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1 1 1
1.1 3 1.1.1 (...) 3 1.1.2 Love 8
1.2 11 1.3 17 1.4 20 1.5 22 1.6 , 24 1.7 Tresca von Mises 27 1.8 Mohr-Coulomb 30 1.9 : Mohr 32
.
. (2008) , . 1 2
1. , 2009
. , Dr-Ing., ..
.. 144, 190-02, http://geolab.mechan.ntua.gr/, I.Vardoulakis@mechan.ntua.gr
. (2008) , . 1 3
1.1 1.1.1 (...) 1
. 1-1:
, 2,3. , 4 (REV). 30 5,6 (. 1-1). . , n
m , ( , )c m n , ( , )m n , nc mck kn n= ,
c . , nc mck kF F= , (. 1-2). () (. 1-3). . , .
1 Bardet, J.-P. and Vardoulakis (2001). The asymmetry of stress in granular media. Int. J. Solids Struct.,38, 353-367. 2 Christoffersen, J., M. M. Mehrabadi, S. Nemat-Nasser (1981), A micromechanical description of granular material behavior, Journal of Applied Mechanics, ASME, Vol. 48, pp. 339-344. 3 Rothenberg, L., and A. P. S. Selvadurai (1981). Micromechanical definition of the Cauchy stress tensor for particulate media, In: Mechanics of Structured Media (edited by A.P.S. Selvadurai), Elsevier, pp. 469-486. 4 . Representative Elementary Volume (REV). 5 . Discrete Element Method (DEM) 6 Cundall P.A., Strack O.D.L. (1979). A discrete numerical model for granular assemblies, Gotechnique 29, No 1, 47-65.
. (2008) , . 1 4
. 1-2: ( )REV .
. 1-3: :
. 1-4:
. (2008) , . 1 5
( )REV , N , ( )REV . ( )REV . ( )REV ,
{ } { }1, , , , 1, , ,a NB a N = = (1.1) ( )REV M ,
{ } { }1, , , , 1, , , ,s MC e e e s M= = (1.2) I C ( )REV , E C ( )REV :
{ }{ }
1
1
, , {1, , }
, { 1, }
,
M
M M
I e e M
E e e M M
I E C I E
+
= == = + = =
(1.3)
a aI a aE a ( )REV ( )REV , a aC ,
,
,
a a a aa B
a aa B a B
C C C I E
I I E E
= = = =
(1.4)
a b { },a b a b c a bE E I I e B = = (1.5) , . ,
0a
aci
c CF
= (1.6)
( ) 0a
c a acijk j j k
c Cx x F
= (1.7)
aix cix
.
. (2008) , . 1 6
a ( ) aiu
ai (. 1-5).
. 1-5: .
(1.6) (1.7) aiu ai ( )REV ,
( )( ) 0a
ac a c a ac ai i ijk j j k i
a c CF u x x F
+ = (1.8)
aC . ,
c ac bci i iF F F= = (1.9) ...
( , ) ( ,int)D ext DW W = (1.10) ( , )D extW ( ,int)DW 7: ) ,
( , )D ext e ei ie
W F u
= (1.11) eiu e
eiF ) :
( ,int)D c ci ic
W F u
= (1.12) ciu c a b (. 1-6),
7 D (. discrete)
. (2008) , . 1 7
( ) ( )( )c b a b c b a c ai i i ijk j k k j k ku u u x x x x = + (1.13)
. 1-6:
. :
a ai i ij ju a b x = + +" (1.14)
a ai i ij jx = + +" (1.15) ,i ija b ,i ij , :
( ) ( ) ( ) ( )( )c b a b a b c b a c ai ij j j j ijk k k jl ijk l k k l k ku b x x x x x x x x x x = + +" (1.16)
( )
( ) ( )e a a e aei i ijk j k k
ae e ae ae e aei ij j ijk j k k ijk jl l k k
u u x x
a b x x x x x x
= + += + + + +
""
(1.17)
aekx a a e .
