Approximate solution of a nonlinear system of equations for two-phase media
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APPROXIMATE SOLUTION OF A NONLINEAR SYSTEM OF EQUATIONS FOR TWO-PHASE MEDIA V. V. Skopetskii,a† O. A. Marchenko,a and T. A. Samoilenkoa
UDC 532.546:539.3
A mixed initial–boundary-value problem for the nonlinear equations describing the dynamic consolidation of water-saturated soils is considered. The error of a time-continuous approximate generalized solution is estimated using the finite-element method. Keywords: nonlinear system, differential model of two-phase soil media, generalized solution, finite-element method, estimate of approximate generalized solution.
A system approach to constructing differential models of water-saturated soils is based on Biot’s model [1] that describes the dynamics of multiphase soil media using the mechanical properties of the soil skeleton, which are necessary to solve specific problems. Since it is difficult to determine the stress-strain state of two-phase soil masses, all the factors influencing the deformation of soil cannot be taken into account yet. The elastic-plastic behavior of a soil can be described by physically nonlinear elastic theory where the bulk strain e u is related to the average (hydrostatic) pressure through the bulk modulus K, and the shear strain intensity e i is defined by both the tangential stress intensity and the hydrostatic pressure in terms of the shear modulus G. An analysis of experimental data shows that there is a certain relationship between the K and G moduli, which depends on the argument e i / e u , and according to some data, it can be assumed linear [2]: K = G ( A - Be i / e u ) . In practice, it seems expedient in many cases not to divide the bulk strain into two parts but to use the general bulk strain modulus. For example, to determine the LamÁ coefficients, elastic theory assumes that l and m are only dependent on 3K (1 - 2n ) 2 the bulk modulus K [3]: m = , K = l + m, where n is Poisson’s ratio. Such an approach is implemented in the 3 2 (1 + n ) present paper in forming the nonlinearity of the equations of dynamic consolidation of water-saturated soils [2, 4, 5]. The system of equations has the following form: r p (1 - m ) - MW
¶ 2 w sk ¶t
2
+ P (w )
¶ ( w sk - w w ) - ( Aw sk ) ( w sk ) ¶t
1- m [(1 - m) grad div w sk + mgrad div w w ] = F1 , m ¶2ww ¶ r wm + P ( w ) ( - w sk + w w ) 2 t ¶ ¶t
- M W [(1 - m ) grad div w sk + m grad div w w ] = F2 ,
(1)
(2)
where w sk ( x, y, t ) = ( u sk ( x, y, t ), u sk ( x, y, t ))T , w w ( x, y, t ) = ( u w ( x, y, t ), u w ( x, y, t ))T , ( x, y, t ) ÎW T , W T = W ´ ( 0, T ] , u sk and u sk are the horizontal and vertical components of the displacement vector for the soil skeleton; u w and u w are a
V. M. Glushkov Institute of Cybernetics, National Academy of Sciences of Ukraine, Kyiv, Ukraine, [email protected]. Translated from Kibernetika i Sistemnyi Analiz, No. 4, pp. 73–88, July–August 2008. Original article submitted November 15, 2007.
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524
1060-0396/08/4404-0524
©
2008 Springer Science+Business Media, Inc.
the corresponding components of the water displacement
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