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Pressure vessel problem for chiral elastic tubes
International Journal of Engineering Science 49 (2011) 411–419
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Pressure vessel problem for chiral elastic tubes D. Iesßan ⇑ Department of Mathematics, ‘‘Al.I. Cuza’’ University and ‘‘O. Mayer’’ Institute of Mathematics, Romanian Academy, 700506 Iasßi, Romania
a r t i c l e
i n f o
Article history: Received 28 January 2010 Received in revised form 7 January 2011 Accepted 23 January 2011 Available online 22 February 2011 Keywords: Chiral tubes Cosserat elastic solids Auxetic materials Mechanics of bone
a b s t r a c t The study of chiral materials is important for the investigation of carbon nanotubes, auxetic materials and bones. The chiral effects cannot be described within classical elasticity. In this paper, the problem of the circular tube under internal and external pressure is solved in the context of the linear theory of chiral Cosserat elasticity. The work is motivated by the recent interest in the using Cosserat elastic solid as model for auxetic materials, bones and chiral carbon nanotubes. First, the generalized plane strain of an isotropic chiral elastic material is investigated. Then, the solution of the pressure vessel problem is established. The salient feature of the solution is that, in the absence of body and surface moments, a pressure acting on the surface of a chiral tube produces a microrotation of the material particles. It is shown that the radial displacement and the radial stress are modified from the values predicted by the theory of achiral materials. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction In recent years the mechanical behavior of chiral materials has been the subject of many investigations (Dyszlewicz, 2004; Hassan, Scarpa, Ruzzene, & Mohamed, 2008; Healey, 2002; Lakes, 2001; Natroshvili, Giorgashvili, & Stratis, 2006). The study of chiral materials is important for the investigation of carbon nanotubes, auxetic materials and bones. A chiral material (also known as noncentrosymmetric, or hemitropic) is a material whose behavior is not invariant with respect to inversion. In the classical elasticity the elasticity tensor is unchanged under an inversion (Lakes, 2001). Consequently, the classical elasticity cannot describe the behavior of the chiral elastic materials. An adequate theory to describe the mechanical behavior of the chiral elastic materials is the theory of Cosserat elasticity (Donescu, Chiroiu, & Munteanu, 2009; Haijun & Zhong-can, 1998; Khurana & Tomar, 2009; Lakes & Benedict, 1982; Sharma, 2004; Teoodorescu, Munteanu, & Chiroiu, 2005). The Cosserat theory studies continua with oriented particles, in which each material point has the six degree of freedom of a rigid body (Ciarletta & Iesßan, 1993; Eringen, 1999; Nowacki, 1981). In this paper we investigate the problem of an isotropic and homogeneous chiral tube subjected to internal and external pressures. Structures such as pipes or tubes capable of holding internal pressure have been very important in the history of science and technology. This work is motivated by real problems. The theory of Cosserat elasticity has been used to study the deformation of various carbon nanotubes (Chandraseker, Mukherjee, Paci, & Schatz, 2009; Guz, Rodger, Guz, & Rushchitsky, 2007) and the mechanical behavior of bones (Fatemi, van Keulen, & Onuck, 2002; Lakes, Yoon, & Katz, 1983; Park & Lakes, 1986). The intended applications of the solution presented in this paper are to nanotubes and to bone mechanics. In the femur bone implants an internal pressure is applied on the interface. Some deformations of the bone can be modeled by using the solution of pressure vessel problem. Examples of chiral nanotubes have been presented by Cao and Chen (2007) and Guz et al. (2007).
⇑ Tel.: +40 232201060; fax: +40 232201160. E-mail address: [email protected] 0020-7225/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijengsci.2011.01.003
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D. Iesßan / International Journal of Engineering Science 49 (2011) 411–419
In the classical theory of elasticity the pressure vessel problem reduces to solving a plane strain problem. To investigate the pressure vessel problem in the context of the linear theory of chiral Cosserat elasticity we have introduced the state of generalized plane strain, in which the displacement and microrotation vectors are independent of the axial coordinate. We have given conditions under which a generalized plane strain is possible. The pressure vessel problem has been investigated as a special generalized plane strain of a chiral elastic tube. This paper is structured as follows. First, we present the basic equations of homogeneous and isotropic chiral Cosserat elastic solids. Then, we define the generalized plane strain problem for a chiral elastic solid. In the case of an achiral Cosserat elastic solid, the basic equations of the two-dimensional theory and the boundary conditions reduce to two uncoupled boundary value problems. A such simplification is not possible for chiral materials. In the following section we solve the generalized plane strain problem which govern the deformation of a chiral tube subjected to internal and external pressures. It is shown that, in the absence of body couples and surface moments, a pressure acting on the surface of an isotropic chiral Cosserat elastic tube, produces a microrotation of the material particles. In the case of an achiral tube subjected to the same loading the microrotation vector vanishes. It is shown that the radial displacement and the radial stress in a chiral tube are modified from the values predicted by the theory of achiral materials. We belive that the solution presented herein will find applications in the theory of nanostructures. 2. Preliminaries In this section we present the basic equations of homogeneous and isotropic chiral Cosserat elastic bodies. Let us consider a body that in the undeformed state occupies the regular region B of euclidean three-dimensional space and is bounded by the surface @B. We refer the deformation of the body to a fixed system of rectangular axes Oxj, (j = 1, 2, 3). Let nj be the outward unit normal of @B. We shall employ the usual summation and differentiation conventions: Latin subscripts (unless otherwise specified) are understood to range over the integers (1, 2, 3) whereas Greek subscripts to the range (1, 2); summation over repeated subscripts is implied and subscripts preceded by a comma denote partial differentiation with respect to the corresponding cartesian coordinate. We suppose that B is occupied by a homogeneous and isotropic chiral Cosserat elastic material. We denote by uj the displacement vector on B. Let uj be the microrotation vector. The strain measures in the linear theory are
eij ¼ uj;i þ ejik uk ;
jij ¼ uj;i ;
ð1Þ
where eijk is the alternating symbol. We denote by tij the stress tensor and by mij the couple stress tensor. The constitutive equations of homogeneous and isotropic chiral Cosserat elastic materials are (Lakes, 2001; Nowacki, 1981)
t ij ¼ kerr dij þ ðl þ jÞeij þ leji þ C 1 jrr dij þ C 2 jji þ C 3 jij ; mij ¼ ajrr dij þ bjji þ cjij þ C 1 err dij þ C 2 eji þ C 3 eij ;
ð2Þ
where dij is the Kronecker delta, and k, l, j, a, b, c and Ck are constitutive constants. The equilibrium equations of Cosserat continua are
tji;j þ fi ¼ 0;
mji;j þ eirs t rs þ g i ¼ 0;
ð3Þ
where fi is the body force and gi is the body couple. The surface force and the surface moment acting at a regular point of @B are defined by
