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Numerical modelling is increasingly supporting the analysis and optimization of manufacturing processes in the production industry. Even if being mostly applied to multistep processes, single process steps may be so complex by nature that the needed models to describe them must include multiphysics. On the other hand, processes which inherently may seem multiphysical by nature might sometimes be modelled by considerably simpler models if the problem at hand can be somehow adequately simplified. In the present article, examples of this will be presented. The cases are chosen with the aim of showing the diversity in the field of modelling of manufacturing processes as regards process, materials, generic disciplines as well as length scales: (1) modelling of tape casting for thin ceramic layers, (2) modelling the flow of polymers in extrusion, (3) modelling the deformation process of flexible stamps for nanoimprint lithography, (4) modelling manufacturing of composite parts and (5) modelling the selective laser melting process. For all five examples, the emphasis is on modelling results as well as describing the models in brief mathematical details. Alongside with relevant references to the original work, proper comparison with experiments is given in some examples for model validation.
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Special Issue Article
Advances in Mechanical Engineering
2018, Vol. 10(5) 1–31
ÓThe Author(s) 2018
DOI: 10.1177/1687814018766188
Multiphysics modelling of
manufacturing processes: A review
Masoud Jabbari
, Ismet Baran
, Sankhya Mohanty
, Raphae
¨l Comminal
Mads Rostgaard Sonne
, Michael Wenani Nielsen
, Jon Spangenberg
Jesper Henri Hattel
Numerical modelling is increasingly supporting the analysis and optimization of manufacturing processes in the produc-
tion industry. Even if being mostly applied to multistep processes, single process steps may be so complex by nature that
the needed models to describe them must include multiphysics. On the other hand, processes which inherently may
seem multiphysical by nature might sometimes be modelled by considerably simpler models if the problem at hand can
be somehow adequately simplified. In the present article, examples of this will be presented. The cases are chosen with
the aim of showing the diversity in the field of modelling of manufacturing processes as regards process, materials, gen-
eric disciplines as well as length scales: (1) modelling of tape casting for thin ceramic layers, (2) modelling the flow of
polymers in extrusion, (3) modelling the deformation process of flexible stamps for nanoimprint lithography, (4) model-
ling manufacturing of composite parts and (5) modelling the selective laser melting process. For all five examples, the
emphasis is on modelling results as well as describing the models in brief mathematical details. Alongside with relevant
references to the original work, proper comparison with experiments is given in some examples for model validation.
Numerical modelling, tape casting, nanoimprint lithography, extrusion, composite, selective laser melting
Date received: 24 May 2017; accepted: 26 February 2018
Handling Editor: Filippo Berto
Numerical modelling is increasingly being used in the
design and optimization of manufacturing processes in
order to increase the quality of the produced parts and
improved production yield. Today, complex manufac-
turing processes are often addressed with multiphysics
models involving numerical heat transfer, computa-
tional fluid dynamics (CFD) and computational solid
mechanics (CSM) as well as thermodynamic and kinetic
Progress in numerical modelling of material
behaviour, efficient computational algorithms and
advances in computer hardware and storage devices
have increased the ability of complex software to be
used for process design and optimization.
Mathematical flow simulations, also known as
CFD, have matured rapidly in the last half-century.
Particularly, in the manufacturing industry, use of
CFD normally leads to reduced design time/cycle and
improved process performance. CFD has been exten-
sively used in metal casting and simulating the flow of
molten metals in moulds.
Reilly et al.
have criti-
cally reviewed the role of CFD in defect entrainment in
WMG, The University of Warwick, Coventry, UK
Faculty of Engineering Technology, University of Twente, Enschede, The
Department of Mechanical Engineering, Technical University of
Denmark, Kongens Lyngby, Denmark
LM Wind Power A/S, Lunderskov, Denmark
Corresponding author:
Masoud Jabbari, WMG, The University of Warwick, Coventry CV4 7AL,
West Midlands, UK.
Creative Commons CC BY: This article is distributed under the terms of the Creative Commons Attribution 4.0 License
( which permits any use, reproduction and distribution of the work without
further permission provided the original work is attributed as specified on the SAGE and Open Access pages (
the shape casting process and summarized different
numerical approaches used in different commercial
packages, that is, MAGMASOFT
and FLOW-3D.
Another example of fluid flow analysis in modelling of
manufacturing processes has been given by Zhang and
in which they investigated the effect of fluid flow
in the weld pool on the numerical simulation accuracy
of the thermal field in hybrid welding.
The use of CSM as another field of interest in
numerical modelling of manufacturing process has also
dramatically increased during the last decades. In most
cases, CSM analysis is coupled with numerical heat
transfer models. As an example, Schmidt et al.
developed an analytical/numerical model for simulating
heat generation in friction stir welding (FSW), which is
based on different assumptions of the contact condition
between the weld piece and the rotating tool surface.
Schmidt and Hattel
have developed a fully coupled
thermomechanical three-dimensional model for FSW
using the arbitrary Lagrangian–Eulerian formulation
combined with the Johnson–Cook (J-C) material law.
In another study, Sonne et al.
studied the effect of
hardening laws and thermal softening on modelling
residual stresses in FSW of aluminium alloys. In their
work, the Thermal Pseudo Mechanical (TPM) model is
sequentially coupled with a quasi-static stress analysis
incorporating a metallurgical softening model.
Multiphysics modelling of welding has recently been
combined with optimization methods to obtain desired
properties of the weld.
Numerical analysis of FSW
has been reviewed in the literature in general
and with
an especial focus on modelling residual stresses.
The basis of most numerical simulation of high tem-
perature processes is a proper thermal model which in
many cases would be coupled with models describing
kinetic and/or thermodynamic phenomena. This is very
much the case for the metal casting industry which still
today is a main provider of a large amount of all manu-
factured parts. In contrast to the traditional experimen-
tal based design of casting,
numerical simulation
holds great potential for increasing the productivity in
the foundry industry by shortening production time.
This field has been active for many years and there is
still increasing effort in further developing the numeri-
cal simulation tools for variety of different casting
In the following, five different examples of manufac-
turing processes and their numerical analysis will be
presented. A short introduction to each process will be
given followed by some important results. The models
for each of the examples will only be explained in brief;
however, proper references to the original work will be
given, in case more specification is needed. The selected
examples are as follows:
1. Modelling of tape casting for thin ceramic
2. Modelling the flow of polymers in extrusion;
3. Modelling the deformation process of flexible
stamps for nanoimprint lithography (NIL);
4. Modelling manufacturing of composite parts;
5. Modelling the selective laser melting (SLM)
The two first examples deal with modelling of fluid
flow, while the following two examples treat thermome-
chanical modelling, and finally, the last example
employs a thermo-metallurgical model.
Modelling of tape casting
Tape casting is one of ceramic processes used to pro-
duce multilayer parts and substrates, like for example,
capacitors, piezoelectric actuators, gas sensors.
this process, the ceramic slurry contains different ingre-
dients, which in general influence on the rheological
behaviour of the ceramic flow by increasing the viscos-
ity magnitude and/or its non-linear dependence on
shear rate (also known as non-Newtonian beha-
However, this does not mean that assuming
Newtonian (linear) explanation for the viscosity of
ceramic is wrong.
Fluid flow analysis in tape casting and especially in
the doctor blade region and the casting reservoir (see
Figure 1) is one of the most important fields of study, as
it directly influences on the produced tapes. This has
been the focus of researchers by developing closed-form
analytical solution for the Navier–Stokes equations
(combined Couette and Poiseuille flow).
such models are limited to predicting flow front (menis-
cus), and hence, tape uniformity as well as thickness var-
iation. Numerical modelling of tape casting instead will
Figure 1. 2D schematic of the tape casting process.
2Advances in Mechanical Engineering
lead to optimizing the process by simulating more
advanced features related to flow analysis.
For fluid flow analysis in tape casting normally the
coupled momentum and continuity equations should be
solved, that is
t+r ruðÞ=0ð2Þ
where ris density (kg/m
), tis time (s), uis velocity vec-
tor, pis pressure (Pa), Tis viscous stress tensor (Pa)
and Fis the contribution from external forces. These
equations can be solved either analytically (one would
have to do quite some modifications/simplifications of
the equations to solve them analytically) or numeri-
cally, in combination with a proper constitutive law
(here the rheological behaviour). In Cartesian co-ordi-
nates, the governing equation of the shear stress (for an
incompressible fluid) can be given as
where mis the dynamic viscosity, tis the shear stress
tensor (Pa) and _
gis the shear rate tensor (Pa s). Here _
is the rate of the strain tensor, _
g=ru+ru>, which is
given by
in which II_
gis the second invariant of _
g,ruis the velo-
city gradient tensor and ru>is its transpose. For the
stress tensor, t, similarly we have
where IItis the second invariant of t. In general, it is
the correlation between shear stress and shear rate that
defines the (rheological) nature of a fluid. Fluids with
linear correlation fall into Newtonian category and
they have constant dynamic viscosity, m, as a slope in
their flow diagram, for example, equation (3). As men-
tioned earlier, assuming Newtonian behaviour will
reduce simulation complexities/efforts yet producing
fair results.