:
( ,int) ( ) ( )D c b a c b aij i j j j ijk i k kc c
W b F x x F x x
= + " (1.18) ( , ) ( )D ext e e ae e e aei i ij i j j ijk i k k
e e eW a F b F x F x x
= + + + " (1.19)
..., . (1.10) :
. (2008) , . 1 8
0 , 0 , 0
0
eij i i i i
ee
ie
b a F a
F
= = = =
(1.20)
. (1.20) ( )REV . ,
0 , 0 , ( )
( )
c b a e aei i ij i j j ij i j ij
c ec b a e ae
i j j i jc e
a b F x x b F x b
F x x F x
= = = =
(1.21)
0 , 0 , ( ) ( )
( ) ( )
c b a e e aei ij j ijk i k k j ijk i k k j
c ec b a e e ae
ijk i k k ijk i k kc e
a b F x x F x x
F x x F x x
= = = =
(1.22)
( )e aek kx x
( )e aek k gx x O R = (1.23)
(1.22) ,
( ) 0c b aijk i k kc
F x x
= (1.24) 1.1.2 Love 8 Cauchy . ( )REV , ,
0ij k REVi
x Vx = (1.25)
ij i j k REVn t x V = (1.26) ( )REV 9: . (1.25) kx ( )REV , ,
8 Froiio, F., Tomassetti, G. and Vardoulakis, I. (2006). Mechanics of granular materials: the discrete and the continuum descriptions juxtaposed, Int. J. Solids Structures, 43, 76847720. 9 L.D. Landau and E.M. Lifshitz, Theory of Elasticity, Vol.7, sect. 2, p.7, Pergamon Press, 1959.
. (2008) , . 1 9
( )0
0REV REV REV
REV REV
REV REV
ij kij kk ij
i i iV V V
ij k i ij kiV V
j k kjV V
x xx dV dV dV
x x x
x n dS dV
t x dS dV
= = =
=
(1.27)
1
REV
ij ijREV V
dVV
= (1.28) . (1.22) (1.27)
( )REV REV
ae e b a ck i ki k i k k i
e cV V
x t dS dV x F x x F
= = (1.29) (REV) 10,
1 ( )b a cij i i jcREV
x x FV
(1.30)
1 c cij i jcREV
FV
A (1.31)
b ai i ix x= A (1.32) . , .(1.31), Love11 . . (1.24) (1.31) ()
0 0c cijk i k ijk kic
F
A (1.33)
312 23 213 32 23 321: 0j = + ... (1.34)
10 e.g. information stemming from a DEM simulation of the mechanical description of a granular medium. 11 A.E.H. Love, A Treatise of the Mathematical Theory of Elasticity, Cambridge University Press, 1927.
. (2008) , . 1 10
ij ji (1.35) Love ,
( )12 c c c cij i j j icREV F FV + A A (1.36)
. 1-7:
1-1: (. 1-7)
Love , . (1.36), : , , [ / ]F kN m , . 1-7. ( a ) , (1) (6). ( 1-1) Love :
( )621
12
c c c cij i j j i
cS S = + A AA (1.37)
(a) :
[ ] 0 1/ 41/ 4 0
F = A (1.38)
. (2008) , . 1 11
1.2
. 1-8:
Cauchy 1 2 3( , , )O x x x (. 1-8.),
[ ] 11 12 1321 22 2331 32 33
= (1.39)
12,
13ij kk ij ij
s = + (1.40)
11 22 33kk = + + (1.41)
( )11 11 22 33 12 121 2 , , . . .3s s = = (1.42) . (;).
1 2 3( , , )O x x x
[ ] [ ]1 12 23 3
0 0 0 00 0 , 0 00 0 0 0
ss s
s
= = (1.43)
12 . deviator
. (2008) , . 1 12
[ ]ijA (;)
3 20 0ij ij A A AA I II III = + = (1.44) AI , AII AIII 3 3[ ] XA , [ ] [ ]ijA A= ( 1,2,3)i i = : 11 22 33 1 2 3AI A A A = + + = + + (1.45)
22 23 11 1311 12 1 2 2 3 3 132 33 31 3321 22
A
A A A AA AII
A A A AA A = + + = + + (1.46)
11 12 13
21 22 23 1 2 3
31 32 33
A
A A AIII A A A
A A A = = (1.47)
13.
. 1-9: Haigh-Westergaard
, , Haigh-Westergaard (. 1-9).
13 Pettofrezzo, A.J., Matrices and Transformations, Dover, 1966
. (2008) , . 1 13
1
2
3
OP
= (1.48)
(). , , 1 2 3 = = , (), , , , . 1 2 3 0 + + = . 1 ,
1 1 2 3kkI = = + + (1.49) ,
1 1 11 , . . .3
s I = + (1.50) 1 ,
1 1 2 3 0s kkJ s s s s= = + + = (1.51) ( 1,2,3)i i = ( 1,2,3)is i = . (;)
3 2 3 0s ss J s J = (1.52) 2 3 ,
( )2 2 22 1 2 31 12 2s ij jiJ s s s s s= = + + (1.53) ( )3 3 33 1 2 31 13 3s ij jk kiJ s s s s s s= = + + (1.54) , , . (1.52) s :
2 2 21 2 32 2 2
cos , cos , cos3 33 3 3
s s ss s s
J J Js s s = = = + (1.55) (. 1-10)
. (2008) , . 1 14
2 3 1 0
2 1 3 0
1 2 3 0
1 3 2 0
3 1 2 0
3 2 1
(1) : 0 / 3 ( ) :(2) : / 3 2 / 3 ( ) : 2 / 3(3) : 2 / 3 ( ) : 2 / 3(4) : 4 / 3 ( ) : 4 / 3(5) : 4 / 3 5 / 3 ( ) : 4 / 3(6) : 5 / 3 2 ( )
s s s
s s s
s s s
s s s
s s s
s
s s ss s s
s s ss s s
s s ss s s
= = + = +
= + = + 0: 2s s = +
(1.56)
. 1-10: . (1.52)
0s 14
30 03/ 22
3 3cos3 , 0 / 32
ss s
s
JJ
= (1.57)
0s Lode15 16 b, 1 :
3 2 0 2 3 11 2 0
sin2 1 2 1 ,sin( / 3 )
s
s
L
= = + (1.58)
( )1 3 0 2 3 11 2 0
sin11 1 1 ( )2 sin( / 3 )
s
s
b L = = + = + (1.59)
14 . stress invariant angle of similarity 15 W. Lode (1926). Versuche ueber den Einfluss der mittleren Hauptspannung auf das Fliessen der Metalle Eisen, Kupfer und Nickel. Z. Physik, Vol. 36, 913-939. 16 b , . Reads & Green, Gotechnique, 26(4), 551-576, 1976. Parry RHG. Ed. Stress-strain behaviour of soils. Proceedings of the Roscoe Memorial Symposium. Cambridge University, March, 1971.
. (2008) , . 1 15
. 1-11: 0s Lode, . (1.58)
. 1-12: b , . (1.59)
1 2 3( , , )O x x x :
1 0 2[ ] 0 2 3
2 3 0ij
=
:
0 0 21 (1 2 0) 1 , [ ] 0 1 33
2 3 1ijp s
= + + = =
. (2008) , . 1 16
2 3 :
2 2 2 2 2 22 11 22 22 33 33 11 12 23 31
2 2 2 2 3 2
1 ( ) ( ) ( )61 (1 2) (2 0) (0 1) 0 3 2 146
sJ = + + + + + = + + + + + =
11 12 13
3 21 22 23
31 32 33
0 0 20 1 3 2(0 2) 42 3 1
s
s s sJ s s s
s s s= = = =
:
0 01.5
3 3 4cos3 0.19839 3 101.442 14s s
= = = D
0 1 233.81 , 153.81 , 273.81s s s = = =D D D
22 2 14 4.32053 3
sJ = = =
( )( )
1
2
3
cos(33.81 ) 3.5897
cos 60 33.81 3.8771
cos 60 33.81 0.2874
s
s
s
= == = = + =
D
D D
D D
: 1 2 3 0s s s+ + =
. (2008) , . 1 17
:
1 1
2 2 2 3 1
3 3
4.592.88 ( )
1.29
p sp sp s
= + == + = = + =
(0 1)b b , ,
3 1
2 1
0.44b = =
ij .
1.3 :
[ ] 10 0
0 00 0
c
c
= (1.60)
2 1
33 1
331
133/ 21
1 ( )32 ( )
27( )3 3 (2 / 3 )cos3 sgn( )
2 (1/ 3)
s c
s c
cs c
c
J
J
= +
= ++= = ++
(1.61)
17
( )1
0 1 2
cos3 1 1 , 12 40 , ,3 3
c s
s s s
L b
< = + = == = = (1.62)
18
17 . axisymmetric extension 18 . axisymmetric compression
. (2008) , . 1 18
( )1
0 1 2
cos3 1 1 , 05, ,
3 3
c s
s s s
L b
< = = + == = = (1.63)
, c 1 , P A (. 1-13)19,
1 cPA
= + (1.64) ( 0P < ), 1 0c < < , ( 0P > ) 1 0c < < (. 1-14).
. 1-13:
. 1-14:
19 . . , , . 3.5, www.geolab.mechan.ntua.gr
. (2008) , . 1 19
. 1-15: Haigh-Westergaard
Haigh-Westergaard (. 1-15). 1 2 , 1( ) / 3OP I = 2( ) 2 sPP J = . o ,
3 1 2 , 5 / 3z c s = = = = (1.65) (. 1-16):
( )( )
1 1 3
2 2 22 1 2 3 3 1
1 3 1
2 3 1
3 3 1
1 1 23 3
1 12 3
2 1cos 53 33 3
2 1cos 53 3 33
2 2cos 53 3 33
s
p I
T J s s s
T Ts
Ts
Ts
= = +
= = + + = = = = = = = + =
(1.66)
. (2008) , . 1 20
. 1-16:
1.4 (REV), . 3 nG ( )REV , . 1-17. E (REV) E . E , , in . E jt ,
EiF , E (REV). jt : Cauchy .
. 1-17: : nG (REV) E . t
G
. (2008) , . 1 21
.. , () (REV); , E E , in ,
n i it t n= (1.67) in ,
np t=< > (1.68)
2
0 0
1 sin4
n nt t d d
< > = (1.69) 1r = , (. 1-18). OE n
= G E 1 2 3sin cos , sin sin , cosn n n = = = (1.70)
. 1-18: : E , .const = , .const = . E .
. (2008) , . 1 22
(;)
( )
( )
2
0 0
ln
1sin3
0
1 13 5 15
0
1 13 5 7 105
i j i j ij
i j k
i j k l ij kl ik jl il jk ijkl
i j k l m
i j k l m n in jklm jn klmi kn lmij mijk mn ijkl ijklmn
n n n n d d
n n n
n n n n
n n n n n
n n n n n n
< > = =< > =< > = + + =< > =< > = + + + + =
(1.71)
1 13 3
nji j i ji j i ij ij kkp t n n n n =< > =< > = < > = = (1.72)
: 1 ,
113
p I = (1.73) , , in .
t ni i it t t n= (1.74) 2
( ) ( ) 225
t tm i i st t J = < > = (1.75)
,
252s m
T J T = = (1.76) 20.
1.5 21 , ( )REV , . , ,
20 . shearing stress intensity. . Kachanov, L.M., Fundamentals of the Theory of Plasticity, Mir Publishers, Moscow, 1974. 21 . shear bands.
. (2008) , . 1 23
(random) (.. , , ). . ( )REV , 22. (), , , (. 1-19).
. 1-19: 23
, ( )sREV ( )rREV , ,
( ) ( )
2 111 22
1 2 1 2
2 22 12 1 21
1 2
1
n
t
t
t
< > = ++ +< > = ++
A AA A A A
A AA A (1.77)
, , 22A ( )rREV 12A , . 2 1/ = A A ,
22 V.V. Novozhilov, Theory of Elasticity, Pergamon Press, 1961. 23 Vardoulakis I. and Graf B. (1985). Calibration of constitutive models for granular materials using data from biaxial experiments. Gotechnique, 35, 299-317.
. (2008) , . 1 24
11 22
2 2
12 21
11 1
11 1
n
t
t
t
< > = ++ + < > = + + +
(1.78)
, 0 , Coulomb
22 21,n tt t < > < > (1.79) , , (. 1-20).
. 1-20:
1.6 , s Lode L , (. 1-21),
3,max 1 2
1,max 2 3
2,max 3 1
/ 2 5(1) & (4) : sin( / 3 )2
/ 2 5(2) & (5) : sin( )2
/ 2 5(3) & (6) : sin( / 3 )2
sm m
sm m
sm m
= = += == =
(1.80)
2 3 1 0
2 1 3 0
1 2 3 0
1 3 2 0
3 1 2 0
3 2 1
(1) : 0 / 3 ( ) :(2) : / 3 2 / 3 ( ) : 2 / 3(3) : 2 / 3 ( ) : 2 / 3(4) : 4 / 3 ( ) : 4 / 3(5) : 4 / 3 5 / 3 ( ) : 4 / 3(6) : 5 / 3 2 ( )
s s s
s s s
s s s
s s s
s s s
s
s s ss s s
s s ss s s
s s ss s s
= = + = +
= + = + 0: 2s s = +
(1.81)
. (2008) , . 1 25
. 1-21:
s . 0s = ( 1)L =
/ 6s = ( 0)L = ,
3,max
3,max
5 3min 1.3692 2
5max 1.5812
m
m
= =
(1.82)
, 2sT J= ,
max30.866 12 T
(1.83)
maxT (1.84) in (;)
2 2 21 1 2 2 3 3
2 2 2 2 2 2 2 2 2 21 1 2 2 3 3 1 1 2 2 3 3( )
n
t
t n n n
t n n n n n n
= + +
= + + + + (1.85)
. (2008) , . 1 26
, . 8 (. 1-22),
{ } { }(1) (2)1 1{ } 1,1,1 , { } 1,1, 1 ,3 3
T Tn n= = (1.86)
. 1-22:
,n toct oct oct octt t = = (1.87)
1 2 3 11 1( )3 3oct
I p = + + = = (1.88)
2 2 2 21 2 3 1 2 32
2 2 21 2 2 3 3 1
2
1 1( ) ( )3 3
1 ( ) ( ) ( )3
2 53 3
oct
s mJ
= + + + +
= + +
= =
(1.89)
(. 1-23)
3,max 3,max5 3 3 5 3min 1.061 , max 1.2252 2 5 2 5oct m
= = (1.90)
. (2008) , . 1 27
. 1-23:
3,max* *3 3 1.061 min 1
2 2 2oct oct oct octT
= = =
(1.91)
* 2 2 21 2 2 3 3 11 ( ) ( ) ( )
2 2oct = + + (1.92)
[ ]1 2 42 2 1 [ ]4 1 3
MPa =
: ) . ) . ) ( )n G (=1,,8). ) . ) s . () , max ( )s mf = . 1.7 Tresca von Mises 24 25 .
24 . yield 25 . failure
. (2008) , . 1 28
, , (. 1-24).
. 1-24: -
.. Y F () (F), -. - , , 26. .. () :
Tresca: ,
max 1 2 2 3 3 11 1 1max , ,2 2 2
Teq = = (1.93)
von Mises: ,
2 2 21 2 2 3 3 11 ( ) ( ) ( )3
Meq oct = = + + (1.94)
( 03 = , .. ) ,
1 2 2 11 1 1max , 0 , 02 2 2
Teq = (1.95)
2 2 2 2 21 2 1 2 11 22 11 22 122 2 3
3 3Meq = + = + + (1.96)
Tresca von Mises : . Y , ,
26 . equivalent stress
. (2008) , . 1 29
1 2,2 3
T Meq Y eq Y = = (1.97)
),( 21 (. 1-25, . 1-26),
1 2 2 11 1 1 1Tresca : max , 0 , 02 2 2 2 Y = (1.98)
2 21 2 1 2v. Mises: Y + = (1.99)
. 1-25: Tresca v. Mises
. 1-26: , (Bisplinghoff, R.L. et al. Statics of Deformable Solids, Dover, 1990)
. (2008) , . 1 30
(1)
10 0 30 3 0 [ ]3 0 2
MPa =
(2)
3 0 00 7 0 [ ]0 0 5
MPa =
, ( ):
(1) (2) ,
) , (1) (2)| | | |oct oct > ) , (1) (2)oct oct > ) , (1) (2)max max > . 1.8 Mohr-Coulomb
. 1-27: Mohr
Mohr-Coulomb (M.-C.) 27 , Mohr (. 1-27),
1 2 2 3 12 1
( ) / 2sin ( )( ) / 2q = + (1.100)
27 . internal friction angle
. (2008) , . 1 31
M.-C. .
