ACTAUNIVERSITATIS

UPSALIENSISUPPSALA

2020

Digital Comprehensive Summaries of Uppsala Dissertationsfrom the Faculty of Science and Technology 1917

On Crack Dynamics in BrittleHeterogeneous Materials

JENNY CARLSSON

ISSN 1651-6214ISBN 978-91-513-0906-4urn:nbn:se:uu:diva-406919

Dissertation presented at Uppsala University to be publicly examined in Polhemsalen,Ångströmlaboratoriet, Lägerhyddsvägen 1, Uppsala, Friday, 15 May 2020 at 09:15 for thedegree of Doctor of Philosophy. The examination will be conducted in English. Facultyexaminer: Associate Professor Ralf Denzer (Lund University).

AbstractCarlsson, J. 2020. On Crack Dynamics in Brittle Heterogeneous Materials. DigitalComprehensive Summaries of Uppsala Dissertations from the Faculty of Science andTechnology 1917. 55 pp. Uppsala: Acta Universitatis Upsaliensis. ISBN 978-91-513-0906-4.

Natural variation, sub-structural features, heterogeneity and porosity make fracture modellingof wood and many other heterogeneous and cellular materials challenging. In this thesis,fracture in such complicated materials is simulated using phase field methods for fracture. Phasefield methods have shown promise in simulations of complex geometries as well as dynamicsand require few additional parameters; only the material toughness and a length parameter,determining the width of a regularised crack, are needed.

First, a dynamic phase field model is developed and validated against experiments performedon homogeneous brittle polymeric materials, wood fibre composites and polymeric materialswith different hole patterns. Then, a high-resolution model of wood is developed and relatedto experiments, this time without considering fracture. Attention is finally focussed on high-resolution numerical analyses of fracture in wood and other cellular microstructures, consideringboth heterogeneity and relative density.

The phase field model is found to reproduce crack paths, velocities and energy release rateswell in homogeneous samples both with and without holes. In more complicated heterogeneousand porous materials, the model is also able to simulate crack paths, but the interpretation ofthe length scale is complicated by the inherent lengths of the micro-structural geometry. Insum, the thesis points to possibilities with the proposed method, as well as limitations in ourcurrent understanding of both quasistatic and dynamic fracture of heterogeneous and cellularmaterials. The findings of this thesis can contribute to an improved understanding of fracturein such materials.

Jenny Carlsson, Department of Materials Science and Engineering, Applied Mechanics, Box534, Uppsala University, SE-751 21 Uppsala, Sweden.

© Jenny Carlsson 2020

ISSN 1651-6214ISBN 978-91-513-0906-4urn:nbn:se:uu:diva-406919 (http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-406919)

List of papers

This thesis is based on the following papers, which are referred to in the textby their Roman numerals.

I Carlsson, J. and Isaksson, P. (2018) Dynamic crack propagation inwood fibre composites analysed by high speed photography and adynamic phase field model. International Journal of Solids andStructures 144–145:78–85.

II Carlsson, J. and Isaksson, P. (2019) Crack dynamics and crack tipshielding in a material containing pores analysed by a phase fieldmethod. Engineering Fracture Mechanics 206:526–540.

III Carlsson, J., Heldin, M., Isaksson, P. and Wiklund, U. (2019)Investigating tool engagement in groundwood pulping: finite elementmodelling and in-situ observations at the microscale. Holzforschung,Epub ahead of print.

IV Carlsson, J. and Isaksson, P. (2020) Simulating fracture in a woodmicrostructure using a high-resolution dynamic phase field model.Accepted for publication in Engineering Fracture Mechanics.

V Carlsson, J. and Isaksson, P. (2020) A statistical geometry approach tolength scales in phase field modelling of fracture and strength ofporous microstructures. Submitted.

Papers I and II are published with open access. Paper III is republished withpermission from the publisher.

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.1 Aims of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Heterogeneous and cellular materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.1 Wood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.1.1 Fracture of wood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2 Other cellular materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2.1 Solid foams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2.2 Bone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3 Effective properties of cellular materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.4 Critical crack length of a cellular microstructure . . . . . . . . . . . . . . . . . . . . . . . . . 15

3 Models of fracture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.1 Griffith’s theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2 Later development of linear elastic fracture mechanics . . . . . . . . . . . . . . 173.3 Crack kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.4 Dynamic fracture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.5 The variational approach to fracture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.6 Phase field models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.6.1 Strain energy splits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.6.2 Analytical solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.7 FE implementation of phase field model for fracture . . . . . . . . . . . . . . . . . . 273.7.1 Time integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.7.2 Newton-Rhapson iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.1 Dynamic fracture (Paper I) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.2 Influence of holes (Paper II) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.3 Interaction between wood and tools (Paper III) . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.4 Fracture in a wood microstructure (Paper IV) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.5 Length scales and continuum modelling (Paper V) . . . . . . . . . . . . . . . . . . . . . 43

5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

Svensk sammanfattning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

1. Introduction

In solid mechanics, materials are typically considered to be continuous andhomogeneous, which means that we assume that if we take an infinitely smallpiece of the material, that piece will have the same properties as the originalspecimen, and also that each small piece will be identical to any other (Cauchy,1823). What is special about heterogeneous materials is that if we were to takea small piece of the material it will neither have the same properties as thelarge-scale material, nor will any piece necessarily be the same as any other.There might be pores in the material, meaning that some such pieces will beempty while others contain material which is both stiffer and heavier than theaverage of the large-scale material. Another source of heterogeneity is when alarge-scale material consists of different materials on the micro-scale, i.e. theyare composites. A small piece of the large-scale material will then consist ofeither one of these materials, and any two pieces will not be alike, nor willthey be like the large-scale material.

In many applications it is sufficient to use the average properties at thelarger scale (Ostoja-Starzewski, 2007). Fracture, however, is a local process,which occurs in a very small region near the crack tip. For this reason, localproperties can play an important role in the fracture behaviour of heteroge-neous materials (Hossain et al., 2014; Kuhn and Müller, 2016). Examples ofheterogeneous materials are wood, bone, glass- and carbon fibre composites,concrete, solid foams etc. The emphasis in this thesis is on wood, but manyresults apply to heterogeneous materials in a more general sense.

