Applied Mathematics and Mechanics

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1 , 22 9 ( ) Appled Mathematcs Mechacs (2001) Ξ 1, 2 3, (11, ; 21, ; 31, ) ( ) 1) ;2), (Homogezato), Mohr-Coulomb, ; ; ; ; O342 A, Mohr-Coulomb ; ;, [1 ],[2 ],[3 ],[4, ], Bassta Gross [5 ] Drago Halm [6 ], [7 ],[8 ], Mohr- [9 Coulomb, ], Mohr-Coulomb [10 ],[11 ] ( ), ( Homogezato), Ξ ; ( ) (1963 ),,,,. 927

2 928 Mohr-Coulomb 1 111, u = u[ x, s ( x), ( x), ( x) ], s ; Clausus-Duhem gs gu ( s,, ) - g, (1) ; ; gs ( ) (1) 5 u u u - gs g - 5 g 5 s 0 (2), g = 0 ; gs = 0 g gs, = 5 u, = 5 u (3) 5 s Y = - 5 u g (4), = u - s = (,, ), (5) = - = (,, ) (6) Clausus-Duhem Y = - g - s + g - g 0 (7) Clausus-Duhem Y = g + - s gs + g 5 s 0 (8) (7) (8) =, s = -, Y 5 ; = - (9) =, s =, Y 5 ; = (10) Y, ( ) - g = 52 g ; g; + 52 g (11) g = 52 g ; g; + 52 g (12) g = 0, = P, = D, (11) (12) g = 52 g P gp + 52 gd = g e + g P + g D, 5 D p D (11a)

3 929 g = 52 g P gp + 52 gd = g e + g P + g D (12a) 5 D (11a) (12a) 3 P = P,= W (, P, D) = 1 2 ( - P ) E ( D) (, - P ), = u (, D) = 1 L ( D) (13) 2 Y D = 1 2 L ( D) ( - P ) 5 E ( D) ( - P ), Y 5 D P = = E ( D) ( - P ), (14) = E ( D) - 1, E Y = g P + Y gd 112 Mohr-Coulomb F P = F P (, D, c 0, 0 ) 0 (16) c 0, 0, D Y F D = F D ( Y, P, 0 ) 0, (17) g P = g P 5 F P, g P > 0, F P = 0, g P F P = 0, (18) gd = g D 5 F D 5 Y, g D > 0, F D = 0, g D F D = 0 (19) 1) ; F P < 0 ; F D < 0 ; 2), F D = gf D = 0, F P < 0 ; 3), F P = gf P = 0, F D < 0 ; 4), F D = gf D = 0, F P = gf P = 0, g P g D gf P = 5 FP 5 FP g + 5 D (15) gd = 0, (20) 5 F - g P FP P E 5 + g 5 F P D 52 FD 5 5 D 5 Y + 5 FP FD 5 5 D 5 Y + 5 FP E g = 0 (21), 5 F - g D P 5 Y 52 5 D P 5 FP + g 5 F D D 5 Y 52 5 D 2 5 FD 5 Y + 5 FD 5 Y 52 g = 0 (22) 5 D g = E g - g P E 5 FP + g 5 2 FD D 5 (23) 5 D 5 Y (23) D 113

4 930, ( ) 0, ( ), [13 Rce ], D d D j = 5 Y j d D kl, (24) kl ( ) d = d = tr (d j) d (25) d v d,,,, 2, 211 D D = 3 =1 g, (26), [2 ],[14 ], gd gd = P + ( ) D, P jkl = Q + kq + jl, Q + = 3 = 1 3 h ( ) p g p, + = p g p, (27) = 1 P +, Q +, h ( ) Heavsde, p [1 ], E E = ( I - gd) E 0 ( I - gd) (28) E 0, gd F D = Y P + Y - < 0 0, (29) < Mohr- Coulomb (Homogezato) [12 ],, 1 1 3, Mohr-Coulomb,,, ;,, d, t, 1, Mohr-Coulomb

