Published in: Altenbach H, Őchsner A (eds), Encyclopedia of Continuum Mechanics, Section: Shells. Springer-Verlag, Berlin et al. 2018
Elastic Shells, Resultant Non-linear Theory
Wojciech Pietraszkiewicz
Faculty of Civil and Environmental Engineering, Gdańsk University of Technology, Gdańsk, Poland
Email: [email protected] Synonyms
Six-field theory of shells; Six-parameter shell theory Definition
Two-dimensional (2D) equilibrium conditions of this general shell model are exact resultant implications of corresponding equilibrium conditions of continuum mechanics. The 2D unique shell kinematics consists of six scalar finite translation and rotation fields. Two surface drilling couples and two work-conjugate surface drilling bendings appear in the description of the shell stress and strain state. These additional fields and the shell drilling rotation become of primary importance when formulating and analyzing problems of irregular shell structures with branchings, intersections or junctions along singular surface curves and /or having connections with beams or columns.
Introduction
In classical linear and non-linear models of shells, the reduction of 3D mechanical problem of a thin solid body to the 2D problem of the shell is usually achieved applying some simplifying assumptions about kinematics of material elements initially normal to the shell base surface. This allows one to construct the approximate 2D kinematic structure of the shell. Then the corresponding 2D non-linear equilibrium conditions are generated by the principle of virtual work postulated on the shell base surface.
In the resultant non-linear theory of shells proposed by Reissner (1974), developed in a number of papers and summarised in monographs by Libai and Simmonds (1983, 1998), Chróścielewski et al. (2004), and Eremeyev & Zubov (2008), dynamics not kinematics is the primary notion. In this formulation, the 2D non-linear equilibrium conditions of forces and couples are derived first by the exact through-the-thickness integration of 3D equilibrium conditions of continuum mechanics. Then, the 2D virtual work identity allows one to construct uniquely the 2D shell kinematics consisting of the translation vector u and rotation tensor Q fields (six independent scalar variables) defined on the shell base surface. Because of this property the resultant shell model is also known as the six-field or 6-parameter (6p) shell model. Here the drilling rotation (about normal to the undeformed shell base surface) remains as the independent kinematic variable, as well as two drilling couples and two work-conjugate drilling bending measures appear in the description of 2D stress and strain state. These additional surface fields extend all classical shell formulations of the Kirchhoff-Love and Timoshenko-Reissner type following from other 3D-to-2D reduction techniques.
The resultant non-linear theory of shells may briefly be called dynamically exact and kinematically unique. The only approximations of this shell model enter through the
2 constitutive equations relating 2D stress and couple resultants with 2D strain measures. But these are material laws based finally on experiments, which are approximate anyway.
Notation
A shell is a three-dimensional (3D) solid body identified in a reference
(undeformed) placement with a region B of the physical space with V as its translation vector space. The shell boundary B consists of three separable parts: the upper M and lower M shell faces, and the lateral shell boundary surface B* . In a given reference frame represented by (O, ),ei i1,2,3, where O and eiV are orthonormal vectors, the position vectors x and y ( )x of any material particle in the reference and deformed placements, respectively, can conveniently be represented by
, ( , )t ( , ) , ( ,0) .
x x n y y x x x 0 (1)
Here x and y are the position vectors of some shell base surface M and N(M) in the reference and deformed placements, respectively, is the distance from M along the unit normal vector n orienting M such that [ h h, ] , hhh is the undeformed shell thickness, is a deviation vector of y from N, while and mean the 3D and 2D deformation functions, respectively. In what follows we use the convention that fields defined on the shell base surface are written by italic symbols, except in a few explicitly defined cases.
Geometry of B can be described in normal curvilinear coordinates
( , ), 1, 2, such that the corresponding base vectors on M and in B are given by (Naghdi 1963; Pietraszkiewicz 2018) 1 3 3 1 1 1 2 1 , , , , , , 2 , , , ( ) , , 1 , ( ) ( 2 ) , ( ) , ( ) , 1 2 , b a b b H H K x g x g g g g g x a = x a a = n = a a a n a = a = = n (2)
where are contravariant components of the permutation tensor on M , b are mixed components of the curvature tensor of M, and (-1) are geometric shifters,
1/ 2
H b is the mean curvature, and K det
b the Gaussian curvature of M . Resultant Equilibrium ConditionsLet ( )b x be the body force vector per unit volume of B, and t xn( ) be the contact force vector per unit area of B . Then the equilibrium of forces and couples applied to any part PB requires that
n n
P dv P da , P dv P da .