ti ¼ t ji nj ;
mi ¼ mji nj ;
ð4Þ
respectively. We assume that the elastic potential is a positive definite quadratic form in the variables eij and jij. The restrictions imposed by this assumption on the constitutive coefficients have been presented by Lakes and Benedict (1982) and Dyszlewicz (2004). We note that
k þ 2l þ j > 0;
2l þ j > 0;
j > 0; c þ b > 0; c b > 0; ðk þ 2l þ jÞða þ b þ cÞ ðC 1 þ C 2 þ C 3 Þ2 > 0:
ð5Þ
The basic equations which govern the equilibrium of isotropic chiral Cosserat elastic materials consist of the geometrical Eq. (1), the constitutive Eq. (2), and the equilibrium Eq. (3). To these equations we must adjoin boundary conditions. In the case of Neumann problem the boundary conditions are
tji nj ¼ ~ti ;
~i mji nj ¼ m
on @B;
~ i are prescribed functions. where ~t i and m 3. Generalized plane strain We assume that the region B from here on refers to the interior of a right cylinder of length h with the open cross section
R and the lateral boundary P. The cartesian coordinate frame is supposed to be chosen in such a way that the x3-axis is par-
D. Iesßan / International Journal of Engineering Science 49 (2011) 411–419
413
allel to generators of B and x1, x2-plane contains one of the terminal cross sections. We assume that the generic cross section R is a regular region with the boundary L. We suppose that the lateral surface of the cylinder B is subjected to surface forces and surface moments which are independent of the axial coordinate. On the lateral surface of B we have the conditions
tai na ¼ ~ti ;
~i mai na ¼ m
on P;
ð6Þ
~ i are given functions. We assume that the body loads are independent of the axial coordinate, where ~ti and m
fi ¼ fi ðx1 ; x2 Þ;
g i ¼ g i ðx1 ; x2 Þ;
ðx1 ; x2 Þ 2 R:
ð7Þ
We say that the cylinder B is in a state of generalized plane strain if the displacement and microrotation vector fields are independent of the axial coordinate,
ui ¼ ui ðx1 ; x2 Þ; ui ¼ ui ðx1 ; x2 Þ;
ðx1 ; x2 Þ 2 R:
ð8Þ
The above restrictions, in conjunction with the geometrical equations (1) and the constitutive Eq. (2), imply that eij, jij, tij and mij are all independent of the axial coordinate. The Eq. (1) reduce to
eai ¼ ui;a þ eiaj uj ; e3i ¼ ei3b ub ;
jai ¼ ui;a ; j3i ¼ 0:
ð9Þ
In a state of generalized plane strain we have the following constitutive equations
t ab ¼ keqq dab þ ðl þ jÞeab þ leba þ C 1 jqq dab þ C 2 jba þ C 3 jab ; t a3 ¼ ðl þ jÞea3 þ le3a þ C 3 ja3 ; t 3a ¼ ðl þ jÞe3a þ lea3 þ C 2 ja3 ; t33 ¼ keqq þ C 1 jqq ; mmg ¼ ajqq dmg þ bjgm þ cjmg þ C 1 eqq dmg þ C 2 egm þ C 3 emg ;
ð10Þ
ma3 ¼ cja3 þ C 2 e3a þ C 3 ea3 ; m3a ¼ bja3 þ C 2 ea3 þ C 3 e3a ; m33 ¼ ajqq þ C 1 eqq : The equations of equilibrium can be written as
tbi;b þ fi ¼ 0;
mbi;b þ eirs t rs þ g i ¼ 0;
ð11Þ
on R. The conditions on the lateral surface become
~i tbi nb ¼ ~ti ; mbi nb ¼ m
on L:
ð12Þ
~ i are functions of class C1. We assume that fi ; g i ; ~ti and m The generalized plane strain problem consists in finding the displacements ui and microrotations ui which satisfy (9)– (11) on R, and the boundary conditions (12) on L. We note that t3i and m3i do not appear in the Eq. (11) and the conditions (12). These functions can be calculated from (2) after the functions uj and uj have been determined. Thus, we can determine the tractions over the bases which maintain the cylinder in equilibrium. In view of (9) and (10), the equations of equilibrium can be expressed in terms of the functions ui and ui,
ðl þ jÞDua þ ðk þ lÞub;ba þ jeab3 u3;b þ C 3 Dua þ ðC 1 þ C 2 Þub;ba þ fa ¼ 0; ðl þ jÞDu3 þ je3ba ua;b þ C 3 Du3 þ f3 ¼ 0; C 3 Dum þ ðC 1 þ C 2 Þuq;qm þ jemb3 u3;b þ cDum þ ða þ bÞuq;qm þ 2ðC 3 C 2 Þemg3 u3;g 2jum þ g m ¼ 0;
ð13Þ
C 3 Du3 þ je3mg ug;m þ cDu3 þ 2ðC 3 C 2 Þe3mg ug;m 2ju3 þ g 3 ¼ 0; on R, where D is the two-dimensional Laplacian. We note that in the case of an achiral material, the system (13) and the boundary conditions (12) reduce to two uncoupled boundary value problems: one for the unknowns ua and u3, and the other for the functions u3 and ua (Dyszlewicz, 2004; Nowacki, 1981). Let us investigate the existence of solution of the boundary value problem (12), (13). The existence theorems in the linear theory of Cosserat elasticity have been presented in various papers (Hlavacek & Hlavacek, 1969; Iesßan, 1970; Kupradze, Gegelia, Bashelishvili, & Burchuladze, 1979). As the generalized plane strain is a special deformation, it requires a separate investigation. The Eq. (13) can be written in the form
Au ¼ f ;
ð14Þ
where
u ¼ ðu1 ; u2 ; u3 ; u1 ; u2 ; u3 Þ;
f ¼ ðf1 ; f2 ; f3 ; g 1 ; g 2 ; g 3 Þ;
ð15Þ
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D. Iesßan / International Journal of Engineering Science 49 (2011) 411–419
and Au has the components
Aa u ¼ ðl þ jÞDua ðk þ lÞub;ba jeab3 u3;b C 3 Dua ðC 1 þ C 2 Þub;ba ; A3 u ¼ ðl þ jÞDu3 je3ba ua;b C 3 Du3 ; A3þm u ¼ C 3 Dum ðC 1 þ C 2 Þuq;qm jemb3 u3;b ða þ bÞuq;qm 2ðC 3 C 2 Þemg3 u3;g þ 2jum ;
ð16Þ
A6 u ¼ C 3 Du3 je3mg ug;m cDu3 2ðC 3 C 2 Þe3mg ug;m þ 2ju3 : We denote by eji(u), jai(u), tji(u) and mji(u) the strain measures, the stress tensor and the couple stress tensor associated to the six-dimensional vector field u. The elastic potential corresponding to u is given by
2WðuÞ ¼ keqq ðuÞemm ðuÞ þ ðl þ jÞeab ðuÞeab ðuÞ þ leba ðuÞeab ðuÞ þ ðl þ jÞea3 ðuÞea3 ðuÞ þ 2lea3 ðuÞe3a ðuÞ þ ðl þ jÞe3a ðuÞe3a ðuÞ þ ajqq ðuÞjmm ðuÞ þ bjgm ðuÞjmg ðuÞ þ cjmg ðuÞjmg ðuÞ þ cja3 ðuÞja3 ðuÞ þ 2C 1 jqq ðuÞemm ðuÞ þ 2C 2 eab ðuÞjba ðuÞ þ 2C 3 eab ðuÞjab ðuÞ þ 2C 2 e3a ðuÞja3 ðuÞ þ 2C 3 ea3 ðuÞja3 ðuÞ:
ð17Þ
We introduce the notation
TðuÞ ¼ ðt i ðuÞ; mi ðuÞÞ;
ð18Þ
where
ti ðuÞ ¼ t ai ðuÞna ;
mi ðuÞ ¼ mai ðuÞna
on L:
ð19Þ
By using the geometrical equations, constitutive equations and the divergence theorem we obtain
Z
uAuda ¼
R
Z
uTðuÞds þ 2
Z
WðuÞda:
ð20Þ
R
K
If v and w are two six-dimensional vectors, then vw is the scalar product of vectors v and w. In what follows we assume that W(u) is a positive definite quadratic form in the variables eib(u), ea3(u) and jaj(u). From (20) we conclude that the solution corresponding to null external data is given by
u0 ¼ ðu0i ; u0i Þ; u01 ¼ b1 b4 x2 ; u2 ¼ b2 þ b4 x1 ; u03 ¼ b3 ;
ð21Þ
u0a ¼ 0; u03 ¼ b4 ;
where bk, (k = 1, 2, 3, 4), are arbitrary constants. The vector field u0 given by (21) can be written as
u0 ¼ b1 uð1Þ þ b2 uð2Þ þ b3 uð3Þ þ b4 uð4Þ ; where
uð1Þ ¼ ð1; 0; 0; 0; 0; 0Þ; uð2Þ ¼ ð0; 1; 0; 0; 0; 0Þ; uð3Þ ¼ ð0; 0; 1; 0; 0; 0Þ; uð4Þ ¼ ðx2 ; x1 ; 0; 0; 0; 1Þ:
ð22Þ
We consider the boundary value problem
Au ¼ f
on R;
TðuÞ ¼ 0 on L:
ð23Þ
Following Fichera (1965) and Iesßan (1973), a solution of the boundary value problem (23) exists if and only if
Z
ðkÞ
fu da ¼ 0;
ðk ¼ 1; 2; 3; 4Þ:
ð24Þ
R
In view of (15) and (22), the conditions (24) reduce to
Z
fk da ¼ 0;
R
Z
ðx1 f2 x2 f1 þ g 3 Þ ¼ 0:
ð25Þ
R
It is easy to see that in the case of the boundary value problem characterized by the Eq. (14) and the boundary conditions (12), the conditions (25) are replaced by
Z R
fk da þ
Z
~t k ds ¼ 0;
L
Z R
ðeab3 xa fb þ g 3 Þda þ
Z
~ 3 Þds ¼ 0: ðeab3 xa~tb þ m
ð26Þ
L
4. Pressure vessel problem In this section we investigate the deformation of a chiral elastic tube under internal and external pressures. We assume that the cylinder B is defined by B ¼ fx : a21 < x21 þ x22 < a22 , 0 < x3 < h}, where a1 and a2 are positive constants (see Fig. 1). Let us denote r ¼ ðx21 þ x22 Þ1=2 . We assume that the tube is subjected to the following loads
D. Iesßan / International Journal of Engineering Science 49 (2011) 411–419
415
x3
Fig. 1. A circular hollow cylinder.
