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On dynamic crack growth in
discontinuous materials
Johan Persson
Supervisors:
Per Gradin
Per Isaksson
Sverker Edvardsson
Faculty of science, technology and media
Mid Sweden University Doctoral Thesis No. 223
Sundsvall, Sweden
2015
ISBN 978-91-88025-26-5
ISSN 1652-893X
Mittuniversitetet
Naturvetenskap, teknik och media
SE-851 70 Sundsvall
SWEDEN
Akademisk avhandling som med tillstånd av Mittuniversitetet framlägges till o�entlig
granskning för avlläggande av teknologie doktorsexamen tisdag den 16 Juni 2015 i
L111, Mittuniversitetet, Holmgatan 10, Sundsvall.
©Johan Persson, Juni 2015
Tryck: Tryckeriet Mittuniversitetet
Abstract
In this thesis work numerical procedures are developed for modeling dynamic fracture of discontinuous materials, primarily materials composed of a load-bearing network. The models are based on the Newtonian equations of motion, and does not
require neither sti�ness matrices nor remeshing as cracks form and grow. They are
applied to a variety of cases and some general conclusions are drawn. The work also
includes an experimental study of dynamic crack growth in solid foam. The aims are
to deepen the understanding of dynamic fracture by answering some relevant questions, e.g. What are the major sources of dissipation of potential energy in dynamic
fracture? What are the major di�erences between the dynamic fracture in discontinuous network materials as compared to continuous materials? Is there any situation
when it would be possible to utilize a homogenization scheme to model network
materials as continuous? The numerical models are compared with experimental
results to validate their ability to capture the relevant behavior, with good results.
The only two plausible dissipation mechanisms are energy spent creating new surfaces, and stress waves, where the �rst dominates the behavior of slow cracks and
the later dominates fast cracks. In the numerical experiments highly connected random �ber networks, i.e. structures with short distance between connections, behaves
phenomenologically like a continuous material whilst with fewer connections the behavior deviates from it. This leads to the conclusion that random �ber networks with
a high connectivity may be treated as a continuum, with appropriately scaled material parameters. Another type of network structures is the ordered networks, such as
honeycombs and semi-ordered such as foams which can be viewed as e.g. perturbed
honeycomb grids. The numerical results indicate that the fracture behavior is di�erent for regular honeycombs versus perturbed honeycombs, and the behavior of the
perturbed honeycomb corresponds well with experimental results of PVC foam.
iii
Sammanfattning
I detta avhandlingsarbete utvecklas två numeriska modeller för att modellera dynamiska brott i diskontinuerliga material, i första hand material med ett lastbärande
nätverk. Modellerna baseras på Newtons rörelseekvationer och kräver varken styvhetsmatris eller ny diskretisering då sprickor uppstår och växer. De appliceras på några
olika brottförlopp och ur det dras några allmänna slutsatser. Arbetet innefattar även
en experimentell studie rörande dynamisk spricktillväxt i skummad cellplast. Målet
med avhandlingen är att fördjupa kunskapen kring dynamiska sprickor genom att
besvara några relevanta frågor. Vilka processer ger betydande bidrag till dissipation
av potentiell energi vid dynamisk spricktillväxt? Vad är de stora skillnaderna mellan
dynamisk spricktillväxt i diskontinuerliga material och kontinuerliga material? Finns
det någon situation då det är möjligt att se nätverksmaterialet som kontinuerligt? De
båda numeriska modellerna jämförs med experimentella resultat för att validera att
de fångar relevanta beteenden, med goda resultat. Det �nns bara två troliga processer för energidissipation i detta fall, de är energi som används för att skapa nya
ytor och energi som omvandlats till stressvågor. Den första källan dominerar beteendet vid långsamma sprickor och de andra vid snabbare sprickor. I de numeriska experimenten visar det sig att oordnade �bernätverk där var �ber binder till många
andra �brer, eller med andra ord där bindningar ligger tätt på �brerna, beter sig
fenomenologiskt som kontinuerliga material medan nätverk med få bindningar per
�ber beter sig annorlunda. Detta leder till slutsatsen att ett oordnade �bernätverk
med många bindningar per �ber kan behandlas som ett kontinuerligt material, med
anpassade materialparametrar. Andra nätverksstrukturer så som honungskakor kan
ses som ordnade och skum-material kan ses som semi-ordnade då de kan beskrivas
väl i grunden som en honungskaka men med en störning i var väggar i strukturen
möts. De numeriska resultaten visar att de ordnade och semi-ordnade strukturerna
beter sig annorlunda och beteendet hos de semi-ordnade nätverken stämmer väl överens med experimentella resultaten från skum-materialet.
