r_tanner_mp

DEPARTMENT OF MECHANICAL ENGINEERING
Report Submitted for the Degree of Engineering June 2016
Fracture of Brittle Lattice
Structures
Robert Tanner
Supervisor: Prof. Martyn Pavier

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Acknowledgements
First of all, thanks to my supervisors Professor David Smith, and Professor Martyn Pavier,
whose advice and support throughout this project made it possible. I acknowledge Andreas
Andriotis for his help in understanding the dimensions required for the test specimens. I also
am also grateful to Guy Pearn for his technical expertise with the testing machines and advice
which informed the specimen design process, and the Light Structures Laboratory for
providing the Instron MJ 6272. Huiyuan Gu was particularly helpful for his knowledge of
Abaqus CAE, as was Paige Spicer for making it possible for me to use the full edition of
Abaqus CAE. I am thankful for the help of Steven Harding and the physics laboratory who
produced the experimental test specimens. Julie Etches and the ACCIS helped me by
supplying a video camera for the experimental phase of the project.
DECLARATION
The accompanying research project report entitled: Fracture of Brittle Lattice
Structures, is submitted in the third year of study towards an application for the
degree of Master of Engineering in Mechanical Engineering at the University of
Bristol. The report is based upon independent work by the candidate. All
contributions from others have been acknowledged above. The supervisors are
identified at the start of the report. The views expressed within the report are
those of the author and not of the University of Bristol.
I hereby declare that the above statements are true.
Signed (author)
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Date
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Author Robert Tanner
Title Fracture of Brittle Lattice Structures
Date of Submission 20th
April 2016
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This dissertation is the property of the University of Bristol Library and may only be used
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Robert Tanner Department of Mechanical Engineering
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Summary
The main objective of this project was to determine the validity of the fracture mechanics
approach to predicting the failure of a large lattice structure. A lattice is a series of repeating
cells making up a framework, and microscopic lattices change the fracture behaviour of
materials they are introduced into. Lattice consisting of two and four unit circular cells were
cut from thick PMMA, to ensure plane strain conditions. Crack of varying lengths and
configurations were introduced into the lattices and then the lattices were loaded under
uniaxial tension. First, the material properties of PMMA were determined by standard
methods and the lattices were observed to see if the cracks would propagate through the
material in the manner predicted by the fracture mechanics approach. Then, the critical
lengths and loads of each crack were measured and used to construct finite element models.
These models were used to estimate the critical stress intensity factor of each crack at the
point of instability, and these were compared to the plane strain fracture toughness of the
PMMA. Ultimately, the critical stress intensity factors of the cracks were found to be close to
or less than the measured fracture toughness of the PMMA, indicating that the fracture
mechanics approach may not be a wholly valid method of predicting the failure of a large
lattice.

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Table of Contents
Acknowledgements.....................................................................................................................i
Declaration of Copyright ...........................................................................................................ii
Summary....................................................................................................................................1
Table of Contents.......................................................................................................................2
List of Figures............................................................................................................................3
List of Tables .............................................................................................................................4
1 Introduction ........................................................................................................................5
1.1 Introduction to fracture mechanics...................................................................................5
1.2 Properties of hierarchical solids.......................................................................................6
1.3 Deformation and fracture of hierarchical solids...............................................................7
1.4 Multiple crack-void interaction........................................................................................8
1.5 Project objectives and scope ..........................................................................................11
2 Materials and methods......................................................................................................12
2.1 Materials....................................................................................................................12
2.2 Methods.....................................................................................................................13
2.2.1 Measurement of material properties ..................................................................13
2.2.2 Testing of lattices...............................................................................................15
3 Results ..............................................................................................................................18
3.1 Measurement of material properties..........................................................................18
3.2 Lattice structures .......................................................................................................20
4 Finite Element Analysis....................................................................................................22
4.1 FE verification................................................................................................................22
4.2 FE analysis of lattice structures......................................................................................24
4.2.1 Two-hole cracked lattice .........................................................................................25
4.2.2 Four-hole cracked lattice .........................................................................................25
5 Discussion.........................................................................................................................26
5.1 Design of lattice specimens.......................................................................................26
5.2 Manufacturing with PMMA......................................................................................27
5.3 Fulfilling project aims and objectives.......................................................................27
6 Conclusions and future work............................................................................................28
6.1 Conclusions...............................................................................................................28
6.2 Future work ...............................................................................................................28
References................................................................................................................................29
Appendix.................................................................................................................................A1

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List of Figures
Figure 1: Hexagonal (L), Triangular (C) and Kagome (R) lattices (Fleck & Qui, 2007)..........7
Figure 2: Multiple pores and cracks in a hypothetical material matrix (Hu et al, 1993)...........9
Figure 3: Variation of S, f(a/W) for a crack extending from a hole (Grandt, 1974).................9
Figure 4: Pore distribution patterns (Cramer & Sevostianov, 2009) .......................................10
Figure 5: Variation of KI for different crack-hole configurations (Hu et al, 1993).................11
Figure 6: Triangular (L) and square (R) periodic arrays (Isida & Nemat-Nassr, 1987)..........11
Figure 7: Transition from slow to fast crack growth in PMMA (Döll & Weidmann, 1976)...13
Figure 8: Dogbone specimen ...................................................................................................14
Figure 9: Tensile test in progress.............................................................................................14
Figure 10: Engineering drawing of a CT specimen.................................................................15
Figure 11: Two-hole (L) and four-hole (R) lattice designs......................................................16
Figure 12: Four-hole cracked lattice........................................................................................17
Figure 13: Two-hole lattice in testing machine .......................................................................18
Figure 14: Failure at loading hole of a four-hole specimen.....................................................20
Figure 15: Two-hole lattice after final fracture........................................................................21
Figure 16: Experimental load-displacement data of a four-hole specimen .............................22
Figure 17: Example of an FE model (L) and mesh (R) used in the verification process.........23
Figure 18: Analytical and FE results for stress intensity factor against a/R for crack extending
from hole in a large plate under uniaxial loading ....................................................................24
Figure 19: FE model of a two-hole lattice ...............................................................................25
Figure 20: FE model of a four-hole lattice...............................................................................25
Figure 21: Significant lattice dimensions ...............................................................................A1
Figure 22: Material properties results sheet............................................................................A2
Figure 23: Tensile test stress-strain curve...............................................................................A2
Figure 24: Fracture toughness test force-displacement curve.................................................A2
Figure 25: Two-hole lattice results sheet................................................................................A3
Figure 26: Two-hole lattice force-displacement curve ...........................................................A3
Figure 27: Four-hole lattice results sheet (1) ..........................................................................A4
Figure 28: Four-hole lattice force-displacement curve (1) .....................................................A4
Figure 29: Four-hole lattice results sheet (2) ..........................................................................A5
Figure 30: Four-hole lattice force-displacement curve (2) .....................................................A5

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List of Tables
Table 1: Results of tensile tests................................................................................................19
Table 2: Results of fracture toughness tests.............................................................................20
Table 3: Critical crack lengths and loads of two-hole specimens............................................22
Table 4: Critical crack lengths and loads of four-hole specimens...........................................22
Table 5: FE model details and calculated values of KIc for the two-hole cracked lattice
specimens.................................................................................................................................25
Table 6: FE model details and calculated values of KIc for the four-hole cracked lattice
specimens.................................................................................................................................26
Table 7: Applied bearing stresses on loading holes at failure .................................................26