However, it is more realistic to
describe the ceramic slurry as a non-Newtonian fluid,
where tis a non-linear function of _
g. The most com-
monly used/reported constitutive models of non-
Newtonian behaviour are the shear thinning power-law
and the Bingham material models,
and they are given as follows
where mais the apparent viscosity (Pa s), nis the power-
law index and kis the consistency for a power-law fluid
(Pa sn), mBis the constant Bingham viscosity (Pa s) and
tyis the Bingham yield point (Pa). Bingham fluids do
not flow until the applied shear stress surpasses ty.
Numerically, the yield stress is treated by introducing a
very high viscosity that is active below a threshold shear
When the stress exceeds the yield stress, ty, the
fluid flows according to the plastic viscosity. In the fol-
lowing, two different numerical examples dealing with
fluid flow analysis will be presented, that is, single-
layered tapes and side-by-side layers, for modelling of
the tape casting process. Moreover, a model for the eva-
poration of water from a thin layer will be presented,
which takes place in the subsequent drying process.
As mentioned earlier, capturing the free surface is
one of the important features in the tape casting pro-
cess. An example of such investigation was first
reported by Loest et al.
Jabbari et al.
developed a
FVM-CFD model capable of tracking the free surface
(using VOF method) combined with Ostwald de
Waele–power-law flow behaviour. These developments
have inherently allowed to study the tape casting pro-
cess more in depth by attempting to simulate the
important intrinsic phenomenon called the ‘side
Side flow by definition shows amount of fluid
which flows in lateral direction as it leaves the doctor
blade region. Jabbari and Hattel
reported first of a
kind in literature where the side flow factor (a) is pre-
dicted numerically. Impact of process parameters like
substrate velocity (v0) and slurry height (H0)onawere,
moreover, investigated in the same work (see Figure 2).
Dinesen et al.
have recently reported an application
of tape casting for producing functionally graded cera-
mics (FGCs) by co-casting of two (or more) ceramic
slurries. With main application in magnetic refrigera-
tion parts, it is important to control the interface
between different layers (see Figure 3(a)) to avoid mix-
ing, and as a result increase the efficiency in a graded
magnetocaloric material.
This interface in its ideal
form has to be a 2D in-plane surface (yz) which is
perpendicular to the substrate peeling belt (xy).
Depending on the density or viscosity of the ceramic
slurry as well as process parameters, the aforemen-
tioned interface can deviate from its ideal shape. This
has been studied numerically by Jabbari et al.
the FVM-CFD model developed previously,
by fur-
ther developing for co-casting of tapes (see Figure 3(b)).
Jabbari et al. 3
Such phenomenon has been verified experimentally by
Bulatova et al.
The second stage after producing tapes is the drying
process (see Figure 4(a)), in which the solvent (water or
any other liquid carrier) is removed using heat and/or
In this stage, there two main mechanisms
which control the rate of drying namely: (1) evapora-
tion rate of the solvent from the top surface of the tape
(in contact with air) and (2) hydraulic conductivity or
the rate of solvent transport to the top surface.
Different kinetics are often covered in drying, that is,
mass diffusion, transport in porous media, evapora-
tion/condensation and viscous deformation, which
influence the overall drying behaviour of the ceramic
Neglecting capillary pressure (and resultant mass
transport in porous media), Jabbari and Hattel
numerically investigated drying of tapes by developing
semi-coupled heat and mass transfer (diffusion), see
Figure 4(b). Assuming water to be the solvent, a mix-
ture of ceramic and water was considered as a represen-
tation of a tape layer. This could then serve as a
relevant model system for analysing the drying process
in tape casting. The results of modelling are shown in
Figure 5(a) for drying tapes with initial water content
of 12% and different thicknesses (d= 400, 300,
200 mm). For the three tapes, there is initial period with
no change on the water content. This is the time period
that the tape temperature is rising without having eva-
poration, and as expected, this time period it the high-
est for the thickest tape.
Three different drying modes, that is, fast, inter-
mediate and slow, were also investigated in Jabbari
et al.
when d=400 mm, as shown in Figure 5(b). The
results showed that fast drying will result in forming an
unsaturated region (solid-like) in upper half of the tape,
which later on acts as a barrier for water transport (by
diffusion). This will eventually reduce the drying rate
for the remaining water in the tape. This may happen
during drying of tapes by high heat input rates. On the
other hand, in the slow drying mode, the water eva-
poration is somehow slow, although hydraulic conduc-
tivity supplies water for the top interface. This
headlights the influence of the drying mode, and the
(a) (b)
Figure 2. Modelling and experimental validation of side flow factor (a), impact of (a) peeling velocity and (b) slurry height.
Figure 3. (a) Schematic illustration of the interface between
the co-cast layers, (b) the influence of changing density and
viscosity of slurry on the interface: (1) r2=2r1and m2=2m1,
(2) r2=2r1and m2=m1, (3) case 1 with doubled velocity and
(4) case 2 with doubled velocity.
4Advances in Mechanical Engineering
competition between the already mentioned drying
mechanisms (the top surface evaporation and the water
diffusion from bottom to the top surface). More inves-
tigations on the drying process with high fidelity simu-
lations using coupled free-flow–porous-media models
can be found in previous works.
Modelling the flow of polymers in
Extrusion is a common process in the plastic industry
to produce long parts with a constant cross-sectional
profile. The plastic material is melted and formed
through a die with the desired cross-section. However,
the extruded material swells at the die exit, because of
the rearrangement of the flow profile after the die exit.
The flow undergoes a transition from the typical para-
bolic profiles inside the die (constrained by the walls),
towards a uniform profile outside the die (with free sur-
faces at equilibrium). This phenomenon is referred to
as the extrudate swelling. For non-axisymmetric pro-
files, the swelling may also produce distortions of the
extrudates. Moreover, the productivity of the polymer
extrusion process is often limited by flow instabilities
occurring at high extrusion speeds.
The rheological properties of the extruded materials
are crucial parameters for their processability (i.e. sta-
bility and flowability). The material inside the die is
subjected to large shear rate deformations, which trig-
ger the viscoelastic behaviours of the polymers due to
the stretching, reorientation and disentanglement of
Figure 4. (a) Overall schematic of the tape casting process and the drying sub-process and (b) schematic illustration of the
simulation domain.
(a) (b)
Figure 5. (a) Evaporation of water from a tape with different tape thicknesses (400, 300, 200 mm) and (b) the different drying
modes for the tape thickness of d=400 mm.
Jabbari et al. 5
polymer chains. In contrast to generalized Newtonian
fluids, elastic liquids build-up normal stress gradients
when they are deformed in shear flows. Outside the die
exit, the stretched polymer chains can recover their ini-
tial configuration, and the force balancing of the nor-
mal stress difference is responsible for an additional
extrudate swelling (as compared to purely viscous
fluids). Thus, taking the elastic effects into account –
although not conventional yet – is desirable, in order to
build accurate models of the polymer extrusion.
Finally, the combination of non-Newtonian flow sol-
vers developed within computational rheology with
optimization algorithms will contribute to the develop-
ment of powerful computer-aided manufacturing soft-
ware, improving the extrusion processes via assisted or
automated die design, according to specified optimiza-
tion strategies (i.e. objective functions).
The extrusion through a capillary die, sketched in
Figure 6, is particularly interesting in spite of its simple
geometry, because it gives an insight into the complex
flow phenomena of the polymeric materials. There are
different flow regimes in polymer extrusion.
Figure 7 represents the typical flow curve and the
regions of instabilities for the extrusion of linear poly-
ethylene. The vertical and horizontal axes show the
wall shear stress (proportional to the pressure inside
the die) and the characteristic shear rate (proportional
to the throughput), respectively.
Stable extrudates with smooth surfaces are obtained
at low extrusion speeds. At moderate shear rates, the
sharkskin instability appears at the surface of the extru-
date. The sharkskin defect produces irregular surfaces
of the extrudate with superficial cracks. There is a con-
sensus that the sharkskin defects occur at the die exit,
where the material near the wall is pulled out by large
tensile stress.
The two mechanisms currently
admitted to explain the sharkskin phenomenon involve
either local fractures of the polymer surface outside of
the die, or a local transition between stretching and dis-
entanglement of absorbed polymer chains inside the
In both cases, stick and partial slip plays
important roles in the sharkskin mechanisms.
