Symmetry Analysis of a System of Anisotropic Plane Plasticity

Alexander Yakhno University of Guadalajara, Mexico

The classical system of plane plasticity with a general yield criterion f(σ, τ) = 0 has the following form [1]

σ_{x_{1}} − 2τ (θ_{x_{1}} cos 2θ + θ_{x_{2}} sin 2θ) = τ_{x_{1}} sin 2θ − τ_{x_{2}} cos 2θ,

(1) σ_{x_{2}} − 2τ (θ_{x_{1}} sin 2θ − θ_{x_{2}} cos 2θ) = −τ_{x_{1}} cos 2θ − τ_{x_{2}} sin 2θ,

where σ is a hydrostatic pressure, τ is a maximal shear stress, θ + π/4 is the angle between the first principal direction of a stress tensor and the ox_{1}-axis, indices mean the derivation with respect to corresponding variables.

This system describes a plastic plane deformation of anisotropic materials, in particular, is applying in the static of soil medias. In the case when τ = τ(θ) the system (1) has the form

σ_{x_{1}} − θ_{x_{1}} [τ sin 2θ]′_{θ} + θ_{x_{2}} [τ cos 2θ]′_{θ} = 0,

σ_{x_{2}} + θ_{x_{1}} [τ cos 2θ]′_{θ} + θ_{x_{2}} [τ sin 2θ]′_{θ} = 0, (2)

and is a hyperbolic one for any form of the function τ, therefore this form of dependence is used widely in the theory of plasticity [2].

In the report we’ll present a group classification of the system (2) using wellknown methods [3]. There were found some specifications of τ, for which we have the extension of a basic group of symmetries. For Lie algebras, corresponding to this specifications the optimal systems of subalgebras were constructed and some invariant solutions were obtained.

This research was supported by PROMEP (Mexico).

References

[1] V.V. Sokolovsky, Statics of Soil Media, London, Butterworth’s Scientific Publications, 1960.

[2] D.D. Ivlev, Theory of Ideal Plasticity, Moscow, Nauka, 1966 (in russian).

[3] L.V. Ovsyannikov, Group Analisis of Diferential Equations, New York, Academic Press, 1982.