This chapter deals with the phenomenological and mathematical modelling of plastic solids according to the schematic classification of Chapter 3: rigid, perfectly plastic solid; elastic, perfectly plastic solid; elastoplastic solid. It can be said that the first scientific work concerning plasticity goes back to Tresca's memoir in 1864 on the maximum shear stress criterion. Although the isotropic flow law was formulated as early as 1871 by St. Venant and Levy, its applications to structures had to wait until about 1950 when limit theorems were discovered. Since 1970 the availability of fast and large computers has led to application of the theory to practical problems. Proportional loading (where principal stresses do not rotate at any point Of a structure) provides a large field of application for theories based on isotropic hardening.

Hardening, as we have seen in Chapter 1, is almost always anisotropia This aspect of plasticity must be considered as soon as the loading is no longer proportional and especially under cyclic loads. Prager, around 1950, gave the first simple formulation of anisotropy, namely kinematic hardening, on which most of the present theories are based.

Domain of validity and use

The theory of plasticity is the mathematical theory of time-independent irreversible deformations; some comments on its physical nature were given in Chapter 1.