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Probabilistic failure criteria for individual microstructural elements: an application to hydrogen-assisted crack initiation in alloy 725

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Abstract

Mechanical failure of polycrystalline metals is often difficult to predict, in part because of the wide range of conditions under which individual elements of the material’s microstructure fail. We describe an efficient method for posing, assessing, and optimizing failure criteria for individual microstructural elements, such as grains, grain boundaries, precipitates, or precipitate–matrix interfaces. Such criteria may lead to improved failure predictions for polycrystalline metals. Our method constructs a failure probability function from a database of individual failure events obtained from experiments on polycrystalline samples. It then uses the Kullback–Leibler (K–L) divergence to compare it to probability densities corresponding to specific hypothesized failure mechanisms. The likeliest failure mechanism is the one that minimizes the K–L divergence with respect to a suitably chosen null hypothesis. As a demonstration, we apply this approach to hydrogen-assisted crack initiation at coherent twin boundaries in Ni-based alloy 725 and deduce a best-fit failure criterion for them.

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Abbreviations

\( D_{{{\text{K}} - {\text{L}}}} \), \( D_{{{\text{K}} - {\text{L}}}}^{i} \) :

Kullback–Leibler (K–L) divergence, K–L divergence for failure model \( i \)

\( p \) :

Experimentally determined failure probability distribution

\( \bar{p} \), \( \bar{p}_{i} \).:

Hypothesized failure probability distribution, indexed by \( i \) when there is more than one

\( \bar{p}_{i}^{e} \) :

Hypothesized failure probability distribution with exponential form

\( \bar{p}_{i}^{F - D} \) :

Hypothesized failure probability distribution with Fermi-Dirac form

\( \varOmega \) :

The domain over which both experimentally determined and hypothesized failure probability distributions are defined

\( \left\{ {Q_{j} } \right\} \) :

Physical parameters upon which failure of the microstructural element of interest depends

\( \left\{ {x_{k} } \right\} \) :

Model parameters upon which hypothesized failure probability distributions depend

\( \tilde{D}_{K - L}^{i} \) :

K–L divergence optimized with respect to the model parameters, \( \left\{ {x_{k} } \right\} \)

\( \left\{ {\tilde{x}_{k} } \right\} \) :

Values of the model parameters, \( \left\{ {x_{k} } \right\} \), at which the K-L divergence is optimized

\( f \) :

Frequency with which the microstructural element of interest is observed to fail in a mechanical test

\( g \) :

Probability of encountering the microstructural parameter of interest in a specific sample or set of samples

\( \sigma_{a} \;\sigma_{ab} \) :

Variance describing all sources of uncertainty in experimental measurements on the ath microstructural element of interest, indexed by \( b \) whenever a class \( b \) of measured parameters has a different uncertainty from other classes

\( \left\{ {Q_{j} } \right\}_{a} \) :

Physical parameters describing the ath microstructural element

\( m \) :

Number of instances of failure of the microstructural element of interest

\( n \) :

Number of instances of the microstructural element of interest—whether failed or not—in the microstructure as a whole

\( P \), \( P_{b} \) :

Any probability distribution peaked at zero, indexed by \( b \) whenever a class \( b \) of measured parameters has a different probability distribution from other classes

\( d \), \( d_{b} \) :

Metric that quantifies the distance between the \( \left\{ {Q_{j} } \right\} \) values where \( f \) or \( g \) are evaluated and the \( \left\{ {Q_{j} } \right\}_{a} \) values corresponding to each of the experimentally determined data points, indexed by \( b \) whenever a class \( b \) of measured parameters has a different metric from other classes

\( \bar{p}_{0} \) :

Hypothesized model representing the null hypothesis

\( C_{\text{norm}} \) :

Factor whose value is to be determined by normalizing the distribution to unity over its domain

\( \hat{n} \) :

Unit vector pointing normal to the coherent twin boundary plane

α:

Angle between \( \hat{n} \) and the tensile axis

β:

Smallest angle between a 〈110〉-type direction in the coherent twin boundary plane and the steepest direction along the coherent twin boundary plane

\( p_{vM} \) :

Un-normalized von Mises distribution with a mean of zero

\( \kappa \) :

Measure of spread in \( p_{vM} \)

\( u \) :

Angular uncertainty in the orientation of the EBSD data with respect to the tensile axis

\( \sigma \) :

Uniaxial tensile stress

\( \sigma_{n} \), \( \bar{\sigma }_{n} \).:

Normal traction perpendicular to the coherent twin boundary plane, normalized by \( \sigma \)

\( \tau_{ \hbox{max} } \), \( \bar{\tau }_{\max} \).:

Maximum shear traction along the coherent twin boundary plane, normalized by \( \sigma \)

\( {\tau }_{\text{rss}},\; \bar{\tau }_{\text{rss}}\) :

Shear stress along the 〈110〉-type crystallographic direction within the coherent twin boundary plane that forms the smallest angle with \( \hat{n} \), normalized by \( \sigma \)

\( \bar{\sigma } \) :

Weighted sum of \( \bar{\sigma }_{n} \), \( \bar{\tau }_{ \hbox{max} } \), and \( \bar{\tau }_{\text{rss}} \) used to predict failure

\( A \), \( B \) :

Weights used in the definition of \( \bar{\sigma } \)

\( \bar{\sigma }^{c} \) :

Critical value of \( \bar{\sigma } \) for failure

\( \bar{\sigma }_{n}^{c} \) :

Critical value of \( \bar{\sigma } \) for failure when \( \bar{\sigma } \) depends solely on \( \bar{\sigma }_{n} \)

\( \Delta \bar{\sigma } \) :

Difference between \( \bar{\sigma }^{c} \) and \( \bar{\sigma } \)

\( s \) :

Prefactor by which \( \Delta \bar{\sigma } \) is multiplied when formulating failure models, \( \bar{p}_{i} \)

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Acknowledgements

This work was supported by the US Department of Energy, Office of Basic Energy Sciences under Award No DE-SC0008926, with the exception of experiments on CTBs in alloy 725 (the “Assembly of an experimental database of failure events for hydrogen-assisted crack initiation in Ni-based alloy 725” section), which were supported by the BP-MIT Materials and Corrosion Center. Access to shared experimental facilities was provided by the MIT Center for Materials Science Engineering, supported in part by the MRSEC Program of the National Science Foundation under award number DMR—0213282. J.P.H. Acknowledges the Department of Energy Office of Science Graduate Fellowship Program (DOE SCGF), made possible in part by the American Recovery and Reinvestment Act of 2009, administered by ORISE-ORAU under contract no. DE-AC05-06OR23100. M.J.D. Acknowledges stimulating discussions with A. Srivastava.

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Authors and Affiliations

  1. Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA, 02139, USA

    M. Seita & S. Gradečak

  2. Department of Nuclear Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA, 02139, USA

    J. P. Hanson

  3. Materials Science and Engineering, Texas A&M University, College Station, TX, 77843, USA

    M. J. Demkowicz

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  1. M. Seita
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  4. M. J. Demkowicz
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Correspondence to M. J. Demkowicz.

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Seita, M., Hanson, J.P., Gradečak, S. et al. Probabilistic failure criteria for individual microstructural elements: an application to hydrogen-assisted crack initiation in alloy 725. J Mater Sci 52, 2763–2779 (2017). https://doi.org/10.1007/s10853-016-0568-2

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