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Meccanica manuscript No.

(will be inserted by the editor)

On failure mode transition: A phase ﬁeld assisted

non-equilibrium thermodynamics model for ductile

and brittle fracture at ﬁnite strain

Ananya Bijaya ·

Shubhankar Roy Chowdhury

Received: date / Accepted: date

Abstract A non-equilibrium thermodynamics model of viscoplasticity coupled

with damage is presented. Keeping in view the experimentally observed failure

mode transitions, e.g. brittle to ductile, under dynamic loading condition, the em-

phasis of the present work has been to endow the formulation with capability of

modelling such transitions in a physically consistent manner. Within a framework

of internal variables, the current formulation tracks the eﬀect of isotropic viscoplas-

ticity with accumulated plastic strain, and a scalar phase ﬁeld variable traces the

degradation of the material caused by either tensile cracking or shear induced fail-

ure. An explicit-implicit strategy is adopted for numerical implementation of the

nonlinear formulation. A series of numerical simulations has been carried out on

notched metallic specimen to demonstrate the predictive ability of the proposed

model and to investigate the inﬂuence of several parameters in governing failure

mode transition.

Keywords Brittle and ductile damage ·Phase ﬁeld model ·Non-equilibrium

thermodynamics ·Failure mode transition

1 Introduction

A myriad of industrial applications of metals calls for developing predictive compu-

tational models to investigate response of metallic components under wide range

of loading rate and temperature. During processes like metal forming, machin-

ing, high speed impact, metals typically undergo large viscoplastic deformation

coupled with degradation caused by micro-crack and micro-void growth and co-

alescence, which ultimately culminates in macroscopic fracture. Over the years

several phenomenological and physics based models have been developed to pre-

dict viscoplastic and damage response of metals. Some prominent examples of vis-

coplasticity modelling could be Johnson and Cook (1983), Zerilli and Armstrong

Shubhankar Roy Chowdhury

Indian Institute of Technology Roorkee, India 247667

Tel.: +91-1332-284912

E-mail: shuvorc@gmail.com; shubhankarcfce@iitr.ac.in

2 Ananya Bijaya, Shubhankar Roy Chowdhury

(1987), Follansbee and Kocks (1988), Estrin (1996), Voyiadjis and Abed (2005),

Gao and Zhang (2012) and that of ductile damage Gurson et al. (1977), Tver-

gaard (1981), Tvergaard and Needleman (1984), Nahshon and Hutchinson (2008),

Xue (2007), Xue and Wierzbicki (2008). These models either directly incorporate

data observed in macroscopic experiments in form of hardening law and damage

surface or establish the formulation based on micro-mechanics, e.g. in term of

dislocation density, micro-crack, void density evolution and interaction. Besides,

in an attempt to incorporate the statistical mechanics aspects of micro-defects

in the thermodynamic formulation, in recent years, a series of models have been

developed in a two-temperature thermodynamics framework. Introducing an inde-

pendent temperature, namely the eﬀective temperature, deviation of which from

the usual kinetic vibrational temperature denotes the extent of non-equilibrium,

these models reﬁne the thermodynamics setup for non-equilibrium process that

metallic system undergoes during deformation. Among these models, Langer et al.

(2010), Roy Chowdhury et al. (2016, 2017a,b) and Roy Chowdhury et al. (2018)

may be noted for viscoplasticity of face centered and body centered cubic metals

and Roy Chowdhury and Roy (2019) and Kar et al. (2020) for ductile damage.

While material degradation happens either through micro-cracking, a phenom-

ena associated with release of elastic strain energy, or shear dominated ductile

damage at localized zones of large plastic deformation, it is the loading rate that

critically inﬂuences the mode of damage at a point in continuum. Often at quasi-

static to moderate rate of loading and under low triaxiality, metals show ductile

damage following predominant plastic deformation caused by mobility of dislo-

cations in polycrystalline metals. However at much higher rate of loading, where

sometime viscous drag dominates in dislocation motion, ﬂow stress may be found

to increase abruptly (Roy Chowdhury et al., 2017a). With limited mobility of

dislocations, rather than plastic deformation metal displays more elastic-like be-

haviour and upon storage of enough strain energy brittle cracks form to release it.

Such transitions are typically known as ductile-to-brittle transition. Rather than

increased rate of loading, lowering of temperature may also lead to this transi-

tion. In a series of impact experiments conducted on notched (both single and

double notched) plates, Kalthoﬀ and Winkler (1988), Kalthoﬀ (2000) and Zhou

et al. (1996b) have observed in a sense a reverse transition than the former, i.e. a

brittle-to-ductile one, when the impact velocity in creased from low to higher val-

ues crossing a critical velocity. In these experiments brittle crack emanates from

the crack tip and a mode-I type (tensile) propagation is observed when the impact

velocity is low. At higher velocity, the pre-notch tip undergoes considerable plastic

deformation and a shear band forms from the tip of notch. At even higher velocity,

material within the shear band region undergoes collapse of strength triggered by

shear dominated ductile damage mechanism and subsequently culminates in duc-

tile fracture propagation in mode-II. In some of these experiments it has also been

observed that depending on impact condition, constitutive property or notch-tip

bluntness, propagating shear band and ductile fracture may also switch to brittle

fracture and alter its direction of propagation.

Since the experimental observation of this dynamic failure mode transition,

several models have been proposed to mimic the experimental ﬁndings and to

understand the competing processes of brittle and ductile damage occurring near

the notch tip. Within the framework of plasticity of porous metals Needleman

and Tvergaard (1995) probed the transition behaviour through enhanced thermal

On failure mode transition 3

softening due to adiabatic heating caused by plastic dissipation. Batra and Lear

(2004) too modelled the constitutive behaviour in a similar manner and simu-

lated the propagation of crack using a node release technique. There are a set

of attempts, for example Zhou et al. (1996a), Zhou et al. (1998) and Li et al.

(2002), where besides usual metal plasticity model, the region inside shear band,

upon damage, is treated as Newtonian ﬂuid in order to replicate the collapse of

shear load carrying capability and model highly mobile plastic ﬂow. While these

models are numerically implemented through Galerkin weakform based ﬁnite el-

ement or mesh-free methods, Liu et al. (2018) used correspondence framework

of non-ordinary state based Peridynamics to model using the similar solid-ﬂuid

constitutive behavior. It is only the brittle fracture, not the ductile crack, that is

modelled with bond breakage, and the constitutive model for material inside shear

band undergoing ductile failure is assigned to ﬂuid ﬂow model. Based on extended

ﬁnite element technique Song et al. (2006) simulated the experiment using super-

posed elements and phantom nodes presenting artiﬁcial shear banding rather than

the physical one.

