100, (J. Bau sch inger, ), (A. Fopp l, ), (L. P randtl, ) (V on. Karm an, ) , ( ) , 1900) ( ,, 1985)

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1 (5): Ch inese J ou rna l of R ock M echan ics and E ng ineering S ep t M ohr-coulom b ( ) ( ) ( ) ( ) 1900 (O. M oh r) 2 (M oh r2coulom b) 100 ( ) ( ) 100 ( ) ( 1985) ( 1991) 100 ( 2 ) TU 451 O 346 A (2000) [ ] [ (M oh r2cou lom b) ] ( ) [ 4 5 (J. Bau sch inger ] ) (C. Bach) (W. V o igt ) (A. Fopp l ) (L. P randtl ) (V on ; ( Karm an ) ) ; ( ) 1882 [ 1 ] [ 6 8 ] [ 2 2 ] ( ) :

2 (Hoek2B row n) [ ] 30 Ρ1 = Ρc + aρ b 3 (3) 60 Σ n 13 = Ρc + aρ b 3 (4) ; 80 Σ13 = Ρc + aρ13 (5) 90 (1) [ 16 ] (R. Bo rker) : Ρ2 100 [ 9 ] ( D. Sandel) 1919 Ρ2 [ 3 ] (Ρ1 - Ρ3) + n (Ρ1 + Ρ2 + Ρ3) = 2Σ0 (6) 1910 (H uber 1904) (M ises 1913) (H encky 1925 ) (Ro s [ 10 ] E ich inger 1927) [ 11 ] 1911 Ρ [ ] (D rucker) (P rager) Ρ2 [ ] (Griffith) 2 ; ;. 3 Ρ (H um pheson2n yalo r) (Zienk iew icz O. C. W. F. Chen) 2 [ ] Σ= f (Ρ) Ρ1 - Ρ3 = f (Ρ1 + Ρ3) (1) 2 [ ] [ 29 ] 2 : 2 Ρ1 - ΑΡ3 = Ρt Σ= Σ0 + ΒΡ (2) : Ρt ; Α Α= ΡtgΡc 1911 (1) Ρ2 60 [ ]

3 M oh r2coulom b Ρ2 15% Ρ2 (Sangha) % [ 3 ] 2 50% [ 17 ] Ρ2 [ ] Ρ2 (M ichelis) Ρ2 [ ] Ρ [ ] [ Ρ2 ] Ρ3 Ρ1 Α F = [ ] Ρ1-2 (Ρ2 + Ρ1 + ΑΡ3 Ρ3 ) = Ρt Ρ2 1 + Α (7a) [ ] F = 1 2 (Ρ1 + Ρ1 + ΑΡ3 Ρ2 ) - ΑΡ3 = Ρt Ρ2 1 + Α 1 1 F ig. 1 L im iting m eridian curves at different stress angles 5 Ρ2 (7b) Π a [ 30 ] 2 F ig. 2 A series of lim it loci of the united strength theo ry

4 [ 39 ] 2 [ 39 [ ] ] [ ] [ ] [ ] ( ) 2 [ ] ( 1985) ( 1991) 100 Α F = Ρ1-1 + b (bρ2 + Ρ ΑΡ3 Ρ3 ) = Ρt Ρ2 1 + Α (8a) 7 (1) F = 1 + b (Ρ1 + Ρ1 + ΑΡ3 bρ2 ) - ΑΡ3 = Ρt Ρ2 1 + Α (2) 1985 (8b) (3) Π [ 36 ] (4) 1991 Π 2 (b = 0) (b = 1) (5) (b < 0 2 ) (b > 1 ) (0 < b < 1 ) (6)

