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1

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.

References

[1] R.O. Ritchie, The conflicts between strength and toughness, Nature Materials 10(11) (2011)

817-822.

[2] C. Makabe, K. Naka, M.S. Ferdous, Method of arresting crack growth for application at a

narrow working space, Mechanical Engineering Journal 1(6) (2014) SMM0058-SMM0058.

[3] T. Marazani, D.M. Madyira, E.T. Akinlabi, Repair of Cracks in Metals: A Review, Procedia

Manufacturing 8 (2017) 673-679.

[4] M.R. Ayatollahi, S.M.J. Razavi, H.R. Chamani, Fatigue Life Extension by Crack Repair Using

Stop-hole Technique under Pure Mode-I and Pure mode-II Loading Conditions, Procedia

Engineering 74 (2014) 18-21.

[5] S. Shkarayev, Theoretical modeling of crack arrest by inserting interference fit fasteners,

International Journal of Fatigue 25(4) (2003) 317-324.

[6] D.-C. Seo, J.-J. Lee, Fatigue crack growth behavior of cracked aluminum plate repaired with

composite patch, Composite Structures 57(1) (2002) 323-330.

[7] B. Graf, A. Gumenyuk, M. Rethmeier, Laser Metal Deposition as Repair Technology for

Stainless Steel and Titanium Alloys, Physics Procedia 39 (2012) 376-381.

[8] D. Stefanescu, Measurement and prediction of fatigue crack growth from cold expanded holes

Part 1: The effect of fatigue crack growth on cold expansion residual stresses, Journal of Strain

Analysis for Engineering Design 39(1) (2004) 25-40.

[9] C.S. Shin, Z.Z. Chen, Fatigue life extension by electroless nickel infiltration plating,

International Journal of Fatigue 23(9) (2001) 777-788.

[10] E. Salvati, S. O'Connor, T. Sui, D. Nowell, A.M. Korsunsky, A study of overload effect on

fatigue crack propagation using EBSD, FIB–DIC and FEM methods, Engineering Fracture

Mechanics (2016).

18

[11] E. Salvati, T. Sui, H. Zhang, A.J.G. Lunt, K.S. Fong, X. Song, A.M. Korsunsky, Elucidating

the Mechanism of Fatigue Crack Acceleration Following the Occurrence of an Underload,

Advanced Engineering Materials 18(12) (2016) 2076-2087.

[12] E. Salvati, H. Zhang, K.S. Fong, X. Song, A.M. Korsunsky, Separating plasticity-induced

closure and residual stress contributions to fatigue crack retardation following an overload, Journal

of the Mechanics and Physics of Solids 98 (2017) 222-235.

[13] W. Xu, J.B. Wintle, C.S. Wiesner, D.G. Turner, Analysis of crack arrest event in NESC-1

spinning cylinder experiment, International Journal of Pressure Vessels and Piping 79(11) (2002)

777-787.

[14] T. Anderson, Fracture mechanics: fundamentals and applications, CRC press2017.

[15] Y. Charles, F. Hild, On crack arrest in ceramic / metal assemblies, International Journal of

Fracture 115(3) (2002) 251-272.

[16] Y. Charles, F.o. Hild, S.p. Roux, Long-Term Reliability of Brittle Materials: The Issue of Crack

Arrest, Journal of Engineering Materials and Technology 125(3) (2003) 333-340.

[17] T. Ohji, Y.K. Jeong, Y.H. Choa, K. Niihara, Strengthening and toughening mechanisms of

ceramic nanocomposites, J Am Ceram Soc 81(6) (1998) 1453-1460.

[18] G.T. Hahn, A.R. Rosenfield, Metallurgical factors affecting fracture toughness of aluminum

alloys, Metallurgical Transactions A 6(3) (1975) 653-668.

[19] D.R. Bloyer, K.T. Venkateswara Rao, R.O. Ritchie, Resistance-curve toughening in

ductile/brittle layered structures: Behavior in Nb/Nb3Al laminates, Materials Science and

Engineering: A 216(1) (1996) 80-90.

[20] R.O. Ritchie, Mechanisms of fatigue-crack propagation in ductile and brittle solids,

International Journal of Fracture 100(1) (1999) 55-83.

[21] J.D. Eshelby, The Determination of the Elastic Field of an Ellipsoidal Inclusion, and Related

Problems, Proceedings of the Royal Society of London. Series A. Mathematical and Physical

Sciences 241(1226) (1957) 376-396.

[22] A.M. Korsunsky, A Teaching Essay on Residual Stresses and Eigenstrains, Elsevier Inc.2017.

