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Математика. Физика
УДК 539.3
ON THE USE OF 6-PARAMETER MULTILAYERED SHELL MODELS IN STRUCTURAL MECHANICS G.M. Kulikov, S.V. Plotnikova
Department of Applied Mathematics and Mechanics, TSTU
Key words and phrases: anisotropy; thickness locking; first-order multilayered shell theory.
Abstract: The new geometrically exact multilayered shell models are considered. These models are based on the objective strain-displacement relationships represented in the local curvilinear coordinates and, therefore, may be used for the formulation of effective curved multilayered shell elements. However, the practical use of such elements require the development of constitutive equations, in order to overcome Poisson thickness and volumetric locking phenomena. For this purpose three types of the material stiffness matrix are studied.
1 Introduction
One of the main requirements of a finite element that is intended for the general analysis of shells is that it must lead to strain-free modes for rigid-body motions. The adequate representation of rigid-body motions is a necessary condition if an element is to have good accuracy and convergence properties. Therefore, when an inconsistent shell theory is used to construct any finite element, erroneous straining modes under rigid-body motions may appear. This problem has been only studied for the Kirchhoff-Love shell theory [1-3] and Timoshenko-Mindlin shell theory [4, 5]. Herein, the more general study on the basis of the first-order multilayered shell theory [6] is considered. As unknown functions six displacements of the bottom and top surfaces of the shell are selected.
It is common knowledge that in some works developing the solid-shell concept [7-9] displacement vectors of the face surfaces are also used and represented in some global Cartesian basis in order to exactly describe rigid-body motions. But in our first-order shell theory selecting as unknowns the displacements of face surfaces of the shell has a principally another mechanical sense and allows us to formulate any curved shell elements on the basis of strain-displacement relationships that are objective, i.e., invariant under all rigid-body motions. In order to circumvent thickness locking the modified laminate stiffness matrix [10, 11] and simplified material stiffness matrices symmetric [5, 7, 8, 12] or non-symmetric [13, 14] corresponding to the generalized plane stress state are employed.
2 Problem formulation
Consider a shell built up in the general case by the arbitrary superposition across the wall thickness of N layers of uniform thickness hk. The kth layer may be defined as a 3D body of volume Vk bounded by two surfaces Sk-1 and Sk, located at the distances 8k-1 and 8k measured with respect to the reference surface S, and the edge boundary surface Qk (Fig. 1). The full edge boundary surface Q = Q1 + Q2 +... + QN is generated by the normals to the reference surface along the bounding curve r c S (with the arc length s) of this surface. It is also assumed that the bounding surfaces Sk-1 and Sk are continuous, sufficiently smooth and without any singularities. Let the reference surface S be referred to an orthogonal curvilinear coordinate system ai and a2, which coincides with the lines of principal curvatures of its surface; e1 and e2 are the tangent unit vectors to the lines of principal curvatures; Aa and ka are the Lame coefficients and principal curvatures of the reference surface. The a3 - axis is oriented along the unit vector e3 normal to the reference surface.
The constituent layers of the shell are supposed to be rigidly joined, so that no slip on contact surfaces and no separation of layers can occur. The material of each constituent layer is assumed to be linearly elastic, anisotropic, homogeneous or fiber reinforced, such that in each point there is a single surface of elastic symmetry parallel to the reference surface. Let p- and p+ be the intensities of the external loading acting on the bottom surface S- = S0 and top surface S + = SN in the ai coordinate directions, respectively; q(k) = qVk) v + q( k )t + q3k) e3 are the external loading vector acting on the
edge boundary surface Qk, where qVk), q(k) and q33k) are the components of its vector in the v, t and a3 directions; v and t are the normal and tangential unit vectors to the bounding curve r. Here and in the following developments the index k = 1, N
identifies the belonging of any quantity to the kth layer; the abbreviation (),a implies the partial derivatives with respect to the coordinate a1 and a2 ; indices i, j, l, m take the values 1, 2 and 3; Greek indices a, p, y, 8 take the values 1 and 2.
