Dual‐mixed hp finite element model for elastic cylindrical shells

Abstract

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.

Dual‐mixed variational formulation and hp finite element method for axisymmetric shell problems in elastodynamics

“…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.…”

“…Thus the number of the unknown functions will be 10 including 6 stress-and 4 displacement components. The degrees of the polynomial space applied to the h-and pversion finite element modeling can be found in [1,2,6]. …”

“…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.…”

“…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.…”

“…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]. In the linearized theory of elasticity the functional of the applied principle can be written for time-dependent analyses as [6] …”