Modelling the microstructural evolution and fracture
of a brittle confectionery wafer in compression
I.K. Mohammed1, M.N. Charalambides1,*, J.G. Williams1, J. Rasburn2
1
Mechanical Engineering Department, Imperial College London, South Kensington, London, SW7 2AZ,
UK
2
Nestec York Ltd., Nestlé Product Technology Centre, Haxby Road, PO Box 204, York YO91 1XY, UK
Keywords: foam; brittle fracture; x-ray micro tomography; scanning electron microscopy; in-situ; finite
element
*Corresponding author. Tel: +44 (0)20 7594 7246; fax:
Email address: m.charalambides@imperial.ac.uk
Abstract
The aim of this research is to model the deformation and fracture behaviour of wafers used in chocolate
confectionery products so as to optimise industrial processes such as cutting as well as aid the development of
product design. Uni-axial compression experiments showed that the mechanical behaviour of the wafer was
characteristic of a brittle foam. The wafer sheet was examined with a Scanning Electron Microscope (SEM) to
determine the wafer dimensions and to observe the internal microstructure. These images visually confirmed
the cellular structure of the wafer and showed that the core of the wafer sheet was more porous than the
dense skins. A finite element (FE) model was used, which employed the actual complex architecture of the
wafer. To attain the wafer architecture, X-ray Micro Tomography (XMT) was used on a sample to produce a
stack of image slices which were reconstructed as a 3D virtual wafer. The microstructure of the volume was
characterised in terms of porosity and then meshed with tetrahedral elements for finite element analysis. The
cell walls of the model were assigned a linear elastic material model and a damage criterion to simulate the
fracture of the cell walls. In-situ SEM and XMT experiments were conducted which allowed the deformation
and fracture of the wafer sheet to be observed simultaneously as the global mechanical response was
recorded. The FE model of the complex architecture was able to predict the brittle response of the wafer in
compression reasonably well.
1 Introduction
Materials with voids are found in both nature and engineering. Manufactured porous materials have
a wide range of applications including lightweight components, impact energy absorption, thermal
insulation, acoustic dampening, vibration suppression and fluid flow control [1,2]. Porous materials
also occur in nature such as wood, coral, bone and foods such as carrots and tomatoes. Foods with
foam structures can also be manufactured and include breads, cakes, cereals and biscuits. The work
presented in this paper uses a baked confectionery wafer.
As the porosity (ratio of void to solid volume) exceeds a fraction of 0.7, a material transitions from a
solid with pores to a cellular structure or foam [3]. Cellular solids consist of a three dimensional
interconnected network of solid struts or plates, which form the edges and faces of cells
respectively. Foams can be classified as open-cell or closed-cell, depending on the nature of the cell
structure. In open-cell foams, the structure is skeletal and neighbouring cells are interconnected to
1
each other. In closed-cell foams, individual cells are enclosed and separated from each other by the
membrane in the cell faces.
Thus in an open cell foam, all of the solid material is contained in the cell edges while in a closed cell
foam, the solid material is situated in both the edges and the faces of the cells. Some foams possess
both an open and closed cell structure. The mechanical properties of the foam are dependent on the
nature of the cellular structure as well as the cell shape, the properties of the parent solid material
and the relative density or volume fraction of the foam. The foam relative density is the ratio of the
density of the foam (ρ*) to the density of the solid cell wall material (ρs). It ranges between 0.05-0.3
for foams and is related to the foam porosity, ε, as expressed in Equation 1.
𝜀 = (1 −
𝜌∗
)
𝜌𝑠
(1)
For any particular cell wall material, the relative density determines the Young’s modulus, yield
strength and energy absorption of the foam. In uni-axial compression, the deformation curves of
foams exhibit three distinct regions. At small strains, the stress-strain response is linear. The linear
region is followed by a plateau in which the strain increases at an almost constant stress. The final
stage is defined as densification in which there is a steep increase in the stress. Foams are
categorized as elastomeric, ductile or brittle based on the mechanical response of the plateau region
in compression. For all three types of foams, the linear elastic region of the stress-strain curve
corresponds to bending of the cell edges and walls, while the densification region is a result of cell
walls collapsing and contacting each other. In elastomeric, ductile and brittle foams, the plateau
arises due to elastic buckling, plastic hinging and brittle fracture of the cell walls respectively.
A number of authors have developed analytical equations to describe the material properties of
foams, some of which are extended from particulate composites [4-12]. The most quoted analytical
models are those developed by Gibson & Ashby [3] which treat the foam as an array of simple cubic
cells. They derived analytical solutions for open and closed cell foams which relate the relative
modulus of the foam to its relative density. The equation for closed cell foams includes terms
accounting for the cell wall stretching and the pressure within the cells. The compressive modulus of
the foam, E*, and the solid modulus of the cell wall material, Es, are related to the relative density of
the foam, ρ*/ρs, by Equation 2 below, where φ represents the proportion of material volume in the
cell edges.
𝐸∗
𝜌∗ 2
𝜌∗
2
(1
= 𝜑 ( ) + − 𝜑) ( )
𝐸𝑠
𝜌𝑠
𝜌𝑠
(2)
The rupture stress of the cell wall material, σs, can be calculated using Equation 3 below, where σ* is
the brittle collapse stress of the foam.
