theoretical and numerical analysis2.0-0045794992902582-main

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Computers Stn~c~wcs Vo. 44, No l/2 pp 381387 1992 00457949192 SS00 +000

Printed in Geat Britan Q 1992 Fw~mon Press Ltd

THEORETICAL AND NUMERICAL ANALYSIS OF

DYNAMIC COMPACTING OF SOIL AROUND A

SPHERICAL SOURCE OF EXPLOSIONW. K. NOWACK I nd B. RANIECKI

Institute of Fundamental Technological Research, Polish Academy of Sciences,00449 Warsaw, ul. Swietokrzyska 21, Poland

Abstract-To account for the effects of finite deformations and high pressure on dynamic compacting ofsoil, the special problem of spherical wave propagation for ‘plastic gas’ is formulated. The numericalmethod for its solution is developed. The results of numerical calculations are discussed.

1 INTRODUCTION pa/p = (det C)‘/* = r2r,,/RZ. (3)

In this paper the theoretical study of explosive

compacting of soil is limited to the analysis of

the solution of the large pressure shock wave

propagation problem where the shock wave is

induced by a spherical charge in an unlimited

space. Large pressure caused by the explosion is

accompanied by plastic volumetric deformations.

The mathematical model includes the nonlinear

physical properties of the soil which is treated with amixture of three phases (quartz -t water + gas) and it

takes into account the geometrical nonlinearity.

2 PROBLEM FORMULATION

spherical cavity in an infinite medium

with the pressure suddenly imposed at its boundary

and then monotonically spherical coor-

dinate system x’ is introduced (Lagrange

t. The initial radius of the

spherical cavity is denoted by & (or rO). Particle

motion of the medium is described by the following

equation

r =r(R, t), 9 = 0, cp =@. (1)

The physical and mechanical properties of the

soil are by the modified three-

component model of Lakhov [I] and Rakhmatulin [2]

in the loading zone (introduced for the first time

in the literature by Rakhmatulin).

insignificant in the process of

the dynamic global compacting of the soil. In

these conditions during the loading process the

soil behaves approximately

incompr~sible

that is

p(R, t) = p,(R) for > t,, (4)

where py is the density of the material at the shockfront. Integrating eqn (3), the specific form of the

function r = r(R, t) is obtained

r3 = 3 ’ C2[p01p.(T)] dT +A’@) for t > t,. (5)

Denoting the defo~ation gradient as F the nonzero In the case of in~mpressible motion, the function

mixed components of the tensor C = FTF are r(R, t) is also in the form r3 = A3(t) + B3 R),

The equation of state of the three-component soil:

Cf = (r,R)2, Ci = C: = r2/R2. (2) quartz $ water + gas is the following

~~~lP2 =@@l+,

1

if p -J/(p)=O, liao ,,\

0, ifp-_l(r(p)<O or

The ratio of the initial density of the medium to the The

density in the current moment is the

381

p-= (p) and @ ~0.to1

function $ (cf. Fig. 1) is determined from

experimental data for the given type of



382 W. K. NOWACKIand B. RANIECKI

__----

lP=gre,P’)

Fig. 1.

soil fl, 9. The equation of motion in Euierian vari-

ables is

-ap/ar =p(av/at +tr +avpr),

where v = A(t)/3r2.

The following initial conditions are assumed

p =p’O’, p = PO, u = 0, r = R for t = 0 (8)

and the boundary condition is

(9)

where the function r = ro t) describes the motion

of the spherical cavity. With the assumed value of

pressure change p = p. (t ) at the boundary of spherical

cavity, the shock wave propagates into an

undisturbed region Do (where p = po, p = p(O),v = 0).

(Fig. 2). At its front radial displacements are equal to

zero, hence at the wave front we have r = R. Denoted

by w(t) the function describing the motion of

the shock wave front: r = 12 = w t). From eqn (5)

we obtain: A(t) = r, t). The pressure, density and

velocity of the material particles at the shock front are

denoted as

p,(t) =plr~o(rf-O~ p,(t)= P lr=W(o-*l

v, t) = ri io/w2. 10)

Kinematic and dynamic continuity conditions have

the following form [4]

P”(Q-v”)=PoQ, Pu-P(o)=P”(~--“)v”, (11)

where R = rit is the velocity of the shock wave propa-

gation and oS= Q(t - ~~~~~). Equation (1) together

with the constitutive eqn (6) form a system of

equations which may be used to derive four unknown

quantities at the front of a shock wave: p., pu v,,and

R. Formula (5) describes the general form of motion

in an incompressible region D. The density pU and r,

are unknown function so far. As the equation is true

at any point of the region D, one obtains for

r = R = o(t) (at the shock wave)

s

o(f)d(t) - 3 ~2[~ol~,(t;)] dc = ri 0) for t > t,=

%

(12)

This is an additional equation defining the new

variable r, t). Calculating the time derivative of eqn

(12), it is found that

fori = o%G[l - pa/p,(t)]. (13)

Equations (6), (11) and (13) form a system of equations

for five unknown functionsp,, p”, vu, o(t) and rdt).

The fifth equation is the equation of motion of the

incompressible zone. Equation (7) takes the form

II.

--=p(‘l*)[~_~]. (14)p tr, t)

&

UNDI~UREED REGION

:p-p(% 9’; To *v-o

@rr-G*+mE

PO(t) p tiaRo R,r

I- “ I r(Fig. 2.



Analysis of dynamic compacting of soil 383

In the adopted model describing the behaviour of

satured soil it is assumed that there exists a lower

boundary of the pressure p =plim = lo6 Pa above

which the proposed equation of state (6) may be

used. Below that limit the compressibility of the soil

considerably depends upon the compressibility of

the soil skeleton. The mechanism of deformation

differs from that described by eqn (6). The condition

u,, - agg = 6 is no longer valid. The general form

of the constitutive equations, first proposed by

Grigorian [5], has the following form [6,7]

Ls=2~D+%@ts-s,), P =$o(P.P*). (1%

where D is the strain rate tensor, s is the deviator

of Cauchy stress tensor, @ = tr(s - sR) D is the

intrinsic dissipation (@ 2 0), u stress in the point

of inversion [6,7] (for the first loading sR = 0,

cr, = -~/(%S,), /J is the Lame parameter, S, the limitshear, w,, the Masing parameter, L the objective

derivative. Function I,& is defined in Fig. 1. At time

t = t,,,,when the pressure at the shock wave decreases

to the value plim the propagation of the shock wave

is assumed to be terminated and transformed into a

spherical strong discontinuity wave. The soil con-

tinues to be compacted. Non-zero components of

the Cauchy stress tensor are bRR, uee = a@. In the

region D, (for t > t,), (Fig. 2), the solution of

the following equation of motion (in Lagrange

variables) must be found: div(Fa) - pOs’= 0 whereII = det(F)F-IoF-’ is the second Piola-Kirchhoff

stress tensor. Also

- (l + u/R)*~,,l +; W + u/R)*a,,

2

- (1 + O(l + u/Rbeel = A , (16)

where u(R, t) is the radial displacement of the

material point at timet.

3, FORMULATION OF THE INITIALBOUNDARY

PROBLEM-REGION D

An initial-boundary problem in the Eulerian de-

scription will now be formulated. Since the equations

include integrals, the mathematical initial-boundary

problems are not easily noticed. The mathematical

problem in the region D is formulated as follows:

(1) Find a functions of two variables: p = p(r. t),

p = p(r, t) defined in the range r,,(t) < r <o(t).

(2) Find two functions of one variable t: r,,(t) andw(t) defining the domain of the function p and p. The

functions p and p must satisfy two partial differential

equations in the region D (r,,(t) < r <<o(t))-

the equation of motion (14) and the incompressibility

condition p = 0. The functions r,,(t) and o(t) satisfy

an ordinary differential equation [the relation

between the velocity of the spherical cavity boundary

CAS4,/1-*-z

and the velocity of the shock wave-eqn (1311. The

function F = 1 -p,,/p, is derived by solving two

equations (6) and (11). The functions r,,(t), w(t)

satisfy the following initial conditions: rO(0) = w(O)

and t(O) = u, . At the origin of coordinates the value

of v, may easily determined provided that pj,=,,+ is

known. The functions p(r, t) and p(r, t) shouldsatisfy the boundary condition (9) and the following

conditions along the curve r = o(t) (shock wave)

Pu-P’“‘=Po(~)2(1 -PoIPA p,=JI p.). (17)

The initial-boundary value problem formulated

here is not a typical one for the system of nonlinear

partial-ordinary differential equations. The lack of

one ordinary equation is substituted by additional

boundary condition (17). If the equation existed, two

of the three conditions would be enough to obtain

the explicit functions p(r, t) and p(r, t). Thus theproblem is properly formulated. At the spherical

cavity r = ro(t) the following boundary condition is

assumed: explosive decompression in the spherical

cavity is polytropic with the generalized polytropy

exponent y

ptr,OLQ(,)=po(t) = r,(t)-“, for 0 < t ( t*,

(18)

where t * > 0 denotes the time when the explosive gas

pressure reaches the initial pressure p(O).The pressure

p. is assumed to be constant for the t > t*;p. = p’0’ = const. Let us integrate eqn (14) with

respect to r. The following boundary condition at the

shock wave is assumed: p(r, t)l,,M,j =p,(t). As a

result we obtain the following equation

-IL

3 4)

p(r,t)=p.(t)+tj3

s

(~le;~) d5

r

(p/t’) dtL (1

where r is defined in the range ro(t) < r f w(t).

3.1. Approximate solution method

Density p appears in the dynamic equilibrium

condition (14). It is an unknown function of the

argument z = r3 - r:(t). In order to avoid solving

the nonlinear partial-ordinary differential equations,

simplified equations are adopted. Two integrals are

approximated by power functions as follows:

I4

(p/r*) d* = 4,rlJ(O s

)

(p/r51dr = R2, (20)row

where

P = ~., z/Ri + 11, n = ln p, t)/p.l )/ln z/R~ + 11,

Pul = JI’[PoWl. (21)



384 W. K. NOWACKI and B. RANIECKI

Let us integrate eqn (13) over the interval

[ro(t). w(t)]. As a result we obtain the following

equation

i,,= -2(1-R,/R,r~)i~r0+[r~3Y-2-r;2p,(t)]/R,.

(22)

Let us solve the system of eqns (12) and (17) with

respect to r,, and w.

h(f) = r;(~)r0(~)/~*(~)[1 - PO/P,(r)1 (23)

2. 2

b”(o-P’O1[l -PolP~~~~llp.=~-‘(~u)=Po .[

(24)

Equations (22) and (23) form the basic system of

differential equations for the functions r, and w. Theformulae are complicated nonlinear functions of i. , r,

and w.

3.2. Numerical method

Introduce the vector Yk(t), (k = 1,2, 3,4,5) which

has the following components

Y, = to(t), Yz= ro(t), Y, = w(t),

y,=p,, y,= P. (25)

The state of medium at time t is determined by the

vector Y,. Three ordinary differential equations and

two algebraic equations determine the changes of the

state. Differential equations have the form:

dY,,,/dt=F,,,(Y,), (k=1,2 ,..., 5, m=l,2,3),

(26)

where F, , F3 are calculated from eqns (22) and (23),

respectively and F2 = Y,

F,(Y,)= -2(1 -R2Y;/R,)Y;/Yz

+ (Y;3’ - YJR, Y:

F2U d = Y,, F,(Y,)= Y, Y:IY:U -PO/Y,) (27)

and

I

rlR, = (x3- Y;+ l)“x-*dx,

YZ

n =ln Ys/ln(Y:- Yi+l)<O

R, =s

“(x3- Y;+ l)“x-5dx.Y2

It is assumed that the vector Yk(fi) is known at time

ti (Fig. 3). The solution of the system of eqns (26)

determines the components of the vector Y,,, or time

ti+ 1= ti At: Y,,,= Y,(ti+ ,). Knowing Y,,,(ti+ ), the

remaining components of the vector yn ti+ I)

n = 4,5) are found from the formula (24) and from

the constitutive equation (17)

(Y,-P’O’) b-b( )I

r: r;”

y __= -y4s= ’ * 3

and Y, = JI -‘(Y,). (28)

Summing up, the basic mathematical problem con-

sists of solving the differential equations (26) and

the algebraic equations (28). We obtain the actual

