31 3 1999 5 ACTA MECHANICA SINICA Vol 31, No 3 May,1999 1) (, 100083), ( d, ) 0 ( d, ) Eucldean,,, :,,,,,,,, [14, ],,,,,, [514 ],,, [511 ], [1214 ] [417 ],,,, 1997-05 - 04, 1998-07 - 14 1) (59425003) (19702021) (97 10202)
3 : 301 1 1 1 : 1, [9, 1113, 1517, ] 1 [15,16 ] ( a ) [15,16 ] Fg 1 Characterstcs of fractal dstrbuton and evoluton of mcro/ meso cracks ( a s the sde length of boundary of the area) R n 2 ( ),E 0 R n,, R n E R n,e R n, ge R n E 0,gE R n 2 Fg 2 Representatve fractal damage element,, R 2 ( ), E 0 a ( 2), E, ge [5, 9, 1214, 18 ], ge :, E 0 \ E E 0 E ge = E 0 - E = E 0 \ E (1) [5,9,1214,18 ], : E Α E 0, ge Α E 0, E ge = E 0 (2) 1 2 Hausdorff 1 2 1 [5,9,1214, 18 ], A < R n,m ( A ) A, R n M, : (a) M ( g) = 0, g ; (b) A, B : A < B,M ( A ) Φ M ( B ) ; (c) A ( = 1,2, s, K) K,: M K = 1 A Φ 6 K = 1 M ( A ) ;
302 1999 31 (d) A Borel,M K A = = 1 6 K M ( A ) = 1, A, B,A B = g;c = A B,C, M ( C) = M ( A ) + M ( B ) (3),, (3), M ( E), M ( ge) M ( E 0 ) R n, E, ge E 0 E, ge, E ge = g, (2) E 0, (3) E 0 = E ge = E ( E 0 \ E) = E ( E 0 - E) (4) M ( E 0 ) = M ( E) + M ( ge) (5) M ( ge) = M ( E 0 ) - M ( E) (6) (5) (6) 1 2 2 Hausdorff Hausdorff, Eucldean (1 ) (2 ) (3 ), E, ge E 0 Hausdorff R n [5,9,1214,18 E,Hausdorff ],{ B } E, E <, B = r / a ( r B, a R 2 a ) ; 0 < < 1,0 < max (Φ Hausdorff, d, E Hausdorff H ( d, E) ) = lm nf C 0 6 ( E) d :{ B } E 2 (7) = 1 d E Hausdorff, C ( E), 2 { B }, ge E 0 Hausdorff H ( gd, ge) = lm nf C 0 6 ( ge) gd :{ B } ge 2 (8) = 1 H ( d e, E 0 ) = lm nf C ( 0 E 0 6 ) d e :{ B } E 0 2 (9) = 1 gd f, d e ge E 0 Hausdorff, C ( ge), C ( E0 ), X Hausdorff H ( d, X) Hausdorff d n [5,9,1214,18, H ( d, X) n Lebesgue ] X n
3 : 303 Lebesgue,n Hausdorff H ( n, E), H ( n, ge) H ( n, E 0 ) E, ge E 0, H ( d, E), H ( gd, ge) H ( d e, E 0 ), 1 3 [14 ] 0 = A / A 0 = 1 - ga / A 0 (10) A, ga, A 0, 0, [19,20, ] [1214 ] : ;, (10),,, H ( d, E), H ( gd, ge) H ( d e, E 0 ), ( d) = H ( d, E) H ( d e, E 0 ) = 1 - H ( gd, ge) H ( d e, E 0 ) (11) H ( d, E), H ( gd, ge) H ( d e, E 0 ) E ge E 0 : d gd Eucldean 2 0, H ( d, E), H ( gd, ge) H ( d e, E 0 ) : H (2, E) = A, H (2, ge) = ga, H (2, E 0 ) = A 0, (11) (10) 0 ( d) Hausdorff d gd Eucldean,,(11), d gd ( d) 1 4, E,, E 2 0 < r l Φ r Φ r u ( r l, r u, r l ), 2 : 0 < l Φ Φ u Φ 1, l = r l / a, u = r u / a R n, Φ 2, d E N ( d) ( ) ; gd ge N ( gd) ( ) ; E 0 N ( de ) ( ),, (7) (9), E, ge E 0