Whether it is the node release technique or extended ﬁnite element method,

modelling crack induced discontinuity requires special treatment as the govern-

ing partial diﬀerential equations (PDEs) no longer remain well deﬁned at the

surface of discontinuity. This is because no valid diﬀeomorphism map exists to

represent the crack-resulting deformation. While Peridynamics, being based on

integral equations rather than PDEs, does not suﬀer from this limitation, design

for bond breakage criterion is still required. Modelling fracture through phase

ﬁeld method (Francfort and Marigo, 1998; Bourdin et al., 2000; Miehe et al., 2010,

2015a; Freddi and Royer-Carfagni, 2014, 2016; Alessi et al., 2018) oﬀers an elegant

solution to this issue. Cracked and uncracked phases in the material are delin-

eated through an internal state variable known as phase ﬁeld. Using phase ﬁeld

the strong discontinuity of crack is represented in a diﬀused manner. Coupled with

gradient enhanced regularized continuum damage modelling, phase ﬁeld method

enables prediction of emergence and propagation of cracks without any additional

enhancement in the ﬁnite dimensional numerical implementation. For example,

Freddi and Royer-Carfagni (2014, 2016); Alessi et al. (2018) arrived at phase ﬁeld

framework for modeling fracture based on minimization of energy functional and

slip-line theory of plasticity. The energy functional consists of three competing

terms– elastic strain energy, activation energy required for initiating fracture and

energy expended by plastic work derived using classical slip-line theory. While

Freddi and Royer-Carfagni (2014, 2016) modeled damage in perfectly plastic ma-

terial, with the latter considering spherical and deviatoric split of elastic strain

energy, (Alessi et al., 2018) extended their approach to model brittle, cohesive and

ductile fractures. The present formulation shares some of the key features of these

models. For example, use of activation or threshold energy for initiation fracture,

and splitting of elastic strain energy to distinguish the eﬀect of local compression

and tension in cracking. However, instead of minimization of energy functional,

our framework exploits thermodynamic principles to derive governing equations.

Moreover, to capture fracture mode transition, in the present formulation the ac-

tivation energy of fracture is assumed to be dependent on triaxiality and plastic

strain rate.

Among several articles which are concerned with phase ﬁeld modelling of brit-

tle, ductile or cohesive fracture, there are only a few, namely Miehe et al. (2015a),

4 Ananya Bijaya, Shubhankar Roy Chowdhury

McAuliﬀe and Waisman (2016) and Chu et al. (2019), that consider modelling

fracture mode transition. Miehe et al. (2015a) have presented a strategy where the

fracture mode transition is attained by considering plastic strain-rate dependent

transition of critical energy release rate from its value at mode-I to mode-II. How-

ever, their model utilizes an artiﬁcial penalty energy to drive ductile fracture and

also degradation of ﬂow stress due to damage is never considered. McAuliﬀe and

Waisman (2016) demonstrate brittle fracture at lower velocity and shear banding

at higher values, but their model does not show shear banding leading to ductile

tearing phenomena. Chu et al. (2019) have presented a uniﬁed phase ﬁeld formu-

lation wherein the transition is modelled by considering the critical energy release

rate to depend on stress triaxiality. While their model could show brittle fracture

under low velocity impact, downward curving path of shear band and subsequent

ductile fracture at higher impact velocity could not be adequately predicted. Be-

sides, thermodynamics consistency of their model is also not discussed.

Addressing the shortcomings of the models discussed above, we present a ther-

modynamically consistent formulation for brittle and ductile fracture with capa-

bility of tracking failure mode transition phenomena. Acknowledging the fact that

critical energy release rate for mode-II fracture is higher than that of mode-I frac-

ture, motivated by Miehe et al. (2015a) and Chu et al. (2019) we introduce a

provision for its dependence on plastic strain rate as well as stress triaxiality. In

modelling damage driving, we introduce a threshold value of energy in such a way

that no damage initiates if the available value of driving is less than this thresh-

old. While deciding the plastic driving too another threshold work is used. For the

regime where accumulated plastic work is yet to reach this value, plasticity does

not drive damage accumulation. Thus during this phase, strength degradation in-

side the shear band region is caused by thermal softening alone. Upon exceeding

that critical threshold, phase ﬁeld starts degrading the strength of the material

and stress collapse gets accelerated within the shear band.

Constitutive closure for mechanics of solids are often sought to be thermo-

dynamically consistent. In order to achieve so, in most cases (e.g. Gurtin et al.,

2010; Fr´emond and Shitikova, 2002; M¨uller and Ruggeri, 2013), a smooth free en-

ergy function dependent on several thermodynamic state variables is introduced.

Upon applying ﬁrst and second laws of thermodynamics, the constitutive rela-

tions are then derived. Speciﬁcally, the non-dissipative forces are found to be the

derivative of the free energy with respect to conjugate state variables and the dis-

sipative forces are presented as history dependent evolution equations. Following

our previous eﬀort (Roy Chowdhury and Roy, 2019), in the present formulation

too we use the speciﬁc internal energy and entropy as basic thermodynamic func-

tions instead of the free energy density. We propose the explicit forms of both

these functions in terms of kinematic and other state variables. However, these

functions are not independent, rather linked with each other through certain ther-

modynamic constraints posed by ﬁrst and second laws. Similar to Fr´emond and

Shitikova (2002) we have described constitutive laws for non-dissipative forces (due

to elastic distortion), and dissipative forces (due to plastic deformation and dam-

age). Non-dissipative forces follow from the internal energy and entropy functions

and our proposition of constitutive laws of dissipative forces are consistent with

non-negativity of mechanical dissipation.

We have organized the rest of the article as follows. In section 2 and 3 we

brieﬂy describe the kinematics and equation of motion for the deformable body

On failure mode transition 5

undergoing ﬁnite deformation elasto-viscoplasticity coupled with damage. Section

4 is dedicated towards the development of the necessary constitutive relations.

Here we describe the thermodynamic restrictions, and specify explicit constitutive

functions and several evolution laws. Numerical implementation strategy and a

set of simulations are then presented in section 5. Finally section 6 concludes the

article.

2 Kinematics

Let a body at time t0be deﬁned by it’s reference conﬁguration, B0⊂R3. Upon

elasto-viscoplastic deformation and damage it occupies the region Btin it’s spacial

conﬁguration at time t > t0. A materials point x∈ B0gets mapped to y∈

Btthrough a continuous invertible deformation map χ:x7→ y. With spatial

displacement uand velocity vdeﬁned as u=y−χ−1(y, t) and v=˙χ=˙u,

the deformation gradient Fand velocity gradient Lmay be found as F=∇xχ

and L=∇yv=˙

FF−1, where ∇x,∇yand overdot are material gradient, spacial

gradient and material time derivative respectively.

To segregate elastic and inelastic contributions from the deformation gradient,

we adopt a multiplicative decomposition as F=FeFp, where Fpis the inelastic

distortion carrying material to a relaxed intermediate space and Fethe elastic

distortion. Consequently, the velocity gradient Lmay be decomposed as L=

Le+FeLpFe−1, where Le=˙

FeFe−1and Lp=˙

FpFp−1. Symmetric parts of Le

and Lpdeﬁne respectively the rate of deformation tensors Deand Dp, and skew

symmetric parts the spin tensors Weand Wp. We will assume the plastic ﬂow to

be irrotational, i.e., Wp=0and volume preserving, i.e. det(Fp) = 1. We deﬁne

the elastic strain as Ee= 1/2 (Ce−I) following the deﬁnition of Green-Lagrange

strain tensor, where Ce=FeTFeand Iis the second order identity tensor.

