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Residual Stress as a Fracture Toughening Mechanism: A
Phase-Field Study on a Brittle Material
Enrico Salvati
Polytechnic Department of Engineering and Architecture (DPIA), University of Udine, Via delle
Scienze 208, Udine 33100, Italy
email: enrico.salvati@uniud.it;
Table of Contents
Abstract ............................................................................................................................. 2
1. Introduction ................................................................................................................ 3
2. Methods ...................................................................................................................... 4
2.1. Eigenstrain ................................................................................................................... 4
2.2. Phase-Field Approach for Brittle Fracture Problem ...................................................... 5
3. Analysis Description .................................................................................................... 6
3.1. 2D Model description and discretisation ...................................................................... 6
3.2. Crack tip position analysis ............................................................................................ 8
3.3. R-curve calculation ....................................................................................................... 9
4. Results and Discussion ................................................................................................ 9
4.1. Stress Distribution due to the Inclusion ...................................................................... 10
4.2. Crack tip inclusion (
𝒙𝟎
=3 mm,
𝒚𝟎
=0) ......................................................................... 10
4.3. Crack tip inclusion (
𝒙𝟎
=20 mm,
𝒚𝟎
=0) ....................................................................... 11
4.4. Inclusion (
𝒙𝟎
=20 mm,
𝒚𝟎
=3 mm) ............................................................................... 12
4.5. General Discussion ...................................................................................................... 13
5. Conclusions .............................................................................................................. 14
Acknowledgments ............................................................................................................. 15
Appendix ......................................................................................................................... 15
References ........................................................................................................................ 17
Keywords: Fracture toughness; Residual Stress; R-curve; Phase-Field; Eigenstrain;
2
Abstract
Recent engineering design practice for materials and structures relies more and more on damage-
tolerant criteria. Such a design approach is attained mainly by employing materials showing a
certain level of fracture toughness.
This work aims to explore a way to generate fracture toughness in materials that intrinsically shows
no toughness at all, i.e. brittle materials. The key idea lies in the introduction of inelastically
deformed sub-regions (e.g. circular inclusions) in the base material, which inevitably generate a
residual stress field.
To accomplish this purpose, the advanced Phase-Field method coupled with the eigenstrain theory
is employed, respectively to simulate the crack propagation behavior and to introduce a residual
stress field in a pre-notched sample. Information about crack propagation and displacement
externally imposed is used to obtain the resistance curve (R-curve) for several configurations.
One of the main findings of this research regards the possibility of originating fracture toughness
in intrinsically brittle materials upon appropriate positioning of one inclusion - containing a certain
amount of inelastic deformation – with respect to a notch tip. This result demonstrates that
accurate design of residual stress is crucial to attaining unprecedented material or structure
performance, and the method shown here represents a valid tool to exploit this advanced design
capability.
3
1. Introduction
Failure of engineering materials is frequently caused by nucleation and subsequent propagation of
one or more cracks. In most cases, failures of materials are triggered by the presence of stresses
exceeding a certain threshold evaluated through an appropriate failure criterion, for the specific
material. The recent practice in engineering design is diverting more and more towards the use of
damage-tolerant materials (or components), which facilitates structural integrity inspection
processes and therefore it prevents catastrophic failures. In other words, materials showing a high
level of fracture toughness are sought, although searching for a good compromise with the
strength of a material is not a trivial task [1]. Ideally, under quasi-static or fatigue loading modes,
cracks should propagate stably so that their detection can occur before complete failure is reached.
After detection, there exist a number of ways to take actions and mitigate the issue, for instance
by replacing the part or, repairing [2, 3] by using the stop-hole technique [4], employing
interference fit [5], introducing patches [6], via Laser Metal Deposition [7], cold expansion [8],
infiltration plating [9] and other methods. Some of these techniques - particularly the latter two
outlined methods - rely on the modification of the mean stress during cyclic loading due to the
enhanced effect of crack closure or residual stress (RS), similarly with what has been extensively
seen in ductile materials when an overload is applied during cyclic loading [10-12].
Unfortunately, not all the engineering materials are intrinsically tough or show a certain degree of
ductility, hence, ensuring continuous monitoring of material damage becomes a rather challenging
task. For example, brittle or semi-brittle ceramics are often used in high-temperature applications
due to the limited number of alternative candidate materials which can perform well under this
loading condition. More examples can be found in some scenarios where material embrittlement
is present, e.g. low-temperature conditions, hydrogen diffusion, etc.
An accepted theory that establishes the conditions to meet in order to achieve crack stop in brittle
propagation is called crack arrest [13], which is valid for a monotonic quasi-static loading test but is
found very useful also for more complex loading manners. This theory simply relies on the
assumptions that a cleaving crack arrests when the crack driving force of the growing crack front
falls below the crack arrest toughness of the material. The great advantage of such a theory is that,
although a crack propagates in an unstable manner, a dynamic analysis of the propagating crack
can be replaced by a static simulation without the introduction of relevant errors [14]. Therefore,
in the context of brittle materials, a propagating crack can be arrested provided that either the
fracture toughness of the material increases locally or the crack driving force is reduced, for
instance, by lowering the stress baseline through the presence of RS. The same concept, but using
a probabilistic approach, has also been proposed [15, 16].
