Available online at ScienceDirect. Procedia Engineering 153 (2016 ) 45 50

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1 Available online at ScienceDirect Procedia Engineering 153 (2016 ) XXV Polish Russian Slovak Seinar Theoretical Foundation of Civil Engineering The solution of the nonlinear probles of elasticity theory for ground assif considering the inhoogeneity caused by soil oisture Vladiir I. Andreev a,**, Ludila S. Polyakova a, Anstoliy S. Avershyev b a Moscow State University of Civil Engineering (National Research University), 26 Yaroslavskoye Shosse, Moscow, , Russia b JSC "Rocket and Space Corporation" Energia "after S.P. Korolev", 4a, Str. Lenin, Korolev, Moscow Region, Russia Abstract The paper deals with the proble for spherical and cylindrical holes in the ground array, heterogeneity is due to changes in huidity. For any soils (clay, loess rock) is characterized by significant influence oisture on the echanical characteristics, particularly in the odulus of elasticity and Poisson's ratio. The calculation also takes into account the non-linear nature of the soil deforation. As the echanical odels are considered thick-walled cylinder (axisyetric proble) and the hollow ball (centrally syetric proble), the inner radius of which is equal to a, and external - b >> a. Nonlinear proble with variable elastic paraeters E 2016 The The Authors. Authors. Published Published by by Elsevier Elsevier Ltd. Ltd. This is an open access article under the CC BY-NC-ND license ( Peer-review under responsibility of the organizing coittee of the XXV Polish Russian Slovak Seinar Theoretical Peer-review Foundation under of Civil responsibility Engineering. of the organizing coittee of the XXV Polish Russian Slovak Seinar Theoretical Foundation of Civil Engineering. Keywords: Pfisical nonlinearity; Inhooheneity; Moisture; Successive approxiations; Sweep ethod. Noenclature E ν odulus of elasticity Poisson coefficient * Corresponding author. Tel.: ; fax: E-ail address: asv@gsu.ru The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license ( Peer-review under responsibility of the organizing coittee of the XXV Polish Russian Slovak Seinar Theoretical Foundation of Civil Engineering. doi: /j.proeng

2 46 Vladiir I. Andreev et al. / Procedia Engineering 153 ( 2016 ) σi stress intensity εi strains intensity E, A, α, paraeters of chart σi( εi) w oisture ρ density p pressure γ specific weight of the soil H hole location depth 1. State of the proble To describe the nonlinear character of deforation (Fig. 1) the following relationship is used: σ i = f( ε i) = Eεi Aε i α, (1) where E, A - variable coefficients that depend on the soil oisture: = ( ), ( ) = ( / A ), ( ) ( / ) Ew ( ) E0 w/ w E s Aw A0 w w s In our case w 0 = 0,12 - natural soil oisture (when r = b), and w s = 0, 22 - oisture saturated soil (when r = a). Based on the works [3-5], the following experiental values for the coefficients of clay were obtained: α w =α 0 w ws α. (2) E 0 = 2, 291 MPa, E = 1, 5785, A 0 = 6, 5137 MPa, (3) A = 0,8427, α 0 = 1,8107, α = 0, In this case w 0 = 0,12 - natural soil oisture (at r = b ), and w s = 0, 22 - oisture saturated soil (at r = a). Effect of huidity on Poisson's ratio of the soil is expressed by dependence [6] Fig. 1. The nonlinear deforation diagra; ρs ν= 0.14 w , (4) E s,ax =σu / ε u secant odulus fracture ρw e point. where w is oisture; ρ s - the density of the soil particles, to clay ρ s = 2.74 g / c 3 ; ρw -density of water ρ w = 1g / c3; e - soil porosity, e=1.05. On the borders of a cylindrical or spherical array with an aperture located on sufficiently great depth the boundary conditions are: r = a, σ r = 0 ; r = b, σ r = p. (5) Here p =γh - earth pressure. In the solution we took b = 10.44a, p = 0.15 MPa. 2. Solution 2.1. Axisyetric proble Moisture distribution along the radius of the cylinder is described by [7,8]: ( a) ws ( b) ln ( ρ / ρ ) w0 ln ρ/ ρ w( ρ ) = ln ρ/ ρ b a, (6)