3 One can directly integrate the relations (3) through the normal coordinate . Then the spatial force vector ( )b x gives rise to the resultant surface force vector f x( ) and the resultant surface couple vector ( )c x , both per unit area of M , defined by
n
n
d | , d | , . d h . d , h
f b t c b t (4)where the quantities (.) mean values of the fields (.) on the upper M and lower M shell faces.
The 2D internal contact stress resultant n and stress couple m vectors, defined at the edge N of an arbitrary part of the deformed base surface N ( ), M, but measured per unit length of the undeformed edge having the outward unit normal vector , are defined by
* * d , d , d , d .
Pn p Pn p n n n m m m (5)Here Ppg p3 g3 is the Piola stress tensor in the shell space, n*g is the
external normal to the reference shell orthogonal cross section P* (see Konopińska and Pietraszkiewicz 2007, A.13), a, and is the tensor product.
With definitions (4) and (5), the 3D equilibrium equations (3) are exactly reduced to the 2D resultant equilibrium equations
* \ * * \ , , f f f f M M M M da ds ds da ds ds
0 0 f n n c m y n m y n (6)where n m*, * are just the external resultant boundary force and couple vectors assigned
along Mf M, which are statically equivalent to distribution of external tractions t*
applied on B*
f
.
It is assumed here that M be a regular geometric surface, so that kinks, branchings and intersections are not allowed. It is also assumed that all surface fields discussed here be smooth on M . Under these conditions one can apply the surface divergence theorem (Pietraszkiewicz 2018) to some terms in (6) leading to
| , | , , | , ds da ds da ds da
n n m m y n y n y n (7)where (.) | is the covariant surface derivative taken in the undeformed surface metric a of M . Then the resultant local 2D equilibrium equations satisfied for any part M are
| | , .
0 0
n f m y n c (8)
The natural static boundary conditions compatible with (8) are
* , * along . f M 0 0 n n m m (9)
4 The resultant local equilibrium conditions (8) and (9) of the shell are exact
resultant implications of the 3D equilibrium equations (3) of continuum mechanics.
Shell Kinematics
Let v and w be two arbitrary vector fields on M . Then the following integral identity must be satisfied:
* *
| | , . f M M da ds
n f v + m y n c w n n v m m w (10)The identity can be transformed with the help of the surface divergence theorem into
* * , , , , f d M M M M da da ds ds
n v y w m w f v c w n v m w n v m w (11) where Md M \Mf .If v and w are interpreted as kinematically admissible virtual translation and rotation vectors such that v w 0 alongMd, then the last integral of (11) identically vanishes. Two integrals in the second row of (11) can then be interpreted as the external virtual work performed by the given surface f c , and boundary n m*, * loads, respectively.
In this context the first surface integral of (11) has the meaning of the internal virtual work, where the expressions ,vy,w and ,w can be interpreted as just virtual changes of appropriately defined shell strain and bending vectors, respectively. Then the formula (11) takes the meaning of the principle of virtual displacements for the shell.
Let the vector field ( )u x represent the work-averaged translations of M and the proper orthogonal tensor field ( )Q x the work-averaged rotations of the shell cross sections. The deformed shell configuration can then be described by the relations
, , ,
y x u t Qa t Qn (12)
where t,t are three directors attached to any point of the deformed base surface ( )
N M .
Let us consider a one-parametric family of shell deformations described by ( , )t ( , )t
y x x u x and ( , )Q x t t( , )x t a t x( , )t n, where t is a scalar (time-like) parameter such that t0 corresponds to the undeformed shell placement and t to the deformed one. Then the vectors v and w in (11) can be interpreted as virtual changes of
u and Q (linear and angular velocities in a real motion) according to
( , ) ( , ) , ax ( , ) ax T , t t t t y x v u x u w Q x Q QQ (13)where δ is the symbol of variation.
The shell strain and bending vectors corresponding to the kinematics (12) and their virtual changes was defined by Chróścielewski et al. (1992) as
5
, , ( ) , 1 ax , , , , 2 T T T E E K K 1 y t u Q a t t Q Q t Q Q t t Q Q t t t t (14) , , , , , c c E E K K v y w t t w = t t t (15)where c(.)Q
[QT(.)]
is the co-rotational variation (the co-rotational time derivative ina real motion), and 1 V V is the metric tensor of the 3D vector space.It follows from (15) and (11) that the surface vector strain measures , are work-conjugate to the respective resultant vector stress measures n m, . As a result, the shell kinematics consisting of the displacements u, Q satisfying (12) and of the strain measures , satisfying (15) is unique and an energetically exact implication of the resultant local equilibrium equations (8). The shell kinematics is valid for unrestricted translations, rotations, strains, and/or bendings of the shell material elements.
