Direct methods for solving the variation problem for multicriteria estimation of the bearing capacity of geomaterials
- Ph.D., Dr.Sci. professor Saint-Petersburg Mining University
Abstract
The article deals with direct methods for solving the variational problem in stresses for multicriteria estimation of the bearing capacity of a geomaterial sample in the current configuration, which can be both reference (undeformed) and actual (deformed). The problem is to minimize the integral quadratic functional from the various stress components in the selected control subdomain on a set of stress fields statically balanced with external influences. For the simplest configurations of the sample, it is proposed to use the method of generalized Fourier series in Hilbert spaces. For complex configurations of a sample with stress concentrators, it is suggested to use finite element approximation with the subsequent minimization of a finite-dimensional quadratic function with linear constraints of equalities. A substantial numerical example is given for estimating the bearing capacity of a sample from a geomaterial under pure compression.
References
- Brigadnov I.A. Multi-criteria assessment of the bearing capacity of geomaterials. Zapiski Gornogo instituta. 2016. Vol. 218, р. 289-295 (in Russian).
- Lur'e A.I. Nonlinear theory of elasticity. Moscow: Nauka, 1980, р.512 (in Russian).
- Nikolaevskii V.N. Geomechanics and Fluid Dynamics. Moscow: Nedra, 1996, р. 447 (in Russian).
- Pal'mov V.A. Elements of tensor algebra and tensor analysis. St. Petersburg: Izd-vo Politekh. un-ta, 2008, р. 109 (in Russian).
- Pozdeev A.A., Trusov P.V., Nyashin Yu.I. Large elastic-plastic deformations. Moscow: Nauka, 1986, р. 232 (in Russian).
- Sea Zh. Optimization. Theory and algorithms. Moscow: Mir, 1973, р. 244 (in Russian).
- Suknev S.V. Application of nonlocal and gradient criteria for the evaluation of fracture of geomaterials in zones of tension
- stress concentration. Fizicheskaya mezomekhanika. 2001. Vol. 14(2), р. 67-75 (in Russian).
- S'yarle F. The finite element method for elliptic problems. Moscow: Mir, 1980, р. 512 (in Russian).
- S'yarle F. Mathematical theory of elasticity. Moscow: Mir, 1992, р. 472.
- Trenogin V.A. Functional Analysis. Moscow: Nauka, 1980, р. 496.
- Cherepanov G.P. Mechanics of brittle failure. Moscow: Nauka, 1974, р. 640.
- Brigadnov I.A. Regularization of non-convex strain energy function for non-monotonic stress-strain relation in the Hencky elastic-plastic model. Acta Mechanica. 2015. Vol. 226. Iss. 8, p.2681-2691.
- Verruijt A. Computational geomechanics. Dordrecht: Springer Science+Business Media, B.V., 1995, p.384.