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1 Introduction
The initial project specification was to simply investigate the fracture of brittle lattice
structures. This was initially inspired by the observation made by Professor Smith that a strip
of acrylonitrile butadiene styrene (ABS) produced by three-dimensional printing with a crack
introduced fractured in a manner other than that predicted by the fracture mechanics approach
to material failure. The crack initially propagated through the ABS for a short distance before
suddenly seeming to take a ninety degree turn and cause the strip to split along the boundary
between two layers of plastic, before taking another turn and propagating through the strip
parallel to the boundary between layers. It was clear that the layered structure of the ABS
making up the strip had caused the material to fail in a manner unlike a solid block of ABS.
The initial project aims were therefore to produce a comprehensive literature review of the
fracture of lattice structures and determine the validity of the fracture mechanics approach to
a lattice structure composed of a brittle material.
This section summarises the research into fracture mechanics, lattice structures and the
superposition of multiple hole-crack interactions in solid matrices carried out in the initial
phases of the project which informed the project objectives and the experiments carried out to
achieve them.
1.1 Introduction to fracture mechanics
Materials can fail under a range of failure mechanisms, however, generally mechanical
failure occurs through either yield or fracture. This subsection will discuss yield and fracture,
as well as the basic relationships and equations that govern fracture behaviour.
A material will fail if the applied stress exceeds the strength, or maximum allowable stress.
Broadly, materials fall into two classes: brittle and ductile. A ductile material will undergo
significant plastic strain before failure, whereas a brittle material will experience little or no
plastic strain prior to failure. Brittleness and ductility are not inherent material properties and
are dependent on external factors such as temperature and sample thickness.
For instance, if a thick walled material specimen is loaded under tension, it will not be able to
contract like a thin walled specimen would, this is a state known as “plane strain”. This leads
to the presence of a tensile stress in the material which opposes contraction. Therefore, the
stress state at the centre of a thick specimen tends towards stress triaxiality, a state where all
normal stresses are effectively equal and yield cannot occur (Roylance, 2001). Since yield is
impossible, if a sharp crack is present in a thick specimen, brittle fracture occurs at the tip.
The study of how the presence of cracks influences the failure behaviour of a material is
known as Fracture Mechanics. Fracture mechanics is a fairly mature field; it has long been
known that the presence of cracks or voids in a material or structure are weak points where
the structure will fail at a lesser than predicted load. In 1910, Hopkinson referenced a cracked
boiler plate which had ruptured under a pressure of 12 Tons per square inch despite being
constructed of a steel which had a fracture strength in excess of 30. In 1913, Inglis
determined that an elliptic hole in a plate under far field loading distorts the local stress field,
with stress being greater at the tips of the ellipse. From the relationships between hole
geometry and stress amplification which Inglis determined, Griffith (1920) was able to
determine the effect of the hole height tending to zero, thereby modelling a sharp-tipped
crack, proposing the crack will grow if the total energy in the plate is increased.
In 1957, Irwin defined the resistance of the plate to new crack growth as 𝑅 and the driving
force of crack growth, or “strain energy release rate” as 𝐺. 𝐺 is dependent on the

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characteristic length of the crack and therefore increases as the crack grows. In intrinsically
brittle materials or materials under plane strain conditions, 𝑅 is effectively a constant and
therefore fracture occurs when 𝐺 exceeds a critical value. When a material specimen
containing a crack is loaded, the crack slowly propagates through the material under a
mechanism known as stable crack growth. Under small loads the crack will experience a
small increase in length and then no longer propagate unless the load is increased. However,
under a critical load or at a critical crack length the crack will transition from slow to fast, or
unstable, crack growth and instantaneously propagate completely through the material, and
final fracture will occur (Broek, 1999).
A sharp-tipped crack distorts the local stress field, amplifying the effective stress in the
material around it. A measure of how much a crack amplifies local stress is the “stress
intensity factor”, 𝐾. For materials under uniaxial tension, known as Mode I loading, the stress
intensity factor 𝐾𝐼 can be calculated as:
𝐾𝐼 = 𝑆 𝜎 √ 𝜋𝑎 𝑓(𝑎 𝑊⁄ ) (1)
Where 𝑆 is a scalar factor, 𝑎 is the characteristic length of the crack and 𝑓(𝑎/𝑊) is a
function of 𝑎 and 𝑊, the characteristic width of the specimen. Tada et al compiled a list of
values for 𝑆 and 𝑓(𝑎/𝑊) in the “Stress Analysis Handbook”.
The stress intensity factor when unstable crack growth occurs is the critical stress intensity
factor, 𝐾𝑐, and is dependent on the thickness of the material, however, as the sample becomes
thicker, 𝐾𝑐 tends to a constant value, the plane strain fracture toughness 𝐾𝐼𝑐, which is a
material rather than a structural property (Janssen et al, 2004).
The test procedure for measuring 𝐾𝐼𝑐 of a material is detailed in Section 2.
1.2 Properties of hierarchical solids
The fracture behaviour of continuous materials, or “continua”, is well understood, but non-
continuous materials are more complex in their behaviour. This subsection explains the
physical properties of non-continuous materials, while the next section will cover their
fracture behaviour.
A continuous material is a uniform, homogenous structure with no microscopic or
macroscopic defects. In reality, however, the vast majority of materials have structure on
multiple length scales, these are known as “hierarchical solids”. The levels of structure in
hierarchical solids may go down to the atomic level. Hierarchical solids come in many forms:
composites composed of fibres embedded in a matrix; metals containing multiple phases;
crystalline polymers; or many natural materials such as wood and bone (Lakes, 1993).
A porous material is a class of hierarchical solid which contains voids within a solid matrix.
The properties of a porous solid are usually defined relative to the material it is made from,
for instance relative density, 𝜌̅, and relative modulus, 𝐸̅. Generally, porous solids could be
thought of as any hierarchical solid with a relative density less than one or a porosity, 𝜑, of
greater than zero (Gibson & Ashby, 1997).
Besides the range of materials containing voids or pockets, a special subset of porous solids is
“cellular solids”; a composite where one phase is a solid material and another is a fluid or
empty space in a naturally occurring lattice. Cellular solids are generally considered to be
porous solids with a porosity of 70% or greater, meaning only 30% of a cellular solid is
composed of a solid material (Gibson & Ashby, 1997). Cellular solids can take the form of
either two-dimensional lattices or three-dimensional polyhedra called foams (Lakes, 1993).

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Honeycombs are generally uniform lattices, whereas foams are generally non-uniform
lattices. Examples of the three main forms of honeycomb lattice structures are shown in
Figure 1.
The properties and failure modes of cellular solids are very different to the properties of the
component material, which will be explored in greater detail in the next subsection.
1.3 Deformation and fracture of hierarchical solids
For simplicity, hierarchical solids are generally modelled as continuous below the relevant
length scale, however, if the hierarchical structure is non-negligible compared to the size of
the material or a defect within it, then it affects the material behaviour (Lakes, 1993). This
section will discuss the methods by which hierarchical structures can fail in a different
manner to continuous materials, with particular emphasis on cellular solids. The fact that
microscopic lattices and material structure affect failure mechanisms, suggest that a large
structure may also affect material failure.
Hierarchical structure can strengthen a material, for example, composites such as MDF
(medium-density fibreboard) and PB (particle board) are often more resistant to fracture than
continuous materials, as wooden fibres bridge the crack in the resin behind the propagating
crack tip (Matsumoto & Nairn, 1992). However, hierarchical structure can also weaken a
material. In 2012, Hooreweder et al performed fracture toughness tests on a specimen of
Ti6Al4V produced by traditional manufacturing means, and a specimen of the same alloy
manufactured by rapid prototyping (RP) with a fine grained microstructure and deposits of
martensite, which proved to have a fracture toughness roughly 75% of the continuous
material.
Cellular solids
The deformation and fracture of cellular solids has been comprehensively investigated, and a
series of equations which describe their physical properties and fracture behaviour. This
branch of fracture mechanics indicates how lattice structures at the microscopic scale alter the
properties of the materials they are composed of and suggest a lattice structure on the
macroscopic scale may also lead to a change in material behaviour.
Cellular solids are composed of unit cells, in honeycombs, unit cells are two-dimensional
geometric shapes, and in foams, unit cells can be either self-contained polyhedral (closed-
cell), or a three-dimensional skeletal structure (open-cell) (Choi & Sanakar, 2003). Unit cells
can be modelled as a pin-jointed truss of strong, stiff and light struts. The cell structure
determines the physical properties of the cellular solid. The structural property that governs
the relationship between cell structure and overall properties is known at the Maxwell
stability criterion, 𝑀, calculated from the number of “struts” and “pins” in the unit cell.
Figure 1: Hexagonal (L), Triangular (C) and Kagome (R) lattices (Fleck & Qui, 2007)