The kinematic structure (12)-(15) of the resultant non-linear shell theory is identical with that of the classical version of the Cosserat surface (Cosserat and Cosserat 1909). The kinematic structure (12)-(15) differs from kinematics of the surface with one or more attached deformable directors, often also called the Cosserat surface (Naghdi 1972;
Altenbach and Zhilin 1988; Rubin 2000). The kinematic structure following from (12)-(15) is also different from kinematics of the nonlinear theories of shells based on the Kirchhoff– Love or Timoshenko–Reissner type kinematic assumptions.
The vectors n,m and f c, appearing in (8) and (9) can naturally be represented through components relative to the rotated base t t, by
, , , N Q M M M M f f c c c c n t t m t t t t t f t + t c t t + t t + t . (16)
The components M mt are usually called the drilling couples while the
work-conjugate components K t are the drilling bendings. These surface stress and strain measures do not appear in any other non-linear shell model.
From (12) and (13) it follows that the displacement boundary conditions which assure vanishing of the last integral of (11) should be
* , * along ,
d
M
0 0
u u Q Q (17)
where u Q*, * are prescribed fields.
Constitutive Equations
In the resultant non-linear shell model discussed above, the displacement fields ,
u Q of the shell base surface are assumed to be the independent field variables of the boundary value equilibrium problem. The fields ( , ,f c n m u Q*, *, *, *) are supposed to be
known in advance. The surface strain measures , are expressed through ,u Q by the kinematic relations (14). To write the equilibrium equations (8) in terms of ,u Q one still
6 needs some constitutive equations relating the resultant surface stress measures n,m in
M with the surface strain measures , .
As noted by Reissner (1974) himself, the problem of constitutive equations in the resultant non-linear shell theory “…may be considered in at least two distinct ways. One of these deals with the problem of devising suitable systems of physical experiments for elements of the two-dimensional continuum in order that a system of two-dimensional constitutive equations be established directly. The other deals with the problem of devising suitable mathematical methods to deduce constitutive equations for the shell as a two-dimensional continuum, as exact or asymptotic, or otherwise rationally approximate consequences of a given system of constitutive equations for the shell considered as a three-dimensional continuum….”.
Within this model of elastic shells, Eremeyev and Pietraszkiewicz (2006)
established the material symmetry group consisting of an ordered triple of tensors which make the 2D strain energy density invariant under change of the reference placement. For the physically linear solid shells, when the density becomes a quadratic function of the shell strain and bending measures, reduced representations of the density were established for orthotropic, cubic-symmetric, and isotropic shells. The reduced representations contain much less independent material constants to be found from experiments.
In what follows an example of establishing the 2D constitutive equations for the geometrically non-linear theory of a homogeneous isotropic elastic shell is formulated from corresponding constitutive equations of 3D geometrically non-linear elasticity. Isotropic Elastic Shell Undergoing Small Strains
Let SF P1 Sijgigj ST,i1, 2,3, be the 2nd Piola-Kirchhoff stress tensor, where FGrady gi gi is the 3D deformation gradient tensor in the shell space. In convected coordinates ( , ) we have F1gk gk and PFSSijgigj, see (Pietraszkiewicz and Badur 1963). In terms of Sij
the 2D resultants n and m appearing in (5) take the form
S i i d , S i i d .
Fg
Fgn m (18)
In shell theory an initially straight and normal material fiber described by xn deforms into a generally spatially curved material fiber described in the deformed placement by the deviation vector , see (1). For what follows it is convenient to utilize after Pietraszkiewicz et al. (2006) the intrinsic deformation vector (e x,) defined by
e e T , Q n n e g (19)
where Qe is a measure of deviation of the deformed curved material fiber, which initially has been straight n, from its approximately linear rotated shape Qn. The representation (19) is purely formal and does not introduce any approximation.
Since in this formulation of shell theory the rotational part of deformation is described by the tensor Q, it is natural to apply here, in place of the usual polar decomposition FRU, the modified one in the form
7 ( ,) ( ) ( ,) ( )[ ( ,)].
Fx Q x x Q x 1 x (20)
In (20) the stretch tensor satisfies det0, T , and the modified relative stretch tensor is not symmetric, in general, ijgigj T.
In the resultant geometrically non-linear shell theory the largest stretch in the shell space is assumed to be small, so that || ||<<1 . Let us also assume here the length of intrinsic deformation vector e to be at least one order smaller than h, so that (| |/ )e h 21. In fact, it was noted (Pietraszkiewicz and Konopińska 2014) that in case of small elastic strains the tangential components of e are of much smaller order. Then omitting the corresponding small terms with respect to the unity, we obtain
3 S d S d S d (S e d N , Q , M , M .
n t n t m t m t (21)The last relation (21)2 indicates that in the geometrically non-linear shell theory M can
be established if S and e are known.
