Chapter 9

Modeling Material Failure

9.6 Linear Elastic Fracture mechanics of interfaces

Many engineering applications require one material to be bonded to another.  Examples include adhesive joints; protective coatings; composite materials; and thin films used in the manufacture of microelectronic circuits.  In all these applications, techniques are required to predict the strength of the bond.

To this end, a great deal of work has been done over the past 20 years to extend linear elastic fracture mechanics to predict the behavior of cracks on, or near, the interface between two dissimilar brittle materials.

9.6.1 Crack Tip Fields for a crack on an interface

The foundation for linear elastic interfacial fracture mechanics is based on an asymptotic analysis of the stress and strain fields near the tip of a crack.

The problem of interest is illustrated in the figure. A semi-infinite crack with a straight front that coincides with the ${x}_{3}$ axis lies on the interface between two linear elastic solids. The material above the crack has shear modulus and Poisson’s ratio ${\mu }_{1},{\nu }_{1}$; the material below the crack has shear modulus and Poisson’s ratio ${\mu }_{2},{\nu }_{2}\text{\hspace{0.17em}}$.  In this section we give the complex variable solution that governs the variation of stress and displacement near the crack tip.   The solid is subjected to static remote loading, and is assumed to deform in plane strain.

The complete stress and displacement fields for an interface crack are given in Section 5.3.6.  The solution is too long to type out here:  Instead, we summarize the key features.

Material parameters for an interface:  The solution is expressed in terms of several additional parameters

1.      Plane strain moduli ${{E}^{\prime }}_{1}=2{\mu }_{1}/\left(1-{\nu }_{1}\right)$, ${{E}^{\prime }}_{2}=2{\mu }_{2}/\left(1-{\nu }_{2}\right)$

2.      Bimaterial modulus $\frac{1}{{E}^{*}}=\left\{\frac{1}{{{E}^{\prime }}_{1}}+\frac{1}{{{E}^{\prime }}_{2}}\right\}$

3.      Dundur’s elastic constants

$\alpha =\frac{{{E}^{\prime }}_{1}-{{E}^{\prime }}_{2}}{{{E}^{\prime }}_{1}+{{E}^{\prime }}_{2}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\beta =\frac{\left(1-2{\nu }_{2}\right)/{\mu }_{2}-\left(1-2{\nu }_{1}\right)/{\mu }_{1}}{2\left(1-{\nu }_{2}\right)/{\mu }_{2}+2\left(1-{\nu }_{1}\right)/{\mu }_{1}}$

Evidently $\alpha$ is a measure of the relative stiffness of the two materials.  It must lie in the range $-1<\alpha <1$ for all possible material combinations: $\alpha =1$ indicates that material 1 is rigid, while $\alpha =-1$ signifies that material 2 is rigid.  The second parameter does not have such a nice physical interpretation $–$ it is a rough measure of the relative compressibilities of the two materials.  For Poisson’s ratios in the range $0<\nu <1/2$, one can show that that $-1<\alpha -4\beta <1$.

4.      Crack tip singularity parameter

$\epsilon =\frac{1}{2\pi }\mathrm{log}\left(\frac{1-\beta }{1+\beta }\right)$

For most material combinations the value of $\epsilon$ is very small $–$ typically of order 0.01 or so.

Crack tip loading parameters The state of stress at the crack tip is characterized by three numbers: an arbitrary characteristic length L (a value of $L=100\mu m$ is often used); the phase angle of the loading $\psi$ and the magnitude of the stress intensity factor $|K|$.  Often, the energy release rate for the crack G is used in place of $|K|$. These are defined as follows

Phase angle $\psi ={\mathrm{tan}}^{-1}\left(\frac{{\sigma }_{12}\left(r=L\right)}{{\sigma }_{22}\left(r=L\right)}\right)$

Stress intensity magnitude  $|K|={e}^{-i\psi }\underset{r\to 0}{\mathrm{lim}}\left({\sigma }_{22}+i{\sigma }_{12}\right){\left(r/L\right)}^{-i\epsilon }\sqrt{2\pi r}$

Energy release rate $G=\frac{{|K|}^{2}}{2{E}^{*}{\mathrm{cosh}}^{2}\pi \epsilon }$

Solutions to interface crack problems are also often expressed in terms of two stress-intensity factor like parameters ${K}_{1}$ and ${K}_{2}$.  These are related to the crack tip parameters by

$\begin{array}{l}\left({K}_{1}+i{K}_{2}\right)=|K|{e}^{i\psi }{L}^{-i\epsilon }\\ |K|=\sqrt{{K}_{1}^{2}+{K}_{2}^{2}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\psi ={\mathrm{tan}}^{-1}\left\{\frac{\mathrm{Im}\left[\left({K}_{1}+i{K}_{2}\right){L}^{i\epsilon }\right]}{\mathrm{Re}\left[\left({K}_{1}+i{K}_{2}\right){L}^{i\epsilon }\right]}\right\}\end{array}$

Interpreting the crack tip fields

The values of $|K|$ and $\psi$ are is determined by the solid’s shape and how it is loaded (the value of  $\psi$ also depends on the choice of the characteristic length L). Once $|K|$ and $\psi$ are known, however, the near tip fields always have the form given by the asymptotic solution.