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If 𝑀 < 0, then the unit cell is defined as a “mechanism”, meaning that the unit cell will
collapse or rotate under loading. Whereas, if 𝑀 = 0, the unit cell is defined as a “structure”,
if loaded it will not collapse and the members will carry stress (Fleck & Qui, 2007).
Cellular solids generally fall into two categories, “stretch-dominated” and “bending-
dominated”. Stretch-dominated structures are stiff and can support greater loads whereas
bending-dominated structures are compliant and can absorb more energy before failing. The
unit cells of stretch-dominated materials are structures, and bending-dominated materials are
composed of mechanisms. Cellular solids are less stiff and weaker than continuous materials
(Ashby, 2006). This indicates that the presence of a lattice structure alters the physical
properties of a material.
Cellular structure also affects fracture behaviour. In honeycombs, cell walls do not all
fracture simultaneously under tension. If one cell wall fractures, then the stress on nearby
walls increases. This may also cause them to fracture and therefore the damage propagates
through the lattice as walls fail, similar to a crack propagating through a continuous material.
A cluster of broken cells in a honeycomb functions in a similar manner to a crack tip in a
continuous material (Gibson & Ashby, 1997). This indicates that the presence of a lattice
structures alters the fracture behaviour of a material. As a result of the lattice structure, the
plane strain fracture toughness of honeycombs can be broadly defined as:
𝐾𝐼𝑐 = 𝑆 𝜌̅ 𝑛
𝜎𝑓 √𝐿 (2)
Where 𝑛 is a scalar exponent, 𝜎𝑓 is the UTS (ultimate tensile stress) of the lattice material and
𝐿 is the characteristic length of the unit cell. The values of 𝑆 and 𝑛 differ with unit cell shape,
indicating that the structure of a microscopic lattice affects the fracture behaviour of a
material (Fleck & Qui, 2007).
In 2003, Choi and Sanakar performed a four-point bending test on a block of open-celled
carbon foam containing a crack extending hallway along its width. They observed that the
crack propagated almost instantly through the foam block, indicating that the material was
significantly weakened by the presence of the multiple voids in the foam.
1.4 Multiple crack-void interaction
This section will cover the deformation and fracture of porous solids containing multiple
large voids and cracks. Virtually all materials contain multiple microscopic voids or pores
which weaken the material significantly, as shown in Figure 2, the actual fracture strength of
a material is often several orders of magnitude less than the stress which would be required to
break the interatomic bonds in a continuous material (Dowling, 2012). However, this section
will cover holes and cracks which are large relative to the size of the material matrix they are
embedded within, rather than the small discontinuities which all materials contain.

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Large isolated pores, or holes, are similar to cracks in that they also redistribute local stresses
towards their boundaries and can acts as sites for crack initiation and failure under loading.
As holes do not have defined tips, there are usually at least two stress concentration factors
for each hole geometry, at the points on the hole parallel and perpendicular to the direction of
loading, for instance, the stress concentration factors at the edge of a circular hole in a plate
under far-field loading are -1.0 and 3.0.
Cracks propagating out of holes also have their own stress intensity factors, which are
generally defined by the ratio of the crack length to the radius of the hole or to the sum of the
crack length and hole radius, which can also be calculated using the superposition method.
The 𝑆, 𝑓(𝑎 𝑊⁄ ) terms, or 𝐾𝐼 𝜎 √ 𝜋𝑎⁄ , of these stress intensity factors are often nonlinear
equations which tend to a constant value as the crack-radius ratio increases, as shown in
Figure 3 (Grandt, 1974). This will be explored in greater detail in Section 4.
In 2009, Cramer and Sevostianov performed uniaxial tensile tests on thin sheets of aluminium
alloy 6061-T6, all containing forty randomly placed pores 2mm in diameter, with no other
cracks or defects present. They compared the fracture behaviour of solid sheets and porous
sheets. The pores could be distributed randomly or form clusters which were circular, elliptic
and oriented perpendicular to the direction of loading, or elliptic and oriented parallel to the
direction of loading, as shown below in Figure 4.
Figure 3: Variation of 𝑆, 𝑓(𝑎/𝑊) for a crack extending from a hole (Grandt, 1974)
Figure 2: Multiple pores and cracks in a hypothetical material matrix (Hu et al, 1993)

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Cramer & Sevostianov determined that the presence of pores significantly reduced the
stiffness of the sheets, but also that the loss of stiffness was “almost insensitive to the actual
distribution of pores”. This is because the specimens all had an equal degree of porosity, and
relative stiffness of a porous material is dependent only on the material’s total porosity. The
exact relationship between 𝐸̅ and 𝜑 of a porous material is non-linear and contains many
higher-order terms, however, a porous material generally becomes less stiff as porosity
increases (Lu et al, 1999).
Cramer & Sevostianov also found that the presence of pores significantly weakened the
sheets, indicated by a decrease in failure stress. This corroborates the analytical work done by
Zheng et al in 1992, who determined that increased porosity reduced the fracture toughness
and failure stress of a porous material.
They observed that, when the aluminium sheets containing the pores fractured, cracks
initiated at holes close to a group of others and near the edge of the sheet, then the crack
propagated to the next closest hole and the crack would travel along the pores similar to a
crack through the unit cells of a honeycomb. It was determined that an elliptic cluster of
pores oriented parallel to the direction loading caused the greatest decrease in fracture
strength, whereas a random arrangement caused the least.
Thus it is clear that, not only does the presence of multiple pores change the strength of a
material, but that the distribution of pores also has a significant effect, as the stress field
around pores and cracks in a material can affect the stress intensity factors of the other pores
and cracks. Not only can they magnify the local stress intensity factors, but also reduce them,
and there has been a significant amount of work to find equations which govern these
relationships. It is possible through the superposition method, where a complex crack-hole
configuration is broken down into a series of individual hole and crack problems with their
own stress intensity factors and then added together, to create a series of deeply complicated
and involved integral equations which allow the calculation of the stress intensity factors at
any crack or hole in a multiple crack-void problem (Hu et al, 1993). These equations are
deeply complicated and beyond the scope of this report, but the solutions of the equations
applied to simple configurations are useful illustrator of the principles, as shown in Figure 5.
Figure 4: Pore distribution patterns (Cramer & Sevostianov, 2009)

