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Математика. Физика
УДК 539.3
LARGE RIGID-BODY MOTIONS AND STRAIN-DISPLACEMENT RELATIONSHIPS OF THE LAYER-WISE SHELL THEORY G.M. Kulikov
Department of Applied Mathematics and Mechanics, TSTU
Key Words and Phrases: finite rotations; large rigid-body motions; layer-wise shell theory.
Abstract: It is considered the relationships for the Green-Lagrange strain tensor in curvilinear orthogonal coordinates exactly representing arbitrarily large rigid-body motions. This consideration is based on the kinematic Timoshenko hypothesis adopted for every layer of the shell (piece-wise linear approximation). As unknown functions 3(N + 1) displacements of the face surfaces of layers are chosen, where N is a number of layers. The deduced relationships may be used in the finite element method for constructing new and efficient elements for multilayered shells.
1 Introduction
One of the main requirements of a finite element that is intended for the general non-linear analysis of shells is that it must lead to strain-free modes for large rigid-body motions. The adequate representation of large rigid-body motions is a necessary condition if a non-linear element is to have the good accuracy and convergence properties. Therefore, when an inconsistent non-linear shell theory is used to construct any finite element, erroneous straining modes under arbitrarily large rigid-body motions may be appeared. This problem has been only studied for the classical Kirchhoff-Love shell theory [1-3] and Timoshenko-Mindlin shell theory [4-8]. Herein, the more general study on the basis of the finite deformation layer-wise Timoshenko-Mindlin (LTM) shell theory taking into account the transverse normal deformation response is considered. As unknown functions 3(N + 1) displacements of the face surfaces of layers are selected [9], where N is a number of layers. Such choice of unknowns allowed to deduce non-linear strain-displacement relationships of the LTM shell theory, which are completely free for arbitrarily large rigid-body motions.
2 Strain-displacement relationships of finite deformation elasticity theory
Let us consider a shell of the uniform thickness h. The shell may be defined as a
three-dimensional body of volume V bounded by two bounding surfaces S- and S +,
located at the distances 8- and 8+ measured with respect to the reference surface S, and the edge boundary surface Q that is perpendicular to the reference surface (Fig. 1). Let the reference surface S be referred to the orthogonal curvilinear coordinate system a1 and a2 , which coincides with the lines of principal curvatures of its surface; e1 and
e 2 denote the tangent unit vectors to the lines of principal curvatures. The a 3 axis is oriented along the unit vector e 3 normal to the reference surface.
The curvilinear components of the Green-Lagrange strain tensor for the finite deformation analysis can be written in a vector form as
f . \ , \
2 1
2s ij =-------u
J H.
e J +-
1
2 H
H
ei +
1
u uiei, Hа Аа (l + kаа3 ),
2 Hi
H 3 = 1,
(1)
where u is the displacement vector; u. (, а 2, а3) are the components of this vector,
which are measured in accordance with the total Lagrangian formulation from the initial configuration (Fig. 1); Аа and k(X are the Lame coefficients and principal curvatures of the reference surface; H а are the Lame coefficients of any surface parallel to the reference surface. Here and in the following developments the abbreviation ( ) . implies the partial derivative with respect to the coordinate а., and indices i, J take the values 1, 2 and 3 while Greek indices а, в, у take the values 1 and 2.
An arbitrarily large rigid-body motion can be defined as
uR = А + (Ф - E)R, (2)
А = ХЛІei, R = r + а3^ i
where R is the position vector of any point of the shell; r (а1, а 2) is the position vector of any point of the reference surface; А is the constant displacement (translation) vector; E is the identity matrix; Ф is the orthogonal rotation matrix defined as
cosGcosу cosGsinу - sin G
Ф = - cos фsin у + sin фsin Gcos у cos фcos у + sin фsin Gsin у sin фcos G , (3)
sin фsin у + cos фsin Gcos у - sin фcos у + cos фsin Gsin у cos фcos 0
where 9, 0 and y are the modified Euler angles [10] that characterize a rotation of the shell around point O (Fig. 1).