3
𝜎∗
𝜌∗ 2
𝜌∗
= 0.2 (𝜑 ) + (1 − 𝜑) ( )
𝜎𝑠
𝜌𝑠
𝜌𝑠
(3)
For a complete understanding of cellular materials, the structure should be studied at two different
length scales: the scale of the microstructure of the constitutive material and the cellular
architecture length scale of the foam itself [13, 14]. The constitutive material governs the global
2
behaviour because it modifies the properties of the solid part of the material. The cellular
microstructure describes the size and morphology of the arrangement of the solid and gaseous
phases in a cellular material. Conventionally, microstructural characterisation is performed in 2D
using an optical or scanning electron microscope [15-20]. These 2D images are sometimes enough to
obtain quantitative information about the microstructure. There are limitations to the use of 2D
images and these may occur when the global geometry is non-uniform or when the connectivity and
size distribution of pores/particles are three-dimensional parameters.
A method for 3D visualisation is computerised X-ray micro tomography (XMT). It is a non-invasive
technique which has the ability to provide high quality 3D images of a wide range of opaque
materials [21-30] thus making it a useful tool for analysing the cellular microstructure of foams. XMT
can be successfully applied to cellular solids due to their low overall absorption of the X-rays which
allows large specimens to be studied. It also allows large deformations to be imaged so the
important buckling, bending or fracture events appearing during the deformation can be visualised.
The experimental implementation requires an X-ray source, a rotation stage and a radioscopic
detector. A complete analysis is made by acquiring a large number of X-ray absorption radiographs
of the same sample from different viewing angles. A final computed reconstruction step is required
to produce a three dimensional map of the local absorption coefficients in the material. The
software uses a filtered back projection algorithm to reconstruct the image slices of the sample
perpendicular to the rotation axis. A detailed description of the X-ray tomography process can be
found in [15, 27].
The 3D image obtained from the X-ray tomography can be analysed in a number of ways depending
on the requirements of the study. In most studies, image analysis is used to highlight the
architecture of the microstructure and determine parameters such as the local volume fraction, cell
size distribution, cell wall thickness and specifically for foams, the local density distribution which
describes the homogeneity of the foam.
In-situ experiments using XMT allow one to monitor the microstructural evolution of the sample [3134]. Any strain localisation can be observed and the deformation mechanism (bending, buckling or
fracture) can be determined. Sometimes the experiments are performed with interruptions, such
that the experiment is halted at different points in time to allow the radiograph to be taken. This
practice is followed because the time needed for a complete scan is larger than the evolution time of
the microstructure or in the case of materials such as polymers, relaxation may introduce blurs in
the reconstruction. Continuous scanning during an experiment is possible however it is limited to
synchrotron fast imaging devices which are not readily available [35, 36]. In addition, the in-situ XMT
data could be used with digital volume correlation software in order to obtain a strain map of the
specimen [37], but this method was not used in this work. Therefore, in cases when in-situ XMT
testing does not allow for continuous deformation and the full strain maps are not available,
simulating the experiment by means of numerical modelling is an alternative method for this
information to be obtained.
A quantitative 3D analysis can be performed by taking advantage of the geometry generated from
the tomography data. The reconstructed images describe the full 3D intricacy of the architecture of
cellular materials. The appropriate tool which allows modelling the deformation of such a complex
geometry is the Finite Element (FE) method [38-46].
3
Conversion of an XMT image to a finite element mesh is not a simple task and many approaches are
available. Three different methods to produce meshes reflecting the architecture of a cellular
material are meshes based on the Voronoi description of microstructure, voxel-element meshes and
tetrahedral meshes [27].
Tetrahedral meshes allow the actual shape of the architecture to be reproduced by implementing
tetrahedral shaped elements. A smooth surface domain of discretised triangles is first generated
before the solid volume is meshed with tetrahedral elements. This is achieved with a marching cubes
algorithm using commercial meshing software. It is the most accurate meshing method because it
reproduces the actual shape of the foam although the calculation time can be longer than a voxelelement mesh of the same resolution. The drawback of using FE computations based on X-ray
tomography is that they are likely to be computations which are time and memory consuming. The
aim is always to find the best tradeoff between the computing time and the accuracy of the results.
In most of the literature reviewed involving FE models generated from XMT data, the foams or
composites were deformed to small global strain values and used either an elastic [41, 42, 44] or
elastic-plastic [23, 33, 39, 40] material model to describe the cell walls or particles-matrix. Wismans
et al [46] extrapolated the XMT data to generate a 2D FE mesh with a hyperelastic material model
but did not account for the contact between adjacent cells at large compressive strains. Zhang et al
[47] used SEM images to obtain the microstructure of their composite while Chen et al [48] used CT
scans and although they both included damage in their FE models, both were simulated at a 2dimensional level. A 3D composite FE model was artificially generated by Segurado et al [49] which
had damage occurring at the interface between brittle particles and the matrix.
In contrast to the studies mentioned above, the numerical modelling of the 3-dimensional XMT
generated wafer in this paper aims to simulate not only the initial linear response but also the brittle
fracture of cell walls as well as the interaction of these cells at high global strains. The wafer material
in this study is described next, followed by the 2D and 3D imaging techniques used: scanning
electron microscopy and x-ray micro tomography. The numerical method is then outlined, including
the geometry generation, boundary conditions and the damage material model. The results from the
in-situ compression experiments are next shown. The analytical methods used to obtain the wafer
material properties are given followed by the numerical results from the finite element simulations.
Finally, the benefits and limitations of the experimental and numerical methods are presented in the
Discussion.