position of the shock wave o(t), values of pressure

and density at the front p,(t) and p,(t) and actual

position of the spherical cavity ro(t) and its velocity

to(t). Now we can calculate the density distribution

p(r, t) in the region D from the formula (21). From

eqn (19) we obtain the distribution of the pressure

p(r, t) and we calculate the velocity of the particles in

the domain D from the formula

u(r, t) = ri(t)io(t)/r2. (29)

4. FORMULATION OF THE INITIAL-BOUNDARY

PROBLEM-REGION D

In the region D,, the region of elastic-plastic

deformations of the medium, we have the followinginitial conditions at the time t = t, Fig. 2)

u R, 1,) = 0, u R, t,) = u, R), ~6 t,,,)=p,(Rh

~,(R tm)=~ee(R hz)=~,(R> (30)

CALCULATE: Y, t, +At ) from dY,/dt= F,JYk) and Y ,., ti +At)

fromtime t i+At -

F Y, ,Y, )=O

i CALCULATE: n, RI , s, p r,t), p r,t), v r,t)

time ti -A

DATA: Yk = { r0 , r,, w , P, , P, } where m=1,2,3,

k=1,2,3,4,5, n=4,5.

Fig. 3.




where the functions u,(R), p,,,(R) and p,,,(R) are

determined from the relations (29), (21) and (19)

respectively, for the time t = r,,, We apply the bound-

ary condition-+qn (18)-for R = rt. From the

constitutive equations (IS) we calculate the time

derivative of stress components aM and uee. In the

Lagrangian coordinate system the derivative is a time

partial derivative with respect to time. From qn (15)

we obtain:

2cl- A,, = ~/LO,, - -

3E,+$,(P,P+)+cV@(e,-

d,, = 2jlD,, - $ E,+$o(P.P+)+@o@(Qe- f&L

(31)

where

S, =trd =a,+2u@,

E,=trD=D,+2D,,=L+-.2lilR

1 +u’ 1 +ulR(32)

We suppose the decomposition of the stress com-

ponents and in the following form

a,=&- G,, am,=&,-G, (33)

where the linear parts of this components are:

u’RR=(II +2r)ut+u , u&9=U+2(1 +p) .

(34)

All nonlinear terms are the comprised in the function

G, and G2. The quation of motion (16) take the

following form

do, 2 at 4 aG 2

,,+_i?(u;-u~,)-~~~=,,+,(G,-G,),

(35)

where the nonlinear terms are the following

expressions:

dr + C,(R),

(36)

where u~=(u,-u,~)/~, ai= 1 +u’ and u2= 1+

u/R. We calculate the functions C,,(R) in eqn (36)

from the initial conditions (30). We obtain

C, R) = C, R) =p R). The boundary condition

for R = rr (for the time t 2 t,,,) must be modified.

This condition for equation of motion (35) take the

following form:

&(R, t) =R(t) +b(r), (37)

where p(r) =p(r)[(l + u/R)‘- l] + G,. It is easy

to generate the approximation of the Lagrange

functionals for qn (35). This is a classical technique.

Denote by M the operator connected with kinetic

energy, K the energy of deformation and by F the

vector associated with density of the volumetric

forces or the forces associated with nonlinear

members of qn (35). The discrete form of the

initial-boundary problem is constructed from thesemi-implicit expression as follows:

3M(u’+ ’ -22u’+u’-‘)+K(u’+l +u’+u’-l)Ar2

=(F+‘+F’+F’-1)Ar2. (38)

Remark. The proposed method of solution can be

easily used for the study of one-dimensional motion

of water saturated soil or of the rocks in the case of

cylindrical or plane shock waves.

5. RESULTS OF NUMERICAL CALCULATIONS

In this section the results of numerical calculations

are discussed.

5.1. Example 1

A single explosion of the spherical charge generates

mechanical compaction of the soil in the spherical

ring of the following dimensions: rr < r <urn

(Fig. 2). It is assumed that a compacted zone is the

region where the soil density, after the dynamic

processes accompanying the explosion are over, is

1% greater than the initial one. According to this

convention, the radius mM corresponds to such a

position of the strong discontinuity wave at which

p = 1 Ol pO.Let us consider the explosive dynamite-

average (initial) pressure of the explosion products

are pUl= 2200 MPa, and initial spherical cavity radius

is R,, = 0.33 m, initial density of different components

of the soil: quartz p:,, = 2.65 x 10-6, water

p$, = 10V6,gas p$, = 0.00125 x 10V6kGs/cm4, initial

volume fractions: water u\‘) = 0.19, gas ui”) = 0.01,

polytropic curve coefficients: y, = 5, y2 = 7, y, = 1.4,speeds of sound propagation (for small disturbances

at p =p’O)): for quartz c ,) = 4500, water co, = 1500,

gas cuj = 330 m/set, and the generalized polytropy

exponent for explosive decompression in the spherical

cavity is y = 3.

After the numerical calculations in the domain D

we obtain the following results [3]: from the condition



386 W. K. NOWACKJnd B. bNIECKl

