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On failure mode transition: a phase field assisted non-equilibrium thermodynamics model for ductile and brittle fracture at finite strain


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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 emphasis 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 viscoplasticity with accumulated plastic strain, and a scalar phase field variable traces the degradation of the material caused by either tensile cracking or shear induced failure. 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.
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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
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; Femond 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 Femond 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
2 Kinematics
Let a body at time t0be defined by it’s reference configuration, B0R3. 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=yχ1(y, t) and v=˙χ=˙u,
the deformation gradient Fand velocity gradient Lmay be found as F=xχ
and L=yv=˙
FF1, 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+FeLpFe1, where Le=˙
FeFe1and Lp=˙
FpFp1. 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 (CeI) 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
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
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:˙
where Teis a second Piola-Kirchhoff type stress tensor and Methe Mandel stress.
They are defied as follows.
Te=JFe1TFe-T and Me=CeTe(4)
where, Tis the Cauchy stress which may be written in terms of first Piola-Kirchhoff
stress tensor as J1PFT. The last line in Eq. (3) follows from the fact that Dpis
deviatoric because of plastic deformation being isochoric and irrotational. Me
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
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.
∂θ ˙
∂d ˙
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
ρ0˙η+x·j0 (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.
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:
∂θ ˙
∂d ˙
θ2q· ∇xθ0 (10)
where we have also used Eq. (9). Combining the first law (Eq. (6)) and the second
(see (10)) we get Clausius–Duhem inequality:
∂θ ρ0e
∂θ ˙
∂d ρ0e
∂d ˙
θq· ∇xθ0
We may now apply standard Coleman-Noll type argument to arrive at the
following thermodynamics restrictions to ensure that the above inequality to hold.
Ee= 0 (12)
∂θ e
∂θ = 0 (13)
Eq. (12) defines the constitutive relation for Teas
Using Eq. (12) and (13) in (11) we get reduced form of the inequality as
Dmech +Dcond 0 (15)
Dmech := θρ0η
∂d ρ0e
∂d ˙
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
| {z }
| {z }
W(p) + Zθ
where e0=e+
0defines the elastic strain energy density of undamaged material.
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 hiininiwhere 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 ¯
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
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=kθxθ, kθ0 (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
One may introduce an effective plastic strain rate νp0 which relates to the
accumulated plastic strain pas
νpds or equivalently ˙p=νp(25)
νpshould be defined in such a way that the following power identity holds
Assuming co-directionality of plastic flow, the flow rule may be assumed to
have the following form so that it also satisfies Eq. (26).
Thus, effective plastic strain rate can be seen to as νp=p2/3|Dp|.
Eq. (16) can thus be rewritten as
Dmech := ¯
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
W(p) + Wdfor WpW0
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
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
πp= (A+B(p)n) 1 + Cln νp
p! 1θθ0
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
𝑑(𝑥1, ҧ𝑥2)
2.5𝑙 2.5𝑙
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
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
GcγldV (34)
where γlis the crack surface density function (Miehe et al., 2015a) given by
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.
W(p) + WddV for Wp< W 0
W(p) + WddV for WpW0
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) = ˜
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.