fi ¼ 0; g i ¼ 0; ~t a ¼ p1 na ; ~t3 ¼ 0; ~t a ¼ p2 na ; ~t3 ¼ 0;
~ i ¼ 0 on r ¼ a1 ; m
ð27Þ
~ i ¼ 0 on r ¼ a2 ; m
where p1 and p2 are prescribed constants. In this case the domain R is bounded by two concentric circles of radius a1 and a2, where a1 < a2. In what follows we study the generalized plane strain problem for B when the external data are given by (34). We note that the conditions (26) for the existence of a solution are satisfied. We seek the solution of the Eq. (13), with fi = 0 and gj = 0, in the form
ua ¼ xa FðrÞ;
u3 ¼ 0;
ua ¼ xa GðrÞ; u3 ¼ 0;
ð28Þ
where F and G are unknown functions. It follows from (28) that we have
ua;b ¼ dab F þ xa xb r 1 F 0 ; uq;q ¼ 2F þ rF 0 ;
ð29Þ
uq;qa ¼ Dua ¼ xa ðF 00 þ 3r 1 F 0 Þ; 0
where f = df/dr. By (9) and (29) we get
eab ¼ eba ¼ ua;b ; eq3 ¼ e3qb xb G ¼ e3q ; e33 ¼ 0; j33 ¼ 2G þ rG0 ; jab ¼ jba ¼ ua;b ;
ð30Þ
ja3 ¼ 0; j3i ¼ 0: In view of (29), the equilibrium Eq. (13), in the absence of body loads become
ðk þ 2l þ jÞðF 00 þ 3r 1 F 0 Þ þ ðC 1 þ C 2 þ C 3 ÞðG00 þ 3r 1 G0 Þ ¼ 0; ðC 1 þ C 2 þ C 3 ÞðF 00 þ 3r1 F 0 Þ þ ða þ b þ cÞðG00 þ 3r 1 G0 s2 GÞ ¼ 0;
ð31Þ
where
s2 ¼
2j : aþbþc
ð32Þ
It follows from (31) that the function G must satisfy the equation
G00 þ 3r 1 G0 q2 G ¼ 0;
ð33Þ
where
q2 ¼ 2jðk þ 2l þ jÞQ 1 ; Q ¼ ðk þ 2l þ jÞða þ b þ cÞ ðC 1 þ C 2 þ C 3 Þ2 : 2
ð34Þ
2
We note that the relations (5) imply that s > 0, Q > 0 and q > 0. Let us introduce the function U by
U ¼ rG:
ð35Þ
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D. Iesßan / International Journal of Engineering Science 49 (2011) 411–419
Then, the Eq. (33) becomes
1 r
U00 þ U0
1 2 þ q U ¼ 0: r2
ð36Þ
In what follows we denote by In and Kn the modified Bessel functions of order n. Then, the general solution of the equation (36) is
U ¼ B1 I1 ðqrÞ þ B2 K 1 ðqrÞ; where B1 and B2 are arbitrary constants. Thus, we find that
G ¼ r 1 ½B1 I1 ðqrÞ þ B2 K 1 ðqrÞ:
ð37Þ
It follows from (31) that the function F satisfies the equation 2
d
dr
þ 2
3 d r dr
!
2
d
dr
þ 2
! 3 d q2 F ¼ 0: r dr
ð38Þ
Clearly, we can write F = F1 + F2 where the functions Fa satisfy the equations
F 001 þ 3r1 F 01 ¼ 0;
F 002 þ 3r 1 F 02 q2 F 2 ¼ 0:
Solving these equations, we find that the function F has the form
F ¼ A1 þ A2 r 2 þ r1 ½D1 I1 ðqrÞ þ D2 K 1 ðqrÞ;
ð39Þ
where Aa and Da are arbitrary constants. We note that the functions F and G, given by (37) and (39), must satisfy the differential system of second order (31). It is easy to see that F and G satisfy the Eq. (31) if we have
D1 ¼ rB1 ;
D2 ¼ rB2 ;
ð40Þ
where
r ¼ ðC 1 þ C 2 þ C 3 Þðk þ 2l þ jÞ1 :
ð41Þ
We conclude that the equilibrium equations are satisfied if the functions F and G are given by
F ¼ A1 þ A2 r 2 rr 1 ½B1 I1 ðqrÞ þ B2 K 1 ðqrÞ; G ¼ r 1 ½B1 I1 ðqrÞ þ B2 K 1 ðqrÞ:
ð42Þ
Let us investigate the boundary conditions. In view of (12) and (27) we obtain the following boundary conditions
t ab na ¼ p1 nb ;
ta3 na ¼ 0; mai na ¼ 0
t ab na ¼ p2 nb ; t a3 na ¼ 0; mai na ¼ 0
on r ¼ a1 ; on r ¼ a2 :
ð43Þ
It follows from (30) and the constitutive Eq. (10) that
t ab ¼ kð2F þ rF 0 Þdab þ ð2l þ jÞðFdab þ xa xb r1 F 0 Þ þ C 1 ð2G þ rG0 Þdab þ ðC 2 þ C 3 ÞðGdab þ xa xb r1 G0 Þ; t a3 ¼ t 3a ¼ je3ab xb G; mmg ¼ að2G þ rG0 Þdmg þ ðb þ cÞðGdmg þ xm xg r1 G0 Þ þ C 1 ð2F þ rF 0 Þ þ ðC 2 þ C 3 ÞðFdmg þ xm xg r 1 F 0 Þ
ð44Þ
mm3 ¼ m3m ¼ ðC 3 C 2 Þe3mb xb G: On the circle r = a1 we have na = xa/a1, and on the circle r = a2 we have na = xa/a2. The boundary conditions (50) can be written as
t ab xa ¼ p1 xb ; ta3 xa ¼ 0; mai xa ¼ 0 on r ¼ a1 ; t ab xa ¼ p2 xb ; ta3 xa ¼ 0; mai xa ¼ 0 on r ¼ a2 :
ð45Þ
In view of (44), we get
t ab xa ¼ xb ½ð2k þ 2l þ jÞF þ ðk þ 2l þ jÞrF 0 þ ð2C 1 þ C 2 þ C 3 ÞG þ ðC 1 þ C 2 þ C 3 ÞrG0 ; t a3 xa ¼ je3ab xa xb G ¼ 0; mab xa ¼ xb ½ð2a þ b þ cÞG þ ða þ b þ cÞrG0 þ ð2C 1 þ C 2 þ C 3 ÞF þ ðC 1 þ C 2 þ C 3 ÞrF 0 ; ma3 xa ¼ ðC 3 C 2 Þe3ab xa xb G ¼ 0: With the help of (46) the boundary conditions (45) reduce to
ð46Þ
D. Iesßan / International Journal of Engineering Science 49 (2011) 411–419
417
ð2k þ 2l þ jÞFða1 Þ þ ðk þ 2l þ jÞa1 F 0 ða1 Þ þ ð2C 1 þ C 2 þ C 3 ÞGða1 Þ þ ðC 1 þ C 2 þ C 3 Þa1 G0 ða1 Þ ¼ p1 ; ð2a þ b þ cÞGða1 Þ þ ða þ b þ cÞa1 G0 ða1 Þ þ ð2C 1 þ C 2 þ C 3 ÞFða1 Þ þ ðC 1 þ C 2 þ C 3 Þa1 F 0 ða1 Þ ¼ 0; ð2k þ 2l þ jÞFða2 Þ þ ðk þ 2l þ jÞa2 F 0 ða2 Þ þ ð2C 1 þ C 2 þ C 3 ÞGða2 Þ þ ðC 1 þ C 2 þ C 3 Þa2 G0 ða2 Þ ¼ p2 ;
ð47Þ
ð2a þ b þ cÞGða2 Þ þ ða þ b þ cÞa2 G0 ða2 Þ þ ð2C 1 þ C 2 þ C 3 ÞFða2 Þ þ ðC 1 þ C 2 þ C 3 Þa2 F 0 ða2 Þ ¼ 0; We introduce the notations
K1 ðrÞ ¼ ðC 1 rkÞr 1 I1 ðqrÞ þ ½C 1 þ C 2 þ C 3 rðk þ 2l þ jÞI01 ðqrÞ; K2 ðrÞ ¼ ðC 1 rkÞr 1 K 1 ðqrÞ þ ½C 1 þ C 2 þ C 3 rðk þ 2l þ jÞK 01 ðqrÞ; N1 ðrÞ ¼ ða rC 1 Þr 1 I1 ðqrÞ þ ½a þ b þ c rðC 1 þ C 2 þ C 3 ÞI01 ðqrÞ;
ð48Þ
N2 ðrÞ ¼ ða rC 1 Þr 1 K 1 ðqrÞ þ ½a þ b þ c rðC 1 þ C 2 þ C 3 ÞK 01 ðqrÞ: By using (42), the conditions (47) can be written in the form
ð2k þ 2l þ jÞA1 ð2l þ jÞa2 1 A2 þ l11 B1 þ l12 B2 ¼ p1 ; ð2C 1 þ C 2 þ C 3 ÞA1 ðC 2 þ C 3 Þa2 1 A2 þ l21 B1 þ l22 B2 ¼ 0; ð2k þ 2l þ jÞA1 ð2l þ jÞa2 2 A2 þ l31 B1 þ l32 B2 ¼ p2 ;
ð49Þ
ð2C 1 þ C 2 þ C 3 ÞA1 ðC 2 þ C 3 Þa2 2 A2 þ l41 B1 þ l42 B2 ¼ 0; where
l11 ¼ K1 ða1 Þ; l12 ¼ K2 ða1 Þ; l21 ¼ N1 ða1 Þ; l22 ¼ N2 ða1 Þ; l31 ¼ K1 ða2 Þ; l32 ¼ K2 ða2 Þ; l41 ¼ N1 ða2 Þ; l42 ¼ N2 ða2 Þ:
ð50Þ
From (49) we find the constants Aa and Ba, 2 2 2 ð2k þ 2l þ jÞða2 1 a2 ÞA1 ¼ p2 a1 p1 a2 b1q Bq ; 2 ð2l þ jÞða2 1 a2 ÞA2 ¼ p2 p1 b2q Bq ;
B1 ¼ ðH22 T 1
1 H12 T 2 Þd1 ;
B2 ¼ ðH11 T 2
ð51Þ
1 H21 T 1 Þd1 ;
where 2 b1q ¼ l3q a2 1 l1q a2 ; b2q ¼ l3q l1q ;
H1q ¼ d2 l2q ð2C 1 þ C 2 þ C 3 Þð2l þ jÞb1q þ ðC 2 þ C 3 Þð2k þ 2l þ jÞa2 1 b2q ; H2q ¼ d2 l4q ð2C 1 þ C 2 þ C 3 Þð2l þ jÞb1q þ ðC 2 þ C 3 Þð2k þ 2l þ jÞa2 2 b2q ; 2 d2 ¼ ð2k þ 2l þ jÞð2l þ jÞða1 1 a2 Þ; d1 ¼ H11 H 22 H 12 H21 ; 2 2 T 1 ¼ ð2C 1 þ C 2 þ C 3 Þð2l þ jÞðp1 a2 2 p2 a1 Þ þ ðC 2 þ C 3 Þð2k þ 2l þ jÞa1 ðp2 p1 Þ; 2 2 T 2 ¼ ð2C 1 þ C 2 þ C 3 Þð2l þ jÞðp1 a2 2 p2 a1 Þ þ ðC 2 þ C 3 Þð2k þ 2l þ jÞa2 ðp2 p1 Þ:
The solution of the pressure vessel problem is given by (28), (42) and (51). In the case of achiral materials we have Ck = 0, and from (41), (48) and (50) we obtain r = 0, l1a = 0 and l3a = 0. Now the system (49) has the solution