iv
Acknowledgements
First I would like to express my gratitude the taxpayers through the Swedish Research Council and the KK-foundation for funding this work. I am grateful for the
�nancial support from SCA and the Bo Rydin foundation. Equally important for the
completion of this work has been the help and support of my supervisors, Per Isaksson, Per Gradin and Sverker Edvardsson, thank you. The experimental part of this
thesis would not have been possible without the support of Sta�an Nyström and Max
Lundström at IMT. Finely I like to extend my deepest and most heartily than you to
my colleagues with a special recognition to Madde, Hanna, Pär, Sara and Amanda,
who make the weeks �y by.
v
Contents
List of Papers
vii
1
General remarks
1
2
Background
3
3
Aims and limitations
4
4
Summary of content and major conclusions
5
5
Concluding remarks
16
Bibliography
16
vi
List of Papers
This thesis is mainly based on the following papers, herein referred by their Roman
numerals:
I Johan Persson, Per Isaksson. A particle–based method for mechanical analyses
of planar �ber–based materials. International Journal for Numerical Methods in
Engineering 93.11 (2013): 1216–1234.
II Johan Persson, Per Isaksson. A mechanical particle model for analyzing rapid
deformations and fracture in 3D �ber materials with ability to handle length effects. International Journal of Solids and Structures 51.11 (2014): 2244–2251.
III Johan Persson, Per Isaksson. Modeling rapidly growing cracks in planar materials with a view to micro structural e�ects. International Journal of Fracture 2015
In press.
IV Johan Persson, Per Isaksson, Per Gradin. Dynamic mode I crack growth in a
notched foam specimen under quasi static loading. Submitted.
vii
Contributions
The work presented in this thesis and the appended papers are results of collaboration. However in all publications the author has done major parts of the programming, experiments, computing analyzing and writing.
viii
Chapter 1
General remarks
Figure 1.1: Some di�erent network materials. Top left: Tissue paper. Top right: Honeycomb composite with paper as both sheet and core, i.e. a multi-scale network material. Lower left: PVC foam.
This thesis concerns a class of materials, including both natural and man- made materials, where the primary load bearing structure is a network, and the deformation
and fracture of such materials. Some examples of materials are shown in Fig. 1.1. In
some cases the load bearing structure are �bers, e.g. paper and �ber glass composites and in others, the structure are walls, e.g. sheet metal in a honeycomb airplane
1
2
General remarks
fuselage or cell walls in a closed cell solid foam. Deformation and fracture in network
materials are very complex processes that depend strongly on the geometric con�guration, the mechanical properties of the material constituents and the rate of applied
load.
The �eld of physics that deals with fracture is known as fracture mechanics, and
is a relatively new science. Although the �rst known systematic experiments were
conducted c. 1500 by Da Vinci, the theoretic work is often considered to begin in
the 1900s with the groundbreaking work by Gri�th and the realization that fracture
is driven by energy and not stress alone [1]. Irwin made signi�cant improvements
to Gri�th’s theory in the 1950s when he introduced the plastic zone, clarifying that
ordinary materials cannot carry an in�nite stress but rather have a maximum stress
known as the yield stress which, for some materials, is constant for a range of strain
values [2]. Some advances towards understanding dynamic fracture consisting of setting up and solving di�erential equations for steady state moving crack tips [3, 4, 5, 6],
these models normally excludes all possible energy dissipations except new surface
creation. In the case of dynamic fracture, some energy is dissipated in the form of
plastic deformation around the crack tip, some energy is consumed in creating new
surfaces, some energy might create new microcracks which does not propagate, and
some energy will cause stress waves [7]. The energy associated with stress waves
can be regarded as e.g. sound and heat, but this distinction is not relevant for our
purposes and all material vibrations are bundled within the concept of stress waves.
