This document describes a new machine learning approach for solving fracture mechanics problems when analytical solutions are not available. The authors develop regression tree and neural network models to predict fracture toughness from microcantilever geometry and loading conditions. They find that both approaches provide accurate results, but neural networks outperform regression trees with predictions within 1.5% of finite element simulations. This demonstrates that machine learning solutions can overcome limitations of empirical approaches and change how engineering problems are solved when analytical solutions cannot be obtained.

1. On the use of machine learning for investigating the
toughness of ceramic nanocomposites
Christos Athanasiou1, Xing Liu1, Nitin Padture1, Brian Sheldon1, Huajian Gao1,2
1 Brown University, USA
2 Nanyang Technological University, Singapore

2. 1997

3. 1997 2017

4. 1997 2017
Elastic modulus
Tensile strength
Density
alumina
370 GPa
70 MPa
3.9 g/cm3
1000 GPa
100 GPa
1.6 - 2.4 g/cm3
graphene

5. σ0
#$% = Yσ ()
2α
geometry
crack length
fracture toughness
σ0
σ = 2σ+
,
-
./0ρ
fracture toughness, KIC: how easy or difficult for a crack to propagate

6. 3.5 – 5.0 MPa.m1/2
Conventional alumina:
N. Koratkar, RPI & E. Corral, University of Arizona
1 μm

7. 3.5 – 5.0 MPa.m1/2
Conventional alumina:
E. Zapata-Solvas, et al. J. Eur. Ceram. Soc, 32, 12 (2012)
Reinforce ceramics w/ nanomaterials for high fracture toughness
5.0 – 8.5 MPa.m1/2
Alumina with graphene:
N. Koratkar, RPI & E. Corral, University of Arizona
1 μm

8. Reinforce ceramics w/ nanomaterials for high fracture toughness
3.5 – 5.0 MPa.m1/2
Conventional alumina:
5.0 – 8.5 MPa.m1/2
Alumina with graphene:
E. Zapata-Solvas, et al. J. Eur. Ceram. Soc, 32, 12 (2012)N. Koratkar, RPI & E. Corral, University of Arizona
1 μm

9. Reinforce ceramics w/ nanomaterials for high fracture toughness?
3.5 – 5.0 MPa.m1/2
Conventional alumina:
5.0 – 8.5 MPa.m1/2
Alumina with graphene:
E. Zapata-Solvas, et al. J. Eur. Ceram. Soc, 32, 12 (2012)W. Curtin, J. Am. Ceram. Soc. 74 ,11 (1991)
What we know: Nanomaterials increase fracture
toughness of ceramic nanocomposites

10. Reinforce ceramics w/ nanomaterials for high fracture toughness?
3.5 – 5.0 MPa.m1/2
Conventional alumina:
5.0 – 8.5 MPa.m1/2
Alumina with graphene:
E. Zapata-Solvas, et al. J. Eur. Ceram. Soc, 32, 12 (2012)W. Curtin, J. Am. Ceram. Soc. 74 ,11 (1991)
What we do not know yet: Perform accurate
mechanical testing of nanocomposites