3 Equation of motion

The governing equation of motion for the body, formulated using balance of linear

momentum, may be expressed in its local form as

∇x·P+b0=ρ0¨χ(1)

Where, Pis the ﬁrst Piola Kirchhoﬀ stress tensor, b0the body force density and

ρ0¨χthe inertial force density with ρ0being the referential density of mass of the

material. The balance of angular momentum concludes PFT=FPT. Using a

constitutive relation for Pand the initial and boundary conditions, deformation

ﬁeld may be obtained by solving Eq. (1).

4 Constitutive relation

In order to provide a closure to the equation of motion, Eq. (1), Pneeds to be

expressed in terms of kinematic quantities and internal state variables that keep

track of the irreversible changes of microstructure caused by viscoplasticity and

damage.

6 Ananya Bijaya, Shubhankar Roy Chowdhury

4.1 Thermodynamic restrictions

Speciﬁcation of constitutive relation must be consistent with laws of thermody-

namics. First law of thermodynamics describes balance of total energy. Taking into

account kinetic energy balance that follows from the equation of motion (Eq. (1)),

ﬁrst law may equivalently be represented as internal energy balance. In its local

form it can be stated in reference coordinate as follows.

ρ0˙e=Pint − ∇x·q(2)

where eis speciﬁc internal energy, Pint the stress power and qthe heat ﬂux vector.

The stress power is given by

Pint =P:˙

F=Te:˙

Ee+Me:Dp=Te:˙

Ee+Me

0:Dp(3)

where Teis a second Piola-Kirchhoﬀ type stress tensor and Methe Mandel stress.

They are deﬁed as follows.

Te=JFe−1TFe-T and Me=CeTe(4)

where, Tis the Cauchy stress which may be written in terms of ﬁrst Piola-Kirchhoﬀ

stress tensor as J−1PFT. The last line in Eq. (3) follows from the fact that Dpis

deviatoric because of plastic deformation being isochoric and irrotational. Me

0is

the deviatoric part of Me.

The internal energy is a constitutive variable. There are several state variables

on which it might depend. For example, it is clearly a function of elastic distortion.

Dislocations that nucleate during plastic deformation will also add to internal

energy. Damage of the medium will reduce it and temperature that governs atomic

vibration will also inﬂuence it. In the present formulation we use equivalent plastic

strain as an indirect measure of dislocation density and a phase ﬁeld variable

to track the extent of damage. Therefore assuming eto be dependent on Ee,

temperature θ, equivalent plastic strain pand phase ﬁeld damage variable d, can

write:

e= ˆe(Ee, θ, p, d) (5)

Using Eq. (5), left hand side (LHS) of the ﬁrst law (see Eq. (2)) may be expanded

to be written as the following temperature evolution equation.

ρ0∂e

∂θ ˙

θ+∇x·q=Te−ρ0∂e

∂Ee:˙

E−ρ0∂e

∂p˙p−ρ0∂e

∂d ˙

d+Me

0:Dp(6)

Note that qis another constitutive variable that must be speciﬁed.

Besides the ﬁrst law, the constitutive relations must also satisfy the second law

of thermodynamics which, in absence of external entropy, is given by the following

inequality.

ρ0˙η+∇x·j≥0 (7)

where ηis the speciﬁc entropy and jthe entropy ﬂux vector. Similar to the con-

stitutive dependence of e,ηis assumed to have the following dependence.

η= ˆη(Ee, θ, p, d) (8)

On failure mode transition 7

As usually considered we assume that entropy ﬂux is given by the ratio of heat

ﬂux and temperature.

j=q

θ(9)

This fact is proven in a more general thermodynamic setup in Roy Chowdhury

et al. (2017b) for isotropic viscoplastic material.

Using Eq. (8) material derivative of ηmay be computed and we can rewrite

the entropy inequality (7) as:

ρ0∂η

∂Ee:˙

Ee+ρ0∂η

∂θ ˙

θ+ρ0∂η

∂p˙p+ρ0∂η

∂d ˙

d+1

θ∇x·q−1

θ2q· ∇xθ≥0 (10)

where we have also used Eq. (9). Combining the ﬁrst law (Eq. (6)) and the second

(see (10)) we get Clausius–Duhem inequality:

ρ0θ∂η

∂Ee+Te−ρ0∂e

∂Ee:˙

Ee+ρ0θ∂η

∂θ −ρ0∂e

∂θ ˙

θ+

ρ0θ∂η

∂p−ρ0∂e

∂p˙p−ρ0θ∂η

∂d −ρ0∂e

∂d ˙

d+Me

0:Dp−1

θq· ∇xθ≥0

(11)

We may now apply standard Coleman-Noll type argument to arrive at the

following thermodynamics restrictions to ensure that the above inequality to hold.

ρ0θ∂η

∂Ee+Te−ρ0∂e

∂Ee= 0 (12)

θ∂η

∂θ −∂e

∂θ = 0 (13)

Eq. (12) deﬁnes the constitutive relation for Teas

Te=ρ0∂e

∂Ee−θρ0∂η

∂Ee=ρ0∂e

∂Ee−θ∂η

∂Ee(14)

Using Eq. (12) and (13) in (11) we get reduced form of the inequality as

Dmech +Dcond ≥0 (15)

where

Dmech := θρ0∂η

∂p−ρ0∂e

∂p˙

p+θρ0∂η

∂d −ρ0∂e

∂d ˙

d+Me

0:Dp(16)

is the mechanical dissipation due to plastic deformation and damage, and

Dcond := −1

θq· ∇xθ(17)

the dissipation associated with heat conduction.

Instead of (15), as is conventionally adopted, we assume that the following two

stronger inequalities hold and hence the second law (i.e, (15)) gets satisﬁed.

Dmech ≥0 and Dcond ≥0 (18)

8 Ananya Bijaya, Shubhankar Roy Chowdhury

4.2 Speciﬁcation of constitutive functions

All the constitutive functions must be speciﬁed in such a way that the restrictions

derived in section 4.1, i.e. Eq. (12), (13) and (18) hold. Let us deﬁne the constitutive

relation for internal energy for the present isotropic case as

ρ0e= ˆg(d)1

2λ(tr(Ee))2

++µEe+:Ee+

| {z }

e+

0

+1

2λ(tr(Ee))2

−+µEe−:Ee−

| {z }

e−

0

+¯

W(p) + Zθ

θ0

c(θ)dθ

(19)

where e0=e+

0+e−

0deﬁnes the elastic strain energy density of undamaged material.

e+

0is the strain energy density due to tensile principal strains, and e−

0for the

compressive principal strains. In order to attain such splitting of energy, following

Miehe et al. (2015a), we use the decomposition E=Ee++Ee−, where Ee+is

the positive part of the elastic strain tensor deﬁned by a spectral decomposition

as Ee+:= P3

i=1 hiini⊗niwhere iand niare respectively the eigenvalues and

eigenvectors of Ee.h·i denotes the Macaulay bracket deﬁned as hxi= (x+|x|)/2.

If tr (Ee) is nonnegative, its value is assigned to (tr (Ee))+, otherwise to (tr (Ee))−.