Wood is a very old – perhaps the oldest – construction material. Woodenwater wells, dating more than seven thousand years back, have been found(Tegel et al., 2012; Rybnícek et al., 2020). But naturally, the use of wood goeseven further back; for example, the numerous founds of axes dating from thestone age imply that wood was being worked regularly in these early societies.Wood is an extremely versatile material, and some other uses include art, tools,clothes, food products and not least paper. In the 19th century, the inventionof the first large-scale process to produce paper from wood, the groundwoodprocess, was the starting point of a rapid decrease in paper prices (Müller,2014). The groundwood process can thus, at least partly, be credited withthe vast increase in information spread during the 20th century: schoolbooks,fiction as well as non-fiction books, letters, newspapers etc. And today, wheninformation to an increasing extent is digital, paper usage in Europe remainsnear the all-time high because of increased demands for paper for packaging(CEPI, 2019).

7

In spite of its widespread use, our understanding of wood is still limited. Apart of this can be attributed to the extreme diversity of plants; plants have beenon Earth for over a billion years and outnumber animals in weight by a factorof one thousand (Jahren, 2017). Trees are youngsters in the plant family, only370 million years, but they still make up the major part of the total biomass onland. This abundance makes for great variation. And so it follows that woodis a natural material with large variability: between different species of wood,between different individuals of the same species and even between differentparts of the same tree.

But our limited understanding of wood is not only due to this so-callednatural variability. Wood, like bone but unlike most man-made materials, isa multiscale heterogeneous material with distinct features from the scale ofnanometers to that of a whole tree (Bodig and Jayne, 1982). The cell wallsconsist of crystalline cellulose microfibrils embedded in hemicellulose andlignin. The proportions of these constituents, the orientations of the cellulosefibrils, the thickness and shape of the cell walls and distribution of cells indifferent directions all influence the macroscopic properties of wood, makingit difficult to model and therefore also difficult to predict its behaviour.

The specific topic of this thesis is modelling of fracture and crack dynam-ics. In short, fracture is the separation of an object or material into two ormore pieces. A more technical definition is that fracture is the process of en-ergy dissipation through the creation of new surfaces; these new surfaces arewhat we refer to as a crack (Griffith, 1921). This latter definition is very closeto the theoretical framework of this thesis, phase field models of fracture basedon the variational principle. In this framework, the energy that is dissipatedthrough fracture is considered as a term of the total energy of the system. To-gether with well-established physical principles, such as the minimisation ofpotential energy or the principle of least action, and mathematical and numer-ical optimisation methods, such as the finite element method (FEM), fractureis predicted solely on the basis of stationary points of the total energy.

Models of fracture are useful for predicting the behaviour of materials with-out having to perform a lot of destructive testing, or at scales not typicallyavailable for mechanical testing. This knowledge is requested by e.g. the con-struction industry; wood construction is a hot topic from a sustainability pointof view. Another useful application is that of mechanical pulping, where woodis repeatedly fractured until what remains is only a mixture of individual fibres(the pulp). The work also provides insights into fracture in general porous andheterogeneous materials, which can be used to improve the integrity of e.g.structural panels (sandwich panels), or in the design of medical implants, e.g.bone implants or bone cements, which require a good understanding of thefracture of the porous cancellous bone structure for optimal strength and dura-bility.

8

1.1 Aims of the thesisThe aim of this thesis is to increase the understanding of dynamic fracture inwood and other heterogeneous materials on the cell scale, and more specifi-cally to identify what features need to be considered when modelling this typeof fracture. This is done by

• proposing a model for simulating dynamic fracture and validating thismodel in comparison with experiments;

• proposing a cell-scale model of wood and relating this model to experi-ments;

• identifying key features in the modelling by investigating the influenceof structural features such as internal voids and interfaces on the modelpredictions; and

• investigating the possibility of generalising the model by going up anddown in scales.

9

2. Heterogeneous and cellular materials

2.1 WoodWood is a cellular, heterogeneous, highly anisotropic material. The woodsprimarily of interest in this thesis are softwoods (or needlewoods). These con-sist almost solely of tracheids (longitudinally oriented cells), which are theprimary structural building blocks of the wood, while at the same transportingfluids and nutrients, Fig. 2.1. This makes the structure somewhat simpler thanthat of hardwoods (or broadleaf woods), which have more rays (radially ori-ented cells) and sap channels (longitudinal cells for fluid transport). Althoughrays exist also in softwoods they are thin (often one cell wide and a few cellsin height) compared to those in hardwoods. Wood without any complicatedfeatures, e.g. knots and rays, is referred to as clear-wood. The specific woodsconsidered in the thesis are Scandinavian softwoods: Norway spruce (PiceaAbies) and Scotch pine (Pinus Sylvestris). These woods have an average rela-tive density (defined as overall density divided by the density of the cell wallmaterial) of around 0.3–0.4, but since they grow at high latitudes, where theclimate is characterised by strong seasonal variation, they also exhibit pro-nounced annual ring structures, meaning that variations in relative density ofbetween 0.1 and 0.6 are common (Thuvander et al., 2000). Both the averageand the local relative density affect the overall properties of the wood, such asstiffness, density and strength.

A typical softwood consists of 90% tracheid cells (also referred to as fibresor simply cells). In softwoods, tracheid cell radii (rcell) are typically around10 – 40 μm (Bodig and Jayne, 1982). Typical fibre lengths are in the orderof a few millimetres. Each tracheid cell consists of several layers, namely theprimary wall (P), the three layers of the secondary wall (S) and a void calledthe cell lumen, Fig. 2.2. The secondary wall, and especially the S2 layer, isthe thickest. Due to the high proportion of cellulose, it contributes most tothe stiffness of the cell wall (Harrington et al., 1998), Table 2.1–2.2. Betweenthe cells is the middle lamella, the “glue” that holds the cells together, whichis weaker due to a high proportion lignin. In addition, it is often difficult todistinguish between the middle lamella and the primary wall, which is whythis compound is often referred to as the compound middle lamella (CML).