At the microscopic length scale it is well known how the presence of inclusions (or sub-domains)
- presenting different properties with respect to the matrix - within the microstructure can modify
materials strength and ductility, and in turns toughness [17-20]. On the other hand, macroscale
inclusions can be thought as of a well-defined sub-domains within a host matrix which have
undergone initial inelastic deformation or, more in general, even having different material
properties. This inelastic deformation that may correspond to plastic deformations, microstructure
phase-change, etc.
First studies on the effect of the presence of inclusions within a homogeneous material can be
undoubtedly attributed to Eshelby [21]. In his works, Eshelby considered a sub-volume
undergoing a uniform permanent deformation also known as eigenstrain [22-27], within a
homogeneous linear elastic solid. As a consequence of this uniform expansion or shrinkage, the
material surrounding the inclusion experiences deformation of elastic nature and therefore
origination of stress. The problem solved by Eshelby was actually the solution for the stress, strain
and displacement field in both the inclusion and matrix. When dealing with failure problems, the
presence of inclusions and thus RS, plays an important role in affecting the crack propagation rate
and stability. Therefore, the mutual interplay between inclusion and propagating cracks is
extremely important [28], particularly when damage-tolerant materials or structures are sought.
More recent studies have focussed their attention on the capability of RS in hampering crack
4
nucleation and propagation in brittle materials, for instance by designing RS profiles to arrest crack
propagation in glass [29] and a combination of RS and interfaces [30].
In the last decades, a lot of effort has been put by the fracture mechanics scientific community on
the development of methods able to model and predict crack propagation through materials. The
main challenge lays in the analysis of materials presenting heterogeneities, for instance: interfaces,
inclusions, residual stress, graded materials. Relying on analytical solutions for this class of
problems is very infrequent [31, 32], while generic numerical approaches seem to be more versatile
in some cases. Indeed, numerical algorithms have been developed for the Stress Intensity Factor
(SIF) of a crack in proximity or in front of an inclusion [33]. Nevertheless, only the use of Finite
Element Methods (FEM)[34], which is itself a numerical approach, can cope with the myriad of
complex problems seen in practical engineering and scientific applications, particularly when
coupled with experimental observations [35]. In particular, to deal with the problem of propagating
cracks, several methods have been developed, for example, node release techniques [36], domain
remeshing [37], discrete eXtended FEM (XFEM) [38-40] & regularised XFEM [41], Cohesive
Zone Model (CZM) [42-44], Peridynamics [45] and Phase-Field [46, 47]. Some of these techniques
present some difficulties associated with the discrete modelling nature of the crack and the
problems related to the intersection of cracks or crack branching [48], and for these reasons,
distinctive attention is lately being put on the development of a variational approach based on the
energetic Griffith theory [49], the Phase-Field Model (PFM). PFM is characterised by its diffusive
description of the crack by a scalar phase-field that discriminates the damaged or broken material
from the undamaged material. Such a diffusive description of the crack is essentially defined by a
length scale
!!
. In this way, there are no particular computational difficulties in dealing with spatial
derivatives. Over the last 20 years, the PFM has been developed for several material behaviours
and is often coupled even with different physics. Starting from the brittle formulation [50-52],
PFM has been implemented for ductile materials [53, 54], functionally graded materials [55],
cohesive fracture [56] dynamics brittle fracture [57], composites [58], fatigue crack growth [59],
hydrogen assisted cracking [60], heterogeneous materials [61], hydraulic fracture [62], phase
transformation and crack interaction [63], three-dimensional problems [64] and others.
In the present paper, the PFM is employed for the first time to account for the presence of RS in
a brittle material. This analysis was made possible thanks to the incorporation of inelastic
deformation (eigenstrain) within a well-defined material sub-domain (inclusion). By using this
novel approach, a calculation framework of the crack growth resistance curve (R-curve) was
developed and the influence of RS on the overall fracture toughness was assessed. Several
eigenstrain magnitudes were prescribed within inclusions placed ahead of a pre-notched sample at
three relevant different positions. The simulation of crack propagation under quasi-static
displacement-controlled loading was performed by assuming identical material properties for both
the matrix and the inclusion regions. The R-curves were obtained through the analysis of the Stress
Intensity Factors (SIFs) for a specified 2D plane-stress sample provided by the literature, from the
displacement vs. crack length information provided by the PFM simulation results. The outcome
of the numerical investigation is thoroughly discussed, and conclusions are drawn with special
attention to those scenarios that revealed beneficial effects on the fracture toughness of the sample.
2. Methods
2.1. Eigenstrain
The eigenstrain method results very useful to prescribe a residual stress field within a solid body.
Its fundamental strength is that the method is self-consistent and therefore both the stress
equilibrium and strain compatibility are satisfied. The term eigenstrain
""
# refers to a permanent
strain that can be arbitrarily prescribed within a solid body, which in practice can be thought of as
temperature change, phase transformation, plastic deformation, etc.
