Drilling couples and refined constitutive equations in the resultant geometrically non-linear theory of elastic shells

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1 Submitted to International Journal of Solids and Strutures on Otober, 01 Revised on February 1, 014 Drilling ouples and refined onstitutive equations in the resultant geometrially non-linear theory of elasti shells W. Pietraszkiewiz 1*, V. Konopińska 1 Institute of Fluid-Flow Mahinery, PASi, ul. Fiszera 14, 80-1 Gdańsk, Poland Gdańsk University of Tehnology, Department of Strutural Mehanis and Bridge Strutures, ul. Narutowiza 11/1, Gdańsk, Poland Abstrat It is well known that distribution of displaements through the shell thikness is non-linear, in general. We introdue a modified polar deomposition of shell deformation gradient and a vetor of deviation from the linear displaement distribution. When strains are assumed to be small, this allows one to propose an expliit definition of the drilling ouples whih is proportional to tangential omponents of the deviation vetor. The onsistent seond approximation to the omplementary energy density of the geometrially non-linear theory of isotropi elasti shells is onstruted. From differentiation of the density we obtain the onsistently refined onstitutive equations for D surfae streth and bending measures. These equations are then inverted for stress resultants and stress ouples. The seond-order terms in these onstitutive equations take onsistent aount of influene of undeformed midsurfae urvatures. The drilling ouples are expliitly expressed by the stress ouples, undeformed midsurfae urvatures, and amplitudes of quadrati part of displaement distribution through the thikness. The drilling ouples are shown to be muh smaller than the stress ouples, and their influene on the stress and strain state of the shell is negligible. However, suh very small drilling ouples have to be admitted in non-linear analyses of irregular multi-shell strutures, eg. shells with branhes, intersetions, or tehnologial juntions. In suh shell problems six D ouple resultants are required to preserve the struture of the resultant shell theory at the juntions during entire deformation proess. Keywords: Drilling ouple, Resultant shell theory, Geometrial non-linearity, Constitutive equations, Complementary energy, Seond approximation 1 Introdution Drilling ouples M are two-dimensional (D) stress ouple fields whih appear in the resultant non-linear model of a shell. Suh shell model was initiated by Reissner (1974), developed in a number of papers for example by Chróśielewski et al. (199, 1997), Ibrahimbegović (1997), Eremeyev and Pietraszkiewiz (006, 011), Pietraszkiewiz (011), Birsan and N (01), and summarized in monographs by Libai and Simmonds (198,1998), Chróśielewski et al. (004), and Eremeyev and Zubov *Corresponding author. Tel (+48) addresses: pietrasz@imp.gda.pl (W. Pietraszkiewiz) viokonop@pg.gda.pl (V. Konopińska)

2 (008), where further referenes are given. The expliit original definition of M proposed in setion of this report reveals that the drilling ouples are generated by nonlinear part of tangential displaement distribution through the shell thikness. This is the reason why M do not appear in most popular non-linear shell models based on kinemati onstraints material fibres, whih are normal to the undeformed shell base surfae, remain straight during shell deformation or their equivalents as well as in the Cosserat type models with one deformable diretor, see for example Naghdi (197), Pietraszkiewiz (1979, 1989), Altenbah and Zhilin (1988), Simo and Fox (1989), Rubin (000), Bishoff et al. (004), Antman (005), or Wiśniewski (010). Also in all lassial linear models of elasti shells the resultant D stress ouple vetor does not have the normal (drilling) omponent by definition, due to identifiation of deformed and undeformed shell geometries, see for example Love (197), Gol denveizer (1961), Naghdi (196), Green and Zerna (1968), or Başar and Krätzig (001). In the non-linear resultant D shell model the loal equilibrium equations are exat impliation of the through-the-thikness integration of D equilibrium equations of nonlinear ontinuum mehanis. Then the D virtual work identity allows one to onstrut uniquely the D shell kinematis onsisting of the translation vetor and rotation tensor fields (six independent omponents) as well as the orresponding twelve D strain measures work-onjugate to the twelve D resultant stress measures, all defined on the shell base surfae. The resultant shell model naturally inludes three parameters of finite rotation as independent field variables and two drilling stress ouples with orresponding two work-onjugate drilling bendings. All these fields beome neessary in analyses of irregular shells with folds, branhings and intersetions (Chróśielewski et al. 1997, Konopińska and Pietraszkiewiz 007), when onneting shell elements between themselves (Pietraszkiewiz and Konopińska 011) and with beams, olumns and stiffeners, as well as in two-dimensional formulation of singular phenomena suh as phase transition (Eremeyev and Pietraszkiewiz 004, 011), rak propagation, disloations (Eremeyev and Zubov 008), wave motion, et. Yet, in almost all theoretial papers and numerial finite-element analyses of geometrially non-linear shell problems the simplest D onstitutive equations of the lassial linear theory of plates of Reissner (1944) type with only modest extensions have been used. Moreover, the onstitutive equations for M are either proposed without derivation in the form analogous to that for shear stress resultants Q only with bending

