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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

journals.sagepub.com/home/ade

Multiphysics modelling of

manufacturing processes: A review

Masoud Jabbari

1

, Ismet Baran

2

, Sankhya Mohanty

3

, Raphae

¨l Comminal

3

,

Mads Rostgaard Sonne

3

, Michael Wenani Nielsen

4

, Jon Spangenberg

3

and

Jesper Henri Hattel

3

Abstract

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.

Keywords

Numerical modelling, tape casting, nanoimprint lithography, extrusion, composite, selective laser melting

Date received: 24 May 2017; accepted: 26 February 2018

Handling Editor: Filippo Berto

Introduction

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

models.

1

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.

2

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.

3–6

Reilly et al.

7,8

have criti-

cally reviewed the role of CFD in defect entrainment in

1

WMG, The University of Warwick, Coventry, UK

2

Faculty of Engineering Technology, University of Twente, Enschede, The

Netherlands

3

Department of Mechanical Engineering, Technical University of

Denmark, Kongens Lyngby, Denmark

4

LM Wind Power A/S, Lunderskov, Denmark

Corresponding author:

Masoud Jabbari, WMG, The University of Warwick, Coventry CV4 7AL,

West Midlands, UK.

Email: M.Jabbaribehnam@Warwick.ac.uk

Creative Commons CC BY: This article is distributed under the terms of the Creative Commons Attribution 4.0 License

(http://www.creativecommons.org/licenses/by/4.0/) 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 (https://us.sagepub.com/en-us/nam/

open-access-at-sage).

the shape casting process and summarized different

numerical approaches used in different commercial

packages, that is, MAGMASOFT

9

and FLOW-3D.

10

Another example of fluid flow analysis in modelling of

manufacturing processes has been given by Zhang and

Wu,

11

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.

12,13

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

14

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.

15

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.

16

Numerical analysis of FSW

has been reviewed in the literature in general

17

and with

an especial focus on modelling residual stresses.

18

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,

19,20

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

processes.

21,22

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

layers;

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)

process.

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.

23,24

In

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-

viour).

25–32

However, this does not mean that assuming

Newtonian (linear) explanation for the viscosity of

ceramic is wrong.

33,34

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).

27–36

However,

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.

35

2Advances in Mechanical Engineering

lead to optimizing the process by simulating more

advanced features related to flow analysis.

26,37–42

For fluid flow analysis in tape casting normally the

coupled momentum and continuity equations should be

solved, that is

r∂u

∂t+uru

=rp+rT+Fð1Þ

∂r

∂t+r ruðÞ=0ð2Þ

where ris density (kg/m

3

), 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

t=m_

gð3Þ

where mis the dynamic viscosity, tis the shear stress

tensor (Pa) and _

gis the shear rate tensor (Pa s). Here _

g

is the rate of the strain tensor, _

g=ru+ru>, which is

given by

_

g

jj

=ﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1

2II_

g

r=1

2_

g:_

g

no

1=2

ð4Þ

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

t

jj

=ﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1

2IIt

r=1

2t:t

no

1=2

ð5Þ

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.

33,34,38

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,

25–27,29–32,35,36,40–43

and they are given as follows

t=ma_

gn,ma=k_

gn1ð6Þ

t=mB+ty

_

gjj

_

gt

jj

.ty

_

gt

jj

tyð7Þ

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

rate.

44

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.

26,37

Jabbari et al.

41

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

flow’.

40

Side flow by definition shows amount of fluid

which flows in lateral direction as it leaves the doctor

blade region. Jabbari and Hattel

40

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.

45

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.

46

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.

47

using

the FVM-CFD model developed previously,

41

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.

48

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

ventilation.

49

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

tapes.

Neglecting capillary pressure (and resultant mass

transport in porous media), Jabbari and Hattel

50

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.

50

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.

40

(a)

(b)

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.

51–56

Modelling the flow of polymers in

extrusion

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.

50

(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.

50

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).

57–63

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.

64,65

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.

65,67–69

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

die.

65,68,69

In both cases, stick and partial slip plays

important roles in the sharkskin mechanisms.

64,65

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

collegues

66,70–73

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

74,75

and theoretical molecular mod-

els

76–78

), 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

Hatzikiriakos

79

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.

66

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,

65,80

which results in a decrease of the wall shear

stress.

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.

81,82

Observations

by particle image velocimetry have shown that there are

correlations between the periodic or chaotic extrudate

distortions and upstream flow instabilities inside the

reservoir.

83–85

The onset of gross melt-fracture can

sometimes be delayed by modifications in the reservoir

geometries,

65,85

without being suppressed, however.

Some experimental and theoretical studies

86,87

also

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.

88–90

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

as

st

ðÞ=hs_

g+tpð9Þ

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

ltp

r+tp=hp_

gð10Þ

where hpis the polymer viscosity, and

tp

r[∂tp

∂t+urtptpru>+rutp

ð11Þ

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.

91–95

However, the Eulerian representation (with a fixed

mesh) seems to be best suited for time-dependent simu-

lations with free surface flows.