The derivatives of the translation and position vectors with respect to the coordinate aj can be written as
A,- = 0, Rj = Hjej. (4)
Taking into account (2)-(4), one obtains the following expression for derivatives:
-1 uRj = Oe- - e-. (5)
Hi
It can be verified by using (5) that curvilinear components of the Green-Lagrange strain tensor (1) are all zero in a general large rigid-body motion, i.e.,
2eR =(Oe - )(Oe j)- e -e j = °. (6)
This conclusion is true because an orthogonal transformation retains the scalar product of the vectors.
3 Strain-displacement relationships of finite deformation LTM shell theory
Let us 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 3-D body of volume Vk bounded by two surfaces Sk-1 and Sk, located at the distances 8k-i and 8k measured with respect to the reference surface S, and the edge boundary surface Qk (Fig. 2). It is assumed that the bounding surfaces Sk-1 and Sk are continuous, sufficiently smooth and without any singularities. The constituent layers of
the shell are supposed to be rigidly joined, so that no slip on contact surfaces and no
Fig. 2 Geometry of the multilayered shell
separation of layers can occur. 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. The a3 axis is oriented along the normal direction. Here and
in the following developments index k = i, N.
The finite deformation LTM shell theory is based on the linear approximation of the displacements in the thickness direction of the kth layer [9]
u(k) = Nk-(a3) v(k-i) + N+ (a3) v(k),
(7)
A1') =
=1
v(1)e,
Nk (a3) = 7— (Sк -a3) > N+ (a3)= -1 (a3-Sk-1) hk hk
where v(1) are the displacement vectors of the face surfaces of the layers; v(1) (i, a2) are the components of these vectors; N±(a3) are the linear shape functions of the kth layer; l = k - i, k.
Substituting displacements (7) into the strain-displacement relationships (i) and taking into account formulas for the derivatives of the unit vector e3 with respect to the curvilinear coordinates [3]
1 ,
-----e3,a = kaea >
Aa
(8)
one may obtain the strain-displacement relationships of the finite deformation LTM theory of shells with thick layers
2p(k)a = 2Sap “
N-(3 )H~ v(a-1) + N>3 H v(a)
Ha Ha
ee +
1
n-(a 3 )H—v (p1+Nk"(a 3)
1
(k)
N- (a3)H v2 1} + N+(a3 )H v(a
Nk (a3 )Hr v(k 1} + Nk (a3 )H-V}
(9a)
tt (k) -1
^ = -TTP(k)ea + “7^ (3 + P(k))+(a3 -5k)—s
— a — a — a
ef3)a = pk fe3 + -p(k)], p(k) = —(v(k) - v(k-i)), vk = -(v(k-i) + v(k)), (9c)
V 2 ] hk 2
where — ak) = ^a(i + ka§k ) are the Lame coefficients of the middle surface of the kth layer; 5k = (5 k-i +5 k) /2 is the distance from the reference surface to the middle surface of the kth layer.
Replacing further the Lame coefficients — a in (9a) by their values on the bottom and top surfaces of the kth layer — a-i) = Aa(i+ka5k-i) and — a) = Aa(i + ka5k), and in (9b) by their values on the middle surface — a) of the kth layer, the strain-
,(k )a J33,a ’
(9b)
displacement relationships of the finite deformation LTM theory of shells with moderately thick layers are obtained
+ N_(a 3) HP» v
P
2r / hk\ I’ 2'
The strain-displacement relationships (10) are more convenient than (9) because they are completely free for arbitrarily large rigid-body motions. The proof of this fundamental fact follows from formulas for large rigid-body motions of the kth layer and their derivatives with respect to the curvilinear coordinates
Sk surfaces of the kth layer and l = k -1, k . It can be verified by using (11), (12) and the above property of the orthogonal transformation (3) that curvilinear components of the Green-Lagrange strain tensor (10) are all zero in an arbitrarily large rigid-body motion:
thickness according to the quadratic law. Taking into account that each layer of the shell is thin, this complication of the LTM shell theory would be unreasonable because of the minor significance of the quadratic terms in most engineering problems. Therefore, more attractive strain-displacement relationships of the finite deformation LTM theory of shells with thin layers can be written as
(12)
where R(1) = r + 8ie3 are the position vectors of points of the bottom Sk_ and top
28 3f* =(Oe3 )(®e3)- e3e3 = 0.