2 Materials & Methods
The wafer sheets provided by Nestlé PTC York were produced by baking. The baking process
consisted of spraying parallel strips of a liquid wafer batter between hot plates in a Haas oven. The
wafer batter ingredients consist primarily of wheat flour and water, with sodium bicarbonate and
trace fat [50, 51]. The plates, which have engravings on them called ‘reedings’, are closed to allow
the batter to spread evenly while baking. The engravings prevent sticking to the plates, and thus
facilitate removal, when the wafer is baked. The plates are heated to around 150°C and the baking
process lasts for 120 seconds. During the baking process most of the moisture evaporates, resulting
in a porous cellular structure which is lightweight and crisp. When viewed under a microscope [52,
53], the baked wafer appears to consist of two regions with different porosities. The sides of the
4
wafer sheet that were in contact with the hot plates are less porous than at the centre of the wafer
sheet. The two denser regions of the wafer sheet are designated in this paper as the ‘wafer skin’
while the less dense region is referred to as the ‘wafer core’.
2.1 Scanning Electron Microscopy
The wafer microstructure was observed using a Hitachi S-3400 scanning electron microscope (SEM).
The SEM operated at an accelerating voltage of 15 kV in the secondary electron mode under a
vacuum pressure of less than 1 kPa. Wafer specimens were coated with a thin layer of gold to obtain
a conductive surface so that high quality images could be obtained.
The Deben Microtest module was designed to be mounted within the scanning electron microscope
to perform in-situ mechanical testing. Testing can be performed at constant speeds between the
range of 0.1 – 1.5 mm/min with a load cell capacity of 300 N. The Deben Microtest V5.2 software has
the capability to plot a load-displacement graph of the deformation and record a video of the
deformation displayed by the SEM visualisation. The resulting graph and video are synchronised
such that the video frame at any time during the deformation can be associated with its
corresponding co-ordinates on the load-displacement plot.
The setup of the Microtest rig was such that it allowed horizontal testing, therefore the face which
showed the entire thickness of the wafer was in the field of view of the SEM lens during the
compression experiments. Uni-axial compression tests were performed because they are simple and
there is no need to grip the specimens [54].
It was assumed that the compressive load was distributed uniformly across the surface of the wafer
in contact with the plates. The wafer is a foam and deforms in the direction of the applied load with
minimal lateral deformation hence justifying the lack of the need for lubrication. Additionally, if a
lubricant was used it would be absorbed by the wafer and change its material properties due to
plasticisation.
Square specimens of 7.5mm length (equivalent to three lines of reedings) were prepared and placed
between two rigid plates on the Microtest rig. A compression speed of 1 mm/min was used to crush
the specimen well into the densification region.
2.2 X-ray Micro Tomography
A Phoenix v|tome|x “s” X-ray tomography system (Phoenix|X-Ray GmbH) was used to scan a square
wafer sample of length 2.5mm, under conditions of an accelerating voltage of 80 kV and a current of
125 μA. A commercial filtered back-projection algorithm, SIXTOS, was used to reconstruct the
tomogram (SIXTOS is a trademark of Phoenix/X-Ray GmbH, Stuttgart, Germany). The stack of image
slices produced was used to generate a 3D volume of the wafer microstructure using the Avizo
software [55]. With this virtual wafer, it was possible to accurately characterise the microstructure,
determine the porous volume fraction and create a meshed volume suitable for quantitative finite
element analysis.
The raw data from the XMT scan was generated in the form of a .vol file which contained each of the
reconstructed image slices. In total there were 512 images, each 720 x 734 pixels with a resolution of
5 μm per voxel (3D pixel). The raw images had a poor contrast and it was difficult to distinguish the
5
wafer material from the background as seen in Figure 1a). It was thus necessary to enhance the
slices using image analysis tools. ImageJ [56] was selected for this purpose.
The brightness and contrast of the stack of images were adjusted so that the wafer material was
more visible as shown in Figure 1b). The drawback of this adjustment was that the background noise
(which look like concentric rings in Figure 1b) in the images was also enhanced. Therefore a noise
filter, which removed outlying pixels based on their size and threshold level, was implemented. In
ImageJ, the noise was reduced using the “Remove Outliers” process with the isolated pixel size of 1
and threshold of 50. This filter analysed the eight pixels surrounding each pixel below the selected
grayscale threshold and if they were all above the threshold value, then the middle pixel (noise)
would be removed. This cleaned the image without losing vital data, ie. pixels belonging to the
wafer. It was not essential to binarise (convert to black and white images) because it was only
necessary to make the wafer material distinct from the background.
The image stack was imported into Avizo to begin the 3D volume reconstruction. The first step was
to segment each slice which meant labelling all pixels that represented the wafer material. A “magic
wand” tool was used for this task. The labelled pixels of a single image slice were then automatically
interpolated through all the slices which generated the voxels of the volume. After the voxels were
all labelled, a 3D volume of the wafer was generated as shown in Figure 1c) and then meshed with a
tetrahedral grid seen in Figure 1d). From the reconstructed 3D volume, it could be seen that some
cell faces appear to be missing which could be due to either the resolution of the scan not being high
enough to capture faces less than 5 μm thick or damage at the edges during the sample preparation.
6
Figure 1 a) single image slice of raw data, b) after image enhancement, c) the 3D volume generated and d) the
meshed tetrahedral grid
An in-situ interrupted compression test was performed within the XMT machine so that the
deformation could be observed in three dimensions. The compression rig consisted of a bottom
platen which was connected to a load cell while the load was applied to the wafer sample by
manually turning a screw thread which was attached to the upper platen [57]. A perspex tube
surrounded the platens and the sample. A single wafer sheet of size 7.5 x 7.5 mm was compressed
sequentially in small displacement increments. At every stage of displacement, the sample was
scanned producing a total of six image stacks inclusive of the initial undeformed state.
