On the characterization of drilling rotation in the 6-parameter resultant shell theory
aa r X i v : . [ m a t h . A P ] M a r On the characterization of drilling rotation in the6–parameter resultant shell theory
Mircea Bˆırsan ∗ and Patrizio Neff † June 27, 2018
Abstract
We analyze geometrically non-linear isotropic elastic shells and prove the exis-tence of minimizers. In general, the model takes into account the effect of drillingrotations in shells. For the special case of shells without drilling rotations we presenta representation theorem for the strain energy function.
The paper is concerned with the geometrically non-linear 6-parameter resultant shell the-ory. This model of shells involves two independent kinematic fields: the translation vectorfield and the rotation tensor field, which have in total 6 independent scalar kinematicvariables. This shell theory was originally proposed by Reissner [11] and was developedconsistently by several authors [6, 3].In Section 2 we briefly present the kinematics of 6-parameter shells, as well as the equationsof equilibrium and the constitutive equations of elastic shells. We formulate the boundary-value problem and prove the existence of minimizers associated to the deformation ofisotropic shells in Section 3. This model is able to describe the effect of drilling rotations inshells. In order to see the difference to the Reissner-type kinematics for shells, we analyzein Section 4 shells without drilling rotations and give a representation theorem for thestrain energy function. Finally, we consider isotropic shells without drilling rotations and ∗ Mircea Bˆırsan, Fakult¨at f¨ur Mathematik, Universit¨at Duisburg-Essen, Campus Essen, Thea-Leymann Str. 9, 45127 Essen, Germany, email: [email protected] ; and Department of Mathe-matics, University “A.I. Cuza” of Ia¸si, 700506 Ia¸si, Romania † Patrizio Neff, Lehrstuhl f¨ur Nichtlineare Analysis und Modellierung, Fakult¨at f¨ur Mathematik,Universit¨at Duisburg-Essen, Campus Essen, Thea-Leymann Str. 9, 45127 Essen, Germany, email:patrizio.neff@uni-due.de, Tel.: +49-201-183-4243
We consider a shell with the base surface S in the reference cofiguration characterized bythe position vector (relative to a fixed point O ) y : ω ⊂ R → R , y = y ( x , x ), andthe structure tensor Q : ω ⊂ R → SO (3), Q = d i ( x , x ) ⊗ e i . Here, ( x , x ) are thematerial curvilinear coordinates on S , { e i } is the fixed orthonormal vector basis of theEuclidean space, and { d i ( x , x ) } is the orthonormal triad of directors which characterizesthe orthogonal tensor field Q [3, 5]. We employ the usual notations: the Latin indexes i, j, ... take the values { , , } , the Greek indexes α, β, ... the values { , } , the partialderivative ∂ α f = ∂f /∂x α , as well as the Einstein summation convention over repeatedindexes. The set ω is assumed to be a bounded open domain with Lipschitz boundary inthe Ox x plane.For the deformed configuration of the shell, we denote by S the base surface, y ( x , x )the position vector and { d i ( x , x ) } the orthonormal triad of directors. The displacementvector is defined as usual by u = y − y and the elastic rotation (between S and S ) by theproper orthogonal tensor field Q = d i ⊗ d i . The orthogonal tensor R = QQ = d i ⊗ e i describes the total rotation from ω to S .Let a α = ∂ α y be the (covariant) base vectors in the tangent plane to S , n = a × a / k a × a k the unit normal to S , and { a α } the reciprocal (contravariant) basis in thetangent plane, with a α · a β = a αβ and a α · a β = δ αβ (the Kronecker symbol). Then theshell deformation gradient tensor is expressed by F = Grad s y = ∂ α y ⊗ a α .We designate by N and M the internal surface stress resultant and stress couple tensorsof the 1 st Piola–Kirchhoff type for the shell, and by f and c the external surface resultantforce and couple vectors