defined as (seen in [4])
ra brs ¼ @brs =@ya þ Cr ab bbs Cb as brb
where Cr ab ¼ ~ hr ~ hab denotes the Christoffel symbols on S. In the coordinate system ðxa ; nÞ, the metric tensor of E3 is given by
~i H ~ j; g ij ¼ H
~ ~ i :¼ @ H : g ik g kj ¼ dij ; where H @xi
3. Formal Taylor expansions of elasticity tensor and strain tensor From Hooke’s law for linearly elastic shell, we have
rij ðuÞ ¼ Aijkl ekl ðuÞ ijkl
ð3-1Þ
where r ðuÞ; A ; ekl ðuÞ is stress tensor, elasticity tensor and strain tensor respectively; u is the displacement of any point in the shell. In order to obtain the asymptotic expansions of tress tensor, we firstly deduce the formal Taylor expansions of elasticity tensor and strain tensor. We assume that the elastic material constituting the shell is isotropic and homogenous and that the reference configuration of the shell is a natural state. The contravariant components of the elasticity tensor are given by ij
Aijkl ¼ kg ij g kl þ l g ik g jl þ g il g jk ; where k P 0 and
l P 0 are Lamé constants.
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Let
8 abrs a ¼ kaab ars þ lðaar abs þ aas abr Þ; > > < abrs ab rs ar bs a s br ¼ kb b þ lðb b þ b b Þ; b > > : abrs rs ab bs ar br as ¼ kðaab b þ ars b Þ þ lðaar b þ abs b þ aas b þ abr b Þ: c and
hðnÞ ¼ 1 2Hn þ Kn2 þ : We give the following expression for the contravariant components of the elasticity tensor
" # 8 4 X > a brs a brs k 4 > a b rs > A ¼h a þ Ak n ; > > > > k¼1 > > > > > < Aabr3 ¼ Aab3r ¼ Aa3br ¼ A3abr ¼ 0;
ð3-2Þ
> Aab33 ¼ A33ab ¼ kg ab ; A3333 ¼ k þ 2l; > > > > > a3b3 > > A ¼ A3a3b ¼ Aa33b ¼ A3ab3 ¼ lg ab ; > > > > : a333 A ¼ A3a33 ¼ A33a3 ¼ A333a ¼ 0; where
8 abrs > A > > 1 > > < Aabrs 2 abrs > > A > 3 > > : abrs A4
¼ 2cabrs 8Haabrs ; abrs
¼ 2ð12H2 KÞaabrs 10Hcabrs þ 4b ¼ 8HðK 4H 2
2
Þaabrs
2 abrs
¼ ð4H KÞ a
þ ð8H
2
2KÞcabrs
þ 2HðK 4H
2
Þcabrs
; abrs
8Hb
;
2 abrs
þ 4H b
:
Note that
8 g ðx; nÞ ¼ aab ðxÞ 2nbab ðxÞ þ n2 cab ðxÞ; > > > ab > > < g a3 ðx; nÞ ¼ g 3a ðx; nÞ ¼ 0; g 33 ðx; nÞ ¼ 1;
ð3-3Þ
ab > > g ab ðx; nÞ ¼ h2 ðaab ðxÞ 2Kb ðxÞn þ K 2 n2 cab ðxÞÞ; > > > : 3a g ðx; nÞ ¼ g a3 ðx; nÞ ¼ 0; g 33 ðx; nÞ ¼ 1:
In order to express the 3D strain tensor in the S-coordinate system, we define:
8 r r 1 k > > > eab ðuÞ ¼ 2 ðaak db þ abk da Þrr u ; > > < 1 eab ðuÞ ¼ ðbak drb þ bbk dra Þrr uk ; > > > > >2 : eab ðuÞ ¼ 12 ðcar dkb þ cbr dkr Þrk ur ; Note that:
8 > > cab ðuÞ ¼ eab ðuÞ bab u3 ; > > > > < 1
1
c3a ðuÞ ¼ 12 ra u3 þ aab @u@nb ;
1
b
cab ðuÞ ¼ eab ðuÞ þ cab u3 rk bab uk ; c3a ðuÞ ¼ bab @u@n ; > > > > > >2 2 1 : cab ðuÞ ¼ eab ðuÞ þ 12 rkc ab uk ; c3a ðuÞ ¼ 1=2cab @u@nb and
qab ðuÞ ¼ ra rb u3 þ bak rb uk þ bbk ra uk cab u3 þ rk bab uk : Where cab and qab are respectively the covariant components of the linearized change of metric tensor and the linearized change of curvature tensor (seen in [5]). From the Codazzi formulae