3 stiffness D and different orreting fator moments also denoted by M or M t, or are derived as for higher-order stress whih meaning is different from the one of drilling ouples as we understand them. In fat, we are not aware of any expliitly derived onstitutive equations for the drilling ouples M available in the literature. In this paper we propose new expliit definition (16) of the drilling ouples for shells undergoing small strains. It reveals that the drilling ouples appear as a result of through-the-thikness integration of tangential stresses ross-multiplied by non-linear part of displaement distribution in the shell spae. This expliit result beomes possible when we apply after Pietraszkiewiz et al. (006) the modified polar deomposition (10) of the shell deformation gradient and isolate in (9) the intrinsi deformation vetor whih desribes the non-linear part of displaement distribution through the thikness. In the resultant shell model D strain measures are defined only on the base surfae, without any relation to D strain measures of non-linear elastiity. Thus, it is not possible to found our disussion here on the D strain energy density W as in many publiations on elasti shells. Instead, we are fored here to begin our disussion of D onstitutive equations from the D omplementary energy density W. Various forms of omplementary energy in D nonlinear elastiity and assoiated variational priniples following from that proposed by Fraeijs de Veubeke (197) were disussed for example by Guo (1980), Atluri (1984), Reissner (1987), Ibrahimbegović (199, 1995) or Wempner (199). In some analogous D shell models onstruted from D ones by thikness kinemati onstraints or D-to-D degeneration the drilling ouples and bendings were not present, see for example Atluri (198), Wempner (1986), Simo and Fox (1989) or Ibrahimbegović (1994). In some other analogous shell models the drilling ouples and bendings were inluded, but the onstitutive equations for them were taken in the form similar as for shear stress resultants, see for example Chróśielewski et al. (199, 1997), Sansour and Bufler (199) or Bishoff et al. (004). The D drilling stress ouples and drilling bendings do not appear by definition in omplementary shell models formulated diretly on the base surfae, suh as in Altenbah and Zhilin (1988), Valid (1989), Gao and Cheung (1990) or Gałka and Telega (199). Brief review of possible forms of W in non-linear elastiity given in setion 4 indiates that even if W is onvex, its dual W obtained by the Legendre transformation need not be unique. For an isotropi elasti material undergoing small strains Koiter (1976)

4 proved that the omplementary energy density W W ( T ), where T is the Jaumann stress tensor, is the unique quadrati funtion provided rotations of material elements are at most moderate. But under small strains T S, where S is the nd Piola-Kirhhoff stress tensor. In our disussion the etive part (7) of W ontaining only tangential S and transverse shear S stresses ating on the shell ross setion is used. For isotropi elasti shells undergoing small strains John (1965) obtained onrete qualitative error estimates for stresses and their derivatives. In partiular, the stresses S were proved to be one order smaller than S. To assure the onsistent approximation to W, distribution of S through the thikness should be approximated up to ubi terms, while for S only quadrati distribution is appropriate. In setion 5 suh ubi approximation (41) of S is onstruted by analogy to refined statially and kinematially admissible stress distributions of the linear Reissner type shell theory, whih were given by Ryhter (1988). Applying the system of error estimates proposed by Koiter (1966, 1980), the through-the-thikness integration of W with refined stress distributions gives the D omplementary energy density in the form of quadrati polynomial (47) of the D resultant stress measures. Two prinipal terms of (47) an be viewed as the onsistent 1 st approximation to the omplementary energy density of the geometrially non-linear isotropi elasti shell. Suh quadrati form of is energetially equivalent to the onsistent 1 st approximation to the elasti strain energy density of the shell, whih within the lassial linear theory of shells was proposed by Koiter (1960). The four seondary terms of (47) provide a onsistent energeti refinement of the two prinipal terms. We all six quadrati terms of the onsistent nd approximation to the omplementary energy density of the geometrially non-linear isotropi elasti shells. This onsistently refined form of is new in the literature. The orresponding refined onstitutive equations (56) - (58) for D strain measures are then obtained by differentiation of appropriate resultant stress measures. with regard to To make the results more readable, in setion 6 we present them in orthogonal lines of prinipal urvatures. It is expliitly shown that the 8 8 matrix of oiients in the onstitutive equations for the surfae strethes and bendings E, K an be divided into two matries 4 4 for whih determinants are alulated. Determinant of the first matrix 4 4 is always positive, while of the seond one is positive provided that the prinipal 4

5 urvatures R1, R of the undeformed middle surfae are not equal. In both ases we are able to solve the set of linear algebrai equations analytially and provide the onsistently refined onstitutive equations for physial omponents of the stress resultants and stress ouples N, M ouples M in terms of E, K and R1, R. Finally, in setion 7 we derive the onstitutive equations (8) - (85) for the drilling following from their definition (16), the onstitutive equations (5) for M, and the quadrati part of displaement distribution through the shell thikness. The drilling ouples are estimated to be very small quantities of negligible order in analyses of regular shells. However, in ase of irregular multi-shells one has to keep these small resultant fields in order to preserve the struture of six-field shell theory at the juntions. Notation and some exat shell relations A shell is a three-dimensional (D) solid body identified in a referene (undeformed) plaement with a region B of the physial spae. The shell boundary onsists of three separable parts: the upper M and lower M shell faes, and the lateral shell boundary surfae B*. The position vetors x and y( x ) of any material partile in the referene and deformed plaements, respetively, an onveniently be represented by B x x n, y y( x) ( x, ). (1) Here x and y are position vetors of some shell base surfae M and N ( M) in the referene and deformed plaements, respetively, is the distane from M along the unit normal vetor n orienting M suh that [ h, h ], h h h is the shell thikness, is a deviation vetor of y from N, while and mean the D and D deformation funtions, respetively. In what follows we use the onvention that fields defined on the shell base surfae are written by itali symbols, exept in a few expliitly defined ases. Geometry of B an be desribed in normal oordinates (, ), 1,, suh that the orresponding base vetors of M and in B are given by (see Naghdi 196, Pietraszkiewiz 1979) 5

6 x 1 a = x,, a a =, n = a a, b a n,, x 1 g x, a, g g =, g ( ) a a, g = g = n, b, ( ) ( b H ), ( ), 1 H ( ), 1 K, () where are ontravariant omponents of the permutation tensor on M, b are mixed omponents of the urvature tensor b of M, H 1 b is the mean urvature and K det b -1 and ( ) are geometri shifters, the Gaussian urvature of M. Within the resultant non-linear theory of shells, formulated in the referential desription and summarised by Libai and Simmonds (1998) and Chróśielewski et al. (004), the respetive D internal ontat stress resultant n and stress ouple defined at the edge m vetors, R of an arbitrary part of the deformed base surfae R ( P), P M, but measured per unit length of the undeformed edge P having the outward unit normal vetor, are defined by h * Pn p h n d n, n d,, * Pn p m d m, m d. () Here P p g p g is the Piola stress tensor in the shell spae, * n g is the external normal to the referene shell orthogonal ross setion P* (see Konopińska and Pietraszkiewiz 007, (A.1)), p p and a. Then the resultant D equilibrium equations satisfied for any part P M are n f 0 m y, n 0, (4) where ( ) is the ovariant derivative in the metri of M, while f and are the external resultant surfae fore and ouple vetors applied at N, but measured per unit area of M. The resultant fields n and m require a unique D shell kinematis assoiated with the shell base surfae M. As it was shown in Libai and Simmonds (198, 1998), Chróśielewski et al. (199, 004), and Eremeyev and Pietraszkiewiz (006), suh D kinematis onsists of the translation vetor u and the proper orthogonal (rotation) tensor 6

