STUDY OF A DEFORMATION LOCALIZATION DIRECTION IN SLOW MOTION OF A GRANULAR MEDIUM Текст научной статьи по специальности «Физика»

DOI: 10.17516/1997-1397-2022-15-5-586-597 УДК 539.37

Study of a Deformation Localization Direction in Slow Motion of a Granular Medium

Olga I. Kuzovatova*

Siberian Federal University Krasnoyarsk, Russian Federation

Received 09.11.2021, received in revised form 18.03.2022, accepted 25.06.2022 Abstract. This paper is devoted to the study of the direction of the deformation localization lines in a slow gravity flow of a granular medium in convergent channels with various geometric characteristics. Variational principles of the theory of limiting equilibrium, established within the framework of a special mathematical model of a material that resist tension and compression differently, are used. Assuming a linear deformation localization zone we obtain safety factors and carry out their comparative analysis. Keywords: variational inequality, materials with different strengths, deformation localization.

Citation: O.I. Kuzovatova, Study of a Deformation Localization Direction in Slow Motion of a Granular Medium, J. Sib. Fed. Univ. Math. Phys., 2022, 15(5), 586-597. DOI: 10.17516/1997-1397-2022-15-5-586-597.

Introduction

The theory of materials with different strengths is one of the most interesting and actively developing branch of mechanics. The field of application of this theory is the problems of mechanics of geomaterials. Such materials have significantly different tensile and compressive strength properties. The range of problems related to the mechanics of geomaterials is diverse. In engineering practice, the analysis of the behavior of geomaterials is important in connection with the issues of mechanical treatment of soils, as well as in relation to the issues of mining, construction of engineering structures etc.

The study of the process of localization of deformations in samples made of a material with different strengths is of constant interest. The importance of solving of such problems is dictated by the fact that in practice in narrow zones of localization of tensile deformations where malleability of the material is significantly higher than in the rest of the sample micro-destructions occur. Therefore, when analyzing the structural design for strength, such zones must be determined. At the same time, the possibilities of constructing exact solutions in such problems are limited, thus the development of computational methods is very relevant.

In the branch of geomechanics related to the study of the behavior of granular media, there is an important problem of analyzing movement of granular media in converging channels. Problems of this kind arise when emptying granular media or geomaterials from storage chambers and bunkers, as well as in many mining technologies. The approximate (engineering) solution of the problem and the results of field experiments are presented in works [1,2]. In the work [3] the problem of a flat slow gravity flow of a granular medium in a converging channel was considered.

* oik17@yandex.ru https://orcid.org/0000-0003-3635-4807 © Siberian Federal University. All rights reserved

For granular sample the safety factor was computed and formulas were obtained for calculating the inclination angle of a narrow linear zone of deformation localization of simple shear deformation with dilatancy. A numerical experiment was also carried out using the finite element method that showed results close to the solution.

The purpose of this work is to construct an approximate solution to the problem of slow gravity flow of a granular medium in converging channels with various geometric structures. During the transition from the static stress-strain state to the movement of a granular medium, the deformation is localized along some surfaces, followed by the movement of the formed blocks. Under assumption of a linear deformation localization zone, it is necessary to calculate the safety factors for various channel samples and conduct the comparative analysis. The solution of the problem will be based on a model that takes into account different strengths of the material [4].

1. Mathematical model

For the description of the stress-strain state of a granular medium as a different strengths material having different tensile and compressive strength limits, we will use a model of a medium with plastic bonds. This model has been developed by V. P. Myasnikov and V. M. Sadovskii in the work [4]. Under compressive or tensile strain lower than the adhesion coefficient (the limit bond strength) such a medium does not deform. As the limit bond strength is approached, the deformation develops according to the theory of linear strain hardening. The rheological scheme of the model is given on Fig. 1 [5]. According to this scheme we have the following

a

Fig. 1. The rheological scheme

additive representation a. = aj + a0. + aj, where afj is the total strain tensor, af. is the rigid

contact component, aj is the cohesion tensor, aj

ij ij Eijklekl is the elastic tensor, e = (eij) is the deformation tensor, Efjkl is the symmetric positively defined elastic modulus tensor (we assume summing in repeating indices). The tensor aj satisfies the variational inequality

— £ij ) ^ 0, £,£ G C,

(1)

where C is the cone of admissible deformations of the form C = [e | ky (e) ^ 0 (e)}, k is the dilatancy parameter, 7 (e) is the intensity of shear, 0 (e) is the volume deformation [5]. In this notation, the inequality (1) takes the form

(Eijkieut — + • (eij — eij) > 0, e,e G C.