|Vh|+Wcfor Wp< W 0
|Vh|+Wcfor WpW0
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) tanh ηtη
c) + 1
c) tanh νpνp
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
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
0.0 0.2 0.4 0.6 0.8 1.0
0 200 400 600 800 1000
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¯
Wdfor Wp< W 0
W0(p)νp+ ˜g0
Wdfor WpW0
Wd= 2Wc1l2xd˙
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
For Wp< W 0
νp= 0 or π=πp(43)
d= 0 or kd˙
02Wc(1 l2xd) (44)
For WpW0
νp= 0 or π= ˜gp(d)πp(45)
d= 0 or kd˙
p2Wc(1 l2xd) (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 ˙
d0, we modify the damage evolution equations as
D(1 d)e+
Wc1(dl2xd)Efor Wp< W 0
D(1 d)e+
Wc1(dl2xd)Efor WpW0
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., ˙
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,
θ+x·(kθxθ) = (π¯
W0)νp+ 2(1 d)e+
In adiabatic condition it gets simplified to
θ= (π¯
W0)νp+ 2(1 d)e+
Presently we do not specify an expression for the cold work density ¯
W. Rather
using Taylor-Quinney coefficient ˜
λthe above equation is recast as
λπνp+ 2(1 d)e+
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
A trial state is considered such that the incremental deformation is elastic, i.e,
∆p= 0. This implies
i+1 = 0, p
i+1 =p
i,Fp(ti+1) = Fp
Based on this trial value, we calculate elastic part of deformation gradient as
i+1 =Fi+1Fp
On failure mode transition 15
Using Fetrial
i+1 , we compute Metrial
i+1 and check if the following holds:
i+1 |<(πp(p
i+1, θi) for Wp(ti)< W 0
(1 di)2πp(p
i+1, θi) for Wp(ti)W0
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 expνp
i+1 (ti+1 ti)Np
where Np
0i|. Elastic part of deformation gradient thus follows
i+1 =Fi+1Fp1
i+1 =Fi+1Fp
i+1∆t Np
where ∆p=p
i+1 p
The yield criterion must be satisfied as:
0)i+1 |=(πp(p
i+1, ν p
i+1, θi) for Wp(ti)< W 0
(1 di)2πp(p
i+1, ν p
i+1, θi) for Wp(ti)W0
0)i+1 = dev (Ce
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
0)i+1 |=(˜
f(∆p, θi) for Wp i < W 0
(1 di)2˜
f(∆p, θi) for Wp i W0
Note that Ce
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,Ti+1 etc.
may be computed and subsequently temperature and phase field may be updated
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 70with 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,
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 s1
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 s1
Strain rate slope parameter δνp(40) 10 s1
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) 108m2/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 l2xd) 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
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
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
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
reported above and other being higher than it. νpis 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 νpis 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
c. This theoretically simulate the case where νpis 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 νpis 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
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
Conflict of interest
The authors declare that they have no conflict of interest.
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notched specimens. International Journal of Plasticity 14(4-5):435–451
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=
c= 904 J/m3,kd= 103Pa-s, and length scale l= 6.64 ×104m. 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 2527 µs, crack
branching occurs around 3436 µs, and crack propagation ends at almost 80 µs. Furthermore,
in our simulation, the crack branching angle is found to be 29which 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.
... To determine constitutive relations, we use the standard Coleman-Noll procedure [38,39,46], based on the fact that any admissible thermodynamic process (i.e., a solution of a system of balance equations) must satisfy the second law of thermodynamics. The procedure is very practical and widely applied [31,37,[47][48][49]. ...
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Usually, Lagrangian or Eulerian–Lagrangian descriptions of motion are adopted to model interaction of fluids and deformable solid bodies. However, in some cases both approaches lead to serious difficulties. An example is a system with large number of solid bodies; another one is the case where topology of the phases can change. In these situations, an Eulerian description is much more convenient. The presented work is devoted to the development of a phase field mathematical model for description of dynamics of multiphase multicomponent system (mixture) with phases having whether liquid or solid rheology. All the phases and interphase boundaries (interfaces) are directly resolved. Main balance laws of the proposed model are formulated using the Eulerian description. The model is of phase field type: the interphase boundary is diffuse and is described by a thin layer of finite thickness. Mass densities of mixture components are used as order parameters. To describe a stress–strain behavior of the solid phase, we assume that the Helmholtz free energy depends on deformation gradient tensor which is defined as a solution of the corresponding evolution equation. Constitutive relations are derived by means of the well-known Coleman–Noll procedure and the second law of thermodynamics. A distinctive feature of the considered model is its preliminary regularization based on the quasi-hydrodynamic technique, which allows one to improve numerical stability properties when an explicit discretization is applied. A new family of quasi-hydrodynamic closures is obtained.
