Statistical failure analysis of brittle coatings by spherical indentation: theory and experiment

Abstract

The mode of failure and failure probability of a brittle coating on a compliant substrate subjected to a static load through a spherical indenter is investigated experimentally and theoretically. We extend our recent study (2003, J Mat Sci 38:1589) of surface crack initiation in a monolithic solid to the layered system, and account for the multi axial stress state of the indentation in the failure probability analysis. Two modes of failure, a Hertzian cone crack initiating from the contacting surface and a half-penny-shaped crack initiating from the interface, are investigated and the probability of failure initiation for both surfaces are theoretically predicted and compared with experimental data.

The effect of interface debonding on failure phenomena is investigated. For a given load the failure probability for debonded specimens is significantly higher than that of well-bonded samples. For the debonded case the theoretical failure probability curve falls within the 90% confidence interval of the experimental data, while the experimental values for the completely bonded case show somewhat lower failure probabilities than that predicted. This may be attributed to the possible bridging effect by the adhesive on interfacial surface defects in the ceramic that is not accounted for in our model.

Appendices

Appendix

Discussion of statistical parameters for indentation contact problem

In an infinitesimal concentric ring bounded by radii r ⁱ and r ⁱ  + δr with constant radial stress σ _rr, the failure probability is given [5, 14] as follows:

$$ F^{i} {\hbox{(}}\sigma _{{rr}} {\hbox{)}} = 1 - {\hbox{exp}}{\left[ { - {\hbox{(}}2\pi r^{i} \delta r{\hbox{)}}k\sigma ^{m}_{{rr}} } \right]} $$

(19)

It is implicitly assumed in Eq. 19 that only the maximum principal stress component (must be tensile) affects the failure probability. The influence of the other principal stress components on the failure probability is neglected. For example, in Fig. 20, a crack is oriented in the direction of the principal stress components σ ₁ (σ ₁ ≥ σ ₂). In the case A, the σ ₂ is compressive and in the case B is tensile. Since Eq. 19 does not account for the differences in stress states between A and B (no dependence on σ ₂), the predicted failure distributions of the two cases are identical. Obviously this is not the case. The crack in case A cannot propagate due to compressive stress applied normal to the crack surface, while the crack in case B may propagate if the magnitude of the applied tensile stress σ₂ satisfies the fracture criterion.

Fig. 20

Illustration of the two different stress states for a crack. In case A crack is under compression and does not propagate, in case B it propagates while Eq. 19 predicts identical results for both cases

In order to overcome this limitation and more accurately reflect the multi-axial stress states, we will use the method proposed by Batdorf et al. [16, 17] (Eq. 6). Equation 6 accounts for the stress components other than the first principal stress in calculation of the failure probability through the solid angle Ω. Within the framework of the probability model (Eq. 6), the probability of crack propagation is proportional to Ω/2π and is measured by the probability of its normal n falling within the solid angle Ω. For example in Fig. 21, the failure will occur when the crack normal n is within the sector Ω, and the failure will not occur when n is outside of this sector. Therefore, the influence of σ ₂ on crack propagation probability is accounted for by calculation [16, 17] of the solid angle Ω for a given stress state (the resulting Ω will depend on stress state, crack size and failure criterion selected). In particular, if a crack in case Fig. 20B satisfies the failure criterion (σ ₂ = σ _c), then Ω is equal to 2π and it will propagate, and for the same crack in Fig. 20A, Ω is less than 2π and it will not propagate.

Fig. 21

Crack under biaxial stress state. Crack propagates when its normal n is in the solid angle Ω

To account for the multi-axial stress state, we have derived from Eq. 6 the modified distribution in the form:

$$ F^{i} {\hbox{(}}\sigma _{{rr}} {\hbox{)}} = 1 - {\hbox{exp}}{\left[ { - {\hbox{(}}2\pi r^{i} \delta r{\hbox{)}}\ifmmode\expandafter\bar\else\expandafter\=\fi{k}\sigma ^{m}_{{rr}} } \right]}{\hbox{,}} $$

(20)

where $\ifmmode\expandafter\bar\else\expandafter\=\fi{k}$ depends on stress state and is given by

$$ \ifmmode\expandafter\bar\else\expandafter\=\fi{k} = \frac{{mk}} {{2\pi }}{\int_0^\infty {\Omega {\left( {\tfrac{{\sigma _{{tt}} }} {{\sigma _{{rr}} }}{\hbox{,}}\tfrac{{\sigma _{c} }} {{\sigma _{{rr}} }}} \right)} \cdot {\left( {\frac{{\sigma _{c} }} {{\sigma _{{rr}} }}} \right)}^{{m - 1}} {\hbox{d}}{\left( {\frac{{\sigma _{c} }} {{\sigma _{{rr}} }}} \right)}} }. $$

(21)

Here σ _tt is the hoop stress and $\Omega {\left( {\tfrac{{\sigma _{{tt}} }} {{\sigma _{{rr}} }}{\hbox{,}}\tfrac{{\sigma _{c} }} {{\sigma _{{rr}} }}} \right)}$ is determined from the actual surface stress state.

Equation 20 will reduce to Eq. 19 for a uniaxial tensile stress state.