Transcript Министерство образования и науки РФ Гос

Lobachevsky State University of Nizhni Novgorod
NATIONAL RESEARCH UNIVERSITY
ABOUT APPLICABILITY OF LOCAL
INTERACTION MODELS FOR DEFINITION
OF FORCES OF RESISTANCE TO
INTRODUCTION OF THE BLUNTED
BODIES OF REVOLUTION IN THE
NONLINEAR-COMPRESSIBLE SOIL
Linnik E.Yu.
Relevance: The ideas of the task used in the study of penetration depth of
the water and ice, the structure and composition of cosmic objects,
radioactive waste disposal, the impact of wells in the ground on an active
volcano, the study of small cosmic bodies. In this regard, the calculation of the
penetration process, optimization of body shape and stability analysis of
motion are fundamentally important and relevant in the present [1-4].
Purpose: definition of common patterns in the study process, impact and
penetration of the bodies of various shapes, taking into account changes in
environmental parameters, shape and material impactors.
Objective: Verification of local interaction techniques as applied to the
study of the impact of the shock-penetrating projectiles of various shapes on
the ground environment.
1. Bogdanov A.V. et al. A method for the study of terrestrial planets // Cosmic Research . 1988. V24, N 4. P.
591-603
2. Zaitsev A.V. et. al. Possibilities of the hypervelocity impact experiment in frames of demonstration
project “Space patrol”//Intern. J. Impact Engng. 1997. V 20. P. 849-860
3. Simonov I.V. et. Al. Before catastrophic state of geophysical objects, triggering effects and penetration
//reports of the Russian Academy of Sciences. 1996 V. 347,№6. P. 811-813
4. Veldanov V.A. et. al. The use of technologies based on the interaction of shock-penetrating // Dual
technology, 1998, №2. P. 1-14
Deformation model of ground
environment Grigoryan
d / dt   ( r , r   z, z )  ( r ) / r ,
d r / dt   rr ,r   rz, z  ( rr    ) / r ,
d z / dt   rz ,r   zz , z  ( rz ) / r ,
D J s ij  s ij  2Geij , (i, j  r , z )
and end relationship
p  f1(  ) H (   0 ) ,
sij s ij  2 f 22 ( p) ,
3
f1  0a 2 /(1   )2 ,   1 -  0 /,
f 2   0   p /(1   p /( M   0 ))
where t – time, d /dt – total time derivative,
– initial and current density,  – velocity vector ,  Cauchy stress tensor, s , e deviatoric stress and strain rate, respectively, H – Heaviside function, р – pressure, DJ – derivative Yaumana, f2=T – yield strength, G
– shear modulus; f1( ) , f 2 ( p) - given functions. Argument  = 0 by elastic deformationпри  > 0, realized if the condition of
plasticity
*
0
и

Grigoryan S.S. An approximate solution of certain problems of soil dynamics // Journal of Applied
Mathematics and Mechanics. 1962. V. 26. № 5. P. 944-946.
Model of deformable media
The problem of expanding spherical
cavity in the ground
Formulation of initial-boundary value
problem *
  r ,r  2 r r  = - d/dt,
 rr, r + 2 rr     r = -dr/dt,
 rr  f1( p) ,  rr    f 2  p .
 r r  R0  V ,  rr r  R  0 ,
 r    rr       0 , R0 t 0  0 .
impact adiabat
с = А+  
 0 A2
f1 
,
f 2  Y  kp,
(1   ) 2
The system of ODE to find the self-similar solutions
**
of plastic flow 0   1 for
u
r
c
,S 

0

r
V0
V0
 rr


,

M
,


,
0
ct
A
c
c2
  2u
u    0


~
Y
2
S  
 u (  u )



u  1 
c A
2
u

u

S
,
,
 1
c
u     0
*
0
Fomin V.M. et. al. High-speed interaction of bodies. Novosibirsk: Publishing house SB RAS 1999. 600 p.
**
Forrestal M.J.et. al.. Penetration into ductile metal targets with rigid spherical-nose rods // Intern. J. Impact Engng.
1995. V. 16. № 5/6. P. 699-710.
The problem of expanding spherical
cavity in the ground
     V02C ,
k  0;
  2 0 ln  ,

   2 0 1   1 , k  0,5;
 1    2k k ,
 0







3 2   4 2 ,
k  0;


C
1 3  2 ln    3 3 ,
k  0,5;

3
2k  1 1 2k k  1 4 2k



,
 (k  2)(2k  1) 2k  1  
k

2




Error in determining the speed of the
wave front and the tension on the
border of the cavity