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Figure 5 shows how the stress intensity factor of a crack perpendicular to the direction of
loading near a hole is greatly magnified by the hole’s proximity, but also shows how a crack
aligned perpendicular to loading between two holes has a significantly reduced stress
intensity factor, compared to a crack in an infinite sheet with no adjacent holes. Other
configurations are more complicated, but they also show how different crack-hole
configurations alter the stress intensity factors. The arrangements analysed are generally only
a small number of cracks and holes, not a regularly repeating lattice (Hu et al, 1993). This
further indicates that stress fields within a porous material are significantly affected by the
presence and distribution of holes and cracks.
Some theoretical analysis of regularly repeating lattices of hole-crack arrangements have
been carried out however. It was concerned with symmetricaly cracked holes in periodic
triangular and square arrays, see Figure 6. The theoretical analysis predicted that, under
compression, only stable crack growth would be possible if the cracked holes were regularly
spaced or closed, and unstable growth would become possible only at greater hole
separations. It also determined that, for regularly spaced holes, the stress intensity factor
would tend to zero as the cracks grew (Isida & Nemat-Nasser, 1987). The arrangements
analysed are similar to those that will be tested in this project differ in several key aspects.
However, they indicate how stress intensity factors are altered by the interactions of cracks
and holes.
1.5 Project objectives and scope
The fracture and deformation behaviour of hierarchical materials and materials with
microscopic lattice structures has been comprehensively investigated and understood, with
the equations for all relevant properties determined and recorded.
Figure 5: Variation of 𝐾𝐼 for different crack-hole configurations (Hu et al, 1993)
Figure 6: Triangular (L) and square (R) periodic arrays (Isida & Nemat-Nassr, 1987)

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Also, there has been significant analytical and theoretical work done on the stress intensity
factors of various regular and irregular crack-hole pattern arrangements under tension and
compression, but very little practical corroboration, and when cracks emanating from holes
were analysed, all cracks were of equal length. The experimental work that has been done
was based on thin specimens with random arrangements of uncracked pores. Therefore, there
has been very little experimental work done in the field of how the fracture mechanics
approach relates to large, thick, macroscopic, regular lattices containing cracks of differing
sizes emanating from circular holes. Therefore, this research project will test the fracture
behaviour of cracked lattices, where the lattice walls are thick and the unit cells are
sufficiently large that only a small number of unit cells are present in a test specimen.
The main objectives of this project will therefore be to design and manufacture appropriate
lattice structures with thick walls and a small number of unit cells present in the specimen,
and then introduce sharp cracks into the lattice structures. Then, to design and carry out the
experiments to observe the failure of those cracked lattice structures, and analyse all relevant
experimental data. Following this, the objective was to use the analysis of the experimental
data to determine if the cracks in the lattice caused the lattice to fail in the manner and at the
approximate loads predicted by the fracture mechanics approach.
The level of success in meting these objectives will be discussed in subsection 5.3.
2 Materials and methods
The purpose of this project was to determine if the presence of a lattice structure in a brittle
material would alter the behaviour of cracks within the material in such a manner that the
fracture mechanics approach would not be an appropriate method of predicting the failure of
the material.
First, a suitable material, in this case PMMA was selected; then some standard measurements
of were carried out to determine the salient material properties of the PMMA; and finally, a
series of regular lattice structures with cracks to acts as sites for failure were manufactured
from PMMA and tested to destruction. This section details the materials and methods used in
the experimental phase of the project.
2.1 Materials
The initial project specification was to investigate brittle fracture of a large lattice structure,
so it was first necessary to select an appropriately brittle material. As previously mentioned in
Section 1, “brittleness” is not an inherent material property, however, a common measure of a
material’s relative ductility is the degree of plastic strain experienced by the material prior to
fracture in a tensile test performed at room temperature, sometimes termed “elongation”.
The material selection software CES Edupack was used to screen out all materials which do
not commonly fracture after experiencing an “elongation” of 5% or less. The remaining
materials were considered to be intrinsically brittle (Granta Design, 2015).

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From the list of intrinsically brittle materials, only PMMA, acrylonitrile butadiene styrene
(ABS), wood and cast iron were readily available or easily obtainable. From this subset of
materials is was necessary to select a single material to manufacture the lattice structures. The
initial choice was cast iron as it was the most brittle; however, after further research PMMA
was instead selected for several reasons: first, as an amorphous polymer, it had no
hierarchical structure unlike wood or iron. Second, PMMA can be shaped by traditional
manufacturing methods unlike ABS, which the workshop could only shape by rapid
prototyping (RP), which would have introduced another level of structure. Finally, when
PMMA transitions from stable to unstable crack growth, there is a clear change in crack
surface morphology from rough to smooth, with a defined boundary as shown in Figure 7,
allowing easy measurement of critical crack length (Döll & Weidmann, 1976).
2.2 Methods
In order to determine the validity of the fracture mechanics approach to a large brittle lattice,
it was first necessary to measure the salient material properties of the PMMA, which were
used in the analytical phase of the project. Then, tensile tests were performed on a series of
lattice structures manufactured from PMMA containing sharp-tipped cracks to determine if
the cracks would propagate in the manner predicted by the fracture mechanics approach.
The material tests were carried out with an Instron MJ 6272, which has a maximum load of
25kN, which was predicted to be sufficient to cause failure in all material specimens. Also,
the machine was the smallest available which had the required test space for the large
material samples used in the tests. Larger machines had significantly more powerful load
cells (80-100kN), but those machines would not have had a sufficiently high resolution to
accurately measure the loads and displacements applied to the specimens.
2.2.1 Measurement of material properties
The salient material properties of the PMMA are the plane strain fracture toughness, Young’s
modulus and yield stress. The yield stress is necessary to determine the plane strain fracture
toughness. The plane strain fracture toughness must be measured so that the stress intensity
factors of the cracks in the lattices at the point of instability can be compared to the plane
strain fracture toughness of the PMMA to determine if the PMMA fractures at the
approximate loads predicted by the fracture mechanics approach. The Young’s modulus is
necessary for the analytical phase of the project. This subsection covers the measurements
conducted to determine these properties.
Figure 7: Transition from slow to fast crack growth in PMMA (Döll & Weidmann, 1976)

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Yield stress and Young’s modulus
In order to determine the Young’s modulus and yield stress of the PMMA, standard tensile
tests were carried out in accordance with ISO 527 (ISO, 1994). A variation on standard “dog-
bone” specimens were cut out of 10mm thick PMMA with a waterjet pressure cutter and then
machined down so that they were approximately 5mm thick at the middle. The extra
thickness at the ends of the sample was to ensure that stress would be a maximum in the
gauge section, and therefore yield and failure would not occur at the ends. An engineering
drawing of the tensile specimens is shown in Figure 8.
The specimens were placed into an MJ 6272 testing machine with standard mechanical
wedge action grips and an extensometer was attached to the specimen, the gauge length of the
extensometer, 𝐿, was also measured and recorded. The test machine was set up so that the
lower grip retracted at a constant rate of 1mm per minute (mm/min), until the tensile
specimens underwent plastic yield and fracture. An example of a tensile test in progress is
shown in Figure 9.
Figure 8: Dogbone specimen
Figure 9: Tensile test in progress

