Chapters 4 and 5 cover delamination in multilayers that have interface crack lengths that are extremely large in comparison to other dimensions, in which case the energy release rate becomes independent of the crack size (i.e., the steady-state limit). However, when the interface crack length is comparable to other dimensions in the problem, the energy release rate generally depends on the interface crack length, and as such, numerical solutions are needed. This chapter describes the relationship between crack length and crack tip parameters, and in so doing provides guidelines that identify crack lengths for which the steady-state solution is accurate.

The focus of coverage is on plane strain (two-dimensional) geometries, with a limited discussion of an interface crack at the corner of a thin film, which is inherently three-dimensional. It will be demonstrated that the two-dimensional results provide the necessary insight for most cases of interest. The two-dimensional results in this chapter were generated using the software described in Chapter 16 and are essentially a reiteration of the results in original papers by Zhuk, Evans, Hutchinson & Whitesides (1998) and Yu, He & Hutchinson (2001). Related coverage of three-dimensional problems is also provided in Chapter 11, which addresses interface cracks between a semi-infinite substrate and patterned lines, that is, thin film strips bonded to very thick substrates.

Interface Edge-Cracks: The Transition to Steady State

Consider the interface cracks located at the edge of a film bonded to a substrate, as illustrated in Figure 9.1. One of the remarkable aspects of thin film mechanics is how quickly a short edge-crack approaches the steady-state limit for a semi-infinite crack. The rapidity of this transition depends on whether the film edge is aligned with the edge of the substrate, as in Figure 9.1(a), or whether the film edge is interior to the edge of the substrate, as in Figure 9.1(b). In this section, we focus on two-dimensional film/substrate systems in plane strain with an infinitely long crack front in the out-ofplane direction. A limited discussion of semicircular crack fronts is provided in the next section, while three-dimensional aspects of debonding of finite width features are considered in Chapter 11.

A significant fraction of the insight and understanding needed to understand mixedmode fracture can be obtained by considering a simple elastic bilayer consisting of two thin layers bonded together, with a semi-infinite crack lying on the interface. The concepts and results generated for the bilayer in this chapter are extended to multilayers consisting of many layers in Chapter 5. Additional aspects of interface fracture in bilayers are discussed throughout subsequent chapters, but notably Chapter 7 (kinking out of the interface), Chapter 9 (finite-sized edge flaws) and Chapter 12 (temperature gradients).

A Basic Solution

A basic solution for a crack on the interface of a bilayer presented in this section illustrates a number of aspects of the mechanics of interface cracks. The solution for the stress intensity factors and energy release to the problem depicted in Figure 4.1 will be presented and applied in this section. The solution was obtained within the context of linear elasticity for an infinitely long bilayer with a semi-infinite crack emerging from the left by Suo & Hutchinson (1990). In this section, we focus on the key results of the bilayer system and their implications: though not shown here, the derivation of the results in this section is identical to that presented in Chapter 5 for a multilayer with many layers.

For the equilibrated load system shown, the stress intensity factors and energy release rate are independent of the position of the crack tip–what is often referred to as steadystate cracking. Within the context of plane stress or plane strain, the energy release rate can be computed exactly using simple energy arguments accounting for the energy difference of the bilayer system far ahead and behind the crack tip. (Here, we focus on the solution for the bilayer and its implications, leaving the underlying derivation for the more general multilayer treatment in Chapter 5.) Determination of the mode mix requires a sophisticated elasticity analysis, which was carried out in the reference cited above and tabulated.

Software accompanying this text generates the full solution for any material combination, using these tabulated values. Alternatively, one can compute the needed results via the finite element code provided with this book, as described in Chapter 16.

For some multilayer systems that experience compression, buckling of the film in an initially debonded region may release energy that becomes available to drive the interface crack and increase the size of the debonded region. This phenomenon, called buckling delamination, occurs in structural composites where a compressed surface layer has debonded from its adjacent layers and buckles when the applied compression is sufficiently large. This surface layer may then delaminate if the interface toughness is not large enough to keep the interface crack from propagating. Buckling delamination is one of the most widely observed failure mechanisms in thin films and coatings on substrates when the stress is compressive. Buckling delamination involves the simultaneous interaction of buckling and interface cracking.

Two of the most commonly observed morphologies of buckling delaminations are seen in Figure 10.1. Both straight-sided and telephone cord delaminations have occurred in Figure 10.1(a) for a multilayered thin film deposited on a glass substrate where the film is subject to equi-biaxial compression. Straight-sided delaminations are rare under equi-biaxial compression. However, they are commonly observed when there is a dominant direction of compression in the film, in which case they propagate perpendicular to the direction of maximum compression. The telephone cord morphology is the mostly commonly observed morphology when the film is under equi-biaxial compression.

The buckle delamination in Figure 10.1(b) has formed under artificially constrained conditions created by patterning on the interface in a long narrowing wedge-shaped region of low adhesion before the film is deposited. The delamination, which takes place only within the low adhesion region, was initiated at the wide end and propagated towards the narrow edge, transitioning from the telephone cord morphology to the straight-sided morphology. Even when the film-substrate interface is not patterned, the delamination arrests along its sides and propagates at the curved front. Figure 10.2 shows a sequence of three closely spaced stages of the front propagation; the upper figures are a numerical solution discussed later, and the lower figures are the experimental observation.

This chapter develops the underlying mechanics of the buckling delamination with particular emphasis on the role of mixed-mode interface toughness, which is essential to understanding the existence, propagation and morphology of the various delamination modes.

A ‘fracture mechanics’ approach to predicting failure essentially boils down to a very simple concept: one calculates a parameter that characterizes the distribution of stresses and strains near the crack tip, and assumes that the crack will extend when this parameter reaches a critical value that depends only on the material or interface properties at hand. As one might expect, the calculated parameter will depend on the geometry of the component, the geometry of the crack, any external loads (be they mechanical or imposed temperature fields) and the constitutive law of the material (i.e., its modulus, Poisson's ratio etc.)

The power of the approach is that while the calculated parameter depends on the problem at hand, the critical value is a material or interface property that does not. The critical parameter is the toughness of the material or interface – it is measured, not predicted. Thus, one can measure the critical value of the fracture parameter in an experiment based on a convenient geometry, but assess the crack stability in a totally different geometry, provided one can calculate the intensity parameter for that geometry. The key to the enormous success of fracture mechanics is this tight connection between the experimental measurement of the critical value of the intensity parameter and the evaluation of the intensity parameter for other structural geometries and loadings of interest.

Fracture can occur in three different modes, which refer to the relative motion of the crack faces very close to the crack tip as the crack advances. These are shown in Figure 3.1; mode I is often referred to as the opening mode, mode II is typically called the shear or sliding mode, and mode III is also a shearing mode often called the tearing mode. In many problems of interest mode III (the tearing mode) does not come into play, but there are exceptions that will be discussed in later chapters. For homogenous materials with isotropic fracture behavior, the nature of the separation process at the crack tip tends to select a mode I trajectory when the crack advances, and this explains the heavy emphasis placed on mode I behavior and mode I toughness in many structural applications. However, the films, coatings and layered materials of primary interest in this book have interfaces which can greatly alter fracture behavior.

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.