304 1999 31 H ( d, E) H ( gd, ge) = lm = lm N ( d) nf C 0 6 ( E) = 1 N ( gd) nf C 0 6 ( ge) = 1 d gd = C ( E) N ( d) ( ) d (12) = C ( ge) N ( gd) ( ) gd (13) H ( d e, E 0 ) = lm nf C ( 0 E 0 6 ) = 1 d e (12) (14) (11), ( d) = C ( E0 ) N ( de ) ( ) d e (14) ( d, ) = C ( E) N ( d) ( ) C ( E0 ) N ( de ) ( ) d - d e = 1 - C ( ge) N ( gd) ( ) C ( E0 ) N ( de ) ( ) gd - d e (15) : E ge Hausdorff Eucldean,d = d e, gd = d e, ( d, ) 0,: ( d, ) d = de gd = d e = 0,, N ( d) ( ), N ( gd) ( ) N ( de ) ( ) 2, (15) ( d, ) = 0 d - d e ( d, ) = 1 - (1-0 ) gd - d e (16), d, gd Hausdorff, d e Eucldean, 0 Hausdorff, d, gd, (16) d (gd ) 2 R 2, ( d, ) = 0 d - 2 ( d, ) = 1 - (1-0 ) gd - 2 (17) 1 5 - d - (16) ( d, ) d ( gd ), d [9,1114,17 ] ( d, ) d,, d ( gd ),, ( d, ) L ( df ) L ( df ) = L 0 1 - d f ( L 0, d f ) [7,9,1114 ], d = d e, (16), ( d, ) = 0,, ( d, ), 0 < < 1, 0 < d Φ d e, (16) : 0 Φ( d, ) = 0 d- d e <
3 : 305, ( d, ),,: 0 < l < < u < 1, l, ; u, ( d, ) 0 Φ 0 = ( d, u ) Φ ( d, ) Φ ( d, l ) Φ 1 (18) ( ), ( d, ), ( d, ), ( d, ) d, 0,, 2 [14 ], [14,17 ], F( Y( d, ) ; ( p,( d, ) ) ) = Y( 2 d, ) 2 m ( d) [1 - ( d, ) ] H ( p - p( d, ) ) (19) : ( d, ) ; Y( d, ), w e ( d, ) : Y( d, ) =, w 1 - ( d, ) e ( d, ) : w e ( d, ) = 1/ 2 e j e kl[1 - ( d, ) ] ; m ( d) ; H( p - p( d, ) ) jkl, p ),: p( d, 1 p Ε p( d, H( p - p( d, ) ) = ), p( d, ) 0 p < p ( d, ) [14,17 ], g( d, ) = 5 F ( Y( d, ) ; ( p, ) ) g( d, ) (20) 5 Y ( d, ) [14,17 ( d, ) ] : g( d, ) = gp [1 - ( d, ) ] g( d, ) (19) g( d, ) = g( d, ) = Y( d, ) m gp H ( p - p( d, ) ) (21) w e ( d, ) gp H ( p - p( d, m [1 - ( d, ) ] ) ) (22) (21) (22) ( d, ) = 0 d - d e, g 0 + gdln 0 = w e ( d, ) d e - d gp H ( p - p( d, m [1 - ( d, ) ] ) ) (23)
306 1999 31 : g, gd, p, d = d e,(23) g 0 = 3 w e m (1 - ) gp H ( p - p ) (24) [14,, ] R n [17 ] F F ( d, ) = ( g- X) eq, ( d, ) - R - y + 3 X 4 X j X j - F( d, ) ( Y ( d, ) : (,( d, ) ) ) (25) ( g- X) eq, ( d, ) - R - y, R, y 1 3, ( j j 2 g- X) eq, ( d, ) = - X 2 1 - ( d, ) j - X 1 - ( d, ) j, j, X j ;, X F ( Y ( d, ) : (,( d, ) ) ), R, Y ( d, ) [14 ] [17 ], j p, ( d, ) = 5 F( d, ) 5 j g( d, ) (26) g( d, ), (25) ( d, ) = 0 d - d e, j p, ( d, ) = 5 52 j 3 j 1-0 d - d e j - X j 1-0 d - d e - X j 1 2 gp (1 - d - d e ) (27) 0 ; d R n ;, ; d e Eucldean R 2 [17 ], = E 1-0 d - 2 (28) d, E,, ( d, ),,, 4 [17, ], 3 4 1 ( 4 ) [17,21 ],
3 : 307 [17 ] ( d, ) ( d, ) = 1-1 - r E r E 0 d - 2 (29) : d, [17,21 ], r ; E, E 0 2 [17 ] 3 Fg 3 Pattern of loads d, 4 [17, 21 ] [17, 21 ] Fg 4 Graphcs of mcro/ meso cracks and ts dstrbuton on cross secton of concrete [17 ], (18) l l, 0 263 = l ΦΦ u = 1 (30) 5 = 1/ 2, 2/ 5, 1/ 3 2/ 7 ( d, ) c / f c, 0 :, 4 2, ( d, ) ( ) = (1 - ( d, ) ) E 0 ( d, ) ( ) = 1-0 d - 2 E 0 (31) 0, [17 ] ; E 0 ; ( d, ) ( ) = (1-0 ) E 0 + (1 - d - 2 ) 0 E 0 (32) ( d, ) ( ) = (1 - d - 2 ) 0 E 0, (32), ( d, ) ( ) = ( ) + ( d, ) ( ) (33) ) ( ), 2 ( d, 2 [ 17, 21 ]