ra bbk ¼ rb bak ; we deduce the following formulae for the linearized 3D strain tensor eij ðuÞ ¼ 12 ð5j ui þ 5i uj Þ:
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8 1 2 > > e ðuÞ ¼ cab ðuÞ þ cab ðuÞn þ cab ðuÞn2 þ ; > < ab 1
2
ð3-4Þ
e3a ðuÞ ¼ ea3 ðuÞ ¼ c3a ðuÞ þ c3a ðuÞn þ c3a ðuÞn2 þ ; > > > 3 : e33 ðuÞ ¼ @u : @n 4. Main results: asymptotic expansions of stress tensor
To derive the asymptotic expansions of stress tensor, i.e., to decompose stress tensor totally into 2-D variable and thickness variable, we have three steps. We call them ‘‘operator splitting’’, ‘‘variables separation’’ and ‘‘dimension splitting’’. First step: operator splitting we can get the expressions of the stress tensor in terms of 2-D variables on S. Theorem 4.1. The contravariant components of the stress tensors associated with the displacement uðx; nÞ are in form of
rij ðuÞ ¼ rij0 ðuÞ þ rij1 ðuÞn þ rij2 ðuÞn2 þ :
ð4-1Þ
where
8 ab abkr c ðuÞ þ kaab @u3 ; > > kr @n > r0 ðuÞ ¼ a < 1 ab a bkr r1 ðuÞ ¼ 2c ckr ðuÞ þ aabkr ckr ðuÞ þ 2kbab @u@n3 ; > > > : rab ðuÞ ¼ ð4babkr þ 6Hcabkr 6Kaabkr Þc ðuÞ þ 2cabkr c1 ðuÞ þ aabkr c2 ðuÞ þ 3cab @u3 ; kr kr kr 2 @n
ð4-2Þ
and
8 3a r ðuÞ ¼ aab c3b ðuÞ; > > < 0 1 r31a ðuÞ ¼ 2bab c3b ðuÞ þ aab c3b ðuÞ; > > 1 2 : 3a r2 ðuÞ ¼ 3cab c3b ðuÞ þ 2bab c3b ðuÞ þ aab c3b ðuÞ;
ð4-3Þ
and
8 r33 ðuÞ ¼ c0 ðuÞ; > > < 0 2 33 3 a r 1 ðuÞ ¼ ð2K 4H Þu u ra H; > > : 33 r2 ðuÞ ¼ 2HðK 4H2 Þu3 þ ua ra ðK 2H2 Þ:
ð4-4Þ
Proof. From Hooke’s law for linearly elastic shell, we have
rij ðuÞ ¼ Aijkl ekl ðuÞ According to (3-2) and (3-4), it follows that
rab ðuÞ ¼ Aabkl ekl ðuÞ ¼ Aabkr ekr ðuÞ þ 2Aab3r e3r ðuÞ þ Aab33 e33 ðuÞ ¼ Aabkr ekr ðuÞ þ kg ab e33 ðuÞ
h
i @u3 1 2 ab ¼ kaab akr þ lðaak abr þ aar abk Þ ckr ðuÞ þ ckr ðuÞn þ ckr ðuÞn2 þ k h2 aab ðxÞ 2Kb ðxÞn þ K 2 n2 cab ðxÞ @n ¼ ra0b ðuÞ þ r1ab ðuÞn þ r3ab ðuÞn2 þ ;
where
ra0b ðuÞ; ra1b ðuÞ; ra2b ðuÞ are defined in Eq. (4-2). and
r3a ðuÞ ¼ A3akl ekl ðuÞ ¼ A3akr ekr ðuÞ þ 2A3a3r e3r ðuÞ þ A3a33 e33 ðuÞ ¼ 2lg ab e3b ðuÞ
h
i 1 2 ab ¼ 2l h2 aab ðxÞ 2Kb ðxÞn þ K 2 n2 cab ðxÞ c3a ðuÞ þ c3a ðuÞn þ c3a ðuÞn2
¼ r30a ðuÞ þ r31a ðuÞn þ r33a ðuÞn2 þ ; where
r30a ðuÞ; r31a ðuÞ; r32a ðuÞ are defined in Eq. (4-3). and
r33 ðuÞ ¼ A33kl ekl ðuÞ ¼ A33kr ekr ðuÞ þ 2A333r e3r ðuÞ þ A3333 e33 ðuÞ ¼ kg ab eab ðuÞ þ ðk þ 2lÞ
@u3 @n
h
i 1 2 @u3 ab ckr ðuÞ þ ckr ðuÞn þ ckr ðuÞn2 þ ðk þ 2lÞ ¼ k h2 aab ðxÞ 2Kb ðxÞn þ K 2 n2 cab ðxÞ @n 2 33 33 ¼ r33 0 ðuÞ þ r1 ðuÞn þ r3 ðuÞn þ :
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33 33 where r33 0 ðuÞ; r1 ðuÞ; r2 ðuÞ are defined in Eq. (4-4). This completes the proof. h