By definition of a projection, this means that

Eijkl (aij — aij

c

i

£ij — nij

here Ei-ll are the components of the inverse tensor, nij the components of the projection of C with respect to the norm |e| = ^JeijEijki£ki-

Consider [3] an element of a construction from a material with different strengths filling a planar domain Q with the boundary dQ = r that consists of two non-intersecting parts T„ and rCT. On the first part displacements are absent and on the second part the distributed load p is given. There hold equilibrium equations in variational form and boundary conditions

ij

Kdj+f)

_ + fi) (Ui - Ui) dQ = 0, (2) \i j

Ui = Ui =0 on r„, aij ■ nj = pi on . (3)

The problem (2)-(3) reduces to the problem of finding the minimum min J (U) = J (u),

ùi€Uc

where

Tf\ Iii0 (dU; 0U2\ 0 du2 0 , . _ _ A

J(u) = II [ox; axx H+ ox;) ai2 + a22- (flUl + f2u2)) dxxdx2-

n

C; \0x2 dx; J UX2

- J (piU; + P2U2 ) dr,

UC = {ui e H1 (Q) \Ui\ru= 0, £ (u) e c}, for the components of the deformation tensor we have kinematic equations

If dui duj,

i dui duj \ j 2 \ dxj dxi J '

A load (f,p) is called safe if u = 0.

Let pi =0, fi = m ■ f0, where m is the loading parameter. A load is safe for m varying from zero to the limit value (safety factor)

¿1 0 f dui du2\0 dv>2 0

(du1 n (du1 du2\ n du2 n \ ,

ff [dx! < n^ + *rj ^ + dxidx2 f x

,,„ — min -¡^---. (4)

(u1,v,2)eUc JJ (fiu + f2u) dxidx2

(«1,«2) = 0 Q

This statement is a formulation of a kinematic theorem on limiting equilibrium from plasticity theory [6].

2. Linear deformation localization zone

In the paper [3] we considered a problem of planar gravity flow of a granular medium in a convergent asymmetric channel with sides inclined at angles a and 3 with the base a, assuming a > ¡, a e ^0; (Fig. 2). The convergent channel fills a planar domain Q with the boundary dQ = r = ru Q rCT. On the boundary ru displacements are absent. The vector p of the distributed load on is equal to zero.

The condition (u1,u2) e UC takes the form

70 < v£0, (5)

Fig. 2. Direction of a narrow linear localization zone

where v = ■sfl/K2 - 4/3, 0 < k < a/3/2. By (4) the safety factor is m1:

mi

Kpga sin a (v sin a — cos a) '

(6)

here ts is the yield point.

In paper [3] we obtained that deformation for a simple shear with dilatancy is localized in a narrow linear zone of thickness h inclined at an angle y:

1

y = a — arcsin

VV^+I

1

or y = a — arctg —.

v

In this case for the angle y we compute

v sin a cos a

sin y :

a/V2"+T

cos y -

v sin a — cos a

VV^TÏ .

(7)

(8)

In this paper we shall consider two problems of planar gravity flow of a granular medium in a convergent asymmetric channel. The geometry of a channel in each case will differ from the one on Fig. 2. Under assumption of linearity of the deformation localization zone we compute the safety factors m* for such channels and compare them. The boundary conditions for the domain Q are analogous to the conditions of the problem (Fig. 2) considered in [3].