Networks may be of many types such as collagen �bers, common paper or nano�bril cellulose paper [8, 9, 10], and one group of special interest is networks formed
by foaming, especially PVC foam. Solid PVC foam has a lower degree of variation
throughout the body than for example paper due to both the more homogeneous
material and the forming process. Therefore, solid foams are suited as model materials when investigating network materials. Foams are sometimes modeled as made
up of regular honeycombs, sometimes as perturbed honeycombs and sometimes with
more advanced unit cells [11, 12, 13, 14]. This is applicable both to open- and closedcell foams, with the di�erence that while a closed cell structure means that all faces
of a cell are solid whilst only the edges of the faces of an open cell foam are solid. The
di�erent unit-cells have some consequences for the mechanical behavior of the foam
which has been studied for static cases, cf. [15, 16, 17, 18].
Network materials are di�erent when viewed at di�erent scales. This is in some
sense true for all materials, but especially so for network materials, since they are
discontinuous at the intermediate scale. To further complicate matters, they are often
used both as skin- and core-material in composites, as the honeycomb panel in Fig.
1.1. Structures built of such composites are strong and sti� in relation to their density,
in tension, bending and compression. When they eventually fail, the failure is most
often caused by local tension, so failure due to tension is of special interest for network
materials.
Chapter 2
Background
Theoretical models of fracture are essential for understanding what causes fracture
and ultimately failure in structures, and understanding is the �rst step towards preventing failure. The theoretical models are of two types, one explaining the micromechanics at and around the crack tip, where Gri�th and Irwin laid the foundation,
and larger scale models concerned with the macro-mechanics and how energy is
transferred to the crack tip area. Macroscopic models can be utilized either to validate/disprove theoretical assumptions, for example to see if the energy spent in plastic deformation is relatively large or small for a certain crack, or they can be used to
make predictions regarding the life of a structure.
In the dawn of fracture mechanics, researchers were limited to linear elastic fracture mechanics and only problems with an analytic solution. Even though some advanced problems might be solved in such ways [19, 20], many interesting problems
lack an analytical solution. There are two possible solutions strategies when the analytic solution is missing; solving the analytic problem numerically which involves a
discrete solution space, or discretizing the problem and solving it numerically. There
are merits to both approaches and they are suited for di�erent problems.
Finite element models (FEM) have been used extensively to model fracture [21, 22,
23], and there are several extensions and specialized elements to cover a wide range of
applications, among the most prominent are the extended FEM (XFEM) [24] which allows a crack to grow through an element, and the cohesive zone model (CZM) [25, 26],
which regards the crack tip not as a point but as a zone with varying material characteristics. All three models are based on a homogenized continuum representation
of the material. The resultant forces are computed based on the weak formulation
of the di�erential equations describing the elements. Another approach is based on
the viewpoint that all matter is composed of particles with attractive and repulsive
interactions. This class of models is known as lattice models (LM) or particle models
(PM), pioneered by Hrenniko� [27]. Whilst FEM/XFEM/CZM are well suited for
static, quasistatic and steady state dynamic cases, LM/PM are better suited for full
dynamic cases. This has prompted the use of hybrid models [28, 29, 30, 31], which utilize the best of the two worlds; the complex elements used in FEM/XFEM/CZM and
the dynamics, inertia, and ease of computation from LP/PM. The model utilized in
this thesis is an example of the latter. FEM has dominated the area of network material modeling, both with the approximation of a homogenized continuum [32, 33, 34]
and by modeling individual structural elements [35, 36, 37, 38, 39]. By modeling
individual structures, accurate models for fracture investigation can be developed
[40, 41, 21, 22, 23]. Recently, particle modeling has reached the �eld of network mechanics, but limited to homogenized continuum and without fracture [42].
3
Chapter 3
Aims and limitations
The overall aim with this thesis is to derive numerical models to bring insight into
the complex phenomenon of dynamic deformations and fractures in network materials. To judge the models several mechanical problems picked from the literature,
together with published and new experimental data, are analyzed and compared
with other published solutions. The in�uence of plasticity or other types of material
non-linearity will not be included in the models.