Since $\psi$ quantifies the ratio of shear to opening stress ahead of the crack tip, it is qualitatively equivalent to the ratio ${\mathrm{tan}}^{-1}\left({K}_{II}/{K}_{I}\right)$, where ${K}_{I}$ and ${K}_{II}$ are the mode 1 and mode 2 stress intensity factors for a crack in a homogeneous solid.

The opening and shear stresses along $\theta =0$ ahead of the crack tip can be calculated from

${\sigma }_{22}+i{\sigma }_{12}=\frac{|K|{e}^{i\psi }}{\sqrt{2\pi r}}{\left(\frac{r}{L}\right)}^{i\epsilon }$

The crack opening displacements behind the crack tip can be calculated from

${\delta }_{2}+i{\delta }_{1}=\frac{4|K|{e}^{i\psi }}{{E}^{*}\left(1+2i\epsilon \right)\mathrm{cosh}\left(\pi \epsilon \right)}\sqrt{\frac{r}{2\pi }}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\left(\frac{r}{L}\right)}^{i\epsilon }$

The complex exponent appearing in these expressions is scary $–$ to understand what it means note that

${r}^{i\epsilon }=\mathrm{exp}\left(i\epsilon \mathrm{log}r\right)=\mathrm{cos}\left(\epsilon \mathrm{log}r\right)+i\mathrm{sin}\left(\epsilon \mathrm{log}r\right)$

so this term indicates that the stresses oscillate near the crack tip.  We will discuss this in more detail below.

Oscillations in the stress and displacement fields

The asymptotic crack tip field for an interface crack is strikingly different to the corresponding solution for a homogeneous solid.  In fact, the results are somewhat disturbing, and have been the cause of much anguish in the fracture mechanics community.

We have already noted that the stress fields are oscillatory near the crack tip.  The stress distributions are plotted on the right as a function of $\mathrm{log}\left(\epsilon r/L\right)$.  Note that the results are shown for an unphysically large value of $\epsilon$ - for practical values the oscillations are so slow it is hard to see them.

Both normal and shear stresses oscillate with increasing frequency as the crack tip is approached.  As a result, it is difficult to unambiguously separate the loading into normal and shear components $–$ an opening stress induces just as much shear near the crack tip as does shear loading, and vice-versa.

Even more disturbingly, the crack opening displacements show the same oscillatory character. This means that the solution predicts that the crack faces overlap near the crack tip, which is clearly unphysical.

It is possible to find a solution that corrects for the overlapping crack faces (Comninou, J. Appl Mech. 44, 631 1977).  This solution predicts that the crack faces touch just behind the crack tip for all combinations of remote load.  There is a square root singularity in shear stress at the crack tip (so it’s strictly always loaded in mode II).  The zone of contact is extremely small, however $–$ typically of the order of a few nanometers for most practical crack sizes and materials, and probably much smaller than the process zone.

The standard procedure in LEIFM (linear elastic interfacial fracture mechanics) is to ignore the overlap between crack faces, and accept the asymptotic field described in the beginning of this sub-section as characterizing the stress and strain fields for an interface crack.  The oscillatory singularity is, after all, no less physical than a square root singularity.  The asymptotic field is expected to represent actual stress and strain fields in an annular region, which is small compared with specimen geometry, and large compared with the process zone.

9.6.2 Phenomenological theory of interface fracture

Phenomenological fracture mechanics for interfaces is based on the same reasoning that is used in fracture mechanics of homogeneous solids.  We anticipate three distinct regions in a plastically deforming specimen containing a crack,

1.      A process zone near the crack tip, with finite deformations and extensive material damge, where the asymptotic field is not accurate;

2.      A K dominant zone, outside the process zone, but small compared with specimen dimensions, where the asymptotic linear elastic field accurately describes the deformation;

3.      The remainder, where stress and strain fields are controlled by specimen geometry and loading.

Material failure (crack growth or fatigue) is a consequence of the failure mechanisms in the process zone.  As usual, we do not attempt to model the failure process in detail, and instead assume that the fields in the process zone are controlled by the fields in the region of K dominance.  In interface fracture, we use the stress state at the reference length L ahead of the crack tip to characterize the loading experienced by the process zone.  As we have seen, this stress is characterized by the energy release rate G (or alternatively the stress intensity magnitude $|K|$ ), together with the phase angle $\psi$

