The finite element method – material nonlinear analysis

Like most physical phenomena, the behavior of materials is predominantly non-linear, and the assumption of linearity is only a special case that is perfectly valid within a well-defined range and scale of observation. When the linearity assumption induces significant deviations from actual behavior, it becomes essential to take non-linear phenomena into account in the analysis methodology.

As early as the 18th century, the notion of irreversible behavior and the ultimate capacity of materials aroused the interest of the scientific community. In the 19th century, numerous experiments on iron revealed the threshold of plasticity, as well as its variability as a function of the manufacturing process. The theory of elastoplasticity was established in the second half of the 19th century. Thanks to advances in numerical methods, notably the finite element method, the practical application of this theory to complex structures only became possible in the 1980s.

Broadly speaking, material non-linearity can be broken down into two main categories:

• non-linear elasticity, which results from the non-proportional relationship between stresses and strains, while ensuring reversibility when the structure is unloaded;

• plasticity, which reflects the dissipation of energy during deformation: mechanical energy is transformed into thermal energy, leading to the irreversibility of the material's behavior; this mechanism also reflects the material's ductility, enabling metals to undergo significant elongation before breaking.

The difficulties of finite element analysis of non-linear material behavior arise from the fact that the response of the structural system (i.e. displacements, strains and stresses) is highly dependent on the loading-unloading history, which must be intrinsically taken into account in the analysis procedure, both in terms of formulation and numerical resolution.

This procedure can only be incremental and iterative. It must satisfy the following three principles:

• compliance with the material's behavior law, throughout the loading history ;

• satisfying the static equilibrium of internal and external forces;

• control of the accuracy of local approximation at the scale of material points and global approximation at the scale of the structure.

In this article, the fundamentals of finite element analysis of nonlinear material behavior are developed and illustrated on simple applications. The focus is on elastoplasticity in homogeneous materials. The extension to the case of nonlinear...