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**On differential properties of extremals of variational problems of the mechanics of viscoplastic media.**
*(English.
Russian original)*
Zbl 0734.73096

Proc. Steklov Inst. Math. 188, 147-157 (1991); translation from Tr. Mat. Inst. Steklova 188, 117-124 (1990).

The author comments that the slow motion of a viscoplastic medium - in which the state is given by velocity vector field and the stress tensor field - may be regarded as a study in relations between two variational problems.

The first problem concerns the velocity field, the second the stress tensor field. This point of view is discussed in the monograph of P. P. Mosolov and V. P. Myasnikov [Mechanics of rigid-plastic media (1981; Zbl 0551.73002)].

Let \(\Omega\) be a precompact set in \(R^ n\), \(I(v,\Omega)=\int_{\Omega}(\frac12)| \epsilon (v)|^ 2+s_ 0| \epsilon (v)| -f\cdot v]d\underset{\tilde{}} x\), where \(S_ 0=(\sqrt{2}/2)(K_*/\mu)\), where \(\mu\) and \(K_*\) are the viscosity and plasticity coefficients.

The author studies the extremals of this and related functionals in two disjoint subsets of \(\Omega\). In some subset the medium moves like a rigid body, while in the second set separated by a free boundary deformation takes place. Hölder continuity of the tensor of velocities of the stress tensor is proved. However no technique is suggested for identifying the free boundary between the two regions.

The first problem concerns the velocity field, the second the stress tensor field. This point of view is discussed in the monograph of P. P. Mosolov and V. P. Myasnikov [Mechanics of rigid-plastic media (1981; Zbl 0551.73002)].

Let \(\Omega\) be a precompact set in \(R^ n\), \(I(v,\Omega)=\int_{\Omega}(\frac12)| \epsilon (v)|^ 2+s_ 0| \epsilon (v)| -f\cdot v]d\underset{\tilde{}} x\), where \(S_ 0=(\sqrt{2}/2)(K_*/\mu)\), where \(\mu\) and \(K_*\) are the viscosity and plasticity coefficients.

The author studies the extremals of this and related functionals in two disjoint subsets of \(\Omega\). In some subset the medium moves like a rigid body, while in the second set separated by a free boundary deformation takes place. Hölder continuity of the tensor of velocities of the stress tensor is proved. However no technique is suggested for identifying the free boundary between the two regions.

Reviewer: V.Komkov (Wright-Patterson)

### MSC:

74S30 | Other numerical methods in solid mechanics (MSC2010) |

74P10 | Optimization of other properties in solid mechanics |

74C10 | Small-strain, rate-dependent theories of plasticity (including theories of viscoplasticity) |

49J50 | Fréchet and Gateaux differentiability in optimization |

58C20 | Differentiation theory (Gateaux, Fréchet, etc.) on manifolds |

49Q20 | Variational problems in a geometric measure-theoretic setting |