λand µare the Lam´e parameters. ˆg(d) is the degradation function and assumed

to have the following quadratic form

ˆg(d) = (1 −d)2(20)

Note that degradation function is not applied to the energy caused by compressive

strain. This would prevent any damage driving under pure compressive loading.

Energy due to dislocations, i.e. the energy of cold work, is presented by ¯

W(p).

The ﬁnal term in Eq. (19) represents the thermal energy where c(θ) is the heat

capacity per unit volume and this may very well be dependent on temperature.

We have denoted the reference temperature by θ0.

Next important constitutive function is the speciﬁc entropy. Ignoring the con-

ﬁgurational entropy (see for example Roy Chowdhury and Roy (2019)) caused by

rearrangements of defects in mesoscale, we consider only thermal entropy (due to

kinetic vibration of atoms) as

ρ0η=Zθ

θ0

c(θ)

θdθ(21)

One may easily verify that the speciﬁed relations for eand ηabide by the ther-

modynamic restriction given in Eq. (13). To meet the thermodynamic restriction

(12), the constitutive relation of Temust be given by Eq. (14). Using the speciﬁc

expressions for eand ηas adopted in the present study (see Eq. (19) and (21) )

we get

Te= ˆg(d)λ(tr(Ee))+I+ 2µEe++λ(tr(Ee))−I+ 2µEe−(22)

Ensuring the nonnegativity of Dcond, heat ﬂux vector is assumed to be given

by Fourier’s law, which for isotropic material may be written as

q=−kθ∇xθ, kθ≥0 (23)

On failure mode transition 9

We are now required to ensure Dmech ≥0. To attain this we ﬁrst deﬁne an

eﬀect stress given by the following

π=r3

2Me

0:Me

0(24)

One may introduce an eﬀective plastic strain rate νp≥0 which relates to the

accumulated plastic strain pas

p=Zt

t0

νpds or equivalently ˙p=νp(25)

νpshould be deﬁned in such a way that the following power identity holds

Me

0:Dp=πνp(26)

Assuming co-directionality of plastic ﬂow, the ﬂow rule may be assumed to

have the following form so that it also satisﬁes Eq. (26).

Dp=r3

2νpMe

0

|Me

0|(27)

Thus, eﬀective plastic strain rate can be seen to as νp=p2/3|Dp|.

Eq. (16) can thus be rewritten as

Dmech := −¯

W0(p)νp−ˆg0(d)e+

0˙

d+πνp(28)

In order to propose evolution laws of phase ﬁeld and equivalent plastic strain

rate consistently with the second law, i.e. the nonnegativity of Dmech, we need

to understand the total mechanical dissipation that occurs during the inelastic

deformation process. We assume that the reference density of total dissipated

work is given by

Wdis =(Wp−¯

W(p) + Wdfor Wp< W 0

p

˜gp(d)Wp−W0

p+W0

p−¯

W(p) + Wdfor Wp≥W0

p

(29)

where Wpis the eﬀective plastic work density function which models the dissipated

plastic work (accumulated over time) plus the energy of cold work of the undam-

aged material. Clearly Wp>¯

W(p). Wpmay be assumed to have the following

form

Wp=Zt

t0

πpνpds(30)

Here πpdenotes the strength or ﬂow stress of the undamaged material and it is

assumed to be given by the following hardening law due to Johnson and Cook

(1983).

πp= (A+B(p)n) 1 + Cln νp

ν0

p! 1−θ−θ0

θm−θ0m!(31)

where A, B, C, n, m, θmand ν0

pare several material parameters. As already dis-

cussed in introduction, there is a critical value of accumulated plastic work density

below which it does not drive damage. In Eq. (29), W0

pdenotes this critical value.

10 Ananya Bijaya, Shubhankar Roy Chowdhury

𝑥1

𝑥2

1

𝑥1

𝑑(𝑥1, ҧ𝑥2)

2.5𝑙 2.5𝑙

5𝑙

Physical

crack

Diffused

phase field

ABPhase field variation along AB line

𝑥2= ҧ𝑥2

Fig. 1: Schematic of phase ﬁeld based diﬀused representation of a physical crack

˜gp(d) is a degradation function that reduces the strength of material as ductile

damage sets in. While ˜gp(d) very well be chosen diﬀerently from ˆg(d) which is

used for degradation of elastic strain energy, in the present case we assume both

of them to be the same. Wdaccounts for the dissipated work density caused by

damage. It is assumed to have the following form.

Wd= 2Wcd+l2

2|∇xd|2+Zt

t0

kd˙

d2ds(32)

Here the second term in the right hand side (RHS) denotes accumulated viscous

dissipation associated with damage. Wcis a material parameter and la length

associated with gradient regularization of damage.

To see how Wcrelates to the experimentally measured critical energy release

rate of fracture (Gc), let us consider cracking in a plane strain specimen schemat-

ically shown in ﬁgure 1 and observe the following. Gcis experimentally measured

as the amount of energy dissipated while creating unit area of cracked surface.

Through phase ﬁeld method of modelling, a fractured surface is delineated as a

region with d= 1 and also the sharp geometry of crack is regularized into a dif-

fused representation (Miehe et al., 2015a). In ﬁgure 1, we have highlighted a region

of unit length and 5lwidth enclosing a crack of unit length. Let us additionally

consider that specimen is of unit thickness. According to (Miehe et al., 2015a), the

crack phase ﬁeld along any line (say AB as in ﬁgure 1) orthogonal to the crack

varies as the following

d(x1,¯x2) = exp (−|x1|/l) (33)

Due to this assumption d(x1,¯x2)≈0 if |x1|>2.5l. Thus beyond the highlighted

region of width 5l, the phase ﬁeld value is practically zero. Energy dissipated to

create this unit crack surface is

Gc≈ZVh

GcγldV (34)

where γlis the crack surface density function (Miehe et al., 2015a) given by

γl=1

2ld2+l2|∇xd|2(35)

On failure mode transition 11

and Vhdenotes the highlighted region of 5lwidth and unit length and unit thick-

ness as shown in ﬁgure 1. We can obtain an alternate expression for this total

dissipated energy from the present formulation by integrating Eq. (29) over the

volume Vh, i.e.

Gc≈(RVhWp−¯

W(p) + WddV for Wp< W 0

p

RVh˜gp(d)Wp−W0

p+W0

p−¯

W(p) + WddV for Wp≥W0

p

(36)

Using Taylor-Quinney coeﬃcient ˜

λthat approximately takes care of the dislocation-

core energy (see Eq. (50) for more details) one may write Wp−¯

W(p) = ˜

λWp.

Substituting for dfrom Eq. (33) into Eq. (32) one can evaluate RVhWddV ≈5lWc.

Note that we have neglected the viscous dissipation contribution in evaluating this

integral. This is justiﬁed since, in the present formulation, the viscous dissipation

only acts as a regularizer and thus its value would be negligible as compared to

the other terms. Eq. (36) may thus be recast as follows.

Gc

5l≈

˜

λRVhWpdV

|Vh|+Wcfor Wp< W 0

p

RVh[Wp(˜gp(d)−(1−˜

λ))+W0

p(1−˜gp(d))]dV

|Vh|+Wcfor Wp≥W0

p

(37)

where |Vh|denotes the volume of the region Vhand its value is 5l.