   0 1   3 ,   V0 / c ,
c  1/ 3  A / 3.
* Kotov V.L., Linnik E. Yu., Makarova A.A., Tarasova A.A. Analysis of
approximate solutions to the problem of expanding spherical cavity in the soil
environment//Problems of Strength and Plasticity: Interuniversity Collection. V..
73. N. Novgorod: publishing house UNN. 2011
The parameters of local interaction
models
Estimate of the
convergence of the
method*

Parameters of LIM
Fz  2  R 2   n sin  cos  d
0
 n /   Vn  Vn
2
Vn  V0 cos
* - Bazhenov V.G. The software package
"Dynamics 2" for the solution of plane and
axisymmetric nonlinear problems of
unsteady interaction of structures with
compressible media// ММ. - 2000. V. 12. №
6. - P. 67-72.
The use of LIM for spherical
projectiles
The distribution of normal stresses along the surface
Time dependence of the resistance
introduction of the spherical
striker in a sandy soil
spherical projectile with , V0=300м/с, S0=R2
 distribution of normal stress at time 1-4
5, 6- distribution of normal stress on LIM-I and modified I’
 n /   Vn 2  Vn   n * / ,
 * - angle flow separation
 n* /   Vn*2  Vn*,
Vn  V0 cos*
*
The use of LIM for spherical
projectiles
The distribution of normal stresses along the surface
Time dependence of the resistance
introduction of the spherical
striker in a sandy soil
spherical projectile with , V0=600м/с, S0=R2
 distribution of normal stress at time 1-4
5, 6- distribution of normal stress on LIM-I and modified I’
 n /   Vn 2  Vn   n * / ,
 * - angle flow separation
 n* /   Vn*2  Vn*,
Vn  V0 cos*
*
Evaluation of applicability of LIM
for spherical projectiles
(F-F1)/F,
Forces of resistance to introduction
F – estimated,
F1 - by LIM
  error in determining the forces of
resistance to introduction
 max value
2 at the quasi-stationary stage of
penetration
Modification of LIM


 ' / 0     * / 0  V02 cos 2   cos 2  *  V0 cos   cos  * .
- angle separation;
- adjustable parameter
The use of LIM, taking into
account Coulomb
  k
Maximum force of resistance:
n
F / 0 S0  AV02  BV0
A   ( A1  kA2 ),
B   ( B1  kB2 )
(1  cos 4 )
A1 
,
2
(  2 sin  cos 3   sin  cos  )
A2 
,
4
2(1  cos 3 )
,
3
2 sin 3 
B2 
.
3
At the quasi-stationary stage of penetration:
B1 
  k 
n
Evaluation of applicability of LIM, taking into
account the friction
(1  cos 2  ) 2
A1 
,
2
(2 sin  cos 3   sin  cos   4 cos 2    )
A2 
,
4
(cos 3   3 cos   2)
B1
,
3
(2 sin 3   3 cos 2  sin   3 cos  )
B2 
,
3
where  – defined earlier angle flow separation
Criterion of applicability of LIM
1-Fmax/Fst ≈0
The ratio of the maximum and quasi-steady force of
resistance
-without the friction force
- taking into account friction force
 - error in determining the resistance force of
resistance
 maximum force of resistance
2 - at the quasi-stationary stage of penetration
The use of LIM for conical
projectiles
F=S (Vn 2 +Vn)
Vn= V0 sin ,
where  semivertex angle.
Evaluation of applicability
of LIM for conical projectiles
Error estimate
Validity criterion
Range of applicability of LIM
Conclusion
1.
2.
3.
1.
A new analytical solution to the problem of expanding spherical
cavity in the assumption of incompressibility of the impact front (as
opposed to the exact numerical solution is less than 5%).
A comparison with the results of axisymmetric calculations of the
introduction of spherical and conical projectiles into the ground. It is
shown that the error derived LIM-based solution to expand a
spherical cavity in the determination of the maximum resistance
force is 10 - 15%.
It is shown that the rate of fall of stress is proportional to the
coefficient of lateral pressure in the soil and reaches a maximum at
K = 1 (k = 0). As a consequence, the ratio of the maximum
resistance force to the quasi-stationary value also increases with a
decrease in the coefficient of internal friction (the highest for K = 1).
It is noted that the criterion of applicability of LIM in the case of the
penetration of spherical and conical projectiles into the soil can serve
as a quasi-stationary and the ratio of the maximum values ​of
resistance force.