7 Q, both desribing the gross deformation (work-averaged through the shell thikness) of the shell ross setion, suh that y x u, t = Qa, t Qn, (5) where t, t are three diretors attahed to any point of N ( M). The vetors n, m and f, an naturally be expressed in omponents relative to the rotated base t, t by n N t Q t, m t M t M t M t M t, f f t + ft, t t + t t + t. (6) The D omponents M m The shell streth t are usually alled the drilling ouples. and bending vetors assoiated with the D shell kinematis (5), whih are work-onjugate to the respetive stress resultant n and stress ouple m vetors, are defined by y, t = u, ( 1 Q) a E t E t, T ax Q, Q t K t K t K t K t, where 1 is the metri tensor of D spae and ax( ) is the axial vetor of a skew tensor (). We all the D omponents K t the drilling bendings. In the numerial analysis it is onvenient to assume a, n to be orthonormal, so that t, t remain orthonormal during shell deformation. (7) Components of stress resultants and stress ouples Let 1 ij T i j i S F P S g g S, 1,,, be the nd Piola-Kirhhoff stress tensor, where i F Grad g g is the D deformation gradient tensor in the shell spae. In i onveted oordinates (, ) we have F g g and P FS S ij gig j, see 1 k k Pietraszkiewiz and Badur (196). Thus, the omponents of P in the mixed tensor basis g g oinide with S ij, although S P. In terms of S ij the D resultants i j appearing in () take the form i i i n and m n S Fg d, m S Fg d. (8) i 7

8 In shell theory an initially straight and normal material fiber desribed by x n deforms into a generally spatially urved material fiber desribed in the deformed plaement by the deviation vetor, see (1). For what follows it is onvenient to utilize after Pietraszkiewiz et al. (006) the intrinsi deformation vetor e( x, ) defined by where T e Q n e e g en, (9) Q e is a measure of deviation of the deformed urved material fiber, whih initially has been straight n, from its approximately linear rotated shape Qn, see Fig. 1. The representation (9) is purely formal and does not introdue any approximation. Figure 1. Deformation of the shell ross setion Sine in this formulation of shell theory the rotational part of deformation is desribed by the tensor Q, it is natural to apply here, in plae of the usual polar deomposition F RU, the modified one in the form F( x, ) Q( x) ( x, ) Q( x)[ 1 ( x, )]. (10) In (10) the modified streth tensor satisfies det 0, T, and the modified relative streth tensor is also not symmetri, in general, i j T ijg g. Let us introdue the referential stress resultant and stress ouple vetors n T i k k i i k Q n S (δ ) g d N a Q n, T i k k i i k m Q m ( n e) S (δ ) g d M a M n. After some transformations the vetor m an also be given in the expanded form (11) 8

9 i S ( e)(δ i i ) e (δ i i ) e (δ i i ) d m a n. The shell stress resultants and stress ouples follow now from (11) 1 and (1) leading to (1) i N n a S ( ) d, i n n S (δ i i Q ) d, i m a S ( e)(δ i i ) e (δ i i ) d M, i M m n S e (δ ) d. i i i The relations (1) and (14) are exat impliations of the through-the-thikness integration of an arbitrary stress distribution in the shell spae. Most shell models are onstruted with the use of kinemati onstraints material fibres initially normal to the shell base surfae remain straight during deformation proess. In suh shell models i (1) (14) e 0 and the drilling ouples (14) disappear by definition. In the resultant geometrially non-linear shell theory the largest streth in the shell spae is assumed to be small, so that <<1. Let us also assume here the length of intrinsi deformation vetor e to be at least one order smaller as ompared with h, so that ( / ) 1 e h. In fat, we shall show in setion 5 that in ase of small elasti strains tangential omponents of e are of muh smaller order. Then omitting the orresponding small terms with respet to the unity, we obtain (S S ) d S d N, Q (S S ) d S d, S ( e) S ( e ) d S d M, M (S e d. The expliit definition (16) of M have beome possible beause in (9) we have introdued expliitly the vetor e and have applied the modified polar deomposition (10) of F. The relation (16) indiates that in the geometrially non-linear shell theory M an be established if S and e are known. We disuss this in more detail in setion 7. Libai and Simmonds (198, 1998) introdued M impliitly as their resultant stress ouple (15) (16) M t as well, but M was defined relative to the deformed, non-material, weighted surfae of mass of the shell, not relative to the deformed material shell base 9

10 surfae N ( M) as in this report. However, in some papers, see for example Naghdi (197), Paimushin (1986), Bishoff et al. (004) or Chróśielewski et al (010), the D fields M or M are defined as the resultants of first moments of shear stresses S d. Suh fields have other mehanial meaning than our drilling ouples M. 4 D omplementary energy density In non-linear elastiity the internal energy of the body is usually desribed by the stored energy density W W( F ) per unit volume of B suh that P W/ F. In our approah the D vetorial stress measures (8) are the primary fields defined by diret through-the-thikness integration of the Piola stress tensor P. Hene, for establishing D onstitutive equations from their D form it is neessary to use the omplementary energy density. The first hoie of suh density W W ( P ), per unit volume of B, would be the one whih is related to the strain energy density W( F ) by the Legendre transformation W ( P) P: F W( F ), (17) T where P: F tr( P F ). Existene of suh W ( P ) ruially depends on whether the stress strain relation P P( F ) an be uniquely inverted to the form F F( P ). Only then from (17) one ould establish uniquely W ( P ) from whih F W/ P. Unfortunately, unique invertibility of the tensor funtion P P( F ) is not assured, beause the salar-valued funtion W W( F ) is not onvex, in general. Only some speial ases were disussed in several papers by Zubov, Koiter, Ogden, Gao, Shield, Wempner, and others. In partiular, in the ase of an isotropi elasti material Zubov (1976) proved that there are four different branhes of suh an inversion. But when the angle of rotation of prinipal axes of strain is suh that os i.e., only one branh of the four is realized. In suh a ase the inverted tensor funtion F F( P ) an be uniquely established, at least in priniple, provided that the tensor T PP has distint eigenvalues at any point of the body. Ogden (1977) independently onfirmed that suh a unique inversion is possible under the latter ondition. These requirements suggest serious diffiulties in onstruting expliitly the unique funtion W ( ) P. 10