The eigenstrain problem consists of searching the elastic strain field that arises as a consequence
of the introduction of the eigenstrain itself; this is also called direct problem [24]. While, if the elastic
field is known a priori and the eigenstrain is the sought quantity, then this problem is called inverse
5
problem [26, 65, 66]. For instance, the inverse problem can be very useful when experimental data
is available concerning the residual stress field, and it is required to check if such a field obeys the
strain compatibility and stress equilibrium conditions. While some analytical solutions are available
in the literature to solve the direct problem in simple geometrical configurations [67-69], for more
complex scenarios the use of FEM is mandatory [70].
In a solid body, the total strain
"#$#
can always be decomposed additively in an elastic
"
and an
inelastic part:
"#$# $𝜀∗
#%&"
(1)
The eigenstrain can also be shown explicitly as:
𝜀∗
#
=
'
"%% "%& "%'
"&% "&& "&'
"'% "'& "''
(
(2)
The analysis described in the present paper embeds the eigenstrain problem within the FEM
simulation concerning the PF modelling of the crack propagation; more details are given in the
following section of the paper.
2.2. Phase-Field Approach for Brittle Fracture Problem
An arbitrary body
)*+(
is considered and shown in Figure 1, where
,
is the geometrical
dimension of the problem. The contour of the body is identified by
&-)
, while internal crack
boundaries in their diffusive description are
.)
. The displacement field of the considered body is
denoted by
/
0
123
4 where
1
is the position vector and
3
is the time dependence. The time-
dependent Dirichlet boundary conditions need to be satisfied by the displacement field, therefore
/*
0
123
4
&
on
&-)(
, as well as the Neumann conditions. In addition, a body force
&5
0
123
4, acting on
the whole body
)
or a subdomain
)+
can be considered, as well as the surface tractions
6*
0
123
4
acting on
&-),
.
Figure 1 Phase-Field diffusive crack approach. (a) Representation of a diffuse crack within a
body in the presence of inclusions. (b) Phase-Field distribution along the transverse coordinate
as a function of the characteristic length,
!!
According to the variational approach proposed by the Griffith’s theory [49] for problems of
fracture, the energy required to create a fracture surface per unit area is equal to the critical energy
density for fracture,
&7-
, also known as the critical energy release rate (ERR). In general, the
potential energy within the body
)
can be expressed as the sum of contributions given by the
elastic energy
&8.
, the energy from fracture of the bulk or fracture of interfaces
8,
and the energy
induced by the external forces
&8/0
. In the present study, no interfaces nor dynamics effects are
6
considered, therefore these contributions are not reported in the total energy balance equation.
Hence, the total energy is:
8#$8.%8,98/0
(3)
The single contributions can also be expressed more comprehensively as:
8#0/2.4$:;.0"4,)
1%:7-,<
29=:5>/&,)
1%: 6>/&,?
31 @
(4)
where
.
defines the crack surface in its discrete representation, while
;.
0
"
4 is the elastic energy
density, which, under the assumption of isotropic homogeneous linear elastic material becomes
[51]:
;.0"4$A
BC"**"44 %D"*4"*4
(5)
where
C
and
D
are the Lamé constant, and
"*4
are the coefficients of the linear strain tensor
"
0
/
4.
For the particular case examined in this paper, because of the eigenstrain presence, from equation
(1):
"$"#$# 9""
#
E
Therefore, the elastic strain tensor can be calculated from the displacement
field as follows:
"*4 $A
B
F
-/*
-14%-/4
-1*
G
9"56
"
H
H
H
(6)
It is worth adding that crack initiation, propagation and branching
.
0
123
4 at the time
3*
0
I2J
4
at a point
1*)
take place when the potential
8#
0
/2.
4 reaches its minimum value and the
irreversible condition
.
0
12K
4
*.
0
123
4
0KL34
is satisfied. The term irreversible refers to the
constrain that do not allow the body to turn in the uncracked configuration once the fracture has
taken place.
In the present work, an anisotropic model proposed by Miehe et al. [71], was employed to split
the elastic energy into compressive and tensile contributions and make sure that
propagation/creation of cracks under compressive loads is suppressed. This decomposition was
accomplished by a spectral decomposition of the strain tensor. Moreover, both the geometric crack
function (GCF) and the energy degradation function (EDF) are quadratic in this study [50]. Details
on the mathematical description of these quantities and a more comprehensive review of
alternative solutions are provided in the Appendix section of this paper.
As far as the solution strategy is concerned, an implicit staggered time integration scheme was
employed. More details about the staggered scheme implemented in this calculation framework
can be found in the work of Zhou et al. [72].
3. Analysis Description
3.1. 2D Model description and discretisation
Numerical analyses of the crack propagation through a brittle material were conducted using a
Single Edge Notched Plate (SENP) geometry under plane stress assumption. The SENP sample
dimensions and other geometrical and loading characteristics are shown in Figure 2. The sample
was loaded under displacement control by varying the position of the sample upper end
/
as a
function of a pseudo-time. It is worth reminding that the loading mode was assumed to be quasi-
static and therefore all the dynamics contributions were considered negligible and therefore did
not accounted for in the computation. Within the SENP sample, the shaded region presented in
Figure 2 represents the inclusion domain in which the inelastic deformation (eigenstrain) was
7
prescribed. The present study considers a hydrostatic eigenstrain applied to the model, therefore
the eigenstrain matrix appears as:
""
#
$
'
""I I
I ""I
I I ""
(
(7)
where
""
is the eigenstrain magnitude that is varied in the parametrical analyses of this research.
