on dynamic crack growth in discontinuous...

On dynamic crack growth indiscontinuous 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

MittuniversitetetNaturvetenskap, teknik och media

ISBN 978-91-88025-26-5 SE-851 70 SundsvallISSN 1652-893X SWEDENAkademisk avhandling som med tillstånd av Mittuniversitetet framlägges till o�entliggranskning för avlläggande av teknologie doktorsexamen tisdag den 16 Juni 2015 iL111, Mittuniversitetet, Holmgatan 10, Sundsvall.©Johan Persson, Juni 2015Tryck: Tryckeriet Mittuniversitetet

AbstractIn this thesis work numerical procedures are developed for modeling dynamic frac-ture of discontinuous materials, primarily materials composed of a load-bearing net-work. The models are based on the Newtonian equations of motion, and does notrequire neither sti�ness matrices nor remeshing as cracks form and grow. They areapplied to a variety of cases and some general conclusions are drawn. The work alsoincludes an experimental study of dynamic crack growth in solid foam. The aims areto deepen the understanding of dynamic fracture by answering some relevant ques-tions, e.g. What are the major sources of dissipation of potential energy in dynamicfracture? What are the major di�erences between the dynamic fracture in discontin-uous network materials as compared to continuous materials? Is there any situationwhen it would be possible to utilize a homogenization scheme to model networkmaterials as continuous? The numerical models are compared with experimentalresults to validate their ability to capture the relevant behavior, with good results.The only two plausible dissipation mechanisms are energy spent creating new sur-faces, and stress waves, where the �rst dominates the behavior of slow cracks andthe later dominates fast cracks. In the numerical experiments highly connected ran-dom �ber networks, i.e. structures with short distance between connections, behavesphenomenologically like a continuous material whilst with fewer connections the be-havior deviates from it. This leads to the conclusion that random �ber networks witha high connectivity may be treated as a continuum, with appropriately scaled mate-rial parameters. Another type of network structures is the ordered networks, such ashoneycombs and semi-ordered such as foams which can be viewed as e.g. perturbedhoneycomb grids. The numerical results indicate that the fracture behavior is di�er-ent for regular honeycombs versus perturbed honeycombs, and the behavior of theperturbed honeycomb corresponds well with experimental results of PVC foam.

iii

SammanfattningI detta avhandlingsarbete utvecklas två numeriska modeller för att modellera dy-namiska brott i diskontinuerliga material, i första hand material med ett lastbärandenätverk. Modellerna baseras på Newtons rörelseekvationer och kräver varken styvhets-matris eller ny diskretisering då sprickor uppstår och växer. De appliceras på någraolika brottförlopp och ur det dras några allmänna slutsatser. Arbetet innefattar ävenen experimentell studie rörande dynamisk spricktillväxt i skummad cellplast. Måletmed avhandlingen är att fördjupa kunskapen kring dynamiska sprickor genom attbesvara några relevanta frågor. Vilka processer ger betydande bidrag till dissipationav potentiell energi vid dynamisk spricktillväxt? Vad är de stora skillnaderna mellandynamisk spricktillväxt i diskontinuerliga material och kontinuerliga material? Finnsdet någon situation då det är möjligt att se nätverksmaterialet som kontinuerligt? Debåda numeriska modellerna jämförs med experimentella resultat för att validera attde fångar relevanta beteenden, med goda resultat. Det �nns bara två troliga pro-cesser för energidissipation i detta fall, de är energi som används för att skapa nyaytor och energi som omvandlats till stressvågor. Den första källan dominerar beteen-det vid långsamma sprickor och de andra vid snabbare sprickor. I de numeriska ex-perimenten visar det sig att oordnade �bernätverk där var �ber binder till mångaandra �brer, eller med andra ord där bindningar ligger tätt på �brerna, beter sigfenomenologiskt som kontinuerliga material medan nätverk med få bindningar per�ber beter sig annorlunda. Detta leder till slutsatsen att ett oordnade �bernätverkmed många bindningar per �ber kan behandlas som ett kontinuerligt material, medanpassade materialparametrar. Andra nätverksstrukturer så som honungskakor kanses som ordnade och skum-material kan ses som semi-ordnade då de kan beskrivasväl i grunden som en honungskaka men med en störning i var väggar i strukturenmöts. De numeriska resultaten visar att de ordnade och semi-ordnade strukturernabeter sig annorlunda och beteendet hos de semi-ordnade nätverken stämmer väl öv-erens med experimentella resultaten från skum-materialet.