With 90% of the cells in the longitudinal direction, wood is naturally an-isotropic. On a local scale (the scale of a few cells), it can be consideredtransversely isotropic with the tangential and radial directions approximatelyequal, Table 2.3. On a global scale, the stiffening effect of rays typically makesthe radial direction somewhat stiffer than the tangential.

10

Figure 2.1. Rendering of CT micrograph showing the structure of wood and the global,trunk-centered coordinate system (uppercase).

Figure 2.2. The different parts of a wood cell. The lowercase coordinate system refersto the local, cell-centered coordinate system.

Table 2.1. Approximate proportions of the constituents of the different cell wall layers(Harrington et al., 1998).

Cellulose Hemicellulose Lignin

S1 wall 37% 35% 28%S2 wall 45% 34% 21%S3 wall 34% 36% 30%CML 11% 14% 75%

11

Table 2.2. Young’s modulus E and ultimate strain εc of the cell wall constituent ma-terials (Gibson et al., 2010).

E εc

Cellulose 165 GPa 20–40%Hemicellulose 9 GPaLignin 3 GPa 3–4%

Table 2.3. Young’s moduli E, Poisson’s ratios ν and shear moduli μ for wood on thecell scale (Harrington et al., 1998). Moduli in GPa. The quantities are given in thecell-centered coordinate system (cf. Fig. 2.2).

El Er Et νlr νlt νrt μlr μlt μrt

S1 wall 53 8 9 0.3 0.3 0.4 3 3 3S2 wall 64 9 10 0.3 0.3 0.4 3 3 3S3 wall 50 8 8 0.3 0.3 0.4 3 3 3CML 18 5 5 0.3 0.3 0.4 2 2 2

2.1.1 Fracture of woodWhen loaded in compression, wood does typically not fracture but collapse(Gibson and Ashby, 1997). This is contrary to most brittle materials, e.g.concrete, which fracture transversely when loaded in compression. In tensionhowever, wood fractures. Due to the highly orthotropic nature of wood, thedifferent fracture directions show different behaviours. Consequently, the dif-ferent crack propagation directions are often referred to as different fracturesystems, Fig. 2.3. The first letter denotes the normal direction of the plane offracture and the second letter the direction of propagation. Sometimes prop-agation in the R direction is divided further into R+ and R−, depending onwhether the crack is propagating away from or towards the pith (the centre ofthe stem). In this thesis, the fracture systems under consideration are RT andT R.

Figure 2.3. Fracture planes of wood. The first letter refers to the direction normal tothe crack faces, the second letter to the direction of propagation.

12

Figure 2.4. Crack propagation in the tangential (RT) direction (left) and in the radial(TR) direction (right). Adapted from Dill-Langer et al. (2002).

As reported by Ashby et al. (1985), when a crack is propagating in high-density wood, such as latewood, crack growth typically occurs by separatingcells in the middle lamella, so-called intercellular fracture. A transition occursat a relative density of around 0.2. Below this value, cracks tend to propagateprimarily through the lumen, i.e. by breaking cell walls, so-called trans-wallfracture; this transition can be related to the relative strengths of the cell walland the middle lamella (Gibson and Ashby, 1997). The transition between cellseparation and cell wall breaking also depends on the architecture of the mi-crostructure. Crack propagation in the pure radial direction is typically straightand occurs by separation of cells in the middle lamella, whereas crack propa-gation in the tangential direction occurs with a combination of cell separationin the middle lamella and cell wall fracture (Dill-Langer et al., 2002), Fig. 2.4.Because of the variation in relative density, cracks propagating in the radial di-rection tend to propagate in a stepwise fashion, with the crack arresting as itapproaches the denser late-wood (Stanzl-Tschegg et al., 2011; Thuvander andBerglund, 2000). With further buildup of available strain energy, the crackwill break through the late-wood, and due to the excess energy this growthmay be unstable. Crack propagation in the RT -direction, on the other hand, istypically stable on a global scale, as the material is more homogeneous in thisdirection.

2.2 Other cellular materialsThe cellular structure of wood, especially the alignment of elongated, stiffcells in a weaker matrix, is similar to that of cortical bone (Gustafsson, 2019).But also without this very specific structure, many cellular materials, e.g. can-cellous bone or solid foams, have features in common with wood, which arenot shared with general homogeneous materials.

2.2.1 Solid foamsSolid foams are typically divided into open-cell and closed-cell foams. Open-cell foams resemble a three-dimensional lattice, or truss structure. Close-cell

13

foams typically resemble a union of bubbles, e.g. bread. For both kinds offoams, stiffness and strength depend strongly on the relative density. Likewood, foams do not fracture under compression, but collapse. Only in tensionis fracture commonly observed. Due to the large possibility for variation infoams, the results of this thesis do not apply to all foams, but can be mostlikely be applied to transversely isotropic, open-cell or thin-walled closed-cell foams (i.e. most foams foamed from liquids) made of relatively brittlematerials.

2.2.2 BoneMost bones are made up of an outer shell of relatively dense cortical bone anda core of porous cancellous, or trabecular, bone. The human cortical bone hasan architecture similar to wood – longitudinally oriented osteons (bone cells)with a central canal, surrounded by a weaker matrix. However, the relativedensity of cortical bone is typically much higher than that of wood (>0.7).

Cancellous bone, on the other hand, has a relative density similar to that ofwood (0.05–0.7). Cancellous or trabecular bone is made up of a network ofconnected rods or plates (trabeculae). Lower-density trabecular bone containsprimarily rods, whereas higher-density trabecular bone contains more plates(Gibson and Ashby, 1997). In general, the behaviour of trabecular bone istypical of cellular materials; stiffness and strength depend strongly on the rela-tive density and it tends to collapse under compression but can fracture intension.