Consider a planar deformed state of a homogeneous sample (Fig. 3). Its geometry differs from that of the one considered in [3] (Fig. 2): the base of the channel is inclined at an angle y (7).

Fig. 3. Cross-section of the sample (a > ¡3) Problem 1. Compute the safety factor m* for the sample (Fig. 3) according to (4).

2TS 1

1

Fig. 4. Geometric constructions

Let u = (u1,u2) be the admissible displacement field describing the deformation localization of simple shear with dilatancy in a narrow linear zone of thickness h inclined at an angle 0 (Fig. 4).

Let us compute some angles. Consider

n

△ABH : ZBAH = y + 0, ZABH = - - (y + 0),

n

△CBH : ZCBH = n - (3 + y + ZABH) = - - (3 - 0), n

ZBCH = - - ZCBH = 3 - 0. In the Cartesian coordinates related to the narrow linear zone

Ui = uo cos (ß — 0), U2 = —u0 sin (ß — 0),

7o = U0 cos (ß — 0). h

£0 = U0 sin (ß — 0) . h

Then we get

(dUi 0 (dUi dUo\ 0 dU2 0 > , 0 „

fe ^ + (T a02 T ^ dxidx2 = £o -0 • S°,

(9)

(10)

where

a0 = —, Sn = hl. k

The 'separating' triangle domain ABC moves as a solid body, therefore

(fiui + /2U2) dxidx2 = f0 • SA,

(11)

where

f0 = pgu0 sinß, Sa = 1 Hl, H = L1 sin(y + 0),

L1 = AB = a (cos y + sin y • ctg (a - y)) = a (cos y + vsin y). substituting values from (8) we get

L1 = a\J 1 + vz sin y.

2 Qir

(12)

Having in mind that (10) and (11), the safety factor m* is equal to

h sin(3 - 0) hl _ 2 ts sin(3 - 0)

* 's

m =

min

mm

Kpg (ui,u2)euc u0 sin3 • 2 Hl Kpg (ui,u2)euc sin3 • Li sin(y + 0) («1,«2) = 0 C«l,«2) = 0

Taking into account (5) and (9), we obtain the relation

COs(3 - 0) ^ v sin(3 - 0),

£0 — —

— = tg(3 - 0) > — or sin(3 - 0) >

70 v

VV^+I '

Then

In this case

0 ^ ß — arctg i or 0 ^ ß — arcsin v

a/V^+I '

2 Ts

sin(ß — 0) ■ min ■

KpgLi sin 3 ^ sin(y + 0) Let us find the minimum of the expression from (13) with respect to 0

d f sin(3 - 0) \ - cos(3 - 0) sin(y + 0) - sin(3 - 0) cos(y + 0)

d0 \ sin(y + 0) /

sin2(y + 0)

sin(y + 0) cos(ß — 0) + cos(y + 0) sin(ß — 0)

sin2 (y + 0) sin (y + 0 + ß — 0) sin (y + ß)

sin2 (y + 0)

sin2 (y + 0)

< 0 y0,

since sin (y + ß) > 0 for ß, y G ^0; •

2

Thus, the minimum of the function (13) in 0 is attained at

0 = 3 - arctg — or 0 = 3 - arcsin v

VV^+I '

Then formula (13) for the safety factor m* assumes the form

2 ts —

m2 =

KpgLi sin ß ^Jv2 + 1 sin(y + 0) '

From (14) we obtain

v sin ß — cos ß v cos ß + sin ß

sin 0 = ---, cos 0 =

v/V2T1 '

VVTT1

(13)

(14)

(15)

(16)

Using geometric constructions (Fig. 4) we find the value of l = L2, which is needed further below.

We have L2 = AC = li + l2. Consider △ABC = △ABH + △CBH. Then

H

△ABH ^ tg(y + 0) = —, H = Li sin(y + 0),

li

H1

△cbh ^ tg(3 - 0) = — = —.

l2 v

We get

H

L2 = Il + l2 = " ;) + h • v = Li • sin(^ + V) •(COS((y + ^ + v)

or

L2 = Li (cos(p + + Vsin(p + , (17)

here L1 is given by (12).