iv

AcknowledgementsFirst I would like to express my gratitude the taxpayers through the Swedish Re-search 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 thecompletion of this work has been the help and support of my supervisors, Per Isaks-son, Per Gradin and Sverker Edvardsson, thank you. The experimental part of thisthesis would not have been possible without the support of Sta�an Nyström and MaxLundström at IMT. Finely I like to extend my deepest and most heartily than you tomy colleagues with a special recognition to Madde, Hanna, Pär, Sara and Amanda,who make the weeks �y by.

v

ContentsList 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 PapersThis thesis is mainly based on the following papers, herein referred by their Romannumerals:

I Johan Persson, Per Isaksson. A particle–based method for mechanical analysesof planar �ber–based materials. International Journal for Numerical Methods inEngineering 93.11 (2013): 1216–1234.

II Johan Persson, Per Isaksson. A mechanical particle model for analyzing rapiddeformations and fracture in 3D �ber materials with ability to handle length ef-fects. International Journal of Solids and Structures 51.11 (2014): 2244–2251.

III Johan Persson, Per Isaksson. Modeling rapidly growing cracks in planar materi-als with a view to micro structural e�ects. International Journal of Fracture 2015In press.

IV Johan Persson, Per Isaksson, Per Gradin. Dynamic mode I crack growth in anotched foam specimen under quasi static loading. Submitted.

vii

ContributionsThe work presented in this thesis and the appended papers are results of collabora-tion. However in all publications the author has done major parts of the program-ming, experiments, computing analyzing and writing.

viii

Chapter 1

General remarks

Figure 1.1: Some di�erent network materials. Top left: Tissue paper. Top right: Hon-eycomb composite with paper as both sheet and core, i.e. a multi-scale network ma-terial. Lower left: PVC foam.

This thesis concerns a class of materials, including both natural and man- made ma-terials, where the primary load bearing structure is a network, and the deformationand fracture of such materials. Some examples of materials are shown in Fig. 1.1. Insome cases the load bearing structure are �bers, e.g. paper and �ber glass compos-ites 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 networkmaterials are very complex processes that depend strongly on the geometric con�gu-ration, the mechanical properties of the material constituents and the rate of appliedload.

The �eld of physics that deals with fracture is known as fracture mechanics, andis a relatively new science. Although the �rst known systematic experiments wereconducted c. 1500 by Da Vinci, the theoretic work is often considered to begin inthe 1900s with the groundbreaking work by Gri�th and the realization that fractureis driven by energy and not stress alone [1]. Irwin made signi�cant improvementsto Gri�th’s theory in the 1950s when he introduced the plastic zone, clarifying thatordinary materials cannot carry an in�nite stress but rather have a maximum stressknown as the yield stress which, for some materials, is constant for a range of strainvalues [2]. Some advances towards understanding dynamic fracture consisting of set-ting 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 surfacecreation. In the case of dynamic fracture, some energy is dissipated in the form ofplastic deformation around the crack tip, some energy is consumed in creating newsurfaces, some energy might create new microcracks which does not propagate, andsome energy will cause stress waves [7]. The energy associated with stress wavescan be regarded as e.g. sound and heat, but this distinction is not relevant for ourpurposes 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 formedby foaming, especially PVC foam. Solid PVC foam has a lower degree of variationthroughout the body than for example paper due to both the more homogeneousmaterial and the forming process. Therefore, solid foams are suited as model mate-rials when investigating network materials. Foams are sometimes modeled as madeup of regular honeycombs, sometimes as perturbed honeycombs and sometimes withmore advanced unit cells [11, 12, 13, 14]. This is applicable both to open- and closed-cell foams, with the di�erence that while a closed cell structure means that all facesof a cell are solid whilst only the edges of the faces of an open cell foam are solid. Thedi�erent unit-cells have some consequences for the mechanical behavior of the foamwhich 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 somesense true for all materials, but especially so for network materials, since they arediscontinuous at the intermediate scale. To further complicate matters, they are oftenused 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 mostoften caused by local tension, so failure due to tension is of special interest for networkmaterials.