In the resultant shell model components of n,m are the primary fields. Hence, for establishing the 2D constitutive equations from their 3D form it is necessary to use the 3D complementary energy density Wc. When strains are small everywhere in the shell space, Wc is the quadratic function Wc 1/ 2K S S ,ijkl ij kl where Kijkl are the elastic compliances which for an isotropic material are
1 K (1 ) g g +g g 2 g g , 2 ijkl ik jl il jk ij kl E (22)with E the Young modulus and the Poisson ratio of the linear elastic material. The resultants n,m are defined only through the stress components S, S3 alone, because only these stresses act on the shell cross section. The stress components S 33 act only on the shell surfaces const parallel to the base surface M . They do not contribute to the effective part Wceff of the 3D complementary energy density associated with the resultants, which is defined by
3 3 3 3 1 W μS μ μS μ 4 μS μS , 2 eff c A A (23)where the following geometric relations have been used:
3 3 3 3
K A , K A . (24) The effective part of 2D complementary energy density may be obtained by direct through-the-thickness integration of (23). This leads to the infinite series of terms of decreasing order. Pietraszkiewicz and Konopińska (2014) estimated orders of all terms of this infinite series applying the concrete qualitative error estimates for stresses, strains and their derivatives obtained by Koiter (1960), John (1965), and Rychter (1988). The outcome of this complex procedure allowed one to distinguish in the series two principal terms and four secondary terms leading to
8
2 2 3 3 3 1 12 W d 2 1 2 1 ( ), eff eff c c s Σ A N N M M h h A b N M N b M M b N A Q Q O Eh h h
(25)where s 5 / 6 is the shear correcting factor. The small parameter in (25) is defined by max , , x M h h L R (26)
where L is the minimal characteristic length of geometric and deformational patterns on M , R is the minimal principal radius of curvature of M , and is the maximal strain in the shell space.
The relation (25) can be called the consistent second approximation to the complementary energy density of the geometrically non-linear homogeneous isotropic elastic shell. The constitutive equations for E, K, E can now be calculated from (25) leading to
3 3 2 1 1 ( ) , 1 12 1 ( ) , eff c eff c Σ E A N b M b M b A M O h h N Σ K A M b N b N b A N O h h h M h (27) 3 3 3 4 ( ) . eff s Σ E A Q O h Q (28)For any particular choice of the surface coordinates the equations (27) can be inverted for N,M provided that determinant of 8 8 matrix coefficients in (27) does not vanish. For the arc-length orthogonal lines of principal curvatures of M such inversion was explicitly performed by Pietraszkiewicz and Konopińska (2014). This resulted in the following constitutive equations for the physical components of 2D shell stress and couple resultants:
11 11 22 11 1 2 22 22 11 22 1 2 11 11 22 11 1 2 22 22 11 22 1 2 1 1 , 1 1 , 1 1 , 1 1 , N C E E D K R R N C E E D K R R M D K K D E R R M D K K D E R R (29)9
12 12 21 12 1 2 21 12 21 21 1 2 12 12 21 12 1 2 21 12 21 21 1 2 1 1 1 (1 ) (1 ) , 2 1 1 1 (1 ) (1 ) , 2 1 1 1 (1 ) (1 ) , 2 1 1 1 (1 ) (1 ) 2 N C E E D K R R N C E E D K R R M D K K D E R R M D K K D E , R R (30)
3 2 2 , 1 12 1 Eh Eh C , D (31)where R1 and R2 are the principal radii of curvatures of M . For the shear stress resultants, by inverting (28) we obtain
1 1 2 2 2(1+ ) 2(1+ ) s s Eh Eh Q E , Q E . (32)
The constitutive equations for the drilling couples take the form
1 1 2 2 4 (1 ) , (1 ) , . 15 d d d M D K M D K (33)
It has been proved that within the error of small strain shell theory the drilling bendings K are entirely expressible through K, 1/R and (E11E22),. Additionally, the M themselves have been estimated to be of negligible order in analyses of regular shells. But in solving non-linear problems of irregular shells with branchings, intersections or junctions with beams and columns, for example, these small fields should be taken into account in order to preserve the structure of the resultant six-field non-linear shell theory. Cross References
Surface Geometry, Elements
Junctions in Irregular Shell Structures
Shell Thermomechanics, Resultant Non-linear Theory References
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