Consider now a planar deformed state of a homogeneous sample (Fig. 5) assuming that

a e (^0; J and a > ¡. The geometry in Problem 2 differs from the geometry in Problem 1: the base of the channel is inclined at an angle The values of the angles p and ^ are computed by formulas (7) and (14), respectively, p < a, ^ < ¡. The boundary conditions are the same as in Problem 1.

Fig. 5. Cross-section of the sample (a > ß)

Problem 2. Compute the safety factor m* for the sample (Fig. 5) according to (4).

Let u = (u1, u2) be the admissible displacement field describing the deformation localization of simple shear with dilatancy in a narrow linear zone of thickness h inclined at an angle $ (Fig. 6).

Fig. 6. Geometric constructions In the Cartesian coordinates related to this zone

U; = — uo cos (a — 0), U2 = —uo sin (a — 0),

uo

7o = ~t cos (a — 0), h

uo

£0 = — sin (a — 0) . h

Then we get

i

Q

where

U (Ü7 "" + (H + dS) ^ + Ü ^ dxidX2 = £0 • Sb, (19)

a° = —, Sb = hl.

k

The 'separating' triangle domain ABC moves as a solid body, hence,

0

(fiUi + f2U2) dxidx2 = f • SA, (20)

where1

f0 = pgu0 sin a, Sa = — Hl, H = L2 sin(0 + $),

here L2 is computed by formula (17).

With (19) and (20), the safety factor m* is

. ts h sin(a - $) hl 2ts sin(a - $) m =- mm —-:-= - min

Kpg (ui,u2)euc u0 sin a • i Hl Kpg («i,«2)euc sin a • L2 • sin(0 + $)

(ui,U2) = 0 (ui,U2) = 0

Taking into account (5) and (18), we get the relation

cos(a - $) ^ vsin(a - $),

£0 1 1

— = tg(a - $) ^ — or sin(a - $) ^

Then

In this case

70 ^ v A/ v2 + 1'

$ ^ a - arctg - or $ ^ a - arcsin

v/V2T1 '

2 Ts . sin(a — 4)

min -—. (21)

KpgL2 sin a 0 sin(0 + $) Let us find the minimum of the expression from (21) with respect to $

d f sin(a - $) \ cos(a - $) sin(0 + y) + sin(a - $) cos(0 + $)

( sin(a — 4)\ Vsin(0 + 4) )

d4 V sin(0 + 4) J sin2(0 + 4)

sin (a — 4 + 0 + 4) sin (a + 0)

sin2(0 + 4) sin2(0 + 4)

< 0 V4,

since sin(a + 0) > 0 for a,0 £ ^0; — j. Hence, the minimum is attained at

tg(a — 4) = 1 or sin(a — 4) =

v v ' Vv2 + 1'

which means that 1 1

$ = a - arctg — or $ = a - arcsin . (22)

v yjv2 + 1

Comparing the expressions (7) and (22) we deduce that the deformation localization zone is inclined at the angle $ = y. Thus, the safety factor (21) takes the form

2 ts__1 (23)

m3 = -z----(23)

KpgL2 sin a^Jv2 + 1 sin (y + 0)

for the values of L2 from (17).

3. Safety factors comparison

Let us compare the safety factors mi, m2, and m3 of the form (6), (15), and (23), respectively.

For that we consider the quotients m2/mi and m3/m2.

By conditions, we know the angles

a > 3, a, ft G (0; 2) , V < a, 0 < /3.

Let us carry out auxiliary computations. Using (8) and (16), we find

sin (p + 0) = sin p cos 0 + cos p sin 0 =

(v sin a — cos a)(v cos 3 + sin ¡) + (v cos a + sin a)(v sin 3 — cos 3) = v2 + 1 .

Modify the numerator of this expression

v2 sin a cos 3 + v sin a sin 3 — v cos a cos 3 — cos a sin ¡+ + v2 cos a sin 3 — v cos a cos 3 + v sin a sin 3 — sin a cos 3 = = (v2 — 1)(sin a cos 3 + cos a sin 3) — 2v(cos a cos 3 — sin a sin 3) = = (v2 — 1) sin(a + ¡) — 2v cos(a + ¡).