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Plane strain fracture toughness
In order to determine a value for 𝐾𝐼𝑐, plane strain fracture toughness tests were performed on
standard CT-specimens of PMMA in accordance with ASTM D5405-14 (2014). The CT
specimens were cut out of 25mm thick sheets of PMMA by waterjet pressure cutter to the
dimensions shown in Figure 10, then a sharp-tipped crack was introduced into the specimen
at the tip of the pointed notch in the centre with a reciprocating fretsaw with a 52TPI piercing
razor saw blade approximately 0.2mm thick. The dimensions chosen allowed the specimen to
fit into the Clevis grips available at the University and ensured that the specimen was under
plane strain conditions. The dimensions were also chosen to satisfy the size criterion (ASTM,
2014), which will be explored in greater detail in Section 4.
Due to manufacturing tolerances, the actual dimensions of the specimens differed from the
engineering drawing specifications, therefore, prior to testing, the thickness, B, and distance
from the centre of the loading holes to the far edge, W, of the CT specimen were measured
and recorded by taking the measurement with digital Vernier callipers at three points and
taking the average of those three values. Then, the specimen was loaded into the Instron MJ
6272 test machine using clevis grips, and was gradually loaded at a rate of 1mm/min until
final fracture occurred. The slow rate of tension was intended to prevent crazes, regions of
low-density, high strength plastically deformed polymer form on the surface of amorphous
polymers under tension, causing the polymer to behave in a manner other than predicted by
the fracture mechanics approach (Kambour, 1973). Crazes cannot spontaneously form in
PMMA (Berry, 1961), but crazing at sharp crack tips is possible.
2.2.2 Testing of lattices
Lattice Development
The circular lattices were designed in two variants; a two-hole variant consisting of two
adjacent bi-axially cracked holes; and a four-hole variant consisting of two evenly spaced
parallel rows of two evenly spaced holes, each containing one hole with a single crack
extending out. An eight-hole variant consisting of four parallel rows of two holes and four
cracks was also designed but it was determined that the eight-hole lattice would be too
complex to provide useful data. Engineering drawings of the holes and crack arrangements in
the lattices are shown below in Figure 11.
Figure 10: Engineering drawing of a CT specimen

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These hole-crack patterns were designed to allow qualitative observation of the validity of the
fracture mechanics approach, as well as the quantitative measurements which will be
explored in Section 4. In the case of the two-hole design, the fracture mechanics approach
predicts that first the centre cracks will become unstable and propagate as the presence of the
two cracks would weaken the material to a greater extent than a single crack, and then the
outer cracks will propagate through and the specimen will split apart completely. This
prediction is based on research carried out by Surendran et al in 2012, which indicated that
the stress intensity factors on the adjacent tips of two collinear cracks were magnified to a
greater degree than the non-adjacent tips, and therefore the cracks would presumably fail at
the adjacent tips.
In the case of the four-hole design, the fracture mechanics approach predicts that the longer,
upper crack will propagate through to the other hole, followed by the shorter, lower crack.
The hole arrangements were chosen so that two expectations could be tested: first, that
collinear cracks in a lattice would fail at the presumed weakest point in the centre and then at
the comparatively stronger points of the other crack tips; and second, that a longer crack in a
lattice would fail before a shorter crack. The use of a two-hole lattice allowed the behaviour
of one set of collinear cracks to be investigated, and a four-hole lattice allowed the
investigation of differing crack lengths. The initial plan was to use analytical methods to
calculate critical stress intensity factors, so simple crack-hole arrangements based on case
studies in the “Stress Analysis of Cracks Handbook” were used to design the lattice.
The cracks were placed at the boundaries of the unit cells (the holes), so that the lattice
structure would have the strongest effect on the stress intensity factors of the cracks, and
extend out horizontally so that the crack propagation direction would be directly horizontal,
allowing for easier analysis. The cracks in the two-hole lattices were placed so that the cracks
would be collinear, and so the region at the centre would be weaker than the regions at the
edges of the specimen. The cracks in the four-hole specimen were placed because it was
initially assumed the cracks would be sufficiently spaced that the stress fields around each tip
would not interfere, therefore the crack-hole arrangement for each row of holes was designed
so they would be symmetrical and identical besides the length of the crack, allowing for
easier analysis and observation of crack behaviour.
Figure 11: Two-hole (L) and four-hole (R) lattice designs

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A detailed engineering drawing of the four-hole cracked lattice specimen is shown in Figure
12. As the drawing shows, the lattice contains pin holes 12.5mm in diameter, allowing it to be
mounted in clevis grips within a test machine. The pin holes are set far enough away from the
lattice holes that the bearing stresses on the loading holes will not affect the stresses at the
crack tips, and set 20mm away from the edges that the pins will not tear out of the specimens,
(Granta Design, 2015). The circular holes are regularly sized and spaced, 20mm in diameter
and 40mm apart, as little work has been done on the presence of regular large lattice
structures in a material. One of the holes in each row of the lattice contains a notch similar to
a CT specimen, where a sharp tipped crack can be introduced to act as a site for fracture. The
notches and therefore cracks are of varying lengths and therefore stress intensity factors.
The lattice structures were cut from 25mm thick PMMA, a thickness chosen so the lattice
would be under plane strain conditions. The lattices were cut using a water jet cutter, and
sharp cracks approximately 2mm long were introduced at the tip of the notches using the
reciprocating fretsaw used to add cracks to the CT specimens. Five specimens of each design
were manufactured. Additionally, two variants of each specimen containing no cracks were
produced so as to provide a control experiment.
Testing method
Due to manufacturing tolerances, the dimensions of the specimens differed from the
specifications. Therefore the specimen dimensions were measured with Vernier callipers and
recorded. This will be covered in greater detail in Sections 3 and 4. The various lattice
structures were mounted in the Instron MJ 6272 in clevis grips, and were subjected to loading
at a rate of 1mm/min, to prevent crazing and allow easy identification of critical loads. A
two-hole specimen under tension is shown in Figure 13.
Figure 12: Four-hole cracked lattice

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As explained in subsection 1.4, the presence of multiple cracks in the specimen means the
stress field around each crack affects the stress field around all the other cracks. This means
that, for the FE models of each specimen to accurately calculate the critical stress intensity
factors of each unstable crack, then the remaining stable cracks must be factored in to the
model. As some stable crack growth may have occurred during the testing, the length of all
cracks in the specimen needed to be monitored. Therefore, each test was filmed with a
camera, so the length of any crack at any point in the test could be estimated by watching the
video and isolating any relevant frames.
3 Results
This section outlines how the salient material properties and the key dimensions of the FE
models discussed in Section 4 were determined.
3.1 Measurement of material properties
This subsection lists the results of the material property measurements of the PMMA.
Yield stress and Young’s modulus
The yield stress and Young’s modulus of the PMMA was determined using the load-
displacement data produced by the extensometer during the tensile tests. The load-
displacement curve was converted to a stress-strain curve, using the calculated area of the
specimen, 𝐴, and the gauge length of the extensometer. Young’s modulus was found as the
gradient of the linear portion of the curve.
The yield stress, 𝜎 𝑦𝑠, was found using the 0.2% offset method (ISO, 1994).
Figure 13: Two-hole lattice in testing machine

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The primary results of the measurements are tabulated below in Table 1, and the raw data; the
completed results sheet, and an example of a stress-strain curve, is shown in the Appendix.
Specimen L (mm) A (mm2) 𝑬 (GPa) 𝝈 𝒚𝒔 (MPa)
1 50 62.0 2.66 27
2 50 61.8 2.89 31
3 50 58.8 2.88 32
4 50 62.0 3.03 35
5 50 61.3 3.14 35
Average 2.92 32
Table 1: Results of tensile tests
The range of values for Young’s modulus is 2.66–3.14 GPa, the mean value is 2.92 GPa and
the standard deviation is 0.16 GPa, all recorded values are within two standard deviations of
the mean and are therefore statistically significant. The measured values for Young’s
modulus also lie within the accepted range (Granta Design, 2015). The range of values for the
yield stress is 27–35 MPa, the mean is 32 MPa and the standard deviation is 3 MPa. The
measured values for the yield stress are within one standard deviation of the mean with one
exception, but are lower than the accepted values by at least a factor of two (Granta Design,
2015), as well as this, while the tensile strengths of the specimens were within the accepted
range, the strain at failure was significantly less than expected. This is likely due to the rough
surface of the specimens, as explained in subsection 5.2.
Fracture toughness
As explained in Section 2, the thickness, 𝐵, and characteristic length, 𝑊, of each CT
specimen was measured and recorded prior to testing, and the critical crack length, 𝑎 𝑐, of
each specimen was measured following final fracture. Each dimension was determined by
taking three separate measurements and calculating the average of those three values.
Following the fracture toughness tests, the critical load of the specimen, 𝑃, the load at which
the crack becomes unstable and propagates through the CT specimen, was determined using
the ASTM standard method of analysing the load-displacement data of a polymer CT
specimen (ASTM, 2014).
Once the significant load was determined, it was used to calculate a trial solution of the plane
strain fracture toughness, 𝐾 𝑄, using the equation:
𝐾 𝑄 =
𝑃
𝐵√𝑊
(2 +
𝑎 𝑐
𝑊
) [0.886 + 4.64 (
𝑎 𝑐
𝑊
) − 13.32 (
𝑎 𝑐
𝑊
)
2
+ 14.72 (
𝑎 𝑐
𝑊
)
3
− 5.6 (
𝑎 𝑐
𝑊
)
4
]
(1 −
𝑎 𝑐
𝑊
)
3
2
(3)
Where 𝑃 is the critical load in kilonewtons, and the dimensions are in centimetres.
Finally, the size criterion was used to verify the trial solutions. As mentioned in Section 2, the
size criterion determines the validity of a fracture toughness test. The equation governing the
size criterion is:
2.5(
𝐾 𝑄
𝜎 𝑦𝑠
)
2
<
𝐵
𝑊
𝑊 − 𝑎 𝑐
(4)
The yield stress determined in the tensile tests was used for the size criterion as, despite being
incorrect, it was a more stringent criterion (ASTM, 2014).