4
Chapter 4
Summary of content and major
conclusions
8
7
6
H35
H60
H80
H200
γ f5
γ0
4
3
2
1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
v/c 2
Figure 4.1: Left: The test specimen clamped in the testing machine. The crack will
grow approximately along the dark line, breaking the electrical circuits. Right: The
normalized crack growth energy versus normalized crack growth speed for specimens with di�erent densities and initial notch lengths varied in the initial w/20 
a0  2w/5 .
Table 4.1: Material parameters for four di�erent Divinycell-H open cell foams. The
data is supplied by DIAB [43].
Density ⇢
Tensile Modulus E
Shear Modulus G
H35
H60
H80
H200
38 k g/m 3
49 MPa
12 MPa
60 k g/m 3
75 MPa
20 MPa
80 k g/m 3
95 MPa
27 MPa
200 k g/m 3
250 MPa
73 MPa
As mentioned in the background, a suitable material for investigations in to dynamic fracture properties of network materials is PVC foam. In the experimental
Paper IV the test specimen is a square with an edge notch, shown in Fig. 4.1. The
specimens are foam panels of Divinycell; H35, H60, H80 and H200, where the di�erence is their e�ective density which varies in the interval 38 to 200 kg/m3 , presented
in Table 4.1. Notches are cut in the specimens with a length in the interval 5 to 40
5
6
Summary of content and major conclusions
mm and the test series consist of separating the clamps with a rate of 1 mm/s while
monitoring the crack growth. When the crack grows, it breaks electrical conductors
painted on the specimen, and the time between consecutive breaks is recorded. From
this time the average crack speed v can be computed. The crack growth speed is several orders of magnitude larger than the end displacement rate, so for all practical
purposes the crack grows in a statically loaded structure. The normalized potential
energy available for surface creation / f versus the resulting crack growth speed is
shown in Fig. 4.1, in this graph a high velocity corresponds to a short initial crack. It
is known from both analytic fracture mechanics and from experiments that a crack
initiated by a smaller notch will grow with a higher speed than cracks from a larger
notch, likely due to the fact that the specimen with the smaller crack can store more
potential energy before crack growth. As can be seen, the general behavior of the
foam is similar to the well-known behavior of continuous materials such as steel (cf.)
Nilsson [44].
To model the dynamic deformation and fracture of network materials, a model
is developed for two- and three-dimensional cases based on geometrically nonlinear engineering beam theory. The formulations are described in Papers I and II and
utilized in Papers I, II and III. The highlights of the model are; the network structure is modeled as mentioned above by geometrically nonlinear engineering beams,
with lumped masses at the ends making up the particle representation. In the twodimensional case, each particle has two translational and one rotational degrees of
freedom; in the three-dimensional case each particle has three translational and three
rotational degrees of freedom. The particles of one structural element may be connected to other particles by linear elastic springs and torsion springs to bind structures together. Both types of interactions may be canceled if an energy criterion is met,
and thus the network may fracture. A semi-implicit Euler method (Abbys method)
is used for integration in time, due to it’s robustness and extreme simplicity [45, 46].
The model could easily be expanded to include other interactions, for example plastic
deformations or stick-slip events. There are four sources of energy dissipation in the
model; through boundaries, breaking of structures, breaking of bonds and through
an arti�cial friction that dissipates energy by scaling velocities in each time-step. The
arti�cial friction is small enough to have a negligible e�ect on the dynamic evolution,
but it acts to increase numerical stability and is essential to dissipate energy prior to
fracture.
Summary of content and major conclusions
10
8
7
10
Analytic solution (Nilsson 2001)
Present study
Nilsson (1974)
8
γf 6
γ0
γf 6
γ0
4
4
2
2
0
0
0.2
0.4
0.6
0.8
Analytic solution (Nilsson 2001)
Present study
1
0
0
0.2
0.4
v/c 2
0.6
0.8
1
v/c 2
Figure 4.2: Both graphs show the normalized crack energy versus normalized crack
speed for a growing crack in notched specimens with notch lengths varied in the
initial w/199  a 0  10w/199 (regular grid) and w/46  a0  30w/46 (honeycomb
grid), where w is the width of the strip. The dashed line is the analytical crack growth
expression from Nilsson (2001). Left: The stars are experimental results from Nilsson
(1974) and the diamonds are the numerical result of computations with a regular
square grid. Right: Diamonds are the numerical result of a regular honeycomb grid.