Then

sin(p + 0)=(^ — 1)sin(a +2ß) — 2V COs(a + ß). (24)

v2 + 1

Analogously,

cos (p + 0) = cos p cos 0 — sin p sin 0 = (v sin a + cos a)(v cos 3 + sin 3) — (v cos a — sin a)(v sin 3 — cos 3) = v 2 + 1 .

After rearranging the numerator

v2 sin a cos 3 + v sin a sin 3 + v cos a cos 3 + cos a sin ¡— — v2 cos a sin 3 + v cos a cos 3 + v sin a sin 3 — sin a cos 3 = = (v2 — 1)(sin a cos 3 — cos a sin 3) + 2v(cos a cos 3 + sin a sin 3) = = (v2 — 1) cos(a + 3) + 2v sin(a + 3),

we obtain

(v2 — 1) cos(a + ß) + 2v sin(a + ß)

Cos(p + 0) =--. (25)

v2 + 1

Problem 3. Find the condition for v for which

m < i.

mi

Taking into account (24), we consider the quotient

2 Ts 1

m2 KpgL; sin ß y/v2 + 1 sin(p + 0) a sin a (v sin a — cos a)

m; 2 Ts__1_ yjv2 + 1 Lx sin ß sin(p + 0)

Kpga sin a (v sin a — cos a)

sin a 1 v sin a cos a 1 v sin a cos a

sin ß (v2 + 1) sin a sin(p + 0) sin ß (v2 — 1)sin(a + ß) — 2v cos(a + ß)

Then

v sin a — cos a . 2

--- < (v2 — 1) sin(a + 3) — 2v cos(a + 3)

sin 3

i.e., we get a quadratic inequality in v:

2 sin a cos a

v2 sin(a + 3) — v 2cos(a + 3) +--— sin(a + 3) +--> 0.

sin 3 sin 3

Solving it, we find

sin a 2 cos a

D = 2 cos(a + 3) + ^— — 4sin(a + 3) — sm(a + 3) + —— =

sin 3 sin 3

2

2 sin a sin2 a 2 cos a

: 4cos2(a + 3) + 4cos(a + 3) ■ -^—¡^ + . 2 o +4sin2(a + 3) — 4sin(a + 3) ■ —

sin 3 sin2 3 sin 3

sin2 a 4

= 4 +--2---(sin(a + 3) cos a — cos(a + 3) sin a) =

sin2 3 sin 3

• 2 A -2 -2

sin2 a 4 sin2 a sin2 a

= 4 +--2---:—- sin(a + 3 — a) = 4 +--2--4 = -,

sin2 3 sin 3 sin2 3 sin2 3

and

sin a sin a

2 cos(a + 3) + —~s ±—~S sin 3 sin 3

V1,2 = 2sin(a + 3) .

Let v1 <v2, namely,

sin a sin a

2cos(a + 3) +--------,

sin 3 sin 3 cos(a + 3)

V1 =-—rir\--—(—i «V = ctg(a + 3), (26)

2sin(a + 3) sin(a + 3)

sin a sin a sin a

2cos(a + 3) + --; + ^—; cos(a + 3) + —-= sin 3 sin 3 = sin 3 (27)

V2 = 2sin(a + 3) = sin(a + 3) . ( )

Thus, the inequality m2 < 1 holds for v < v1 and v > v2 . If v1 < v < v2 then m2 > m1 and

m1

the second fragment does not move. Problem 4. Find the condition for v for which

mi < 1.

m2

Consider the relation

2 ts 1

m3 _ KpgL2 sin a yjv2 + 1 sin(p + 0) _ Lx sin ß _ 1 sin ß

m2 2 Ts__1_ L2 sin a cos(p + 0) + v sin(p + 0) sin a

KpgLi sin ß ^v2 + 1 sin(p + 0)