2.3 Numerical Modelling
A Finite Element (FE) model with the geometry of the actual wafer architecture was generated from
the XMT image slices using the 3D meshing software, Avizo. A surface consisting of triangles was
rendered to encase the volume and then a tetrahedral grid was generated to fill the interior of the
surface model. The meshed grid contained node co-ordinates and element numbers which were
saved in a text file and then exported to an Abaqus input file for performing the numerical
simulation of the deformation of the wafer during compression. Models with different levels of
7
mesh refinement between 230,000 and 2,400,000 linear tetrahedral elements were generated using
the Avizo software. The dimensions of the wafer model were approximately 2.5 mm in the x and z
axes (See Figure 1). Two analytical rigid body parts were created above and below the wafer to act
as the compression plates.
For the boundary conditions, each rigid plate was displaced an equal and opposite amount in the yaxis. A ramp amplitude was applied to give the plates a uniform displacement and thus constant
speed. The four vertical faces of the wafer were each given symmetric boundary conditions. Since
the geometry of the mesh represented approximately 2.5mm of the wafer, applying symmetric
boundary conditions replicated a 7.5mm x 7.5mm wafer specimen. This dimension was significant
because both the in-situ SEM and in-situ XMT compression tests were performed on specimens of
this size.
Beyond the elastic region of compression, there were interactions between adjacent cell walls of the
wafer and thus it was necessary to model this correctly in the finite element calculations. Contact
was assigned to the entire assembly of rigid plates and wafer so as to prevent inter-penetration
between elements during the deformation. General contact selected all exterior surfaces in the
entire model and was also capable of spanning unconnected regions. The model assumed that the
normal and tangential contacts were hard (impenetrable) and frictionless respectively.
The Abaqus 6.11 Explicit solver was used instead of the Implicit solver for the simulations because
the wafer deformation consisted of a combination of a complex architecture, surface contact
interactions and a progressive damage material model which is described in the next section. In
order to simulate quasi-static conditions, a long enough step time was required in order to ensure
that the dynamic effects were damped and the inertial effects were negligible. To ensure that the
simulation produced a quasi-static response, the kinetic energy of the entire model was not allowed
to exceed 10% of its internal energy throughout the time history output.
It was desirable to predict the wafer material response beyond the linear elastic region and thus a
progressive damage and failure model was used to simulate the fracture of the cell walls. In Abaqus,
this material model is called “Ductile Damage” which is a subcategory of “Damage for Ductile
Metals”. The stress-strain curve of an element can be divided into three parts as represented in
Figure 2. The linear elastic region is described by a-b, the plastic region by b-c and the evolution of
degradation by c-d. The initial yield stress (σ0) is at point b, the damage initiation criterion (σy0)
occurs at point c and the element deletes at point d. Element deletion implies that the stiffness of
the element has fully degraded to zero and is thus no longer used in the finite element calculations.
8
Figure 2 The stress vs strain response of the ductile damage material model
The damage evolution law describes the rate of the material stiffness degradation of the elements
after the damage initiation criterion has been satisfied. At any given time in the analysis, the
material stress tensor is given by Equation 4 where σun is the undamaged stress tensor which would
exist in the absence of damage and D is the overall damage variable. Thus D = 0 prior to damage
initiation (point c in Figure 2) and D=1 at element deletion (point d in Figure 2).
𝜎 = (1 − 𝐷)𝜎𝑢𝑛
(4)
The onset of damage was determined by an equivalent plastic fracture strain (εpl0) criterion while the
damage evolution was determined either by a fracture energy dissipation (Gf) or a failure
displacement (uplf) criterion. The fracture energy per unit area, as defined in Abaqus, is given by:
𝑝𝑙
𝐺𝑓 = ∫
𝜀𝑓
𝑝𝑙
𝑝𝑙
𝐿𝑒𝑙 𝜎𝑦0 𝑑𝜀 𝑝𝑙 = ∫
𝜀0
𝑢𝑓
0
𝜎𝑦0 𝑑𝑢𝑝𝑙
(5)
The introduction of the characteristic element length (Lel) reduces the mesh dependency of the
damage model and implies that the damage evolution is characterised by a stress-displacement
response. Also note that Gf corresponds to the area under the stress-displacement graph during the
damage evolution stage. In the case of a linear evolution, the energy (Gf) and displacement at
initiation (upl0) and failure (uplf) are related by Equation 6. Figure 3a graphically describes the fracture
energy while Figure 3b shows the variation in the damage variable as an element is deformed until
complete failure.
𝑢𝑓𝑝𝑙 − 𝑢0𝑝𝑙 =
2𝐺𝑓
𝜎𝑦0
(6)
9
Figure 3 a) graphical representation of the fracture energy and b) the damage variable evolution
It was assumed that the plastic damage model described above could be modified to simulate brittle
fracture of the wafer by effectively eliminating the plastic region of the stress-strain curve and by
setting the damage evolution stage as a steep, almost vertical line. Ultimately, if a very small fracture
strain and zero fracture energy were used as material parameters in the FE model, the resulting
deformation would be linear elastic followed by almost instantaneous deletion of the element thus
simulating a brittle material model, as shown by the dotted line at σ0 in Figure 3a. This progressive
damage model was advantageous over other failure methods such as cohesive zone models [58],
since it was not required to preselect the crack paths, which would be impossible with such a
complex architecture. All simulations were run using four processor cores (Intel i7 CPU at 3.2 GHz) of
an HP Z-200 workstation with a 64 bit operating system with 8GB of RAM.
