This book relies heavily on concepts from strength of materials, continuum mechanics and the finite element method. Excellent books on these topics are abundant. The book on solid mechanics by Bower (2010) is a particularly complete reference that meshes well with much of the coverage here. This chapter is merely meant to provide a convenient reference for concepts used frequently in the rest of the book.

The majority of analytical solutions relevant to thin films and multilayers correspond to two-dimensional (planar) idealizations, which in one way or another represent a slice through a specific (x, y) plane in Figure 1.1. Even more narrowly, the mechanics review presented here is focused on results used to analyze blanket thin films, in which the width of the layers in these slices (i.e., the dimension in the x-direction) is much greater than their thickness. In this scenario, the films behave as plates whose deformation is uniform in the z-direction. Often the term ‘beam’ is used with the understanding that the behavior in the z-direction may not correspond to plane stress.

In this chapter, isotropic linear elastic constitutive descriptions are reviewed first, which are used exclusively throughout the book. The reader is referred elsewhere for generalizations to orthotropic and/or nonlinear constitutive relationships, for example, Bower (2010). Then, the mechanics of beams and plates are reviewed; the majority of problems addressed in this book involve small deformations and the corresponding linear strain-displacement relationships. As there are a few important problems in coatings that require nonlinear strain-displacement relationships (e.g., buckling of coatings subject to compressive stresses), a brief introduction to moderate rotation beam/plate theory is provided. The extension of these results to multilayers, that is, the analysis of individual layers bonded together, is left for future chapters. Finally, this chapter concludes with a section on unidirectional heat transfer, which is invoked in later chapters to determine temperature distributions through multilayers, which generate stresses that drive failure.

Review of Linear Isotropic Elasticity

The foundation of much of the subject matter covered in this book rests on linear elasticity theory. Many solutions are two-dimensional (2D) idealizations of three-dimensional (3D) problems, and extensive use of plate and beam theory is made for modeling purposes.