[23] T. Mura, Micromechanics of Defects in Solids, (1987).

[24] E. Salvati, A.M. Korsunsky, A Simplified FEM Eigenstrain Residual Stress Reconstruction for

Surface Treatments in Arbitrary 3D Geometries, International Journal of Mechanical Sciences

(2018).

[25] M. Achintha, D. Nowell, Eigenstrain modellingof residual stresses generated by arraysof LSP

shots, Procedia Engineering 10 (2011) 1327-1332.

[26] E. Salvati, A.J.G. Lunt, S. Ying, T. Sui, H.J. Zhang, C. Heason, G. Baxter, A. Korsunsky,

Eigenstrain Reconstruction of Residual Strains in an Additively Manufactured and Shot Peened

Nickel Superalloy Compressor Blade, Computer Methods in Applied Mechanics and Engineering

(2017).

[27] M. Benedetti, F. Berto, M. Marini, S. Raghavendra, V. Fontanari, Incorporating residual

stresses into a Strain-Energy-Density based fatigue criterion and its application to the assessment

of the medium-to-very-high-cycle fatigue strength of shot-peened parts, International Journal of

Fatigue 139 (2020) 105728.

[28] P. Lipetzky, S. Schmauder, Crack-particle interaction in two-phase composites Part I: Particle

shape effects, International Journal of Fracture 65(4) (1994) 345-358.

[29] D.J. Green, R. Tandon, V.M. Sglavo, Crack Arrest and Multiple Cracking in Glass Through

the Use of Designed Residual Stress Profiles, Science 283(5406) (1999) 1295.

[30] M. Meindlhumer, L.R. Brandt, J. Zalesak, M. Rosenthal, H. Hruby, J. Kopecek, E. Salvati, C.

Mitterer, R. Daniel, J. Todt, J. Keckes, A.M. Korsunsky, Evolution of stress fields during crack

growth and arrest in a brittle-ductile CrN-Cr clamped-cantilever analysed by X-ray nanodiffraction

and modelling, Materials & Design 198 (2021) 109365.

[31] Z. Li, Q. Chen, Crack-inclusion interaction for mode I crack analyzed by Eshelby equivalent

inclusion method, International Journal of Fracture 118(1) (2002) 29-40.

19

[32] C. Atkinson, The interaction between a crack and an inclusion, International Journal of

Engineering Science 10(2) (1972) 127-136.

[33] J. Helsing, Stress intensity factors for a crack in front of an inclusion, Engineering Fracture

Mechanics 64(2) (1999) 245-253.

[34] J.S. Novak, D. Benasciutti, F. De Bona, A. Stanojević, P. Huter, Thermo-Mechanical Finite

Element Simulation and Fatigue Life Assessment of a Copper Mould for Continuous Casting of

Steel, Procedia Engineering 133 (2015) 688-697.

[35] P.C. Savalia, H.V. Tippur, A Study of Crack–inclusion Interactions and Matrix–inclusion

Debonding Using Moiré Interferometry and Finite Element Method, Experimental Mechanics

47(4) (2007) 533-547.

[36] A. Materna, V. Oliva, Elastic-plastic FEM investigation of the thickness effect on fatigue crack

growth, Procedia Engineering 10 (2011) 1109-1114.

[37] P.O. Bouchard, F. Bay, Y. Chastel, Numerical modelling of crack propagation: automatic

remeshing and comparison of different criteria, Computer Methods in Applied Mechanics and

Engineering 192(35) (2003) 3887-3908.

[38] T.P. Fries, T. Belytschko, The extended/generalized finite element method: An overview of

the method and its applications, International Journal for Numerical Methods in Engineering 84(3)

(2010) 253-304.

[39] Q.Z. Xiao, B.L. Karihaloo, Improving the accuracy of XFEM crack tip fields using higher

order quadrature and statically admissible stress recovery, International Journal for Numerical

Methods in Engineering 66(9) (2006) 1378-1410.

[40] S. Kumar, I.V. Singh, B.K. Mishra, XFEM simulation of stable crack growth using J-R curve

under finite strain plasticity, Int. J. Mech. Mater. Des. 10(2) (2014) 165-177.

[41] E. Benvenuti, A regularized XFEM framework for embedded cohesive interfaces, Computer

Methods in Applied Mechanics and Engineering 197(49-50) (2008) 4367-4378.

[42] D.S. Dugdale, Yielding of steel sheets containing slits, Journal of the Mechanics and Physics

of Solids 8(2) (1960) 100-104.