At larger shear rates, the extrusion experiences a
stick–slip instability (or spurt phenomenon), character-
ized by an alternation of smooth and rough surfaces of
the extrudate, and oscillations of the pressure inside the
die. The numerical analyses presented in Georgiou and
have shown that the self-sustained
pressure oscillations of stick–slip instability require a
non-monotonic slip law (which is consistent with both
experimental data
and theoretical molecular mod-
), and either the compressibility or viscoelasticity
of the polymer melt in the reservoir. Thus, the slip
behaviour of the molten polymer on the surface of the
die is a crucial phenomenon in both the sharkskin and
the stick–slip instabilities. The recent review of
draws a complex picture of the wall
slippage of molten polymers, with two distinct slip
mechanisms: flow-induced chain desorption from the
wall (weak slip), and chain disentanglement of the bulk
from a monolayer of absorbed chains (strong slip).
Moreover, the slip mechanisms of molten polymers are
Figure 6. Schematic view of extrusion through a capillary die.
Figure 7. Typical flow curve and regions of instabilities for the
extrusion a linear polyethylene (reproduced from Achilleos
et al.
with permission).
6Advances in Mechanical Engineering
time dependent, because of the thermal degradation of
absorbed polymer chains. Consequently, the inclusion
of realistic slip laws in the macroscale models and the
prediction of the onsets and shapes of the sharkskin
and stick–slip instabilities with dynamical flow simula-
tions are still challenging tasks. Nevertheless, the shark-
skin and stick–slip instabilities can be eliminated or
minimized by enhancing the slippage of the polymer,
by means of chemical additives (processing aids) in the
polymer formulation and/or coating on the surfaces of
the die,
which results in a decrease of the wall shear
At larger throughput, the flow of molten polymers is
subjected to the gross melt-fracture instability, which is
characterized by distortions of the extruded volume. At
the onset of the gross melt-fracture, the extrudate devel-
ops regular undulated or helical shapes. The distortion
of the extrudate gradually looses its periodicity and
eventually evolves into a chaotic regime, when the
throughput is further increased. Unlike the sharkskin
and stick–slip phenomena, the gross melt-fracture origi-
nates from the bulk of the molten material and is due
to the viscoelasticity of the polymers.
by particle image velocimetry have shown that there are
correlations between the periodic or chaotic extrudate
distortions and upstream flow instabilities inside the
The onset of gross melt-fracture can
sometimes be delayed by modifications in the reservoir
without being suppressed, however.
Some experimental and theoretical studies
show that the gross melt-fracture can be due to a non-
linear subcritical instability of the viscoelastic Poiseuille
flows, independent of the flow in the reservoir. Indeed,
the elastic flow instabilities are intrinsic phenomena of
the viscoelastic flows at low Reynolds numbers, which
manifest themselves in various geometries.
In viscoelastic flows, the relative effect of the elasti-
city versus the viscous forces is characterized by the
Weissenberg number: a non-dimensional quantity relat-
ing the relaxation time lof the material and the charac-
teristic deformation rate of the flow. In an idealized
extrusion through a capillary die, the Weissenberg num-
ber is defined as
Wi =lU=Rð8Þ
where Uis the average velocity in the capillary die and
Ris the radius of the die. Thus, the elastic effects are
magnified when the extrusion speed is increased, or
when the geometry of the die is downscaled.
The constitutive behaviour of viscoelastic materials
does not only depend on the instantaneous deformation
rate _
g, but also the history of the deformations (mem-
ory effects); thus, time is a key parameter. The stress
response s(t) of molten polymers can be decomposed
where the first term on the right-hand side corresponds
to the instantaneous (purely viscous) stress response,
and tpis an extra-stress tensor, which is typically gov-
erned by an integral or a partial–differential model. The
extra-stress accounts for elastic effects, such as stress
relaxation, elastic recoil, normal stress differences. For
the sake of simplicity, we assume linear viscoelasticity
and use the well-known Oldroyd-B model
where hpis the polymer viscosity, and
is the upper-convected time derivative, which takes into
account the advection and rotation of the material in a
fixed vector base. This model gives G=hp=las the
elastic modulus, and h0=hs+hpas the apparent
steady shear viscosity.
The viscoelastic constitutive model for the extra-
stress tensor is coupled with the continuity equation
and the momentum conservation. These governing
equations are solved with a numerical method that dis-
cretizes the geometry, for instance with finite volumes
or finite elements. The Lagrangian representation of
the flow (with a mesh mapped onto the deformed geo-
metry) has demonstrated its ability to predict the steady
state of stable extrusions, see for instance.
However, the Eulerian representation (with a fixed
mesh) seems to be best suited for time-dependent simu-
lations with free surface flows.
Eulerian models have
been used to simulate extrudate swelling in the stable
flow regime, where the free surface outside the die was
represented via an explicit surface-tracking technique,
see for instance.
Recent results of Tome
´et al.
show that Tanner’s theory,
which provides
approximate analytical solutions of the extrudate swel-
ling from capillary dies, did not predict the correct rela-
tion of the swelling ratio with the Weissenberg number.
At large extrusion rates, the numerical simulations
show a linear relation, while Tanner’s theory predicted
a cubic-root dependency. Nonetheless, for Weissenberg
numbers below unity, Tanner’s theory provides accep-
table results which fit experimental data well.
Despite promising results, the numerical simulation
of viscoelastic liquids is prone to an inherent numerical
instability, referred to as the high Weissenberg number
problem, which manifests itself by the breakdown of
the simulation (i.e. blow-up of numerical values), when
the Weissenberg number reaches a critical value.
This has been a long-standing challenge of computa-
tional rheology, and the origins of the problem have
Jabbari et al. 7
been understood only recently. The simulations crash
when the conformation tensor
a non-dimensional internal variable representing the
strain of the polymer chains, which should be sym-
metric positive definite by definition,
looses its posi-
tive definiteness.
Moreover, the loss of positive
definiteness happens because of numerical errors dues
to under-resolution of the spatial stress profiles,
which may have steep gradient variations and/or be
exponential, near the boundary layers and geometrical
A seminal reformulation of the differen-
tial constitutive model (3) has been proposed by Fattal
and Kupferman.
The problem is removed by a
change of variable using the matrix-logarithm of the
conformation tensor
C= log cðÞ ð13Þ
In terms of c, the exponential stress profiles become
linear and are easily resolvable. An evolution equation
for the log-conformation tensor is derived as
t+urCVC CVðÞ2E=1
lexp CðÞI½
where Vand Eare anti-symmetric (pure rotation) and
symmetric traceless (pure extension) matrices that com-
pose the velocity gradient ru, see Fattal and
for more details. Finally, the recovery of
the conformation tensor by the matrix-exponential
operation c= exp (C) automatically enforces the posi-
tive definiteness. The log-conformation representation
was a breakthrough that has opened new possibilities
in the simulations of the viscoelastic flows dominated
by elastic effects, like in the gross melt-fracture for
instance, which were not accessible before.
To the knowledge of the authors, the recent work of
is the only attempt of direct numerical simula-
tions of the gross melt-fracture instability that have
been reported in the literature. Kwon simulated the
flow of an elastic liquid through a two-dimensional
extrusion die, using the log-conformation representa-
tion with the Leonov constitutive model.
The geome-
try of the simulation included two rectilinear sections
with an abrupt contraction, representing the die and
the extruder reservoir. Kwon’s model neglects the iner-
tia and the slip of the polymer melt on the die’s wall
(i.e. no-slip boundary condition). The free surface of
the extrudate was tracked with the Level-Set method.
The results presented different types of unstable extru-
sions, where some are qualitatively similar to the gross
melt-fracture defect. The simulations showed that these
extrusion instabilities originate from flow instabilities
emanating from various locations within the die and
reservoir, and which are only attributable to elastic
A similar modelling approach has been employed by
to simulate two-dimensional viscoelastic
flows at the exit of a slit die. Comminal et al.
the same hypothesis as Kwon of an inertialess flow and
no-slip at the walls, but used the log-conformation rep-
resentation with the Oldroyd-B model and a viscosity
ratio hp=8hs. Moreover, Comminal et al. utilized a
pressureless reformulation of the conservation laws in
terms of streamfunctions,
defined as the vector
potential of the incompressible velocity field
(Helmholtz decomposition)
In two dimensions, the in-plane velocity components
are solely defined by the out-of-plane component fzof
the streamfunction vector
and fzis governed by the following evolution equation
(derived from the curl of the momentum equations)
The streamfunction formulation removes the pres-
sure unknown and automatically fulfils the mass con-
servation, by construction, in virtue of the following
vector calculus identities
r r3FðÞ=0,8F2R3ð18Þ
The pressureless formulation removes potential flaws
due to the coupling of the pressure with the velocity
and the extra-stress. Numerical investigations in the lid-
driven cavity have shown an enhancement of the accu-
racy and robustness of the viscoelastic simulations.
