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The two-field dual-mixed Fraeijs de Veubeke variational formulation of three-dimensional elasticity serves as the starting point of the derivation of a dimensionally reduced shell model presented in this paper. The fundamental variables of this complementary energy-based variational principle are the not a priori symmetric stress tensor and the skew-symmetric rotation tensor. The tensor of first-order stress functions is applied to satisfy translational equilibrium, while the rotation tensor plays the role of a Lagrange multiplier to ensure rotational equilibrium. The volumetric locking-free shell model uses unmodified three-dimensional constitutive equations, and no classical kinematical hypotheses are employed during the derivation. The numerical performance of the related low-order h-, and higher-order p-version finite elements developed for axisymmetrically loaded cylindrical shells is investigated by two representative model problems. It is numerically proven that no negative effect can be experienced when the thickness is small and tends to zero.

In this paper a dimensionally reduced cylindrical shell model based on the dual-mixed variational principle of Fraeijs de Veubeke will be presented. The fundamental variables of this variational principle are the not a priori symmetric stress tensor and the skew-symmetric rotation tensor. The tensor of first-order stress functions is applied to satisfy translational equilibrium. A shell model derived in this way makes the application of the classical kinematical hypotheses unnecessary, and enables us to use unmodified three-dimensional constitutive equations. On the basis of this shell model, a new dual-mixed cylindrical shell finite elements capable of both h-and p-approximation can be derived. The Fraeijs de Veubeke variational principleThe developments of the new dimensionally reduced cylindrical shell model and related dual-mixed finite elements are motivated by the results of the first-order stress function-based finite element formulations addressed in [1,2]. Cylindrical shell models using the equilibrium hypothesis (the definition is given in Section 3) have been developed and investigated assuming axisymmetric problems in [3][4][5].The functional of the two-field dual-mixed Fraeijs de Veubeke variational principle [6] can be derived from the complementary energy functional by adding a Lagrange multiplier term which ensures the symmetry of the stress tensor:where σ pq , D k pq , ϕ,ũ, ∈ qpc and n q are, respectively, the stress tensor, the fourth-order elastic compliance tensor, the rotation vector, the prescribed displacement, the covariant permutation tensor and the outward normal of the bounding surface of the three-dimensional elastic body. Applicability of (1) requires that the stress tensor σ pq satisfies a priori the translational equilibrium equationsand the stress boundary conditions σ k n =p k on S p , where q k is the density of the body forces andp k is the prescribed surface tractions on the surface S p . The displacement boundary condition on surface S u is imposed weakly in (1).By introducing the first-order stress function tensor Ψ k .p , a stress field that fulfills (2) can be obtained:whereσ k is a particular solution to (2). Geometric descriptionIn the Cartesian frame of reference x a , we consider a three-dimensional cylindrical shell with curvilinear coordinate system ξ k . The parametric form of the middle surface is defined by the set of equations, where R is the radius of the middle surface. Let L denote the length and d be the uniform thickness of the shell. Then it occupies the region V = {r(ξ2 }, and the inner and outer surfaces3 About the new cylindrical shell modelIn the development of the dimensionally reduced cylindrical shell model, all the variables are expanded into power series with respect to the thickness coordinate around the shell middle surface. For example: