, 22 9 (2001 9 ) Appled Mathematcs Mechacs 1000-0887(2001) 09-0927-07 Ξ 1, 2 3, (11, 110023 ; 21, ; 31, 100084) ( ) 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 2000-08-19 ; 2001-04-08 (1999 363) (1963 ),,,,. 927
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 + - 5 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 + 52 5 ; g; + 52 g (11) g = 52 g + 52 5 ; g; + 52 g (12) g = 0, = P, = D, (11) (12) g = 52 g + 52 5 P gp + 52 gd = g e + g P + g D, 5 D p D (11a)
929 g = 52 g + 52 5 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
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) < 0 212 Mohr- Coulomb (Homogezato) [12 ],, 1 1 3, Mohr-Coulomb,,, ;,, d, t, 1, Mohr-Coulomb
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 3 + 3 (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
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, 1990. [ 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) 359 389. [ 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) 333 363. [ 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) 1761 1796. [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) 487 509. [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, 321 336. [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) 703 727. [ 8 ] Yazda S, Karawat S. A costtutve theory for brttle solds wth applcato to cocrete [J ]. It J Da ma Mech, 1996, 5 (1) 93 110. [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, 1974. [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) 807 813. [11 ] Hoek E, Brow E T. Practcal estmato of rock mass stregth [J ]. It J Rock Mech M Sc, 1997, 34 (8), 1165 1186. [12 ] Muller D, Kratochvl J,Berveller M. Nolocal versus local elastoplastc behavor of heterogeeous materals [J ]. It J Plastcty, 1993, 9 (3) 633 645. [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) 433 455. [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) 183 203.
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 110023, P R Ch a ; 21 IFTR, Polsh Acemy of Sceces, Wa rsaw 00-049, Pola d ; 31 Departmet of Eg eer g Mecha cs, Ts ghua Uversty, Bej g 100084, 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