Chapter VIII - Hencky’s and Deformation Theories of Plasticity
Section snippets
Timoshenko Beam in Hencky’s Model with Uncertain Yield Function
We start with the Timoshenko model of beam bending (see also Section 8 and Section 9) in combination with a simple model of plasticity, namely Hencky’s. Passing to a dual variational formulation in terms of bending moments and shear forces, we obtain an analogy of the Haar-Kármán principle (see (Nečas and Hlaváček, 1981), for example), i.e., a modification of the Castigliano principle of minimum complementary energy. Thus the well-known compatibility method (see (Neal, 1964), for instance) can
Torsion in Hencky’s Model with Uncertain Stress-Strain Law and Uncertain Yield Function
In this section, we consider an orthotropic twisted bar of an arbitrary cross-section under the classical Saint-Venant hypotheses; see (Nečas and Hlaváček, 1981), for instance. The constitutive law suggested by Hencky ((Duvaut and Lions, 1976), for example) is one of the simplest mathematical models describing the elasto-plastic behavior of solid bodies. By employing this law, we can solve the torsion problem in terms of stresses on the basis of the Haar-Kármán principle (Lanchon, 1970).
What is
Deformation Theory of Plasticity
One of the simplest models of elasto-plastic bodies is represented by the deformation theory of plasticity; see (Kachanov, 1959), (Langenbach, 1976), or (Nečas and Hlaváček, 1981). It can be identified with an elastic model obeying special nonlinear stress-strain relations. From the mathematical point of view, the model is advantageous, being formulated by means of potential and strongly monotonous operators. The crucial role in the model is played by a material function, which has to satisfy
Bibliography and Comments on Chapter VIII
Hencky’s model of elastoplasticity was introduced by (Hencky, 1924) to solve static or quasi-static problems without taking into consideration the history of deformations. See (Duvaut and Lions, 1976) for an analysis and applications of this model. A numerical analysis of torsion problems is presented in (Glowinski et al., 1976), (Falk and Mercier, 1977), and (Hlaváček, 1981), for instance.
The model based on deformation theory also originates in (Hencky, 1924). It is analyzed in (Kachanov, 1959