11. Measuring fracture toughness of coatings using
focused-ion-beam-machined microbeams
D. Di Maio and S.G. Roberts
Department of Materials, University of Oxford, Oxford OX1 3PH, United Kingdom
(Received 23 July 2004; accepted 10 November 2004)
Measuring the toughness of brittle coatings has always been a difficult task. Coatings
are often too thin to easily prepare a freestanding sample of a defined geometry to use
standard toughness measuring techniques. Using standard indentation techniques gives
results influenced by the effect of the substrate. A new technique for measuring the
toughness of coatings is described here. A precracked micro-beam was produced using
focused ion beam (FIB) machining, then imaged and loaded to fracture using a
nanoindenter.
Determining the mechanical properties of coatings can
be very difficult due to their thinness (typically a few
microns), effects of the substrate, effects of adhesion, and
residual stresses. Using a nanoindenter, it is possible to
determine some properties of coatings (hardness and the
elastic modulus).1
Some other properties, however, are
still very difficult to measure. One of them is the fracture
toughness. The classical method for measuring fracture
toughness is to fracture a pre-notched sample with well-
defined geometry. From the critical load it is then pos-
sible to determine the fracture toughness KIc. However,
in the case of thin coatings, it is difficult to manufacture
a sample of the coating material alone: normal fracture
toughness specimens are several tens of millimeters in
size or larger.In the case of brittle ceramic materials, toughness is
very often measured using indentation techniques. The
size of cracks around the indent is measured and then
related to fracture toughness using a specific model.2,3
This method is difficult to use for thin coatings because
typical indentation crack sizes are several tens of microns
to several hundreds of microns; the effect of the substrate
will dominate. If the substrate is not brittle it will be
difficult to obtain a well-defined fracture geometry. This
is also the case for fracture toughness testing using
Hertzian indentation.4Some other methods more spec
been proposed. Thedeterm
with those obtained with other techniques. Micro-beams
and similar micro-mechanical elements have been pre-
pared for many years in silicon, mostly by etching meth-
ods.8–11
These methods are not easily applicable to other
materials (though silicon-etching has been used to pre-
pare Al and Au beams by Son et al.12
). Variants on these
micro-beam testing methods have included focused ion
beam (FIB) machining of cantilevers of Al-coated Si
from material polished to 10 ␮m thickness,13
and use of
FIB to put V-notches in Si specimens produced by etch-
ing methods.14
The technique used here involves the preparation and
testing of pre-cracked micro-beams from “bulk” speci-
mens, using FIB (FEI Company, Hillsboro, OR). The
method is first demonstrated using monolithic silicon and
then applied to produce beams of a thin (10 ␮m) WC
coating on a bulk steel substrate. Specimens are imaged
and loaded to fracture using a nanoindenter.
The operating principle of a FIB system is similar to
that of a scanning electron microscope (SEM), the major
difference being the use of a rastered gallium ion (Ga+
)
beam instead of an electron beam. The
moves material from the su
ing).15
The secto
D. DiMaio, S. Roberts, J. Mater. Res, 20 (2005)

12. D. DiMaio, S. Roberts, J. Mater. Res, 20, (2005)

13. D. DiMaio, S. Roberts, J. Mater. Res, 20, (2005)

14. D. DiMaio, S. Roberts, J. Mater. Res, 20, (2005)
Fracture toughness – processing and mechanical
testing at small scales

15. D. DiMaio, S. Roberts, J. Mater. Res, 20, (2005)
Step 1
Fracture toughness – processing and mechanical
testing at small scales

16. D. DiMaio, S. Roberts, J. Mater. Res, 20, (2005)
Step 2
Step 1
Fracture toughness – processing and mechanical
testing at small scales

17. D. DiMaio, S. Roberts, J. Mater. Res, 20, (2005)
Step 2
Step 1
Step 3
Fracture toughness – processing and mechanical
testing at small scales

18. D. DiMaio, S. Roberts, J. Mater. Res, 20, (2005)
! =
#$%&
' ()
=
* %, , %&
- %, ,
.
)
%
- %, , =
,%&
12
+ * −
%
2
3
%, +
,4
288
+
,
6
+ % − *
3 ,3
4
* %, , =
%3
,
2 +
,3
4 (% +
,
6)
%, +
,3
4
.
)
%
= 1.85 − 3.38
)
%
+ 13.24
)
%
3
− 23.26
)
%
&
+ 16.8
)
%
4
Empirical solution for Y
Fracture toughness – processing and mechanical
testing at small scales

19. D. DiMaio, S. Roberts, J. Mater. Res, 20, (2005)
Measuring fracture toughness via finite elements
Empirical solutions
FEM

20. Using finite elements to prepare the dataset
KI/P
!/# = %. '~%. )
*/# = '. %~+. %
,'/# = %. '~%. -
,./# = .. %~/. %
Small P (Linear Elastic Fracture Mechanics)
Simulation #Finite element software input
Finite element software output
35
100
10
10
Total number of simulations: 439,956
16 CPUs: 120 sec per simulation

21. Measuring fracture toughness via regression trees

22. Measuring fracture toughness via regression trees
v

23. v
Measuring fracture toughness via neural nets

24. v
Measuring fracture toughness via neural nets

25. v
Measuring fracture toughness via neural nets
ML as good as FEM

26. 7 μm
12 μm
length: 12 μm width: 5 μm thickness: 6 μm
5 μm
Microcantilever testing via neural nets
geometry load

27. v
hint1412.github.io/XLiu.github.io/SIF/
Portable deployment
https://hint1412.github.io/XLiu.github.io/SIF/
Output: KIC
Inputs

28. Summary
Analytical solutions
not available/accurate
1
New materials & testing
methods
2 Regression trees
(<2% max APE)
3
4
Neural nets
(<1.5% max APE)
5
Portable
deployment

