1. Some Mathematical Tools. - 1. 1 Measure and Integration. - 1. 2 Function Spaces. - 1. 3 Nonlinear Equations. - 1. 4 Ordinary Differential Equations. - 2. Hysteresis Operators. - 2. 1 Basic Examples. - 2. 2 General Hysteresis Operators. - 2. 3 The Play Operator. - 2. 4 Hysteresis Operators of Preisach Type. - 2. 5 Hysteresis Potentials and Energy Dissipation. - 2. 6 Hysteresis Counting and Damage. - 2. 7 Characterization of Preisach Type Operators. - 2. 8 Hysteresis Loops in the Prandtl Model. - 2. 9 Hysteresis Loops in the Preisach Model. - 2. 10 Composition of Preisach Type Operators. - 2. 11 Inverse and Implicit Hysteresis Operators. - 2. 12 Hysteresis Count and Damage Part II. - 3. Hysteresis and Differential Equations. - 3. 1 Hysteresis in Ordinary Differential Equations. - 3. 2 Auxiliary Imbedding Results. - 3. 3 The Heat Equation with Hysteresis. - 3. 4 A Convexity Inequality. - 3. 5 The Wave Equation with Hysteresis. - 4. Phase Transitions and Hysteresis. - 4. 1 Thermodynamic Notions and Relations. - 4. 2 Phase Transitions and Order Parameters. - 4. 3 Landau and Devonshire Free Energies. - 4. 4 Ginzburg Theory and Phase Field Models. - 5. Hysteresis Effects in Shape Memory Alloys. - 5. 1 Phenomenology and Falk's Model. - 5. 2 Well-Posedness for Falk's Model. - 5. 3 Numerical Approximation. - 5. 4 Complementary Remarks. - 6. Phase Field Models With Non-Conserving Kinetics. - 6. 1 Auxiliary Results from Linear Elliptic and Parabolic Theory. - 6. 2 Well-Posedness of the Caginalp Model. - 6. 3 Well-Posedness of the Penrose-Fife Model. - 6. 4 Complementary Remarks. - 7. Phase Field Models With Conserved Order Parameters. - 7. 1 Well-Posedness of the Caginalp Model. - 7. 2 Well-Posedness of the Penrose-Fife Model. - 8. Phase Transitions in Eutectoid Carbon Steels. - 8. 1 Phenomenology of the Phase Transitions. - 8. 2 The MathematicalModel. - 8. 3 Well-Posedness of the Model. - 8. 4 The Jominy Test: A Numerical Study. Language: English