2.3 Effective properties of cellular materialsThe effective, or macroscale, properties of porous materials typically dependon the relative density of the material. If the solid material making up the wallsof the porous material is considered transversely isotropic and linearly elastic,and if the architecture of the material is isotropic, then on the macroscale (con-tinuum scale), we have the macroscale stress σ = Eε , and on the microscalewe have the microscale stress σμ = Eμεμ , where E is Young’s modulus andε the strain and the index μ is used to indicate microscale properties. Ac-cording to Gibson and Ashby (1997), wood with its elongated circular, hexag-onal or rectangular cells, can be considered as a slightly perturbed honey-comb material. For a honeycomb material, the effective stiffness and stresson the macroscale depend on the relative density ρ∗ (defined as the averagedensity of the material divided by micro-scale density, i.e. ρ∗ = ρ/ρμ ) asE3rd order ∝ (ρ∗)3Eμ . Numerical experiments performed in Paper V suggestthat a quadratic scaling is more accurate for the materials considered in thisthesis, i.e.

E2nd order ∝ (ρ∗)2Eμ .

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Table 2.4. Scaling relationships for stiffness E, toughness Gc, strain ε and stress σdepending the relative density. Numerical experiments performed in Paper V showthat the second order model gives the most accurate predictions for the materialsstudied in this thesis.

E Gc ε σ

Second-order (ρ∗)2Eμ ρ∗Gcμ εμ/ρ∗ ρ∗σμ

The strain energy postulate, stating that the strain energy of the macro- and mi-croscale models coincide, gives expressions for macroscale stress and strain.In addition, the energy dissipated by fracture must be equal on the macro- andmicroscales. Since the dissipated energy is proportional to the surface area ofthe crack on the macro scale, and the surface area is scaled by ρ∗, the frac-ture toughness must scale as Gc ≈ ρ∗Gcμ . The scaling relationships for thesecond-order stiffness-scaling model are summarised in Table 2.4.

2.4 Critical crack length of a cellular microstructureCellular materials are typically less sensitive to small cracks than continuousmaterials are. A random porous microstructure can be described by a spatialPoisson point process (Okabe et al., 2000), Fig. 2.5. The average number ofcell mid-points in an area A is a spatial Poisson point process with intensityparameter (i.e. number of occurrences per unit area) λA. The probability thatthe number of points n occupying an area A is exactly equal to k is

P{n = k}= λ kA

k!e−λAA.

A lot of useful results have been derived for the Poisson Voronoi tessella-tion, such as the expected total edge length per unit area, LA = 2

√λA (Okabe

et al., 2000).A typical question that occurs when dealing with fracture of random cellular

materials is: How large must a crack or other defect be in order to affect thegeometry at all? A straight crack can be described by a sectional diagram ofthe Poisson Voronoi tessellation, Fig. 2.5. A line sectional diagram of a 2DPoisson Voronoi tessellation is a Poisson line process with intensity λL. Here,λL is the average number of cells encountered in a unit line distance along anydirection (Okabe et al., 2000),

λL =2LA

π=

4√

λA

π.

The probability of encountering k cell walls along a distance L is thus

P(n = k) =λ k

Lk!

e−λLL.

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Figure 2.5. A Poisson Voronoi tesselation and its sectional diagram. Crosses indicatethe seeds of the spatial process, the average number of seeds per unit area is λA. Theline process is given as a sectional diagram of the original tessellation. For the lineprocess, the intensities of walls (rings) and centres are the same, λL.

Suppose we introduce a crack of length a somewhere in the material. Theprobability that this crack does not intersect any cell wall is then

P{n = 0}= e−aλL .

By setting P= 1/2 we can find a critical crack length ac, for which it is equallylikely that an introduced crack will interfere with the structure as it is that itwill not,

ac =− 1λL

ln(

12

). (2.1)

This is true only when the width of the cell walls is negligible. If this is notthe case, then in order for a crack not to intersect any cell wall, it has to be at adistance wπ/4 away from any cell wall, where w/2 is the width of half of thecell wall, and the factor 2/π is the mean of sine over [0,π]. This is equivalentto stating that

P{x = 0}= e−(a+wπ/4)λL .

Setting P = 1/2, again, we find a critical crack length,

ac =− 1λL

ln(

12

)− wπ

4.

The critical crack length can be rewritten as a function of the relative densityρ∗ using the expected total edge length per unit area LA, as,

ac =− 1λL

ln(

12

)− ρ∗π

8√

λA. (2.2)

This critical length can be related to the notch sensitivity of a porous material,as demonstrated in Paper V.

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3. Models of fracture

3.1 Griffith’s theoryAccording to the theory developed by Griffith (1921), fracture is a process ofenergy dissipation through creation of new surfaces. Griffith found that duringstable crack propagation, there is no net change in total energy. Let Πe denotepotential energy in the material, stored in the form of strain energy, and Πs thesurface energy of the crack. Moreover, let F be work done by external forcesand let a denote the crack length. Griffith concluded that as the crack advancesa distance da, the change in total energy is zero, i.e.

d(Πe +Πs −F )

da= 0. (3.1)

Since the derivative in (3.1) is equal to zero, the energy in the numeratorhas a possible extremum with respect to crack length. Griffith thus concludedthat “[t]he ‘Theorem of minimum potential energy’ may be extended so as tobe capable of predicting the breaking loads of elastic solids, if account is takenof the increase in surface energy which occurs during the formation of cracks”(Griffith, 1921).

Griffith also showed that the product of the breaking strength σc and thesquare root of the crack length is constant,

σc√

a ≈√

2γEπ

, (3.2)

The quantity 2γ is twice the surface energy of the material, i.e. the energyrequired to create two surfaces. It can further be noted that zero crack lengthpredicts infinite breaking strength.

