The design and optimization of sheet metal forming operations is aided by tools and techniques that have been developed and refined over several decades. This chapter covers many of these methods and practices and explains where and how they are used. It begins by showing how the stress state at any point in a material can be expressed in different ways for different purposes. It then compares and contrasts some of the more widely used yield criteria and demonstrates the use of flow rules. It also explains how to calculate power, energy, and effective strain and strain rate and how hardening laws are used to predict strain-hardening behaviors.

This chapter explains why it is sometimes necessary to separate inelastic from elastic strains and how to do it using one of two methods. It first discusses the direct calculation of strain-range components from experimental data associated with large strains. It then explains how the method can be extended to the treatment of very low inelastic strains by adjusting tensile and compressive hold periods and continuous cycling frequencies. The chapter then begins the presentation of the second approach, called the total strain-range method, so named because it combines elastic and inelastic strain into a total strain range. The discussion covers important features, procedures, and correlations as well as the use of models and the steps involved in predicting thermomechanical fatigue (TMF) life. It also includes information on isothermal fatigue, bithermal creep-fatigue testing, and the predictability of the method for TMF cycling.

This chapter discusses the concept of mean stress and explains how it is used in fatigue analysis and design. It begins by examining the stress-strain response of test samples subjected to cyclic forces and strains, noting important features and what they reveal about materials and their fatigue behaviors. It then discusses the challenge of developing hysteresis loops for complex loading patterns and accounting for effects such as ratcheting and stress relaxation. The sections that follow provide a summary of the various ways mean stress is described in the literature and the methods used to calculate or predict its effect on the fatigue life of machine components. The discussion also sheds light on why tensile mean stress is detrimental to both fatigue life and ductility, while compressive mean stress is highly beneficial.