5 M oh r2coulom b (7) M oh r O. U ber die D arstellung des Spannungszustandes und des D efo rm ationszustands eines Ko rperelem entes [ M ]. Zivilingenieur: [ s. n. ] M ohr O. W elche Um stande bedingen die Elastizitatsgrenze und den bruch eines m aterials[j ]. Ingenieure : Zeitschrift des V ereins D eutscher 3 M ohr O. A bhandlungen aus den Gebiete der Technischen M echanik [M ]. V erlag von W ilhelm: Ernst & Sohn 1905 T h ird ed. 1928; : [M ].. : W ittke W. Rock M echanics: T heo ry and A pp lications w ith Case H isto ries[m ]. Berlin: Sp ringer2v erlag 1990 :.. : [M ]. : [M ]. : (11) : [M ]. : T imo shento S P. H isto ry of Strength of M aterials [M ]. N ew Yo rk: M cgraw 2H ill V on Karm an T. Festigkeitsversuche unter allseitigem [J ]. Z. V ereins D eutscher Ingenieure : Boker R. D ie M echanik der B leibenden Fo rm anderung in Kristallinisch A ufgebauten Ko rpern [ J ]. Berlin : 1 51 Fo rschungsarbeiten 12 A ndreev G. B rittle Failure of Rock M aterials: Test Results and Constitutive M odels[m ]. Ro tterdam: A. A. Balkem a Paul B. M odification of the Coulom b2m ohr theo ry of fracture [J ]. J. A pp l. M ech : Sheo ryey P R. Emp irical Rock Failure C riteria[m ]. Ro tterdam: A. A. Balkem a [M ]. : 39 Yu M H Zhao J Guan L W. Strength theo ry fo r rock and H arkness R M. A n essay on M oh r2coulom b [A ]. In: Parry R H G ed. Stress2Strain of So il[c ]. O xfo rd: Foulis & Co [J ]. [J ] ( ) : (4) : [J ]. associated and non2associated constitutive model and its 1979 (2) : M ogi K. Effect of interm ediate p rincipal stress on rock failure [J ]. J. Geophys Res : M ogi K. Failure and flow of rocks under high triaxial comp ression[j ]. J. Geophys Res : M ichelis P. Po lyaxial yielding of granular rock [J ]. J. Engng. M ech. A SCE (8) : M ichelis P. T rue triaxial cyclic behavio r of concrete and rock in comp ression[j ]. Inṫ J. P lasticity : [J ] (1) : [R ]. 25. [A ]. :. [C ]. : Zienk iew icz O C Pande G N. Som e useful fo rm s of iso trop ic yield surfaces fo r so il and rock m echanics [A ]. ed. In: Gudehus G F inite Elem ents in Geom echanics[c ]. London: John W iley & Sons L td Chen W F Baladi G Y. So il P lasticity: Theo ry and Imp lem entation[m ]. Am sterdam: Elsevier Science Pub [A ]. : [C ]. : [M ]. : [J ] (1) : [J ]. A (12) : [M ]. : Yu M H H e L N Song L Y. Tw in shear theo ry and its generalized [ J ]. Scientia Sinica ( Science in China) Series A 33. [J ] (3) : [J ] (50) : [R ] Yu M H H e L N. A new model and theo ry on yield and failure of m aterials under comp lex stress state [A ]. In: M echanical Behaviour of M aterials2g [C ]. O xfo rd: Pergamon P ress : [J ] (2) : [M ]. : 1999 concrete: histo ry p resent situation and developm ent [ J ]. P rogress in N atural Science (Chinese Edition) (6) : ; (English Edition) (4) : Yu M H Yang S Y Fan S C et al. U nified elasto2p lastic engineering app lications [ J ]. Computer and Structures : [J ] (2) : [J ] (2) : [J ].

6 Guow ei M Iw asaki S M iyano to Y. P lastic lim it analyses of circular p lates w ith respect to unified yield criterion [J ]. Inṫ J. M ech. Sci (10) : [J ]. arbitrary load[j ]. J. A pp lied M echanics A SM E (2) : (1) : [J ] (6) : M a G H ao H. U nified p lastic analyses of circular p lates under AD VANCES IN STRENGTH THEORY OF ROCK IN 20 CENTURY 100 Y EARS IN M EMORY OF THE MOHR-COULOM B STRENGTH THEORY Yu M aohong 1 Zan Yuew en 1 Fan W en 2 Zhao J ian 3 Dong Zhengzhu 4 ( 1 S chool of C iv il E ng ineering &M echanics X i an J iaotong U niversity X i an Ch ina) ( 2 Op en L aboratory of Geotechnical E ng ineering Chang an U niversity X i an Ch ina) ( 3 S chool of C iv il & S tructu ral E ng ineering N any ang T echnolog ical U niversity S ing ap ore) ( 4 Ch ina U niversity of M ining and T echnology X uz hou Ch ina) Abstract It has been 100 years since P rofesso r O. M oh r estab lished the w ell2know n M oh r2cou lom b strength theo ry (failu re criterion ) in A great am oun t of effo rts w ere dedicated to the study and developm en t of st reng th theo ry o r fa ilu re criteria of rock under the com p lex st ress sta te and to their co rrelation w ith test data. T en s of failu re criteria of rock under the com p lex stress state w ere p ropo sed. A con siderab le am oun t of experim en tal w o rk w ere done in connection w ith the failu re of rock under the p lane stress and strain asymm etric tri2ax ial and true tri2ax ial stress state. A su rvey of the m ain researches and developm en t of strength theo ry on rock are p resen ted since T he advances in strength theo ry of rock from the single2shear strength theo ry (M oh r 1900) to the tw in2 shear strength theo ry (Yu 1985) and to the un ified strength theo ry (Yu 1991) are summ arized. T he single2shear strength theo ry of M oh r2cou lom b is the low er ( inner) bound of all the convex lim it su rface; the tw in2shear strength theo ry is the upper (ou ter) bound of the convex lim it su rface. T he lim iting loci of variou s em p irical criteria in Πp lane are situated betw een the single2shear and tw in2shear strength theo rieṡ T he un ified strength theo ry is a system of strength theo ry. It gives a series of new failu re criteria estab lishes a relation sh ip am ong variou s failu re criteria and encom passes p reviou s yield criteria (T resca M ises) failu re criteria (M oh r2cou lom b Tw in2shear etc. ) and o ther sm oo th criteria o r em p irical criteria as special cases o r linear app rox im ation ṡ It can be m atched w ith experim en tal resu lts over a w ide range of stress states of m any k inds of m aterialṡ T he un ified st reng th theo ry ha s been genera lized and app lied in som e field ṡ It is ea sy to u se fo r analytical so lu tion s engineering app lication and im p lem en ting in to com puter codeṡ T he un ified strength theo ry m ay have good p ro spects in the 21st cen tu ry. Key words rock strength theo ry (failu re criterion) single2shear strength theo ry (M oh r2cou lom b failu re criterion) tw in2shear strength theo ry un ified strength theo ry

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