29. Materials & mechanics enthusiasts
Christos E. Athanasiou
Brian Sheldon Nitin Padture
Xing Liu
Huajian Gao

30. Thank you !
www.ceathanasiou.com
christos_edward
Full length article
A machine learning approach to fracture mechanics problems
Xing Liua
, Christos E. Athanasioua
, Nitin P. Padturea
, Brian W. Sheldona,
*, Huajian Gaoa,b,
*
a
School of Engineering, Brown University, Providence, RI 02912, USA
b
College of Engineering, College of Science, Nanyang Technological University, Singapore
A R T I C L E I N F O
Article History:Received 3 February 2020Revised 9 March 2020Accepted 12 March 2020Available online 18 March 2020
A B S T R A C T
Analytical and empirical solutions to engineering problems are usually preferred because of their conve-
nience in applications. However, they are not always accessible in complex problems. A new class of solu-
tions, based on machine learning (ML) models such as regression trees and neural networks (NNs), are
proposed and their feasibility and value are demonstrated through the analysis of fracture toughness meas-
urements. It is found that both solutions based on regression trees and NNs can provide accurate results for
the specific problem, but NN-based solutions outperform regression-tree-based solutions in terms of their
simplicity. This example demonstrates that ML solutions are a major improvement over analytical and
empirical solutions in terms of both reliable functionality and rapid deployment. When analytical solutions
are not available, the use of ML solutions can overcome the limitations of empirical solutions and substan-
tially change the way that engineering problems are solved.© 2020 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
Keywords:
Machine learningAnalytical methodsMechanical properties testing
Fracture
1. Introduction
Engineers often seek analytical solutions for simplicity and reli-
ability, which bring great convenience to engineering applications
such as materials characterization, structural analysis and design.
However, analytical solutions cannot always be obtained. An accept-
able compromise is the empirical solution which relies on engineers’
understanding and generalization of experimental and numerical
data. For example, in ASTM standard C1421 for determination of frac-
ture toughness of advanced ceramics, empirical solutions are pro-
vided to evaluate the plane-strain stress intensity factor at the crack
tip. Both analytical and empirical solutions are known for their rapid
deployment and reliability. However, some complicated engineering
problems may involve a nonlinear and complex relationship among
higher-dimensional data, and empirical solutions cannot be readily
obtained. In the case where neither analytical nor empirical solutions
are feasible the following question can be raised: is there any other
possible way to obtain a solution? Machine learning (ML) algorithms
can be informed directly by experiments and simulations [1À3], and
thus provide “machine learning solutions”. These ML solutions are a
promising substitute for analytical and empirical solutions if they can
provide rapid and accurate results. In this context, the initial study
presented here, summarized in Fig. 1, uses ML solutions for materials
characterization, taking fracture toughness measurements of micro-
fabricated brittle ceramic microcantilevers as a relevant example.
2. Experimental
This investigation is based on a mechanical test that is used to
measure the mode-I fracture toughness, KIC, in small specimens
[4À7]. In this method, a pentagonal cross-section microcantilever is
cut with focused ion beam (FIB) micromachining in Helios NanoLab
450 system (FEI, Oregon, USA), as shown in Fig. 1b. To induce a well-
defined controlled fracture event, a sharp pre-notch is milled at a dis-
tance, L0, from the fixed end of the cantilever. The pre-notch is
required to be sufficiently sharp to guarantee the validity of the frac-
ture toughness measurement. Therefore, a two-step strategy is
adopted, i.e., a coarse cut with a moderate current followed by a sec-
ond finer cut with a lowest possible current, 1 pA. This strategy
restricts the notch radius well below 50 nm and eliminates the arti-
facts caused by ion-implantation damage around the notch tip during
FIB milling. In this way, the notch tip is sharp enough to behave like
an ideal crack and ensure the validity of the measurement [8,9]. After
the microcantilever fabrication, a nanoindenter (Hysitron TI 900, Tri-
boindenter, Minneapolis, USA) equipped with a Berkovich tip
to apply a controlled load at the free end
load-displacement respospecim
* Corresponding author.E-mail addresses: xing_liu@brown.edu (X. Liu),
christos_edouardos_athanasiou@brown.edu
nitin_padture@brown.eduhuajia
Acta Materialia 190 (2020) 105À112
Contents lists available at ScienceDirect
Acta Materialiajournal homepage: www.elsevier.com/locate/actamat
A machine learning approach to fracture mechanics problems
Acta Materialia,190 (2020) 105-112