The first-order multilayered shell theory is based on the linear approximation of displacements in the thickness direction [13, 15]
where u is the displacement vector; ui (a1, a2 , a3) are the components of this vec-
thickness of the shell. It is important that displacement vectors (1b) are represented in the local reference surface frame ei that allows one to reduce the costly numerical integration by deriving the stiffness matrix.
Substituting displacements (1a) into a vector form of the 3D strain-displacement relationships [5] and replacing Lame coefficients by their values on the bottom and top
surfaces Aa and middle surface Aa in corresponding expressions for the in-plane and transverse shear components, the following equations are obtained
Note that equations for the transverse shear strains (2) and (3b) differ from similar equations [5] and are more convenient for the finite element implementation. Strain-displacement relationships (2) and (3) are very attractive because they are objective, i.e., invariant under rigid-body motions. This may be readily proved by using a technique [5].
3 Shell kinematics
u = N (a3 )v + N + (a3 )v+ ,
(1a)
(1b)
N~ (a3) = h (5+ -“3), N + (a3) = h(a3 - S'),
(1c)
tor; v± are the displacement vectors of surfaces S± ; v± (a1, a2) are the components of these vectors; N± (a3) are the linear through-the-thickness shape functions; h is the
eai = N (a3 ) eai + N + (a3 ) 4 , e33 = ^ (2)
where eap and ea 3 are the in-plane and transverse shear components of the strain tensor of face surfaces S ± defined by
(2)
(3b)
P = h (v+- v )’ A±= ^a^a ’ A“= A“Z“ ’ Z“= 1 + kaS± , Za= 1 + kaS, S = ^2(S-+S+)
(3c)
4 Hu-Washizu variational equation
The first-order multilayered shell theory developed is based on the assumed approximations of displacements (1) and displacement-dependent strains (2) in the thickness direction. Additionally, one should adopt the similar approximation for the assumed displacement-independent strains
= n~ (аэ )E- + n+ (a3 )E+
pAS = E S33 = E33-
(4)
Substituting approximations (1), (2) and (4) into the Hu-Washizu mixed variational principle [16] and accounting for that metrics of all surfaces parallel to the reference surface are identical and equal to the metric of the middle surface, one can derive
jj[(H - DE)T 5E + (E - e)T 5H - HT5e + PT5v
A1 A2da^a 2 +
+ ^НГ 5vp (l + kn sjds = 0. Г
(5)
Here, matrix notations are introduced
D
D00 M111 D-1 D00 M122 D01 M122 D00 1112 D01 1112 0 0 0 0 D1133
D01 M111 DU11 D01 M122 DU22 D01 1112 -^112 0 0 0 0 D+133
D00 M211 D01 M211 D00 2222 D2222 D00 2212 D2212 0 0 0 0 D2233
D01 M211 ^2>211 D01 2222 D2222 D01 2212 D2212 0 0 0 0 D+233
D00 M211 D01 M211 D00 M222 D1222 D00 1212 D01 1212 0 0 0 0 D1233
D01 1211 D01 1222 D222 D01 1212 —1^12 0 0 0 0 D+233
0 0 0 0 0 0 D00 1313 D01 1313 D00 1323 D1323 0
0 0 0 0 0 0 D01 1313 D113113 D01 1323 D113123 0
0 0 0 0 0 0 D00 2313 D2313 D00 2323 D01 2323 0
0 0 0 0 0 0 D01 2313 D11 d2313 D01 2323 «11 d2323 0
—311 ^3+311 D3322 D+322 D3312 D+312 0 0 0 0 D3333
(6)
- + - + - + T - + - + - +
v = V1 V1 V2 V2 V3 V3 , vr = Vv Vv Vf Vf V3 V3
e =
e11 e1+1 e22 e+2 2 e12 2 e1+2 2 e13 2 e13 2 e23 2 e+3 e33
E =1 E-1 E+1 E-2 E+2 2 E-2 2 E+2 2 E- 2 E+ 2 E-3 2 E2+3 £33
H Г
H = |_H-1 Hi+1 H22 H+ H-2 H+2 H-3 H+3 H2-3 Я2+3 Я33
T
H- H+ H - H + H-3 H+3
P = [-P1 P1+ - P2 P2 - P3 P+]
where D is the constitutive stiffness matrix whose components are defined in the next section; v±, v± and v± are the components of displacement vectors of face surfaces in the coordinate system v , t and a3 (Fig. 1); kN is the normal curvature of the reference bounding curve r ; r is the bounding curve belonging to the middle surface S ; Hap,
Ha3 and H33 are the stress resultants; H±v , H±t and H±3 are the external load resultants defined as
sk sk
Ha =Z f CT0/} N±(a3 )da3, H33 =Z f CT33)da3, (7)
5k-1 k
k-1
H±£E =Z f ?®k)N±(a3 )da3 (s = v, t and3).