3.2 Later development of linear elastic fracturemechanics

In the 1950’s, Irwin (1957) developed the concept of linear elastic fracturemechanics. Irwin uses the concept of stress intensity factors, which dividesfracture into three modes, Fig. 3.1. For each mode, the stress at the crack tipis given by

σi j =

(KI/II/III√

2πr

)fi j(θ)

17

Figure 3.1. The three modes of fracture, from left to right: Mode I (opening), ModeII (in-plane shear) and Mode III (out-of-plane shear).

where KI, KII and KIII are the stress intensity factors of the three modes, thefunction fi j(θ) is a trigonometric function of the angle relative to the crackplane (Williams, 1957), and r is the distance from the crack tip. Irwin andOrowan (1955) also divided the energy dissipated by fracture into two parts,Gc = 2γ +Gp where Gp is energy dissipated by plasticity. As long as theplastic zone, or process zone, remains small, it is possible to replace 2γ in(3.2) by Gc.

The Griffith-Irwin relationship is

G =K2

IE

,

where the stress intensity factor KI and energy release rate G are replaced bythe critical stress intensity factor KIc and the toughness Gc at the initiation ofcrack propagation.

In later work on plastic fracture Rice and Rosengren (1968), Cherepanov(1967) and Hutchinson (1968) independently developed the concept of the J-integral which is the equivalent of G.

3.3 Crack kinematicsAn issue regarding the aforementioned theories is the inability to predict crackkinematics. In a body Ω ⊆ R

n, a crack Γ ⊂ Ω is a subset of the body, typi-cally of dimension of n− 1. This means that a crack can be parametrised byn−1 parameters, but require n functions of these parameters. Griffith’s equa-tions are scalar-valued, thus the system is under-determined for n > 1. Irwin’sequations rely on the assumption that a crack is in one of the three modes.And while there exist expressions for mixed-mode cracks, these do not predictcrack propagation direction. So where will the crack propagate?

One hypothesis is that the crack will propagate in the direction of the max-imum hoop stress. This hypothesis is usually credited to Erdogan and Sih(1963) but had already been used by Yoffe (1951) to predict crack branchingin dynamic fracture. It follows from the assumption that cracks will orientthemselves in the mode I direction.

Another common hypothesis is that the crack will propagate in the direc-tion of the maximum energy release rate, G. For the simplest case of pure

18

mode II loading, the two assumptions give slightly different predictions on thedirection of crack propagation.

3.4 Dynamic fractureIn dynamic fracture, crack growth is not stable, i.e. the energy release rate isnot necessarily equal to the change in elastic energy, cf. (3.1). The energysurplus becomes kinetic energy, causing stress waves to propagate through thespecimen. Under some conditions, for example when slowly loading speci-mens without or with only a very short initial crack, it is possible to load moreelastic energy into the structure than can be released by stable fracture; theresulting fracture must therefor be unstable, or dynamic. Moreover, the en-ergy cannot be dissipated instantaneously, as crack surfaces cannot extend atunlimited velocity. A speed limit for mode I crack propagation is the Rayleighvelocity cR; theoretically, a mode I crack can propagate with velocity v ei-ther below cR or above the pressure wave velocity cp (Broberg, 1989), i.e. therange cR ≤ v ≤ cp is “forbidden”. In mode I crack propagation, the velocitytypically relates to the energy release rate G as (Freund, 1990),

GGc

≈ 1− vcR

. (3.3)

Equation (3.3) is true only for relatively low velocities, approximately v ≤0.6cR. At higher velocities, the crack becomes unstable. Instead of accelerat-ing, it is energetically favourable for the crack to branch into two or more cracktips, each propagating with velocities around 0.6cR. This relation between en-ergy release rate and branching was first suggested by Eshelby (1999) and hasalso been verified numerically using phase field models (Bleyer et al., 2017).

Broberg (1989) also showed theoretically that a mode II (shear) crack canpropagate with a velocity under cR or over the shear wave velocity cs. This wasverified experimentally by Rosakis et al. (1999); their experiments have beenreproduced using the phase field method by Schlüter et al. (2016). Instanta-neous mode I crack propagation velocities as high as 0.9cR have been observedexperimentally (Sharon and Fineberg, 1999), supersonic mode I cracks havebeen observed only in atomistic simulations (Buehler et al., 2003).

3.5 The variational approach to fractureThe phase field models used in this thesis are based on the variational ap-proach to fracture of Francfort and Marigo (1998). The variational approachto fracture is closely related to the works of Griffith, but resolves most of thedrawbacks related to crack initiation and direction of propagation. In the vari-ational approach to fracture, a family of possible crack paths is considered.

19

For each member of the family, there is an associated energy Π, depending onthe displacement uuu and the crack Γ,

Π(Γ,uuu) = Πe(Γ,uuu)+Πs(Γ), (3.4)

where the elastic energy

Πe(Γ,uuu) =∫

Ωψe(Γ,∇uuu)dxxx,

and the surface energy

Πs(Γ) = Gc

∫Γ

ds.

Throughout this thesis, Π (regardless of index) denotes total energies, inte-grated over the entire body Ω whereas ψ denotes energy densities. Also, ∇denotes the spatial differential operator and dxxx indicates integration over spa-tial coordinates whereas ds indicates integration along line segments given byΓ.

According to Francfort and Marigo (1998), the evolution of the crack andenergy must follow some conditions. One such condition is that broken mate-rial does not heal, by requiring that the crack surface is non-decreasing. An-other condition is that the “real” crack must give a minimum with respect toenergy compared to all other admissible cracks (i.e. cracks for which the pre-vious cracks are enclosed in the current crack).

The crack state is found as the infimum of (3.4), but this infimum is gener-ally not trivial to find, in fact, it is most often impossible. Bourdin et al. (2000)noted the similarity between the variational fracture functional and the Mum-ford and Shah (1989) functional used for image segmentation. Thus, the firstnumerical experiments in the variational approach to fracture used this simi-larity to apply the already known Ambrosio and Tortorelli (1990) functionalapproximating the Mumford-Shah functional to fracture.

