In this chapter, the framework to analyze a multilayer stack of blanket films is presented. Emphasis is placed on stacks with piecewise-linear distributions of misfit, or ‘eigenstrain’, strain variations within each layer allowing for the possibility of discontinuities from layer to layer. In addition to being applicable to thermal problems with steady-state thermal distributions through the multilayer, the formulation encompasses layers with residual processing strains which vary from layer to layer and possibly within layers. The principle focus of this chapter is on computing the steady-state energy release rate for a semi-infinite crack. The results of the analysis are algebraic but nevertheless will usually require some computation. As the framework is applicable to any number of layers, it provides the basis to derive the results presented in Chapter 4 for bilayers.

In Section 5.1, basic results for the stresses and strains in a multilayer subject to overall stretching and bending, together with internal misfit strains, are derived. Section 5.2 makes use of the basic results to derive general results for delamination energy release rates in the absence of any misfit strains. Section 5.3 gives an alternative derivation which can be used to predict the energy release rate under general conditions, including misfit strains, and for the special case where the stresses prior to cracking arise under equi-biaxial conditions but the delamination occurs under plane strain conditions. Section 5.4 discusses the computation of the mode mix for the general multilayer. The derivation in Section 5.3 also leads to a bilayer approximation in Section 5.5, which can be used to estimate both the energy release rate and mode mix. Illustrative examples of multilayer analysis are presented in Section 5.6.

Software that implements the specific framework described in this chapter is described in Chapter 14, which provides additional examples beyond those in this chapter. Computation of the mode mix for general multilayer problems requires finite element analysis, which is described in Chapter 16. However, several illustrative examples of mode mix in multilayers are provided here in this chapter, providing some insight regarding the role of layer properties.

The previous chapter dealt with steady-state, time-independent temperature distributions which induce stress in a body due, for example, to interruption of heat flow across a preexisting crack or to thermal expansion and/or conductivity mismatches across material interfaces. This chapter deals with problems where transient (time-dependent) temperature distributions induce stresses in a body and, in turn, how these stresses can cause cracking or interface delamination.

A simply connected solid with uniform properties which is unconstrained at its boundaries experiences no stress under temperature distributions imposed on its surface once the temperature distribution ceases to change and steady-state conditions are attained. However, during the transient period while the temperature is changing, stresses will generally be induced, and the solid may be susceptible to cracking.

The first set of examples considered in this chapter deals with an unconstrained uniform semi-infinite solid having an initial uniform temperature which is suddenly subject to a temperature change imposed on its entire free surface. Cold shock, with cooling imposed, induces transient tensile stresses acting parallel to the surface with the potential to causemode I cracking perpendicular to the surface, as well as mixed-mode delamination cracking on a weak interface parallel to the surface. When a sudden increase in surface temperature is imposed, the resulting hot shock produces compressive stresses parallel to the surface such that subsurface mixed-mode delamination is again a possibility if weak interfaces exist parallel to the surface.

The second example again considers a uniform semi-infinite solid at an initial uniform temperature, but in this case localized hot shock is considered with a higher temperature suddenly imposed over a local region on its surface. The stress field is more complicated in this case with both tensile and compressive components such that various modes of cracking must be considered.

A ‘generic’ system consisting of bonded layers of different materials and illustrations of various failure modes are shown in Figure 1.1. Arguably, the two dominant technological applications with this type of geometry are microelectronic devices and protective coatings in extreme environments (e.g., thermal barrier coatings). These applications involve layers with very disparate properties and are subjected to rather aggressive external stimuli. Cracking happens either between the layers (interface delamination or debonding) or within a layer (tunneling or channeling cracks). Delamination can occur regardless of whether the stresses are tensile or compressive, while buckling-driven delamination occurs only in layers experiencing compressive stress, and channeling or tunneling cracks require tensile stress in the layers.

Failure can be driven by a variety of factors, but it is probably fair to state that the integrity of the vast majority of multilayered devices is controlled by the layers’ tendency to expand at different rates in response to thermal, mechanical or chemical stimuli. In essence, when left by their lonesome, the layers expand differently in response to temperature fields, mechanical loading and so forth. However, in the multilayer component they are not alone: they are constrained to experience conformal deformation where they are bonded. This constraint generates stresses (both tensile and compressive) and stored elastic energy that drive system failure. Crudely speaking, the layers would be happiest and in their lowest energy state as separate pieces, and they seek to return there – even if it means splitting themselves into pieces and leaving part of themselves stuck to another layer.

As with the development of any predictive framework, the initial challenge is to reduce the complexity of the actual system to produce a model that captures the salient features of the system, while ignoring those details that have little effect on the behavior of interest. This can be a critical step in the development of multilayered systems, because their geometrical and multimaterial complexity can make full numerical representations extremely costly. Make no mistake: this is an art.1 Further, it can be highly problem specific. Nonetheless, there are some central idealizations that are widely applied to thin film systems that have served to generate considerable general insight regarding failure.

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.

As described in the previous chapter, the computation of energy release rates and stress intensity factors using the finite element method requires specification of various numerical parameters. These parameters relate to both the FE model itself (basically, the mesh) and the calculations performed during postprocessing of the FE results (basically, the calculation of the stiffness derivitive). In this chapter, the powerful automation of the LS-FEA framework is exploited to conduct extensive parametric studies, which illustrate the impact of these parameters on computational accuracy. The outcomes of these studies are used as the basis to establish recommended practices and default values for the associated numerical parameters, such as those in Table 16.1.

Convergence of Crack Tip Parameters

Numerical convergence in the present context is conceptually very simple; when the values of interest no longer change upon increases in mesh density (i.e., decreases in element size, often referred to as refinement), the model is said to have converged. In practice and especially for fracture problems, convergence can be more nuanced, as the behavior of different outputs will converge at different levels of mesh density. For example, when considering the potential energy of the system, and in many cases the energy release rate, convergence is achieved for much coarser meshes than when considering the crack tip stress intensity factors.

Such differences can create headaches if the spatial distribution of elements changes with refinement. For example, this would occur if the number of elements inside the focused fan region were increased, without changing the number of elements outside the focused region. Strictly speaking, mesh refinement to test convergence should involve a uniform increase in mesh density, such as would be achieved by splitting every element in themodel into four smaller elements. This can be prohibitively expensive, particularly for components that require dense meshes in multiple locations: for example, bending of a multilayer with one particularly thin layer.

In this chapter, we consider mesh refinements associated with increasing the number of elements around the crack tip in the focused region and in the fill regions. Practically speaking, the crack tip parameters become independent of the mesh upon increase of all or even some of these parameters.