96

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.

97–99

Recent results of Tome

´et al.

99

show that Tanner’s theory,

100,101

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.

102

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.

103,104

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

c=tp

G+Ið12Þ

a non-dimensional internal variable representing the

strain of the polymer chains, which should be sym-

metric positive definite by definition,

105

looses its posi-

tive definiteness.

105–107

Moreover, the loss of positive

definiteness happens because of numerical errors dues

to under-resolution of the spatial stress profiles,

108,109

which may have steep gradient variations and/or be

exponential, near the boundary layers and geometrical

singularities.

110

A seminal reformulation of the differen-

tial constitutive model (3) has been proposed by Fattal

and Kupferman.

107,108

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

∂C

∂t+urCVC CVðÞ2E=1

lexp CðÞI½

ð14Þ

where Vand Eare anti-symmetric (pure rotation) and

symmetric traceless (pure extension) matrices that com-

pose the velocity gradient ru, see Fattal and

Kupferman

107

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

Kwon

111

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.

112

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.

113

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

effects.

A similar modelling approach has been employed by

Comminal,

114

to simulate two-dimensional viscoelastic

flows at the exit of a slit die. Comminal et al.

115

applied

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,

115–117

defined as the vector

potential of the incompressible velocity field

(Helmholtz decomposition)

u=r3Fð15Þ

In two dimensions, the in-plane velocity components

are solely defined by the out-of-plane component fzof

the streamfunction vector

ux=∂fz

∂y,uy=∂fz

∂xð16Þ

and fzis governed by the following evolution equation

(derived from the curl of the momentum equations)

r∂

∂tr2fz

=hsr4fz+r3rtp

ð17Þ

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

r3rpðÞ=0,8p2Rð19Þ

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.

118

In contrast to Kwon, Comminal et al.

118

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

scheme.

120

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.

122

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.

90

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

instabilities.

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.

123

The sci-

entific literature is rich in examples of advanced

materials emulating the well-known super-hydrophobic

effect of the lotus leaf

124

and adhesive surfaces of the

gecko’s feet.

125

Structural colours and iridescence are

most often observed in invertebrates such as butterflies

(see Figure 9) and beetles, but also in the feathers of

birds.

128,129

Since the early observations of functional

nanostructures in nature, engineers have tried to repli-

cate these nanostructures in order to functionalize sur-

faces.

130–132

Recently, Christiansen et al.

127

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

–1

; reproduced from Kalika and Denn

122

with

permission. (d) Transition regime of the stick–slip instability of a LLDPE extrudate with an apparent shear rate of 985 s

–1

;

reproduced from Kalika and Denn

122

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.

133

In the study conducted by Sonne et al.,

134

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.

134–136

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.

137

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

138

). (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

process.

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.

139

This is problematic since

even very small changes in the nanostructures geometry

will change the wanted overall functionality.

140

The

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.

126

(b) Diffraction gratings with different orientations and periods can be

combined, resulting in a glitter colour effect.

127

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,

141–144

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

145

) as benchmark

1

h2=1

h2

0

+2p

hs2ð20Þ

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.

146

(2) The material behaviour is

non-linear and is for the polymer material described by

a viscoelastic–viscoplastic constitutive behaviour.

147,148

(3) With respect to the contact conditions, changes in

frictional behaviour on the different length scales

(macroscale to nanoscale) have to be taken into

account

149–152

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

model.

153

In the work done by Sonne et al.

134

a standard con-

tinuum mechanics was applied with the overall goal of

solving the three static equilibrium equation

sij,i+pj=0ð21Þ

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

154–156

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

155

by a modified version of Hooke’s law represented by

the elastic stiffness tensor

Cve

ijkl =Emod

1+n

1

2dik djl +dildjk

+n

12ndijdkl

ð22Þ

Emod =Emax +E2,Emax =E1

E1Dt

h+1ð23Þ

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.

130

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

f=s2

es2

y=0ð24Þ

where seis the equivalent stress. The yield stress syis in

this work given by the J-C viscoplastic model,

157

which

from previous work has been shown to give a reason-

able correlation between strain, strain rate, temperature

and stresses for this material.

158

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,

130

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.

130

Figure 12. Comparison of measured and calculated maximum

principal strains along two different paths on the deformed

flexible stamp.

130

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.

159

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,

160,161

filament wind-

ing,

162,163

resin transfer moulding (RTM),

164–166

vacuum infusion (VI),

167

vacuum-assisted resin transfer

moulding (VARTM),

168

to name a few.

169

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-

trol.

170–172

Furthermore, avoiding process-induced

shape distortions and residual stress build-up remains a

challenge in many applications.

173

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

rcp

∂T

∂t+u∂T

∂x

=kx

∂2T

∂x2

+ky

∂2T

∂y2

+kz

∂2T

∂z2

+_

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

174

proposed a

modified linear elastic model incorporating tempera-

ture dependency which also allows thermal soften-

ing.