A1 ¼ ðp2 a22 p1 a21 Þ½ða22 a21 Þð2k þ 2l þ jÞ1 ; A2 ¼ ðp2 p1 Þa21 a22 ½ða22 a21 Þð2l þ jÞ1 ; Ba ¼ 0: Thus, if the tube is made of an isotropic achiral Cosserat elastic material, then the microrotation vector field vanishes. The salient feature of the solution (28) is that, in the absence of the body couples and surface moments, a pressure acting on the surface of a chiral tube produces a microrotation of the material particles. This is a new chiral effect. 5. Conclusions The mechanical properties of chiral materials are not invariant with respect to inversions. In the classical elasticity the elasticity tensor is unchanged under an inversion. It is therefore common to use Cosserat elasticity to introduce chirality in the context of elasticity. In this paper the pressure vessel problem is solved in the context of the theory of chiral Cosserat elasticity. This work is motivated by real problems. The Cosserat elasticity was used to study the deformation of chiral nanotubes and the mechanical behavior of bones. The intended applications of the solution presented in this paper are to nanotubes and bone mechanics. In the femur bone implants an internal pressure is applied on the interface. The solution of the pressure vessel problem can also be used to study the mechanical behavior of tubes made of chiral composites.
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D. Iesßan / International Journal of Engineering Science 49 (2011) 411–419
To investigate the pressure vessel problem in the context of the theory of chiral Cosserat elasticity, the state of generalized plane strain was introduced. Conditions have been found under which the generalized plane strain problem is possible. The pressure vessel problem has been investigated as a special generalized plane strain of a chiral elastic tube. It is shown that, a pressure acting on the surface of an isotropic chiral elastic tube, in contrast with the case of achiral materials, produces a microrotation of the material particles. It follows from the solution presented in Section 4 that the radial displacement for a chiral tube is given by
1 u¼ d2
(
1 1 2 þ rI1 ðqrÞ ð2l þ jÞðp2 a2 1 p1 a2 Þr þ ð2k þ 2l þ jÞðp2 p1 Þ B1 ½ð2l þ jÞb11 r þ ð2k þ 2l þ jÞb21 r r ) 1 B2 ½ð2l þ jÞb12 r þ ð2k þ 2l þ jÞb22 þ rK 1 ðqrÞ : r
ð52Þ
In isotropic linear Cosserat elasticity, chirality introduces three additional parameters for a total of nine constitutive coefficients. The chiral behavior of the body is controlled by the material parameters C1, C2 and C3 which appear in the constitutive equations. The achiral isotropic Cosserat elastic solid requires six elastic constants, k, l, a, b, c and j, for its description. In the case of achiral materials the parameters Ck are equal to zero and we find from (52) that the radial displacement has the form
u ¼
1 p2 a22 p1 a21 ðp2 p1 Þa21 a22 : r þ ð2l þ jÞr a22 a21 2k þ 2l þ j
In the theory of classical elasticity an isotropic solid has two constitutive coefficients, the Lamé moduli k and l. In this theory the radial displacement is given by
u0 ¼ ½ðp2 a22 p1 a21 Þrðk þ lÞ1 þ ðp2 p1 Þa21 a22 l1 =½2ða22 a21 Þ: The solution of the pressure vessel problem for a chiral elastic tube yields the following radial stress
S ¼ ð2k þ 2l þ jÞA1 ð2l þ jÞr 2 A2 þ B1 K1 ðrÞ þ B2 K2 ðrÞ;
ð53Þ
where A1, A2, B1 and B2 are given by (51), and the functions K1 and K2 are defined in (48). It follows from (53) that the radial stress in a tube made of an achiral material is given by
S ¼
1 ½p a2 p1 a21 ðp2 p1 Þa21 a22 r2 : a22 a21 2 2
The radial stress and radial displacement are modified from values predicted by the theory of achiral materials. Acknowledgements This paper was supported by CNCSIS, Project PN II-IDEI 89/2008. I express my gratitude to the referee for his helpful suggestions. References Cao, G., & Chen, X. (2007). The effects of chirality and boundary conditions on the mechanical properties of single-walled carbon nanotubes. International Journal of Solids and Structures, 44, 5447–5465. Chandraseker, K., Mukherjee, S., Paci, J. T., & Schatz, G. C. (2009). An atomistic-continuum Cosserat rod model of carbon nanotubes. Journal of Mechanics and Physics of Solids, 57, 932–958. Ciarletta, M., & Iesßan, D. (1993). Non-classical elastic solids. Harlow: Longman Scientific & Technical. Donescu, S., Chiroiu, V., & Munteanu, L. (2009). On the Young’s modulus of an auxetic composite structure. Mechanics Research Communications, 36, 294–301. Dyszlewicz, J. (2004). Micropolar theory of elasticity. Berlin, New York: Springer. Eringen, A. C. (1999). Microcontinuum field theories, I: Foundations and solids. Berlin, New York: Springer. Fatemi, J., van Keulen, F., & Onuck, P. R. (2002). Generalized continum theories: Application to stress analysis of bone. Meccanica, 37, 385–396. Fichera, G. (1965). Linear elliptic differential systems and eigenvalue problems. Lecture notes in mathematics (Vol. 8). Berlin: Springer-Verlag. Guz, I. A., Rodger, A. A., Guz, A. N., & Rushchitsky, J. J. (2007). Developing the mechanical models for nanomaterials. Composite Part A: Applied Science and Manufacturing, 38, 1234–1250. Haijun, Z., & Zhong-can, O. (1998). Bending and twisting elasticity: A revised Marko–Sigga model of DNA chirality. Physical Review E, 58, 4816–4821. Hassan, M. R., Scarpa, F., Ruzzene, M., & Mohamed, N. A. (2008). Smart shape memory alloy chiral honeycomb. Materials Science and Engineering: A, 481–482, 654–657. Healey, T. J. (2002). Material symmetry and chirality in nonlinearly elastic rods. Mathematics and Mechanics of Solids, 7, 405–420. Hlavacek, I., & Hlavacek, M. (1969). On the existence and uniqueness of solutions and some variational principles in linear theories of elasticity with couple stresses. Aplikace Matematiky, 14, 387–401. Iesßan, D. (1970). Existence theorems in the micropolar elasticity. International Journal of Engineering Science, 8, 777–791. Iesßan, D. (1973). On the second boundary value problem in the linear theory of micropolar elasticity. Atti Della Accademia Nazionale Dei Lincei RendicontiClasse Di Scienze Fisiche-Matematiche & Naturali Series VIII, 55, 456–459. Khurana, A., & Tomar, S. K. (2009). Longitudinal wave response of a chiral slab interposed between micropolar solid spaces. International Journal of Solids and Structures, 46, 135–150. Kupradze, V. D., Gegelia, T. G., Bashelishvili, M. O., & Burchuladze, T. V. (1979). Three dimensional problems of mathematical theory of elasticity and thermoelasticity. Amsterdam: North-Holland Publ..
