68 research outputs found
A Variational Formulation for Finite Elasticity with Independent Rotation and Biot-Axial Fields
No full text
Computational Shell Mechanics by Helicoidal Modeling, II: Shell Element
No full textThe virtual work of stresses developed in Part I for the helicoidal shell model and then reduced to the material surface is taken as one term of a variational principle stated on a two-dimensional domain. The other terms related to the external loads and to the boundary constraints are added here and include a weak-form treatment of the constraints, which becomes necessary in the context of helicoidal modeling. All terms are cast in incremental form and yield a linearized variational principle of the virtual work type for two-dimensional continua, endowed with an internal constraint conjugate to an extra stress field that is able to control the drilling degree of freedom. The virtual functional and the virtual tangent functional are approximated by the finite element method, using helicoidal interpolation for the kinematic field (which ensures objectivity and path independence) and a uniform representation for the extra stress field. A low-order four-node shell element is obtained, with 6 degrees of freedom per node and a unique stress-vector discrete unknown per element. Several test cases demonstrate the performance of the element and its outstanding locking-free behavior
Computational Shell Mechanics by Helicoidal Modeling, I: Theory
No full textStarting from recently formulated helicoidal modeling in three-dimensional continua, a low-order kinematical model of a solid shell is established. It relies on both the six degrees of freedom (DOFs) on the reference surface, including the drilling DOF, and a dual director - six additional DOFs - that controls the relative rototranslation of the material particles within the thickness. Since the formulation pertains to the framework of the micropolar mechanics, the solid shell mechanical model includes a workless stress variable - the axial vector of the Biot stress tensor, referred to as the Biot-axial - that allows us to handle nonpolar materials. The local Biot-axial is approximated with a linear field across the thickness and relies on two vector parameters. On the reference surface, the dual director is condensed locally together with one Biot-axial parameter, leaving the surface strains and the other Biot - axial parameter as the basic variables governing the two-dimensional internal work functional. The continuum-based shell mechanics are cast in weak incremental form from the beginning. They yield the two-dimensional nonlinear constitutive law of the shell in incremental form, built dynamically along the solution process. Poisson thickness locking, related to the low-order kinematical model, is prevented by a dynamical adaptation of the local constitutive law. No hypotheses are introduced that restrict the amplitudes of displacements, rotations, and strains, so the formulation is suitable for computations with strong geometrical and material nonlinearities, as shown in Part II
Consistency Issues in Shell Elements for Geometrically Nonlinear Problems
No full textSome singular concepts and non-standard practices in the FEM solution of geometrically nonlinear shell problems are highlighted and discussed. In particular, four issues are addressed. (i) The question of the drilling rotation: a shell is essentially a non-polar medium in its tangent plane, so the drilling rotation is a redundant d.o.f. to be defined by an extra stress field, and the latter ought to hold as a primary unknown field of the surface mechanics. It is shown that a proper constitutive characterization and a sound variational formulation lead to a full micropolar setting of the shell mechanics with a true three-parametric rotation tensor. (ii) The interpolation of the orientation field on the shell surface. It is shown that an interpolation scheme firmly abiding by the rules of the SO(3) group leads naturally to frame-invariant and path-independent finite elements. (iii) The linearization of the virtual functional. Again, an approach fully complying with the special orthogonal group allows an easy and correct resolution of the mixed virtual-incremental variation variables that issue in nonlinear variational formulations involving finite rotations. (iv) The question of a good discrete representation of curved surface geometries. It is shown that a pole-based kinematics built on an integral orthogonal oriento-position field leads to a fair approximation of curved geometries and allows to build low-order finite elements that are naturally locking-free
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