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IT is generally believed that deformation twinning, as one of the two principal modes of plastic deformation in crystalline solids, becomes prevalent only at high strain rates a n d / o r low temperatures? In hcp metals and alloys, however, deformation twinning plays an important role in maintaining a generalized plastic flow of polycrystals at all temperatures. The role of deformation twinning in fracture strength and ductility in hexagonal metals has been the subject of a n u m b e r of investigations.2 ~0Experimentally, it is observed that deformation twinning can effectively strengthen a material under some circumstances and weaken it under others. This complexity results mainly from the intricate interrelationships between slip, twinning, and fracture processes of hcp metals. The purpose of the present paper is to review the role of deformation twinning in fracture of hcp metals from a theoretical point of view. Questions of general interest are, " W h y is beryllium inherently so brittle?" and "What makes titanium and its alloys so ductile?" The immediate problem of our concern is to what extent twinning influences the answers to these questions. The approach adopted for the present review study is to develop a systematic analysis of the role twinning plays in deformation and fracture processes based on the intrinsic physical and mechanical properties, such as the axial ratio, the cohesive energy, the surface energy, the twin boundary energy, the elastic constants, and so forth. Theoretical strength and ductility are discussed in Section 2. Slip and twin systems, as the independent modes necessary for a compatible polycrystalline deformation, are reviewed in Section 3. In Section 4, the formation of twins and cleavage cracks is treated by a dislocation model, and the competitive role of twinning vs crack extension is analyzed. Finally, in Section 5, the current results are summarized in view of the varying degrees of ductility reported for hexagonal metals, and M. H. YOO is Senior Research Staff Member, Metals and Ceramics Division, Oak RxdgeNational Laboratory, Oak Ridge, TN 38730. This paper is based on a presentation made at a symposium on "The Role of Twinning in Fracture of Metals and Alloys" held at the annual meeting of the AIME, St. Louis, Missouri, October 1519, 1978, under the sponsorship of the Mechanical Metallurgy Committee of The Metallurgical Society of AIME. METALLURGICAL TRANSACTIONSA

future studies necessary for a better understanding of the subject matter are recommended. 2. S T R E N G T H A N D D U C T I L I T Y 2.1 Theoretical Strength Some pertinent physical properties are listed in Table I for eighteen elemental hexagonal metals. They are listed in order of the simple metals, the three transition metal series, and the rare-earth elements. The atomic numbers are shown in parentheses. The axial ratio, c / a , the lattice constant, a, and the density, p, are the values at room temperature. The melting point, T m, is given together with the phase transition temperature, ( T 0, in parenthe