k sk-1
Mixed variational equation (5) may be used for constructing non-conventional assumed stress-strain four-node curved shell elements.
5 Constitutive equations
In this section four types of the constitutive equations are discussed. We consider first an orthotropic ply and then study the more general case of monoclinic symmetry.
5.1 Complete constitutive equations Consider the kth orthotropic layer of the shell and denote its axes of symmetry as
’, a(k )and a3 2 J
Hooke’s law will be
a(k), a2k )and a3. In these axes of symmetry the equations of the complete 3D
(k) (k)
CT( k) v21 CT( k) v31 ст(к)
(8)
1 1 k’ k’ 2'2' k’ 33
M 2 3
( k ) ( k )
s2,2, — V*) + _L_^
E(k) 1 1 E( ’ 2 2 E( ’
^1 ^2 -^3
2B1,2, — _±_v(k2), 281-3 — —a(,k3), 2s2-3 —-^-v%.
1 2 G (k) 1 2 ’ 1 3 G( k) 1 3’ 2 3 G( k) 2 3'
G12 G13 G23
v(k) v(k) ,
„ - V13 J. k) V 23 (k) , 1 (k)
S33 —----■—v , ,------------■—v , , +-----—,
E( k) 1 1 E( k) 2 2 E( k)
^1 2 ^3
where v( k1), V,) , v33) and v( f’, v( k3), vf are the normal and shear components
of the stress tensor in the a(, ), a^, ’ and a3 coordinate system; E| ’, e2 ’ and
E,(k) are the elastic moduli; G^’, G^’ and g23’ are the shear moduli; v j ’ are Poisson's ratios. From reasons of symmetry, we have
vJ ’ E j ’ — vj ’ e( k ’ for i * j.
lJ J J1 1 J
In coordinate directions a1, a2 and a3 , when a case of monoclinic symmetry is realized, the equations of the 3D Hooke’s law can be represented in the more general form
ej=z j vim), (9)
l ,m
where Ajm are the components of the material tensor of the kth layer depending on
engineering constants E( k ’, gJ ’ and vj ’ whose expressions can be found in many
books (see e.g. [16]) and are not displayed here.
Solving first three equations (9) for the in-plane stresses, one can find
(k) = vQ(k) p _M(k) ~(k) (10)
ap = ZQapy5PyS Map33CT33 ’ (10)
Y ,5
where
M(k) = z Q(k) A(k) (11)
^ap33_Zj ^apY^YS 33’ (11)
Y ,5
Q(,) = _L( A(,) A(,) _ A(,) A(,) \ Q(,) = wA(,) A(,) _ A(,) A(,) \
Ak V2222 1212 ^2.212^1222 ]’ *1122 a, V 1112 1222 ^1122^1212 ^
Q(k) = _L/A(k) A(k) _ A(k) A(k) \ Q(k) = W A(k) A(k) _ A(k) A(k) \
2 A, V 1122 2212 1112 2222 y ^2222 a, I 1111 1212 HI2 12H/’
Q(k) = _L(Ak) A(k) _ A(k) A(k) \ Q(k) = /',(k) A(k) _ A(k) A(k) \
^2212 2A, V 1112 2211 WW 22\2y *1212 4a, \ HH 2222 1122^*2211 ^
A = A(k’ (A(k’ A(k’ _A(k’ A(k’ )+ A(k’ (A(k’ Ak’ _A(k’ A(k’ ) +
^k - 1211y 1122 2212 1112 2222/ 1222 y 1112^211 "1111^*2212^
+Ak) (A(k) A(k) _ A(k) A(k) )
"1212 ^"1111^*2222 1122 2211 y
Substituting in-plane stresses (10) in the last equation (9) and solving for the transverse normal stress, one obtains
43) Z ^33 -)5Py5 + (12)
Y ,5
where
C( k) =- 1 M( k) C( k) = 1 (13)
33ap . Г33 ap’ ^3333 . ’
Ak Ak
m33ap = ZeYSdpA33kYs , Ak=-ZA33YsmYS33 + A33k3)3 •
Y,5 Y,5
By using the transverse normal stress (12) into formula (10) yields
^ = ZCapy5SYS + Cakp)33S33 , (14)
Y ,5
where
C( k) = Q( k) + 1 M( k) M( k) C( k) =- 1 M( k) (15)
apY^^apY^Ak ^ap33^33y5 ’ ap33 _ д ^ap33 •
Finally, one can derive from Hooke’s law (9) the remaining equations for the transverse shear stresses as
(16)
where
Г (k) = — A(k) 4313 _ j 2323 ; dk
Г(k) =- _L A(k)