applied to points of S , but measured per unit area of S . Theequilibrium equations for 6-parameter shells are [5]Div s N + f = , Div s M + axl( N F T − F N T ) + c = , (1)where Div s is the surface divergence, ( · ) T denotes the transpose, and axl( · ) represents theaxial vector of a skew–symmetric tensor. We consider boundary conditions of the type[10] N ν = n ∗ , M ν = m ∗ along ∂S f , y = y ∗ , R = R ∗ along ∂S d , (2)where ν is the external unit normal vector to the boundary curve ∂S (lying in the tangentplane) and { ∂S f , ∂S d } is a disjoint partition of ∂S .2ccording to [5, 3], the elastic shell strain tensor E e and the bending–curvature tensor K e in the material representation are E e = Q T F − Grad s y = (cid:0) Q T ∂ α y − a α (cid:1) ⊗ a α K e = axl( Q T ∂ α Q ) ⊗ a α . (3)Under the hyperelasticity assumption, N and M are expressed by the constitutive equa-tions N = Q ∂ W∂ E e , M = Q ∂ W∂ K e , (4)where W = W ( E e , K e ) is the strain energy density of the elastic shell. The boundary–value problem describing the deformation of non-linear elastic shells consists of the equa-tions (1)-(4). We assume the existence of a function Λ( y , R ) representing the potentialof external surface loads f , c , and boundary loads n ∗ , m ∗ [4]. This loading potential canbe decomposed additively asΛ( y , R ) = Λ S ( y , R ) + Λ ∂S f ( y , R ) , Λ S ( y , R ) = Z S f · u d S + Π S ( R ) , Λ ∂S f ( y , R ) = Z ∂S f n ∗ · u d l + Π ∂S f ( R ) , where the load potential functions Π S , Π ∂S f : L ( ω, SO (3)) → R are assumed to becontinuous and bounded operators. Corresponding to the deformation of elastic shells,we consider the following two–field minimization problem: find the pair (ˆ y , ˆ R ) in theadmissible set A which realizes the minimum of the functional I ( y , R ) = Z S W ( E e , K e ) d S − Λ( y , R ) (5)for ( y , R ) ∈ A , where A := (cid:8) ( y , R ) ∈ H ( ω, R ) × H ( ω, SO (3)) (cid:12)(cid:12) y | ∂S d = y ∗ , R | ∂S d = R ∗ (cid:9) . Here, the boundary conditions are to be understood in the sense of traces, H denotes as usual the Sobolev space, and L the Lebesgue space. The variational principleassociated to the total energy of elastic shells (5) has been proved in [4]. In case of physically linear isotropic shells, the strain energy density is assumed as thequadratic form2 W ( E e , K e ) = α (cid:0) tr E e k (cid:1) + α tr (cid:0) E e k (cid:1) + α tr (cid:0) E e,T k E e k (cid:1) + α ( n E e ) + β (cid:0) tr K e k (cid:1) + β tr (cid:0) K e k (cid:1) + β tr (cid:0) K e,T k K e k (cid:1) + β ( n K e ) , (6)3here E e k = E e − n ⊗ n E e , K e k = K e − n ⊗ n K e . The eight coefficients α k , β k candepend in general on the initial structure curvature tensor K = axl (cid:0) ∂ α Q Q ,T (cid:1) ⊗ a α .For the sake of simplicity, we assume in our discussion that α k and β k are constant. Theorem 1.
Assume that the initial position vector y is continuous and injective and y ∈ H ( ω, R ) , Q ∈ H ( ω, SO (3)) ,∂ α y ∈ L ∞ ( ω, R ) , det (cid:0) a αβ ( x , x ) (cid:1) ≥ a > , where a is a constant. The external loads and boundary data are assumed to satisfy theconditions f ∈ L ( ω, R ) , n ∗ ∈ L ( ∂ω f , R ) , y ∗ ∈ H ( ω, R ) , R ∗ ∈ H ( ω, SO (3)) . If the constitutive coefficients satisfy the conditions α + α + α > , α + α > ,α − α > , α > , β + β + β > ,β + β > , β − β > , β > , (7) then the minimization problem (5) admits at least one minimizing solution pair (ˆ y , ˆ R ) ∈A . Proof.
In view of the inequalities (7) we can deduce that there exists a constants C > W ( E e , K e ) ≥ C (cid:0) k E e k + k K e k (cid:1) . Moreover, in this case the strain energy density W ( E e , K e ) is a strictly convex functionof its arguments. Then, according to Theorem 6 from [1], we derive the existence ofminimizers. The proof is based on the direct methods of the calculus of variations. (cid:3) Remark 2.
