Solving Variation Problems

Solving Variation Problems-86
When assessing the bearing capacity of a deformable solid with respect to specified external influences, it is necessary to take into account the features of the topology of the current configuration, since the most critical are the stress concentration zones (holes, grooves, cuts, etc.). Therefore, we single out a bounded Lipschitz subdomain ra e Q, containing a stress concentrator. 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.

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In this case the functional K(ra, T) is strictly convex and coercive on L2(ra, M3) and therefore the existence and uniqueness of the solution of the problem (3) is guaranteed [6, 9].

Thus, in the control subdomain ra we consider the weakest stress fields, universally balanced in current configuration of the body Q, i.e.

Graphs can also be used to represent direct variation, in which case the graph must be a straight line and pass through the origin.

If the graph is a straight line, but does not pass through the origin, then the relationship it represents cannot be a direct variation.

This constant ratio is called the rate of change, or slope: This is different from the condition imposed by direct variation in that the quantities themselves have a constant ratio.

Finally, note that direct variation is sometimes called direct proportionality, in which case the constant of variation is called the constant of proportionality.^ 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]. Direct variation should not be confused with linearity.Two quantities are linearly related if they have a constant ratio of change.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]. The influence of the remaining part of the body on the selected (further control) subdomain is replaced by the following boundary conditions: a section of the subdomain boundary y1 e r1 is fixed, the rest of the boundary y2 = dra/y1 has a preset external force n T0, where n is the external in relation to a control domain normal to the boundary y2. Note that the fixed section of the boundary y1 may be absent. 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.


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