3 Results
3.1 Experimental
A SEM image of the cross-section of the wafer along its side face is shown in Figure 4.
10
Figure 4 Scanning electron micrograph of the cross-section of the wafer
It was observed that the wafer core and skins were visibly distinct regions. The dense skins appeared
to follow the contours of the reedings on the baking plates and possessed small pores. Therefore,
the skin had an approximately constant thickness and followed the shape of the reedings while the
core had a variable thickness which was dependent on its position relative to the reeding. The core
of the wafer had much larger pores which appeared to be closed cell in nature since most of the cell
faces were visible. Various measurements of different geometric features were recorded and are
summarised in Figure 5. It should be noted that due to the three dimensional grid of reedings on the
wafer (see Figure 7a), it was not possible to determine the overall porosity from a single SEM image
cross-section.
Figure 5 Measurements of the cross-section of the wafer
The pores within the core varied in diameter between 0.5 – 1.1 mm. It was also observed that the
larger pores tended to be located at the centre of the core while the smaller pores were closer to
the skin regions. While the majority of the cells were closed in nature, there were a few which
11
looked open and were interconnected to neighbouring cells. These tended to occur amongst the
largest cells where the cell walls were the thinnest. The cell wall thickness within the core was very
small at the centre of the core and varied between 5 – 25 μm. Figure 6a shows a magnified region of
the core in which the variation of the wall thickness can be seen. At this level of magnification,
micropores were observed to exist within the solid material of the wafer. Due to their small size,
they were ignored in this study and the solid wafer material was assumed to be homogenous. The
pores located within the skins were all closed in nature and were separated from each other by very
thick walls, which gave the skins their dense appearance as highlighted in Figure 6b. The cells
themselves were much smaller than within the core and varied in size between 0.05 – 0.15 mm.
Figure 6 Magnified region of the a) wafer core and b) wafer skin
SEM images were also taken of the surface of the wafer which clearly showed the skin with its grid of
reedings. Some wafer samples possessed imperfectly formed reedings as seen in Figure 7a. For these
particular reedings, a single large pore spanned almost the entire length between parallel lines of
reedings. They were most likely formed as the wafer was removed from the baking plates. The
square surface between adjacent reedings was not smooth and contours could be seen in Figure 7b
which indicated slight variations in the skin topography. Small pores were visible throughout the
entire surface of the skin and varied in size between 20 – 60 μm.
12
Figure 7 Scanning electron micrograph of the surface of the wafer showing a) the large pores on some
reedings and b) the contours and micropores on the wafer surface
Figure 8 shows the stress-strain curve of the in-situ wafer compression and the fifteen labelled
points indicate the wafer deformation at 0.05 intervals of compressive strain while Figure 9 shows
the corresponding micrographs. The stress (σ) and strain (ε) were calculated from Equations 7 and 8
respectively for all experiments using load (F) -displacement (δ) data and the dimensions of the
sample length (L) and height or thickness (H). The sample height was measured from reeding peak to
peak [53].
𝜎=
𝐹
𝐿2
(7)
𝜀=
𝛿
𝐻
(8)
Figure 8 A typical stress-strain curve of a wafer sheet obtained from in-situ SEM compression
The compressive stress-strain curves were characteristic of a brittle foam displaying the three
distinctive stages of the deformation. The first stage (1-3) was linear elastic until the brittle collapse
stress, at which point there was a sudden drop in the stress due to the initial fracture of the cell
walls. This was followed by the jagged plateau (4-11) which was characteristic of a brittle foam. The
final stage (12-15) showed a rapid increase in the stress, typical of foam densification. In addition to
the three stages, a small initial non-linear deformation was observed before the linear elastic region
which was attributed to the geometric imperfections and uneven surface of the wafer [52].
As a control, compression tests were conducted using an Instron 5543 mechanical testing machine
under environmental conditions of 21 °C and 51% humidity. These experiments showed that the
vacuum environment did have an effect on the material properties of the wafer as seen by the curve
in Figure 8. The Gibson & Ashby analytical equations for foams [3] do include the pressure of the
fluid within the cells but assume that its effect is negligible if the fluid is air. However in the case of
13
the wafer, the air has a noticeable effect on the deformation since the wafer fractures at very low
stresses, comparable to atmospheric pressure [52]. It is also possible that electron irradiation
damage may be a possible source of the discrepancy between the in-situ SEM and Instron
compression curves. The deformation response was still characteristic of a brittle foam but was
stiffer and more fracture resistant when tested in standard environmental conditions. The foam
modulus, E*, and brittle collapse stress, σ*, of the wafer were calculated to be 4.32 ± 1.05 MPa and
0.38 ± 0.07 MPa respectively, using the data from the experiments performed under atmospheric
pressure and controlled environmental conditions.
Figure 9 In-situ SEM images of the wafer sheet at different stages of the compression
14
It was difficult to see any cell wall bending in the linear elastic region (1-3), by visual inspection the
SEM images. Beyond the apparent fracture point (4), cracks were visibly propagating along the cell
walls. In some experiments, there was no visible damage despite the deformation plot indicating
that the fracture point had been reached. This was because cell wall fracture was occurring
somewhere within the structure that was not observable on the 2D plane of sight. Within the
plateau region (5-11), cell walls fractured and progressively collapsed. The pores diminished in size
and fractured pieces of material filled the voids. Cracks propagated to the skins and they eventually
failed along the reeding lines. The densification stage (12-15) began at this stage as the wafer
material was being fully compacted. As expected, the initial fracture occurred within the cell walls of
the core while the skins remained undamaged until the core had fully collapsed.