3.6 Phase field modelsIn gradient-regularised phase field methods, a sharp crack is represented bya diffuse damage field. The distribution of this damage field is regularisedby the gradient of the damage variable, thus dividing the surface energy intotwo terms, one local and one non-local (gradient-dependent) (Bourdin et al.,2000; Ambrosio and Tortorelli, 1990; Braides, 1998). The damage variablerepresents the degree of damage in a region of the material, and typically de-termines the degree of stiffness loss.

Phase field methods for fracture have quickly become popular, which canbe explained by their extreme versatility. Phase field models have been suc-cessfully applied to dynamics (Bourdin et al., 2011; Larsen et al., 2010; Bor-den et al., 2012; Hofacker and Miehe, 2012; Schlüter et al., 2016), anisotropic

20

Figure 3.2. Regularised representation of a crack. A sharp crack Γ (left) is representedby a crack density function γ(d,∇d) as a diffuse damage field d (right).

surface energy (Teichtmeister et al., 2017; Li et al., 2015), heterogeneous ma-terials (Hossain et al., 2014; Kuhn and Müller, 2016), plasticity (Hesch et al.,2017; Alessi et al., 2015) and fatigue (Alessi et al., 2018).

Consider a problem domain Ω⊂R2, with exterior ∂Ω, part of which ∂ΩT is

subject to natural boundary conditions (prescribed stress TTT ) and part of which∂Ωu is subject to Dirichlet boundary conditions (prescribed displacement UUU),with a discrete crack Γ, Fig. 3.2. The total free energy of the system can bewritten as (Larsen et al., 2010; Larsen, 2010; Bourdin et al., 2011)

Π(uuu,d) =∫

Ωψk(uuu)dxxx−

∫Ω

ψe(∇uuu,d)dxxx−Gc

∫Γ

ds. (3.5)

Here ψk is the kinetic energy density. Through the phase field implementation,the discrete crack Γ is represented by a crack density function γ(d,∇d), whered is a diffuse regularised crack field. Throughout this thesis, it is assumedthat d = 1 represents broken material and d = 0 represents intact material. Inthe works of this thesis, two of the most common crack density functions, theso-called Ambrosio-Tortorelli 1 (AT1) and Ambrosio-Tortorelli 2 (AT2), areused. The AT1 model is linear in d and quadratic in ∇d (Pham and Marigo,2009; Pham et al., 2011)∫

Γds ≈

∫Ω

γAT1(d,∇d)dxxx =∫

Ω

38l(d + l2∇d ·∇d)dxxx. (3.6)

The constant l is a regularisation parameter, a kind of characteristic length,determining the width of the regularised crack. The AT2 model is quadratic inboth d and ∇d (Bourdin et al., 2000; Miehe et al., 2010),∫

Γds ≈

∫Ω

γAT2(d,∇d)dxxx =∫

Ω

12l

(d2 + l2∇d ·∇d

)dxxx. (3.7)

A number of different crack density functions exists, each one gives a some-what different fracture behaviour, (cf. e.g. Braides, 1998; Pham et al., 2011;Marigo et al., 2016).

The evolution of damage must fulfill some conditions, closely related tothe conditions stipulated by Francfort and Marigo (1998), presented in the

21

previous section. Specifically, damage must be irreversible, the “real” stateof the damage must be a minimum of the action integral (or the total energyfunctional) and the energy balance must always hold.

The kinetic energy is

Πk(uuu) =∫

Ωψk(uuu)dxxx =

12

∫Ω

ρ uuu · uuudxxx,

where uuu, and later occurring uuu, denotes first and second derivatives of displace-ment with respect to time t, and ρ is the density. In the case of quastistaticsimulations, the velocity and acceleration terms are considered to be so smallthat they can be neglected, and this term is then zero.

The kinetic energy is unaffected by the crack phase field d, but d locallydegrades the elastic stiffness of the material. A degradation function of thetype (1− d)2 is used in both the AT1 and AT2 models. Other degradationfunctions give a somewhat different response (cf. Marigo et al., 2016; Kuhnet al., 2015; Sargado et al., 2018).

In order to prevent crack growth under compression, and to account forcrack surface over-closure, the strain energy is typically split into a positivepart, which is degraded as damage increases, and a negative part, which isunaffected, ψe(∇uuu,d) = (1− d)2ψ+

e (∇uuu)+ψ−e (∇uuu) (cf. Amor et al., 2009;

Miehe et al., 2010). There are many different ways to assign the positive andnegative strain energy, two are given in Section 3.6.1.

For the AT1 model, the action integral over time t ∈ Θ = [0, tend ] is

J(uuu,d) =∫

Θ

∫Ω

(12

ρ uuu · uuu− (1−d)2ψ+e −ψ−

e − 3Gc

8l(d + l2(∇d ·∇d))

)dxxxdt,

(3.8)and for AT2 model

J(uuu,d) =∫

Θ

∫Ω

(12

ρ uuu · uuu− (1−d)2ψ+e −ψ−

e − Gc

2l

(d2 + l2∇d ·∇d

))dxxxdt.

(3.9)To account for irreversibility of the crack evolution, a history field is used

in place of ψ+e (Miehe et al., 2010), so that H (xxx, t) is the maximum positive

strain energy experienced at each point xxx in the time 0 ≤ τ ≤ t,

H (xxx, t) = maxτ∈[0,t]

ψ+e (uuu(τ)).

The history field ensures that it is the largest strain energy experienced in thematerial during the simulation history that determines the present stiffness.This kind of history field is known to induce a small error in the simulation,especially for small l (Bourdin et al., 2008; Linse et al., 2017). Stresses arehowever reversible, and are always evaluated as σσσ = ∂ψe/∂εεε , where the lin-earized strain εεε = 1/2(∇uuu+∇T uuu). Taking a second derivative gives the con-sistent tangent stiffness tensor C= ∂ 2ψe/∂εεε2.

