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Chapter 6 Geometry of Deformation and Work-Hardening

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Common metalworking methods. (a) Rolling. (b) Forging (open and closed die). (c) Extrusion (direct and indirect). (d) Wire drawing. (e) Stamping. Common Metal Working Methods

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Stress–strain curves (schematic) for an elastic, ideally plastic; a work-hardening; and a work-softening material. Work-Hardening of a Material

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Engineering-stress– engineering-strain curves for nickel. (a) Nickel subjected to 0, 20, 40, 60, 80, and 90% cold-rolling reduction. (b) Nickel cold rolled to 80%, followed by annealing at different temperatures. (From D. Jaramillo, V. S. Kuriyama, and M. A. Meyers, Acta Met. 34 (1986) 313.) Engineering Stress-Strain Curves for Nickel

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Stress–strain curves for annealed polycrystalline TiC deformed in compression at the temperatures indicated (ε = 1.7 × 10 −4 s −1 ). (Adapted from G. Das, K. S. Mazdiyasni, and H. A. Lipsitt, J. Am. Cer. Soc., 65 (Feb. 1982) 104.) Compression Tests on TiC at Different Temperatures

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Shear stress τ vs. shear strain γ for prism plane slip in Al 2 O 3 at various temperatures; έ = 3.5 × 10 −4 s −1 for the solid curves, έ = 1.4 × 10 −4 s −1 for the dashed curves. (Courtesy of T. E. Mitchell.) Shear Stress-Shear Strain Response of Al 2 O 3

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(a) Representation of crystallographic directions and poles (normals to planes) for cubic structure. (b) Standard [100] stereographic projection. (Reprinted with permission from C. S. Barrett and T. B. Massalski, The Structure of Metals, 3d ed. (New York: McGraw-Hill, 1966), p. 39.) Stereographic Projections

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Standard [001] stereographic projection divided into 24 triangles. Standard Stereographic Projection

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Relationship between loading axis and slip plane and direction. Slip Plane and Slip Direction-Schmid Law

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Comparison of Schmid’s law prediction with experimental results for zinc. (Adapted with permission from D. C. Jillson, Trans. AIME, 188 (1950) 1120.) Effect of orientation on the inverse of Schmid factor (1/M) for FCC metals. (Adapted with permission from G. Y. Chin, “Inhomogeneities of Plastic Deformation,” in The Role of Preferred Orientation in Plastic Deformation (Metals Park, OH: ASM, 1973), pp. 83, 85.) Schmid’s Law and Schmid Factor

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Stereographic projection showing the rotation of slip plane during deformation. Direction P 1, inside stereographic triangle, moves toward P 2 on boundary [100]–[111]. Then, P 2 moves toward [211]. Plastic Deformation- Rotation of Slip Plane

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Shear-stress vs. shear-strain curves for Nb (BCC) monocrystals at different crystallographic orientations; arrows indicate calculated strain at which conjugate slip is initiated. (From T. E. Mitchell, Prog. App. Matls. Res. 6 (1964) 117.) Shear-Stress vs. Shear-Strain Curve for Nb (BCC)

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Generic shear-stress–shear-strain curves for FCC single crystals for two different temperatures. Model of cross-slip. Cross-Slip

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Generic shear-stress–shear-strain curves for FCC single crystals for two different temperatures. Shear Stress-Shear Strain Curves for FCC Single Crystals

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Cross-Slip

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Average dislocation density ρ as a function of the resolved shear stress τ for copper. (Adapted with permission from H. Wiedersich, J. Metals, 16 (1964) p. 425, 427.) Work-Hardening in Polycrystalline Cu

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Relationship between flow shear stress and dislocation density for monocrystalline sapphire (A1 2 O 3 ) deformed at different temperatures. (Adapted from B. J. Pletka, A. H. Heuer, and T. E. Mitchell, Acta Met., 25 (1977) 25.) Work-Hardening in Polycrystalline Alumina

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Taylor model of interaction among dislocations in a crystal. Taylor Model of Work Hardening

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Development of substructure of Nickel-200 as a function of plastic deformation by cold rolling. (a) 20% reduction. (b) 40% reduction. (c) 80% reduction. Dislocation Cells

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Schematic representation of dislocation cells of size L, with activation of dislocation sources from the cell walls and bowing out of loops into the cell interior. (Courtesy of D. Kuhlmann–Wilsdorf.) Kuhlmann-Wilsdorf’s Work Hardening Theory

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Typical load deformation curve for concrete under uniaxial compression; the specimen was unloaded and reloaded at different stages of deformation. (From G. A. Hegemier and H. E. Reed, Mech. Mater., 4 (1985) 215; data originally from A. Anvar.) Load-Deformation Curve fro Concrete

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(a) Compressive true-stress–true-strain curves for titanium at different strain rates; notice the onset of softening at the arrows. (Adapted from M. A. Meyers, G. Subhash, B. K. Kad, and L. Prasad, Mech. Mater., 17 (1994) 175.) (b) Schematic linear shear-stress–shear-strain curves for titanium at different temperatures, with superimposed adiabatic curve constructed from isothermal curves by incrementally converting deformation work into heat (and a consequent rise in temperature.) (Adapted from M. A. Meyers and H. - R. Pak, Acta Met., 34 (1986) 2493.) Work Softening

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Shear bands in titanium. (a) Optical micrograph, showing band. (b) Transmission electron micrograph, showing microcrystalline structure, with grain size approximately equal to 0.2 μm. The original grain size of the specimen was 50 μm. Shear Bands in Titanium

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Perspective view of microstructure of Nickel-200 cold rolled to a reduction in thickness of 60%. Rolling Texture

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Theoretical bounds on the Young’s modulus E of steel. Orientation dependence of yield strength and strain to fracture of a rolled copper sheet. Texture Strengthening

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Common Wire and Sheet Textures

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[111] pole figure of a rolled-brass sheet. Rolled-Brass Sheet

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