For isotropic shells, the simplest expression of W ( E e , K e ) corresponds tothe form (6) with α = Cν, α = 0 , α = C (1 − ν ) ,α = α s C (1 − ν ) , β = Dν, β = 0 ,β = D (1 − ν ) , β = α t D (1 − ν ) , (8)where h is the thickness of the shell, E the Young modulus, ν the Poisson ratio of thematerial, C = Eh/ (1 − ν ) is the stretching (in-plane) stiffness, D = E h / − ν ) isthe bending stiffness, and α s , α t are two shear correction factors. Note that the conditions(7) are fulfilled for the coefficients (8). (cid:3) Without loss of generality, one can choose the directors { d i } such that d = n is theunit normal to S . In what follows, we assume that d = n .4n the 6-parameter shell theory the drilling rotations are taken into account. The drillingrotation in a given point S can be interpreted as the rotation about the director d . Thegeneral form of rotations about d is R θ = d ⊗ d + cos θ ( − d ⊗ d ) + sin θ ( d × ) , where θ = θ ( x , x ) is the rotation angle and = d i ⊗ d i is the unit tensor.Let us describe next shells without drilling rotations in the framework of the 6-parametershell theory. In case of shells without drilling rotations the strain energy density W must be insensibleto the superposition of rotations R θ about d . This means that W ( E e , K e ) is assumedto remain invariant under the transformation Q −→ R θ Q . (9)In view of the definitions (3), this is equivalent to W ( E e , K e ) = W (cid:0) [ Q T R Tθ ∂ α y − a α ] ⊗ a α , axl[ Q T R Tθ ∂ α ( R θ Q )] ⊗ a α (cid:1) (10)for any angle θ ( x , x ). The following result gives a characterization of shells withoutdrilling rotation. Theorem 3.
Assume that the strain energy function W is invariant under the trans-formation (9) . Then, W can be represented as a function of the arguments W = f W (cid:0) F T F , d F , F T Grad s d (cid:1) . (11) Conversely, any function W of the form (11) is invariant under the superposition ofdrilling rotations (9). Proof.
Firstly, it is clear that the function (11) is invariant under the drill rotation,since F = Grad s y and Grad s d are both independent of rotations about d . Conversely,let us assume that W is invariant under the transformation (9). If we denote by d θ = R θ d = cos θ d + sin θ d and d θ = R θ d = − sin θ d + cos θ d , then we find R θ Q = d θi ⊗ d i . Inserting the last relation into equation (10) and imposing the conditions thatthe derivative of (10) with respect to θ and ∂ α θ are zero, we obtain the equations ∂W∂ E e · c ( E e + a ) + ∂W∂ K e · c ( K e + K ) = 0and ∂W∂ ( n K e ) = , (12)5here we have used the notations a = a α ⊗ a α = d α ⊗ d α and c = d ⊗ d − d ⊗ d .We interpret the relation (12) as a first order linear partial differential equation for theunknown function W ( E e , K e ), which depends on 12 independent scalar arguments (the12 components of E e and K e in the tensor basis { d i ⊗ a α } ). According to the theoryof differential equations (see e.g., [12], Chap. 6), to solve equation (12) we determine 11first integrals of the associated system of ordinary differential equationsd E e d s = c ( E e + a ) , d K e d s = c ( K e + K ) . (13)We observe that the functions U k are first integrals: U = F T F = ( E e + a ) T ( E e + a ) , U = n E e = d F , U = n K e . (14)Indeed, we have d U d s = (cid:16) d E e d s (cid:17) T ( E e + a ) + ( E e,T + a ) d E e d s = , d U d s = n d E e d s = n [ c ( E e + a )] = , in view of (13) , and analogously for U . Finally, another independent first integral isthe function U = F T Grad s d = ( E e + a ) T c ( K e + K ) , (15)since dd s U = by virtue of relations (13). The functions (14) and (15) represent in total11 scalar independent first integrals. Then, the general solution of the first order partialdifferential equation (12) is W = f W (cid:0) F T F , d F , n K e , F T Grad s d (cid:1) , which in view of (12) reduces to (11). (cid:3) Remark 4.