171,175,176

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)

177

Er=

E0;TTC1

Aeexp (KeT);TC1\T\TC2

E1+TTC2

TC3TC2

(E‘E1);TC2\T\TC3

E‘;TC3T

8

>

>

>

>

>

<

>

>

>

>

>

:ð26Þ

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 E‘are the viscous and

glassy state elastic modulus, respectively. Tgcan be

defined as

Tg=T0

g+aTgað27Þ

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.

178

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.

179

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.

180

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.

181,182

Some of the thermo-chemical

computational analyses can be found in previous

works.

183–188

Baran et al.

119,189,190

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.,

191

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.

119

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.

177

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.

192

The model predicted the stress–strain

development as well as the nodal displacements.

192

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.

119

14 Advances in Mechanical Engineering

In Baran,

193

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.

194

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.

195,196

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

process.

197–200

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.

175,176

The

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.

175,176

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.

119

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

119

).

Figure 17. Transverse shear stress distribution (t23) for different Tpreheat values after the pultrusion process.

193

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.

175

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.

171

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.

171

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

representation.

Thermo-metallurgical modelling of SLM

Modelling of thermal conditions during manufacturing

processes and resultant metallurgical features is a well-

established field of research.

201–208

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

209–212

clearly

indicates the scope and need of thermo-metallurgical

modelling studies.

213,214

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.

SLM

215

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.

171

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

centre.

171

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

216–224

from experimental and modelling

perspective, including current practices and future chal-

lenges regarding raw materials, processing and post-

processing.

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

2

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

2

with

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

rCp

∂T

∂t=∂

∂xkxx

∂T

∂x

+∂

∂ykyy

∂T

∂y

+∂

∂zkzz

∂T

∂z

+F

... 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

k∂T

∂h=hT

amb TðÞ+seT4T4

amb

ð29Þ

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-

niques.

226–230

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

231–233

wherein

a combination of the emissivity of the particles and the

emissivity of the cavities in the powder bed is used

e=Aheh+1Ah

ðÞesð30Þ

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.

225

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

Ah=0:908f2

0

1:908f2

02f0+1ð31Þ

eh=

es2+3:082 1f0

f0

2

es1+3:082 1f0

f0

2

+1ð32Þ

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

Zehner–Schlu

¨nder–Damko

¨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

k

kf

=1ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1f0

p

1+f0

kr

kf

+ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1f0

p2

1kf

ks

1

1kf

ks

ln ks

kf

1

!

+kr

kf

!

ð33Þ

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

kr=4FesBT3

pDpð34Þ

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

kr=

4esBT3

pDp

10:132eð35Þ

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

228

Aeffective =3

4

1v+3a

1+2a

ð36Þ

a=12

3v+1

3v2

1

2

ð37Þ

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

Ix,yðÞ=2aP0

pv2

0

e

2xx0

ðÞ

2+yy0

ðÞ

2

ðÞ

v2

0

ð38Þ

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

qsx,yðÞ=2aP0

pv2

0

3ðððe

2xx0vxt

ðÞ

2+yy0vyt

ðÞ

2

ðÞ

v2

0

dxdydt

ð39Þ

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.

228,230,234–237

The internal state variable approach

238

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

below.

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

239

is most popular

wherein the nucleation distribution is described by a

Gaussian distribution. The grain density at a particular

undercooling is given by

nDT

ðÞ

=Ð

DT

0

nmax

DTsﬃﬃﬃﬃ

2p

p

exp 1

2

DT0DTN

DTs

2

dDT0ð40Þ

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.

240

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

n,i

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

Lt

n=ð

t

tn

qDTt0

n

dt0ð41Þ

where qis the velocity of the dendrite tip (i.e. interface

velocity). The total undercooling DTis decomposed

into a sum of two different terms

DT=DTt+DTrð42Þ

with the contribution of thermal undercooling DTtand

curvature undercooling DTrbeing expressed as

DTt=DHf

Cr

IvPt

ðÞ ð43Þ

DTr=2G

Rð44Þ

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.

240

20 Advances in Mechanical Engineering

the radius of the dendrite tip and Ivis the Ivantsov

function expressed as

IvxðÞ=xexð

inf

x

ez

zdz ð45Þ

The radius of curvature of the dendrite tip is a func-

tion of the thermal Peclet number and is given by

R=4p2GDHf

Cr

Ptjt

1

ð46Þ

jt=11

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1+2p

Pt

2

rð47Þ

Pt=qR

2að48Þ

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

228

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

dnDTðÞ

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

follows:

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

Lt

n=lcos u+ sin u

jj

ðÞð50Þ

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

Lt

m, right =Lt

n, right ð51Þ

Lt

m, left =Lt

n,left ð52Þ

Lt

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

undercooling.

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

formation.

237

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.

237

Parameters Value

Maximum nucleation density (nmax)10

16

m

–3

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.

237

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

isotherm.

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

model.

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.

Conclusion

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.

237

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

article.

Funding

The author(s) received no financial support for the research,

authorship and/or publication of this article.

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