Chapter 2

Background

Theoretical models of fracture are essential for understanding what causes fractureand ultimately failure in structures, and understanding is the �rst step towards pre-venting failure. The theoretical models are of two types, one explaining the microme-chanics 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 istransferred to the crack tip area. Macroscopic models can be utilized either to vali-date/disprove theoretical assumptions, for example to see if the energy spent in plas-tic deformation is relatively large or small for a certain crack, or they can be used tomake predictions regarding the life of a structure.

In the dawn of fracture mechanics, researchers were limited to linear elastic frac-ture mechanics and only problems with an analytic solution. Even though some ad-vanced problems might be solved in such ways [19, 20], many interesting problemslack an analytical solution. There are two possible solutions strategies when the an-alytic solution is missing; solving the analytic problem numerically which involves adiscrete solution space, or discretizing the problem and solving it numerically. Thereare 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 ofapplications, among the most prominent are the extended FEM (XFEM) [24] which al-lows 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 char-acteristics. All three models are based on a homogenized continuum representationof the material. The resultant forces are computed based on the weak formulationof the di�erential equations describing the elements. Another approach is based onthe viewpoint that all matter is composed of particles with attractive and repulsiveinteractions. 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 forstatic, quasistatic and steady state dynamic cases, LM/PM are better suited for fulldynamic cases. This has prompted the use of hybrid models [28, 29, 30, 31], which uti-lize the best of the two worlds; the complex elements used in FEM/XFEM/CZM andthe dynamics, inertia, and ease of computation from LP/PM. The model utilized inthis thesis is an example of the latter. FEM has dominated the area of network mate-rial modeling, both with the approximation of a homogenized continuum [32, 33, 34]and by modeling individual structural elements [35, 36, 37, 38, 39]. By modelingindividual structures, accurate models for fracture investigation can be developed[40, 41, 21, 22, 23]. Recently, particle modeling has reached the �eld of network me-chanics, 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 intothe complex phenomenon of dynamic deformations and fractures in network mate-rials. To judge the models several mechanical problems picked from the literature,together with published and new experimental data, are analyzed and comparedwith other published solutions. The in�uence of plasticity or other types of materialnon-linearity will not be included in the models.

4

Chapter 4

Summary of content and majorconclusions

0 0.1 0.2 0.3 0.4 0.5 0.60

1

2

3

4

5

6

7

8

v/c2

γf

γ0

H35

H60

H80

H200

Figure 4.1: Left: The test specimen clamped in the testing machine. The crack willgrow approximately along the dark line, breaking the electrical circuits. Right: Thenormalized crack growth energy versus normalized crack growth speed for speci-mens 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. Thedata is supplied by DIAB [43].

