A unifying formulation for nonlinear solid mechanics and Finite Element Analysis

Abstract

A unifying formulation for nonlinear solid mechanics and Finite Element Analysis The finite element method is a well-known technology that allows to obtain an approximation to the real structural behaviour of a continuum solid media subjected to external forces. Its use is widely extended in civil engineering and many other fields, such as naval or aerospace engineering. This formulation can be derived under the linear or the nonlinear analysis framework. If the displacements and their corresponding gradients are assumed to be small, the analysis is considerably simplified, and it turns out to be carried out under the assumptions of the linear theory. However, if the displacements and/or the displacement gradients become large, the nonlinear analysis arises. As both analyses are based on different assumptions, they lead to completely different structural responses. And the accuracy of the results depends on the precisión of the assumptions made. That is, if the structure does not experiment small displacements or small displacement gradients, the linear analysis leads to unacceptable results that significantly differ from the real behaviour. Before running a structural simulation, the engineer has to decide, based on its experience and intuition, if the linear assumptions are correct. If the real structural response does not verify the linear assumptions, the linear analysis must be discarded and a nonlinear one should be carried out in order to obtain accurate results. Therefore, the assumptions made about the magnitude of both the displacements and the displacement gradients are quite important, since they define the theoretical framework of the structural analysis. The implications of each assumption have to be clearly defined. Most references in the existing literature do not clearly identify the implications of these assumptions. Therefore, one of the main aims of this work is to clearly identify them and to properly define both the linear and the nonlinear mathematical models that governs the structural behaviour according to each analysis. To accomplish this goal, a unifying formulation of both the linear and nonlinear solid mechanics complete and detailed is proposed. This formulation allows to completely describe and understand the deformation that an elastic solid experiments over time. A novel, simple, and clear nomenclature is proposed, in order to properly state the solid mechanics principles and the strictly necessary equations that describe this deformation process. Once the mathematical models are well-posed, the finite element method can be applied. A complete original derivation in both linear and nonlinear theory is presented. The linear one is also performed in order to compare this well-known derivation with the nonlinear version. One of the main differences between both formulations lies in the application of the external forces. In general, the linear formulation leads to a linear behaviour, whereas the nonlinear one drives to a nonlinear one. As long as the response is linear, the total load can be applied in only one step, and the load superposition principle usually applied in linear theory holds. However, this principles can no longer be applied when dealing with a nonlinear behaviour. If the response is nonlinear, a given load state has multiple possible solutions. Therefore, the total load can not be applied in only one step, and the load history has to be taken into account to reach the correct solution. To overcome these inconveniences, the external loads are usually applied according to an incremental loading process. This incremental strategy is actually a suitable procedure, since the structural response corresponding to each load step has to be solved iteratively. This procedure needs to start iterating from a close approximation to the solution. If the incremental loads are small enough, the result of the previous load step can be adopted to start the iterative procedure, and the convergence should be guaranteed. Many reference textbooks and research papers address the derivation of the nonlinear finite element formulations. Nevertheless, there is no consensus about a common nomenclature and notation. Moreover, the hypotheses made along these derivations are no clearly specified or are not even stated. Therefore, to completely comprehend the underlying physics and the essence of the proposed algorithms, a detailed overview which clarifies this knowledge becomes necessary. In this thesis, a great effort is made to clearly identify the intermediate hypotheses, and extensively analyse the origin and composition of the matrices that arise in nonlinear analysis. A detailed guideline that facilitates the deep comprehension of this powerful technology is proposed. This work states a unifying, clear and complete formulation for the nonlinear analysis field, so the extension of some research lines that have been carried out in linear theory until now becomes possible.