10
8
12
Nilsson (1974)
r̃ ∆ = 0.16l 0
10
Analytic solution (Nilsson 2001)
Present study
8
γf 6
γ0
γf
γ0 6
4
4
2
0
0
2
0.2
0.4
0.6
v/c 2
0.8
1
0
0
0.2
0.4
0.6
0.8
1
v/c 2
Figure 4.3: Both graphs show the normalized crack energy versus normalized crack
speed for a growing crack in notched specimens with notch lengths varied in the
initial w/199  a 0  10w/199 (regular grid) and w/46  a0  30w/46 (honeycomb
grid), where w is the width of the strip. The dashed line is the analytical crack growth
expression from Nilsson (2001). Left: Stars are experimental results from Nilsson
(1974) and the horizontal bars represent plus/minus one standard deviation to the
mean value of the numerical result of computations with a perturbed square grid.
Right: the horizontal bars represent plus/minus one standard deviation to the mean
value of the numerical result of computations with a perturbed honeycomb grid.
The experiment of Nilsson [44] is repeated as a computational experiment in Paper III both for a regular square grid which approximates a perfect continuous ma-
8
Summary of content and major conclusions
terial, a perturbed square grid, a regular honeycomb and a perturbed honeycomb.
The test specimen is a strip with one side notched, as in Paper IV. The edges are
slowly displaced until a crack grows. For a large enough notch, the crack will come
to rest before the ultimate failure, and for any notch smaller than this the crack will
propagate dynamic until reaching the end of the strip. The potential energy available for surface creation at this pivot point f is used to normalize the results. Fig.
4.2 shows the normalized potential energy available for surface creation / f versus
the resulting crack growth speed for a regular square grid, and a regular honeycomb
grid. There is a noticeable di�erence in the shape of the graphs in that the honeycomb grid has a steeper slope and there is almost a discrete step from stable static
crack growth to dynamic crack growth with a speed above 0.2c 2 . Note that c2 is the
shear wave speed of the material, not of the network for the honeycomb. It is also
noteworthy that although the model does not include any plastic deformation, it still
captures the behavior of steel well.
Honeycomb grids respond di�erently to perturbations of node locations than regular grids do, as is evident in Fig. 4.3, whilst perturbations in a regular grid cause
only a slight perturbation in the fracture response. With a slight perturbation, the
model captures the spread of the experimental data of Nilsson [44]. Similar perturbations in a honeycomb grid cause not only perturbations to the response but also
a signi�cant shift towards higher crack speeds, in some cases up to three times that
of the regular honeycomb. The / f versus v/c2 graph is similar for the perturbed
honeycomb grid and the experimental data in Fig. 4.1, con�rming that it is adequate
to model a foam material as a perturbed honeycomb, and illustrating that although it
is adequate to use a regular honeycomb model to model static conditions for foams,
it is inadequate when dealing with dynamics.
2
L cr ack/(B − a 0 )
L cr ack/(B − a 0 )
2
1.5
1
0.5
0
0
10
20
a o [mm]
30
40
1.5
1
0.5
0
0
50
100
150
200
250
Density[kg/m 3 ]
Figure 4.4: Normalized crack length, L crack is the length of the crack whilst B a0 is
the shortest possible crack length, i.e. the distance from the notch tip to the end of the
specimen. Left: Crack length versus notch length, the line shows the mean value for
all specimens with a certain notch length (and di�erent densities) and the vertical bar
shows the standard deviation. Right: Crack length versus e�ective density, the line
shows the mean value for all specimens with a certain density (and di�erent notch
lengths) and the vertical bar shows the standard deviation.
Summary of content and major conclusions
9
L cr ack/(B − a 0 )
2
1.5
1
H35
H60
H80
H200
0.5
0
0
0.1
0.2
0.3
0.4
0.5
0.6
v/c 2
Figure 4.5: Normalized crack length, L crack is the length of the crack whilst B a0 is
the shortest possible crack length, i.e. the distance from the notch tip to the end of the
specimen, versus crack speed. Note that there is no clear relation between the two.