H35 H60 H80 H200

Density ⇢ 38 k g/m3 60 k g/m3 80 k g/m3 200 k g/m3

Tensile Modulus E 49 MPa 75 MPa 95 MPa 250 MPa

Shear Modulus G 12 MPa 20 MPa 27 MPa 73 MPa

As mentioned in the background, a suitable material for investigations in to dy-namic fracture properties of network materials is PVC foam. In the experimentalPaper IV the test specimen is a square with an edge notch, shown in Fig. 4.1. Thespecimens are foam panels of Divinycell; H35, H60, H80 and H200, where the di�er-ence is their e�ective density which varies in the interval 38 to 200 kg/m3, presentedin 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 whilemonitoring the crack growth. When the crack grows, it breaks electrical conductorspainted on the specimen, and the time between consecutive breaks is recorded. Fromthis time the average crack speed v can be computed. The crack growth speed is sev-eral orders of magnitude larger than the end displacement rate, so for all practicalpurposes the crack grows in a statically loaded structure. The normalized potentialenergy available for surface creation �/�

f

versus the resulting crack growth speed isshown in Fig. 4.1, in this graph a high velocity corresponds to a short initial crack. Itis known from both analytic fracture mechanics and from experiments that a crackinitiated by a smaller notch will grow with a higher speed than cracks from a largernotch, likely due to the fact that the specimen with the smaller crack can store morepotential energy before crack growth. As can be seen, the general behavior of thefoam 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 modelis developed for two- and three-dimensional cases based on geometrically nonlin-ear engineering beam theory. The formulations are described in Papers I and II andutilized in Papers I, II and III. The highlights of the model are; the network struc-ture is modeled as mentioned above by geometrically nonlinear engineering beams,with lumped masses at the ends making up the particle representation. In the two-dimensional case, each particle has two translational and one rotational degrees offreedom; in the three-dimensional case each particle has three translational and threerotational degrees of freedom. The particles of one structural element may be con-nected to other particles by linear elastic springs and torsion springs to bind struc-tures 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 plasticdeformations or stick-slip events. There are four sources of energy dissipation in themodel; through boundaries, breaking of structures, breaking of bonds and throughan arti�cial friction that dissipates energy by scaling velocities in each time-step. Thearti�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 tofracture.

Summary of content and major conclusions 7

0 0.2 0.4 0.6 0.8 10

2

4

6

8

10

v/c2

γf

γ0

Analytic solution (Nilsson 2001)

Present study

Nilsson (1974)

0 0.2 0.4 0.6 0.8 10

2

4

6

8

10

v/c2

γf

γ0


Present study

Figure 4.2: Both graphs show the normalized crack energy versus normalized crackspeed for a growing crack in notched specimens with notch lengths varied in theinitial w/199 a0 10w/199 (regular grid) and w/46 a0 30w/46 (honeycombgrid), where w is the width of the strip. The dashed line is the analytical crack growthexpression from Nilsson (2001). Left: The stars are experimental results from Nilsson(1974) and the diamonds are the numerical result of computations with a regularsquare grid. Right: Diamonds are the numerical result of a regular honeycomb grid.

0 0.2 0.4 0.6 0.8 10

2

4

6

8

10

v/c2

γf

γ0

Nilsson (1974)

r̃∆ = 0.16l0

0 0.2 0.4 0.6 0.8 10

2

4

6

8

10

12

v/c2

γf

γ0


Present study

Figure 4.3: Both graphs show the normalized crack energy versus normalized crackspeed for a growing crack in notched specimens with notch lengths varied in theinitial w/199 a0 10w/199 (regular grid) and w/46 a0 30w/46 (honeycombgrid), where w is the width of the strip. The dashed line is the analytical crack growthexpression from Nilsson (2001). Left: Stars are experimental results from Nilsson(1974) and the horizontal bars represent plus/minus one standard deviation to themean value of the numerical result of computations with a perturbed square grid.Right: the horizontal bars represent plus/minus one standard deviation to the meanvalue of the numerical result of computations with a perturbed honeycomb grid.