It is clear from here that experimentally observed Gctakes larger value if plastic

deformation precedes cracking than its value corresponding to brittle fracture. It

may be observed from Eq. (37) that in absence of plasticity, 5lWc=Gc, i.e.

Wcis the true energy density associated with new surface creation since in brittle

fracture there is no other source of dissipation. However, when plastic deformation

happens before cracking, experiments record surface energy plus some additional

average plastic work done within the diﬀused damaged zone as reported in Eq.

(37) as Gc.

Currently Wcis assumed to vary between two values based on the state of

stress triaxiality (ηt= tr(Me)/|Me

0|) and equivalent plastic strain rate as follows

(cf. Miehe et al., 2015a; Chu et al., 2019).

Wc= max Wη

c, W νp

c(38)

where,

Wη

c=Wd

c−1

2(Wd

c−Wb

c)−1

2(Wd

c−Wb

c) tanh ηt−η∗

t

δη(39)

and,

Wνp

c=Wb

c+1

2(Wd

c−Wb

c) + 1

2(Wd

c−Wb

c) tanh νp−νp∗

δνp(40)

Here Wb

ccorresponds to surface energy density associated with surface generated

by brittle crack and Wd

cdenotes the same for surface originating from ductile frac-

ture. Wd

cis typically of higher value than Wb

c.η∗

t, νp∗, δ ηand δνpare parameters.

See table 1 for their values. A typical plot of Wcas a function of ηtand νpis

shown in ﬁgure 2. The plastic work threshold W0

pis assumed to be a constant.

12 Ananya Bijaya, Shubhankar Roy Chowdhury

0.0 0.2 0.4 0.6 0.8 1.0

η

0

200

400

600

800

1000

νp

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

×107

(a)

0.0 0.2 0.4 0.6 0.8 1.0

η

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

Wc

×107

(b)

0 200 400 600 800 1000

νp

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

Wc

×107

(c)

Fig. 2: The Wcfunction: (a) Dependence on strain rate (νp) and triaxiality (η), (b)

variation of Wcwith triaxiality at νp= 200 and (c) Wcas a function of equivalent

plastic strain rate at 0.9 triaxiality. Parameters used to generate these plots are

given in table 1.

Phase ﬁeld evolution and yield condition:

Taking the material time derivative of Eq. (29), one may get an expression for the

rate at which the mechanical work is dissipated. Denoting it by Rdis we may write

the following

Rdis =(πpνp−¯

W0(p)νp+˙

Wdfor Wp< W 0

p

˜gp(d)πpνp−¯

W0(p)νp+ ˜g0

p(d)Wp−W0

p˙

d+˙

Wdfor Wp≥W0

p

(41)

where

˙

Wd= 2Wc1−l2∆xd˙

d+kd˙

d2(42)

The symbol ∆xdenotes the Laplacian and is given by ∆x=∇x· ∇x.

On failure mode transition 13

Now equating Dmech from Eq. (28) with Rdis, we get the following sets of

equations.

For Wp< W 0

p:

νp= 0 or π=πp(43)

˙

d= 0 or kd˙

d=−ˆg0(d)e+

0−2Wc(1 −l2∆xd) (44)

For Wp≥W0

p:

νp= 0 or π= ˜gp(d)πp(45)

˙

d= 0 or kd˙

d=−ˆg0(d)e+

0−˜g0

p(d)Wp−W0

p−2Wc(1 −l2∆xd) (46)

Clearly the above equations govern yielding and damage. By satisfying the con-

dition of yielding, νpcan be determined. In order to introduce irreversibility of

damage, i.e ˙

d≥0, we modify the damage evolution equations as

¯

kd˙

d=

D(1 −d)e+

0

Wc−1−(d−l2∆xd)Efor Wp< W 0

p

D(1 −d)e+

0+Wp−W0

p

Wc−1−(d−l2∆xd)Efor Wp≥W0

p

(47)

where ¯

kd=kd/2Wcand we have used the explicit quadratic expression for degra-

dation functions in deriving these. It may be observed from Eq. (47) that unless

elastic plus plastic driving exceed Wc, damage evolution does not occur, i.e., ˙

d

remains zero. Thus Wcacts as a threshold energy and driving below this value can

not increase damage in solid.

Temperature evolution equation:

Finally from Eq. (6) the temperature evolution equation for the present case may

be written as,

c(θ)˙

θ+∇x·(−kθ∇xθ) = (π−¯

W0)νp+ 2(1 −d)e+

0˙

d(48)

In adiabatic condition it gets simpliﬁed to

c(θ)˙

θ= (π−¯

W0)νp+ 2(1 −d)e+

0˙

d(49)

Presently we do not specify an expression for the cold work density ¯

W. Rather

using Taylor-Quinney coeﬃcient ˜

λthe above equation is recast as

c(θ)˙

θ=˜

λπνp+ 2(1 −d)e+

0˙

d(50)

14 Ananya Bijaya, Shubhankar Roy Chowdhury

5 Numerical simulation and details on implementation strategy

We simulate the Kalthoﬀ-Winkler’s experiment on double notched plate under

impact to demonstrate the capability of the current model to predict appropriate

fracture response and mode transition phenomena. The simulations are carried

out under plane-strain idealization. Proposed constitutive equations including the

evolution equations of internal state variables are incorporated in a commercial

ﬁnite element software ABAQUS R

through a user deﬁned subroutine in an ex-

plicit dynamics setup. The domain is discretized into a uniform, structured mesh

of 2D plane strain element ‘CPE4R’ (a four noded quadrilateral with reduced in-

tegration) of approximate size 0.5 mm . The equation of motion is integrated over

a time step by inbuilt code of ABAQUS and the presently developed constitutive

model is implemented through ABAQUS-explicit material subroutine VUMAT.

VUMAT provides as input the value of deformation gradient, say Fi+1, at current

time instant ti+1 and other state variables at previous instant ti. Our goal is to

compute the value of Cauchy stress Ti+1 and other state variables at ti+1 instant.

This requires the solution of a set evolution equations of the internal state vari-

ables and satisfaction of yield condition. We adopt a staggered explicit-implicit

strategy to solve these. Fixing all the state variables, e.g. damage, temperature,

the accumulated plastic strain and the equivalent plastic strain rate are solved ﬁrst

using an implicit return mapping algorithm. This is followed by explicit update

of temperature, damage and stress. Note that the evolution equation for phase

ﬁeld (see Eq. (47)) is a PDE and therefore can not be handled through VUMAT

appropriately. However, currently as a work around, we use a ﬁnite diﬀerence ap-

proximation of ∆xdto integrate the equation. Phase ﬁeld variable evaluated at

integration points are saved as a thread safe global array after each time step.

This array is then used to compute an approximation of ∆xdat every integration

point using values of din a set of neighbouring points. To demonstrate that such

ﬁnite diﬀerence approximation can adequately model crack propagation, we have

carried out simulation of a standard dynamic brittle fracture test and compared

the prediction with result available in literature. See Appendix A for details.