In contrast to Kwon, Comminal et al.
modelled a
slit die without reservoir (no contraction geometry).
Moreover, the symmetric boundary condition was
applied at the mid-plane of the die, to limit the compu-
tational costs. Hence, the simulations of unstable extru-
sions are constrained to solutions with a symmetric
mode. The free surface of the extrudate was tracked
with a recently developed volume-of-fluid advection
The results present stable extrudates, for
low throughputs only. At the moderate Weissenberg
number Wi =1:5, the extrudate is unstable but has a
surface with wavy undulations coming from regular
fluctuations in the stress boundary layer inside the die,
8Advances in Mechanical Engineering
see Figure 8(a) and (b). Another type of instability is
visible at the larger Weissenberg number of Wi =6,
with a more complex shape of the extrudate, see Figure
8(c). In spite of the enforced no-slip boundary condi-
tion, elastic stress instabilities emulate cohesive failures
that are qualitatively similar to those of the stick–slip
phenomenon. As a result, the simulated free surface
resembles that of the transition regime of the stick–slip
instability observed in Figure 8(d), except that the
simulation does not produce the sharkskin surface.
Moreover, the present model cannot truly reproduce
the helicoidal instabilities as observed in real extru-
sions, since these are asymmetric and three-dimen-
sional. Nonetheless, the present simulations indicate
that the unstable extrusion can be caused by purely
elastic instabilities arising from the parallel flow inside
the die, independent of the reservoir. Noticeable fluc-
tuations in the normal stress difference near the wall
give rise to oscillations in the velocity profile, which
eventually result in distortions of the extrudate outside
the die. The simulated instability seems to have a peri-
odicity, but additional investigations are necessary to
confirm this observation. Finally, since the governing
equation for the extra-stress tensor is hyperbolic, its
solution depends on the initial and boundary values
imposed at the upstream of the flow’s characteristics.
Thus, potential instabilities in the elastic stress are also
influenced by arbitrary choices made for the extra-
stress values at the inflow boundaries.
Moreover, the
use of pre-computed, fully developed, steady-state
stress profiles at the inflow boundaries may be ques-
tionable, given the proneness of viscoelastic flow
towards elastic turbulence.
In conclusion, it has been shown that direct numeri-
cal simulations of the gross melt-fracture phenomenon
are achievable, provided that the high Weissenberg
number problem is removed, for instance with the log-
conformation representation. However, further work is
necessary to refine the models. To date, the simulations
do not include all the complex phenomena present in
polymer extrusion that are attributable to non-linear
slip laws, inertia and three-dimensional effects.
Nevertheless, the numerical simulations remain power-
ful tools to discover the underlying mechanism of the
gross melt-fracture and its possible link with elastic
Modelling the deformation process of
flexible stamps for NIL
Functional nanostructures on double curved surfaces
have attracted increasing attention in industry.
Examples of functional nanostructures are well known
from nature, where organisms and plants possess opti-
cal, adhesive and self-cleaning capabilities.
The sci-
entific literature is rich in examples of advanced
materials emulating the well-known super-hydrophobic
effect of the lotus leaf
and adhesive surfaces of the
gecko’s feet.
Structural colours and iridescence are
most often observed in invertebrates such as butterflies
(see Figure 9) and beetles, but also in the feathers of
Since the early observations of functional
nanostructures in nature, engineers have tried to repli-
cate these nanostructures in order to functionalize sur-
Recently, Christiansen et al.
developed a
surface consisting of 1D gratings with varying wave
length and orientation, resulting in a glitter colour
effect, where the colour appearance is angle dependent,
see Figure 9(a). The nanostructures are in first place
created in a clean room, where technologies from the
Figure 8. Simulated free surfaces of the extrudate and first normal stress differences inside the die, (a) for Wi =1:5 and (c) for
Wi =6. (b) Wavy distortion of HDPE extrudate with an apparent shear rate of 500 s
; reproduced from Kalika and Denn
permission. (d) Transition regime of the stick–slip instability of a LLDPE extrudate with an apparent shear rate of 985 s
reproduced from Kalika and Denn
with permission.
Jabbari et al. 9
semiconductor industry, such as electron beam litho-
graphy (EBL) and standard ultraviolet (UV)) lithogra-
phy, are applied in order to transfer the nanostructures
to a silicon wafer. From there, the nanostructures are
transferred to the surface of the material where the
functionality is wanted via a technology called NIL a
technology invented in 1995.
In the study conducted by Sonne et al.,
the objec-
tive was to upgrade existing injection moulding produc-
tion technology for manufacture of plastic components
by enhancing the lateral resolution on free-form sur-
faces down to micrometre and nanometre length scales.
This will be achieved through the development of a
complete NIL solution for structuring the free-form
surface of injection moulding tools and tool inserts.
This will enable a cost-effective and flexible nanoscale
manufacturing process that can easily be integrated
with conventional mass production lines. The proposed
technology enables functionality of plastic surfaces by
topography instead of chemistry with some of the appli-
cations as describe above (e.g. colour or hydrophobic
effects). A manufacturing process resembling some of
the same methods as presented here, though with a
nickel foil used for transferring the nanostructures, has
been shown in the literature.
Furthermore, other
manufacturing processes associated with additive man-
ufacturing (AM) such as direct laser writing (DLW)
allow for creating sub-100 nm structures with similar
functionalities as mentioned above.
However, the
method presented in this work allows for resolutions as
achieved in the semiconductor industry and the wafer-
based platform makes the process competitive to the
AM methods in a mass production environment. The
anticipated pattern transfer method from a silicon
wafer to final injection moulded plastic product is
visualized in Figure 10 (this technology has been tested
at pilot scale and implemented industrially
). (1) A
planar microstructured and nanostructured master is
prepared by existing microfabrication and nanofabrica-
tion techniques such as EBL and photolithography. (2)
A flexible stamp is designed, replicated from the planar
master. (3) The injection moulding tool insert is coated
with a polymer resist for NIL. (4) Stamping equipment
is employed to imprint the flexible stamp (by a blow-
moulding like process) into the double curved (free-
form) injection moulding tool insert by use of com-
pressed air. The flexible stamp is imprinted into the
resist on the insert. (5) The injection moulding tool
insert surface is patterned by means of electroplating or
etching, using the imprinted polymer resist as a masking
layer. (6) The nanostructured injection moulding tool
insert is used in a conventional injection moulding
For NIL on injection moulding tool inserts (and on
double curved surfaces in general), the deformation
and stretch of the flexible stamp can be up to 50% and
in the millimetre range.
This is problematic since
even very small changes in the nanostructures geometry
will change the wanted overall functionality.
deformations are therefore important to take into
account, when the planar silicon masters are designed.
Here simulation tool such as the finite element method
Figure 9. (a) Green swallowtail, an explanation of colours.
(b) Diffraction gratings with different orientations and periods can be
combined, resulting in a glitter colour effect.
10 Advances in Mechanical Engineering
(FEM) is an appropriate way of predicting those defor-
mations. Numerical models for simulating the NIL pro-
cess in general have been shown in the literature,
with the main purpose of solving for the flow of resist
and the local stamp bending, with the classical squeeze
flow model (the Stefan equation
) as benchmark
where his the height of the resist, pis the pressure, his
the viscosity and sis the permeability.
The Stefan equation is however not directly suitable
for the NIL on curved surfaces due to a number of dif-
ferent aspects that complicates the overall physics: (1)
Because of the large deformations and hence strains in
the flexible stamp, non-linear geometry has to be taken
into account in order to get accurate results from the
numerical simulations.
(2) The material behaviour is
non-linear and is for the polymer material described by
a viscoelastic–viscoplastic constitutive behaviour.
(3) With respect to the contact conditions, changes in
frictional behaviour on the different length scales
(macroscale to nanoscale) have to be taken into
and add yet another nonlinearity to the
system of equations that have to be solved. Here, multi-
scale modelling is an approach where the global and
local conditions can be taken into account in the same
In the work done by Sonne et al.
a standard con-
tinuum mechanics was applied with the overall goal of
solving the three static equilibrium equation
where pjis the body force at any point within the geo-
metry and sij is the stress tensor. The flexible stamp is
in this case made of polytetrafluoroethylene (PTFE)
which due to its thermal resistance is suitable for NIL
which normally is performed at elevated temperatures
(when applying a non-UV curing resist). The material
behaviour has to a large extend been shown in the liter-
ature, and different corresponding constitutive models
have been proposed
These different contributions
are in overall agreement on how the PTFE material by
a 1D rheological representation can be described. In
the present work, the overall approach is to include the
viscoelastic behaviour represented by a Zener body
by a modified version of Hooke’s law represented by
the elastic stiffness tensor
ijkl =Emod
2dik djl +dildjk
Emod =Emax +E2,Emax =E1
where his the viscosity of the dashpot, E1and E2are
Young’s moduli for the springs and Dtis the time
Figure 10. A flexible nanoimprint solution for nanostructuring an injection moulding tool insert.
Jabbari et al. 11
increment size. It is assumed that J2flow theory applies
during the viscoplastic deformation of the PTFE mate-
rial. Hence, the material has to satisfy the von Mises
yield condition
where seis the equivalent stress. The yield stress syis in
this work given by the J-C viscoplastic model,
from previous work has been shown to give a reason-
able correlation between strain, strain rate, temperature
and stresses for this material.
The model has been verified through a series of
experiments, where NIL on a double curved tool insert
for injection moulding (Figure 11(a)) was performed
with different process parameters. The outcome of the
model is first of all contour plots of the maximum prin-
cipal strain field on the geometry of the deformed flex-
ible stamp (see Figure 11(b)). This will give an
indication of how much the nanostructures have been
stretched during the nanoimprinting process.
Furthermore, the transient pressure distribution
between the mould and flexible stamp can be examined
and used to optimize the process parameters in terms
of imprinting temperatures and pressures and the thick-
ness of the flexible stamp.
For the application shown in Figure 11, good agree-
ment between simulations and experimental results in
terms of maximum principal strain was found, see
Figure 12.
In general, both experiments and simulations have
shown a mismatch between the defined and measured
nanostructures as a result of stretching of the flexible
stamp. This clearly indicates the necessity for numerical
models in order to take this deformation into account
when the nanostructures are designed in the first step
of the process. The developed models have shown to
predict the stretch of the nanostructures with a maxi-
mum error of 0.5% defined as the difference between
measured and simulated wave lengths of the nanostruc-
tures on the surface of the deformed flexible stamp,
indicating their ability to capture the essential physics
of this manufacturing process.
Modelling manufacturing of composites
Fibre-reinforced polymer (FRP) matrix composite
materials have found widespread use in the manufac-
ture of large energy critical structural applications such
Figure 11. (a) The used flexible stamps with the glitter effect nanostructure on top of the steel tool insert after nanoimprint. (b)
Numerical model of the same setup where it is possible to study the strain field within the flexible stamp in combination with the
actual imprinting pressure underneath the flexible stamp.
Figure 12. Comparison of measured and calculated maximum
principal strains along two different paths on the deformed
flexible stamp.
12 Advances in Mechanical Engineering
as wind turbine blades where low weight and high
strength are crucial. In the wind energy industry, the
use of FRP composites in large wind turbine blades has
surpassed all other materials for both the load-carrying
and airfoil sections of the blades.
Composites offer a
low weight-to-stiffness ratio and user-defined high dur-
ability and strength. Application-defined properties are
also more readily obtained by tailoring the composite
lay-up, thicknesses and fibre/matrix combinations
according to the mechanical, thermal, electrical or aes-
thetic requirements. During the manufacture of thermo-
set polymer composites in general, the reinforcement
fibre material is moulded into a desired shape by
impregnation and curing of the fibres with a liquid
polymer resin, typically a thermosetting epoxy or polye-
ster. Upon curing a structural unified solid combination
of these materials is achieved, resembling the mould
geometry. Many different variations of this basic pro-
cess exist, for instance pultrusion,
filament wind-
resin transfer moulding (RTM),
vacuum infusion (VI),
vacuum-assisted resin transfer
moulding (VARTM),
to name a few.
In this arti-
cle, focus is on two different processing methods: pul-
trusion and VARTM, see Figures 13(a) and (b),
respectively. The combination of thick geometries, cure
cycles at elevated temperatures and the highly non-
linear resin phase transition characteristics and release
of latent heat can make the process complex to con-
Furthermore, avoiding process-induced
shape distortions and residual stress build-up remains a
challenge in many applications.
Only the main features of the needed theory for
modelling of composite manufacturing will be outlined
here. In order to analyse the thermal conditions during
processing, the energy equation must be solved
Q000 ð25Þ
Note here that the material flow during processing is
taken into account via the advection on the left-hand
side, where uis the pulling speed in the x-direction for
pultrusion, and u=0for VARTM. Other material
flow than this is not considered here. _
Q000 is determined
by the total reaction enthalpy of the matrix material
and the cure rate which in turn depends on the degree
of cure and temperature in a highly non-linear manner,
often described by the Kamal and Sourour autocataly-
tic kinetic expression.
Thermoset resins also exhibit volumetric shrinkage
during curing, sometimes as high as 9% for polyesters.
Moreover, as the resin develops from the viscous state
to the solid state, large changes in the thermal and
mechanical properties occur. Johnston
proposed a
modified linear elastic model incorporating tempera-
ture dependency which also allows thermal soften-
This model which is often denoted the
model cure hardening instantaneous linear elastic
(CHILE) model is used in the present work. This indi-
cates that with an increase in degree of cure, the modu-
lus increases monotonically. User-defined subroutines
are developed in ABAQUS for the constitutive material
models. The corresponding expression for the CHILE
model is seen in equation (26)
Aeexp (KeT);TC1\T\TC2
where Trepresents the difference between the instanta-
neous glass transition temperature (Tg) and the resin
temperature T, that is, T=TgT.Aeand Keare the
constants for the exponential term. TC1,TC2and TC3are
defined as the critical temperatures and E0,E1and E
are the corresponding elastic modulus values, respec-
tively. More specifically, E0and Eare the viscous and
glassy state elastic modulus, respectively. Tgcan be
defined as
where T0
gis the glass transition temperature at a=0
and aTgis a fitting constant.
Figure 13. (a) Schematic view of the traditional pultrusion process and (b) schematic illustration of VARTM.
Jabbari et al. 13
The effective mechanical properties as well as ther-
mal and chemical shrinkage strains of a laminate are
calculated using a micromechanics approach, for exam-
ple, the self consistent field micromechanics (SCFM)
approach which is a well-known and documented tech-
nique in the literature.
During composites processing, manufacturing-
induced strains are known to develop as a result of the
matrix material chemical cure shrinkage in combination
with thermal gradients as well as mismatch in thermal
expansion properties.
More specifically, knowing the
isotropic resin shrinkage strain, the incremental longitu-
dinal and transverse chemically induced shrinkage can
be determined for a composite material taking micro-
mechanics into account. Thermal strain increments are
calculated in a similar manner based on the temperature
change and the composite effective thermal expansion
coefficient. Since strains are in general small, linear
strain decomposition can be used such that the process-
induced strains are expressed as the sum of the thermal
strains and chemical strains and finally the total strains
are hereafter found from adding the mechanical strains
and process-induced strains. At each time step, the
properties of the rein are updated. Using the SCFM
approach, the effective properties of the composite
laminate are then calculated in which the fibre proper-
ties are assumed to remain constant.
Numerical modelling of pultrusion
Pultrusion is a cost-effective production process for
manufacturing FRP composite profiles. Constant
cross-sectional profiles are produced in a continuous
manner. The resin material impregnated the fibre rein-
forcements in a resin bath system while being pulled by
a pulling mechanism. Curing takes place inside the hea-
ter pultrusion die. A saw is often used to cut the profile
into desired length. A schematic illustration of the pro-
cess is seen in Figure 13.
Pultrusion process has been analysed using various
thermo-chemical numerical models since 1980s in order
to control the curing and temperature distribution dur-
ing processing.
Some of the thermo-chemical
computational analyses can be found in previous
Baran et al.
have made the first thermo-
chemical-mechanical model of pultrusion in which a
coupled thermo-chemical-mechanical is developed
using ABAQUS and used to model the pultrusion of
the L-shaped profile as shown in Figures 14 and 15. In
Baran et al.,
a more comprehensive mechanical
model was proposed to analyse the process in 3D by
calculating the stresses and shape deformation. A glass/
polyester was considered in Baran et al.
for the UD
as well as the CFM layers, and for the die, a chrome
steel was used. As shown in Figure 15, the heating
regions were 275 mm long and 60 mm wide and they
were placed 150 mm apart from each other. The length
of the die was 1000 mm and the post-die region was
5000 mm long. The surfaces of the composite at the
post-die region were exposed to convective cooling
boundary condition. Similar boundaries were also
defined for the die surfaces except the heater regions.
The polyester resin entering the die inlet was assumed
to be at uncured viscous state and the curing took place
with the help of heaters. The necessary material proper-
ties were characterized in Baran et al.
such as curing
kinetics, temperature and cure-dependent elastic modu-
lus and viscosity.
The generalized plane strain elements CPEG8R
available in ABAQUS were used in the 2D quasi-static
mechanical analysis. Figure 15 shows the 2D cross-
section of the L-shaped profile. A Lagrangian frame-
work was applied in the 2D mechanical model such that
the temperature and cure distributions calculated in the
3D thermo-chemical model as well as the mechanical
boundary conditions were updated at each quasi-static
time step.