11 For our purpose it is more onvenient to use, after Koiter (1976), the stored energy density W W( ), where tensor U1 T is the relative streth tensor with the right streth T U U following from the polar deomposition F density W( ) we obtain RU. Differentiating the where W 1 ij T SU US T gig j, (18) T T T is the Jaumann stress tensor. But even in this ase inversion of T () T is still omplex for anisotropi elasti materials, beause then T and are not oaxial, in general. Only in the ase of an isotropi elasti material, when T and beome oaxial, one an invert in priniple the stress-strain relation to ( T), and applying the Legendre transformation one an onstrut expliitly W ( T ) suh that W/ T, provided that rotations of material elements are at most moderate, see Koiter (1976). The elasti range of many engineering materials is restrited to small strains suh that <<1 and the onstitutive equations are governed by the Hooke law. In suh ase W( ) beomes the positive definite, homogeneous, onvex, quadrati funtion of the form where L ijkl 1 ijkl ijkl jikl ijlk klij W( ) L ijkl, L =L =L =L, (19) are omponents of the 4 th -order tensor of elasti moduli. The linear onstitutive ij ijkl equations T W/ L an now be easily inverted to obtain K T kl, where ij kl K ijkl are omponents of the 4 th -order tensor of elasti omplianes, whih satisfy the relation klpq 1 p q q p KijklL i j i j. (0) The orresponding omplementary energy density follows from the Legendre transformation and takes the form 1 ij kl W ( T) T: ( T) W[ ( T)] Kijkl T T. (1) It an easily be seen from (18) that within small strains T S, so that also their ij ijkl omponents T ij S ij in the undeformed tensor base gig j. For an isotropi elasti material the D elasti moduli and omplianes are E L = g g +g g + g g (1+ ijkl ik jl il jk ij kl () 11

12 1 K ijkl (1 ) gikg jl +gilg jk g ijg kl, E () with E the Young modulus and the Poisson ratio. The restrition to small elasti strains used in (19) and (1) does not redue the non-linear elastiity to the linear theory of elastiity, beause the rotational part of deformation 1 R= FU is still allowed to be moderate, see Koiter (1976). Taking into aount symmetries of K ijkl and S ij, the quadrati expression (1) an be written as the sum of four separate terms eah representing a part of D omplementary energy density alulated from the stresses S, S S and where S, so that 1 W K S S +K S +S S +S K S S +K S S 1 A μs μ μs μ 4A μs μs A ( S )( S ) A ( S )( S ), S S, K A, K A, S = S K K, A, A. (4) (5) In partiular, for an isotropi linearly elasti solid 1 1 A (1 ) a a + a a a a, A a, E E (6) 1 A a, A. E E However, definitions (1) - (16) of the resultant surfae stress measures are given through the stress omponents S, S alone, beause only those stress omponents at on the shell ross setion and their resultants enter the resultant shell equilibrium equations (4). The stress omponent S ats on the shell surfaes onst parallel to the base surfae M. Although S ontributes to the D omplementary energy density (4), it does not enter the resultant D equilibrium equations and does not ontribute to the etive part W of W assoiated with the resultants (1) - (16). Thus, 1 W A μs μ μs μ 4A μs μs. (7) This etive part of W will be used to derive the onstitutive equations of elasti shells. 1

13 Let us assume, for definiteness, that the base surfae M is taken as the middle surfae of the shell in the undeformed plaement, that is h h h/. This partiular hoie of M will onsiderably simplify all transformations given below. We also assume, for simpliity, that there are no surfae fores applied at the upper and lower shell faes M, and no body fores applied in the internal shell spae (otherwise these loads would appear expliitly in D onstitutive equations, whih we do not want). Then the exat redution of D stress field to its D resultants defined by (15) and (16) means that to within bulk terms distribution of pseudo-stresses in the shell spae an, in fat, be approximately represented up to ubi terms in the thikness diretion aording to 1 1 S N M Q ( ) C ( ), h h (8) 1 4 μs Q f ( ), f ( ) 1, h h where Q ( ) are quadrati and C ( ) are ubi polynomials of whih should satisfy the relations Q ( ) Q ( ), Q ( ) d 0, Q ( ) d 0, C ( ) C ( ), C ( ) d 0, C ( ) d 0. The approximately ubi tangential stress distribution (8) 1 with quadrati and ubi parts having properties (9) satisfy definitions (15) 1 for N and (16) 1 for M. This distribution an be used to derive the approximate expressions for the drilling ouples following from (16). Unfortunately, we are not aware of any disussion in the literature of possible forms of Q ( ) and C ( ) in the geometrially non-linear theory of elasti shells. Looking for suggestions as to appropriate forms of Q ( ) and C ( ), let us reall some results available in the linear shell theory. (9) 5 The linear theory of shells In the linear theory of shells not only strains in the shell spae are small, but also translations and rotations are assumed to be small, max u, 1, (0) xm 1