5.1 Return mapping

The return mapping procedure (Simo and Taylor (1986); Matzenmiller and Tay-

lor (1994); Simo and Hughes (2006)) implemented in VUMAT is brieﬂy outlined

below. Given the values of d,θetc. from the tth

iinstant, we proceed to obtain accu-

mulated plastic strain increment ∆pfrom tito ti+1 through a predictor-corrector

technique.

A trial state is considered such that the incremental deformation is elastic, i.e,

∆p= 0. This implies

νp

i+1 = 0, p

i+1 =p

i,Fp(ti+1) = Fp

i(51)

Based on this trial value, we calculate elastic part of deformation gradient as

Fetrial

i+1 =Fi+1Fp

i

−1(52)

On failure mode transition 15

Using Fetrial

i+1 , we compute Metrial

i+1 and check if the following holds:

r3

2|(Me

0)trial

i+1 |<(πp(p

i+1, θi) for Wp(ti)< W 0

p

(1 −di)2πp(p

i+1, θi) for Wp(ti)≥W0

p

(53)

If (53) is true then trial update values are true updates. Otherwise a plastic cor-

rection strategy, described below, is followed.

Integrating the ﬂow rule in Eq. (27) we may write

Fp(ti+1) = Fp

i+1 ≈expνp

i+1 (ti+1 −ti)Np

iFp

i(54)

where Np

i=p3/2Me

0i/|Me

0i|. Elastic part of deformation gradient thus follows

as

Fe

i+1 =Fi+1Fp−1

i+1 =Fi+1Fp

i

−1exp−νp

i+1∆t Np

i

=Fi+1Fp

i

−1exp{(−∆pNp

i)}(55)

where ∆p=p

i+1 −p

i.

The yield criterion must be satisﬁed as:

r3

2|(Me

0)i+1 |=(πp(p

i+1, ν p

i+1, θi) for Wp(ti)< W 0

p

(1 −di)2πp(p

i+1, ν p

i+1, θi) for Wp(ti)≥W0

p

(56)

where

(Me

0)i+1 = dev (Ce

i+1Te

i+1) (57)

Since p

i+1 =p

i+∆pand νp

i+1 =∆p/∆t, RHS of Eq. (56) may be written as

function of ∆pand thus

r3

2|(Me

0)i+1 |=(˜

f(∆p, θi) for Wp i < W 0

p

(1 −di)2˜

f(∆p, θi) for Wp i ≥W0

p

(58)

Note that Ce

i+1,Te

i+1 or (Me

0)i+1 are determined through Fe

i+1 given in Eq.

(55) as a function of the unknown ∆p. Clearly, therefore in (58), both RHS and

LHS are functions of only one unknown ∆pand can be solved using suitable

nonlinear iterative solver. Method of bisection may be a good one to use as it

does not require jacobian computation. Once ∆pis found, Fe

i+1,Me

i+1,Ti+1 etc.

may be computed and subsequently temperature and phase ﬁeld may be updated

explicitly.

5.2 Simulation of Kalthoﬀ-Winkler’s experiment

When impacted by a cylindrical projectile, a thin steel plate with two symmetri-

cally placed pre-notches shows intriguing failure pattern depending on the velocity

of impact. At lower impact velocity, a brittle crack emerges from each pre-notch

tip and propagates at an approximate angle of 70◦with respect to the horizontal

direction of pre-notch. For velocity higher than a threshold value, the failure pat-

tern changes and horizontal shear-bands appear from the tip of notches and sub-

sequently bend towards the central horizontal line. Ductile fracture occurs along

these shear bands. See Kalthoﬀ (2000) for more details on this experiment.

16 Ananya Bijaya, Shubhankar Roy Chowdhury

Fig. 3: Geometry of the symmetric half of Kalthoﬀ-Winkler experiment specimen.

AB is the line of symmetry. Dimensions are in meters. Radius of the pre-notch tip

is 0.3 mm.

In order to numerically simulate this experiment through the model proposed,

we consider only the symmetric upper half of the specimen due to the symmetry

of structure and loading; the geometry of the chosen half of the specimen is shown

in ﬁgure 3. Mirror image about the line of symmetry (edge AB) creates the other

half. For the symmetric half shown in ﬁgure 3, AC is the edge where the projectile

impact occurs normal to AC and towards the positive 1 axis. Using an equivalent

velocity boundary condition applied at AC, the eﬀect of impact may be replicated

in the simulation (cf. Roy Chowdhury and Roy, 2019; Miehe et al., 2015a). Velocity

is considered only in the direction of positive 1 axis. It is applied in a way that it

reaches linearly from zero to the peak value in small rise time of 1 µs, upon which

it is held constant over the remaining duration of simulation. Symmetry boundary

conditions, i.e. u2= 0 and ∂d/∂x2= 0, are applied at AB. The other parts of the

boundary are considered traction free and we carry out a plane strain analysis.

Material parameters, given in table 1, are representative of structural steel.

While the values of shear modulus, density, Poissons ratio and Taylor-Quinney co-

eﬃcient are adopted from Miehe et al. (2015b); McAuliﬀe and Waisman (2016), the

parameters for the isotropic hardening model are chosen in a way that the harden-

ing behaviour closely follows the ones used in these reference articles. Keeping in

view 5lW b

c=Gcfor brittle fracture, our choice of Wb

cvalue corresponds to energy

release rate of the same order as used in (McAuliﬀe and Waisman, 2016). Also,

Wd

cfor ductile shear fracture is set notably higher than brittle tensile fracture,

consistent with the ﬁndings of Kalthoﬀ (2000) which concludes that shear bands

require signiﬁcantly higher energy than tensile cracks for initiation and propaga-

tion. The chosen value of WP

0does not allow any contribution of plastic work done

in damage driving until certain amount (≈0.08) of plastic strain is accumulated.

On failure mode transition 17

It should be noted that some of these parameters can be more accurately deter-

mined by directly calibrating with experimental data. However for the present

article we have kept this parameter estimation problem out of scope and used the

representative data as discussed above for the numerical simulations.

Name Symbol Eq. Value Unit

Shear modulus µ(19) 77 GP a

Density ρ(19) 7830 kg/m3

Poisson ratio ν(2) 0.3 -

Reference temperature θ0(31) 298 K

Melting temperature θm(31) 1793 K

Reference yield strength A(31) 2 GP a

Strain hardening constant B(31) 0.51 GP a

Strain rate hardening coeﬃcient C(31) 0.014 -

Reference strain rate νp

0(31) 1.01 s−1

Strain hardening exponent n(31) 0.75 -

Thermal sofening exponent m(31) 1.03 -

Taylor-Quinney coeﬃcient ˜

λ(31) 0.9 -

Critical triaxiality η∗

t(39) 0.8 -

Triaxiality slope parameter δη(39) 0.01 -

Critical strain rate νp∗(40) 450 s−1

Strain rate slope parameter δνp(40) 10 s−1

Critical surface energy density (brittle) Wb

c(38) 107J/m3

Critical surface energy density (ductile) Wd

c(38) 5 ×107J/m3

Plastic work density threshold Wp

0(29) 1.5×108J/m3

Viscous regularizer ¯

kd(47) 10−8m2/N

Fracture length scale l(32) 0.001 m

Heat capacity c (21) 3.51 ×106J/(m3K)

Table 1: Values of several material parameters used for the simulation

5.2.1 Low impact velocity leading to brittle fracture

We replicate the eﬀect of low velocity impact by applying 20 m/s velocity at

the edge AC. Impact induced compressive wave travels to the free end and gets

reﬂected as tensile wave. This tensile wave upon reaching the notch tip increases

the triaxiality to an extent that the crack resistance governed by Wcfalls below

the elastic crack driving. Thus brittle damage starts at the tip and with time

it develops macroscopic crack which propagates approximately at an angle 70◦.