The centre position of such an inclusion is identified through the coordinates (
1!2M!
) with respect
to the sharp notch tip position.
Figure 2 Sample geometry, boundary conditions and relevant parameters
As the material properties are concerned, a linear elastic material was chosen with elastic modulus
and Poisson’s ratio (
N
and
O
) as shown in the summary Table 1, along with the most relevant
dimensions and parameters involved in the simulations.
Table 1 Sample dimensions and parameters of the analysis
Dimension /
Parameter
Description
Value
P
Sample width
120 mm
Q
Half sample height
50 mm
R!
Pre-crack length
20 mm
1!
Inclusion centre x-coordinate
variable
M!
Inclusion centre y-coordinate
variable
S*78
Inclusion radius
6 mm
""
Eingenstrain magnitude
variable
/
Imposed displacement of the sample upper end
variable over pseudo-time
3
!!
PF characteristic length
1 mm
N
Young’s modulus
70 GPa
T
Poisson’s ratio
0.3
78
Fracture Toughness
4096 N/m
8
The FEM implementation of this analysis was carried out using COMSOL package. The original
implementation proposed by Zhou et. al. [72, 73] was employed and modified to account for the
sole elastic strain part when computing the elastic strain energy density, which is necessary when
an inelastic strain is prescribed to the model, i.e. eigenstrain. Moreover, the eigenstrain
implementation followed the procedure reported by the author in a recent publication [24].
Great care was taken to discretise the sample. In fact, it is well known how an inappropriate mesh
element size can alter the outcome of the simulation. To this end, a good compromise between
the computational cost and the element size
U
was found by setting the latter:
U$!!VW
. To obtain
this value, a convergence analysis was carried out on the sample presenting &
""$I
, i.e. no inclusion effect, and the value of load reached the instant of crack nucleation was
monitored while varying the element size,
U
. To ensure the same accuracy during the whole crack
propagation process, a sub-domain of the sample in which the crack was supposed to propagate
through was discretised using elements of the same size. 4-nodes quadrilateral elements were used
to mesh the whole domain. It is worth mentioning that recently some mesh adaptive methods
have been developed to speed up the computation process, particularly for unknown a priori crack
propagation paths [74].
3.2. Crack tip position analysis
A problem related to the use of a diffusive model such as PFM, for the simulation of crack
propagation, is the search for the exact crack tip location. This information is of great importance
when it comes to the reconstruction of the R-curves since the crack length is involved. One natural
way to find the crack tip would be to localise the point where the PF parameter
X
assumes
maximum magnitude before dropping down rapidly along the direction in which the crack length
is thought to evolve. Although this method turned out to be efficient in the presence of both a
single inclusion or absence of inclusions for the low magnitude of eigenstrain, it did not show the
expected robustness when dealing with multi-inclusion problems. To cope with this issue, the
author could ascertain that the analysis of the gradient of the PF parameter along the propagation
direction provides a more accurate and solid method to crack tip localisation.
To this end, a parameter
Y
is calculated based on a filtered value of the PF gradient
9:
9;
, according
to the sample and coordinate system illustrated in Figure 2. The purpose of this filtering is the
discrimination of unwanted peaks that may originate due to the presence of large tensile residual
stress. Therefore,
Y
is evaluated as:
Y012M2/4$
Z
[
\
[
]
I2 -X
-1^_-X
-1`#<
a-X
-1a2 -X
-1b_-X
-1`#<
(8)
where c
9:
9;
d
#<
is the gradient threshold that was chosen as a cut-off value to identify the peak. Once
the PF gradient field is defined, the crack length can be determined as a sum of three contributions:
1) the initial crack length
R!
; 2) a constant parameter
R=
that accounts for the shift of the PF
gradient peak position with respect to the actual crack tip, and it has been calibrated and it is
assumed equal to 0.3 mm; 3) the peak position of the integral of the
Y
parameter over a reduced
domain height
Q"
of the sample; the domain height
Q"
was chosen to make sure that it embedded
the whole crack path, even in circumstances where the crack diverted.
R0/4$R!%R=%RSefR1
;>?@"ABC@"D: Y012M2/4,M
<∗
(9)
9
3.3. R-curve calculation
For the evaluation of the R-curves, the J-integral parameter is considered. The problem itself
presented in this paper does not involve any plastic deformation, so in principle a simple SIF
evaluation would have been sufficient. Nevertheless, in order not to lose the generality of the
approach, still the J-integral was considered.
For the case study presented here, the calculation of the J-integral can be simply and conveniently
done by conversion from the SIF. In general, it is important to highlight that the J-integral
calculated directly through the integration operation [14], rather than using the SIF, might be
slightly affected by the presence of internal interfaces or discontinuities, if present. In any case,
this aspect is thought to be negligible in the present study.