... Surana and Carranza [5] demonstrate the influence of internal rotations on the dynamic response of solid continua. In connection with the phase-field method [1], Bijaya and Chowdhury [6] address fracture problems considering finite strains. The next group of papers addresses novel numerical methods in computational mechanics including the variational multiscale method (VMS) for incompressible flows by Kang and Masud [7], the virtual element method (VEM) for nearly incompressible materials by Park et al. [8], the generalized finite element method (GFEM) for fracture of composites by Alves et al. [9], and the Carrera Unified Formulation (CUF) for stress analysis by Filippi and Carrera [10]. ...
In this work, the hybrid peridynamic and classical continuum mechanical model based on the element‐type numerical discretization is used to study the dynamic fracture problem of brittle materials. The weak form of the motion equation for the hybrid model is derived by applying Hamilton's principle. An element‐based spatial discretization scheme is set up using the peridynamics‐based finite element method (PeriFEM) (Han and Li). The ‐convergence and m ‐convergence of the hybrid model are analyzed to verify the numerical convergence and the mesh dependency. The influence of the area of the peridynamic model subdomain and the hybrid model subdomain on the calculation efficiency and accuracy is discussed. And the relationship between the crack propagation speed and the crack branching phenomenon is also investigated. Good agreements are obtained by comparing the crack path and the crack speed profile predicted by the hybrid model with those in the literature. Numerical results verify the ability and efficiency of the hybrid model for dynamic fracture problem.
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Fracture is one of the most commonly encountered failure modes of engineering materials and structures. Prevention of crack-induced failure is, therefore, a major constraint in engineering designs. Computational modelling of fracture constitutes an indispensable tool not only to predict the failure of cracked structures but also to shed insights into understanding the fracture processes. Mathematically, crack evolution is the growth of a sharp discontinuity over time, which is a very complicated process. Phase field method (PFM) is a promising computational tool where the sharp crack is not treated as a discontinuity; instead, it is approximated as a thin band of damage with a bandwidth proportional to the considered length scale parameter. With this approximation, PFM can seamlessly simulate the crack evolution involving crack initiation, propagation, branching and fragmentation. However, the conventional phase field fracture models are primarily developed to simulate brittle fracture, and are not suitable for cohesive fracture problems. Also, these models do not explicitly consider the tensile strength as a material parameter, which attributes to their length scale sensitive mechanical response. Recently, Wu (JMPS, 103, 2017); Wu and Nguyen (JMPS, 119, 2018) have proposed a length scale insensitive PFM to describe the brittle and cohesive fracture of linear elastic solids by incorporating the material traction-separation law (TSL) in the phase field theory. This model is dubbed as phase field regularised cohesive zone model (PF-CZM). The present thesis aims to investigate and extend PF-CZM to various fracture problems with focus on fracture dynamics, hyperelasticity, anisotropy, and environment assisted multi-physics fracture. Understanding the usefulness, limitations, and scope of improvements of the PFMs are the key aspects of this thesis. The PF-CZM is found to provide a suitable framework to incorporate chemical concentration or temperature dependent TSLs into the phase field theory. It facilitates phase field fracture modelling of fibre-reinforced composites considering the distinct softening behaviour of the matrix and the fibres. PF-CZM is extended to hyperelastic fracture considering the energetic equivalence between the hyperelastic material model with linear elasticity. Also, the is found to be suitable for numerous dynamic fracture problems. Detailed numerical simulation of several benchmark problems using the developed PF-CZM, implemented in the framework of finite element method, are in good agreement with previous findings and independent to the discretisation resolution. Most importantly, they are independent to the incorporated length scale parameter, which is distinctive to the previous PFMs. Moreover, the numerical results show that the is as efficient as, if not more than, previous models. Phase field fracture simulations are found to be computationally expensive, which is even worse for multi-physics fracture. Herein, this issue is addressed using a monolithic quasi-Newton BFGS solver for the thermal fracture equation. The BFGS solver is found to be 4-5 times faster than the conventional stagger solver for multi-physics PFM. The limitations of PFM are also highlighted here, namely, rate-dependence in dynamic fracture formulation, numerical convergence issues of finite deformation problems, damage band widening in anisotropic fracture, etc.
