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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) r˜∆ = 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. 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