# Solving Variation Problems

Then, according to the principle of virtual power, the basic equation of statics is valid [2, 5]: n(Q, T0, v) = A(Q, f, g, v), Vv e F (Q), (1) where n(Q, T0,v) = JT0 --(V® v)Td Q and A(Q, f, g, v) = J f ■ v d Q J g ■ v dy - powers of internal Q Q G2 stress T0 and external forces (f, g) at permissible speeds of displacements v e F(Q) = respectively. Here and below, the dot denotes the scalar product of vectors, and two consecutive dots - the double scalar product (convolution); V = d/dx is the symbolic Hamiltonian operator; the symbol ® is a tensor (dyadic) product; the superscript T denotes the transpose operation [4]. Graphs can also be used to represent direct variation, in which case the graph must be a straight line and pass through the origin.

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Let us introduce into consideration a set of permissible velocities of displacements in ra: V(ra) = and a quadratic functional on the Hilbert space of stresses L2(ra, M3) in the form of K(ra,r) = -"^| f T--B-Td Q, (2) 2 ra J I I ra where B - a tensor of rank 4, depending only on the coordinate x. Depending on the choice of the tensor B the functional of K(ra, T) has different physical meaning: 1/2 • if B = E is the first unit tensor of rank 4 [4], then K is the root-mean-square intensity of the stresses in ra; 1 1/2 • if b = E — I ® I, where I is the unit tensor of rank 2 [4], then K is - the root-mean-square 3 intensity of the shearing stresses in ra; 1 1/2 • if b = — I ® I, then K is the root-mean-square intensity of the hydrostatic pressure in ra; • if b = 1 e —— I ® 11 is the tensor of material hardness, the inverse tensor of elasticity, 2^ 1 v ) where ^ - shear modulus, and v - Poisson's ratio [2, 3, 5, 9], then K is the average specific internal energy of the deformed solid in ra .

In general, the choice of the control subdomain and the tensor are determined by engineering and technical considerations.

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. In [1] the author proposed an original approach, based on a multicriteria estimation of the bearing capacity of a final sample from a geomaterial in a reference (undeformed) or actual (deformed) configuration.

For the simplest configurations of the sample, it is proposed to use the method of generalized Fourier series in Hilbert spaces. Within the framework of this approach, a variational problem for stresses in a given subdomain was set, where, depending on engineering concepts, the root-mean-square values of various stress components are estimated, then based on their combination the bearing capacity of the current body configuration relative to the specified external influences is assessed.

More specifically, two variables x and y vary directly if there is a nonzero constant k such that Y = K . The constant k is called the constant of variation.

Examples of direct variation: Directly varying quantities are commonly represented by statements, graphs, or tables. For example, given that x and y vary directly, the following statement provides enough information to find the constant of proportionality:y is 20 when x is 10.it is assumed that in ra the solid body resists the external impact in the weakest way. In reality the body can balance much more intense ex- 1/2 ternal forces. Key words: geomaterial; variation problem in stresses; bearing capacity; multicriteria estimation; generalized Fourier series; finite-element approximation How to cite this article: Brigadnov I. Direct Methods for Solving the Variation Problem for Multicriteria Estimation of the Bearing Capacity of Geomaterials. The body is in an equilibrium stress-strain state under the action of external stationary effects: a fixed displacement U is given on a non-empty section of a boundary r1 with a Lebesgue measure I T1! Regularization of non-convex strain energy function for non-monotonic stress-strain relation in the Hencky elastic-plastic model. 0, at Q the volume force with a density f is applied, and at the section of a boundary T2 = 5Q\GT1 a surface force is applied with a density g. A substantial numerical example is given for estimating the ﻿^ Igor A Brigadnov Direct Methods for Solving the Variation Problem UDC 539.3 DIRECT METHODS FOR SOLVING THE VARIATION PROBLEM FOR MULTICRITERIA ESTIMATION OF THE BEARING CAPACITY OF GEOMATERIALS Igor A. BRIGADNOV Saint-Petersburg Mining University, Saint-Petersburg, Russia 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 (unde-formed) and actual (deformed). Rocks and concrete are some of the basic construction materials, and therefore the assessment of their bearing capacity is a very urgent scientific and technical problem [3, 13]. 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. ^ Igor A Brigadnov Direct Methods for Solving the Variation Problem From the mathematical point of view, in this problem we search for the minimum of a quadratic functional on a linear affine manifold in a Hilbert space of stresses L2(ra, M3). Further we will consider only tensors B, satisfying the strict condition of Coleman-Noll, for which there exists a constant a a Q|2 for any Q e M3 and almost all x era [2, 9].

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