Superplasticity is the ability of polycrystalline materials under certain conditions to exhibit extreme tensile elongation in a nearly homogeneous/isotropic manner. Historically, this phenomenon was discovered and systematically studied by metallurgists and physicists. They, along with practising engineers, used materials in the superplastic state for materials forming applications. Metallurgists concluded that they had the necessary information on superplasticity and so theoretical studies focussed mostly on understanding the physical and metallurgi­ cal properties of superplastic materials. Practical applications, in contrast, were led by empirical approaches, rules of thumb and creative design. It has become clear that mathematical models of superplastic deformation as well as analyses for metal working processes that exploit the superplastic state are not adequate. A systematic approach based on the methods of mechanics of solids is likely to prove useful in improving the situation. The present book aims at the following. 1. Outline briefly the techniques of mechanics of solids, particularly as it applies to strain rate sensitive materials. 2. Assess the present level of investigations on the mechanical behaviour of superplastics. 3. Formulate the main issues and challenges in mechanics ofsuperplasticity. 4. Analyse the mathematical models/constitutive equations for superplastic flow from the viewpoint of mechanics. 5. Review the models of superplastic metal working processes. 6. Indicate with examples new results that may be obtained using the methods of mechanics of solids.

1 Phenomenology of Superplastic Flow.- 1.1 Historical.- 1.2 Mechanical Behaviour of Superplastics.- 1.2.1 Mechanical Tests.- 1.2.2 Typical Experimental Results.- 1.2.3 Conditions for Superplastic Flow.- 1.3 Strain Rate Sensitivity of Superplastic Flow.- 1.3.1 Strain Rate Sensitivity Index, m.- 1.3.2 `Universal¿ Superplastic Curve.- 1.3.3 Stability of Uniaxial Superplastic Flow.- 1.4 Superplasticity from the Point of View of Mechanics.- 1.4.1 On the Definition of Superplasticity.- 1.4.2 On Experimental Studies Concerning Superplasticity.- 1.4.3 On the Presentation of Results Obtained.- 1.4.4 On Some Parameters of Superplastic Flow.- 1.4.4.1 Range of Optimal Flow.- 1.4.4.2 Mechanical Threshold.- 1.4.4.3 Activation Energies.- 1.4.4.4 Structure and Mechanical Response.- 1.4.5 On Stability of Superplastic Flow.- 2 Mechanics of Solids.- 2.1 The Subject.- 2.2. Basic Concepts.- 2.2.1 Concept of a Continuum.- 2.2.2 Stress, Strain and Strain Rate States.- 2.3 General Laws and Boundary Value Problems.- 2.4 Mathematical Models of Materials.- 2.4.1 Typical Models for Describing Mechanical Behaviour.- 2.4.2 Mechanical Models/Analogues.- 2.4.3 Theories of Plasticity.- 2.4.4 Theories of Creep.- 2.4.4.1 Phenomenology of Creep.- 2.4.4.2 Internal Variable Approach.- 2.5 Experiments in Mechanics.- 2.5.1 Mechanical Tests on Materials.- 2.5.2 Influence of Testing Machine.- 3 Constitutive Equations for Superplastics.- 3.1 Basic Requirements of Constitutive Equations.- 3.2 Phenomenological Constitutive Equations.- 3.2.1 Standard Power Law.- 3.2.2 Polynomial Models.- 3.2.3. Mechanical Modelling.- 3.2.3.1 Generalised Maxwell Body.- 3.2.3.2 Generalised Bingham Body.- 3.2.3.3 Mechanical Threshold: Analyses of Karim and Murty.- 3.2.3.4 Smirnov¿s Mechanical Analogue.- 3.2.3.5 Models of Murty¿Banerjee and Zehr¿Backofen.- 3.2.3.6 Combinations of Non-Linear Viscous Elements.- 3.2.4 Smirnov¿s Model.- 3.2.5 Anelasticity.- 3.2.6 Kinks on the Load Relaxation Curves.- 3.2.7 Mechanistic Model.- 3.2.8 Activation Energies.- 3.3 Physical Constitutive Equations.- 3.3.1 Classical Models.- 3.3.2 Modern Theories.- 3.3.2.1 Model of Ghosh.- 3.3.2.2 Model of Hamilton.- 3.3.2.3 The Model of Pschenichniuk¿Astanin¿Kaibyshev.- 3.3.2.4 The Model of Perevezentsev et al.- 3.4 Construction of Constitutive Equations.- 3.4.1 Common Scheme.- 3.4.2 Model of Padmanabhan and Schlipf.- 3.5. Constitutive Equations in Tensor Form.- 3.5.1 Non-Uniaxial Stress¿Strain States.- 3.5.2 Some Tensor Constitutive Equations.- 3.6 Material Constants from Technological Tests.- 3.6.1 Inverse Problems.- 3.6.2 Constant Pressure Forming of a Rectangular Membrane.