In Fig. 4.2, 4.3 and 4.1 a high energy and speed corresponds to a short initial notch,
and vice versa. Fig. 4.4 shows the length of the running cracks after complete failure,
plotted versus the initial notch length and the e�ective density of the material. With
reference to the fact that a small notch results in a fast running crack, and a large
notch results in a slow running crack, it is possible to draw the conclusion that the
crack speed is not related to the crack length, also illustrated in the slightly chaotic
Fig. 4.5. With reference to the fact that the material density does not signi�cantly affect the crack speed, it is noteworthy that the density a�ects the crack length. When
the potential energy is not spent making new surfaces (a longer crack) and not to plastic deformation which does not exist in the model, and not to micro-cracks around
the major crack which does not exist in the model, there are only a few options left.
The duration of the crack growth is too short for the arti�cial friction to make any
signi�cant contribution. For the �rst half of the growth, no energy has had time to
travel from the crack tip to the boundaries. The only plausible explanation is that the
potential energy released has gone to stress waves and vibrations.
Figure 4.6: Illustration of a square specimen with a central mode I notch. The zoom
bubble illustrates the �ber network microstructure on a smaller length scale.
10
Summary of content and major conclusions
1
0.8
0.6
∆x = 0.45
0.4
0.2
0
−4
−2
0
2
4
normalized potential energy
normalized potential energy
In Paper I, the focus is on wave propagation in random �ber networks. A random
�ber network is the structure formed by a number of randomly- placed and oriented
identical �bers. Some experiments are carried out with a square specimen, with a
small initial central mode I notch, which is loaded axially in tension as illustrated
in Fig. 4.6. The boundaries are clamped and the specimen is stretched with a constant rate and then brought back to their original positions at the same rate, so as to
load the specimen in an impulse-like manner. Tension waves form at the upper and
lower boundaries and travel through the network and when they meet in the center
at the notch, the crack grows. Later, when the following compressive wave reaches
the center, the crack stops.
1
0.8
0.6
∆x = 0.85
0.4
0.2
0
−4
−2
0
y/a 0
2
4
y/a 0
Figure 4.7: Graphs shows the normalized potential energy, summed for all x-values
R
W/2
and normalized to a maximum of unity i.e. U ( y, t ) ⇤
u ( x, y, t ) dx/max (U )
W/2
with the potential energy u ( x, y, t ) at time t ⇤ 3 µs (left) and t ⇤ 4 µs (right). They
illustrate how the initial Heaviside step function tension wave has changed character
while traveling through the specimens. The left graph represents a square grid. The
right graph represents a random �ber network with a lenght ratio L/l s ⇤ 14, where L
is the mean �ber length and l s is the mean distance between bonds.
normalized derivative
∆U
∆y
Summary of content and major conclusions
1
l s /L
l s /L
l s /L
l s /L
0.9
0.8
0.7
=
=
=
=
11
1/12
1/17
1/25
1/62
0.6
0.5
0.4
0.3
0.2
0.5
1
1.5
2
2.5
3
3.5
4
time µs
Figure 4.8: The left graph shows the maximum slope of the normalized potential
energy graph, for example 4.7, the normalization is such that the largest value is unity.
The fraction l a /a0 is the normalized mean distance between �ber bonds.
A �rst observation is that the dispersive properties are signi�cantly more pronounced for a random �ber network than for a regular grid, illustrated in Fig. 4.7
and the dispersive properties is greater for a less dense network, i.e. a network with
a larger average distance between two �ber-�ber bonds l s , illustrated by the slope of
the energy graph shown in Fig. 4.8. It is plausible to explain this by the many ways
to walk a network. The fastest way is through stretching the �bers oriented parallel
to the path, and the considerable slower way through bending �bers oriented more
orthogonal to the path and anything in between. From a structure integrity point of
view, this is a good thing, since it implies that stress waves traveling in sparse networks will blur, causing the peek amplitude to be lower, and the material thus gains
a small protection against stress waves.