Figure 4 shows the phase ﬁeld at diﬀerent time instants. It is clear from the

triaxiality plots (see ﬁgure 5(a)) that always high triaxiality prevails at the crack

tip and this aids in its propagation in tensile mode (i.e. mode-I). We have also

shown the resistance to damage, using Wc(1 −l2∆xd) as a measure of it (see Eq.

(44), (46)), in ﬁgure 5(b). Resistance stays low at the crack tip through out crack

propagation. In order to conﬁrm that cracking that takes place is indeed brittle, we

show in ﬁgure 4(b) plots of accumulated plastic strain. It can be seen that plastic

strain only accumulates at the surrounding of the tip of pre-notch and the crack

path is completely free from any plastic deformation. The fully cracked specimen

is shown in ﬁgure 6(a).

18 Ananya Bijaya, Shubhankar Roy Chowdhury

Fig. 4: Contour plots of (a) phase ﬁeld and (b) accumulated plastic strain for

simulation under applied velocity of 20 m/s at diﬀerent time instants.

Fig. 5: Time history of (a) triaxiality and (b) crack resistance for specimen with

applied velocity of 20 m/s.

On failure mode transition 19

(a) (b)

Fig. 6: Deformed conﬁgurations of the cracked plates: (a) Brittle crack at low

impact velocity and (b) ductile fracture at high impact velocity.

5.2.2 Shear band and ductile tearing at high impact velocity

To simulate high velocity impact, edge AC is applied with 40 m/s velocity. Because

of higher rate of loading, the notch tip undergoes considerable plastic deformation

and plastic strain starts to accumulate over a narrow band, known as shear band,

from the notch tip approximately in a horizontal direction. Because of high plas-

tic strain rate and lower triaxiality value as compared to the low velocity impact

case, the resistance to damage remains larger than the elastic driving. Accordingly

brittle crack could not appear from the notch tip. However plastic work surplus to

W0

pcontinues to degrade material within the shear band. As may be seen from ﬁg-

ure 7(b), the shear band initially propagates horizontally and subsequently bends

towards the horizontal line of symmetry. Continuous plastic driving ultimately

leads to tearing of material through this band. This is clear from the phase ﬁeld

plots shown in ﬁgure 7(a). In ﬁgure 7(b) we have also reported the temperature

distribution. It is evident that temperature rises considerably within the shear

band. Deformed conﬁguration of the plate upon complete ductile tearing is shown

in ﬁgure 6(b).

5.3 Eﬀect of critical triaxiality and strain rate, and more on transition of failure

mode

We have shown that the proposed formulation can adequately model the brittle-to-

ductile transition observed in Kalthoﬀ-Winkler’s experiment. The variable crack

resistance parameter Wcis found to have signiﬁcant impact on the response charac-

teristic. Motivated byChu et al. (2019) and Miehe et al. (2015a), we assumed Wc

to depend on both triaxiality and plastic strain rate– Wcbasically transits form

one value to other depending on some critical values of these states, i.e. η∗

tand

νp∗. In order to assess the eﬀect of these critical values on the fracture response,

we carried out a few more simulations choosing diﬀerent combinations of values of

these parameters.

We ﬁrst consider two cases where we use two diﬀerent values for critical tri-

axiality η∗

t, 0.7 and 0.9, one being less than what is used (0.8) for the simulations

20 Ananya Bijaya, Shubhankar Roy Chowdhury

Fig. 7: Contour plots of (a) phase ﬁeld, (b) accumulated plastic strain and (c)

temperature for simulation with 40 m/s velocity at diﬀerent time instants.

Fig. 8: Phase ﬁeld distribution under 20 m/s at 50µs. (a) Corresponds to η∗

t= 0.7

and shows the usual brittle cracking; (b) corresponds to η∗

t= 0.9 with no crack.

On failure mode transition 21

Fig. 9: Phase ﬁeld distribution under 40 m/s at 40µs. (a) Corresponds to η∗

t= 0.7,

showing a ductile to brittle transition; (b) for η∗

t= 0.9, it shows the usual shear

banding

reported above and other being higher than it. νp∗is kept at 450. For simulations

with η∗

t= 0.7, it may be seen from ﬁgure 8(a) that under applied velocity 20

m/s the usual brittle crack appears. However at velocity 40 m/s, while shear band

emanates from the notch tip, after propagating horizontally for some time ductile-

to-brittle damage transition takes place and inclined brittle crack starts to develop

from the tip of the shear band. This can be seen from ﬁgure 9(a). Figure 10 shows

the distribution of accumulated plastic strain to conﬁrm that the included crack

formed is indeed of brittle nature. To demonstrate why this transition happens, we

have shown in ﬁgure 11(a) the elastic driving and in ﬁgure 11(b) the corresponding

crack resistance. It is evident from this set of ﬁgures that near the tip of shear

band, elastic driving exceeds the resistance which assumes a low value when triax-

iality exceeds 0.7. Adequate lowering of resistance does not happen for η∗

t= 0.8 or

greater and accordingly no such transition towards brittle crack formation takes

place. Simulating with η∗

t= 0.9, we see that under 20 m/s, no damage occurs at

all (see ﬁgure 8(b)). However for 40 m/s, as experimentally observed, shear band

forms and propagate with out deviating to create a brittle crack.

Next, we consider cases where νp∗is varied keeping η∗

t= 0.8. In one simulation,

we incorporate νp∗= 100 which is less than 450 that has been considered so far.

In another simulation we altogether remove the dependence of νpfrom Wc, i.e

Wc=Wη

c. This theoretically simulate the case where νp∗is extremely large.

Transition of Wcat lower value of νp, i.e. at νp∗= 100, aﬀects adversely the low

velocity (20 m/s) impact response. As may be seen from ﬁgure 12(a), no damage

occurs in this case. Similar observation was also made for simulation with high

ηt∗. However for impact at 40 m/s similar shear banding and ductile tearing as

observed in experiment is attained. See ﬁgure 13(a). For very large value of νp∗

or in case Wcis independent of νp, simulation results for both 20 m/s and 40

m/s impact cases are akin to experimental observations. It may be veriﬁed from

the phase ﬁeld plots shown in ﬁgures 12(c) and 13(c). This observation implies

that unlike the value of η∗

t,Wcis not that critically sensitive to the value of νp∗

as long as it is reasonably large. While this might suggest that one can model

Wcas a function of ηtalone, the following set of observations keeps the question

about the dependence of Wcon νpopen. The value of νp∗is found to inﬂuence

22 Ananya Bijaya, Shubhankar Roy Chowdhury

Fig. 10: Accumulated plastic strain for impact velocity of 40 m/s at 40µs for

simulation with η∗

t= 0.7. It conﬁrms ductile-to-brittle transition, i.e. the upward

crack is of brittle nature.