Under the plane-stress condition, the relation between the SIF (K) and the J-integral is:
g$h&
N
(10)
It is very important to state that for the particular case of linear elastic material, J value is coincident
with the ERR (
7
), hence
g$7
. For the chosen geometry (SENP) and loading configuration
(displacement control), which is a widely adopted experimental design for crack propagation
studies, it is common to find accurate analytical solutions in terms of SIF. In general, the SIF can
is analytically expressed as a function of the applied stress
i
, the crack length
R
and a shape
function
j
that depends on the sample geometry and loading mode.
h$i&j&kl&R
(11)
According to the SIF functions provided by the literature and obtained by means of FEM
calculations [75], the shape function
j
is offered in the form of a 6th order polynomial function:
j$m!%m%R
P%m&cR
Pd&%m'cR
Pd'%mEcR
PdE%mFcR
PdF%mGcR
PdG
(12)
and the
m*
coefficients are reported in the table below.
Table 2 Beta coefficient for h/w=0.42, plane stress and displacement control
nH
nI
nJ
nK
nL
nM
nN
1.1200
0.2492
-21.2426
89.222
-168.3852
154.2504
-55.6114
The
j
shape function can be used provided that the applied stress is calculated as the nominal
stress due to the application of the
/
displacement, therefore invoking Young’s modulus and the
height of the sample:
i$ /
B&QN
(13)
4. Results and Discussion
This section reports essentially the outcome of three inclusion geometrical configurations and
associated sub-parametrical analysis in which the eigenstrain magnitude spans from positive to
negative values. For the sake of normalising the results and facilitating the comparison between
the inclusion models and the fully homogeneous sample
0""$I4
, the latter was computed first.
10
Following, the load at which crack nucleation occurred
o8
for the homogeneous sample was taken
as a reference and all the loading values reported in this study were scaled by it. Analogously, also
the displacement and the J-integral at the simulated critical load, respectively
/8
and
g8
, were used
to normalise the displacement and J-integral quantities. It is important to note that the simulation
for the fully homogeneous sample showed crack initiation at a critical value
g8
slightly higher than
the fracture toughness imposed in the model
78
(
pAA2Wq
higher). This divergence is possibly due
to the limited capability of the chosen EDF to capture the actual crack initiation instant.
Nevertheless, for the specific sake of this study, this is a marginal aspect.
For each inclusion geometrical configuration, the eigenstrain magnitude assumed the following
values:
""$9IEIIAW
,
""$IEIIAW
,
""$IEIIrI
, while the fully homogeneous sample was
simulated only once by imposing
""$I
.
4.1. Stress Distribution due to the Inclusion
With the purpose of gaining an understanding of the origin of fracture toughness modification
induced by the presence of RS, in Figure 3 some distribution of the crack opening stress (i.e.
&iO
)
are reported, with no external forces nor displacement applied externally to visualise the actual RS
field. An example of full-field distribution of stress within and surrounding the inclusion for
positive eigenstrain is shown in Figure 3(a), highlighting the tensile stress at the notch tip and the
compressive residual stress that the inclusion is subjected to, when a positive hydrostatic
eigenstrain magnitude is applied.
Line profiles of the notch opening residual stress component are depicted in Figure 3(b). An
additional, but intuitive, observation can be made about the effect of negative eigenstrain, which
in this case, leads to compressive RS at positions very close to the notch tip, followed by a steep
gradient as soon as the x position moves towards the inclusion, especially for the inclusion placed
at
1!
=6.
Figure 3. Stress field examples (no external loads applied). (a) Contour plot of
iO
at the inclusion
and surrounding matrix for
1!
=6 and
""$IEIIAW
. (b) Line profiles of
iO
for inclusion
configurations
1!
=6 and
1!
=20 both at
M!
=0 along the notch bisector normalised position
(x/w), for two significant levels of eigenstrain magnitude.
4.2. Crack tip inclusion (
sH
=6 mm,
tH
=0)
For this configuration, the inclusion boundary curve is placed so that it corresponds to the crack
tip position. From the Load vs. Displacement curves Figure 4(a) it can be promptly seen that
positive eigenstrain magnitudes increase the “apparent” critical load (sharp knee), up to 1.4 times
with respect to the homogeneous sample, for
""$IEIIrI
. However, it important to note that
the apparent critical load occurs only when the crack fully propagated through the inclusion; to
help understanding this, the Crack Length vs. Displacement or the J-R curve can be observed,
11
respectively in Figure 4(b) and (c). In fact, from the Figure 4(b), it is clearly visible that the crack
nucleates way earlier than the apparent critical load. Although this behaviour might seem
counterintuitive after observing the RS distribution in Figure 3, it is evident that the sign of
maximum peak at the notch tip does not play a relevant role, instead, the surrounding stress field
does. To better comprehend the fracturing behaviour and stability, the J-R curves are very useful.
