Number of independent slip systemsNumber of independent slip systemsInvariants of stress tensors and their relationship with yield criteriaExamples of limit analysisNotched bar in plane bendingC12 PART IIA and Part IIB C12 MATERIALS SCIENCE AND METALLURGY Course C12: Plasticity and Deformation Processing KMK/LT06 9 Lectures + 2 Examples Classes KMK Plasticity and plastic flow in crystals (11/2 lectures) Plastic flow and its microscopic and macroscopic descriptions. Yield in crystals: derivation of the stress tensor for glide on a general plane in a general direction. Definition of an independent slip system. Need for five independent slip systems to accommodate general strain. Examples in metals and ceramics. Continuum plasticity (2 lectures) Stress-strain curves of real materials. Definition of yield criterion. Concept of a yield surface in principal stress space. Yield criteria: Tresca, von Mises, Coulomb, pressure-modified von Mises. Physical interpretation of Tresca (maximum shear stress) and von Mises (strain energy density, octahedral shear stress) yield criteria. Experimental test of yield criteria (Taylor and Quinney). Yield criteria applicable to metals, polymers and geological materials. Plastic strain analyses (11/2 lectures) Levy-Mises equations. Example: pressurised thin-walled cylinder. Deformation in plane stress: yielding of thin sheet in biaxial and uniaxial tension. Plane stress deformation: Lüders bands. Plane strain deformation: derivation of stress tensor and separation into hydrostatic and deviatoric components. Equivalence of Tresca and von Mises yield criteria in plane strain. Slip line field theory (1 lecture) Physical interpretation of slip lines. Slip line fields for compression of slab, comparison with shear pattern in transparent polymers. Hencky relations. Examples: indentation by flat punch, yield in deeply notched bar. Estimation of forces for plastic deformation (3 lectures) Stress evaluation and work formulae. Application to rolling and wire-drawing. Upper and lower-bound theorems. Limit analyses for indentation, extrusion, machining and forging. Velocity-vector diagrams: hodographs. Classical theory of the strength of soldered and glued joints. Use of finite-element methods in analyses of metalforming operations.C12 − 2 − C12 Book List P.A. Benham, R.J. Crawford and C.G. Armstrong, Mechanics of Engineering Materials, 2nd edn, Prentice-Hall, 1996 AB181 C.R. Calladine, Plasticity for Engineers, Ellis Horwood, 1985 Kc38 G.E. Dieter, Mechanical Metallurgy, McGraw-Hill, 1988 Ka62 W.F. Hosford and R.M. Caddell, Metal Forming, 2nd. edn., Prentice Hall, 1993 Ga171b A. Kelly, G.W. Groves and P. Kidd, Crystallography and Crystal Defects, Addison-Wesley, 2000 NbA84 W. Johnson and P.B. Mellor, Engineering Plasticity, Van Nostrand Reinhold, 1983 Kc29 Other books in the Departmental library that you might find useful are: W.A. Backofen, Deformation Processing, Addison-Wesley, 1972 Ga96 G.W. Rowe, Principles of Industrial Metalworking Processes, Edward Arnold, 1977 Ga106 G.W. Rowe, Elements of Metalworking Theory, Edward Arnold, 1979 Ga124 G.W. Rowe, C.E.N. Sturgess, P. Hartley and I. Pillinger, Finite-element Plasticity and Metalforming Analysis, CUP, 1991 Kc42 In addition there are three teaching and learning packages relevant to this course in the Plasticity and Deformation Processing part of the Teaching and Learning Packages Library at http://www.doitpoms.ac.uk and these should help to reinforce ideas that you will meet in this course about stress, strain, yield, yield criteria, metal forming processes and energy estimates for deformation processes.C12 – 3 – C12 Strain tensor for glide on a general plane in a general direction Define unit vectors: n normal to slip plane ββββ in slip direction Slip moves point P to P' where PP' = r' - r = γ (r.n) ββββ and γ is the angle of shear due to glide n has components n1 n2 n3 ββββ has components β1 β2 β3 r has components x1 x2 x3 For small γ the deformation tensor eij is obtained by differentiating the displacements, e.g. () ()113322111111111111)('βγ=++∂∂γβ=β⋅γ∂∂=−∂∂=∂∂=nnxnxnxxxxxue nrrr Similarly for all eij: βββββββββγ=βγ=∂∂=333231232221131211 nnnnnnnnnenxueijijjiij This deformation tensor can be separated into a symmetrical strain tensor εij and an antisymmetric rotation tensor ωij, where ββ+ββ+ββ+βββ+ββ+ββ+ββγ=ε332332211331213223212212212131132121122111)()()()()()(nnnnnnnnnnnnnnnij and β−ββ−ββ−ββ−ββ−ββ−βγ=ω0)()()(0)()()(0 233221133121322321122121311321211221nnnnnnnnnnnnij Or.nrr'PP'nββββC12 − 4 − C12 Independent glide systems for different crystal structures Cubic ( mm3 ) Slip systems Number of independent slip systems Crystal structure < 0 1 1 > {111} 5 c.c.p. metals < 1 1 1 > {110} 5 b.c.c. metals < 0 1 1 > {110} 2 NaCl structure <001> {110} 3 CsCl structure Hexagonal (basal plane slip) Slip systems Number of independent slip systems Crystal structure < 0 2 1 1 > {0001} 2 h.c.p. metals For further examples of slip systems see Kelly, Groves and Kidd, pages 188-189.C12 – 5 – C12 Independent glide systems in c.c.p. metals: proof of five independent slip systems Slip systems which produce different pure strains are said to be independent. Therefore, we need only pay attention to the forms of the symmetrical strain tensor quoted on page 3 for the various possible combinations of slip plane and slip direction for c.c.p. metals. There are 12 slip systems of the form <0 1 1 > {111}. Considering each of these in turn, we can choose to look at slip directions within a given slip plane, and the strain tensors they produce. Example: slip along [ 011 ] on the (111) slip plane: If the amount of slip is of magnitude γ, where γ is the angle of shear due to glide, −−−γ=011120102 621ε since =31,31,31n and −= 0,21,21β . We can therefore determine the 12 strain tensors produced by the 12 slip systems for slip of magnitude γ, designating the slip systems A – L. Label Slip system Label Slip system A [ 0 1 1 ](111) G [1 1 0](1 1 1 ) B [ 1 1 0 ](111) H [110](1 1 1 ) C [ 1 0 1 ](111) I [1 0 1](1 1 1 ) D [ 0