14 where i is the linearised rotation vetor, with the angle of rotation about the rotation axis desribed by the eigenvetor i of Q, i.e. In omponents we have Qi i. u u a wn, n( a ) n a n. (1) Sine for small rotations Q 1, following Chróśielewski et al. 004, Chapter.8, we an linearize the kinemati relations (7) with regard to u and to obtain E u, a n u b w w, E u, n a w, b u, K K, a b,, n, b. () () Then linearization of omponent form of equilibrium equations (4) yields N b Q f 0, Q b N f 0, M b M Q 0, M ( N b M ) + 0. (4) Please note that within suh resultant linear shell theory twelve linear kinemati relations () and () involve the drilling rotation and the drilling bendings K while six linear equilibrium equations (4) inlude also the drilling ouples M. This was expliitly shown already by Reissner (1974). In lassial linear shell theories of Kirhhoff- Love and Timoshenko-Reissner types the omponents, K and M do not appear in analogous shell relations, see for example Love (197), Naghdi (197), Basar and Kratzig (001), Ciarlet (005). However, even within suh extended six-field linear theory of shells we are not aware of any disussion on possible forms of Q ( ) and C ( ) available in the literature. Leaving suh a disussion for future work, for the purpose of this report we shall use some results available for a simpler version of the linear shell theory. Analysing auray of the linear Reissner-type shell theory, Ryhter (1988) onstruted onsistently refined D displaement and stress fields in the shell as polynomials of D shell solutions. The refined D fields were then ompared to unknown solutions of linear elastiity in energy norm using the hypersphere theorem of Prager and Synge (1947) (see also Synge 1957) and appropriate inequalities to obtain refined global error estimates. It was found, in partiular, that the onsistently refined kinematially 14

15 admissible tangential omponents of D displaement field are (see Ryhter 1988, Eqs. (6a), (7a,b,,d)) û (, ) u ( ) ( ) ( ) ( ) q h h 5 h (5) h h 5 q D E( ),, D K( ), he, D L / L 0 a, where symbols uˆ, x, t,, h,, d, C, ( ),, ( ) of Ryhter (1988) have been hanged here into the respetive symbols û,, / h, u, h, q,, D, E( ), E, K( ) used in this report. Within the onsistent approximation (5) the D omponents u ( ) and ( ) of the linear theory of shells of Reissner type an be interpreted through the kinematially admissible D omponents û (, ) of linear elastiity by u ( ) u ˆ (, ) d, ( ) u ˆ (, ) d. (6) The orresponding onsistently refined statially admissible tangential pseudostresses of the linear shell theory of Reissner type take the form (see Ryhter 1988, eqs. (0a), (6) ) where h h h h 5 h E (1 ) N M H q ( ) ( ) H L L L / L 0 a a a a a a are statially admissible omponents of D symmetri stress tensor of linear elastiity. It is easy to hek that the stress field (7) 1 is ompatible with definitions N and M following from linearization of (15) 1 and (16) 1, N d, M d., (7) (8) The eqn. (5) 1 suggests that within the linear shell theory of Reissner type omponents e ( ) of the intrinsi deviation vetor e introdued in (9) an be onsistently approximated by the following quadrati and ubi polynomials: k q g k g e ( ) ( ) ( ), ( ), ( ), (9) h h 5 h whih satisfy the relations (9). In partiular, the funtion k( ) is even while g( ) is odd with regards to, k( ) k( ), g( ) g( ). (40) 15

16 M Sine orders of q, following from (5), seem to be very small, the values of alulated from (16) would be very small as well in the linear six-field shell model. This suggests that onsistently refined D tangential displaement and stress fields (5) 1 and (7) 1, whih are appropriate for the Reissner type linear shell model, should also be adequate for the resultant six-field linear shell model. 6 The geometrially non-linear theory of shells The bulk distribution of S given in (8) 1 does ontain only intrinsi resultant D variables. Thus, when strains are small everywhere in the shell spae, the pseudostresses an still be refined by quadrati and ubi terms analogous to those appropriate for the Reissner type linear shell model in (7) 1, 1 1 S N M H k( ) q ( ) g( ) ( ). h h (41) For thin isotropi elasti shells undergoing small strains John (1965) obtained onrete quantitative error estimates for stresses and their derivatives in the ase of vanishing surfae and body fores. With additional physially motivated estimates proposed by Koiter (1960, 1966, 1980), we an estimate orders of some fields appearing in suh small-strain shell theory as follows: 1,, S, S, A A E E E E h h h L min l, LE, LK, max,,,, 1, xm xm L d R a 1, b b H, K, R h R h where means of the order of, l is the harateristi length of geometri patterns of M, L E and L are the harateristi lengths of extensional and bending deformation patterns K on M, respetively, d is the distane of internal shell points to the shell boundary B, and is the ommon small parameter. Assuming that the stresses S entering definitions (15) 1 of N and (16) 1 of M are of the same order, from (8) and (4) we obtain the estimates (4) N Eh, M Eh, Q Eh. (4) 16

17 We have assumed above (15) that in ase of small strains ( / ) 1 e h. This means that in terms of defined in (4) we have assigned orders of the amplitudes in (9) to be q h. But now from (5), and (4) follow muh stronger estimates for these amplitudes q, h and their surfae derivatives q ( ) / L,( ). These estimates together with H E and k( ) g( ) h indiate that the quadrati and ubi terms in (41) are of the relative order of with two prinipal terms. and The D etive omplementary energy density h, respetively, as ompared Σ of the shell an now be obtained by diret through-the-thikness integration of the orresponding D density, Σ W d. (44) Taking into aount that 1/ 1 b (4 H K)... and introduing (8) and (41) into (7), we an express the integrand of (44) by infinite series of the resultant stress measures, urvatures of M, material parameters, as well as polynomials of n, n 0,1,,..., suh that 1 W 1 b... A 1 1 b N M hh k( ) q( ) g( ) ( ) h h 1 1 b N M hh k( ) q( ) g( ) ( ) h h b... A b Q f ( ) b Q f ( ). h h n h and With the estimates (4), (4) and those given below (4) we are able to estimate the order of any term appearing in the infinite series (45). To speed-up the analysis, one should note that only even terms ontaining (45) n with n 0,,4,... in (45) should be integrated in (44), beause integrals of odd terms of (45) ontaining n with n 1,,5,... vanish identially in (44) for our symmetri bounds of integration h/. Performing integration in (44) with (45) we should also take into aount the following relations: 17