Using (24) and (25), we obtain the value of the expression

cos (y + 0) + v sin (y + 0) =

(v2 - l) cos(a + ß) + 2v sin(a + ß) (v2 - l) sin(a + ß) - 2v cos(a + ß)

= V^TT +v V^TT

(v2 - l - 2v2) cos(a + ß) + (2v + v3 - v) sin(a + ß)

= v 2 + l =

- (v2 + l) cos(a + ß) + v (v2 + l) sin(a + ß)

Then

v2 T l sin ß

= - cos(a + ß) + v sin(a + ß).

sin a

or

< v sin(a + /3) — cos(a + ¡3)

. n. sin 3 cos(a + ¡3) +--

v > vo = -. ( + 3fn a • (28)

sin(a + ¡3)

m3

Thus, the inequality — < 1 holds for v > v0, if v < v0 then m3 > m2, and the third fragment m-2

does not move.

Let us compare the obtained values vi < v2 and v0 from (26), (27), and (28).

We have sin a > sin 3, or Sm > 1, since a > 3 and a, 3 e (0; ^. Therefore v1 <v0 <v2.

sin 3 V 2 /

m2 m3

Consequently, for v > v2 of the form (27) the inequalities — < 1 and — < 1 hold

mi m-2

simultaneously.

Thus, depending on the value of the coefficient v that characterize the dilatancy of the medium the deformation zones are localized differently.

Conclusion

In this paper we use the model of a granular medium with different strengths by V. P. Myasnikov and V. M. Sadovskii to study a slow gravity flow of a granular medium in convergent channels. Assuming a linear deformation localization zone we obtain an approximate value of the safety factor and formulas for the slope of a narrow linear zone of the deformation localization for a simple shear with dilatancy. A comparative analysis of the obtained factors is carried out.

This work is supported by Krasnoyarsk Mathematical Center and financed by the Ministry of Science and Higher Education of the Russian Federation in the framework of the establishment and development of regional Centers for Mathematics Research and Education (Agreement no. 075-02-2022-873).

References

[1] S.V.Lavrikov, A.F.Revuzhenko, Localized flows of friable material in radial channels, Soviet Mining, 26(1990), 1-6.

[2] A.F.Revuzhenko, S.B.Stazhevskii, E.I.Shemyakin, Asymmetry of plastic flow in converging axisymmetric channels, Dokl. Akad. Nauk SSSR, 246(1979), no. 3, 572-574 (in Russian).

[3] O.I.Kuzovatova, Numerical modelling of slow motion of a granular medium, Journal of Siberian Federal University. Mathematics and Physics, 14(2021), no. 1, 21-27.

DOI: 10.17516/1997-1397-2021-14-1-21-27

[4] V.P.Myasnikov, V.M.Sadovskii, Variational principles of the theory of the limiting equilibrium of media with different strengths, Journal of Applied Mathematics and Mechanics, 68(2004), no. 3, 437-446.

[5] O.Sadovskaya, V.Sadovskii, Mathematical modeling in mechanics of granular materials, Heidelberg, Springer-Berlin-Heidelberg, 2012. DOI: 10.1007/978-3-642-29053-4

[6] A.A.Gvozdev, Analysis of bearing capacity of structures by Limit State Method. Part 1. The essence of the method and its foundations. Gosstroyisdat, Moscow, 1949 (in Russian).

Исследование направления локализации деформации при медленном движении сыпучей среды

Ольга И. Кузоватова

Сибирский федеральный университет Красноярск, Российская Федерация

Аннотация. В статье исследуется направление линий локализации деформации при медленном движении сыпучей среды под действием собственного веса в сходящемся канале с разной геометрической структурой. Используются вариационные принципы теории предельного равновесия, установленные на основе специальной математической модели материала, по-разному сопротивляющегося растяжению и сжатию. В рамках предположения о линейной зоне локализации деформации вычислены коэффициенты безопасности и проведен их сравнительный анализ.

Ключевые слова: вариационное неравенство, разнопрочная среда, локализация деформации.