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Lakes, R. S., & Benedict, R. L. (1982). Noncentrosymmetry in micropolar elasticity. International Journal of Engineering Science, 29, 1161–1167. Lakes, R. S., Yoon, H. S., & Katz, I. L. (1983). Slow compressional wave propagation in wet human and bovine cortical bone. Science, 220, 513–515. Lakes, R. (2001). Elastic and viscoleastic behaviour of chiral materials. International Journal of Mechanical Science, 43, 1579–1589. Natroshvili, D., Giorgashvili, L., & Stratis, I. G. (2006). Representation formulae of general solutions in the theory of hemitropic elasticity. Quarterly Journal of Mechanics and Applied Mathematics, 59, 451–474. Nowacki, W. (1981). Theory of asymmetric elasticity. PWN, Warszawa. Park, H. C., & Lakes, R. S. (1986). Cosserat micromechanics of human bone: strain redistribution by a hydration-sensitive constituent. Journal of Biomechanics, 19, 385–397. Sharma, P. (2004). Size-dependent elastic fields of embedded inclusions in isotropic chiral solids. International Journal of Solids and Structures, 41, 6317–6333. Teoodorescu, P. P., Munteanu, L., & Chiroiu, V. (2005). On the wave propagation in chiral media. In M. Mihailescu-Suliciu (Ed.), New trends in continuum mechanics (pp. 303–310). Bucharest: Thetha Foundations.
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International Journal of Engineering Science journal homepage: www.elsevier.com/locate/ijengsci
Pressure vessel problem for chiral elastic tubes D. Iesßan ⇑ Department of Mathematics, ‘‘Al.I. Cuza’’ University and ‘‘O. Mayer’’ Institute of Mathematics, Romanian Academy, 700506 Iasßi, Romania
a r t i c l e
i n f o
Article history: Received 28 January 2010 Received in revised form 7 January 2011 Accepted 23 January 2011 Available online 22 February 2011 Keywords: Chiral tubes Cosserat elastic solids Auxetic materials Mechanics of bone
a b s t r a c t The study of chiral materials is important for the investigation of carbon nanotubes, auxetic materials and bones. The chiral effects cannot be described within classical elasticity. In this paper, the problem of the circular tube under internal and external pressure is solved in the context of the linear theory of chiral Cosserat elasticity. The work is motivated by the recent interest in the using Cosserat elastic solid as model for auxetic materials, bones and chiral carbon nanotubes. First, the generalized plane strain of an isotropic chiral elastic material is investigated. Then, the solution of the pressure vessel problem is established. The salient feature of the solution is that, in the absence of body and surface moments, a pressure acting on the surface of a chiral tube produces a microrotation of the material particles. It is shown that the radial displacement and the radial stress are modified from the values predicted by the theory of achiral materials. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction In recent years the mechanical behavior of chiral materials has been the subject of many investigations (Dyszlewicz, 2004; Hassan, Scarpa, Ruzzene, & Mohamed, 2008; Healey, 2002; Lakes, 2001; Natroshvili, Giorgashvili, & Stratis, 2006). The study of chiral materials is important for the investigation of carbon nanotubes, auxetic materials and bones. A chiral material (also known as noncentrosymmetric, or hemitropic) is a material whose behavior is not invariant with respect to inversion. In the classical elasticity the elasticity tensor is unchanged under an inversion (Lakes, 2001). Consequently, the classical elasticity cannot describe the behavior of the chiral elastic materials. An adequate theory to describe the mechanical behavior of the chiral elastic materials is the theory of Cosserat elasticity (Donescu, Chiroiu, & Munteanu, 2009; Haijun & Zhong-can, 1998; Khurana & Tomar, 2009; Lakes & Benedict, 1982; Sharma, 2004; Teoodorescu, Munteanu, & Chiroiu, 2005). The Cosserat theory studies continua with oriented particles, in which each material point has the six degree of freedom of a rigid body (Ciarletta & Iesßan, 1993; Eringen, 1999; Nowacki, 1981). In this paper we investigate the problem of an isotropic and homogeneous chiral tube subjected to internal and external pressures. Structures such as pipes or tubes capable of holding internal pressure have been very important in the history of science and technology. This work is motivated by real problems. The theory of Cosserat elasticity has been used to study the deformation of various carbon nanotubes (Chandraseker, Mukherjee, Paci, & Schatz, 2009; Guz, Rodger, Guz, & Rushchitsky, 2007) and the mechanical behavior of bones (Fatemi, van Keulen, & Onuck, 2002; Lakes, Yoon, & Katz, 1983; Park & Lakes, 1986). The intended applications of the solution presented in this paper are to nanotubes and to bone mechanics. In the femur bone implants an internal pressure is applied on the interface. Some deformations of the bone can be modeled by using the solution of pressure vessel problem. Examples of chiral nanotubes have been presented by Cao and Chen (2007) and Guz et al. (2007).
⇑ Tel.: +40 232201060; fax: +40 232201160. E-mail address: [email protected] 0020-7225/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijengsci.2011.01.003
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D. Iesßan / International Journal of Engineering Science 49 (2011) 411–419
In the classical theory of elasticity the pressure vessel problem reduces to solving a plane strain problem. To investigate the pressure vessel problem in the context of the linear theory of chiral Cosserat elasticity we have introduced the state of generalized plane strain, in which the displacement and microrotation vectors are independent of the axial coordinate. We have given conditions under which a generalized plane strain is possible. The pressure vessel problem has been investigated as a special generalized plane strain of a chiral elastic tube. This paper is structured as follows. First, we present the basic equations of homogeneous and isotropic chiral Cosserat elastic solids. Then, we define the generalized plane strain problem for a chiral elastic solid. In the case of an achiral Cosserat elastic solid, the basic equations of the two-dimensional theory and the boundary conditions reduce to two uncoupled boundary value problems. A such simplification is not possible for chiral materials. In the following section we solve the generalized plane strain problem which govern the deformation of a chiral tube subjected to internal and external pressures. It is shown that, in the absence of body couples and surface moments, a pressure acting on the surface of an isotropic chiral Cosserat elastic tube, produces a microrotation of the material particles. In the case of an achiral tube subjected to the same loading the microrotation vector vanishes. It is shown that the radial displacement and the radial stress in a chiral tube are modified from the values predicted by the theory of achiral materials. We belive that the solution presented herein will find applications in the theory of nanostructures. 2. Preliminaries In this section we present the basic equations of homogeneous and isotropic chiral Cosserat elastic bodies. Let us consider a body that in the undeformed state occupies the regular region B of euclidean three-dimensional space and is bounded by the surface @B. We refer the deformation of the body to a fixed system of rectangular axes Oxj, (j = 1, 2, 3). Let nj be the outward unit normal of @B. We shall employ the usual summation and differentiation conventions: Latin subscripts (unless otherwise specified) are understood to range over the integers (1, 2, 3) whereas Greek subscripts to the range (1, 2); summation over repeated subscripts is implied and subscripts preceded by a comma denote partial differentiation with respect to the corresponding cartesian coordinate. We suppose that B is occupied by a homogeneous and isotropic chiral Cosserat elastic material. We denote by uj the displacement vector on B. Let uj be the microrotation vector. The strain measures in the linear theory are
eij ¼ uj;i þ ejik uk ;
jij ¼ uj;i ;
ð1Þ