Note. After this step we don’t completely decompose stress tensor into 2-D variable and thickness variable because uðx; nÞ still depends on thickness variable n. Second step: variables separation We apply the new asymptotic analysis method to the displacement variable uðx; nÞ. As we all know, 3-D boundary value problem (see [3]) in linearly elastic shell is as follows:
8 rj rij ðuÞ ¼ f i > > > < ui ¼ 0 > rij ðuÞnj ¼ 0 > > : ij r ðuÞnj ¼ g i
in Xe ; on C0 ; on C1 ; on Ct [ Cb :
Consequently, the 3-D variational equations:
8 < Find u 2 VðXe Þ :¼ fv 2 ðH1 ðXÞÞ3 ; v jC ¼ 0:g such that 0 R R R i j pffiffiffi : Aijkl ekl ðuÞeij ðv Þpffiffiffi g dx þ Ct [Cb g ij g i v j ds; g dx ¼ g f v X X ij
for all
v 2 VðXe Þ:
ð4-5Þ
Equivalently, the Ritz minimization problem:
8 Find u 2 VðXe Þ such that > > < JðuÞ ¼ inf v 2VðXÞ Jðv Þ where hR i > R R ijkl pffiffiffi pffiffiffi > : Jðv Þ ¼ 1 A ekl ðv Þeij ðv Þ g dx X g ij f i v j g dx þ Ct [Cb g ij g i v j ds : 2 X where ds denotes the area element along the surface:
ds ¼
pffiffiffi 1 2 adx dx ;
on Ct [ Cb :
P i Different from the asymptotic analysis method uðx; nÞ ¼ ui e proposed by Ciarlet (seen in [6–8]),we suppose that the solution uðx; nÞ to (4-2) exists a formal asymptotic expansion as follows:
uðx; nÞ ¼ u0 ðxÞ þ u1 ðxÞn þ u2 ðxÞn2 þ ¼ u0 ðxÞ þ u1 ðxÞn þ u2 ðxÞn2 þ u3 ðx; n Þn3 ; where
ð4-6Þ
q
@ u 2 ½"; "; uq ðxÞ ¼ q!@ q j¼0 . Note that uðx; nÞ is expanded with respected to the thickness variable n instead of thickness e. Since the problem is linear, we may assume, without loss of generality, that the leading term u0 ðxÞ is of order 0 with respect to n. The successive terms u0 ; u1 ; . . . ; etc, are independent of n. The first term u0 is called the leading term, and uq ; q P 0, is called the term of order q. Dots ‘‘. . .’’ account for the fact that the number of successive terms which will be eventually needed is left unspecified (seen in [9]). Theorem 4.2. Assume that the solution u to the variational problem (4-2) satisfies (4-3), and that the body and surface forces satisfy
f ¼
1 X
fi ni ;
g¼
i¼1
1 X
g i ni :
i¼1
Then there exist two mappings
P1 : V K ðxÞ ! V 1 :¼ fv 2 ðH1 ðxÞÞ3 ; v jc ¼ 0g and
P2 : V K ðxÞ ! V 2 :¼ fv 2 ðL2 ðxÞÞ3 g; such that
u1 ðxÞ ¼ P1 u0 ðxÞ;
u2 ðxÞ ¼ P2 u0 ðxÞ;
At the same time u0 is the solution to 2-D variational equations as follows
Find u0 2 V 2 ðxÞ such that
Z
x
pffiffiffi 2K1 ðu0 ; v 0 Þe þ 2=3K3 ðu0 ; v 0 Þe3 adx ¼< P;
v 0 >; 8 v 0 2 V 2 ðxÞ
S. Xiaoqin et al. / Applied Mathematical Modelling 37 (2013) 7964–7972
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where
8 > < P1 u0 ¼ aak rk u30 ðxÞ~ ea k0 c0 ðu0 Þ~ n; > : P2 u0 ¼ 12 rb ðcab ðu0 Þ þ 2k0 aab c0 ðu0 ÞÞ bab rb u30 ~ ea þ k20 ðD u30 m1 ðu0 Þ ð1 þ 3k0 ÞHc0 ðu0 ÞÞ þ k4 b0 ðu0 Þ ~ n;
2l c0 ðu0 Þ; b0 ðu0 Þ; K1 ðu0 ; v 0 Þ; K3 ðu0 ; v 0 Þare defined in [3], and k ¼ 1 k0 ¼ kþ2 l.