12
Summary of content and major conclusions
3
1/12
1/17
1/25
1/62
4
x−a 0
a0
x−a 0
a0
fracture events
3.5
=
=
=
=
fracture events
l s /L
l s /L
l s /L
l s /L
4
2.5
2
1.5
1
0.5
0
4
3.5
3
2.5
2
1.5
1
0.5
0
5
6
7
time µs
8
9
−0.5
4
5
6
7
8
9
10
time µs
Figure 4.9: The two graphs show how di�erent cracks have grown from the initial
notch. The factor x a0a0 is a unitless measure of the distance from the notch tip to a
fractured element, measured along the notch line. In the left graph the crack has
grown by fracturing �bers, in the right graph the crack has grown by breaking �ber�ber bonds (connections). Note that there is a signi�cant di�erence between the two
graphs in the crack propagation speed (the slope of the graphs) but only a slight
variation within the graphs.
Paper I also aims to illustrate the di�erence when a network fails due to walls
fracturing or walls loosing connections to other walls. This is illustrated for four different networks in Fig. 4.9. At the time of publication the most likely explanation
to the considerable di�erence in crack growth speed was considered to be the di�erence in energy consumption to drive the crack, i.e. the energy to create new surface
(break bonds). However, after closer examination, this explanation does not hold up.
The assumption was that in the most energy consuming case, three time as much energy would be spent to advance the crack past a �ber, but a more realistic assumption
would be on average half that, since a randomly placed �ber on average would have
a quarter of its length on one size of a idealized crack line, and three quarters on the
other side. This small energy di�erence is not likely to account for the di�erence in
speed. However, it is possible that the spreading of the e�ective crack tip might be
responsible.
Micro-fracture localization ratio
Summary of content and major conclusions
13
1
0.8
0.6
0.4
0.2
0
0
0.05
0.1
0.15
0.2
0.25
0.3
l s /L
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.05
0.1
0.15
l s /L
0.2
0.25
Macroscopic crack width [2∆y/L]
Microscopic crack width [2∆w/L]
Figure 4.10: Left: Fraction of network where new crack growth is located near the
existing notch tip for networks normalized mean free length l s /L where L is the average length of a �ber and l s is the average distance between �ber–�ber bonds. Right:
Illustration of the crack width, where y is the orthogonal distance from the original
crack plane to a fractured segment, used co compute the macroscopic crack width
and w is the orthogonal distance from the �tted line to a fractured segment utilized
to compute the microscopic crack width.
3
2.5
2
1.5
1
0.5
0
0
0.05
0.1
0.15
0.2
0.25
l s /L
Figure 4.11: Left: Microscopic crack width. Right: Macroscopic crack width. The
illustrative dashed lines are least-square �tted lines, the square markers are the estimated average values while the vertical bars are the standard deviations.
Summary of content and major conclusions
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0
0
Macroscopic crack growth [v c /c]
Microscopic crack growth [v c /c]
14
0.1
0.05
0.1
0.15
l s /L
0.2
0.25
0
0
0.05
0.1
0.15
0.2
0.25
l s /L
Figure 4.12: Left: Microscopic crack growth velocity. Right: Macroscopic crack
growth velocity. The horizontal dashed line is a �t for the whole dataset in each
�gure and the lines connecting the bars running through squares shows the mean
values of each subset. The vertical bars show the standard deviation.
Paper II is concerned with the question of for what cases it is necessary to treat
a structure like a �ber network and when it would be possible to treat the problem
as a homogenized continuum. The test specimen is once again square with a central
mode I notch. The specimen is rapidly stretched, until a complete failure occurs. The
rapid stretch is an order of magnitude lower than the crack growth speed; it is thus
still a case of dynamic growth with quasi-static boundaries, even though the events
leading to the crack growth are not quasi-static.
One aspect that is of major importance is whether or not new cracks form as extensions of the existing notch, or if they form at arbitrary points in the bulk material.
In continuous materials strain localization will always cause the notch to grow rather
than new cracks to form, and this is also the case for highly connected �ber networks,
as is evident in Fig. 4.10. Sparse �ber networks do not recognize the existence of a
notch, and new cracks are likely to form at arbitrary points in the material, and at
boundaries. The same relation is present in the resultant crack width, whether it is
measured from an expected crack plane or the actual crack plane, illustrated in Fig.