It should be mentioned that tangential strains 80p)6 are distributed over the layer
24* _ N- (“3)
H
(k-1)
v (a-1) eR +-^- v (k-i) e„ +-
e h (k-1) ,e a h(k-1)h (k-1) c
k-1) v (k-1)
+N+(a3)
A ^
1 (k) , 1 (k) . 1 (k) (k
--------v en +-----------v ~ e +--------------------v vV
h (k)v,a ep+h ek) vp ea+h ak) h ek)v ,a v ,p
(13a)
28 ^ = P (k)e a + ^T(k) (e3 + P (k) ) + ( _8k ^ 8 3^, (13b)
H a H a
83f = P(k) (e3 + -2P(k)'j, P(k) = —(v(k) - v(k-1)), v(k) = -2-(v(k-1) + v(k)). (13c)
V 2 j hk 2
These strain-displacement relationships are also invariant under all large rigid-body motions because
8 (k )cR = 0 bj - 0 .
It is apparent that deduced strain-displacement relationships (9), (10) and (13) satisfy the following coupling conditions:
8 a? (81 )=8 ak,is (8, )=8 » (8, )=£«,
8 a? (st )=8 a? (8k)=8 ®c (st )=e«,
where E^ are the tangential strains of the face surfaces of layers; E^ are the transverse shear strains of the middle surface of the kth layer and 1 = k -1, k . This statement is illustrated in Fig. 3.
Finally, we represent in a scalar form the strain-displacement relationships (13) that have a great importance for the finite element method
8akp)c = N-(a3)E(kp-1) + N+ (a3)E(kp), 8ak3)c = (a3)E£)- + N+ (a3^ , (14)
where
,(k)c _ E(k) - e(k) + n(k)
"33 _ E33 _ K33 'r Ч33 ;
E(l) _ e(l) +n(l) E(k)± _ e(k)± +n(k)± Eap _ eap+rlap, Ea3 _ ea3 + Па3 ’
(15)
(l) _ _L^ (l) 2e(l) _ — ro(l) +—ro(l)
’ 2e12 z(l)Ю1 +z21) “2 ’
'aa z (l) za
2e
(k)- _ a3
1 -
ka h
\
ank
2Z(k)
^Sa У
eak) -1
(k-1) 2e(k)+ _
(k )”a ’ a3
e(k) _ e(k) 33 3
1 + ka hk 2Z (k) •^ba У
eak) -
—0(k)
Z(k) a ! Sa
_■
1 laa
;(z (l))‘
(>. «)2+(«af))2+(0
in
2n<« ________1___(x «»«> +x (Vl> +0<«0 «)
^12 z(i)z(l) V 1 2 +л2 Ш1 +°1 2 )
Z1 Z 2
2ng- ^ P« (+P<k» ^-1) -e3k) 0ak-1)),
Ca
2nak3» +_4- (e ak»x ak >+P <*»^»-,
Ca
n(k) _1 П33 2
(e(k))2+(P2k))2+( p3k*)2
xa f) _ — v (la + Bv(f>+к v<l) ®a l) _ — v(f ) - в v( 1)
Aa ~ , va,a + Byvy +kav3 , wa _ , vY.a BYva ,
Aa
^y,a ^y^a
0 (l) L v (l) + k v (l) R (k) _ _1 (v (k) _ v (k-1))
0a _ , V3,a+kaVa , e _ hj[Vr vi У
Aa
Zal) _ 1 + ka51, zak) _ 1 + ka§k,
A A.
1^2
'Ay,a (y^a;l _ k -1, k).
Fig. 3 Distribution of tangential (a) and transverse shear (b) strains over the shell thickness
4 Conclusions
Principally new strain-displacement relationships (14), (15) of the finite deformation LTM shell theory are developed. These strain-displacement relationships are very attractive because they are objective, i.e., invariant under all large rigid-body motions. So, they can be used for the formulation of the efficient curved finite elements in local curvilinear coordinates for the multilayered shells undergoing finite deformations.
References
1. Cantin G. Strain displacement relationships for cylindrical shells // AIAA Journal. - 1968. - Vol. 6. - Pp. 1787-1788.