308 1999 31,6 2, [(30) ], = 1/ 315, 2 : 2 2, 5 Fg 5 5 6 Fractal damage of concrete and ts evoluton 2 Fg 6 Stress - stran responses of concrete wth fractal damage,, : 1) 0 ( d, ) = 0 d - d e ( d, ) = 1 - (1-0 ) gd - d e 0 ( d, ) d gd Eucldean, 2) ( d, ) d (gd ),, 3), : 1 Kachanov LM Introducton to Contnuum Dama ge Mechancs The Netherlands : Martnus Njhoff Dordrecht, 1986 2 Lematre J, Chaboche JL Mechancs of Sold Materals Cambrd ge : Cambrdge Unversty Press, 1990 3 Lematre J A course on damage mechancs Berln : Sprngs2Verlag, 1992 4 :,1990 ( Xe HP Damage Mechancs of Rocks and Concrete Xuzhou : Chna Unverst y of Mnng and Technology Publsher, 1990 (n Chnese) )
3 : 309 5 Mandelbrot BB The Fractal Geometr y of Nature New York : W H Frecman, 1982 6 Ostoja2Starzewsk M Damage n a random mcrostructure : sze effects, fractals and entro py maxmzaton A ppl Mech Rev, 1989, 42 : 52025212 7,, 1990, 20 (4) : 468 477 (Lu CS, Ba YL Fractal behavors of damage and fracture of materals A dvances n Mechancs, 1990, 20 (4) : 468477 (n Chnese) ) 8 Panagotopoulos PD Fractal geometry n solds and structures Inter J Solds & St ruct, 1992, 29 : 21592175 9 Xe H P Fractals n rock mechancs Rotterdam, Netherlands : A Balkema Publsher, 1993 10 Underwood EE Fractals n materals research Acta Stereogca, 1994, 13 : 269279 11, 1995, 17 (2) : 7582 ( Xe HP Fractal damage n brttle materals J Mech St rength, 1995,17 (2) : 7582 (n Chnese) ) 12, 1995, 25 (2) : 174185 ( Xe HP Mathematcal fundamentals to fractal mechancs A dvances n Mechancs, 1995, 25 (2) : 174185 (n Chnese) ) 13 :, 1997 ( Xe HP Introducton to Fractal Rock Mechancs Bejng : Scence Press, 1997 (n Chnese) ) 14, :, 1997 ( Xe HP, Xue XQ Mathematcal Fundamentals and Approaches to Applcaton of Fractal Theory Bejng : Scence Press, 1997 (n Chnese) ) 15 Nolen2Hoeksema RC, Gordon RB Optcal detecton of crack patterns n the openng2mode fracture of marble Inter J Rock Mech Mn Sc, 1987, 24 :135144 16 Botss J, Kunn B On self2smlarty of crack layer Inter of Fract ure, 1987, 35 : 5156 17 : 1997 (J u Y A Study on Theory of Fractal Damage Mechancs of Concrete Postdoctoral Research Thess, Chna Unverst y of Mnng and Technology (Bejng Campus), Bejng, 1997 18 Falconer Fractal geometry : mathematcal foundaton and applcaton New York : John Wly & Sons, 1990 19 Truesdell C Ratonal Thermodynamcs New York : McGraw2Hll Book Company, 1969 20 Erngen AC Nonlnear theory of contnuous meda New York : McGraw2Hll, 1962 21 Shah SP, Sanker R Inter crackng and stran2softenng response of concrete under unaxal compresson A CI Materal Journal, 1987, 84 (3) :200212 A STUD Y OF DAMA GE M ECHAN ICS THEOR Y IN FRACTIONAL DIM ENSIONAL SPACE 1) Xe Hepng J u Yang ( Chna U nversty of Mnng and Technology, Bejng Cam pus, Bejng 100083, Chna ) Abstract Of most mportance n contnuum damage mechancs s how to properly defne a damage varable t hat s avalable for descrbng damage degree and t s evoluton It plays a key role n correlatng macro mechancal responses to t her nternal mcro/ meso damage effect s n materals As one of wdely2accepted effectve approaches to defne a damage varable, macro p