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All the calculated values of 𝐾 𝑄 proved valid, and therefore were considered accurate
calculations of 𝐾𝐼𝑐. The calculated values of plane strain fracture toughness are tabulated in
Table 2. An example of the load-displacement curve of a specimen, as well as the dimension
measurements can be found on page A2 of the Appendix.
Specimen 𝑩 (mm) 𝑾 (mm) 𝒂 𝒄 (mm) 𝑷 (kN) 𝑲 𝑰𝒄 (MPa√m)
1 25.08 50.49 19.5 0.81 1.01
2 24.66 50.38 20.33 0.87 1.15
3 25.22 50.88 20.51 0.93 1.20
4 24.82 50.36 19.59 0.94 1.19
5 24.96 50.53 19.68 0.90 1.14
Average 1.14
Table 2: Results of fracture toughness tests
The range of values found for fracture toughness is 1.01–1.2 Mpa√m, the mean value is 1.14
MPa√m and the standard deviation is 0.07 MPa√m. All values are within two standard
deviations of the mean and the accepted range of values (Granta Design, 2015).
3.2 Lattice structures
This subsection details the observations made during the tests on the lattice structures, and
the quantitative measurements which were used to construct the FE models to determine the
validity of the fracture mechanics approach to brittle lattice structures.
Observations
Under tension all cracks in the two hole lattices failed simultaneously rather than, as
predicted, the central cracks propagating first followed by the outer cracks. It is possible that
it did occur at a speed the camera could not detect, but since only one critical load was
detected by the test machine, it is unlikely. In two specimens, one of the central cracks did
not propagate through the material. In that case, the cracks were not totally collinear and the
other crack simply propagated past and over it. Presumably the adjacent crack reduced the
stress at the tip of the stable crack and prevented it from becoming unstable.
The four hole lattices, however, failed in a manner that was not predicted at the planning
stage. In none of the five specimens tested did Crack 2 (see Figure 11) become unstable and
propagate through to the other hole. As well as this, in two specimens Crack 1 (see Figure 11)
did not become unstable either. In all the four-hole lattice specimens, final failure occurred
when a crack initiated at the loading hole, causing the pin to tear out of the specimen,
rendering it unable to be leading back in the grips and ending the experiments. An example of
a specimen that has failed in such a manner is shown in Figure 14.
Figure 14: Failure at loading hole of a four-hole specimen

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This was despite predictions in the specimen design phase that the bearing stresses exerted on
the loading holes would be significantly less than the compressive strength of PMMA.
Attempts to avert this phenomenon by lengthening the cracks or changing the rate of loading
proved ineffective.
Two variant specimens of each lattice design which did not contain pre-machined notches or
cracks were also manufactured and loaded with clevis grips. They were observed to fail in the
same manner as the four-hole lattices, with a crack forming from the loading holes.
Measurements
As mentioned in Section 2, due to manufacturing tolerances the dimensions of the lattice
specimens differed from the design dimensions. In order to ensure the accuracy of the FE
models, the dimensions of the specimens were measured with Vernier callipers. A diagram
detailing the dimensions which were measured can be found on page A1 in the Appendix.
Also, the lengths of any stable cracks were estimated and recorded at the point of instability
of any cracks that reach their critical length. This information was recorded on a results sheet
and used to construct an FE model to estimate the stress intensity factor at the point of the
cracks. These results sheets are reproduced on pages A3-A5 in the Appendix.
Once all the cracks had propagated through the specimen it was necessary to measure the
critical crack length of each crack for the analysis phase. As with the CT specimens, this was
done by measuring the distance from the edge of the circular hole to the rough-smooth
transition boundary at three separate points with digital Vernier callipers and averaging those
values, as mentioned in subsection 2.1, when crack growth in PMMA transitions from stable
to unstable it is marked by a clearly defined rough-smooth boundary. Figure 15 below is an
annotated photograph of the crack surface of a two-hole lattice after all cracks have
propagated, it indicates the rough-smooth transition boundary for each crack.
As explained in Section 2, the loads at the point of fracture for the cracks were measured by
observing the load-displacement data from the testing machine. At the point of fracture for a
crack there is a noticeable drop in applied load, the load at the peak before this drop is 𝑃𝑐,1,
the critical load for Crack 1 (see Figure 11), an example of this is shown in Figure 16.
Figure 15: Two-hole lattice after final fracture

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The critical loads and crack lengths for each crack in each individual specimen are tabulated
below in Table 3 and Table 4. Examples of the results sheets and load-displacement curves of
the lattices can be found on page A3-A5 in the Appendix.
Specimen
𝒂 𝒄,𝟏
(mm)
𝑷 𝒄,𝟏
(kN)
𝒂 𝒄,𝟐
(mm)
𝑷 𝒄,𝟐
(kN)
𝒂 𝒄,𝟑
(mm)
𝑷 𝒄,𝟑
(kN)
𝒂 𝒄,𝟒
(mm)
𝑷 𝒄,𝟒
(kN)
1 10.9 6.84 - - 6.7 6.84 4.8 6.84
2 9.4 7.84 - - 7.3 7.84 6.2 7.84
3 9.5 7.19 7.2 7.19 6.8 7.19 5.9 7.19
4 6.8 6.29 5.5 6.29 5.9 6.29 6.0 6.29
5 8.4 6.49 5.4 6.49 6.9 6.49 7.3 6.49
Table 3: Critical crack lengths and loads of two-hole specimens
Specimen 𝒂 𝒄,𝟏 (mm) 𝑷 𝒄,𝟏 (kN) 𝒂 𝒄,𝟐 (mm) 𝑷 𝒄,𝟐 (kN)
1 12.4 8.57 - -
2 - - - -
3 13.3 9.26 - -
4 - - - -
5 7.6 10.99 - -
Table 4: Critical crack lengths and loads of four-hole specimens
This experimental data was used to construct FE models which calculated the stress intensity
factors at the tips of each of the cracks at the critical length and load. The details of this
analytical phase are covered in Section 4.
4 Finite Element Analysis
This section is concerned with the analysis of the experimental results data.
4.1 FE verification
Before the analytical phase of the project could begin, it was first necessary to select an
appropriate FE software package and verify it as accurate in a comparable scenario. The
package selected was Abaqus CAE, as it is a widely used and well-documented program
commonly used to model cracks in materials, and which has the built-in capability to
calculate stress intensity factors at crack tips.
In order to verify Abaqus for use in the analytical phase, a series of FE models of a hole in
the centre of a large square plate with a crack propagating out, under a uniaxial loading of
Figure 16: Experimental load-displacement data of a four-hole specimen