From Theorem 3 follows that the strain energy (11) can be alternativelyexpressed as a function of the following arguments W = c W ( E , γ , Ψ ) , γ = d F = n E e , E = ( F T F − a ) = E e,T E e + sym E e k , Ψ = ( F T Grad s d − Grad s n ) − E Grad s n = ( E e,T + a ) cK e + [ E e,T E e + skew E e k ] b , (16)where we denote by b = − Grad s n . The tensor E is a second order symmetric tensoraccounting for extensional and in-plane shear strains, γ is the vector of transverse sheardeformation, and Ψ is a second order tensor for the bending and twist strains. (cid:3) E and γ coincide with those given in [14], but nevertheless thebending-twist tensor Ψ is different. In this respect, Zhilin proposed the tensor Φ = ( E e,T + a ) K e k − [ E e,T E e + skew E e k ] c b = [ F T ( d × Grad s d ) + n × b ] + E ( n × b ) . (17)We consider that the definition of the bending-twist tensor in the form (16) is moreappropriate since the relation (17) introduces an additional (unnecessary) rotation ofGrad s d in the plane { d , d } .From (16) and (17) we see that in the linearized theory these deformation tensors reduceto: E . = sym( a Grad s u ) , γ . = n Grad s u + c ψ , Ψ . = c Φ . = c Grad s ( a ψ ) + [skew( a Grad s u )] b , where ψ is the vector of small rotations. One can easily see that E , γ and Ψ are indepen-dent of the drilling rotation ( ψ · n ). In this case one gets the Reissner-type kinematicsof shells [13, 9] with 5 degrees of freedom. The isotropic shells without drilling rotations have been investigated in details by Zhilin[14], who determined the form of the strain anergy density W as a quadratic functionof its arguments ( E , γ , Φ ). Suggested by these results, we consider the following strainenergy function for elastic shells without drilling rotations (for the simplified case whenthe coefficients are independent of K )2 c W ( E , γ , Ψ ) = C [(1 − ν ) k E k + ν (tr E ) ] + C (1 − ν ) κ γ + D [ (1 − ν ) k Ψ k + (1 − ν ) tr( Ψ ) + ν (tr Ψ ) ] , (18)where κ is a shear correction factor. If we insert the expression (16) of E , γ and Ψ into(18), then we find the form of W in terms of the strain tensors ( E e , K e ). We observe thatthe resulting energy density W ( E e , K e ) is a super-quadratic function of its arguments. Inthe case of physically linear shells, when only the quadratic terms in ( E e , K e ) are kept,we obtain the simplified expression of the energy density2 W ( E e , K e ) = C [ ν (tr E e k ) + − ν tr( E e k ) + − ν tr( E e,T k E e k )] + C − ν κ k n E e k + D [tr( K e,T k K e k ) − − ν (tr K e k ) − ν tr( K e k ) ] . (19)Finally, if we compare the relation (19) with the general form of the strain energy densityfor isotropic shells (6) we find the following values for α k and β k α = C ν, α = α = C − ν , α = C − ν κ,β = D ν − , β = − Dν, β = D, β = 0 . (20)7 emark 5. The coefficients α k and β k given in (20) for shells without drilling rotationsare different from the values (8) corresponding to shells with drilling rotations. Thisindicates that the two types of shells will have different mechanical responses. (cid:3) Remark 6.
The conditions (7) which insure the existence of minimizers are not satisfiedby the coefficients (20) corresponding to shells without drilling rotations since α − α = 0 ,2 β + β + β = 0, and β = 0. In this case, the strain anergy function (6) is not uniformlypositive definite, and therefore the proof of existence of minimizers is more difficult (inthis respect, see [8]). This is in accordance to the results presented by Neff [7, 8] for aplate model derived directly from the 3D equations of Cosserat elasticity. The comparisonbetween the 6-parameter resultant shell theory and the model developed in [7, 8] has beenpresented in [2, 1]. (cid:3) Acknowledgements.
The first author (M.B.) is supported by the german state grant: “Pro-gramm des Bundes und der L¨ander f¨ur bessere Studienbedingungen und mehr Qualit¨at in derLehre”.
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