4323 _ j 4323 ; dk
Г(k) = A(k)
°2323 4313 ;
dk
d, = 4 (A(k) A(k) - A(k) A(k) )
“k ^ 1313 2323 1323 2313) '
(17)
Unfortunately, such shell formulation on the basis of the complete 3D constitutive law (12)-(17) is deficient because so-called thickness locking [9, 17] can occur. This phenomenon occurs in bending dominated shell problems when Poisson’s ratios are not equal to zero. In order to avoid thickness locking the effective remedies may be used.
5.2 Modified constitutive equations
It is well-known that in the first-order multilayered shell theory the constitutive equations for transverse stresses are not satisfied pointwise [13] but may be fulfilled in an integral sense. In particular, for the transverse normal component the following integral equation [13] should be adopted
Sk
У J
k Sk_1
CT
(k) 33
Zn( k) n( k)
Г33 y5SyS С3333833
Y ,S
d a3 = 0.
Taking into account relations (2) and introducing notations
Sk Sk
^( k) d a3
H(k) = 33
CT
33
nk = J N +(a3 )da3 :
(18)
(19)
one can rewrite equation (18) as follows
У
H
(k) _V Г( k)
33
Y ,S
33 Ys \nkeYb + nk elS
~hkC3333e33
= 0.
(20)
This allows us to assume that the transverse normal stress is independent on the thickness coordinate a3 that may be appreciated as a good remedy [10, 11] for overcoming the thickness locking phenomenon. So, following this idea, the transverse normal stress is assumed to be constant in the thickness direction for every layer of the shell
CT
(k) = _L H( k) = in e
h H33 h У Г33 yS lnkeYS
k y ,S
< k)
33
h
k
+ n+ e+5 ) + ^333}3e33 .
(21)
Substituting further thickness stress (21) in Eq. (10) and integrating these modified equations together with remaining constitutive equations (16) and (21) across the shell thickness with account for relations (2), (6), (7) and (13), one derives
H = De.
(22)
k
Here, D denotes a constitutive stiffness matrix introduced in Section 4 whose components are defined as
( p q (k) (k) ^
pqQ(k ) + npnq ^33^5
n QaeY5 hk Л
-1+ _ V 1 + (k) ГЧ+ _ V 1 + (k)
ap 33 = у. nk ^ap33’ D33 ap = y. nk ^33a|
k Л k k Л k
^3 =Уnpqr„k3> 3, 0,333 =E7Lhk,
k k k
5k
p = J [N_(a3 )] 2_P_q [N+(a3 )] P+q da3
°k-1
where instead of (19) the more suitable notations n° = n- and n^ = n+ have been used,
and throughout this section superscripts p, q take the values 0 and 1.
It should be mentioned that this modified constitutive law is quite efficient for most engineering problems. But for incompressible or nearly incompressible materials [9] some difficulties leading to volumetric locking can occur.
5.3 Reduced constitutive equations
In order to circumvent simultaneously thickness and volumetric locking, we invoke a standard engineering assumption ct3^) . This implies that coefficients
Axp33 should be formally set to zero in equations of Hooke’s law (9) for the in-plane strains. At the same time the last equation for the thickness strain is left unchanged, i.e., Alap ^ 0 . Allowing for this assumption and using a technique described in Section 5.1, one arrives at constitutive equations (10), (12) and (16), where
1^33 = 0 and A k = A3333
in accordance with relations (11) and (13).