[43] M. Kuna, S. Roth, General remarks on cyclic cohesive zone models, International Journal of

Fracture 196(1) (2015) 147-167.

[44] Z.J. Yang, A.J. Deeks, Fully-automatic modelling of cohesive crack growth using a finite

element–scaled boundary finite element coupled method, Engineering Fracture Mechanics 74(16)

(2007) 2547-2573.

[45] G. Fang, S. Liu, M. Fu, B. Wang, Z. Wu, J. Liang, A method to couple state-based

peridynamics and finite element method for crack propagation problem, Mechanics Research

Communications 95 (2019) 89-95.

[46] G.A. Francfort, J.J. Marigo, Revisiting brittle fracture as an energy minimization problem,

Journal of the Mechanics and Physics of Solids 46(8) (1998) 1319-1342.

[47] J.-Y. Wu, V.P. Nguyen, C.T. Nguyen, D. Sutula, S. Sinaie, S.P.A. Bordas, Chapter One - Phase-

field modeling of fracture, in: S.P.A. Bordas, D.S. Balint (Eds.), Advances in Applied Mechanics,

Elsevier2020, pp. 1-183.

[48] D. Sutula, P. Kerfriden, T. van Dam, S.P.A. Bordas, Minimum energy multiple crack

propagation. Part I: Theory and state of the art review, Engineering Fracture Mechanics 191 (2018)

205-224.

[49] A.A. Griffith, The Phenomena of Rupture and Flow in Solids, Philosophical Transactions of

the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character

221 (1921) 163-198.

[50] B. Bourdin, G.A. Francfort, J.J. Marigo, Numerical experiments in revisited brittle fracture,

Journal of the Mechanics and Physics of Solids 48(4) (2000) 797-826.

[51] C. Miehe, M. Hofacker, F. Welschinger, A phase field model for rate-independent crack

propagation: Robust algorithmic implementation based on operator splits, Computer Methods in

Applied Mechanics and Engineering 199(45) (2010) 2765-2778.

20

[52] V.I. Levitas, H. Jafarzadeh, G.H. Farrahi, M. Javanbakht, Thermodynamically consistent and

scale-dependent phase field approach for crack propagation allowing for surface stresses,

International Journal of Plasticity 111 (2018) 1-35.

[53] F.P. Duda, A. Ciarbonetti, P.J. Sánchez, A.E. Huespe, A phase-field/gradient damage model

for brittle fracture in elastic–plastic solids, International Journal of Plasticity 65 (2015) 269-296.

[54] R. Alessi, J.-J. Marigo, S. Vidoli, Gradient damage models coupled with plasticity: Variational

formulation and main properties, Mechanics of Materials 80 (2015) 351-367.

[55] Hirshikesh, S. Natarajan, R.K. Annabattula, E. Martínez-Pañeda, Phase field modelling of

crack propagation in functionally graded materials, Composites Part B: Engineering 169 (2019)

239-248.

[56] M. Paggi, J. Reinoso, Revisiting the problem of a crack impinging on an interface:A modeling

framework for the interaction between the phase field approach for brittle fracture and the

interface cohesive zone model, Computer Methods in Applied Mechanics and Engineering 321

(2017) 145-172.

[57] M.J. Borden, C.V. Verhoosel, M.A. Scott, T.J.R. Hughes, C.M. Landis, A phase-field

description of dynamic brittle fracture, Computer Methods in Applied Mechanics and Engineering

217-220 (2012) 77-95.

[58] R. Alessi, F. Freddi, Phase-field modelling of failure in hybrid laminates, Composite Structures

181 (2017) 9-25.

[59] P. Carrara, M. Ambati, R. Alessi, L. De Lorenzis, A framework to model the fatigue behavior

of brittle materials based on a variational phase-field approach, Computer Methods in Applied

Mechanics and Engineering 361 (2020) 112731.

[60] E. Martínez-Pañeda, A. Golahmar, C.F. Niordson, A phase field formulation for hydrogen

assisted cracking, Computer Methods in Applied Mechanics and Engineering 342 (2018) 742-761.

[61] T.T. Nguyen, J. Yvonnet, Q.Z. Zhu, M. Bornert, C. Chateau, A phase-field method for

computational modeling of interfacial damage interacting with crack propagation in realistic

microstructures obtained by microtomography, Computer Methods in Applied Mechanics and

Engineering 312 (2016) 567-595.