The model predicted the stress–strain
development as well as the nodal displacements.
The predicted spring-in angle as a function of the
pulling distance is depicted in Figure 16 (left). It is seen
that the deformation process of the profile continues
until around 6 m away from the die-exit during the
cooling stage in air. The final spring-in angle as a func-
tion of different pulling speeds from 500 to 1000 mm/
min is depicted in Figure 16 (right). It was found that
the spring-in angle increased with pulling speed.
Moreover, the final spring-in for parts produced with a
pulling speed of 600 mm/min was measured and good
agreement with the predicted value was found.
Figure 14. Pultruded L-profile.
14 Advances in Mechanical Engineering
In Baran,
the process-induced residual stress
development was described for a 100 3100 mm pul-
truded square profile made of glass/polyester. The tem-
perature and degree of cure distributions were
calculated for three different preheating temperatures.
The nonuniform internal constraints in the part yielded
in an internal shear deformation during the process.
The transverse shear stress and compressive normal
stress levels decreased significantly as compared with
the tensile normal stresses with an increase in preheat-
ing temperature. Predicted transverse shear stress distri-
butions are depicted in Figure 17.
The mechanical properties of the fibre-reinforced
composite materials might vary significantly due to the
manufacturing-induced effects.
The possible causes
for this variation can be the nonuniform distribution of
fibre/matrix, fibre misalignment, residual stresses at
micro and meso level, dry spots and voids due to poor
impregnation and so on. The probabilistic and reliabil-
ity analyses of the pultrusion process were studied in
Baran et al.
The variability in the product proper-
ties and their effects on the curing and temperature
were investigated. In addition, the process conditions
were optimized in several studies for the pultrusion
Modelling process-induced strains and stresses in a
thick laminate plate
The internal strain development during curing of a
thick laminate plate at elevated temperatures using a
long cure cycle was analysed by Nielsen et al.
objective with the study was to evaluate how accurate
the aforementioned CHILE constitutive model is at
predicting the internal strain development, in what is
essentially a viscoelastic problem. 3D numerical model
total strain predictions are compared with experimen-
tally determined in situ strains within the laminate part,
using embedded optical fibres with fibre Bragg grating
(FBG) sensors. Hence, a direct comparison can be
made with model predictions, capturing how thermal
expansion and chemical shrinkage strains develop
within the laminate throughout curing during the VI
moulding process, as well as the influence of tooling.
A glass/epoxy UD laminate was manufactured using
the VI techniques. The dimension of the laminate was
400 3600 mm consisting of 52-layer UD E-glass fibre
mats. A tempered glass of 10 mm thickness was
employed as a mould.
Prior to infusion, sensors
were embedded within the dry preform; including ther-
mocouples and optical fibres (FBGTop, FBGCenter
and FBGBottom), each consisting of three FBG sen-
sors interspaced along the optical fibre, see Figure
18(a). The optical fibres were placed transversely to the
main reinforcement fibre direction in order to capture
the matrix-driven process strains. The laminate was
vacuum infused and cured at 40°C on a heated plate.
A sequentially coupled thermomechanical numerical
analysis was subsequently conducted. Symmetry condi-
tions were assumed why only half of the laminate plate
and tool was modelled (Figure 18(b)). In the thermal
step, Dirichlet boundary conditions (known tempera-
tures) were prescribed on the laminate plate top and
bottom surfaces, as determined experimentally.
Moreover, the heat flux through the symmetry plane
was neglected. In the mechanical step, a tied condition
Figure 15. Pultrusion setup of L-shaped profile with dimensions indicated.
Jabbari et al. 15
Figure 16. Left: Predicted spring-in angle of L-shaped profile as a function of the pulling distance. Right: Predicted spring-in angles
for different pulling speeds (compared with experimentally found value for 600 mm/min
Figure 17. Transverse shear stress distribution (t23) for different Tpreheat values after the pultrusion process.
Figure 18. Schematic of (a) exploded view showing the embedded sensor placement, (b) laminate and tool plane of symmetry and
(c) mechanical contact conditions assumed.
16 Advances in Mechanical Engineering
at the tool/part interface was prescribed, mimicking
perfect bonding between the tool and laminate during
the process. This condition is a simplification of real-
life sliding, sticking and other cohesive interfacial con-
tact conditions. In the model, it was assumed that the
preform is perfectly impregnated through the use of a
constant fibre volume fraction resembling the final
state of the part after processing.
Demoulding is
modelled after curing at ambient temperature by simply
suppressing all mechanical constraints between tool
and part.
Figure 19 presents the experimentally determined
centre plane laminate temperature (T2), compared with
model predictions of temperature and corresponding
cure degree model predictions. Upon commencement
of the curing reaction, an increase in temperature is
seen as a result of heat generation during the release of
latent heat upon matrix cross-linking. After the tem-
perature peak, a steady-state temperature is upheld cor-
responding to approximately 40°C, after which the
laminate cools to ambient temperature. It is seen that a
substantial amount of curing develops prior to when
the exothermic peak temperature is reached. This
relates to the autocatalytic equation for the cure rate
being self-catalysed by increases in temperature.
Figure 20 shows the comparison of the predicted
transverse strains with the measured ones at the centre
of the laminate (mid-plane). The total strains are seen
to relate somewhat to the thermal profile presented in
Figure 19, deferring mainly due to the influence of
chemical cure shrinkage. Good agreement between the
model predictions and the experimentally measured
strains is observed, with the best agreement early in the
process. As the resin cures, the strain development goes
from being driven by the tool expansion due to the tied
contact, to being driven by the induced thermal and
chemical shrinkage laminate strains.
The influence of the tool (tempered glass plate) is
visible upon cooling, that is, at time of 1050–1300 min.
It is seen that a smaller gradient is present on the
predicted total strain curves as compared to the experi-
mentally determined strains. This may be due to the
lower coefficient of thermal expansion (CTE) of the
tempered glass plate (9:031068C1) as compared with
the transverse CTE of the laminate plate in the glassy
(cured) state (46:07 31068C1). The tool–part interac-
tion may cause the difference in strain between the
model and experiments during cooling down stage, for
example, stick–slip friction contact behaviour. In order
to accurately model this behaviour at the tool/part
interface, information of contact properties such as the
orthotropic friction coefficients, along with the maxi-
mum effective contact shear stress, is needed, as a func-
tion of temperature and cure degree. Hence, the tied
contact approximation is a reliable simple
Thermo-metallurgical modelling of SLM
Modelling of thermal conditions during manufacturing
processes and resultant metallurgical features is a well-
established field of research.
In the case of SLM,
however, the application of these techniques of micro-
structure predictions is still limited. Nonetheless, the
plethora of experimental studies in the literature focus-
ing on characterizing and empirically predicting the
microstructure during SLM of materials
indicates the scope and need of thermo-metallurgical
modelling studies.
In this section, a brief overview
of the work carried out on thermo-metallurgical model-
ling of SLM is presented. Subsequent to a brief outlin-
ing of the theory associated with the thermo-
metallurgical model, the evolution of centre-line micro-
structure during single melt track formation is simu-
lated and discussed.
process was developed in Fraunhofer
Institute for Laser Technology in Aachen, Germany
Figure 19. Experimental laminate mid-plane temperature and
model prediction. Also shown is the corresponding modelled
cure degree development.
Figure 20. Process-induced total transverse strain comparison
between model predictions and experimental data. Also seen is
the predicted cure degree development at the laminate plate
Jabbari et al. 17
and it has since been a prominent centre for research in
SLM. Together with selective laser sintering and elec-
tron beam melting, it is among the most prevalent form
of metal AM. Potential target areas providing signifi-
cant scope for SLM include aerospace applications,
automobiles and medical implants. Due to the clear
advantages of such an AM process, SLM has been the
subject of intense research in the last decade. SLM and
similar metal AM processes have been reviewed in pre-
vious works
from experimental and modelling
perspective, including current practices and future chal-
lenges regarding raw materials, processing and post-
The SLM process begins after a sliced 3D CAD
model is received by the machine. The sliced model con-
tains information about the zones to be melted in differ-
ent layers. A powder scrapper moves some powdered
material from the feed container onto the build plat-
form, distributing it more or less uniformly. Then the
laser beam source is turned on. Typically, a Nd:YAG
or CO
laser having a Gaussian energy distribution is
used as a power source for selectively melting certain
zones in a layer of powdered material, which eventually
give rise to a 3D structure (Figure 21). The generated
beam is deflected by scanner mirrors which control the
movement of the laser beam over the powder surface.
The beam is focused by the help of a lens that can alter
the focusing distance and the divergence of the beam.