2. Dawe D.J. Rigid-body motions and strain-displacement equations of curved shell finite elements // Int. J. Mech. Science. - 1972. - Vol. 14. - Pp. 569-578.
3. Gol'denveiser A.L. Theory of elastic thin shells. - Pergamon Press, Oxford,
1961.
4. Kulikov G.M., Plotnikova S.V. Finite element formulation of straight composite beams undergoing finite rotations // Trans. Tambov State Tech. Univ. - 2001. -Vol. 7, No. 4. - Pp. 617-633.
5. Kulikov G.M., Plotnikova S.V. Efficient mixed Timoshenko-Mindlin shell elements // Int. J. Numer. Methods Engrg. - 2002. - Vol. 55, No. 10. - Pp. 1167-1183.
6. Kulikov G.M., Plotnikova S.V. Simple and effective elements based upon Timoshenko-Mindlin shell theory // Comp. Methods Appl. Mech. Engrg. - 2002. -Vol. 191, No. 11-12. - Pp. 1173-1187.
7. Kulikov G.M., Plotnikova S.V. Non-linear strain-displacement equations exactly representing large rigid body motions. Part I. Timoshenko-Mindlin shell theory // Comp. Methods Appl. Mech. Engrg. - 2003. - Vol. 192, No. 7-8. - Pp. 851-875.
8. Kulikov G.M., Plotnikova S.V. Finite deformation plate theory and large rigid body motions // Int. J. Non-Linear Mech. - 2004. - Vol. 39, No. 7. - Pp. 1093-1109.
9. Kulikov G.M. Non-linear analysis of multilayered shells under initial stress // Int. J. Non-Linear Mech. - 2001. - Vol. 36, No. 2. - Pp. 323-334.
10. Washizu K. Variational methods in elasticity and plasticity, 3rd edition. -Pergamon Press, Oxford, 1982.
Большие перемещения оболочки как твердого тела и деформационные соотношения многослойных оболочек
Г. М. Куликов
Кафедра «Прикладная математика и механика», ТГТУ
Ключевые слова и фразы: большие перемещения твердого тела; конечные повороты; многослойная оболочка.
Аннотация: Получены соотношения для тензора деформаций Грина-Лагранжа в криволинейных ортогональных координатах, точно представляющие произвольно большие перемещения оболочки как жесткого тела, на основе кинематической гипотезы Тимошенко, принятой для каждого слоя (гипотеза ломаной линии). В качестве искомых функций выбраны 3(Ж+1) перемещений лицевых поверхностей слоев, где N - число слоев. Выведенные соотношения могут быть с успехом использованы в методе конечных элементов при построении новых и эффективных элементов многослойных оболочек.
GroBe Htillenbewegungen als Hartkorper und Deformationskorrelationen der vielschichtigen Htillen
Zusammenfassung: Es sind die Korrelationen fur den Deformationstensor von Grin-Lagrange in den krummlinigen orthogonalen Koordinaten erhalten. Sie zeigen absichtlich groBe Hullenbewegungen als Hartkorper auf Grund der fur jede Schicht ub-lichen kynematischen Hypothese von Timoschenko (Hypothese der gebrochenen Linie). Als gesuchte Funktionen sind 3 (N + 1) der Bewegungen von AuBenseiten der Schich-ten gewahlt, wo N die Zahl der Schichten ist. Die erhaltene Korrelationen konnen in der Methode der Endelemente beim Aufbau von neuen und effektiven endlichen Elementen der vielschichtigen Hullen erfolgreich benutzt werden.
Grands transferts de l’enveloppe comme un corps solide et rapport de deformation des enveloppes a plusieurs couches
Resume: Sont regus les rapports pour le tenseur de deformation de Grin-Lagrange dans les coordonnees orthogonales curvilignes representant exactement les transferts arbitrairement grands de l’enveloppe du corps rigide a la base de l’hypothese cinematique de Timochenko prise pour chaque couche (hypothese de la ligne brisee). En qualite des fonctions recherchees on a choisi les 3(N + 1) transferts des surfaces des couches ou N est le nombre des couches. Les rapports regus peuvent etre utilises avec succes dans la methode des elements finis pour la construction de nouveaux elements finis efficaces des enveloppes a plusieurs couches.