Proof. The proof can be seen in [3]. h Note. The solution of 3-D variational equations is as follows:
uðx; nÞ ¼ u0 ðxÞ þ P1 u0 ðxÞn þ P2 u0 ðxÞn2 :
ð4-7Þ
Third step: dimension splitting Thus combining Theorem 4.1 and Theorem 4.2, we can decompose the 3-D stress tensor into 2-D variable and thickness variable: Theorem 4.3. If uðx; nÞ satisfies (4-3), then the contravariant components of the stress tensors have the following asymptotic expansions:
rij ðuÞ ¼ Rij0 ðu0 Þ þ Rij1 ðu0 Þn þ Rij2 ðu0 Þn2 þ ; where
8 ab R ðu0 Þ ¼ aabkr ckr ðu0 Þ; > > > 0ab < R1 ðu0 Þ ¼ 2cabkr ckr ðu0 Þ aabkr qkr ðu0 Þ 2kk0 bab c0 ðu0 Þ;
> 1 2 > ab > : R2 ðu0 Þ ¼ 4babkr þ 6Hcabkr ckr ðu0 Þ 2cabkr qkr ðu0 Þ þ aabkr ckr ðu0 Þ þ ckr ðu0 Þ þ ckr ðu0 Þ 6K ckr ðu0 Þ ;
ð4-8Þ
and
8 3a R ðu0 Þ ¼ 0; > > > 0 < R31a ðu0 Þ ¼ 12 rb ðcab ðu0 Þ þ k0 aab c0 ðu0 ÞÞ; >
> > : R3a ðu Þ ¼ 1 cab r 3 1 a krml cml ðu0 ÞÞ 12 bam abl þ k0 bab aml rb cml ðu0 Þ; 0 b ðu0 Þ þ 2l bk rr ða 2 2
ð4-9Þ
and
8 33 > > > R0 ðu0 Þ ¼ c0 ðu0 Þ; > > > < R33 ðu Þ ¼ D u3 þ 2k Hc ðu Þ uk r H þ ð2K 4H2 Þu3 ; 0 0 0 0 1 0 0 k 0 > 33 1 a ab c ðu ÞÞ r ðbab r u3 Þ þ aab r u3 r H þ ua r ðK 2H2 Þ > > R ðu Þ ¼ r r ð c bðu Þ þ k a 0 a 0 0 0 a a 0 b b b 0 > 2 0 a 0 2 > > : þð4H2 2KÞk0 c0 ðu0 Þ þ 2HðK 4H2 Þu30 :
ð4-10Þ
Proof. From Eqs. (4-2) and (4-7), it’s clear that
R0ab ðu0 Þ ¼ aabkr ckr ðu0 Þ: From Eqs. (4-3) and (4-7), we have
3a X 1 @ua ðu0 Þ ¼ aab c3b ðu0 Þ ¼ aab rb u30 þ aba 0 ¼ 0: 2 @n 0 From Eqs. (4-4) and (4-7), it’s clear that
R33 0 ðu0 Þ ¼ c0 ðu0 Þ: 33 Similarly, we can get expressions of Ra1b ðu0 Þ; Ra2b ðu0 Þ; R31a ðu0 Þ; R32a ðu0 Þ; R33 1 ðu0 Þ; R2 ðu0 Þ by large quantity computations according to 4-1, 4-2, 4-3, 4-4 and 4-7, 4-8, 4-9, 4-10. The proof can be completed. h
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S. Xiaoqin et al. / Applied Mathematical Modelling 37 (2013) 7964–7972
Fig. 1. The shell with vertical applied forces.
(a)
(b)
(c)
Fig. 2. The displacements of the middle surface in new asymptotic method.
(a)
(b)
(c)
Fig. 3. The displacements of the middle surface in classical asymptotic method.