where eijk is the alternating symbol. We denote by tij the stress tensor and by mij the couple stress tensor. The constitutive equations of homogeneous and isotropic chiral Cosserat elastic materials are (Lakes, 2001; Nowacki, 1981)
t ij ¼ kerr dij þ ðl þ jÞeij þ leji þ C 1 jrr dij þ C 2 jji þ C 3 jij ; mij ¼ ajrr dij þ bjji þ cjij þ C 1 err dij þ C 2 eji þ C 3 eij ;
ð2Þ
where dij is the Kronecker delta, and k, l, j, a, b, c and Ck are constitutive constants. The equilibrium equations of Cosserat continua are
tji;j þ fi ¼ 0;
mji;j þ eirs t rs þ g i ¼ 0;
ð3Þ
where fi is the body force and gi is the body couple. The surface force and the surface moment acting at a regular point of @B are defined by
ti ¼ t ji nj ;
mi ¼ mji nj ;
ð4Þ
respectively. We assume that the elastic potential is a positive definite quadratic form in the variables eij and jij. The restrictions imposed by this assumption on the constitutive coefficients have been presented by Lakes and Benedict (1982) and Dyszlewicz (2004). We note that
k þ 2l þ j > 0;
2l þ j > 0;
j > 0; c þ b > 0; c b > 0; ðk þ 2l þ jÞða þ b þ cÞ ðC 1 þ C 2 þ C 3 Þ2 > 0:
ð5Þ
The basic equations which govern the equilibrium of isotropic chiral Cosserat elastic materials consist of the geometrical Eq. (1), the constitutive Eq. (2), and the equilibrium Eq. (3). To these equations we must adjoin boundary conditions. In the case of Neumann problem the boundary conditions are
tji nj ¼ ~ti ;
~i mji nj ¼ m
on @B;
~ i are prescribed functions. where ~t i and m 3. Generalized plane strain We assume that the region B from here on refers to the interior of a right cylinder of length h with the open cross section
R and the lateral boundary P. The cartesian coordinate frame is supposed to be chosen in such a way that the x3-axis is par-
D. Iesßan / International Journal of Engineering Science 49 (2011) 411–419
413
allel to generators of B and x1, x2-plane contains one of the terminal cross sections. We assume that the generic cross section R is a regular region with the boundary L. We suppose that the lateral surface of the cylinder B is subjected to surface forces and surface moments which are independent of the axial coordinate. On the lateral surface of B we have the conditions
tai na ¼ ~ti ;
~i mai na ¼ m
on P;
ð6Þ
~ i are given functions. We assume that the body loads are independent of the axial coordinate, where ~ti and m
fi ¼ fi ðx1 ; x2 Þ;
g i ¼ g i ðx1 ; x2 Þ;
ðx1 ; x2 Þ 2 R:
ð7Þ
We say that the cylinder B is in a state of generalized plane strain if the displacement and microrotation vector fields are independent of the axial coordinate,
ui ¼ ui ðx1 ; x2 Þ; ui ¼ ui ðx1 ; x2 Þ;
ðx1 ; x2 Þ 2 R:
ð8Þ
The above restrictions, in conjunction with the geometrical equations (1) and the constitutive Eq. (2), imply that eij, jij, tij and mij are all independent of the axial coordinate. The Eq. (1) reduce to
eai ¼ ui;a þ eiaj uj ; e3i ¼ ei3b ub ;
jai ¼ ui;a ; j3i ¼ 0:
ð9Þ
In a state of generalized plane strain we have the following constitutive equations
t ab ¼ keqq dab þ ðl þ jÞeab þ leba þ C 1 jqq dab þ C 2 jba þ C 3 jab ; t a3 ¼ ðl þ jÞea3 þ le3a þ C 3 ja3 ; t 3a ¼ ðl þ jÞe3a þ lea3 þ C 2 ja3 ; t33 ¼ keqq þ C 1 jqq ; mmg ¼ ajqq dmg þ bjgm þ cjmg þ C 1 eqq dmg þ C 2 egm þ C 3 emg ;
ð10Þ
ma3 ¼ cja3 þ C 2 e3a þ C 3 ea3 ; m3a ¼ bja3 þ C 2 ea3 þ C 3 e3a ; m33 ¼ ajqq þ C 1 eqq : The equations of equilibrium can be written as
tbi;b þ fi ¼ 0;
mbi;b þ eirs t rs þ g i ¼ 0;
ð11Þ
on R. The conditions on the lateral surface become
~i tbi nb ¼ ~ti ; mbi nb ¼ m
on L:
ð12Þ
~ i are functions of class C1. We assume that fi ; g i ; ~ti and m The generalized plane strain problem consists in finding the displacements ui and microrotations ui which satisfy (9)– (11) on R, and the boundary conditions (12) on L. We note that t3i and m3i do not appear in the Eq. (11) and the conditions (12). These functions can be calculated from (2) after the functions uj and uj have been determined. Thus, we can determine the tractions over the bases which maintain the cylinder in equilibrium. In view of (9) and (10), the equations of equilibrium can be expressed in terms of the functions ui and ui,
ðl þ jÞDua þ ðk þ lÞub;ba þ jeab3 u3;b þ C 3 Dua þ ðC 1 þ C 2 Þub;ba þ fa ¼ 0; ðl þ jÞDu3 þ je3ba ua;b þ C 3 Du3 þ f3 ¼ 0; C 3 Dum þ ðC 1 þ C 2 Þuq;qm þ jemb3 u3;b þ cDum þ ða þ bÞuq;qm þ 2ðC 3 C 2 Þemg3 u3;g 2jum þ g m ¼ 0;
ð13Þ
C 3 Du3 þ je3mg ug;m þ cDu3 þ 2ðC 3 C 2 Þe3mg ug;m 2ju3 þ g 3 ¼ 0; on R, where D is the two-dimensional Laplacian. We note that in the case of an achiral material, the system (13) and the boundary conditions (12) reduce to two uncoupled boundary value problems: one for the unknowns ua and u3, and the other for the functions u3 and ua (Dyszlewicz, 2004; Nowacki, 1981). Let us investigate the existence of solution of the boundary value problem (12), (13). The existence theorems in the linear theory of Cosserat elasticity have been presented in various papers (Hlavacek & Hlavacek, 1969; Iesßan, 1970; Kupradze, Gegelia, Bashelishvili, & Burchuladze, 1979). As the generalized plane strain is a special deformation, it requires a separate investigation. The Eq. (13) can be written in the form
Au ¼ f ;
ð14Þ
where
u ¼ ðu1 ; u2 ; u3 ; u1 ; u2 ; u3 Þ;
f ¼ ðf1 ; f2 ; f3 ; g 1 ; g 2 ; g 3 Þ;
ð15Þ
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D. Iesßan / International Journal of Engineering Science 49 (2011) 411–419
and Au has the components
Aa u ¼ ðl þ jÞDua ðk þ lÞub;ba jeab3 u3;b C 3 Dua ðC 1 þ C 2 Þub;ba ; A3 u ¼ ðl þ jÞDu3 je3ba ua;b C 3 Du3 ; A3þm u ¼ C 3 Dum ðC 1 þ C 2 Þuq;qm jemb3 u3;b ða þ bÞuq;qm 2ðC 3 C 2 Þemg3 u3;g þ 2jum ;
ð16Þ
A6 u ¼ C 3 Du3 je3mg ug;m cDu3 2ðC 3 C 2 Þe3mg ug;m þ 2ju3 : We denote by eji(u), jai(u), tji(u) and mji(u) the strain measures, the stress tensor and the couple stress tensor associated to the six-dimensional vector field u. The elastic potential corresponding to u is given by
2WðuÞ ¼ keqq ðuÞemm ðuÞ þ ðl þ jÞeab ðuÞeab ðuÞ þ leba ðuÞeab ðuÞ þ ðl þ jÞea3 ðuÞea3 ðuÞ þ 2lea3 ðuÞe3a ðuÞ þ ðl þ jÞe3a ðuÞe3a ðuÞ þ ajqq ðuÞjmm ðuÞ þ bjgm ðuÞjmg ðuÞ þ cjmg ðuÞjmg ðuÞ þ cja3 ðuÞja3 ðuÞ þ 2C 1 jqq ðuÞemm ðuÞ þ 2C 2 eab ðuÞjba ðuÞ þ 2C 3 eab ðuÞjab ðuÞ þ 2C 2 e3a ðuÞja3 ðuÞ þ 2C 3 ea3 ðuÞja3 ðuÞ:
ð17Þ
We introduce the notation
TðuÞ ¼ ðt i ðuÞ; mi ðuÞÞ;
ð18Þ
where
ti ðuÞ ¼ t ai ðuÞna ;
mi ðuÞ ¼ mai ðuÞna
on L:
ð19Þ
By using the geometrical equations, constitutive equations and the divergence theorem we obtain
Z
uAuda ¼
R
Z
uTðuÞds þ 2
Z
WðuÞda:
ð20Þ
R
K
If v and w are two six-dimensional vectors, then vw is the scalar product of vectors v and w. In what follows we assume that W(u) is a positive definite quadratic form in the variables eib(u), ea3(u) and jaj(u). From (20) we conclude that the solution corresponding to null external data is given by
u0 ¼ ðu0i ; u0i Þ; u01 ¼ b1 b4 x2 ; u2 ¼ b2 þ b4 x1 ; u03 ¼ b3 ;
ð21Þ
u0a ¼ 0; u03 ¼ b4 ;
where bk, (k = 1, 2, 3, 4), are arbitrary constants. The vector field u0 given by (21) can be written as
u0 ¼ b1 uð1Þ þ b2 uð2Þ þ b3 uð3Þ þ b4 uð4Þ ; where
uð1Þ ¼ ð1; 0; 0; 0; 0; 0Þ; uð2Þ ¼ ð0; 1; 0; 0; 0; 0Þ; uð3Þ ¼ ð0; 0; 1; 0; 0; 0Þ; uð4Þ ¼ ðx2 ; x1 ; 0; 0; 0; 1Þ:
ð22Þ
We consider the boundary value problem
Au ¼ f
on R;
TðuÞ ¼ 0 on L:
ð23Þ
Following Fichera (1965) and Iesßan (1973), a solution of the boundary value problem (23) exists if and only if
Z
ðkÞ
fu da ¼ 0;
ðk ¼ 1; 2; 3; 4Þ:
ð24Þ
R
In view of (15) and (22), the conditions (24) reduce to
Z
fk da ¼ 0;
R
Z
ðx1 f2 x2 f1 þ g 3 Þ ¼ 0:
ð25Þ
R
It is easy to see that in the case of the boundary value problem characterized by the Eq. (14) and the boundary conditions (12), the conditions (25) are replaced by
Z R
fk da þ
Z
~t k ds ¼ 0;
L
Z R
ðeab3 xa fb þ g 3 Þda þ
Z
~ 3 Þds ¼ 0: ðeab3 xa~tb þ m
ð26Þ
L
4. Pressure vessel problem In this section we investigate the deformation of a chiral elastic tube under internal and external pressures. We assume that the cylinder B is defined by B ¼ fx : a21 < x21 þ x22 < a22 , 0 < x3 < h}, where a1 and a2 are positive constants (see Fig. 1). Let us denote r ¼ ðx21 þ x22 Þ1=2 . We assume that the tube is subjected to the following loads
D. Iesßan / International Journal of Engineering Science 49 (2011) 411–419
415
x3
Fig. 1. A circular hollow cylinder.