The experiment of Nilsson [44] is repeated as a computational experiment in Pa-per III both for a regular square grid which approximates a perfect continuous ma-


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 areslowly displaced until a crack grows. For a large enough notch, the crack will cometo rest before the ultimate failure, and for any notch smaller than this the crack willpropagate dynamic until reaching the end of the strip. The potential energy avail-able 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

versusthe resulting crack growth speed for a regular square grid, and a regular honeycombgrid. There is a noticeable di�erence in the shape of the graphs in that the honey-comb grid has a steeper slope and there is almost a discrete step from stable staticcrack growth to dynamic crack growth with a speed above 0.2c2. Note that c2 is theshear wave speed of the material, not of the network for the honeycomb. It is alsonoteworthy that although the model does not include any plastic deformation, it stillcaptures the behavior of steel well.

Honeycomb grids respond di�erently to perturbations of node locations than reg-ular grids do, as is evident in Fig. 4.3, whilst perturbations in a regular grid causeonly a slight perturbation in the fracture response. With a slight perturbation, themodel captures the spread of the experimental data of Nilsson [44]. Similar pertur-bations in a honeycomb grid cause not only perturbations to the response but alsoa signi�cant shift towards higher crack speeds, in some cases up to three times thatof the regular honeycomb. The �/�

f

versus v/c2 graph is similar for the perturbedhoneycomb grid and the experimental data in Fig. 4.1, con�rming that it is adequateto model a foam material as a perturbed honeycomb, and illustrating that although itis adequate to use a regular honeycomb model to model static conditions for foams,it is inadequate when dealing with dynamics.

0 10 20 30 400

0.5

1

1.5

2

ao[mm]

Lcr

ack/(B

−a0)

0 50 100 150 200 2500

0.5

1

1.5

2

Density[kg/m3]

Lcr

ack/(B

−a0)

Figure 4.4: Normalized crack length, L

crack

is the length of the crack whilst B � a0 isthe shortest possible crack length, i.e. the distance from the notch tip to the end of thespecimen. Left: Crack length versus notch length, the line shows the mean value forall specimens with a certain notch length (and di�erent densities) and the vertical barshows the standard deviation. Right: Crack length versus e�ective density, the lineshows the mean value for all specimens with a certain density (and di�erent notchlengths) and the vertical bar shows the standard deviation.


0 0.1 0.2 0.3 0.4 0.5 0.60

0.5

1

1.5

2

v/c2

Lcr

ack/(B

−a0)

H35

H60

H80

H200

Figure 4.5: Normalized crack length, L

crack

is the length of the crack whilst B � a0 isthe shortest possible crack length, i.e. the distance from the notch tip to the end of thespecimen, 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. Withreference to the fact that a small notch results in a fast running crack, and a largenotch results in a slow running crack, it is possible to draw the conclusion that thecrack speed is not related to the crack length, also illustrated in the slightly chaoticFig. 4.5. With reference to the fact that the material density does not signi�cantly af-fect the crack speed, it is noteworthy that the density a�ects the crack length. Whenthe potential energy is not spent making new surfaces (a longer crack) and not to plas-tic deformation which does not exist in the model, and not to micro-cracks aroundthe 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 anysigni�cant contribution. For the �rst half of the growth, no energy has had time totravel from the crack tip to the boundaries. The only plausible explanation is that thepotential energy released has gone to stress waves and vibrations.

Figure 4.6: Illustration of a square specimen with a central mode I notch. The zoombubble illustrates the �ber network microstructure on a smaller length scale.


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 orientedidentical �bers. Some experiments are carried out with a square specimen, with asmall initial central mode I notch, which is loaded axially in tension as illustratedin Fig. 4.6. The boundaries are clamped and the specimen is stretched with a con-stant rate and then brought back to their original positions at the same rate, so as toload the specimen in an impulse-like manner. Tension waves form at the upper andlower boundaries and travel through the network and when they meet in the centerat the notch, the crack grows. Later, when the following compressive wave reachesthe center, the crack stops.