18 f ( ) d h, k( ) d 0, g( ) d 0, h h 1 f ( ) d, k( ) d, g( ) d h, ( ), ( ), ( ), k( ) d h, k( ) g( ) d h, et., f d h k d h g d h 4 5 The outome of suh an elementary but involved estimation and through-thethikness integration proedures, whih we do not reprodue here for brevity of (46) presentation, gives the following two prinipal terms Eh and four seondary terms Eh : Σ A N N M M b N h h M N b M M b N 1 1 A Q Q O( Eh ), h s (47) where O (.) means of the order of and the shear orreting fator 5/ 6. One would expet that within the higher auray O( Eh ) of s there should also appear some terms ontaining the quadrati and ubi distributions of stresses through the thikness. But it is ease to hek that terms of (47) with even funtions k( ) and g( ) disappear during the integration proess aording to formulae (46),. Thus, higher order terms of the stress distribution (41) do not appear in the refined form of. Two first terms in the first row of (47) take into aount the prinipal ingredients of the D shell omplementary energy density, 1 1 ( ) ( ) 1 ( ) ( ) Σ A N N M M O( Eh ), h h where due to symmetries of A only symmetri parts of D stress measures (48) N and M ( ) ( ) equations are present. The eq. (48) leads to the orresponding onstitutive Σ ( ) ( ) 1 ( ) E AN O( ), N h Σ 1 ( ) ( ) ( ) K AM O. M h h (49) 18

19 To invert (49) for ( ) N and ( ) M one has to find omponents H of a D 4 th - order surfae elastiity tensor whih are dual to the omplianes A in the sense 1 H A. (50) For the isotropi linearly elasti material with omplianes (6) 1 it is easy to find that H satisfying (50) are E H a a a a a a (1 ) 1 Then we an invert the onstitutive equations (49) and obtain ( ) ( ) h ( ) ( ). (51) N hh E O( Eh ), M H K O( Eh ). (5) 1 Please note that H in (5) do not oinide with elastiities A alulated on M diretly from () under the ondition 0. The elastiities H orrespond to the plane stress state in the shell spae as disussed in Pietraszkiewiz (1979), setion 6.1. In the present approah the plane stress state is automatially indued by the invertibility requirement (50). The geometrially non-linear theory of thin isotropi elasti shells based on (48) an be alled the onsistent first approximation to the omplementary energy density of the geometrially non-linear isotropi elasti shells. Within the error O( Eh ) the density (48) provides the onstitutive equations only for symmetri parts N and M ( ) ( ) of D stress resultants and stress ouples. The drilling ouples M an be alulated from (8) with auray to the skew part [ ] M whih should satisfy the third salar moment equilibrium equation following from (4). Then Q an be established solving two tangential salar moment equilibrium equations following from (4). The virtual work identity based on remaining three fore equilibrium equations requires the translation vetor u to be the only kinemati field variable, while the rotation tensor Q beomes entirely expressible through u. This version of shell theory an be shown to be energetially equivalent to the one based on the onsistent first approximation to the shell strain energy density developed in many historial papers, onviningly presented for the lassial linear theory of shells by Koiter (1960) and summarised within the geometrially non-linear theory of thin elasti shells in Pietraszkiewiz (1989). In FEM numerial analyses suh -field shell model (of the Kirhhoff-Love type) requires 1 C 19

20 interelement ontinuity and seond derivatives of the translations appear as nodal variables, so that suh finite elements beome omplex and numerially iniient. The remaining four seondary terms in (47), whih are Eh, provide the onsistent energeti refinement, ompatible with the estimates (4) and (4), of the first two prinipal terms of (47). These seondary terms take into aount additional omplementary energies following from the transverse shear stress resultants Q as well as from oupling between the stress resultants N and stress ouples M due to the undeformed midsurfae urvature. We an all (47) the onsistent seond approximation to the omplementary energy density of the geometrially nonlinear isotropi elasti shell. Within the error O( Eh ) the shell theory based on (47) annot be regarded as equivalent to the one based on the onsistent seond approximation to the elasti strain energy density of the shell proposed by Pietraszkiewiz (1979) and extensively disussed by Badur (1984). In these works shell kinematis was first simplified by assuming the linear distribution of displaements through the shell thikness. The error of suh an assumption annot be preisely estimated. Then D strain measures were defined on M from expansion of D Green strain tensor 1 T E= ( F F 1 ) in the thikness diretion. The resulting D strain measures were regarded as the primary fields. In the seond approximation to the shell strain energy density there appeared also seond-order D strain measures and orresponding seond-order ouple stress fields K work-onjugate to, while D strains and bendings at M were eliminated through the additional assumption of plain stress. In our present approah leading to (47), only twelve omponents of D stress measures N, Q, M, M ating on the shell ross setion are the primary fields, while twelve D strain measures E, E, K, K are onstruted uniquely only on M through u and Q as D fields work-onjugate to the orresponding D stress measures. Additionally, by definition (16), M are expressible through N, M and amplitudes q, of the quadrati and ubi tangential omponents of the intrinsi deviation vetor e. Thus, within the seond-order auray of the shell omplementary energy density the resultant D stress and strain measures introdued in this paper are not the same as those 0