fi ¼ 0; g i ¼ 0; ~t a ¼ p1 na ; ~t3 ¼ 0; ~t a ¼ p2 na ; ~t3 ¼ 0;
~ i ¼ 0 on r ¼ a1 ; m
ð27Þ
~ i ¼ 0 on r ¼ a2 ; m
where p1 and p2 are prescribed constants. In this case the domain R is bounded by two concentric circles of radius a1 and a2, where a1 < a2. In what follows we study the generalized plane strain problem for B when the external data are given by (34). We note that the conditions (26) for the existence of a solution are satisfied. We seek the solution of the Eq. (13), with fi = 0 and gj = 0, in the form
ua ¼ xa FðrÞ;
u3 ¼ 0;
ua ¼ xa GðrÞ; u3 ¼ 0;
ð28Þ
where F and G are unknown functions. It follows from (28) that we have
ua;b ¼ dab F þ xa xb r 1 F 0 ; uq;q ¼ 2F þ rF 0 ;
ð29Þ
uq;qa ¼ Dua ¼ xa ðF 00 þ 3r 1 F 0 Þ; 0
where f = df/dr. By (9) and (29) we get
eab ¼ eba ¼ ua;b ; eq3 ¼ e3qb xb G ¼ e3q ; e33 ¼ 0; j33 ¼ 2G þ rG0 ; jab ¼ jba ¼ ua;b ;
ð30Þ
ja3 ¼ 0; j3i ¼ 0: In view of (29), the equilibrium Eq. (13), in the absence of body loads become
ðk þ 2l þ jÞðF 00 þ 3r 1 F 0 Þ þ ðC 1 þ C 2 þ C 3 ÞðG00 þ 3r 1 G0 Þ ¼ 0; ðC 1 þ C 2 þ C 3 ÞðF 00 þ 3r1 F 0 Þ þ ða þ b þ cÞðG00 þ 3r 1 G0 s2 GÞ ¼ 0;
ð31Þ
where
s2 ¼
2j : aþbþc
ð32Þ
It follows from (31) that the function G must satisfy the equation
G00 þ 3r 1 G0 q2 G ¼ 0;
ð33Þ
where
q2 ¼ 2jðk þ 2l þ jÞQ 1 ; Q ¼ ðk þ 2l þ jÞða þ b þ cÞ ðC 1 þ C 2 þ C 3 Þ2 : 2
ð34Þ
2
We note that the relations (5) imply that s > 0, Q > 0 and q > 0. Let us introduce the function U by
U ¼ rG:
ð35Þ
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D. Iesßan / International Journal of Engineering Science 49 (2011) 411–419
Then, the Eq. (33) becomes
1 r
U00 þ U0
1 2 þ q U ¼ 0: r2
ð36Þ
In what follows we denote by In and Kn the modified Bessel functions of order n. Then, the general solution of the equation (36) is
U ¼ B1 I1 ðqrÞ þ B2 K 1 ðqrÞ; where B1 and B2 are arbitrary constants. Thus, we find that
G ¼ r 1 ½B1 I1 ðqrÞ þ B2 K 1 ðqrÞ:
ð37Þ
It follows from (31) that the function F satisfies the equation 2
d
dr
þ 2
3 d r dr
!
2
d
dr
þ 2
! 3 d q2 F ¼ 0: r dr
ð38Þ
Clearly, we can write F = F1 + F2 where the functions Fa satisfy the equations
F 001 þ 3r1 F 01 ¼ 0;
F 002 þ 3r 1 F 02 q2 F 2 ¼ 0:
Solving these equations, we find that the function F has the form
F ¼ A1 þ A2 r 2 þ r1 ½D1 I1 ðqrÞ þ D2 K 1 ðqrÞ;
ð39Þ
where Aa and Da are arbitrary constants. We note that the functions F and G, given by (37) and (39), must satisfy the differential system of second order (31). It is easy to see that F and G satisfy the Eq. (31) if we have
D1 ¼ rB1 ;
D2 ¼ rB2 ;
ð40Þ
where
r ¼ ðC 1 þ C 2 þ C 3 Þðk þ 2l þ jÞ1 :
ð41Þ
We conclude that the equilibrium equations are satisfied if the functions F and G are given by
F ¼ A1 þ A2 r 2 rr 1 ½B1 I1 ðqrÞ þ B2 K 1 ðqrÞ; G ¼ r 1 ½B1 I1 ðqrÞ þ B2 K 1 ðqrÞ:
ð42Þ
Let us investigate the boundary conditions. In view of (12) and (27) we obtain the following boundary conditions
t ab na ¼ p1 nb ;
ta3 na ¼ 0; mai na ¼ 0
t ab na ¼ p2 nb ; t a3 na ¼ 0; mai na ¼ 0
on r ¼ a1 ; on r ¼ a2 :
ð43Þ
It follows from (30) and the constitutive Eq. (10) that
t ab ¼ kð2F þ rF 0 Þdab þ ð2l þ jÞðFdab þ xa xb r1 F 0 Þ þ C 1 ð2G þ rG0 Þdab þ ðC 2 þ C 3 ÞðGdab þ xa xb r1 G0 Þ; t a3 ¼ t 3a ¼ je3ab xb G; mmg ¼ að2G þ rG0 Þdmg þ ðb þ cÞðGdmg þ xm xg r1 G0 Þ þ C 1 ð2F þ rF 0 Þ þ ðC 2 þ C 3 ÞðFdmg þ xm xg r 1 F 0 Þ
ð44Þ
mm3 ¼ m3m ¼ ðC 3 C 2 Þe3mb xb G: On the circle r = a1 we have na = xa/a1, and on the circle r = a2 we have na = xa/a2. The boundary conditions (50) can be written as
t ab xa ¼ p1 xb ; ta3 xa ¼ 0; mai xa ¼ 0 on r ¼ a1 ; t ab xa ¼ p2 xb ; ta3 xa ¼ 0; mai xa ¼ 0 on r ¼ a2 :
ð45Þ
In view of (44), we get
t ab xa ¼ xb ½ð2k þ 2l þ jÞF þ ðk þ 2l þ jÞrF 0 þ ð2C 1 þ C 2 þ C 3 ÞG þ ðC 1 þ C 2 þ C 3 ÞrG0 ; t a3 xa ¼ je3ab xa xb G ¼ 0; mab xa ¼ xb ½ð2a þ b þ cÞG þ ða þ b þ cÞrG0 þ ð2C 1 þ C 2 þ C 3 ÞF þ ðC 1 þ C 2 þ C 3 ÞrF 0 ; ma3 xa ¼ ðC 3 C 2 Þe3ab xa xb G ¼ 0: With the help of (46) the boundary conditions (45) reduce to
ð46Þ
D. Iesßan / International Journal of Engineering Science 49 (2011) 411–419
417
ð2k þ 2l þ jÞFða1 Þ þ ðk þ 2l þ jÞa1 F 0 ða1 Þ þ ð2C 1 þ C 2 þ C 3 ÞGða1 Þ þ ðC 1 þ C 2 þ C 3 Þa1 G0 ða1 Þ ¼ p1 ; ð2a þ b þ cÞGða1 Þ þ ða þ b þ cÞa1 G0 ða1 Þ þ ð2C 1 þ C 2 þ C 3 ÞFða1 Þ þ ðC 1 þ C 2 þ C 3 Þa1 F 0 ða1 Þ ¼ 0; ð2k þ 2l þ jÞFða2 Þ þ ðk þ 2l þ jÞa2 F 0 ða2 Þ þ ð2C 1 þ C 2 þ C 3 ÞGða2 Þ þ ðC 1 þ C 2 þ C 3 Þa2 G0 ða2 Þ ¼ p2 ;
ð47Þ
ð2a þ b þ cÞGða2 Þ þ ða þ b þ cÞa2 G0 ða2 Þ þ ð2C 1 þ C 2 þ C 3 ÞFða2 Þ þ ðC 1 þ C 2 þ C 3 Þa2 F 0 ða2 Þ ¼ 0; We introduce the notations
K1 ðrÞ ¼ ðC 1 rkÞr 1 I1 ðqrÞ þ ½C 1 þ C 2 þ C 3 rðk þ 2l þ jÞI01 ðqrÞ; K2 ðrÞ ¼ ðC 1 rkÞr 1 K 1 ðqrÞ þ ½C 1 þ C 2 þ C 3 rðk þ 2l þ jÞK 01 ðqrÞ; N1 ðrÞ ¼ ða rC 1 Þr 1 I1 ðqrÞ þ ½a þ b þ c rðC 1 þ C 2 þ C 3 ÞI01 ðqrÞ;
ð48Þ
N2 ðrÞ ¼ ða rC 1 Þr 1 K 1 ðqrÞ þ ½a þ b þ c rðC 1 þ C 2 þ C 3 ÞK 01 ðqrÞ: By using (42), the conditions (47) can be written in the form