22

Using the principle of least action, the Euler-Lagrange equations of (3.8–3.9) are obtained for uuu and d as

⎧⎨⎩

∇ ·σσσ = ρ uuu (3.10a)

2(1−d)H − 3Gc

8l+

3Gcl4

∇2d = 0 (3.10b)

⎧⎨⎩

∇ ·σσσ = ρ uuu (3.11a)

2(1−d)H − Gc

l(d − l2∇2d) = 0. (3.11b)

For quasistatic cases, we have uuu ≈ 0 and the governing equations are stillobtained as the Euler-Lagrange equations of Eqs. (3.8–3.9). Assuming Πk = 0and minimising the total energy rather than the action functional producesthe same result. The equations of motion also require boundary and initialconditions (nnn is the normal to the boundary),

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

uuu =UUU on ∂Ωu (3.12a)σσσnnn = TTT on ∂ΩT (3.12b)∇d ·nnn = 0 on ∂Ω (3.12c)uuu(uuu, t) = u0(xxx) in Ω (3.12d)uuu(uuu, t) = v0(xxx) in Ω. (3.12e)

3.6.1 Strain energy splitsA common strategy in phase field modelling is to split the strain energy den-sity such that only tensile strain energy contributes to the damage. This way,materials do not fail under pure compression, and stiffness is kept in the caseof crack closure (Bourdin et al., 2008; Amor et al., 2009; Miehe et al., 2010).Many different split variants have been suggested, (Ambati et al., 2015; Strobland Seeling, 2016). In the various works of this thesis, two of the most com-mon splits are being used, the spectral split of Miehe et al. (2010) and thehydrostatic-deviatoric split of Amor et al. (2009).

The spectral split (Miehe et al., 2010) uses the fact that for an isotropicmaterial, the principal strain system gives independent modes of deformationwhich can be split into positive and negative parts,

ψ+e =

α2

λ ′tr(εεε)2 +μtr(εεε2+),

ψ− =1−α

2λ ′tr(εεε)2 +μtr(εεε2

−).

23

Figure 3.3. 1D traction model.

where α = 1 if tr(εεε) > 0 and α = 0 otherwise and μ and λ ′ are the Laméparameters, where λ ′ = λ in plane strain and λ ′ = λ (1−2ν)/(1−ν) in planestress. The positive and negative strain contributions are given by a spectraldecompostion of the strain tensor,

εεε± =2

∑1

ei ±|ei|2

nnniiinnnTiii , (3.13)

where ei are the eigenvalues and nnniii the eigenvectors of the strain tensor εεε .Another possibility is to use the hydrostatic and deviatoric deformation

modes to assign the strain energy to the positive and negative parts (Amoret al., 2009),

ψ+e =

α2

κ ′tr(εεε)2 +μεεεdev : εεεdev.

ψ−e =

1−α2

κ ′tr(εεε)2,

where κ ′ = λ ′+μ is the bulk modulus (in 2D) and εεεdev = εεε −1/2tr(εεε)III is thestrain deviator (and III the identity matrix).

3.6.2 Analytical solutionsFor some simple quasistatic problems, such as traction of a one-dimensionalbar, as well as for the transverse direction across a crack, it is possible to deriveanalytical solutions.

1D traction of a bar

Consider the bar Ω ⊂R of cross sectional area S, subject to end displacementsU(0) = 0 and U(L) = tL (Carlsson, 2019), Fig. 4.17. The stiffness is thescalar stiffness E(1− d)2 and stress σ and strain ε are also scalar-valued. Itis possible to drop the nabla notation and instead write d′, as d only has onedirectional derivative. Note also that ε = u′ = t.

Letting γAT1(d,d′) = 3/(8l)(d + l2d′2) (cf. Eq. 3.6), i.e. the AT1 model,gives

Π(u,d) = S∫ L

0

[12

E(1−d)2t2 +3Gc

8l

(d + l2d′2)]dx.

24

The Euler-Lagrange equations with respect to u and d are (cf. (3.10))

− ddx

E(1−d)2t = 0,

3Gc

8l−E(1−d)t2 − 6Gc

8ld′′ = 0.

The second derivative of d must be zero due to constant strain, and since d ≥ 0,the Euler-Lagrange equation for d is unfulfilled until t = tc =

√3Gc/(8lE).

The rod thus remains damage-free with d = 0 with constant stiffness E untilt = tc. At this time, the stress is

σc =

√3GcE

8l. (3.14)

After tc, damage is evolving and the Euler-Lagrange equation is fulfilled,which gives the damage as

d(t) = max(

0,1−( tc

t

)2),

and also the stress,

σ(t) =

{σc

ttc

if t ≤ tc,

σc( tc

t

)3 otherwise.

The stress (or the load-carrying capacity) falls quickly and approaches zero,Fig. 3.4.

Letting γAT2(d,d′) = 1/(2l)(d2 + l2d′2) (cf. Eq. 3.7), i.e. the AT2 model,gives

Π(u,d) = S∫ L

0

[12

E(1−d)2t2 +Gc

2l

(d2 + l2d′2)]dx.

The Euler-Lagrange equation with respect to d is

Gc

ld −E(1−d)t2 −Gcld′′ = 0.

Damage evolves from t = 0 as

d(t) =Elt2

Gc +Elt2 ,

i.e. there is no elastic phase in which d = 0. The stress is

σ(t) = E(1−d(t))2

√Gcd(t)

lE(1−d(t)).

25

0 1 2 3 4

Time/tc

0

1/

c

0 1 2 3 4

Time/tc

0

1

/c

Figure 3.4. Stress plotted versus strain (time) for the AT1 model (left) and for the AT2model (right). The stresses and strains are normalised by the critical stress and criticalstrain of the respective models.

The stress takes on a maximum value when the derivative with respect to straint is zero. This happens when d = 1/4, which means that

σc =

√27EGc

256l. (3.15)

At this point, tc =√

Gc/(3lE). Unlike the previous model, the Euler-Lagrangeequation is satisfied during the whole evolution. The stress depends on dis-placement as shown in Fig. 3.4.

Equations (3.14) and (3.15) can be used to determine appropriate values ofl, given a known ultimate tensile stress (Pham et al., 2011; Tanné et al., 2018).