−4 −2 0 2 40

0.2

0.4

0.6

0.8

1

y/a0

normalizedpotentialenergy

∆x = 0.45

−4 −2 0 2 40

0.2

0.4

0.6

0.8

1

y/a0

normalizedpotentialenergy

∆x = 0.85

Figure 4.7: Graphs shows the normalized potential energy, summed for all x-valuesand normalized to a maximum of unity i.e. U (y , t) ⇤

RW/2�W/2 u(x , y , t)dx/max(U)

with the potential energy u(x , y , t) at time t ⇤ 3 µs (left) and t ⇤ 4 µs (right). Theyillustrate how the initial Heaviside step function tension wave has changed characterwhile traveling through the specimens. The left graph represents a square grid. Theright 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.


0.5 1 1.5 2 2.5 3 3.5 40.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

time µs

normalizedderivative

∆U

∆y

ls/L = 1/12ls/L = 1/17ls/L = 1/25ls/L = 1/62

Figure 4.8: The left graph shows the maximum slope of the normalized potentialenergy 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 pro-nounced for a random �ber network than for a regular grid, illustrated in Fig. 4.7and the dispersive properties is greater for a less dense network, i.e. a network witha larger average distance between two �ber-�ber bonds l

s

, illustrated by the slope ofthe energy graph shown in Fig. 4.8. It is plausible to explain this by the many waysto walk a network. The fastest way is through stretching the �bers oriented parallelto the path, and the considerable slower way through bending �bers oriented moreorthogonal to the path and anything in between. From a structure integrity point ofview, this is a good thing, since it implies that stress waves traveling in sparse net-works will blur, causing the peek amplitude to be lower, and the material thus gainsa small protection against stress waves.


4 5 6 7 8 90

0.5

1

1.5

2

2.5

3

3.5

4

time µs

fractureevents

x−a0

a0

ls/L = 1/12ls/L = 1/17ls/L = 1/25ls/L = 1/62

4 5 6 7 8 9 10−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

time µsfractureevents

x−a0

a0

Figure 4.9: The two graphs show how di�erent cracks have grown from the initialnotch. The factor x�a0

a0is 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 hasgrown 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 twographs in the crack propagation speed (the slope of the graphs) but only a slightvariation within the graphs.

Paper I also aims to illustrate the di�erence when a network fails due to wallsfracturing or walls loosing connections to other walls. This is illustrated for four dif-ferent networks in Fig. 4.9. At the time of publication the most likely explanationto the considerable di�erence in crack growth speed was considered to be the di�er-ence 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 en-ergy would be spent to advance the crack past a �ber, but a more realistic assumptionwould be on average half that, since a randomly placed �ber on average would havea quarter of its length on one size of a idealized crack line, and three quarters on theother side. This small energy di�erence is not likely to account for the di�erence inspeed. However, it is possible that the spreading of the e�ective crack tip might beresponsible.


0 0.05 0.1 0.15 0.2 0.25 0.30

0.2

0.4

0.6

0.8

1

ls/L

Micro-fracturelocalizationratio

Figure 4.10: Left: Fraction of network where new crack growth is located near theexisting notch tip for networks normalized mean free length l

s

/L where L is the aver-age 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 originalcrack plane to a fractured segment, used co compute the macroscopic crack widthand �w is the orthogonal distance from the �tted line to a fractured segment utilizedto compute the microscopic crack width.

0 0.05 0.1 0.15 0.2 0.250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Microscopic

crackwidth[2∆w/L]

ls/L0 0.05 0.1 0.15 0.2 0.25

0

0.5

1

1.5

2

2.5

3

Macroscopic

crackwidth[2∆y/

L]

ls/L

Figure 4.11: Left: Microscopic crack width. Right: Macroscopic crack width. Theillustrative dashed lines are least-square �tted lines, the square markers are the esti-mated average values while the vertical bars are the standard deviations.