The porosity of the wafer could be determined using the data from the XMT scan. After the
segmentation process was performed on Avizo, the total number of labelled voxels was counted
which represented the solid volume of the wafer. The process of labelling the images was repeated
again, this time filling all the holes thus creating a volume with no pores. These voxels represented
the bulk volume of the wafer. The ratio of the solid and bulk volume voxels gave the porosity of the
wafer which was 0.735 and corresponded to a relative density of 0.265 (see Equation 1). The
porosity value was close to the experimentally obtained value of 0.714 which was calculated using
separate measurements from helium pycnometry and glass bead displacement methods to
determine the solid density and foam density respectively [52].
One advantage of measuring the relative density using imaging techniques over experimental
methods was that it is a non-destructive technique. Therefore, it was possible to determine the
porosity of the wafer skin and core individually without having to physically separate the two
regions. Given the brittle nature of the material, the geometry of the wafer and small scale of the
sample, it proved impossible to produce separate skin and core specimens. By manually sectioning
the skin and core regions of the image slices, and then using the labelling method described above,
the porosity of the skin and core was found to be 0.4 and 0.85 respectively corresponding to relative
densities of 0.6 and 0.15 respectively. These values conformed to the range of values for porous
solids (< 0.7) and cellular foams (0.7 – 0.95) respectively [3].
Another advantage of the XMT scans over the SEM images was that the volume of the pores could
be measured as compared to the 2D measurements of the pore diameter. This was performed by
labelling voxels belonging to the pores. Within the skins of the wafer, the pore volumes varied
between 0.0008-0.016 mm3. However, the interconnectivity between some cell walls in the core
made this a tedious task and no further characterisation of the pore volumes in the core was
performed.
Figure 10 shows XMT 2D image slices (left) and 3D radiographs (right) at a cross-section in the
middle of the wafer sample at the different stages of the in-situ compression. At the final
displacement, the sample was strained to a value of 0.209, which suggested that the wafer had
fractured based on the compression stress-strain curve of Figure 8.
15
Figure 10 XMT image slices and radiographs at different compressive strains
The individual image cross-sections at different locations were compared with each other to see how
the internal microstructure changed during the deformation. There was no noticeable damage
occurring within the first two increments of strain. Evidence of broken cell walls was first seen in the
fourth scan, at a strain of 0.091. Some of these cell walls are circled in Figure 10. In the final two
scans, cell walls within the core were collapsing and contacting each other indicating that the
deformation was now within the plateau region. The in-situ XMT compression test was static and
thus a continuous deformation plot could not be obtained. However, the load was recorded for each
displacement increment and used to find the corresponding stress and strain (which is shown in
Section 3.3).
3.2 Analytical
As already outlined in Section 3.1, the foam modulus, E*, and brittle collapse stress, σ*, were
measured experimentally. It was then desired to estimate the solid wall modulus, Es, and rupture
stress, σs, using analytical modelling because these material properties are needed for the finite
element simulations of the foam compression presented in Section 3.3. For the analysis of the solid
part of the wafer, it was assumed that the material properties were the same throughout the entire
structure. This assumption was justified since the image stack obtained from the XMT scan showed
that the voxels belonging to the wafer were of uniform contrast which implied that the material was
the same throughout the wafer.
The foam modulus of the wafer, E*, was determined experimentally from compression tests to be
4.32 MPa (see Section 3.1). The relative density was measured experimentally and verified using
image analysis of the reconstructed wafer volume to be 0.265 (see Section 3.1). Upon inspection of
Equation 2, the only unknown therefore (apart from Es) was the proportion of wafer material in the
16
cell edges, φ. Due to the irregularity of the shape of the pores in the wafer, it was difficult to
accurately determine this value from the SEM and XMT images. A parametric study was performed
to study the effect of φ on Es. The parameter φ was varied between 0 and 1, and the results are
plotted in Figure 11.
From the in-situ SEM and XMT compression tests, it was visually deduced that the wafer core was
mainly responsible for the deformation in compression and hence the foam behaviour. Thus the
value of Es might be more accurate if the relative density of the core (0.15) was used in Equation 2
instead of the relative density of the entire wafer (0.265). The solid modulus was estimated for
values of φ between 0 and 1 for both values of relative densities with the results shown in Figure 11.
Figure 11 The effect of φ on the calculation of the solid modulus and rupture stress of the wafer and the core
using the Gibson-Ashby foam model
Many closed-cell foams, produced from liquids and with very thin cell faces have been analysed as
open-cell foams [1]. In the case of the wafer under study here, the SEM and XMT images showed
that the cell walls and cell faces in the wafer core were indeed very thin. This indicated that the
foam possessed a φ value close to 1. The results in Figure 11 showed that as the value of φ tended
towards 1, and thus the foam approached an open cell structure, the value of Es was heavily
influenced by the relative density of the foam. For a completely open celled foam (φ=1), the solid
modulus was 61.5 MPa and 199.9 MPa using the relative density of the entire wafer (0.265) and the
wafer core (0.15) respectively.
The brittle collapse stress, σ*, of the foam was measured from the experimental stress-strain
compression curves. As with the solid modulus, the rupture stress, σs, of the cell wall was calculated
from Equation 3 using the relative densities of the entire wafer (0.265) and the wafer core (0.15).
These values were found to be 13.9 MPa and 33.7 MPa respectively when φ was set equal to 1. A
summary of the measured porosities and calculated relative densities, solid moduli and rupture
stresses are given in Table 1.