21 introdued in Pietraszkiewiz (1979), and both sets of surfae measures annot be identified. The onstitutive equations for D strain measures should now follow from differentiation of (47) with regard to appropriate resultant D stress measures. To perform derivative of the tensor funtion FN ( ) with regard to N let us reall the general rules of differentiation of tensor funtions given in Pietraszkiewiz (1974) aording to whih 0 F( N ) d B F N B for any Β TxM TxM, R. (5) N d 1 For the linear tensor funtion F N A b N M appearing as the fifth term of h (4) we obtain 1 F( ) A b N B M, h (54) df( ) A b B M b A M B, d h h so that FN ( ) 1 b AM, (55) N h with similar formula for derivative of the fourth term in (47) with regard to M. The onstitutive equations for E, K and E differentiating (47) with (55), an now be alulated by Σ 1 1 E A N b M b M b A M O( ), N h h Σ K A M b N b N b A N O, M h h h h Σ 4 E A Q O Q h s. (56) (57) (58) The onstitutive equations (58) an easily be solved for Q with the help of (0) and (6), whih leads to E (1 ) Q shc E O( Eh ), C L 0 a. The relations (56) and (57) onstitute the set of eight linear inhomogeneous algebrai equations for eight non-symmetri omponents N (59) and M. It seems to be diffiult to solve them analytially for an arbitrary system of surfae oordinates, although suh solution an always be performed numerially for any partiular hoie of 1

22 oordinates does not vanish. provided that determinant of 8 8 matrix of oiients of (56) and (57) 7 Constitutive equations in lines of prinipal urvatures To be more speifi, let the surfae oordinates lines of prinipal urvatures of M. Then be ar lengths of orthogonal 1 1 a a 1, a 0, a 1, b, b, b b 0, R1 R 1 1 A1111 A, A11, A11 A11 A, A111 A1 0, E E E where R 1 and R are prinipal radii of urvatures of M, and other values of A follow from symmetries of the surfae elasti omplianes. In suh oordinate system the ovariant and ontravariant omponents beome indistinquishable. Then partiular omponents of E and K following from (56), (57) and (60) are (60) E11 N11 N M11 O( ), Eh R1 R E1 N1 N1 M1 O( ), Eh R1 R E1 N1 N1 M 1 O( ), Eh R1 R E N N11 M O( ), Eh R1 R (61) (6) (6) (64) K11 M11 M N11+ ( ), Eh Eh R1 R h 6(1+ ) K1 M1 M 1 N1 O( ), Eh Eh R1 R h 6(1+ ) K1 M1 M 1 N1 O( ), Eh Eh R1 R h K M M11 N O ( ). Eh Eh R1 R h (65) (66) (67) (68)

23 One should note that E1 E1 in (6) and (6) as well as K1 K1 in (66) and (67). Thus, within the onsistent seond approximation to the D strain measures are defined as non-symmetri surfae fields on M. where The onstitutive equations (61) - (68) an be written in the matrix form D CS, (69) T D [ E, E, E, E, K, K, K, K ], T S [ N, N, N, N, M, M, M, M ], (70) C Eh Eh Eh R1 R Eh Eh Eh R1 R Eh Eh Eh R1 R Eh Eh Eh R1 R Eh R1 R Eh Eh (1+ ) 6(1+ ) Eh R1 R Eh Eh (1+ ) 6(1+ ) Eh R1 R Eh Eh Eh R1 R Eh Eh (71) The matrix C given above is symmetri with non-zero values of elements along the main diagonal. This matrix is non-singular if its determinant does not vanish. In suh ase there exists an inverse matrix inverted to the form 1 C suh that the redued onstitutive equations (69) an be 1 S C D. (7) To reveal when the matrix C may be singular, let us note that the eight linear algebrai equations (69) an be written as two separate sets of four linear algebrai equations, where D D D AS, D BS, (7) 1 1 T T E E K K S N N M M T E E K K S N N M M,,,,,,,, ,,,,,,,, T (74)

24 Eh Eh Eh R1 R Eh Eh Eh R1 R A, (75) Eh R1 R Eh Eh Eh R1 R Eh Eh Eh Eh Eh R1 R Eh Eh Eh R1 R B, (76) (1+ ) 6(1+ ) 0 Eh R1 R Eh Eh (1+ ) 6(1+ ) 0 Eh R1 R Eh Eh Determinant of A is det A 1 1. (77) Eh h h R1 R R1 R Hene, the matrix A is non-singular for any geometry of M. Thus, using the Cramer rule we an alulate analytially elements of S 1 with the seond-order auray leading to 1 1 N11 C E11 E D K11 O Eh R1 R ( ), 1 1 N C E E11 D K O Eh R1 R 1 1 M11 DK11 K D E11 O Eh R1 R ( ), 1 1 M DK K11 D E O Eh R1 R ( ), ( ), (78) Determinant of B is Eh Eh C, D (79) 4

25 det B Eh. 1 R R (80) Hene, the matrix B is non-singular provided that R1 R. If R1 R the prinipal terms of the inverted onstitutive equations are given by (5) only for the symmetri omponents of the resultant D stress and D strain measures. In order to refine them by the onsistent seondary terms proportional to h / R h / R 1 BB I O( ), where I is the identity we require 1 B to be suh that matrix. Then the refined onstitutive equations for mixed omponents of the resultant D stress measures are N C E E D K O Eh 1 (1 ) 1 1 (1 ) 1 ( ), R1 R N C E E D K O Eh 1 (1 ) 1 1 (1 ) 1 ( ), R1 R M D K K D E O Eh 1 (1 ) 1 1 (1 ) 1 ( ), R1 R M 1 D(1 ) K1 K1 D(1 ) E1 O( Eh ). R1 R The onstitutive equations for shear stress resultants (59) beome (81) 1 1 Q1 sc(1 ) E1, Q sc(1 ) E. (8) The refined onstitutive equations (78) and (81) as well as (8) are partiularly suitable for development of numerial FEM odes for analyses of omplex shell strutures, see Chróśielewski et al. (004). It is worth noting that up to the prinipal first-order terms our onstitutive equations (78) and (81) agree with those proposed in main lassial linear models of an isotropi elasti shell, see for example Koiter (1960) and Naghdi (196). Koiter (1960) proposed to treat various linear shell models with different additional seondary terms in the onstitutive equations as to be equivalent within the onsistent 1 st approximation to the shell elasti strain energy density. We have derived our onstitutive equations (78) and (81) from the onsistent nd approximation to the shell elasti omplementary energy (47). This has allowed us to selet, among various possible seondary terms, only those whih are onsistent with the higher auray of (49). 8 Constitutive equations for drilling ouples 5