Fig. 11: (a) Elastic driving and (b) crack resistance (at 31µs) corresponding to the

ductile-to-brittle damage transition state triggered in simulation with η∗

t=−0.7

under 40 m/s velocity.

Fig. 12: Phase ﬁeld distribution of cracked specimen under 20 m/s impact velocity.

(a) Corresponds to ν∗

p= 100 (at 90 µs); (b) Corresponds to ν∗

p= 450 (at 83.6 µs);

(c) Corresponds to Wcwithout νpdependence and (at 83.6 µs)

On failure mode transition 23

Fig. 13: Phase ﬁeld distribution at the instant when complete ductile tearing is

attained under 40 m/s impact velocity. (a) Corresponds to ν∗

p= 100 (at 72 µs);

(b) Corresponds to ν∗

p= 450 (at 72 µs); (c) Corresponds to Wcindependent of

νp(at 62.7 µs); Sub-ﬁgures (d), (e) and (f) show the zoomed in portions of the

regions marked with boxes in sub-ﬁgures (a), (b) and (c) respectively. Sub-ﬁgures

(f) shows a prominent brittle damage region branching out from the shear band.

the ductile crack speed. As may be seen from ﬁgures 13(a), 13(b) and 13(c) that

complete ductile tearing takes place at 72 µs for cases with νp∗= 100 and 450, but

for νpindependent Wc, this happens much earlier, at around 63 µs. Apart from

crack speed, one might also observe in ﬁgure 13(f) a prominent region of brittle

damage branching out from the shear band in case of νpindependent Wc. On the

other hand ﬁgures 13(d) and 13(e) both show such brittle damage branching is

signiﬁcantly suppressed for νp∗= 100 and 450.

6 Conclusion

In order to model brittle-to-ductile fracture transition phenomenon observed in a

set of experiments conducted on notched plate under impact loading, we have laid

out a thermodynamically consistent formulation of viscoplasticity coupled with

phase ﬁeld fracture. Present formulation overcomes some of the issues faced by

other recent eﬀorts based on phase ﬁeld method to simulate the same problem,

and it successfully predicts the transition. Two important material parameters,

namely a threshold value of plastic work done and the critical value of true surface

energy density, of our model signiﬁcantly inﬂuence the fracture process. While

the critical surface energy density is assumed to be dependent on stress triaxial-

ity and equivalent plastic strain rate, the threshold value of plastic work done is

taken to be constant. Eﬀect of triaxiality and plastic strain rate dependence of

critical surface energy density has been systematically investigated to gain further

24 Ananya Bijaya, Shubhankar Roy Chowdhury

insight about their role in fracture mode switching. For the present simulations,

it has suﬃced to consider a two-state type nature of the critical surface energy

density and the threshold plastic work as constant. However, the threshold value

of plastic work might very well be triaxiality dependent in a such a way that

below certain negative triaxiality it assumes very high value so that practically

no ductile damage is possible under such circumstances. Dependence of the crit-

ical surface energy density on triaxiality and plastic strain rate may be expected

to be more complex than the simplistic two-state type assumed in the present

formulation. Consideration of the lode angle parameter might also be necessary.

This set of hypotheses follows from many experimental observations pertaining to

strain-to-fracture in metallic components. Notwithstanding these simpliﬁcations

of the material parameters, our formulation predicts the signiﬁcant traits of the

fracture propagation and mode transition observed in experiments. However, for

more quantitative congruence with experimental data further exploration on this

set of hypothesis should be carried out. Besides, investigating the eﬀects of in-

troduction of topological features, e.g. inhomogeneity, perforations, in inﬂuencing

damage mode are some of the additional scope of this work that may be pursued

subsequently.

Conﬂict of interest

The authors declare that they have no conﬂict of interest.

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A Validation of ﬁnite diﬀerence approximation based phase ﬁeld

evolution in predicting dynamic fracture

As a validation of the proposed ﬁnite diﬀerence approximation based phase ﬁeld evolution, we

present here a comparison of results given in literature and result obtained using our framework,

for a benchmark problem of dynamic brittle fracture. This exercise not only establishes the

validity of the ﬁnite diﬀerence approximation, it also demonstrates the correctness of the

relation 5lW b

c=Gc, as derived in the present article, for brittle fracture.

Fig. 14: Geometry of specimen used for dynamic crack branching problem. Dimen-

sions are in meters. Radius of pre-notch tip is 0.3 mm

A dynamic crack branching problem, previously reported in several articles (e.g Borden

et al., 2012; Roy Chowdhury and Roy, 2019; Li et al., 2016), of brittle fracture– dynamic crack

growth and branching– in a pre-notched plate specimen subjected to tensile loading is chosen as

the benchmark problem. The geometry of the plate specimen is given in ﬁgure 14. Edge AB is

subjected to 1 MPa traction force along negative 2 direction (refer to the 1-2 axes in ﬁgure 14),

and edge CD is subjected to 1 MPa traction force along 2 axis. For numerical implementation

through VUMAT subroutine in Abaqus, the domain is discretized into a uniform, structured

mesh of 2D plane strain element ‘CPE4R’ (a four noded quadrilateral with reduced integration)

of approximate size 0.23 ×0.23 mm2. The material parameters used for the simulation are

adopted from Borden et al. (2012). They are– mass density ρ0= 2450 kg/m3,shear modulus

µ= 13.3 GPa and Poisson’s ratio = 0.2. To simulate damage evolution, we consider Wc=

Wb

c= 904 J/m3,kd= 10−3Pa-s, and length scale l= 6.64 ×10−4m. The value of Wcis

computed using Wc=Gc/(5l), with Gc= 3 J/m2taken taken from Borden et al. (2012).

The phase ﬁeld distribution at diﬀerent time instances are presented in ﬁgure 15. From the

simulation results, it can be observed that crack initiates from the notch tip and propagates

28 Ananya Bijaya, Shubhankar Roy Chowdhury

Fig. 15: Phase ﬁeld contour plots from dynamic crack branching simulation. The

time instants are noted along the side of the plots

horizontally for a while. It then begins to widen at around 27 µs. Crack branching occurs at

34 µs, and crack propagation is completed around 81 µs. These predicted values are in line

with the simulation results reported by Borden et al. (2012). From ﬁgure 10 (c) of Borden et al.

(2012), it can be noted that in their simulations crack widening starts around 25−27 µs, crack

branching occurs around 34−36 µs, and crack propagation ends at almost 80 µs. Furthermore,

in our simulation, the crack branching angle is found to be 29◦which is within the range

of values (22◦- 34◦) observed in theoretical and experimental studies on crack branching.

Thus, this quantitative validation exercise demonstrates the feasibility of using ﬁnite diﬀerence

approximation based phase ﬁeld evolution in modelling fracture. Correctness of the derived

relation between Wb

cand Gcis also conﬁrmed.