## On locking materials.(English)Zbl 0594.73037

Locking materials are hyperelastic materials for which the strain tensor is constrained to stay in some convex set. The strain energy density of these materials is convex. This paper is concerned with the boundary value problem of locking materials. The boundary value problem is formulated as a dual variational problem. The author gives the dual variational principles governing the equilibrium state of an elastic- locking material. Two minimum theorems of Prager are extended and inf-sup equality is proved by means of a penalty method.
The locking limit analysis is presented and discussed. The upper and lower bound formulae for the safety factor are obtained by the simultaneous use of the two variational principles (statical and kinematical methods).
The results of convex analysis are applied to get the mathematical proof of the duality relations and of the equality inf-sup. The appendices of the paper deal with the stress concentrations in a problem of torsion and with the limited compressibility. In limited compressibility, only volume changes are constrained. These materials lead to simpler mathematical problems. The existence of a stress solution which satisfies the boundary conditions is proved for the limited compressibility material.
Reviewer: I.Ecsedi

### MSC:

 74R20 Anelastic fracture and damage 49J40 Variational inequalities 74S30 Other numerical methods in solid mechanics (MSC2010) 49J20 Existence theories for optimal control problems involving partial differential equations 74C99 Plastic materials, materials of stress-rate and internal-variable type 74G70 Stress concentrations, singularities in solid mechanics
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### References:

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