1 2 1 21 1( ) 0 , ( ) 02 2M M
= + < = > (1.101)
sin MMq
= (1.102)
sin cos , tanM M mc c q = + = (1.103) c 28 .
(M.-C.) (. 1-28)
2 12cos 1 sin
1 sin 1 sinc
+= + (1.104)
22 11 sin2 , tan (45 / 2)1 sinp p p
K c K K += + = = +
D (1.105)
pK
. 1-28: Mohr-Coulomb 1 2( , ) 3
28 . cohesion
. (2008) , . 1 32
1.9 : Mohr , 20 Otto Mohr (1835-1918). ,
n n , ( , )P x y () dA ,
() n, Ox (. 1-29).
( ) ( )( )
1 1 cos 2 sin 22 21 sin 2 cos 22
n xx yy xx yy xy
n xx yy xy
= + + +
= + (1.106)
. 1-29: ( , )P x y .
xx , yy xy yx = n n , . Error! Reference source not found., ,
( )( )
n n
n n
== (1.107)
nO nO , Mohr .
Otto Mohr (1835-1918)
. (2008) , . 1 33
() ( , )n nO , , . . , ( , )O x y . ( , )n nO , . (1.107) . Mohr . (1.107) , Mohr , .
( , )P x y . , ( , )P x y . ( , )O x y ( , )n nO . :
10 3[ ]
3 5xx xy
yx yykPa
=
. (1.106) . ( )n n = ( )n n = , . (1.106) xx , yy xy yx = , FORTRAN (. 1-30).
. (2008) , . 1 34
c PROGRAM Mohr.for
c Mohr Circle of Stresses
IMPLICIT DOUBLE PRECISION (a-h,o-z)
OPEN (UNIT=1, FILE='Mohr.IN1', STATUS='UNKNOWN')
OPEN (UNIT=2, FILE='Mohr.OU1', STATUS='UNKNOWN')
c Input
pi=4.d0*datan(1.d0)
write(*,*) 'sigma-xx=? kPa'
read(*,*) sxx
write(1,100) 'sigma-xx [kPa]= ', sxx
write(*,100) 'sigma-xx [kPa]= ', sxx
write(*,*) 'sigma-yy=? kPa'
read(*,*) syy
write(1,100) 'sigma-yy [kPa]= ', syy
write(*,100) 'sigma-yy [kPa]= ', syy
write(*,*) 'sigma-xy=? kPa'
read(*,*) sxy
write(1,100) 'sigma-xy [kPa]= ', sxy
write(*,100) 'sigma-xy [kPa]= ', sxy
syx=sxy
sM=0.5d0*(sxx+syy)
write(1,100) 'sigma-M [kPa]= ', sM
write(*,100) 'sigma-M [kPa]= ', sM
tM=dsqrt(((sxx-syy)/2.d0)**2+sxy**2)
write(1,100) 'tau-M [kPa]= ' , tM
write(*,100) 'tau-M [kPa]= ' , tM
100 format(1x,a20,F10.3,1x)
sigma1=sM+tM
sigma2=sM-tM
write(1,100) 'sigma-1[kPa]=',sigma1
write(*,100) 'sigma-1[kPa]=',sigma1
write(1,100) 'sigma-2[kPa]=',sigma2
write(*,100) 'sigma-2[kPa]=',sigma2
write(*,*) 'phi=? in [deg]'
read(*,*) phi
write(1,100) ' phi in [deg]=', phi
. (2008) , . 1 35
write(*,100) ' phi in [deg]=', phi
phi=phi*pi/180.d0
sxi=0.5d0*(sxx+syy)+0.5d0*(sxx-syy)*dcos(2.d0*phi)
# +sxy*dsin(2.d0*phi)
sxieta=-0.5*(sxx-syy)*dsin(2.d0*phi)+sxy*dcos(2.d0*phi)
seta=0.5d0*(sxx+syy)-0.5d0*(sxx-syy)*dcos(2.d0*phi)
# -sxy*dsin(2.d0*phi)
write(1,100) 'sigma-xi-xi[kPa]=',sxi
write(*,100) 'sigma-xi-xi[kPa]=',sxi
