International Journal of Rock Mechanics & Mining Sciences 39 (2002) 975–989 A Unified Strength criterion for rock material Mao-Hong Yu a, *, Yue-Wen Zan a , Jian Zhao b,c , Mitsutoshi Yoshimine d a Department of Civil Engineering, Xi’an Jiaotong University, 710049 Xi’an, China b School of Civil and Environmental Engineering, Nanyang Technological University, Singapore c College of Architecture and Civil Engineering, China University of Mining and Technology, Xuzhou, China d Department of Civil Engineering, Tokyo Metropolitan University, Japan Accepted 6 September 2002 Abstract Anon-linearUnifiedStrengthcriterionforrockmaterialispresented.ItisthedevelopmentoftheUnifiedStrengthTheory(in:M. Jono,T.Inoue(Eds.),MechanicalBehaviourofMaterials-VI(ICM-6),Pergamon,Oxford,1991,pp.841–846)andthemodification of the Hoek–Brown strength criterion (Underground Excavations in Rock, The Institution of Mining and Metallurgy, London, 1980). The effect of intermediate principal stress on rock strength is considered in the non-linear Unified Strength criterion. The Hoek–Brown criterion is a single-shear strength criterion that forms the lower bound, and the non-linear twin-shear strength criterion forms the upper bound in the deviatoric plane. All the failure criteria ranging from the Hoek–Brown criterion (lower bound) and the non-linear twin-shear criterion (upper bound) and a series of criteria ranging between these two bounds may be introduced by the non-linear Unified Strength criterion. The theory can also be generalized to rock mass strength. r 2002 Published by Elsevier Science Ltd. Keywords: Strength criterion; Rock material; Single-shear strength criterion (Mohr–Coulomb strength criterion); Twin-shear strength criterion; Unified Strength Theory; Hoek–brown criterion; Non-linear behaviour 1. Introduction Rockisageotechnicalmaterialwithhighdifferenceof tensile and compressive strengths. Its properties vary with rock type. The rock in underground engineering is often acted under complex multiaxial stress state, and sometimes yield and failure in extreme circumstances. The main objectives of the study of strength theory (yield and failure criteria) are to understand the materials’ intrinsic response to loads, to identify the failure conditions in different stress states, and ulti- mately,toformulatenecessarydesigncriteriaforproper development of these construction materials. Numerousstrengthcriteria(failurecriteria)havebeen developedinthelastCentury[1–61],thougheachoneis generally suitable only for a certain type of material. Varioustestsofrockunderthecomplexstressstatehave been conducted, and most of these tests are axisym- metric triaxial, plane strain and plane stress tests, with some truly triaxial tests, which revealed multiaxial mechanical behaviour. The proposed criteria in the 20th Century are too many, and it is difficult to classify them. One method we used for representing these criteria is to use the principal shear stresses t 13 ; t 12 ; t 23 ; and the normal stress s 13 ; s 12 ; s 23 acting on the same planes, respectively. Thestrengthcriteriamaybeclassifiedintothreemain series (Fig. 1): the single-shear stress series of strength criteria (Tresca–Mohr–Coulomb theory, 1864–1900), octahedral-shear stress series of strength criteria (or the three-shear stress series of strength criteria, Huber- von Mises-Schleicher and Drucker–Prager criteria, 1904–1913–1928–1952), and the twin-shear stress series of strength criteria (Yu, 1961–1985) [60,61]. The limit locionthedeviatoricplaneofthesethreeseriesoffailure criteria are as shown in Fig. 1. They are the lower bound,upperboundandanintermediateseriesbetween lower bound and upper bound. The isotropy was considered in these studies. A Unified Strength Theory that covers all the regions fromthelowerboundtoupperboundwasproposedby Yu in 1991–1994 [62–64]. The Unified Strength Theory encompasses the Mohr–Coulomb strength criterion *Correspondingauthor.SchoolofCivilEngineeringandMechanics, Xi’an Jiaotong University, 710049 Xi’an, China. Fax: +86-29-323- 7910. E-mail address: mhyu@mail.xjtu.edu.cn (M.-H. Yu). 1365-1609/02/$-see front matter r 2002 Published by Elsevier Science Ltd. PII:S1365-1609(02)00097-7