Depending on the input from the sliced 3D model, the
laser beam melts out a shape in the powder layer. Upon
completion of the laser treatment, the build platform
moves down by a fixed amount and another layer of
powder is scrapped onto it. The process is repeated until
the complete 3D object is created. The entire process is
carried out in an inert atmosphere of argon or N
continuous gas flow through the chamber.
In SLM process, the melt pool size (typically in the
range of 100–500 mm) is small enough compared to the
size of the component being manufactured that conduc-
tion becomes the primary mode of heat transfer in the
entire domain. In such a case, the spatial and temporal
distribution of temperature is governed by the heat con-
duction equation, which can be expressed as
... ð28Þ
where Tis the temperature, tis the time, (x,y,z) are the
spatial co-ordinates, kxx,kyy and kzz are the thermal con-
ductivities in the different directions, ris the density,
Cpis the specific heat and F
is the heat source term.
However, at a local scale, several interacting physical
phenomena such as Marangoni flow in the melt pool,
keyhole formation and vapour expulsion, inter-particle
radiative heat transfer, temporary plasma formation
and collapse can occur depending upon the process
parameters. Using a conductive heat transfer based
model to simulate the SLM process, thus, requires sev-
eral modifications and calculation of equivalent mate-
rial properties. For instance, SLM is usually carried
out in a chemically inert gaseous environment under
continuous flow, which can be modelled via a
thermal interaction at the components exterior bound-
aries following the Newtonian cooling and Stefan–
Boltzmann law
amb TðÞ+seT4T4
where his the heat transfer coefficient, Tamb is the
temperature of the gaseous environment, sis the
Stefan–Boltzmann constant and eis the emissivity of
the material. The above thermal equations can then be
solved using any of the different numerical tech-
One of the most common material property for
which an equivalent value is required when using a con-
ductive heat transfer formulation is the emissivity of the
powder bed. The current work is based on the predic-
tive models proposed by Sih and Barlow
a combination of the emissivity of the particles and the
emissivity of the cavities in the powder bed is used
where eis the effective emissivity of the powder bed, es
is the emissivity of the bulk material, ehis the emissivity
of the cavities and Ahis the area fraction of surface
Figure 21. Schematic representation of heat transfer during
selective laser melting.
18 Advances in Mechanical Engineering
occupied by the cavities. The morphology as well as
density of the powder bed packing influences the effec-
tive emissivity via the area fraction, Ah. For a randomly
packed powder bed of porosity (f), the area fraction
and the emissivity of the cavities are given by
es2+3:082 1f0
es1+3:082 1f0
Similarly, effective thermal conductivity values also
need to be found for the powder bed as well as for mol-
ten/vaporized materials. A modified version of the
¨hler model found in the lit-
erature, which describes a randomly packed powder
bed formed of mono-sized spherical powder particles,
can be used to calculate the equivalent thermal conduc-
tivity as
ln ks
where kis the effective thermal conductivity of the pow-
der bed, f0is the porosity of the powder bed, ksis the
thermal conductivity of the bulk material, kfis the ther-
mal conductivity of the gaseous environment and kris
the equivalent thermal conductivity arising due to inter-
particle radiation given by
where sBis the Stefan–Boltzmann constant, Tis the
mean absolute temperature and Dpis the diameter of
powder particles. Fis called the view factor and can be
chosen as a function of the emissivity of the powder
bed leading to
Similarly, for properties related to laser–material
interaction such as albedo, scattering coefficient, extinc-
tion coefficient, equivalent values need to be calculated.
While the involved physics can typically only be mod-
elled via differential and integral equations, simple
empirical models drawing upon experimental results do
exist in the literature. For instance, the extinction coef-
ficient (a) and scattering albedo (v) of bulk material
can be used to approximate an effective absorption
coefficient for the powder bed with an assumption of
diffusive reflections on powder particles
Aeffective =3
Apart from calculation of equivalent material prop-
erties, the heat conduction equation also needs to be
provided with appropriate heat source term F
in order
to account for heat input via the laser beam. In case of
a simple laser beam with a Gaussian distribution pro-
file, the irradiance is given by
where P0is the power of the laser beam, v0is the beam
1=e2radius and ais the absorptivity of the laser beam
in the material. For a laser beam moving on the powder
bed with a velocity (vx,vy), the thermal flux is thus com-
puted as
where qsis the thermal flux, (x0,y0) is the initial location
of the laser beam and tis the time. The calculated ther-
mal flux then needs to be converted into an equivalent
volumetric heat generation to be put into the heat con-
duction equation as a source term. The above govern-
ing equations and constitutive models are sufficient to
develop a continuum model of the thermal conditions
during SLM.
The internal state variable approach
is well suited
to the development of models for non-isothermal
microstructural evolution. In general, a microstructure
may be defined by different state variables such as
grain size, volume fraction of grains, fraction of solid
in solidification. For the current case of solidification,
two internal state variables can be used to describe the
microstructure evolution – namely temperature and
fraction of solid. The usage of these two state variables
for modelling solidification microstructure is described
The interaction of the two state variables (tempera-
ture and fraction of solid) determines the type of micro-
structure formed during solidification. All grains
produced during solidification are assumed to be
equiaxed in nature, with columnar grains considered to
be similar to elongated equiaxed grain.
Jabbari et al. 19
The equiaxed solidification begins with nucleation in
the regions of melt pool just below liquidus temperature
and along the boundary of the melt pool. Although
nucleation is a randomized phenomenon, statistical
models exist for characterizing the overall nucleation in
a melt pool. The Oldfield model
is most popular
wherein the nucleation distribution is described by a
Gaussian distribution. The grain density at a particular
undercooling is given by
exp 1
where nmax is the maximum possible nucleation density,
DTNis the mean undercooling and DTsis the standard
deviation of the grain density distribution. Typically,
these three parameters need to be obtained experimen-
tally for a given material and condition.
Nucleation can occur in a homogeneous or hetero-
geneous manner; however, the latter is much more
common due to less energy requirements. The heteroge-
neous nucleation in the above model is assumed to be
instantaneous and dependent on the characteristic
undercooling. Once nucleated, the grains start to grow
outwards. This results in the fraction of solid in the
local melt pool area increasing, and there is corre-
sponding release of latent heat of fusion which might
increase the local temperature(this phenomenon is
called recalescence). The grain nucleation is also ran-
dom with respect to the orientation of the grain. Thus,
to distinguish between different grains, a characteristic
misorientation angle is attributed to each grain with
respect to one of the principal direction (Figure 22).
Dendritic grain growth is assumed to be primarily
driven by thermal undercooling and curvature under-
cooling. The kinetic undercooling and solute under-
cooling are ignored for the present work, although they
are known to be important especially when in case of
liquids with high Peclet number. The adopted grain
growth model is similar to that described by Gandin
and Rappaz.
The model focuses on the envelope outlining the
dendrite tip positions. In grains with four dendritic
arms as in (corresponding to \11.direction), the
envelope can be approximated as a square. The nuclea-
tion site for the grain shown in Figure 22 is labelled as
n, and the grain has already been growing for time tas
shown by the (inner) square envelope. The current
length of the dendrite tips at time tis shown by Lt
where i=1,2,3,4are the four directions of growth.
For the purpose of grain growth, the temperature inside
the square envelope is assumed to be locally uniform.
Typically, nucleation is spherical and a small incuba-
tion time is associated with each nucleus necessary to
generate the instability leading to dendritic growth. The
incubation time is neglected here and the grains are
assumed to be dendritic from start.
The growth of the grain is associated with the length
of the dendrite tip which in turn depends upon the
growth kinetics. For a grain nucleated at time tnand
experiencing an undercooling of DT, the length of den-
drite tip is given by
where qis the velocity of the dendrite tip (i.e. interface
velocity). The total undercooling DTis decomposed
into a sum of two different terms
with the contribution of thermal undercooling DTtand
curvature undercooling DTrbeing expressed as
ðÞ ð43Þ
where DHfis the latent heat of fusion, Cris the specific
heat, Ptis thermal Peclet number, Gis the Gibbs–
Thomson coefficient defined as the ratio of the solid–
liquid interfacial energy to the entropy of fusion, Ris
Figure 22. Schematic diagram illustrating the growth algorithm
used in the cellular automata model for a dendritic grain whose
\10.direction is misoriented by an angle uwith respect to the
horizontal axis of cellular automata network.
20 Advances in Mechanical Engineering
the radius of the dendrite tip and Ivis the Ivantsov
function expressed as
zdz ð45Þ
The radius of curvature of the dendrite tip is a func-
tion of the thermal Peclet number and is given by
where ais the thermal diffusivity. The velocity of the
dendrite tip and the radius of dendrite tip, in this case,
are expressed as functions of the undercooling.
