Material Flaw Populations and Component Strength Distributions in the Context of the Weibull Function

Abstract

A clear relationship between the population of brittle-fracture controlling flaws generated in a manufactured material and the distribution of strengths in a group of selected components is established. Assumptions regarding the strength-flaw size relationship, the volume of the components, and the number in the group, are clarified and the contracting effects of component volume and truncating effects of group number on component strength empirical distribution functions highlighted. A simple analytical example is used to demonstrate the forward prediction of population → distribution and the more important reverse procedure of empirical strength distribution → underlying flaw population. Three experimental examples are given of the application of the relationships to state-of-the-art micro- and nano-scale strength distributions to experimentally determine flaw populations: two on etched microelectromechanical systems (MEMS) structures and one on native and oxidized silicon nanowires. In all examples, the minimum threshold strength and conjugate maximum flaw size are very well estimated and the complete flaw population, including the minimum flaw size, are very poorly estimated, although etching, bimodal, and oxidation effects were clearly discernible. The results suggest that the best use of strength distribution information for MEMS manufacturers and designers might be in estimation of the strength threshold.

Appendix: Inhomogeneous Loading

For a component composed of M discreet sub-volumes V_i, such that $\sum \limits_i^M{V}_i=V$ where i is a sub-volume index, and each sub-volume consists of k_i elemental volumes such that V_i = k_iΔV and thus $\sum \limits_i{k}_i=k$, the independent probability assumption gives the component ccdf as

$$ {\overline{F}}_V\left(\sigma \right)=\overline{F}{\left(\sigma \right)}^{k_1}\overline{F}{\left(\sigma \right)}^{k_2}\dots \overline{F}{\left(\sigma \right)}^{k_M} $$

If each sub-volume is held to a separate strength exceedance, σ_i, the overall component exceedance is

$$ {\overline{F}}_V\left({\sigma}_i,{k}_i\right)=\overline{F}{\left({\sigma}_1\right)}^{k_1}\overline{F}{\left({\sigma}_2\right)}^{k_2}\dots \overline{F}{\left({\sigma}_M\right)}^{k_M} $$

for the specified configuration. The exceedance can thus be written as a logarithmic sum

$$ {\displaystyle \begin{array}{c}\ln \left[{\overline{F}}_V\left({\sigma}_i,{k}_i\right)\right]=\sum \limits_i^M\ln \left[\overline{F}\left({\sigma}_i\right)\right]{k}_i\\ {}=\lambda \sum \limits_i^M\ln \left[\overline{F}\left({\sigma}_i\right)\right]{V}_i\end{array}} $$

where the second line makes clear that the sum is over the volume of the component and λ = 1/ΔV is the number density of elements (and thus flaws) per volume.

If V_i and thus ΔV are reduced in size to the infinitesimal limit (and λ is still defined) such that the component can be regarded as a continuum, each point can be held to separate strength exceedance, σ(x, y, z), where (x, y, z) is a point coordinate in the component. The sum above thus becomes an integral

$$ \ln \left({\overline{F}}_V\right)=\lambda \underset{V}{\int}\ln \left[\overline{F}\left(\sigma; x,y,z\right)\right]\mathrm{d}V $$

where ${\overline{F}}_V$ is understood to be configuration dependent. If the exceedance is constant, the integral collapses to a product and gives the oft-cited result F_V = 1 - exp [λVln(1 – F)], [42, 52, 54] recognizing that $F=1-\overline{F}$. The full equation can be reduced in complexity considerably: (1) If a plane (say, x-y) of fixed area is held to a separate exceedance such that σ = σ(z) only and dV = dxdydz = A_xydz, where A_xy is the area of the plane. Thus

$$ {\overline{F}}_V=\exp \left[\mu \underset{Z}{\int}\ln \left[\overline{F}\left(\sigma; z\right)\right]\mathrm{d}z\right], $$

where μ = λA_xy is the number density of flaws/length along z in the plane; (2) If a simple form is selected for $\ln \left(\overline{F}\right)$, say the Weibull distribution with σ_th = 0 such that $\ln \left(\overline{F}\right)=-{\left(\sigma /{\sigma}_0\right)}^m$. Thus

$$ {\overline{F}}_V=\exp \left[-\frac{\mu }{\sigma_0^m}\underset{Z}{\int }{\left(\sigma; z\right)}^m\mathrm{d}z\right] $$

and, (3) If a simple form is selected for σ, say the outer-fiber tensile stress in a built-in cantilever beam of length L, supporting a weight w at the free end, such that σ = wz/Z, where Z is the section modulus of the beam perpendicular to z. Thus

$$ {F}_L(w)=1-\exp \left[-\mu {\left(\frac{w}{Z{\sigma}_0}\right)}^m\underset{0}{\overset{L}{\int }}{z}^m\mathrm{d}z\right]=1-\exp \left[-\frac{\mu L}{\left(m+1\right)}{\left(\frac{wL}{Z{\sigma}_0}\right)}^m\right] $$

similar to an earlier derivation [55]. The analogous expression for a homogeneous rod of cross section A, uniformly loaded in tension by weight w, is

$$ {F}_L(w)=1-\exp \left[-\mu LA{\left(\frac{w}{A{\sigma}_0}\right)}^m\right] $$

The group cdf F_L(w) gives the proportion of beams or rods of length L that fail under weight w; the probability of failure is now expressed in terms of the extensive variables of failure weight and component size, rather than the intensive variable of strength σ.

In both inhomogeneous and homogeneous configurations, the variation of the cdf with the failure variable w is identical. The form of the distribution is unaffected by the mode of loading. However, in the case of inhomogeneous loading, (unsurprisingly) there is a much greater dependence on the geometry of the component: In the case of the beam in bending, an additional dependence on beam length and details of the shape of the cross-section appear in the cdf, whereas for the tensile rod, only the area of the cross section appears. Note also that the flaw-population exponent appears twice in the inhomogeneous cdf. The magnitude of the distribution is thus strongly affected by the geometry of the component in inhomogeneous loading. The extreme opposite to the simple case considered above is a stochastic distribution of stress superposed on the stochastic distribution of strengths [51].