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We present a physically enhanced ductile damage model applicable for body centered cubic (BCC) metals. The current proposition extends the authors’ recent work on thermo-viscoplasticity based on two-temperature thermodynamics and physics of disparate types of dislocation densities. The description of the thermodynamic system involves primarily two types of variables (or degrees of freedom, DOFs) representing several micro/meso-scopic processes occurring in two separable time-scales during ductile damage. Processes of rearrangement and movement of defects, namely dislocations, voids, micro-cracks, take place in a time scale much slower than that of the vibration of atoms about their equilibrium positions in the lattice. Consequently, they appear in the thermodynamic theory in terms of slow configurational DOFs and the fast kinetic vibrational DOFs respectively. While we consider physics based internal variables, e.g., mobile and forest dislocation densities, for modeling viscoplasticity alone, material degradation due to ductile damage is treated in a phenomenological fashion taking recourse to the framework of continuum damage mechanics. In order to assess the performance of our proposal, numerical experiments on boundary value problems of viscoplasticity with or without damage are carried out and validated against available experimental evidence.
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Posed within the two-temperature theory of non-equilibrium thermodynamics, we propose a model for thermoviscoplastic deformation in metals. We incorporate the dynamics of dislocation densities–mobile and forest—that play the role of internal state variables in the formulation. The description based on two temperatures appears naturally when one recognizes that the thermodynamic system undergoing viscoplastic deformation is composed of two weakly interacting subsystems, viz. a kinetic-vibrational subsystem of the vibrating atomic lattices and a configurational subsystem of the slower degrees of freedom relating to defect motion, each with its own temperature. Starting with a basic model that involves only homogeneous deformation, a three-dimensional model for inhomogeneous viscoplasticity applicable to finite deformation is charted out in an overstress driven viscoplastic deformation framework. The model shows how the coupled evolutions of mobile and forest dislocation densities, which are critically influenced by the dynamics of configurational temperature, govern the strength and ductility of the metal. Unlike most contemporary models, the current proposal also affords a prediction of certain finer details as observed in the experimental data on stress–strain behaviour of metals and this in turn enhances the understanding of the evolving and interacting dislocation densities. Graphical Abstract
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Plasticity and damage are two fundamental phenomena in nonlinear solid mechanics associated to the development of inelastic deformations and the reduction of the material stiffness. Alessi et al. [5] have recently shown, through a variational framework, that coupling a gradient-damage model with plasticity can lead to macroscopic behaviours assimilable to ductile and cohesive fracture. Here, we further expand this approach considering specific constitutive functions frequently used in phase-field models of brittle fracture. A numerical solution technique of the coupled elasto-damage-plasticity problem, based on an alternate minimisation algorithm, is proposed and tested against semi-analytical results. Considering a one-dimensional traction test, we illustrate the properties of four different regimes obtained by a suitable tuning of few key constitutive parameters. Namely, depending on the relative yield stresses and softening behaviours of the plasticity and the damage criteria, we obtain macroscopic responses assimilable to (i) brittle fracture à la Griffith, (ii) cohesive fractures of the Barenblatt or Dugdale type, and (iii) a sort of cohesive fracture including a depinning energy contribution. The comparisons between numerical and analytical results prove the accuracy of the proposed numerical approaches in the considered quasi-static time-discrete setting, but they also emphasise some subtle issues occurring during time-discontinuous evolutions.
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A physics based model for dislocation mediated thermo-viscoplastic deformation in metals is proposed. The modelling is posited in the framework of internal-variables theory of thermodynamics, wherein an effective dislocation density, which assumes the role of the internal variable, tracks permanent changes in the internal structure of metals undergoing plastic deformation. The thermodynamic formulation involves a two-temperature description of viscoplasticity that appears naturally if one considers the thermodynamic system to be composed of two weakly interacting subsystems, namely, a kinetic vibrational subsystem of the vibrating atomic lattices and a configurational subsystem of the slower degrees of freedom of defect motion. Starting with an idealized homogeneous setup, a full-fledged three-dimensional continuum formulation is set forth. Numerical exercises, specifically in the context of impact dynamic simulations, are carried out and validated against experimental data. The scope of the present work is, however, limited to face-centered cubic metals only.