[62] S. Zhou, X. Zhuang, Phase field modeling of hydraulic fracture propagation in transversely

isotropic poroelastic media, Acta Geotechnica 15(9) (2020) 2599-2618.

[63] H. Jafarzadeh, V.I. Levitas, G.H. Farrahi, M. Javanbakht, Phase field approach for nanoscale

interactions between crack propagation and phase transformation, Nanoscale 11(46) (2019) 22243-

22247.

[64] J.-Y. Wu, Y. Huang, H. Zhou, V.P. Nguyen, Three-dimensional phase-field modeling of mode

I + II/III failure in solids, Computer Methods in Applied Mechanics and Engineering 373 (2021)

113537.

[65] F. Uzun, A.M. Korsunsky, On the identification of eigenstrain sources of welding residual

stress in bead-on-plate inconel 740H specimens, International Journal of Mechanical Sciences 145

(2018) 231-245.

[66] T.S. Jun, A.M. Venter, A.M. Korsunsky, Inverse Eigenstrain Analysis of the Effect of Non-

uniform Sample Shape on the Residual Stress Due to Shot Peening, Experimental Mechanics 51(2)

(2011) 165-174.

[67] A.M. Korsunsky, T. Sui, E. Salvati, E.P. George, M. Sebastiani, Experimental and modelling

characterisation of residual stresses in cylindrical samples of rapidly cooled bulk metallic glass,

Materials and Design 104 (2016) 235-241.

[68] L. Ma, A.M. Korsunsky, The principle of equivalent eigenstrain for inhomogeneous inclusion

problems, International Journal of Solids and Structures 51(25–26) (2014) 4477-4484.

[69] A.M. Korsunsky, Eigenstrain analysis of residual strains and stresses, Journal of Strain Analysis

for Engineering Design 44(1) (2009) 29-43.

21

[70] F. Uzun, J. Everaerts, L.R. Brandt, M. Kartal, E. Salvati, A.M. Korsunsky, The inclusion of

short-transverse displacements in the eigenstrain reconstruction of residual stress and distortion

in in740h weldments, J. Manuf. Processes 36 (2018) 601-612.

[71] C. Miehe, F. Welschinger, M. Hofacker, Thermodynamically consistent phase-field models of

fracture: Variational principles and multi-field FE implementations, International Journal for

Numerical Methods in Engineering 83(10) (2010) 1273-1311.

[72] S. Zhou, T. Rabczuk, X. Zhuang, Phase field modeling of quasi-static and dynamic crack

propagation: COMSOL implementation and case studies, Advances in Engineering Software 122

(2018) 31-49.

[73] S. Zhou, X. Zhuang, T. Rabczuk, A phase-field modeling approach of fracture propagation

in poroelastic media, Engineering Geology 240 (2018) 189-203.

[74] Hirshikesh, E. Martínez-Pañeda, S. Natarajan, Adaptive phase field modelling of crack

propagation in orthotropic functionally graded materials, Defence Technology 17(1) (2021) 185-

195.

[75] R. Evans, A. Clarke, R. Gravina, M. Heller, R. Stewart, Improved stress intensity factors for

selected configurations in cracked plates, Engineering Fracture Mechanics 127 (2014) 296-312.

[76] S.J. Turneaure, Y.M. Gupta, Inelastic deformation and phase transformation of shock

compressed silicon single crystals, Applied Physics Letters 91(20) (2007).

[77] V.I. Levitas, A.M. Roy, Multiphase phase field theory for temperature- and stress-induced

phase transformations, Physical Review B 91(17) (2015) 174109.

[78] E. Salvati, A.J.G. Lunt, C.P. Heason, G.J. Baxter, A.M. Korsunsky, An analysis of fatigue

failure mechanisms in an additively manufactured and shot peened IN 718 nickel superalloy,

Materials & Design 191 (2020) 108605.

[79] M. Meindlhumer, N. Jäger, S. Spor, M. Rosenthal, J.F. Keckes, H. Hruby, C. Mitterer, R.

Daniel, J. Keckes, J. Todt, Nanoscale residual stress and microstructure gradients across the cutting

edge area of a TiN coating on WCCo, Scripta Materialia 182 (2020) 11-15.

[80] J.-Y. Wu, A unified phase-field theory for the mechanics of damage and quasi-brittle failure,

Journal of the Mechanics and Physics of Solids 103 (2017) 72-99.

[81] K. Pham, H. Amor, J.-J. Marigo, C. Maurini, Gradient Damage Models and Their Use to

Approximate Brittle Fracture, International Journal of Damage Mechanics 20(4) (2010) 618-652.