Table 1 Comparisons of two methods. Methods
Classical asymptotic
New asymptotic
True solution
u10 (h ¼ 0)
9.85146e006
9.86917e006
9.87135e006
u20 (h ¼ 0)
1.46135e005
1.46407e005
1.46441e005
u30 (h ¼ 0)
1.01874e005
1.02056e005
1.02079e005
u10 (h ¼ p2 )
1.10631e043
6.64541e043
3.33379e043
u20 (h ¼ p2 )
1.98923e031
9.97023e032
4.98659e032
u30 (h ¼ p2 )
3.96794e032
1.94933e032
9.73378e033
5. Numerical simulations Assume that the middle surface S of shell is a hemispherical surface whose reference equation is given by the mapping ~ r defined by
~ rð/; hÞ ¼ ðr cos / sin h; r sin / sin h; r cos hÞ; where r > 0 is a constant, 0 6 / 6 2p is longitude and 0 6 h 6 p=2 is colatitude. The distance between an arbitrary point of shell and the origin is d ¼ r þ n, where e 6 n 6 e. Let the radius of the middle surface S be r ¼ 1 m , the thickness be 2e ¼ 0:0005 m and Lam e parameter be k ¼ 2 105 MPa, l ¼ 1 106 MPa. Suppose the equator surface of shell is fixed and the applied forces are vertical (seen in Fig. 1). We let p1;0 ¼ p2;0 ¼ 0; p3;0 ¼ 100 MPa. We use the FEM (finite element method) to do the numerical experiments. The integration domain w ¼ ½0; 2p ½0; p=2 is partitioned into triangles. The mesh is 40 5. We use the P1-element in every unit.
S. Xiaoqin et al. / Applied Mathematical Modelling 37 (2013) 7964–7972
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Fig. 4. The distributions of the stress fields.
Fig. 5. The distributions of the norm of the stress in middle surface.
Firstly, we can get the distribution graphs of the displacements in middle surface (see Fig. 2). Fig. 2 includes (a), (b), (c) 3 parts, where (a) shows the displacements of u10 , (b) shows the displacements of u20 , (c) shows the displacements of u30 . According to Fig. 2, we can find that the displacement of the point in the middle surface in the same colatitude and different longitude is the same. It means the distribution of the displacements is axial-symmetric, which only depends on the colatitude. Furthermore, the length of displacements is smaller as the colatitude is bigger.Thus the new asymptotic method is validate in applicant. In order to compare the new asymptotic method with the classical method, we do the numerical experiments for the classical method in the same conditions (see Fig. 3). Some information is included in the Table 1, from which we can see that it’s better to approximate the true solutions for the new asymptotic method than old. Secondly, we derive the distribution graphs of the stress fields in middle surface by linear Hooke laws (see Fig. 4). In order to get the more intuitional results we define a new norm of stress fields krk :¼ detðrij ðuÞÞ. Fig. 5 shows that the norm of stress fields in the middle surface is axis-symmetric, too. At the same time, the norm of stress fields is smaller as the colatitude is bigger, which is consistent with displacements in middle surface. 6. Conclusions In this paper we have proposed the asymptotic expansions of stress tensor for linearly elastic shell by using new asymptotic analysis method, which is different from the classical asymptotic analysis. The new asymptotic analysis method has two distinguishing features: one is that the displacement is expanded with respect to the thickness variable n of the middle surface not to the thickness e; another is that the first order term u1 and the second order term u2 of displacement variable can be algebraically expressed by the leading term u0 . To decompose stress tensor totally into 2-D variable and thickness variable, we have three steps operator splitting, variables separation and dimension splitting. Finally, a numerical experiment of special hemispherical shell by FEM (finite element method) is provided. We derive the distribution of displacements and stress fields of the middle surface in new asymptotic method. Numerical experiment shows that the distribution of displacements and stress fields in the middle surface is axis-symmetric, i.e., the length of displacements and the norm of stress in the same colatitude and different longitude is the same. At the same time, as the colatitude is bigger, the length of displacements and the mode of stress is smaller, which is consistent with the practice and thus proves the validity of our model.
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In the future, we should go on the works and move it forward to solve the real-world engineering problems. It require us to consider multiple aspects. Acknowledgement I’m very grateful for Professor P.G.Ciarlet’s direction during my visiting to City University of Hong Kong. In addition, this paper is supported by National Natural Science Foundation of China (NSFC 11101330, NSFC 11202159, NSFC 61004122) and Natural Science Foundation of Shaanxi Province (2011JQ1007). References [1] [2] [3] [4] [5] [6]
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