henomenologcal defnton performs a great advantage of beng easly utlzed n analyzng macro damage mechancal responses of materals and st ruct ures Receved 4 May 1998, revsed 14 J uly 1998 1) The project s supported by Natonal Dstngushed Youth s Scence Foundaton of Chna, Trans2century Tranng Programme for Talents by State Educaton Commsson of Chna, Natonal Natural Scence Foundaton of Chna and Coal Scence Foundaton of Chna Nevert heless, t he defnton cannot reflect nt rnsc mechansm of damage propertes The majorty of p henomenologcal defntons are
310 1999 31 defcent n p hyscal meanng, and most of proposed mechancal models for damage evoluton and constt utve equaton are emprcal ones t hat have very lmted applcablty For applyng damage mechancs to accurately explanng and elucdatng damage and rupt ure p henomena n materals and structures, t s essental to determne a damage varable that can not only quanttatvely state damage nt rnsc mechansm but also be easly adopted n macro damage analyses In the present paper,a fractal damage varable ( d, ) s proposed, whch can not only reflect nternal damage mechansm but also be ftted to macro damage mechancs analyses Furt hermore, t he f ractal expressons of damage evoluton laws and damage constt utve equaton are deduced n terms of defnton of f ractal damage varable As an example, damage mechancal behavors of concrete under unaxal compressve st ress have been dscussed by means of t he proposed met hod It s shown that fractal damage varable ( d, ) to be defned at fractonal dmensonal space actually s a generalzed case of the apparent damage varable 0 whch s defned at Eucldean space The fractal damage varable( d, ) wll be the same as 0 when the fractal dmenson d s equal to Eucldean nteger dmenson The concept of damage varable n contnuum damage mechancs has been extended f rom Eucldean space to f ractonal dmensonal space The dscusson also ndcates that fractal damage varable ( d, ) depends on f ractal dmenson d and measure scale The dependency on measure scale s a hallmark of fractal damage varable to be dffered from the apparent damage varable 0 The analyses of fractal damage of concrete mples that f ractal damage s hgher t han apparent damage f t he f ractal effect s of damage s consdered Fractal damage ncreases when t he measure scale decreases Addtonally, t he nvestgaton shows t hat f ractal models of damage evoluton law and damage constt utve relatonshp for concrete are better n agreement wt h t he act ual damage evoluton and st ress2st ran responses The models quanttatvely manfest t he dependence of macro mechancal behavors on t her nternal mcro/ meso damage effect s Key words damage, fractal, fractonal dmensonal space, fractal damage varable, fractal damage evoluton law, f ractal damage constt utve equaton