Table 1 The numerical values of the material model parameters
Entire wafer
Ε
ρ*/ρs
Es [MPa]
(open)
Es [MPa]
(closed)
σs [MPa]
(open)
σs [MPa]
(closed)
0.735
0.265
61.5
16.3
13.9
1.4
17
Wafer core
0.85
0.15
199.9
29.4
33.7
2.6
It should be noted here that the analytical equations for Es and σs assume a simple repetitive cellular
structure which is not the case in the wafer. Thus these values are not exact but serve as a guide to
use as material properties of the wafer cell wall in the finite element model which will be described
in the next section.
3.3 Numerical
A mesh sensitivity analysis was first performed to determine the appropriate level of mesh
refinement which would ensure accurate results. Models with mesh sizes between 230,000 and
2,400,000 linear tetrahedral elements (C3D4) were used in this study. Initially, the wafer material
was assigned a solid modulus (Es) of 200 MPa and Poisson’s ratio of 0.3 with no damage criteria.
Results from a parametric study (not shown here) suggested that the Poisson’s ratio had a negligible
effect [52]. The reaction force acting on the rigid plates and their corresponding displacements were
used to calculate the apparent stress and strain respectively. The resulting foam modulus (apparent
stress divided by apparent strain) and corresponding computing times for each mesh are shown in
Figure 12. It should be noted that the most refined mesh was completed in over 24 hours and hence
not plotted in Figure 12. Apart from the coarsest mesh, the foam modulus showed minimal deviation
as the mesh was refined. Additionally, the two coarsest models were converted to meshes with
quadratic elements. The 230, 000 and 400, 000 elements simulations were completed after 10 and
56 hours respectively, both resulting in values of E* which were approximately 4 MPa and less stiff
than the equivalent linear elements. In order to minimise computational resources, all future
simulations were performed using the wafer mesh with approximately 400,000 linear elements. This
was equivalent to almost 55 voxels per element.
Figure 12 The variation in E* and computing times for various mesh densities
18
The value of the wafer solid modulus was unknown and had to be estimated, based on calculations
in Section 3.2. A parametric analysis was performed by varying the Es value. The linear part of the
predicted stress-strain curve was used to determine the foam modulus E*. The results showed that
the relationship between the solid and foam moduli is proportional with a gradient of 0.0251, in
agreement with Equation 2. Using the experimentally measured average foam modulus of 4.32 MPa,
the solid modulus was estimated to be almost 172 MPa. Using this value and inspecting Figure 11
showed that φ was close to 1 when using the relative density of the core but there was no sensible
value when using the relative density of the entire wafer. This further supports the earlier argument
regarding the use of the relative density of the core rather than the one of the entire wafer in
Equation 2 (See Figure 11).
Having established the value of the cell wall modulus, Es, the next step is to determine the cell wall
yield stress in the damage model. The latter was assumed to be equivalent to the rupture stress, σs,
of the cell wall material. The solid modulus and Poisson’s ratio values were kept constant at 200 MPa
(see Table 1) and 0.3 respectively. The parameters of the damage criterion were set to εpl0 = 0.001
and Gf = 0. In order to be able to determine the collapse stress, it was assumed that the
experimental collapse stress for a brittle foam was the same in tension as well as in compression [3].
The model was loaded in tension, instead of compression, by displacing the nodes on the reedings in
the y-axis until there was complete fracture. By loading the model in tension it was relatively simple
to determine the collapse point, as this was when some cell walls had broken completely resulting in
a drop in the applied load. The apparent stress at initial fracture was recorded for the various
rupture stresses and a proportional relationship with a gradient of 0.0107 was obtained which
agrees with Equation 3. For the experimentally measured brittle collapse stress of 0.38 MPa, the
rupture stress is 36 MPa. Using this value and inspecting Figure 11 showed that φ was approximately
1 when using the relative density of the core but there was no sensible value when using the relative
density of the entire wafer. This was in agreement with the observation regarding the modulus of
the cell wall.
The numerical relative modulus, E*/Es, and relative fracture stress, σ*/σs, were compared to the
analytical predictions of the Gibson & Ashby analytical models, Equation 2 and Equation 3
respectively. The results are shown in Figure 13. The numerical predictions for the relative modulus
and the relative fracture stress were closer to the open cell analytical value (φ = 1) using the relative
density of the wafer core (ρ*/ρs = 0.15). Of note is the fact that the Gibson & Ashby calculations for
fully closed cell foams were far from the finite element output. The FE simulation validated the
assumption that the relative density of the wafer core should be used to estimate the solid modulus
and rupture stress of the cell wall material. The numerical and analytical values of E*/Es were 0.0251
and 0.0216 respectively while the equivalent values of σ*/σs were 0.0107 and 0.0113 respectively.
Additionally, the FE results implied that the cell wall membranes in the core were thin enough to
represent the wafer core as an open cell structure.
19
Figure 13 Comparison of the numerical results to analytical predictions for relative modulus and relative
fracture stress
The material parameters which were ultimately used in the wafer model are summarised in Table 2
and were selected based on the wafer core open cell calculations (Table 1). This final simulation
included the material model with a damage function, contact between cell walls, two rigid bodies as
the compression plates and an appropriate step time to obtain a quasi-static response.
Table 2 The numerical values of the material model parameters
Young’s Modulus
[E] (MPa)
200
Yield Stress
[σ0] (MPa)
35
Poisson’s Ratio
[ν]
0.3
Fracture Strain
[εpl0]
0.001
Fracture Energy
[Gf] (kJ/m2)
0
As the deformation progressed, elements in the core deformed and eventually degraded to zero
stiffness. These elements were 'deleted' from the model thus simulating fracture in the wafer. This
was quite similar to the deformation that was observed from the in-situ compression experiments.