Fracture Criterion The critical condition for an interface crack to propagate is therefore given by a fracture criterion of the form

$G={G}_{i}\left(\psi \right)$

where ${G}_{i}\left(\psi \right)$ is the fracture toughness of the interface and $\psi$ is the phase angle, defined using some appropriate choice of length L.    The fracture toughness is a function of phase angle, just as the fracture toughness of a homogeneous solid subjected to mixed mode loading is a function of ${K}_{II}/{K}_{I}$

The fracture resistance of the interface ${G}_{i}\left(\psi \right)$ must be measured experimentally.  Several specimens are available for this purpose.  Examples include sandwich specimens (e.g. Leichti & Knauss, Exp. Mech. 22 383 1982; see also Suo & Hutchinson Mat. Sci & Eng A107 1989 135) and 4 point bend specimens (e.g. Charalambides et al Mechanics of Materials 8 269, 1990).  Experiments show that ${G}_{i}$ increases rapidly with phase angle: the typical variation of fracture toughness with phase angle is sketched in the figure. In fact, many experimental data seem to be fit by ${G}_{i}={G}_{0}/\mathrm{cos}\left(\psi \right)$.

To apply LEIFM, then, it is necessary (i) to measure the fracture resistance of the interface as a function of phase angle; (ii) calculate energy release rate and phase angle for the interface crack in the structure or component of interest, and (iii) apply the fracture criterion to assess the load bearing capacity of the component.

9.6.3 Stress intensity factors for some interface cracks

The solutions to interface crack problems are most conveniently expressed in terms of the stress intensity factors ${K}_{1},{K}_{2}$, because these are linear in stress and can therefore be superposed.  Stress intensity factors have been computed for many standard specimen geometries (usually using a numerical technique).  A few examples are shown below.

 STRESS INTENSITY FACTORS FOR INTERFACE CRACKS ${K}_{1}+i{K}_{2}=\left({\sigma }_{22}+i{\sigma }_{12}\right)\sqrt{\pi a}\left(1+2i\epsilon \right){\left(2a\right)}^{-i\epsilon }$ ${K}_{1}+i{K}_{2}=\sqrt{\frac{2}{\pi }}\mathrm{cosh}\pi \epsilon \frac{{F}_{2}+i{F}_{1}}{{b}^{1/2+i\epsilon }}$ ${K}_{1}+i{K}_{2}=\sqrt{\frac{2}{\pi }}\mathrm{cosh}\pi \epsilon \frac{{F}_{2}+i{F}_{1}}{{\left(2a\right)}^{1/2+i\epsilon }}{\left(\frac{a+b}{a-b}\right)}^{1/2+i\epsilon }$

The point force solutions can be used to calculate stress intensity factors for interface cracks that are subjected to nonuniform stress fields, following the procedure given in Section 9.3.3.  The energy release rate and phase angle can be calculated using the formulas given in the preceding section.  For example, for the slit crack with length 2a subjected to uniform stress far from the crack, the energy release rate and phase angle are

$\begin{array}{l}\psi ={\mathrm{tan}}^{-1}\left\{\frac{\mathrm{Im}\left[\left({\sigma }_{22}+i{\sigma }_{12}\right)\left(1+2i\epsilon \right){\left(L/a\right)}^{i\epsilon }\right]}{\mathrm{Re}\left[\left({\sigma }_{22}+i{\sigma }_{12}\right)\left(1+2i\epsilon \right){\left(L/a\right)}^{i\epsilon }\right]}\right\}\\ G=\frac{\left({\sigma }_{22}^{2}+{\sigma }_{12}^{2}\right)\left(1+4{\epsilon }^{2}\right)\pi a}{2{E}^{*}{\mathrm{cosh}}^{2}\left(\pi \epsilon \right)}\end{array}$

Note that, for a fixed value of L, the phase angle depends on the size of the crack.   This is a general feature of interface cracks: the mode mixity depends on specimen size.  However, the value of $\epsilon$ for most material pairs is very small, so the variation with specimen size is very weak.

9.6.4 Crack Path Selection

A final issue that is of great interest in interfacial fracture is the question of crack path selection.  An interface crack can either propagate along the interface, or deflect into one of the two materials adjacent to the interface.  In addition, a crack approaching transverse to an interface may be deflected along it $–$ this is a mechanism for trapping cracks in composite materials.

A rather involved stress analysis is required to answer these questions, but the results are simple. A crack approaching perpendicular to an interface (as shown on the left above) will deflect along the interface as long as

${G}_{i}\left(\pi /2\right)<{G}_{cm}/4$

where ${G}_{i}\left(\pi /2\right)$ is the interface toughness for a phase angle of 90 degrees, and ${G}_{cm}$ is the fracture toughness of the material on the far side of the interface.  If this condition is satisfied, the crack remains trapped in the interface and will not kink out of it.