t

time f ps]

Fig. 4.

p(r,,,) = p& we obtain the time r,,,= 9.2 x 1O-3 set,

and after that we obtain: radius of the cavity at

time tm: rf = 1.44 m, radius of the cavity at time

tm: of” = 5.03 m, volume of compacted zone:

VP = 520.57 m’, average compaction level: z = 1.024.

The average compaction level z, defined as

z =i pm/p,,, where pm is the average density of the

region r? < r d corn, may be determined from theformula

2 = pm/p* N [I - r; l/Wm)3]-1.

From the formula (21),, we can calculate the

density ~st~butio~ p r, t) in the region 11 and from

eqn (19) the pressure dist~bution p{r, t)-for the

time 0 d t G r,,,.For example the density distribution

at time t, is presented in the Fig. 4. The solution

in the region L), gives: final radius of the cav-

ity: rt:= 1.58 m, final radius of compacted zone:

wM = 6.65 m.

The diagram shown in Fig. 5 presents the variation

of the pressure at the front of the strong discontinuity

wave.

The final density in the sphere r < rt after the

explosion cannot be determined theoretically. The

phenomena occurring in this zone at the time of

explosion and after it are very complex and dif5cult

to formulate. The final density is influenced by the

process of soil degradation in the neighbourhood of

the cavity resulting from the negative pressure, and

by the process of filling the hole due to the gravityforces after decomposition of the explosive fuses is

finished. Si~ifi~t influence upon the final density

in the range r < r: may result from the dynamicwater titration when the negative pressure exists in

the hole. The phenomenon of water filtration in the

direction of the cavity during the explosion also

cannot be ruled out.

plP”,6

-us 5.03 6.0 Uk6.65

Fig. 5.

0 20 40 60 80

Fig. 6.

radius [m1

Analysis of the adopted model of the saturated

soil behaviour shows that the function p =p@) is

very sensitive to the change of the 4 parameter.

Even small contents of the gas in the soil results in

significant changes of the p =p@) function value. It

has a significant influence upon the decay of the

shock wave front.

5.2. Example 2

We consider now a single explosion of the spherical

charge situated in the volcanic tuff. In this case we

can apply the same ~nstitutive eqns (6). Numerical

calculations were performed for the following values:

the initial volume fractions of water and gas

u*=O.Ol, u,= 0.2, also the initial density of the

volcanic tuff is p. = 2.1037 x 10e6 [g/en?]; initial

pressure of the order of put = 8 x 10’ MPa-the same

as that in the ex~~ment ‘BLANCA’[8] experiment

with nuclear charge of the capacity * 22 kT situated

at a depth of 300 m in the volcanic tuff. The growth

of the radius of the compacted zone as a function

of time is presented in the Fig. 6-measured fromthe experiment ‘BLANCA’[S] dashed line (2)

in the Fig. 6. In the same figure is indicated the

presumable change of the radius of spherical cavity as

a function of the time--calculated from the empirical

formula [Q-dashed line (1). By the line (3) is shown

the final radius (measu~) of the cavity. In the same

diagram the solution obtained by the proposed

method-solid lines--is presented. The accordance

of the numerical solution with the experiment is

satisfactory.

REFERENCES

1. C. M, Lakhov, The Pri ncipl esofDynamic Expiosims n

e Soils and F&id Media. Nicdra, Moskva W64).2. KB A. Ra~atuIi~, On the wave pro~~ti~~ in

multicemponcnt solids. PMM 33, 11 1969).3. W. K. Nowaeki and B. Raniecki, Theoretical analysis of

dynamic compacting of soil around a spherical sourceof explosion. Arch. Mech. 39, 4 1987).




4. J. Mandel, int roducti on of la M ecanique des M il ieux

Continw Deformables.PWN, Warsaw (1974).5. S. S. Grigorian, On fundamental presentation of

dynamics of soils. PMM 24, 6 (1960).6. P. Guelin and W. K. Nowacki, Remarques sur les ondes

d’acceleration dans un continu elastoplastique avechysteresis. Arch. Me&. 36, 1 (1984).

7. P. Pegon, W. K. Nowacki, P. Guelin, D. Favier and

B. Wack, Hysteresis with coupling effects in cyclicbehaviour of granular materials: constitutive scheme,modes of bifurcations and numerical methods. IBF-KGPG Workshop, Gdansk, 26-30 Sept (1989).

8. G. W. Johnson and G. H. Higgins, Engineering appli-cation of nuclear explosives. Project Plowshare. Reportfor III United Nations Intern. Conf. on the PeacefulUses of Atomic Energy, May (1964).

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