In fact, Figure 4(c) shows what the actual critical ERR (
g8
), for the inclusion at
""$IEIIrI
, is
superior to that experienced by the homogeneous sample. This critical value is found by searching
for a constant force-controlled J-R curve (iso-load), from equation (11), that is tangent the J-R
curve. As it can be seen from the plot, the J-R curve and the iso-load curve intersect at a value of
around twice as big as the critical load of the homogenous sample. In other words, it means that
the introduction of an inclusion in this configuration and this eingenstrain magnitude allows for a
propagation of a crack in a stable manner monotonically increasing the external force across about
half of the inclusion diameter; at the same time fracture toughness twice as high as the
homogenous sample can be achieved. This is valid provided that the J of the propagating crack
falls under the crack arrest toughness value of the specific material. Another interesting
observation derives from the J-R curve trend after the crack goes past the inclusion domain. It is
important to stress that the J-R curves shown in this work are evaluated through a monotonically
increasing displacement load. Therefore, the energetic level reached as soon as the crack past the
inclusion domains is such high that it does not enable to capture the actual toughness of the
material. Indeed, once the crack overtakes the inclusion, the fracture toughness of the parent
material should be obtained. To observe this falling R-curve effect, periodic unloading should be
simulated once the crack reaches the instability levels. Anyway, this was not the aim of the present
study.
The Sample at
""$IEIIAW
adds further insight on the toughening mechanisms induced by the
residual stress, i.e. it shows how only above a certain level of eigenstrain it is possible to propagate
under a force-controlled loading condition. In fact, with such a small magnitude the crack does
not propagate stably under force-control.
As far as the negative eigenstrain magnitude is concerned, for this configuration no relevant
improvement of fracture toughness was observed, rather, a decrease of its performance was seen,
i.e. critical ERR
uIEv&g8
.
Figure 4 Crack propagation results for an inclusion centred at
1!
=6 mm and
M!
=0. (a)
Normalised Load vs. Displacement. (b) Normalised Crack Length vs. Displacement. (c)
Normalised J-R (shaded in grey the region occupied by the inclusion)
4.3. Crack tip inclusion (
𝒙𝟎
=20 mm,
𝒚𝟎
=0)
12
By shifting the inclusion position further away from the sharp notch tip, the effect of the residual
stress within the matrix becomes more apparent, as the crack must propagate through this region
first. Differently from the previous case just seen, the inclusion undergoing negative values of
eigenstrain experiences higher critical ERR (ERR
uAEA&g82
for
""$9&IEIIAW
). This effect is
explained by the compressive residual stress generated around the inclusion, arising from the fact
that strain compatibility gives rise to tensile residual stress within the inclusion and, as a
consequence, compressive residual stress takes place in the matrix to obey the force equilibrium.
On the other hand, as soon as the crack goes past the beneficial compressive RS field, the crack is
forced to propagate through a tensile stress field present in the inclusion, which accelerates its
growth. Nonetheless, this deduction in practice is not very useful since the crack propagation
becomes suddenly unstable under load control as soon as the critical ERR is reached.
No evident benefit is observed relative to the positive eigenstrain cases for force-controlled
propagation. For this instance, in spite of a rise of the J-R curve while the crack passes through
the inclusion, this effect does not play a role in the force-controlled scenario since the curve is
always below the iso-load curve of the critical ERR. Instead, for displacement-controlled
propagation a benefit is achieved due to the higher energetic level (J) that the crack tip has to reach
in order to continue its propagation.
Figure 5 Crack propagation results for an inclusion centred at
1!
=20 mm and
M!
=0. (a)
Normalised Load vs. Displacement. (b) Normalised Crack Length vs. Displacement. (c)
Normalised J-R (shaded in grey the region occupied by the inclusion)
4.4. Inclusion (
𝒙𝟎
=20 mm,
𝒚𝟎
=3 mm)
The last case study concerned an inclusion having centre slightly shifted relative to the
notch bisector line. This configuration aimed to understand whether the crack path could be
deflected under the effect of heterogeneous and asymmetric residual stress field. In Figure 6 are
highlighted the crack paths simulated for two different magnitudes of eigenstrain, i.e.
w"$IEIIAW
and
w"$9IEIIAW
. As it is possible to observe, the positive eigenstrain created a tensile RS field
around the inclusion that attracted the crack tip, so the crack diverted and crossed the inclusion
which was subjected to a compressive RS instead. In contrast, when negative eigenstrain was
applied, the compressive RS generated around the inclusion forced the crack to search for a more
efficient way to overcome the obstacle and make the propagation possible: diverting towards
regions that undergone less severe compressive RS.
13
Figure 6 Phase-Field contour plot close-up for a propagated crack. Colouring indicates the PF
magnitudes (red is
X$A
and blue is
X$I
). The dashed circle depicts the inclusion placed at
1!
=20 mm and
M!
=3 mm. (a)
""$IEIIAW
. (b)
""$9IEIIAW
.
Regarding the J-R curves, no relevant discrepancies were found compared to those considering an
inclusion centred with the notch bisector. Although the crack diverts under the effect of an
asymmetric RS field and the actual crack length becomes greater than that observed in the
homogeneous sample, this effect does not seem to play a relevant role, at least in this configuration
of inclusion geometry and position relative to the notch tip (see Fig. 7).
Figure 7 Crack propagation results for an inclusion centred at
1!
=20 mm and
M!
=3 mm. (a)
Normalised Load vs. Displacement. (b) Normalised Crack Length vs. Displacement. (c)
Normalised J-R (shaded in grey the region occupied by the inclusion)
4.5. General Discussion
Overall, the results showed here prove that residual stress can indeed be used as a toughness
mechanism, if appropriately designed. For instance, many works are present in the literature in
which it is well known how residual stress affects crack propagation [30], nevertheless, up to date,
no such a robust predictive method was available to account for this effect while determining
fracture resistance of components and materials.