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1MPa were generated and the stress intensity factor at the tip of the crack was calculated for a
small range of different ratios of crack length to hole radius. The plate material was modelled
as a continuous material with a Young’s modulus of 3 GPa and Poisson’s ratio of 0.4, the
approximate properties of PMMA. Cracks were modelled as partitions within the plate, which
were defined as cracks at the interaction stage, the cracks were assumed to propagate purely
in the x-direction when defining the crack growth direction by q-vectors. In order to simplify
the modelling, the plate was modelled as a two-dimensional object and the finite elements
were defined as being under plane strain conditions rather than modelling a thick, three-
dimensional object. The stress intensity factors were calculated using the contour integral
method, whereby ten circular regions were drawn around the crack tip and the motion of this
conceptual block of material was measured and used to estimate the J-integral, which is in
turn used to calculate an approximation of the stress intensity factor (Dassault Systèmes,
2007). J is essentially a generalisation of G which can be applied to materials past the point of
plastic yield (Anderson, 2005), calculated as a path-independent line integral of the strain
energy density field around a crack tip. It is only valid when the strain tensor is effectively
two-dimensional, for instance in plane strain conditions (Rice, 1968).
In each case the mesh was generated using Quad-dominated, structured elements as other
element shapes cannot be used in contour integral evaluations (Dassault Systèmes, 2007). A
structured element is an element in a structured mesh, meaning a mesh based on a regular
shape such as a square is translated to fit over more complex geometry (Dassault Systèmes,
2013). Abaqus calculates the 𝐽-integral through an iterative process. Ten contours were used
in the calculation of the stress intensity factor, meaning that ten circular regions of decreasing
size were used to find 𝐾𝐼𝑐. An example of one of the models is shown in Figure 17.
Due to the symettry of the problem geometry, a half- or quarter-model would have been
possible, although it would have been modelling a symmetricaly cracked hole, such as those
in the two-hole specimens. However, attempting a quarter-model proved difficult to mesh as
the crack would have laid on the boundary of the model, which Abaqus had difficulty
processing, and, since a hole and single crack was the desired scenario, a half-model was not
used.
The stress intensity factors determined by the FE models were then compared to the
analytical solution of stress intensity factors of a hole with a crack propagating out under
Figure 17: Example of an FE model (L) and mesh (R) used in the verification process

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uniaxial loading. The analytical solutions were calculated using equations found in the Stress
Analysis of Cracks Handbook.
The comparison between the analytical and FE solutions are shown in Figure 18, which
shows the analytical and FE results are sufficiently similar that Abaqus is an appropriate
choice for the analysis of the lattice specimens.
4.2 FE analysis of lattice structures
Using the accumulated experimental data from the experimental phase of the project, as
shown in Section 3, stress intensity factors were calculated at the critical crack length of each
crack using FE models generated for each scenario in Abaqus. The FE models were 2D
approximations in plane strain, with the thickness defined in the section assignments.
In each scenario, ten contours were used in the calculation of 𝐾𝐼𝑐 and the specimens were
modelled as two-dimensional objects with the elements defined as being in plane strain
conditions. The specimen material was modelled as continuous with a Young’s modulus of
2.92 GPa, and a Poisson’s ratio of 0.4. The calculated values of 𝐾𝐼𝑐 at each crack tip are
tabulated in subsections 4.3.1 and 4.3.2.
The loading pins were modelled as circular partitions within the specimen, 12.5mm in
diameter with the properties of steel. Opposing concentrated forces were applied at the centre
of each of the pins, which were meshed with Quad-dominated sweep elements, allowing a
small circular shape to be modelled. All FE models used Quad, structured elements with a
characteristic size of 2.5mm
Figure 18: Analytical and FE results for stress intensity factor against a/R for crack
extending from hole in a large plate under uniaxial loading

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4.2.1 Two-hole cracked lattice
An example of an FE model for a two-hole lattice is shown below in Figure 19.
The calculated values of 𝐾𝐼𝑐 for each crack in the lattice structures are tabulated in Table 5.
All models used Quad, structured elements with a characteristic size of 2.5mm. As all cracks
propagated instantaneously at the same moment and the same load, or near enough as to
make no difference, they were all modelled with the same FE model and with the same
applied load.
Specimen
𝑲 𝑰𝒄 (MPa√m)
Crack 1 Crack 2 Crack 3 Crack 4
1 0.66 - 0.74 0.60
2 0.68 - 0.66 0.82
3 0.67 0.57 0.75 0.72
4 0.49 0.45 0.65 0.91
5 0.56 0.47 0.68 0.28
Average 0.61 0.50 0.70 0.67
Table 5: FE model details and calculated values of 𝐾𝐼𝑐 for the two-hole cracked lattice specimens
In each scenario, the critical stress intensity factors at the tip of each crack lie outside the
range of values for plane strain fracture toughness, and the majority lie outside the accepted
range (CES Edupack, 2015). There is an average discontinuity of 46% between the plane
strain fracture toughness and the mean critical stress intensity factors of the cracks. For each
crack, the calculated values of 𝐾𝐼𝑐 all lie within two standard deviations of the mean. This
level of significance suggests that these results are accurate.
4.2.2 Four-hole cracked lattice
An example of an FE model for a four-hole lattice is shown in Figure 20.
Figure 19: FE model of a two-hole lattice
Figure 20: FE model of a four-hole lattice

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The calculated values of 𝐾𝐼𝑐 for each crack are tabulated in Table 6.
Specimen
𝑲 𝑰𝒄 (MPa√m)
Crack 1 Crack 2
1 0.96 -
2 - -
3 1.07 -
4 - -
5 0.92 -
Average 0.98 -
Table 6: FE model details and calculated values of 𝐾𝐼𝑐 for the four-hole cracked lattice specimens
Only one critical stress intensity factor for Crack 1 (see Figure 11) of the four-hole lattice lies
within the experimentally determined range of values for plane strain fracture toughness,
there is a discontinuity of 14% between the mean stress intensity factor and the fracture
toughness, although all lie within the accepted range (Granta Design, 2015). All the values lie
within two standard deviations of the mean, and appear to be statistically significant.
5 Discussion
5.1 Design of lattice specimens
The lattice specimens were designed to include a regular lattice of circular unit cells, into
which cracks of varying lengths could be introduced. The plan was to observe different
cracks becoming unstable as the load increased. However, the final design proved inadequate
in some respects, particularly the four-hole lattice.
The use of clevis grips was chosen from necessity. The MJ 6272 had no pneumatic clamp
grips which could hold a specimen 25mm thick, and the use of loading pins simplified the FE
models. However, the loading holes proved to be more of a weak point than initially
predicted, and the four-hole lattice specimens failed when the loading pins tore out of the
specimen. The applied bearings stresses at the point of failure are shown in Table 7.
Specimen
Applied bearing stresses (MPa)
1 2 3 4 5 Average
Two-hole 37.5 45.5 - - - 41.5
Four-hole 51.9 57.6 - - - 54.8
Four-hole (cracked) 36.2 20.9 46.7 26.9 45.8 35.3
Table 7: Applied bearing stresses on loading holes at failure
Table 7 shows that the applied stresses at failure are less than the accepted range of
compressive strength of PMMA (Granta Design, 2015). The lattice structures evidently
redistributed the stresses to the loading hole to a degree not predicted in the design phase.
Initial estimates of the failure loads of the four-hole lattices predicted that loads on the order
of 7-21kN would be required. Although the failure loads of Crack 1 (see Figure 11) fall
within that range, an FE simulation of a four-hole lattice where Crack 1 has already
propagated estimates that a load of 25-30kN would be required to cause Crack 2 (see Figure
11) to fail. This load is greater than the MJ 6272 could exert.
A more powerful machine would also have had a greater amount of test space. The test
specimens were so large that they were on the threshold of what the MJ 6272 could