The stress contours on the wafer model showed that the initial stress concentrations occurred
within the core of the wafer and thus it was the site at which element deletion initiated. The stress
contours for the wafer at different stages of the compression is shown in Figure 14. In the images it
can be seen that there are some stresses in parts of the wafer which are in contact with the plates,
however the magnitude of the stresses are much less than the critical stress needed for element
deletion.
20
Figure 14 The damaged wafer at different stages in the compression (stress in MPa)
The stress-strain output is plotted in Figure 15. The predicted stress-strain curve had an initial linear
elastic region, followed by a drop in the stress at a strain of approximately 0.1. A jagged region
typical of a brittle foam with a rising trend in the stress as the strain increased ensued. The element
deletion and contact between adjacent cell walls accounted for the jagged region. The deformation
curve predicted from the finite element simulation was compared to what was observed
experimentally from the Instron and in-situ XMT compression tests. For clarity, only two
experimental stress-strain plots are shown in Figure 15 and the range of data is indicated by the
shaded region. The finite element output and the results obtained from the experiments were in
close agreement to each other. The initial slope was observed to be not perfectly linear which was
attributed to using an explicit solver. The curve obtained using the explicit solver and the linear
elastic model (Es = 200 MPa) with no damage is also shown as a comparison which indicates that
there is minimal difference. The fracture stress from the simulation was below the average
experimental value but was still near to the lower measured limit. It should be noted that the
material parameters used in this model (400,000 elements) were also implemented in a more
refined mesh of approximately 800,000 elements. The stress-strain output was very similar, however
the computing time was 8 hours and 95 hours for 400,000 and 800,000 elements respectively.
21
Figure 15 Comparison of the FE output to the Instron and in-situ XMT compression results
4 Discussion
The microstructure of the SEM, XMT and FE wafer at equivalent global strains were analysed using
Figure 9, Figure 10 and Figure 14 respectively. From the FE model it could be seen that at global
strains less than 5% there was deformation in the cell walls but no fracture, as was the case with the
SEM and XMT images. The contour plot of the model (Figure 14c) showed that internal stresses were
developing within the cell walls of the core. When the wafer model was deformed to approximately
10% strain, some elements within the core reached the maximum stress and were then deleted,
representing the initial fracture in the cell walls (Figure 14d). As the model was further compressed,
more elements from the simulation progressively deleted and interactions between adjacent cell
walls were apparent (Figure 14f), representing the brittle plateau region. The simulation predicted
the core being damaged while the denser skins were relatively unaffected.
However the model in its present state cannot be used to simulate crushing the wafer well into the
densification region. Firstly, as the rigid plates continued compressing the model, elements would
continue being damaged and hence deleting. Physically, this implies that wafer material is
“disappearing” which is not realistic. Secondly, contact was only implemented on the faces of
elements which were on the initial surfaces of the wafer. Thus when these elements were deleted,
the elements adjacent to them did not offer any resistance and interpenetration would occur. In the
future, this will be rectified by updating the interior faces of elements progressively as the simulation
progresses.
By comparison to the in-situ experiments, it could be seen that the finite element simulation
predicted well the deformation of the wafer. It should be noted that each of these in-situ methods
possessed associated drawbacks. In-situ SEM experiments give only 2D information relevant to the
cross-section being imaged but it does give a synchronised and continuous load-displacement trace.
To capture a video, the resolution of the individual frames must be sacrificed as compared to the
static micrographs. The in-situ XMT data allowed the internal 3D microstructure to be observed.
22
However the test is interrupted and deformation occurs in increments, thus it cannot capture the
exact point of initial fracture and brittle collapse.
The FE model is a 3D volume simulation and any cross section can be viewed. It can be used to
determine stress or strain contours on cell walls as well as produce a continuous load-displacement
graph and a complete microstructural evolution of the foam. The drawback is the computational
time required. The FE model presented in this paper included a damage parameter which would
allow the deformation to be simulated beyond the linear elastic region. The deformation plot from
the FE analysis was compared to the experimental stress-strain curves and was shown to have quite
similar trends with an initial linear region followed by a jagged plateau. The brittle collapse stress
was slightly underpredicted by the FE model, but the plateau region matched the experimental data
quite well. It is important to note that the FE model represents only the architecture as obtained
from a single XMT experiment, which would also account for discrepancies between numerical and
experimental results.
5 Conclusions
The FE model’s ability in predicting the compressive response of the wafer to a high level of accuracy
both qualitatively and quantitatively was demonstrated at large global strains. The loading
conditions can be varied and thus the model can be used in the future to simulate biting for sensory
perception studies or other industrial processes such as cutting. The load deformation predicted by
the numerical model could be correlated to texture and help in determining the ‘crispness’ of
various confectionery wafer geometries which would remove the need to physically bake different
products. A cutting simulation would allow multiple parameters such as blade thickness, tip
sharpness, cutting angle and cutting speeds to be varied easily therefore saving time and money
needed to perform real experiments. The method described in this paper is generic and can
therefore be applied to any cellular material, including foams for structural applications.
Acknowledgements
The financial support of the EPSRC is greatly appreciated for the Deben Microtest used to complete this
research and special thanks to Nestlé for partially funding the studentship and supplying materials for testing.
We also wish to express our thanks to Prof. Peter Lee and Richard Hamilton for performing the CT-scanning.
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