ð2k þ 2l þ jÞA1 ð2l þ jÞa2 1 A2 þ l11 B1 þ l12 B2 ¼ p1 ; ð2C 1 þ C 2 þ C 3 ÞA1 ðC 2 þ C 3 Þa2 1 A2 þ l21 B1 þ l22 B2 ¼ 0; ð2k þ 2l þ jÞA1 ð2l þ jÞa2 2 A2 þ l31 B1 þ l32 B2 ¼ p2 ;
ð49Þ
ð2C 1 þ C 2 þ C 3 ÞA1 ðC 2 þ C 3 Þa2 2 A2 þ l41 B1 þ l42 B2 ¼ 0; where
l11 ¼ K1 ða1 Þ; l12 ¼ K2 ða1 Þ; l21 ¼ N1 ða1 Þ; l22 ¼ N2 ða1 Þ; l31 ¼ K1 ða2 Þ; l32 ¼ K2 ða2 Þ; l41 ¼ N1 ða2 Þ; l42 ¼ N2 ða2 Þ:
ð50Þ
From (49) we find the constants Aa and Ba, 2 2 2 ð2k þ 2l þ jÞða2 1 a2 ÞA1 ¼ p2 a1 p1 a2 b1q Bq ; 2 ð2l þ jÞða2 1 a2 ÞA2 ¼ p2 p1 b2q Bq ;
B1 ¼ ðH22 T 1
1 H12 T 2 Þd1 ;
B2 ¼ ðH11 T 2
ð51Þ
1 H21 T 1 Þd1 ;
where 2 b1q ¼ l3q a2 1 l1q a2 ; b2q ¼ l3q l1q ;
H1q ¼ d2 l2q ð2C 1 þ C 2 þ C 3 Þð2l þ jÞb1q þ ðC 2 þ C 3 Þð2k þ 2l þ jÞa2 1 b2q ; H2q ¼ d2 l4q ð2C 1 þ C 2 þ C 3 Þð2l þ jÞb1q þ ðC 2 þ C 3 Þð2k þ 2l þ jÞa2 2 b2q ; 2 d2 ¼ ð2k þ 2l þ jÞð2l þ jÞða1 1 a2 Þ; d1 ¼ H11 H 22 H 12 H21 ; 2 2 T 1 ¼ ð2C 1 þ C 2 þ C 3 Þð2l þ jÞðp1 a2 2 p2 a1 Þ þ ðC 2 þ C 3 Þð2k þ 2l þ jÞa1 ðp2 p1 Þ; 2 2 T 2 ¼ ð2C 1 þ C 2 þ C 3 Þð2l þ jÞðp1 a2 2 p2 a1 Þ þ ðC 2 þ C 3 Þð2k þ 2l þ jÞa2 ðp2 p1 Þ:
The solution of the pressure vessel problem is given by (28), (42) and (51). In the case of achiral materials we have Ck = 0, and from (41), (48) and (50) we obtain r = 0, l1a = 0 and l3a = 0. Now the system (49) has the solution
A1 ¼ ðp2 a22 p1 a21 Þ½ða22 a21 Þð2k þ 2l þ jÞ1 ; A2 ¼ ðp2 p1 Þa21 a22 ½ða22 a21 Þð2l þ jÞ1 ; Ba ¼ 0: Thus, if the tube is made of an isotropic achiral Cosserat elastic material, then the microrotation vector field vanishes. The salient feature of the solution (28) is that, in the absence of the body couples and surface moments, a pressure acting on the surface of a chiral tube produces a microrotation of the material particles. This is a new chiral effect. 5. Conclusions The mechanical properties of chiral materials are not invariant with respect to inversions. In the classical elasticity the elasticity tensor is unchanged under an inversion. It is therefore common to use Cosserat elasticity to introduce chirality in the context of elasticity. In this paper the pressure vessel problem is solved in the context of the theory of chiral Cosserat elasticity. This work is motivated by real problems. The Cosserat elasticity was used to study the deformation of chiral nanotubes and the mechanical behavior of bones. The intended applications of the solution presented in this paper are to nanotubes and bone mechanics. In the femur bone implants an internal pressure is applied on the interface. The solution of the pressure vessel problem can also be used to study the mechanical behavior of tubes made of chiral composites.
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To investigate the pressure vessel problem in the context of the theory of chiral Cosserat elasticity, the state of generalized plane strain was introduced. Conditions have been found under which the generalized plane strain problem is possible. The pressure vessel problem has been investigated as a special generalized plane strain of a chiral elastic tube. It is shown that, a pressure acting on the surface of an isotropic chiral elastic tube, in contrast with the case of achiral materials, produces a microrotation of the material particles. It follows from the solution presented in Section 4 that the radial displacement for a chiral tube is given by
1 u¼ d2
(
1 1 2 þ rI1 ðqrÞ ð2l þ jÞðp2 a2 1 p1 a2 Þr þ ð2k þ 2l þ jÞðp2 p1 Þ B1 ½ð2l þ jÞb11 r þ ð2k þ 2l þ jÞb21 r r ) 1 B2 ½ð2l þ jÞb12 r þ ð2k þ 2l þ jÞb22 þ rK 1 ðqrÞ : r
ð52Þ
In isotropic linear Cosserat elasticity, chirality introduces three additional parameters for a total of nine constitutive coefficients. The chiral behavior of the body is controlled by the material parameters C1, C2 and C3 which appear in the constitutive equations. The achiral isotropic Cosserat elastic solid requires six elastic constants, k, l, a, b, c and j, for its description. In the case of achiral materials the parameters Ck are equal to zero and we find from (52) that the radial displacement has the form
u ¼
1 p2 a22 p1 a21 ðp2 p1 Þa21 a22 : r þ ð2l þ jÞr a22 a21 2k þ 2l þ j
In the theory of classical elasticity an isotropic solid has two constitutive coefficients, the Lamé moduli k and l. In this theory the radial displacement is given by
u0 ¼ ½ðp2 a22 p1 a21 Þrðk þ lÞ1 þ ðp2 p1 Þa21 a22 l1 =½2ða22 a21 Þ: The solution of the pressure vessel problem for a chiral elastic tube yields the following radial stress
S ¼ ð2k þ 2l þ jÞA1 ð2l þ jÞr 2 A2 þ B1 K1 ðrÞ þ B2 K2 ðrÞ;
ð53Þ
where A1, A2, B1 and B2 are given by (51), and the functions K1 and K2 are defined in (48). It follows from (53) that the radial stress in a tube made of an achiral material is given by
S ¼
1 ½p a2 p1 a21 ðp2 p1 Þa21 a22 r2 : a22 a21 2 2
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