4.10 and 4.11. In all tree cases the shift towards continuum behavior occurs around
a normalized mean free length l s /L ⇡ 0.1 and is complete at l s /L ⇤ 0.05. Finally, the
speed of crack propagation is presented in Fig 4.12, with the result that the speed is
signi�cantly less than for a continuum material with the same base material, and it
is insensitive to the mean free length of the network. Put together, these three results
indicate that highly connected network materials might be described as a continuum,
but with signi�cantly lower speed of crack propagation than might be expected.
Summary of content and major conclusions
15
Figure 4.13: A growing crack in a random �ber network with time progressing from
left to right. Left: The existing central notch has opened. Center: Cracks have grown
both left and right from the central notch. Right: Complete failure throughout the
specimen.
A random �ber network under tension buckles out of plane in what is called
fringes, especially visible around the major pores. One example is shown in Fig. 4.13
for a network with growing cracks taken from paper II. In some cases, this fringing
may be helpful in �nding pores in the material, and thus potential weak spots. However, as is evident in Fig. 4.13 a crack does not necessarily follow the path with the
largest pores. In this case, the left crack splits where one path follows a number of
smaller pores slightly below the notch, and the second crack grows from the notch to
the largest pore. Even though both fringes and pores in some cases may be indicators
of weak spots, neither of them are reliable. This behavior is at least to some extent
speci�c for random �ber networks and is not to be seen as a general network material
characteristic, since random �ber networks have stronger than average regions which
are not visible in this test but may play an important role.
Chapter 5
Concluding remarks
In this thesis work, two numerical procedures are developed for modeling fracture
of network materials. They are applied to a variety of cases and some general conclusions are drawn. The work also includes an experimental study of dynamic crack
growth in solid open cell foams. A �rst observation which is of importance to fracture
is the considerable inherent dispersive properties for random �ber networks. Stress
waves traveling in a random �ber network are more e�ciently dispersed than stress
waves traveling in a continuous material, which in the case of modeling is represented
by a regular grid. Fiber networks that fracture by breaking �ber-�ber bonds have an
ability to blunt at the crack tip and divide stress waves generated by crack growth,
since several smaller events are associated with the crack growing past a �ber.
Random �ber networks where each �ber is connected to many �bers (a low mean
free length l s ) have more predictive characteristics than networks where �bers have
few connections. For networks with normalized mean free length l s /L  0.05 the
fracture behavior resembles that of a continuous material, but with a slightly wider
crack path, and slightly lower speed of crack growth. It is therefore possible to regard
highly connected networks as a continuum but with e�ective parameters. This result
agrees well with the static deformation study by Åslund and Isaksson [39].
In the models mentioned above, there are no plastic deformation, but still they
capture the dynamic fracture behavior of steel, both in static and dynamic loading
conditions. Whilst this is not an actual proof that energy dissipation due to plastic
deformation around the crack tip is a small contributor, it is certainly an indication.
It is more likely that the increased energy dissipation of fast moving cracks is due to
stress waves, possible originating from the speed variation of the crack tip.
In experiments with open cell foam, it is evident that foam has the same phenomenological behavior as steel in the relation crack speed versus potential energy.
It is therefore interesting to note that while the crack speed is dependent on the notch
length, the actual crack length is not dependent on the notch length. While the crack
speed is not dependent on the e�ective density of the foam, the actual crack length
is. With this in mind, and with reference to Fig. 4.5 it seems that the crack speed,
measured as the average speed over a short distance of the mean crack path, i.e. not
along the actual crack path, is not dependent on the actual crack length. If the fraction of energy dissipated by formation of new surfaces is large, it is expected that the
crack length will a�ect the e�ective crack speed. If, on the other hand, the energy
dissipated by new surfaces is small, this e�ect should not be visible. The result in
Fig. 4.5 is an indicator that the actual energy dissipated to form new surfaces is small
in comparison to the total energy dissipation. The best candidate to explain what
happens to the energy is stress waves.
16
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