0 0.05 0.1 0.15 0.2 0.250

0.1

0.2

0.3

0.4

0.5

ls/LMicroscopic

crackgrowth[v

c/c]

0 0.05 0.1 0.15 0.2 0.250

0.1

0.2

0.3

0.4

0.5

ls/LMacroscopic

crackgrowth[v

c/c]

Figure 4.12: Left: Microscopic crack growth velocity. Right: Macroscopic crackgrowth 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 meanvalues 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 treata structure like a �ber network and when it would be possible to treat the problemas a homogenized continuum. The test specimen is once again square with a centralmode I notch. The specimen is rapidly stretched, until a complete failure occurs. Therapid stretch is an order of magnitude lower than the crack growth speed; it is thusstill a case of dynamic growth with quasi-static boundaries, even though the eventsleading to the crack growth are not quasi-static.

One aspect that is of major importance is whether or not new cracks form as ex-tensions 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 ratherthan 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 anotch, and new cracks are likely to form at arbitrary points in the material, and atboundaries. The same relation is present in the resultant crack width, whether it ismeasured 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 arounda normalized mean free length l

s

/L ⇡ 0.1 and is complete at l

s

/L ⇤ 0.05. Finally, thespeed of crack propagation is presented in Fig 4.12, with the result that the speed issigni�cantly less than for a continuum material with the same base material, and itis insensitive to the mean free length of the network. Put together, these three resultsindicate that highly connected network materials might be described as a continuum,but with signi�cantly lower speed of crack propagation than might be expected.


Figure 4.13: A growing crack in a random �ber network with time progressing fromleft to right. Left: The existing central notch has opened. Center: Cracks have grownboth left and right from the central notch. Right: Complete failure throughout thespecimen.

A random �ber network under tension buckles out of plane in what is calledfringes, especially visible around the major pores. One example is shown in Fig. 4.13for a network with growing cracks taken from paper II. In some cases, this fringingmay be helpful in �nding pores in the material, and thus potential weak spots. How-ever, as is evident in Fig. 4.13 a crack does not necessarily follow the path with thelargest pores. In this case, the left crack splits where one path follows a number ofsmaller pores slightly below the notch, and the second crack grows from the notch tothe largest pore. Even though both fringes and pores in some cases may be indicatorsof weak spots, neither of them are reliable. This behavior is at least to some extentspeci�c for random �ber networks and is not to be seen as a general network materialcharacteristic, since random �ber networks have stronger than average regions whichare 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 fractureof network materials. They are applied to a variety of cases and some general con-clusions are drawn. The work also includes an experimental study of dynamic crackgrowth in solid open cell foams. A �rst observation which is of importance to fractureis the considerable inherent dispersive properties for random �ber networks. Stresswaves traveling in a random �ber network are more e�ciently dispersed than stresswaves traveling in a continuous material, which in the case of modeling is representedby a regular grid. Fiber networks that fracture by breaking �ber-�ber bonds have anability 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 meanfree length l

s

) have more predictive characteristics than networks where �bers havefew connections. For networks with normalized mean free length l

s

/L 0.05 thefracture behavior resembles that of a continuous material, but with a slightly widercrack path, and slightly lower speed of crack growth. It is therefore possible to regardhighly connected networks as a continuum but with e�ective parameters. This resultagrees well with the static deformation study by Åslund and Isaksson [39].

In the models mentioned above, there are no plastic deformation, but still theycapture the dynamic fracture behavior of steel, both in static and dynamic loadingconditions. Whilst this is not an actual proof that energy dissipation due to plasticdeformation 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 tostress 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 phe-nomenological 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 notchlength, the actual crack length is not dependent on the notch length. While the crackspeed is not dependent on the e�ective density of the foam, the actual crack lengthis. 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. notalong the actual crack path, is not dependent on the actual crack length. If the frac-tion of energy dissipated by formation of new surfaces is large, it is expected that thecrack length will a�ect the e�ective crack speed. If, on the other hand, the energydissipated by new surfaces is small, this e�ect should not be visible. The result inFig. 4.5 is an indicator that the actual energy dissipated to form new surfaces is smallin comparison to the total energy dissipation. The best candidate to explain whathappens to the energy is stress waves.

16

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