The Mechanical and Thermodynamical Theory of Plasticity by Mehrdad Negahban
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The Mechanical and Thermodynamical Theory of Plasticity by Mehrdad Negahban

The author would like to acknowledge the help of many people who provided insight, sometimes unknowingly, and people who helped in the preparation of this book. These include colleagues and students. Many of the students taking my classes pointed out corrections and provided suggestions, which was very much appreciated. Particular thanks go to Ashwani Kumar Goel, Lili Zhang, Kyle Strabala, Benjamin Polly, and Saeed Eghtedar Doust, who each read part or all of the text and provided corrections and comments, and Yenan Wang for doing the steel and aluminum tests.

I owe a particular debt of gratitude to Ashwani Kumar Goel, who did most of the simulations, and Lili Zhang, who carefully read and checked all the equations and representations; they were true partners in this endeavor. he one-dimensional theory of straight bars. This approach will allow us to concentrate on the ideas central to continuum mechanics and plasticity without needing to deal with the many complications that appear when dealing with multidimensional theories.

With this in mind, the one-dimensional theory presented in this chapter is developed on the same theoretical foundations that the multidimensional theory is based on. As such, we will avoid introducing simplifications or developments that do not easily generalize to multiple dimensions. Once we have become familiar with the basic concepts central to the development, then in the following chapters, we will study the multidimensional theory. It is my hope that this method of presentation will aid the reader to better understand plasticity.

By following the presentation in this chapter, the reader also should develop a strong understanding of how plasticity fits into continuum mechanics and continuum thermodynamics, which then should help the reader in understanding the structure of the developments in the later chapters. Even though plasticity normally refers to “rate-independent and temperatureindependent plasticity,” we will not limit ourselves to this.

After we introduce the basic concepts for a bar, we will look at the mechanical theory of rate-independent plasticity, followed by temperature-dependent plasticity, and then rate-dependent plasticity. To complete the presentation, we will look at the balance laws and jump conditions and consider specific problems that use these laws in their solution process. Before we proceed to look at the theory of bars, let us take a look at plasticity.

Plasticity refers to a specific type of material response normally characterized, for example, by the cyclic loading and unloading response of the aluminum shown in Figure 1.1. The schematic loading diagram for elastic plastic response is shown in Figure 1.2. The figure shows the loadextension plot for a uniform bar undergoing a homogeneous deformation.

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