Introduction to the Finite-Element Method for Elastic and Elasto-Plastic Solids
This introduction to the finite-element method is offered to Earth Scientists with an interest in numerical methods, continuum mechanics and the theory of plasticity. No previous exposure to this material is required. Starting with the theorem of minimum
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finite-element (FE) method is now a well established technique for solving complex boundary value problems. The continuum unknowns, such as the displacements or the temperature, are replaced by a set of nodal unknowns thanks to the introduction of a spatial interpolation. A weak form (integral) of the governing equations or the minimization of an energy provides the set of linear or non-linear equations that are satisfied by these nodal unknowns.
Y. M. Leroy et al. (eds.), Mechanics of Crustal Rocks © CISM, Udine 2011
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Y.M. Leroy
The power of modern computers and the availability of commercial codes, such as Abaqus and Nastran or of more academic codes, such as Cesar (LCPC) and Castem (CEA) in France, render possible the rapid construction of the solutions to coupled non-linear problems. The teaching of the finite-element method reflects this potential and has evolved over the last twenty years. The objectives early on were to train developers which could ”dive” into finite-element codes and implement new classes of elements, solvers and constitutive relations. The teaching is now geared towards researchers who want to make the best use of the above codes, which required hundred of man-years of development. This chapter is certainly part of the second trend since no code development is proposed. It emphasizes, however, the fundamentals of the method for elastic and elasto-plastic materials, from the algorithmic point of view. It is hoped that this chapter will help the researcher in making educated choices while using the standard finite-element packages. There is an impressive number of books and papers which present the finite-element method and the following lines do no pretend to contain an exhaustive list but only reflect the author’s own experience, as a student, as a researcher and then as a teacher in France and in the USA. The first book on the finite-element method I came across is certainly most appropriate for engineers and is due to Zienkiewicz (1977). The book by Hughes (1987) will complement this first reading. The book by Becker (1981) was instrumental in the preparation of a series of lectures. The finite-element method has attracted the attention of applied mathematicians and the book by Reddy (1991) is an introduction to functional analysis readable by engineers. The concise contribution of Johnson (1987) is an interesting second reading in this direction and the book by Ciarlet (1978) for elliptic problems (typically linear elasticity) is of an advanced level. A very original look on the finiteelement method and its reliability is found in the work of Babu´ska and Strouboulis (2001). This chapter also introduces many concepts of continuum mechanics and of the theory of plasticity. Although it has been written to be self-contained, the reader interested to brush up his understanding of continuum mechanics should read the concise book by Chadwick (1976). A more detailed presentation is proposed by Malvern (1969). The presentation of continuum mechanics in this contribution, including the presentation of
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