- 3.6.3 Constant Pressure Forming of a Circular Membrane.- 3.6.4 Model of Padmanabhan and Schlipf.- 4 Boundary Value Problems in Theory of Superplastic Metalworking.- 4.1 General Formulation of the Boundary Value Problem for Metalworking Processes.- 4.1.1 Basic Concepts and Principal Equations.- 4.1.2 Initial and Boundary Conditions.- 4.1.3 Damage Accumulation.- 4.2 Model Boundary Value Problems in Mechanics of Superplasticity.- 4.2.1 Couette Flow of Superplastics.- 4.2.1.1 Newtonian Viscous Liquid.- 4.2.1.2 Shvedov¿Bingham Plastic.- 4.2.1.3 Non-Linear Viscous Material.- 4.2.2 Combined Loading of a Cylindrical Rod by Axial Force and Torque.- 4.2.3 Free Bulging of Spherical and Cylindrical Shells.- 4.2.3.1 Free Forming of a Sphere.- 4.2.3.2 Free Forming of an Infinite Cylindrical Shell.- 4.3 Numerical Solving of Boundary Value Problems in Superplasticity.- 4.3.1 Features of Boundary Value Problems in Mechanics of Superplasticity.- 4.3.2 Finite Element Modelling of Superplastic Metalworking Processes.- 4.3.3 Numerical Models of Superplastic Sheet Forming Processes.- 4.3.3.1 Principal Equations of Membrane Theory.- 4.3.3.2 Numerical Solutions of the Principal Equations of Membrane Theory.- 5 Mathematical Modelling of Superplastic Metalworking Processes.- 5.1 Modelling of Superplastic Bulk Forming Processes.- 5.1.1 General Comments.- 5.1.2 Compression of a Disc using Platens.- 5.1.3 Forging of a Disc by Rotating Dies.- 5.1.3.1 Formulation of the Simplified Boundary Value Problem.- 5.1.3.2 Solving the Simplified Boundary Value Problem.- 5.1.3.3 Analysis of the Solution Obtained.- 5.1.4 Extrusion.- 5.1.5 Die-less Drawing.- 5.1.6 Roll Forming Processes.- 5.1.7 Clutching.- 5.2 Modelling of Sheet Metal Processes.- 5.2.1 Simplifications in Modelling SPF and SPF/DB Processes.- 5.2.2 Main Challenges in Modelling SPF and SPF/DB Processes.- 5.2.3 SPF of Hemispherical Domes.- 5.2.3.1 Finite Strain Behaviour.- 5.2.3.2 Jovane¿s Model.- 5.2.3.3 Geometric /Kinematic Models.- 5.2.3.4 Model of Cornfield¿Johnson and its Modifications.- 5.2.3.5 Holt¿s Model and its Modifications.- 5.2.4 Free Forming of Spherical Vessels.- 5.2.4.1 Description of the Process.- 5.2.4.2 Mathematical Model.- 5.2.4.3 Wrinkling in Superplastic Forming.- 5.2.5 SPF of a Long Rectangular Membrane.- 5.2.5.1 Thickness Distribution.- 5.2.5.2 Pressure ¿ Time Cycle.- 5.2.5.3 Comparison with Experimental Results.- 5.2.6 Estimating Strain in SPF and SPF/DB Processes.- 5.3 Deformation Processing of Materials.- 5.3.1 General Notes.- 5.3.2 Torsion under Pressure and ECA Extrusion.- 5.3.3 Thermomechanical Conditions for Grain Refinement.- 5.3.4 On Some Principles of Structure Refinement.- 6 Problems and Perspectives.- 6.1. Influence of Strain History on Evolution of Structure.- 6.2. Constitutive Equations Including Structural Parameters.- 6.3. The Concept of Database `TMT¿Structure¿Properties¿.- 6.4. Challenges in Mechanics of Superplasticity.- 6.4.1. Experimental Superplasticity.- 6.4.2. Constitutive Equations.- Appendix A: Finite Strain Kinematics of Solids.- A.1 Basic Concepts.- A.2 Theory of Deformations.- A.2.1 Strain Tensors.- A.2.2 Geometrical Sense of Strain Tensor Components.- A.2.3 Method of Determining the Principal Components of a Strain Tensor.- A.2.4 Volumetric and Deviatoric Parts of Strain Tensors.- A.3 Strain Rate Tensor.- A.3.1 Covariant Components of Strain Tensor.- A.3.2 Distortion and Spin Tensors.- A.3.3 Strain Rate Tensor Invariants.- A.3.4 Volumetric and Deviatoric Parts of the Strain Rate Tensor.- A.3.5 On Some Scalar Characteristics of a Deformed State.- Appendix B: Kinematics of Some Simple Deformation Modes.- B.1 Tension/Compression of a Cylindrical Rod.- B.2 Simple Shear.- B.3 Pure Shear.- B.4 Bulging of a Sphere.- B.5 Finite Strain Kinematics under Combined Loading of a Cylindrical Rod by Axial Force and Torque.- Appendix C: On Dimensional Analysis.- C.1 Basic Concepts.- C.2 Viscous Flow.- C.3 Non-Newtonian Flow.- C.4 Superplastic Flow.- C.5 Dimensionless Parameters for the Boundary Value Problem of Superplasticity.- C.6 Physical Modelling of Superplastics.- Appendix D: Group Properties of Thermoviscoplasticity.- D.1 About Single-Parameter Groups of Transforms.- D.2 Applications of Group Methods in Superplasticity.- References.