Utilizing a non-equilibrium thermodynamic setting that involves two temperatures, we present a model for ductile and brittle damage. The thermodynamic system consists of two interacting subsystems- configurational and kinetic-vibrational. While the kinetic-vibrational subsystem describes fast degrees-of-freedom (DOFs) of ordinary thermal vibration, the configurational subsystem includes the slower DOFs pertaining to a slew of configurational rearrangements that characterize elasto-visco-plasticity and damage, e.g. dislocation motion, lattice stretching, void nucleation, void growth and micro-crack formation. Following statistical mechanics, an expression for the entropy of a plastically deforming metal with growing voids and micro-cracks is derived. Subsequent application of the first and second laws of thermodynamics, suitably modified for the two-temperature system, yields coupled evolution rules for dislocation density, void volume fraction, micro-crack density etc. A modified flow rule for dilatant plasticity and evolution equations for the two temperatures are also derived. Even when the two subsystems are strongly coupled, we show that a splitting of energy and entropy is feasible and that the notion of two temperatures conforms with such splitting. We conduct numerical experiments on both brittle and ductile damage to assess the predictive features of the model and validate the results against available experimental evidence. Finally, a generalized fluctuation relation is put forth for deformations with extremely high strain rates. This leads to an entirely new procedure for constitutive closure, providing valuable insights into the emergent pseudo-inertial aspects of the evolving thermodynamic states.
This study presents an approach to model dynamic failure mode switching and shear band propagation using the correspondence framework of state-based peridynamics. To effectively model spontaneous shear-band-to-crack switching phenomenon, which is of intrinsic complexity, a combined peridynamic nonlocal and classical local damage model for failure prediction is proposed. The concept of bond failure, often employed in the bond-based peridynamic modeling of brittle materials, is extended to the state-based peridynamics to model cleavage failure in elasto-viscoplastic materials so that the directional and progressive nature of damage can be readily handled. The classical constitutive relation for viscous fluid is adapted for use in non-ordinary state-based peridynamics to model stress collapsing state in shear bands. To show the effectiveness of the proposed approach, a comprehensive numerical study of a pre-notched plate subjected to asymmetric impact loading is conducted, including the phenomenon of shear-band-to-crack switching under an intermediate impact velocity and ductile failure under high impact velocity. A zero-energy mode suppression with an upper bound to avoid over-correction is introduced. By using the proposed modeling approach, dynamic failure mode switching and shear band propagation path are accurately captured.
A rigorous derivation of the relation between the entropy flux and the heat flux in a recently developed two-temperature thermodynamic model of metal thermoviscoplasticity is presented. The two-temperature model exploits the internal variable theory of thermodynamics, wherein thermodynamic restrictions on the constitutive functions are based on the second law written in a form similar to the classical Clausius-Duhem (CD) inequality. Here, the weakly interacting thermodynamic subsystems, e.g., configurational and kinetic vibrational subsystems, enable defining their own temperatures, heat fluxes, and entropy fluxes. The CD-type inequality is then constructed with the assumption, as in rational thermodynamics, that entropy fluxes equal heat fluxes divided by respective absolute temperatures. Validity or otherwise of this restrictive assumption is however an open question in the context of two-temperature thermomechanics, and there are, indeed, known materials for which this assumption fails to hold. To settle this important point, we start with a detailed analysis based on a general entropy inequality, whose thermodynamic consequences are extracted using M€uller-Liu procedure of Lagrange multipliers, and subsequently, appeal to material frame-indifference, material symmetry groups for additional constitutive restrictions. We conclude that, for isotropic-viscoplastic materials, subsystem entropy fluxes are indeed given by the respective heat fluxes divided by their own temperatures.