It has been demonstrated how the presence of an inclusion in proximity of the crack tip can
theoretically affect the crack propagation mechanism. Same underlying idea applies also to blunt
notches where the crack is not present but, as soon as the crack nucleates due to the excessive
load, this region becomes suddenly a sharp notch and all the concepts seen here become valid.
Therefore, the presence of inclusions can be designed either close to the crack tip or close to the
location where the crack is supposed to nucleate.
14
It is worth highlighting that the methodology proposed here is limited to materials showing a
purely brittle behaviour, therefore a generalisation is still required.
In order to check the robustness of the model in real applications, an experimental validation will
be devised in future works. As outlined earlier, several ways exist to introduce inelastic deformation
in materials to replicate the numerical tests presented in this paper. One may be the use of
interference fits which are practically obtained by either cooling or heating one of the domains to
couple, inclusion or matrix. A second option may exploit material microstructural phase-change
due to thermal or mechanical effects. In this area several works have been proposed which attempt
to capture the associated inelastic deformation [76, 77], and thus the linked RS, which show the
feasibility of this experimental route. It is worth mentioning that a crack propagation PF model
considering martensitic phase transformation at the nano-scale was proposed lately [63], which
motivates even more the present study and highlights the importance of a homogenisation
procedure to account for this effect at the macro-scale. Further experimental approaches may
involve some versatile and advanced manufacturing processes to introduce RS, for instance
additive manufacturing [78], chemical vapour deposition [79] and many others.
An important aspect that was noticed in this study was that the damage zone width identifying the
crack sub-domain depends on the crack propagation rate, as visible in the example reported in
Figure 6. As a matter of the fact, the damage width within the inclusion in Figure 6(a) for instance
is smaller than that outside it, exactly at the zone where a reduced crack propagation rate was
experienced for that case-study. Conversely, outside the inclusion the crack showed a pronounced
tendency to propagate unstably due to its high crack propagation rate, and actually the damage
zone width was larger. So eventually, it seems to exist a causal link between the crack propagation
rate and the width of the damage zone which is certainly of interest and worth further investigation.
5. Conclusions
The combination of the Phase-Field method and the eigenstrain theory allowed for a more
advanced revisitation of the residual stress influence on the fracture toughness of materials at the
continuum mechanics scale, for a brittle material originally showing a flat R-curve. The presented
approach paves a new way to study material fracturing behavior under the effect of pre-existing
RS fields. Although the outcome of this study is restricted to a well-defined class of materials
(brittle) and loading manner (quasi-static monotonic), the mechanisms affecting crack propagation
are believed to be predominant also for materials showing some degrees of ductility and different
loading manners, especially for high-cycle crack fatigue propagation where the effect of plasticity
can be neglected.
By means of the eigenstrain concept, inelastic deformation was prescribed within a circular domain
to generate RS fields of different magnitudes and signs. The R-curves analysis enabled the
determination of crack propagation stability under both displacement and force controls. Thanks
to this, several scenarios were simulated and some of these configurations were seen to actually
benefit the overall fracture toughness of the examined sample. In fact, it was experienced that
when the inclusion boundary is coincident to the notch tip in one point – under the effect of
positive eigenstrain, it is possible to produce a “rising” R-curve. This means that a stable quasi-
static crack propagation even under force control is theoretically achievable. It is therefore possible
to conclude that the RS, under specific circumstances, is capable of originating fracture toughness
even in intrinsically brittle materials.
The challenge, especially for brittle materials, is to find ways to introduce RS in real applications
given the very limited possibility to plastically deform the material. Nevertheless, plastic
deformation is not the sole source of RS, in fact different routes are possible, e.g. interference fits,
thermal expansion mismatch during cooling, microstructure phase-change etc. These examples
may be put into practice in future experimental analyses to verify what degree of fracture toughness
enhancement is realistically achievable through this very promising approach.
Aside from the key results, the proposed simulation framework will have a significant impact on
the way RS can be effectively designed in order to maximise its beneficial effect on the reliability
of engineering structures and materials, at different scales and in multiphysics problems.
15
Acknowledgments
Enrico Salvati would like to express his gratitude to Ivan Moro who conducted preliminary PF
analyses during his master’s thesis in Mechanical Engineering at the University of Udine. Further
acknowledgments are due to Prof. Francesco De Bona for his support in providing the necessary
computational tools to conduct this research. Dr. Roberto Alessi is also acknowledged for the
fruitful discussions we had on this topic.
Appendix
Phase-Field representation
A scalar parameter (Phase-Field)
x01234&y&zI2A{
can be introduced in equations (3) and (4) to help
representing the fracture surface
.
in a diffusive manner, i.e.
.)
. Therefore, the surface energy
associated to the fracture of the bulk
8,
becomes:
8,|: 78&}0X2~X4&,)
1
(14)
where
}0X2~X4
is the crack surface density functional, which is expressed as a function of the
Phase Field
x
and its gradient
~X
. The generic form of the
}
can be written as [80]:
}0X2~X4$A
U!•A
!!€0X4%!!•~X•&‚
(15)
in which
€0X4
is called geometric crack function and
U!
is a scaling parameter necessary to ensure
that (14) is valid, i.e. the energy of the sharp crack remains the same as that of the diffuse crack.