1217762 27
accommodate, and the clevis grips used by the Light Structures Laboratory needed to be cut
down in size to allow the specimens to be mounted in the machine. More test space could
have allowed the holes of the four-hole lattice to be set further away from the loading holes,
which might possibly have prevented the pins tearing out.
5.2 Manufacturing with PMMA
The choice to manufacture the test specimens from PMMA was a logical decision based on
what material properties were required. However, this decision was made before a full
appreciation of the difficulty of working PMMA was understood.
The thick sheets of PMMA needed to ensure plane strain conditions could not be machined
on a laser cutter, and the use of saws would have been impractical- a vertical band saw,
which easily cuts through PMMA, could not cut the shapes required, and the thin blades used
by a reciprocating fretsaw would have been too delicate. Introducing sharp-tipped cracks into
the test specimens destroyed five saw blades. Also, as was shown in Figure 15, the blades
bent while cutting the PMMA, resulting in a bowed initial crack front, and furthermore the
crack front at final fracture was also curved. As Abaqus models all crack fronts as perfectly
straight, this was another potential source of inaccuracy in the FE models which could
explain the discrepancy between the fracture toughness of the PMMA and the critical stress
intensity factors of the cracks.
Due to the difficulty of machining the hard and brittle PMMA through other methods, a
waterjet pressure cutter was used. The outer surfaces cut by waterjet pressure cutters are
rough. In general, smooth surfaces are more likely to yield whereas rough surfaces are more
likely to fracture, as the peaks and troughs on the surface act as stress concentrators, this is
likely the cause of the low failure strain and premature yield of the tensile specimens. Due to
technical problems with the cutter, test specimens were completed late in the year, there was
not sufficient time to grind and polish the specimens to the required smoothness. Also,
Abaqus models surfaces as smooth, so the FE models were not totally accurate.
5.3 Fulfilling project aims and objectives
The project objective of completing a comprehensive literature review of the fracture of
lattice structures was successfully completed and makes up Section 1 of this report. Test
specimens were designed but, as explained in subsection 5.1, they were not totally
appropriate to determining the validity of the fracture mechanics approach. The experiments
designed for this project, and the resultant analytical phase, were appropriate to achieving the
aim as they were completed successfully barring unforeseen circumstances that were due to
faults in the specimen design, and may have successfully provided more useful information
with different specimens.
However, the initial project aim was not fully completed. As will be explained in subsection
6.1, based on the results of this project, it seems that the fracture mechanics approach was
sufficient to predict the failure of the lattice structures. However, due to the limited scope and
unforeseen failures of this project, further work must be done.

1217762 28
6 Conclusions and future work
6.1 Conclusions
The fracture mechanics approach was able to predict the propagation direction of the cracks,
all cracks propagated horizontally in the same direction at the initial saw cut. However, it was
not able to predict the failure loads of the cracks in each lattice.
The fracture mechanics approach was somewhat valid in predicting the failure of the four-
hole lattices. In cases where Crack 1 (see Figure 11) propagated, it did so when the stress
intensity factor at the tip of the crack was within the range of accepted values of plane strain
fracture toughness, but not the range determined by experiment. Crack 2 (see Figure 11) did
not propagate in any specimen, and the stress intensity factor was never equal to the fracture
toughness, which corresponds to the fracture mechanics approach.
The initial prediction for two-hole lattices was that Cracks 1 and 2 (see Figure 11) would
propagate first, followed by Cracks 3 and 4 (see Figure 11). However, in the experiments all
the cracks were observed to propagate together. The critical stress intensity factors of each
crack were found to be less than the values for plane strain fracture toughness determined by
experiment, when the fracture mechanics approach predicts that a crack in a thick material
specimen will only propagate when the stress intensity factor of the crack is equal to the
plane strain fracture toughness of the material. This appears to suggest that the fracture
mechanics approach was not totally valid in predicting the failure of the two-hole lattices.
However, the discontinuities between the critical stress intensity factors and the plane strain
fracture toughness of the PMMA could possibly be explained by potential sources of error in
the experiments such as the bowed saw cuts and the limitations of FE analysis. Were a
significantly greater number of specimens produced and tested, and if the crack could be
controlled so that the crack front is perfectly straight through the experiments, the
discontinuities may have averaged to zero.
6.2 Future work
First, studies must be carried out to determine how the presence of large, regular lattice
structures with circular unit cells in specimens loaded with clevis grips alters the stress field
around the loading holes. Furthermore, this study should determine what distance between
the lattice and loading holes is required for the bearing stresses on the loading holes to not be
intensified by the lattice structure.
Second, another study must be done to determine at what thickness a sample of PMMA is
effectively in plane strain and the critical stress intensity of a crack is equal to the plane strain
fracture toughness.
Third, a new series of lattice specimens should be designed using the information provided
by these studies. The new specimens should be 25mm thick at the ends, and be at least the
minimum thickness required for plane strain conditions at the centre. The lattice pattern must
be set at least the minimum distance from the loading holes. Also, new dogbone and CT
specimens should be manufactured from PMMA, and all surfaces ground and polished to
smoothness.
Finally, the experiments and analysis detailed in this report should be repeated on these new
specimens, in a testing machine with a maximum load of at least 40kN.

1217762 29
References
1 American Society for Testing and Materials (2014) ASTM D5405-14, Standard Test
Method for Plane-Strain Fracture Toughness and Strain Energy Release Rate of Plastic
Materials, Pennsylvania: ASTM
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Florida: CRC Press.
3 Ashby, M. F. (2006), “The Properties of Foams and Lattices”, Philosophical Transactions
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4 Berry, J. P. (1961), “Fracture Processes in Polymeric Materials I & II”, Journal of
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International Journal of Solid Structures, 30 (11), 1473-1489

1217762 30
20 International Organization for Standardization (1994) ISO 527, Determination of tensile
properties of plastics Part 4, Switzerland: ISO
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1217762 A1
Appendix
This appendix shows charts and other raw data not included in the body of the report.
Figure 21 shows the dimensions of the test specimens that were measured and used to
construct FE models.
Figure 22 is a scan of the results sheet for the material properties measurements.
Figure 23 is a stress-strain curve from a tensile test.
Figure 24 is a load-displacement curve from a fracture toughness test.
Figure 25 is a scan of the results sheet for a test on a two-hole specimen.
Figure 26 is the load-displacement curve of a test on a two-hole specimen.
Figure 27 is a scan of the results sheet for a successful test on a four-hole specimen.
Figure 28 is the load-displacement curve of a successful test on a four-hole specimen.
Figure 29 is a scan of the results sheet for a failed test on a four-hole specimen.
Figure 30 is the load-displacement curve of a failed test on a four-hole specimen.
Figure 21: Significant lattice dimensions

1217762 A2
Figure 23: Tensile test stress-strain curve Figure 24: Fracture toughness test force-displacement curve
Figure 22: Material properties results sheet

1217762 A3
Figure 25: Two-hole lattice results sheet
Figure 26: Two-hole lattice force-displacement curve

1217762 A4
Figure 27: Four-hole lattice results sheet (1)
Figure 28: Four-hole lattice force-displacement curve (1)

1217762 A5
Figure 29: Four-hole lattice results sheet (2)
Figure 30: Four-hole lattice force-displacement curve (2)

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