5 F P t = t - t t - c t = 0, (30) ; c t 1,, Mohr-Coulomb Fd P = d - d d - c d = 0 (31) 1,, Mohr-Coulomb 1 F P = - - c = 0 (32), 1, = t (1 - - d ) + d d, (33) = t (1 - - d ) + d d (34) = gd j j, d = ( D j - gd j ) j 0,, = 0, t = d = (1 - ) - 1 (35) Mohr-Coulomb = c (36), 3 = (1 - ) [ t (1 - - d ) + d d ], (37) c 3 = c t (1 - - d ) + c d d (38), Mohr-Coulomb -, v, ( = 1, N), N (37) = c 3 + 3, (39) g P = 3 5 g P F P, (40) = 1 g P g F P = F P, (21) (22) ( (26) - (40) ), 931

6 932 3, 1), 2), ( Homogezato), Mohr-Coulomb, Mohr-Coulomb,, 3), [ ] [1 ] Lematre J. A Course o Da mage Mecha cs. 2d ed[m ]. Berl Sprger, [ 2 ] Hase N R,Schreyer H L. A thermodyamcally cosstet framework for theores of elastoplastcty coupled wth damage [J ]. It J Solds Str uct ures, 1994, 31 (2) [ 3 ] Hayakawa K,Murakam S. Thermodyamc modelg of elastc-plastc damage expermetal val2 dato of damage potetal [J ]. It J Da ma Mech, 1997, 6 (2) [ 4 ] Marott de Scarra F. A ew varatoal theory a computatoal algorthm for coupled elastoplas2 tc damage models [J ]. It J Solds Str uct ures, 1997, 34 (9) [5 ] Bassta M, Gross D. The sldg crack model of brttle deformato a teral varable approach [J ]. It J Solds Str uct ures, 1998, 35 (3) [6 ] Drago A, Halm D. A mesocrack damage frcto coupled model for brttle materals [ A ]. I Voyjs G Z, J u J W, Chaboche J L Eds. Da mage Mecha cs Eg eer g Materals [ C ]. Ams2 terdam Elsever Scece, 1998, [7 ] Meschke G, Lacker R, Mag H A. A asotropc elastoplastc- damage model for pla cocrete [J ]. It J Numer Mech Egg, 1998, 42 (3) [ 8 ] Yazda S, Karawat S. A costtutve theory for brttle solds wth applcato to cocrete [J ]. It J Da ma Mech, 1996, 5 (1) [9 ] Vutukur V S, Lama R D, Saluja S S. Ha dbook o Mecha cal Propertes of Rocks [ M ]. Vol. 1. Berl Tras Tech Publsher, [10 ] Duveau G, Shao J F. A modfed sgle plae of weakess theory for the falure of hghly stratfed rocks [J ]. It J Rock Mech M Sc, 1998, 35 (6) [11 ] Hoek E, Brow E T. Practcal estmato of rock mass stregth [J ]. It J Rock Mech M Sc, 1997, 34 (8), [12 ] Muller D, Kratochvl J,Berveller M. Nolocal versus local elastoplastc behavor of heterogeeous materals [J ]. It J Plastcty, 1993, 9 (3) [13 ] Rce J R. Ielastc costtutve relatos for solds a teral varable theory ts applcat to metal plastcty [J ]. J Mech Phys Solds,1971, 19 (2) [14 ] Swoboda G, She X P, Rosas L. Damage model for joted rock mass ts applcato to tuel2 g [J ]. Comp uter a d Geotechcs, 1998, 22 (3/ 4)

7 933 Cos tt utve The ory of Plas tct y Couple d Wt h O rt hot r opc Da mage f or Ge oma t e rals SHEN X-p u 1, Zeo Mroz 2, XU Bg- ye 3 (11 Depa rtmet of Archtect ural Eg eer g, Sheya g Uversty of Techology, Sheya g , P R Ch a ; 21 IFTR, Polsh Acemy of Sceces, Wa rsaw , Pola d ; 31 Departmet of Eg eer g Mecha cs, Ts ghua Uversty, Bej g , P R Ch a) Abs t ract Costtutve theory of plastcty coupled wth orthotropc damage for geomaterals was es2 tablshed the framework of rreversble thermodyamcs. Prme results clude 1) evoluto laws are preseted for coupled evoluto of plastcty orthotropc damage ; 2) the orthotropc damage tesor s troduced to the Mohr- Coulomb crtero through homogezato. Both the degrato of shear stregth degrato of frcto agle caused by damage are cluded ths model. The d2 latacy s calculated wth the so- called damage stra. Key words damage ; plastcty ; couplg ; dlatacy ; geomateral

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