There exist a number of geometric crack functions (GCF)
€0X4
adopted in the literature, for
instance:
€$X
[81],
€$X&
[50] ,
€$A9
0
A9x
4
&
[54],
€$ƒx%
0
A9ƒ
4
x&
[80].
In the present work, the chosen geometric crack function is
€$X&
, which has a scaling parameter
U!
=2. Therefore, for this geometric crack function, the crack density functional is:
}0X2~X4$X&
B&!!%!!•~X•&
(16)
It is useful to note that, if one wonder what the shape of the phase field is across the crack is, that
the functional (16) upon its minimisation gives the following Ordinary Differential Equation:
}X014
!!9&!!-&X
-ƒ&$I
(17)
where
ƒ
is a linear coordinate normal to the crack path, as depicted in Figure 1, lying on the plane
and having origin at the actual crack jump in the discrete representation. Therefore, the distribution
of
X
across the crack can be found upon solution of this equation provided that the following
boundary conditions are applied:
X$A
at
&ƒ$I
and
X„I
for
ƒ„…
. The solution turns out
to be:
X0ƒ4$†CPQP
0"
(18)
A plot showing its profile is reported in Figure 1(b). Since the crack evolution is driven by
the elastic energy that is partially converted into crack surface energy, it is important to separate
16
the contributions given from the tensile and compressive parts. This aspect is necessary in order
to attribute the correct weight to the tensile part in contributing to crack propagation with respect
to the compressive counterpart. It is also worth highlighting that this decomposition problem is
still not solved completely, and different approaches can be found in the literature. In the present
work, an anisotropic model proposed by Miehe et al. [71] was employed, which completely
suppresses the propagation/creation of cracks under compressive loads. In order to compute the
decomposition, it is found useful to start by performing the spectral decomposition of the elastic
strain tensor into a compressive and tensile part, respectively
"C
and
"R
:
Z
[
\
[
]
"R$‡ˆ"@‰R&Š@‹Š@
(
@S%
"C$‡ˆ"@‰C&Š@‹Š@
(
@S%
(19)
Where
"@
is the principal elastic strain and
Š@
is the direction vector. Also, ˆ
Œ
‰
R
and ˆ
Œ
‰
C
are
respectively defined as [51]:
&
0
Œ%
•
Œ
•4
VB
and 0
Œ
•
Œ
•4
VB
. Form these two tensors, the two contributions
of elastic strain energy density can be promptly obtained:
•8.
R0"4$CBˆ3Sˆ"‰‰R
&%D&3S&ˆ"R
&‰
8.
C0"4$CBˆ3Sˆ"‰‰C
&%D&3S&ˆ"C
&‰
(20)
The PF and the structural problems are coupled thanks to the Energetic Degradation Function
(EDF)
e0X4
that links the elastic energy
8.
to the PF as follows:
8.0"4$z0A9Ž4e0X4%Ž{8.
R0"4%8.
C0"4
(21)
where
Ž
is a positive small coefficient to ensure numerical stability.
In this work, the EDF used is of quadratic type,
e0X4$
0
A9X
4
&
as proposed by Bourdin et al.
[50]. However, several other formulations have been proposed and used in other studies.
Equations of the Problem
The problem consists of solving the differential equations relative to the linear elastic problem
coupled with that of the PF. As far as the PF problem is concerned it is possible to write the
Lagrangian energy functional as the total energy (4) but with the introduction of the PF
approximation:
•$9: z0A9Ž40A9X4&%Ž{8.
R0"4%8.
C0"4&,)
19: 7-&=X&
B&!!%!!•~X•&@&,<%
2#: 5>/&,)
1%: 6>/&,?
31
(22)
In order to make sure the irreversibility condition is enforced, and therefore a monotonically
increasing PF, the positive strain energy density
8.
R0"4
is replaced by
•$‘’“
T>?!A#D 8.
R0"012K44
By
imposing that the variation of the functional
•
must be null, the following governing equations are
derived:
17
Z
[
\
[
]
-i*4
-14%5*$I
=B&!!&0A9Ž4•
7-%A@X9!!
&-&X
-1*
&$B&!!0A9Ž4•
7-
(23)
where
i*4
are the components of the stress Cauchy stress tensor. This tensor can also be rewritten
as a function of the elastic strain energy density as:
i$z0A9Ž40A9X4&%Ž{zCˆ3Sˆ"‰‰R”%BD&"R{%Cˆ3Sˆ"‰‰C”%BD"C
(24)
where
”
is the identity matrix.
Last but not least, the boundary conditions need to be applied to solve the system of differential
equations. In this case, for instance, the boundary conditions can be the external forces applied to
the model for the mechanical problems, and Neumann conditions for the PF:
•i*4 f4$6*
-X
-14f4$I
(25)
In which
f4
is the outward-pointing normal vector on the boundary. It is also worth mentioning
that initial cracks (pre-cracking) can be modeled directly by applying the unitary value of the PF as
X012I4$A
on the domain.
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