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