Modélisation numérique des solides par éléments finis volumiques basés sur le concept SFR (Space Fiber Rotation).

Abstract

The main objective of this thesis is to develop a new concept to enrich the 3D low-order finite elements. The major application of proposed models is the numerical modeling of solid mechanics and three-dimensional structures problems. In this context, in a first part, a new family of solid finite elements, with three translational and three rotational degrees of freedom per node has been presented. This family, named SFR, is based upon SFR (Space Fiber Rotation) concept. Using the rotation of a material fiber in 3D space, the SFR approach allows to get a more accurate displacement field, which becomes quadratic without changing the number of nodes of the linear elements. Based upon the SFR concept, eight-node brick element SFR8 and six-node wedge element SFR6 are proposed. In addition to that, a non-conforming version of SFR8, named SFR8I, is developed to overcome the Poisson’s ratio locking. The SFR8I formulation includes three incompatible modes in the natural space of the element that are then eliminated by a static condensation technique. A reduced integration technique is used to integrate the SFR solid finite elements in order to avoid locking effects and to achieve an attractive, low-cost formulation. All remaining zero energy modes, resulting from the reduced integration and the equal rotation modes in both elements are efficiently controlled using special stabilization techniques. In a second part, two different modeling approaches are used for analysis of thick composite structures. A common feature in both approaches is to use the SFR concept. The first approach is to use one solid element per layer. For the second approach, the multilayered solid elements which can represent different material layers with varying fiber angles. By defining several layers with different materials and ply orientation inside one layered solid, number of elements through the thickness is remarkably reduced. These elements use two steps to calculate the full stress tensor. In the first step the in-plane stresses are computed from the material law using a displacement approximation, and then the transverse stresses are calculated from the 3D equilibrium equation. In a third part, the application of SFR elements is extended to include geometric nonlinear problems. The formulation of the SFR elements for nonlinear problems in elasticity is presented. A total Lagrangian approach is adopted for the element formulation. The set of nonlinear equilibrium equations, obtained by appropriate energy minimization, is solved using the Newton-Raphson method. All these models are implemented in the finite element code REFLEX. To illustrate the capacities of these elements, its performances are evaluated on varied patch-tests in linear or non-linear configurations, which are used in the literature to test the finite elements of solid type. The new elements pass the patch tests for solid element and have the proper rank. Numerical results show that the SFR elements are noticeable in low sensitivity to mesh distortion and in high-accuracy of stresses. The SFR models prove to be an interesting alternative with regards to classical solid finite elements models. In order to verify the accuracy of the SFR elements for composite multilayer structures, a several problems of laminate composite are solved. The present solutions are compared with those obtained using three-dimensional elasticity theory and those available in literature. The analysis gives accurate values for displacements and stresses compared to other formulations developed by other researchers. The use of layered solid elements offers a possibility to model thick composite layups in detail with acceptable times as well as an acceptable model size.