The modelling of microstructure for SLM is per-
formed using a sequentially coupled 3D FVADI ther-
mal model
and a 2D Cellular Automata
microstructure model. The cellular automata model
and implementation of grain nucleation and grain
growth models are explained below.
A cellular automata is a discrete model consisting of
a regular grid of cells, each in one of a finite number of
states. For each cell, a set of cells called its neighbour-
hood is defined relative to the specified cell. The two
most common descriptions of neighbourhood are
Moore’s neighbourhood (all eight cells surrounding a
square cell) and the von Neumann neighbourhood (the
top, bottom, left and right adjacent cells). An initial
state is selected by assigning a state for each cell. A new
generation is then created according to some fixed rule
that determines the new state of each cell in terms of
the current state of the cell and the states of the cells in
its neighbourhood. Typically, the rule for updating the
state of cells is the same for each cell and does not
change over time.
The grain nucleation phenomenon is modelled in the
cellular automata model in a probabilistic manner.
Each cell of the cellular automata model is prescribed a
random probability of nucleation which indirectly cor-
responds to a random distribution of critical under-
cooling. For nucleation to take place in a cell n
nVCAvProb vðÞ ð49Þ
where VCA is the volume of the cellular automata and
Prob(v) is the randomly generated probability at cell n.
In the current implementation, the mathematical
function (rule) governing the cellular automata is based
on the growth of grain, more specifically the growth of
the dendritic arms of the crystal. The rules can be set as
Rule 1: The dendritic arm of a cell can start to grow
only if the cell has been nucleated or captured by an
adjacent grain.
Rule 2: The dendritic arm of a cell in a direction can
grow only if the adjacent cell in that direction is
below the liquidus temperature and has not already
been nucleated or captured by another grain.
Rule 3: Once captured by a dendritic arm from an
adjacent cell, a cell gets the same state (misorienta-
tion) as the cell capturing it and is assumed to pos-
sess dendritic arms of same length as the capturing
cell in the orthogonal direction to that of the captur-
ing dendritic arm.
For the purpose of capturing an adjacent cell, the
grain envelope must reach the centre of the adjacent
cell. For the grain shown in Figure 22, this corresponds
to the situation when
n=lcos u+ sin u
where uis the misorientation angle of the capturing cell
and lis the distance between the cells. Once captured,
the length of the dendrite arm in captured cell is reset
to match the dendrite lengths in orthogonal direction
of the capturing cell. For example, if an adjacent cell m
is captured by the top dendrite arm of cell n, the den-
drite arm lengths in mare given by
m, right =Lt
n, right ð51Þ
m, left =Lt
n,left ð52Þ
m,top =Lt
n, bottom =0ð53Þ
As per the cellular automata rules, the growth of cell
nin the top direction would then stop and the grain
growth in cell mwill develop based on the local under-
cooling in that cell. Thus, the non-isothermal condi-
tions of grain growth are captured while ensuring the
propagation of grain misorientation.
In the particular implementation, only equiaxed
grains are considered, and therefore, the growth of den-
drite arms in the four directions for a cell is the same
and only based on the local undercooling. In case of
columnar grains, however, the growth might be pre-
ferred along certain direction over the others.
The 3D FVADI and 2D CA models are coupled
sequentially with the 3D FVADI model providing the
temperature field. To take into account the release of
latent heat, equivalent specific heat capacity values are
used during calculations. The temperature field calcu-
lated from the 3D FVADI model thereby governs the
Jabbari et al. 21
undercooling occurring at each cell in the 2D CA
model. As the thermal calculations are performed at a
larger spatial and temporal discretization, the values
are interpolated in the corresponding domains to
match the requirements of the cellular automata. As a
thumb rule, the spatial discretization of the CA model
is near the order of the radius of dendritic tip, and the
time resolution is such that the dendrite tip can only
grow till the next adjacent cell at a maximum chosen
The centre-line microstructure of a single melt track
is modelled using the 3D FVADI–2D CA model. The
parameters used for the thermal model are shown in
Table 1 and the parameters for the microstructural
model in Table 2.
The thermal model was used on a domain of
1mm 31mm 31:5mm with elements of 10 mm edge
length in each direction. As the scan speed of the laser
was high, the thermal model was solved at a time inter-
val of 7:8125 3107ms which corresponds to 1/16th of
the time taken by centre of laser beam to move from
one element to the adjacent one. The 2D cellular auto-
mata model had square cells with edge length of
1:25 mm, thereby dividing each element from thermal
model into eight cells in the microstructural model. The
time steps for the microstructural was selected as
2:232 3107s based on the requirements for grain
growth as discussed before.
Figure 23 shows the temperature field obtained from
the thermal model at four time instances during the
simulation of single melt track formation. The black
rectangular borders shown on the figures correspond to
the area selected for microstructure modelling using
cellular automata. Figure 24 shows the solidification
microstructure at the corresponding times. The grains
are initially nucleated at the solid–liquid interface as
can be seen in Figure 24. Although simulated on a
Table 1. Parameters for thermal model of single melt track
Parameters Value
Power 120 W
FWHM 1003106m
Powder bed porosity 0.35
Powder diameter 303106m
Scan speed 0.8 m/s
Chamber temperature 200 8C
Table 2. Parameters for microstructural model of single melt
track formation.
Parameters Value
Maximum nucleation density (nmax)10
Mean undercooling (DTm)25K
Standard nucleation density distribution (DTs)5K
Number of grain misorientation states 48
Unit thermal undercooling (DHf=Cp) 523 K
Figure 23. Centre-line temperatures at four time points during single track formation with SLM.
22 Advances in Mechanical Engineering
structured grid, the grains maintained their misorienta-
tion during growth which resembles real grain growth
mechanism. Most of the grains are seen to be oriented
at a large angle to the direction of laser movement (also
commonly observed phenomena), which is a result of
the grains growing normal to the solidification
For simulation of microstructures which can be vali-
dated against experimentally generated single track, the
model would require several parameters which need to
be determined experimentally such as the maximum
nucleation density, the mean undercooling, the radius
of unit thermal undercooling. In addition, the model
would need to be extended to include solute diffusion
and kinetic undercooling.
For comparison with single track which is allowed
to cool under processing conditions, the model would
also require to include the recrystallization occurring at
btransus temperature from bto a martensitic a0phase
followed by the decomposition into Widmanstatten
plate-like or basketweave acicular aphase. As such
modelling of solid-state phase transformation is an
extensive research field in itself – wherein the challenge
of predicting process-driven microstructure is tackled
at different length and time scales. Along with cellular
automata–based techniques, methodologies such as
molecular dynamics, phase-field modelling, Monte
Carlo Potts modelling and crystal plasticity have been
used by researchers to predict microstructures across
micro- to meso- to macro-length and time scales.
Figure 25 shows prediction of a single-phase equili-
brium microstructure (from bto a0phase) through
Monte Carlo simulations of two-dimensional Potts
The thermo-metallurgical model was able to predict
the representative microstructure observed during sin-
gle melt track formation on a loose powder bed using
SLM. The small size of melt pool and the high cooling
gradient, especially in case of low Peclet number materi-
als, allows neglecting certain otherwise important terms
from the grain growth models. However, it would be
necessary to include them in case of high Peclet number
material such as 316L steel. In addition, the evolution
of solidification microstructure from SLM into the final
a+bstate will be the subsequent direction of work.
Modelling and simulation of manufacturing processes
is a truly interdisciplinary field spanning mathematics,
computer science, heat transfer, fluid and solid
mechanics as well as materials science and manufactur-
ing engineering. Models allow us to create mathemati-
cal representations of the system under investigation
and simulate its properties and behaviour over time
and under different conditions. This is an extremely
powerful approach in planning and understanding
experiments and producing predictions in designing
manufacturing systems and increasing their efficiency.
Modelling and simulation techniques are used in all
Figure 24. Evolution of centre-line microstructure at four time points during single track formation with SLM.
Jabbari et al. 23
realms of science and engineering and find endless
industrial applications.
The present paper presents five cases of modelling
different aspects of modern manufacturing processes
ranging from fluid flow analysis (tape casting of cera-
mics and extrusion of polymers) over thermomechani-
cal analysis (NIL of features on the surface of injection
moulding dies and composite manufacturing) to
thermo-metallurgical analysis in SLM. For some of
these areas, selected results from numerical modelling
have been compared with corresponding experimental
findings and good agreement has been found. It is evi-
dent that models and simulations like the ones pre-
sented here will further increase their use in the future
as an important part of analysing the effect of selected
materials and process parameters on the subsequent
performance of manufactured parts.
Declaration of conflicting interests
The author(s) declared no potential conflicts of interest with
respect to the research, authorship and/or publication of this
The author(s) received no financial support for the research,
authorship and/or publication of this article.
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