26 The onstitutive equations for drilling ouples M an now be formulated diretly from definition (16) in whih the stress distribution (41) should be introdues together with estimates (4), (4) and q following relation:. To within the relative error of this leads to the 1 1 M M b q 1 O( ) O Eh. 45 (8) h From this estimate it is apparent that the influene of M on the stress distribution through the shell thikness is muh ( / h times) smaller than the influene of M Eh. Introduing (5) with (51) and (5) into (8), we an represent M in the more expliit and onise form, where M D(1 ) K, (84) d 4, ( ) ( h ( d K K a K ) b E ), O ( / h ) (85) Sine within small strains M is expressible through M ( ), b and q, our drilling bendings K in (85) are not derivable from rotations but are defined intrinsially through bendings K( ), urvatures b and surfae derivatives of the strain invariant ( E. As a ) result, the order of K / h is muh smaller (again / h times) than the order of K. Suh small quantity of M an always be omitted in numerial analyses of regular shell strutures. However, in ase of irregular multi-shells (eg. with branhes, intersetions or juntions with beams), when six ouple resultants are required at any interfae, one has to keep these very small values of M in order to preserve the struture of six-field shell theory at the juntions. oordinates If ar-length orthogonal lines of prinipal urvatures are taken as the surfae, then the relations (84) and (85) lead to M D(1 ) K, M D(1 ) K, (86) 1 d 1 d h 1 h 1 1 K K K E E, K K E E,, (1 ) R 4 1 R1 h 1 h 1 1 K K K E E, K K E E, (1 ) R1 4 1 R (87) 6

27 The onstitutive equations in the form (86) were first proposed by Chróśielewski et al. (199) with d t 1 and K t, where were understood as expressed in rotations aording to (7). In our ase d 4 /15 follows from the result through-the-thikness integration of k( ), whih is then multiplied by h / 45 of 1 / h standing in front of the onstitutive equation (5). Within the geometrially non-linear theory of elasti shells the K in (87) are not independent surfae bendings, but are expressible entirely through K, 1/ R, and ( E11 E),. 9 Conlusions We have disussed several problems arising in the resultant, six-field, geometrially non-linear model of isotropi elasti shells. Our approah has been based on the D omplementary energy density of geometrially non-linear elastiity undergoing moderate rotations. Among new results obtained here let us point out the following: 1. Expliit definition (16) of the drilling ouples M.. The tangential stress distribution (41) through the shell thikness onsistently refined by quadrati and ubi terms.. The onsistent nd approximation (47) to the D omplementary energy density of the geometrially non-linear isotropi elasti shells. 4. The refined onstitutive equations (56) and (57) for D strains E and bendings K. 5. The refined onstitutive equations (61) - (68) for E and K expressed in orthogonal lines of prinipal urvatures, and their inverted forms (78) and (81) for the stress resultants N and stress ouples M. 6. The expliit onstitutive equations (8) - (87) for drilling ouples and their estimates as very small quantities of negligible order in analyses of regular shells. These theoretial results should be of interest to speialists of the non-linear theory of elasti shells and those developing omputer FEM software for analyses of omplex non-linear problems of irregular multi-shell strutures. 7

28 Aknowledgements The researh reported in this paper was supported by the National Centre of Siene of Poland with the grant DEC 01/05/D/ST8/098. Referenes Altenbah, H., Zhilin, P.A., General theory of elasti simple shells. Advanes in Mehanis 11(4), Antman, S.S., 005. Nonlinear Problems of Elastiity, nd ed. Springer. Atluri, S.N., 1984.Alternative stress and onjugate strain measures, and mixed variational formulations involving rigid rotations, for omputational analyses of finitely deformed solids, with appliations to plates and shells I. Theory. Comput. & Strut. 18, Badur, J., Non-Linear Analysis of Elasti Shells Aording to Seond Approximation to the Strain Energy (in Polish). Institute of Fluid-Flow Mahinery PASi, Gdańsk. Başar, Y., Krätzig, W.B., 001. Theory of Shell Strutures, nd ed. VDI Verlag GmbH, Düsseldorf. Birsan, M., N, P., 01. Existene of minimizers in the geometrially non-linear 6- parameter resultant shell theory with drilling rotations. Math. Meh. Solids ( in print), DOI: / Bishoff, M., Wall, W.A., Bletzinger, K.-U., Ramm, E., 004. Models and finite elements for thin-walled strutures, in: Stein, E., de Borst, R., Hughes, T.J.R. (Eds.), Enylopedia of Computational Mehanis, Vol., Chapter. Wiley, Chiester, pp Chróśielewski, J., Makowski, J., Pietraszkiewiz, W., 004. Statis and Dynamis of Multi-Shells: Nonlinear Theory and Finite Element Method (in Polish). IFTR PASi Press, Warsaw. Chróśielewski, J., Makowski, J., Stumpf, H., 199. Genuinely resultant shell finite elements aounting for geometri and material non-linearity. Int. J. Num. Meth. Engng. 5, Chróśielewski, J., Makowski, J., Stumpf, H., Finite element analysis of smooth, folded and multi-shell strutures. Comp. Meth. Appl. Meh. Engng. 41, Eremeyev, V.A., Pietraszkiewiz, W., 004. The non-linear theory of elasti shells with phase transitions. J. Elastiity 74, Eremeyev, V.A., Pietraszkiewiz, W., 006. Loal symmetry group in the general theory of elasti shells. J. Elastiity 85, Eremeyev, V.A., Pietraszkiewiz, W., 011. Thermomehanis of shells undergoing phase transitions. J. Meh. Phys. Sol. 59, Eremeyev, V.A., Zubov, L.M., 008. Mehanis of Elasti Shells (in Russian). Nauka, Mosow. Fraeijs de Veubeke, B., 197. A new variational priniple for finite elasti displaements. Int. J. Engng. Si. 10, Gałka, A., Telega, J.J., 199. The omplementary energy priniple as a dual problem for a speifi model of geometrially nonlinear elasti shells with an independent rotation vetor General results. Eur. J. Meh. A Sol. 11,