Integrating these reduced constitutive equations across the shell thickness (see Section 5.2), the following expressions for components of the constitutive stiffness matrix D are obtained:
Dpq =V nPqQ(k) D ± = 0 D ± = - V 1 n ±11(k)
apyfi / j nk Mapy5’ ^ap33 “ ^33ap _ Zj a(3) k M*33ap’
k k ^3333
^3=y»pqrse3. D3333 =y
,pq =y „Pqr<kj):
k k ^3333
The reduced constitutive law was proposed in works [13, 14] for overcoming thickness locking and showed a good performance in case of using the Timoshenko-Mindlin shell theory. It should be mentioned that this approach yields the non-symmetric material matrix and, as a result, more computational efforts have to be made.
k
5.4 Simplified constitutive equations
When a shell is undergone pure bending, one half of the shell body in the thickness direction is under tension and the other half is under compression, i.e., the thickness strain according to the complete 3D Hooke’s law would be zero due to the limitation of the linear displacement approximation (1). So, a shell will be in the plane strain state instead of the expected plane stress state. In order to circumvent these difficulties, the simplified constitutive stiffness matrix can be employed
This is due to the plane stress enforcement which is done by decoupling the transverse normal stress with all other stresses in the 3D Hooke’s law [5, 7, 8, 12], i.e., it is supposed that
It is apparent that the simplified constitutive law leads to the symmetric constitutive stiffness matrix D but it is slightly deficient for the thick anisotropic shells. So, allowing for a simplicity of such approach it may be recommended for an analysis of composite thin-walled structures.
The present research was supported by Russian Fund of Basic Research (Grant No. 04-01-00070).
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k
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Об использовании 6-параметрических моделей многослойных оболочек в механике конструкций
Г.М. Куликов, С.В. Плотникова
Кафедра «Прикладная математика и механика», ТГТУ
Ключевые слова и фразы: анизотропия; заклинивание по толщине; теория многослойных оболочек первого порядка.
Аннотация: Рассмотрены новые геометрически точные модели многослойных оболочек. Эти модели основаны на объективных деформационных соотношениях, представленных в локальных криволинейных координатах, и поэтому могут быть использованы для построения эффективных криволинейных элементов многослойных оболочек. Однако практическое использование таких элементов требует развития соотношений упругости, для того чтобы преодолеть Пу-ассоновское и объемное заклинивания. С этой целью изучаются и сравниваются три типа модифицированной материальной матрицы жесткости.
Uber die Benutzung der 6-parametrischen Modelle der vielschichtigen Mantel in der Konstruktionsmechanik
Zusammenfassung: Es sind neue geometrisch genaue Modelle der vielschichtigen Mantel untersucht. Diese Modelle sind auf die objektiven in lokalen krummlinigen Koordinaten vorgestellten Deformationsverhaltnisse gegrundet. Deshalb konnen sie fur die Konstruktion der effektiven krummlinigen Elemente der vielschichtigen Mantel verwendet werden. Doch fordert die praktische Anwendung solcher Elemente die Ent-wicklung der Elastizitatsverhaltnisse, um Puasson- und Raumfestklemmen zu uberwin-den. Zu diesem Zweck werden 3 Arten der abgeanderten materiellen Matrix der Steif-heit studiert und verglichen
Sur l’utilisation des modeles multicouches a 6-parametres pour les enveloppes dans la mecanique des constructions
Resume: Sont envisages de nouveaux modeles multicouches pour des enveloppes qui sont presis geometriquement. Ces modeles sont fondes sur les relations deformationnelles presentees dans les coordonnes locales cuvilignes, c’est pourquoi ils peuvent etre utilises pour la construction des elements efficaces des enveloppes multicouches. Toutefois l’emploi pratique de tels elements demande un developpement du rapport de l’elasticite pour surmonter le coingage Poisson et celui de volume. Dans ce but sont etudies et compares trois types de la matrice materielle modifiee de la rigidite.