write(1,100) 'sigma-xi-et[kPa]=',sxieta
write(*,100) 'sigma-xi-et[kPa]=',sxieta
write(1,100) 'sigma-et-et[kPa]=',seta
write(*,100) 'sigma-et-et[kPa]=',seta
pause
if (phi.eq.pi/2.d0) goto 20
phi=pi/2.d0
20 phi0=0.d0
dphi0=1.d0
do 1000 i=1,361
phi=phi0*pi/180.d0
sn=0.5d0*(sxx+syy)+0.5d0*(sxx-syy)*dcos(2.d0*phi)
# +sxy*dsin(2.d0*phi)
tn=-0.5*(sxx-syy)*dsin(2.d0*phi)+sxy*dcos(2.d0*phi)
write(*,200) phi0,sn,tn
write(2,200) phi0,sn,tn
200 FORMAT(1x,F7.2,1x,2(F10.4,1x))
phi0=phi0+dphi0
1000 continue
STOP
END
. (2008) , . 1 36
. 1-30: ( , )n n . Mohr Mohr ( , )n nO . 0 = . (1.106) :
100 :
3n xx
n xy
kPakPa
= == = =
( 0 )A = D Mohr (. 1-30). , 90 = D . (1.106) :
590 :
3n yyo
n yx
kPa
kPa
= == = =
( 90 )B = D Mohr.
. (2008) , . 1 37
, Mohr . ( , )n n G G , Mohr . , ( , )n n G G , Mohr
. 1-31:
, nO . Mohr. M Mohr () ,
( )1( ) 2M xx yyOM = = + (1.108) 7.5M kPa = . Mohr ( ) ( ') ,
22( ) ( )
2xx yy
xy
= = = + (1.109)
3.91M kPa = . M , . 1 2 ,
. (2008) , . 1 38
12
M M
M M
= += (1.110)
( ) 2 21/ 2 12 2xx yyxx yy xy = + + (1.111) Mohr , . , 1 1( )x = , , , 1 . n,
cos , sinx yn n = = (1.112) n 40 = D Ox . . (1.106) :
10.89 , 1.94n nkPa kPa = = Mohr ( 40 ) = D (. 1-32). Mohr :
( ) ( )1 1
1 1
( ) sin 2 sin 2
11 2( ) cos 2 cos 22
xyxy M
M
xx yyxx yy M
M
= = =
= = =
(1.113)
1( ) cos 2( )M M = + (1.114) ,
1 1 1cos 2( ) cos 2 cos 2 sin 2 sin 2 = + (1.115) . (1.108) ,
( )( ) ( )
12( ) cos 2 sin 2
1 1 cos 2 sin 22 2
( )
xx yy xyM M M
M M
xx yy xx yy xy
= + +
= + + + =
(1.116)
Mohr :
. (2008) , . 1 39
1( ) sin 2( )M = (1.117) ,
1 1 1sin 2( ) sin 2 cos 2 cos 2 sin 2 = (1.118) . (1.113) ,
( )( )
12( ) sin 2 cos 2
1 sin 2 cos 22
( )
xx yy xyM M
M M
xx yy xy
= +
= + =
(1.119)
Mohr , n, . (1.112), O . ,
10.889
4.111
1.941
kPa
kPa
kPa
===
. 1-32: Mohr
Mohr n , , . : ( )
. (2008) , . 1 40
nO Mohr , nO . , ( )x + = (1.120)
O , n, , , ( ), ( )n n n n = = . : ( ) Mohr nO . , n ( O ), Mohr . .
O nO , Mohr , n - Ox 90 = +D . : () ( , )O ( , )O x y . ( ) , (H) Mohr , :
( ) 2AM = (1.121)
( ) 2(90 ) ( ) 180AM AM = + = +D D (1.122) . (. 1-33).
Mohr . Mohr (. 1-34), , .
. (2008) , . 1 41
. 1-33: .
. 1-34:
. (2008) , . 1 42
: .
()
xx xy
yx yy
cos sin cos sinsin cos sin cos
xx xy
yx yy
= (1.123)
.. 1 = . (1.123) ,
1
2
00
=
, () Mohr . Mohr .
. (2008) , . 1 43
. Mohr .
. (2008) , . 1 44