3 Vladiir I. Andreev et al. / Procedia Engineering 153 ( 2016 ) where ρ= r /, ρ b = b /. Fro Boundary conditions ρ= 1, w s = 0.22 ; ρ= 10.44, w 0 = 0.12 we obtain the following relationship between the relative oisture of the radius: w( ρ ) = w s (1 0,1938 ln ρ, ) (7) Then the functions E, A and take the for: Er ( ) 0 1 0,1938 ln E = E ( ρ ), Ar ( ) = A ( 1 0,1938 ln ρ A ), ( r) ( ln ) 0 α =α α 0 ρ, (8) Thus diagra of soil deforation varies with r (Figure 1.) and is described by the relation: A α0( ln ρ) α σ = f ε = E ρ ε A ρ ε (9) ( ) ( ln E ) ( ln ) i i 0 i 0 i On the basis of (4) and (6) Poisson's ratio is a function of radius: ν ( r) = , 0156 ln ρ (10) Solution of the proble for the nonlinear aterial was carried out by successive approxiations. At the zero stage solution is sought to the proble of linear-elastic aterial. The resolving equation in ters of stresses for axisyetric proble in the absence of ass forces and forced deforation has the for [2]: σ r +ϕ() r σ r +ψ() r σ r = 0, (11) where 1 E 2νν ϕ () r = 3 r r r E 2 ; 1 ν (12) 1 1 2ν E ( 1+ 4ν) ν Fig. 2. The distribution of oisture along the radius ψ () r = + r 1 E 2. of the cylinder ν 1 ν By solving Eq. (11) with conditions (5) by sweep ethod with variable pitch will get stresses σ r. The sweep ethod used the variable step along the radius. In this first step equal 0.2a and each following should be increased 1.2 ties. Thus, the section (a, b) broken down into 25 intervals. Thickening of the grid near the hole iproves the accuracy of calculations in the area of stress concentration. In its turn at great value b we can use the second boundary condition (5). Fro the equilibriu equation we have: σ θ = r σ r +σ r, (13) In the proble of plane strain state deforations are equal: () r 1+ν ε z = 0, ε θ = ε r = ( 1 ν() r ) σr ν() r σθ Er (). (14) As a result for deforation intensity, we obtain the expression: ( +ν r ) ( ( 1 ( r) ) ( r) ) ( ) ε i = ε r = σr ν σθν. (15) 3 3 Er ( ) Knowing intensity of the strain can be deterined secant odulus at each point of the body:

4 48 Vladiir I. Andreev et al. / Procedia Engineering 153 ( 2016 ) σi f ( εi) α () ()[ ] ( r ) E 1 c = = = E r A r εi. (16) εi εi In the next stages of approxiation equation (11) is solved again in which the functions and have the for: where 1 E1 2νν () 3 1 ϕ r = r r 1 r E ν1 3 EEr c ( ) E1 = 3 Er ( ) + Ec 2 Ecν( r),, ( 1 4 ) 1 1 2ν E1 + ν1 ν 1 ψ () r = + r 1 E ν ν1, (17) 3E1 ν 1 = 1. (18) 2E The expression (18) includes a secant odulus E calculated in the previous step. Each subsequent stage of the iterative process akes solution close to the exact. Fig. 3 shows the curves of the stress distribution in the nonlinear aterial obtained in the eighth approxiation, and, for coparison, the stress distribution in the linear-elastic aterial. Fig. 3. Diagras of stresses in the cylindrical shell: 1 - linear-elastic inhoogeneous aterial; 2 - nonlinear elastic inhoogeneous aterial 2.2. Centrally syetric proble Moisture distribution along the radius of the sphere (Figure 5) is described as follows [9]: [9]: ( ) ρb ws w0 1 ws ρb w0 w( ρ ) =, (19) ρb 1 ρ ρb 1 Fro conditions ρ= 1, w s = 0.22 ; ρ= 10.44, w 0 = 0.12 relative oisture of the radius: we obtain the following relationship between the w( ρ ) = w s + 0, 497 ρ, (20) Then the functions E, A and take the for:

5 Vladiir I. Andreev et al. / Procedia Engineering 153 ( 2016 ) Er ( ) / 0,497 E = E ( ρ+ ), Ar ( ) = A ( / ρ+ 0,497) A, r ( ) In turn Poisson's ratio is described by equality: 0.04 νρ ( ) = ρ 0 α ( ) =α / ρ+ 0,497 α. (21) By analogy with p. 2.1 solution was obtained by the ethod of successive approxiations using the sweep ethod. The calculation results are shown in Fig. 4. (22) Fig. 4. Diagras of stresses in the spherical shell: 1 - linear-elastic inhoogeneous aterial; 2 - nonlinear elastic inhoogeneous aterial According to calculations inhoogeneity of Poisson's ratio has little effect on the state of stress. For exaple, for a spherical array with ν = 0,41 axiu stress greater by 2.8% as copared with the calculation for the function ν(), r and for the cylindrical array stress is less by 1.7%. The validity of the results in both probles is confired by a test of static balance half of the array with a hole under the influence of external pressure p and stresses σ θ (Fig. 5). For a cylindrical array equilibriu condition is: b π Y = 0; 2 σ θdr = p sin θ bdθ= 2 pb, a 0 and for spherical array: b π 1 2 Y = 0; σ θrdr = psin θ bcos θ bdθ= pb. Fig. 5. To condition of static equilibriu. 2 a 0 3. Conclusions The nonlinear behavior typical for any aterials, such as soil, concrete, polyers and coposites based on the, etc. The account in the calculation of non-linearity leads to a reduction of stresses [10, 11, etc.]. As for the results of this work, it should be noted that reducing stresses near the holes in the cylindrical and the spherical array copared with the linear aterial was about 31-32%. This paper investigates the effect of non-linearity on the stress in an inhoogeneous aterial. This is the novelty of the research. The feature of this work is the fact that the diagra σi εi different at different points of the body. This requires additional studies of physical and echanical properties of aterials. The first results of the authors on the application of analytical and nuerical ethods to the solution a considered proble, published in [12-14].

6 50 Vladiir I. Andreev et al. / Procedia Engineering 153 ( 2016 ) Acknowledgeent This work was financially supported by the Ministry of Russian Education (state task #2014/14). 4. References [1] V.I. Andreev, Y.N. Malashkin, Calculation of thick-walled pipes fro the nonlinear elastic aterial.building echanics and calculation of constructions. 6 (1983) [2] V.I. Andreev, Soe probles and ethods of echanics inhoogeneous bodies. ASV Publ. House, Moscow, [3] O.R. Golly, Using the laws of swelling clay soils in construction. Reconstruction of cities and Geotechnical Engineering, 8 (2004) [4] ODN Design nonrigid road clothes, [5] V.I. Andreev, A.S. Avershyev, Nonstationary proble oisture elasticity for nonhoogeneous hollow thick-walled cylinder. Transactions of International Conference on Fluid Structure Interaction, Gran Canaria, April, WITpress (2013) [6] E.V. Pioro, Deforation and acoustic properties of clay soil in results of laboratory geotechnical and ultrasound exaination. PhD thesis. Loonosov Moscow State University (2014), p.162 [7] V.I.Andreev, A.S. Avershyev, Stationary probles of oisture-elasticity for inhoogeneous thick-walled shells. Advanced Materials Research (2013) [8] V.I.Andreev, A.S. Avershyev, About influence of oisture on stress state of soil taking into account inhoogeneity, International Journal for Coputational Civil and Structural Engineering. 9 #3 (2013) [9] V.I. Andreev, A.S. Avershyev, Axisyetric proble oisture elasisity in inhoogeneous spherical array. International Journal for Coputational Civil and Structural Engineering, Vol. 10, Iss. 1 (2014) [10] L.U. Stupishin, A.G. Kolesnikov, Geoetric Nonlinear Orthotropic Shallow Shells Investigation//«Applied Mechanics and Materials» (2014) [11] L.U. Stupishin, K.E. Nikitin, Mixed finite eleent of geoetrically nonlinear shallow shells of revolution // «Applied Mechanics and Materials» (2014) [12] V.I. Andreev, Equilibriu of a Thick-walled Sphere of Inhoogeneous Nonlinear-elastic Material. Applied Mechanics and Materials Vols (2013) [13] V.I. Andreev, L.S. Polyakova, Analytic solution of a physically nonlinear proble for an inhoogeneous thick-walled cylindrical shell. Vestnik MGSU, 11 (2015) [14] V.I. Andreev, L.S. Polyakova, Calculation of a Thick-walled Inhoogeneous Cylinder of Nonlinear-elastic Material. Advanced Materials and Structural Engineering Taylor & Francis Group, London, (2016)