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Key words Dimensional reduction, cylindrical shell model, axisymmetric load, dual-mixed locking-free hp finite element.A dimensionally reduced cylindrical shell model using a three-field complementary energy-based Hellinger-Reissner's variational principle of non-symmetric stresses, rotations, and displacements is presented. An important property of the shell model is that the classical kinematical hypotheses regarding the deformation of the normal to the shell mid-surface are not applied. A dual-mixed hp finite element model with stable polynomial stress-and displacement interpolation and C 0 continuous normal components of stresses is constructed and presented for the bending-shearing problem, using unmodified three-dimensional inverse stress-strain relations for linearly elastic materials. It is shown through an example that the convergences in the energy norm as well as in the maximum norm of stresses and displacements are rapid for both h-extension and p-approximation, not only for thin but also for moderately thick shells loaded axisymmetrically, even if the Poisson ratio is close to the incompressibility limit of 0.5.

Three types of multi-field dual-mixed variational principles using a priori non-symmetric stress field will be derived for dynamic problems of linearly elastic solids. Starting from the functional of complementary Hamilton's principle it is shown how this variational formulation is modified with applying the Lagrange multiplier technique, to obtain a new four-field dual-mixed functional. The independent fields of this functional will be the displacement vector, the non-symmetric stress tensor, the skew-symmetric rotation tensor and the momentum vector. Then we present how to treat appropriately the six different types of initial conditions in a weak and/or strong sense by adding a new integral expression to the four-field functional. Modifying this functional through a Legendre transformation results in a new complementary energy-based five-field dual-mixed functional which allows the momentum-and velocity vectors to be independently approximated. A complementary energy-based three-field dual-mixed variational formulation is obtained by eliminating the momentum-and velocity fields through the strong enforcement of the kinematic equation and the impulse-velocity relation from the five-field principle.

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A new dimensionally reduced axisymmetric shell model is presented briefly for modeling time-dependent problems. This is based on the extended version of the three-field dual-mixed variational formulation of elastostatics [1,2] to linear elastodynamics, the independent fields of which are the non-symmetric stress tensor, the displacement-and the rotation vector. An important property of the related shell model is that the classical kinematical hypotheses regarding the deformation of the normal to the shell middle surface are not used, i.e., unmodified three-dimensional constitutive equations are applied. The computational performance of the new h-and p-version axisymmetric shell finite elements is tested through a representative cylindrical shell problems. The development presented in this paper has been motivated by the fact that efficient dual-mixed hp plate and shell finite elements were managed previously to be developed for elastostatics by [1][2][3][4][5]. the fundamental variables of which are the a priori non-symmetric stress tensor σ k , the displacement vector u k and the rotation vector φ s . Furthermore V denotes the volume of the body in the undeformed configuration, S = S σ ∪S u (S σ ∩S u = ∅) defines the bounding surface of V , k s is the covariant permutation tensor and u k is the prescribed displacement vector on the surface part S u with outward unit normal n , as well as b k and ρ stand for, respectively, the density of the body forces and the material, and t ∈ [t 0 , t 1 ] defines a closed time interval. The fourth-order tensor C pqk with symmetry properties C pqk = C pq k = C k pq is the elastic compliance tensor. The solution of the linear elastodynamic problem can be sought as the stationary point of functional (1). The subsidiary conditions to (1) are the stress boundary conditionswhere p k are prescribed surface tractions on S σ , as well as the initial conditions2 Basic assumption of the shell modelLet us consider an axisymmetric shell of length L as a three-dimensional body. We assume axisymmetric boundary conditions, as well as homogeneous and isotropic material properties, in this case the variables depend only on the meridian coordinate ξ 1 , the thickness coordinate −d/2 ≤ ξ 3 ≤ d/2 and the time t. The shell-thickness d is considered to be constant. The fundamental variables, i.e., the stresses σ kλ , σ k3 , the displacements u k and the rotations φ s are approximated by polynomials of first-and second-degree in ξ 3 :see the details in [1,2,6]. It is important to note here that this is the only hypothesis used in the derivation of the dimensionally reduced shell model. Then the number of the independent stress components is reduced by a priori satisfaction of the timedependent prescribed surface loads on the inner and outer surfaces of the shell and by the elimination of the rotations. Our investigations is restricted to bending-shearing problems of the axisymmetic shell. Thus the number of the unknown functions will be 10 including 6 stress-and 4 displacement components. The degree...