Damage profiles

Optimal damage profiles can be obtained by minimising the damage func-tional, i.e. the integral of the crack density function over the domain Ω. For aone-dimensional bar described only by the spatial coordinate x (where x = 0 isnow assumed to be in the interior of the domain), for which damage localisa-tion occurs, the crack phase profile is obtained by minimisation of Eqs. (3.6)and (3.7) with Ω ⊂ R. For the AT1 model (3.6),∫

ΩγAT1(d,∇d)dx =

∫Ω

38l(d + l2d′2)dx,

the Euler-Lagrange equation with respect to d is (1−2l2d′′) = 0, for which

d(x) =(

1− |x|2l

)2

is a solution. For the AT2 model (3.7),∫

ΩγAT2(d,∇d)dx =

∫Ω

d2

2l+

l2

∇d′2dx,

26

-4 -2 0 2 4

x/l

0

1

d

AT1

AT2

-1 -0.5 0 0.5 1

x/lch

0

1

d

AT1

AT2

Figure 3.5. Damage profiles for the two models used in this thesis, normalised by theirrespective length parameter l (left) and the material characteristic length lch =EGc/σ2

c(right).

the Euler-Lagrange equation is d − l2d′′ = 0, for which

d(x) = e−|x|/l

is a solution. By using relations (3.14) and (3.15), the damage profiles canalso be related to the (model-independent) material characteristic length, lch =EGc/σ2

c . The solutions are shown in Fig. 3.5.

3.7 FE implementation of phase field model for fractureEquations (3.10–3.11) are solved using FEM. To write Eq. (3.10b) on theweak form, the equation is multiplied by a test function δd and integratedover the domain Ω. The product rule for derivatives, the divergence theoremand the boundary conditions (Eq. 3.12c) then give the weak form (Ottosenand Petersson, 1992; Zienkiewicz and Taylor, 2000)

∫Ω

(2H dδd +

3Gcl4

∇d∇(δd))

dxxx =∫

Ω

(2H − 3Gc

8l

)δddxxx. (3.16)

For Eq. (3.11b), the weak form is obtained in the same way as∫

Ω

((2H +

Gc

l

)dδd +Gcl∇d∇(δd)

)dxxx =

∫Ω

2H δddxxx. (3.17)

To discretise Eq. (3.16–3.17), let d(xxx) = N(xxx)d, ∇d(xxx) = B(xxx)d, δd(xxx) =N(xxx)δd and ∇δd(xxx)=B(xxx)δd where d are the nodal values of the crack phasefield d and N and B = ∂N/∂xxx are some standard FE basis functions and theirfirst spatial derivative. Since the test function is arbitrary, it can be dropped,and the discretised equation for the crack phase field is

Kdd = fd , (3.18)

27

where

Kd =∫

Ω

(2H NT N+

3Gcl4

BT B

)dxxx,

fd =∫

Ω

(2H − 3Gc

8l

)NT dxxx,

for the AT1 model, and

Kd =∫

Ω

((2H +

Gc

l

)NT N+GclBT B

)dxxx,

fd =∫

Ω2H NT dxxx,

for the AT2 model.Likewise, for Eqs. (3.10–3.11a) multiplication by a test function δuuu, inte-

gration over Ω, using the product rule for derivatives, the divergence theoremand the boundary conditions in Eq. (3.12), gives the weak form of Eq. (3.10–3.11 a),

∫Ω

[(∇δuuu)T ((1−d)2

C++C

−)∇uuu+δuuuρ uuu]

dxxx =∫

∂ΩT

δuuuTTT d(∂ΩT ),

(3.19)where C

+ and C− are the parts of the stiffness matrix related to postive and

negative strain energy.To discretise Eq. (3.19), let uuu(xxx)=N(xxx)u, ∇uuu(xxx)=B(xxx)u, δuuu(xxx)=N(xxx)δu

and ∇δuuu(xxx) = B(xxx)δu where u are the nodal values of the displacement fielduuu. Since the test function is arbitrary, the discretised equation for the crackphase field is

Mu+Ku = f, (3.20)

where

M =

∫Ω

NT ρNdxxx, (3.21)

K =∫

ΩBT ((1−d)2

C++C

−)Bdxxx, (3.22)

and

f =∫

∂ΩT

NT TTT d(∂ΩT ). (3.23)

In all works in this thesis that involve dynamic simulation, a lumped massmatrix has been used, obtained by using a quadrature only involving the nodalpoints (only for M) (Zienkiewicz and Taylor, 2000).

28

3.7.1 Time integrationIn dynamic simulations, Eq. (3.20) is solved using explicit time integration,using an explicit Newmark time stepping scheme (Newmark, 1959). First, foreach time step, the displacement is solved explicitly based on the damage ofthe previous time step. The fracture phase field, Eq. (3.18) is then solved,while keeping displacements constant.

In the Newmark scheme, the displacement of the new time step k + 1 ispredicted by

uk+1 = uk +Δtuk +12(1−β2)(Δt)2uk,

where Δt is the length of the time step, and uk, uk and uk are the displacement,velocity and acceleration of the previous time step k (Zienkiewicz and Tay-lor, 2000). The predicted displacement, uk+1, is used to explicitly solve thedynamic system of equations for the displacement problem (Eq. 3.20) by

uk+1 =−[

M+12

β2(Δt)2K

]−1

(fk+1 +Kuk+1) , (3.24)

where M, K and fk+1 are the mass- and stiffness matrices and load vector (ofthe new time step k+1) of Eqs. (3.21 – 3.23). The acceleration (Eq. 3.24) isthen used to calculate the velocity uk+1 and updated displacement uk+1 as

uk+1 = uk +(1−β1)Δtuk +β1Δt ¨uk+1,

uk+1 = uk+1 +12

β2(Δt)2uk+1.

The time step Δt is always chosen so that it is less than or equal to the timerequired for a pressure wave to propagate the edge length h of the smallestelement (Courant et al., 1928)

Δt ≤ hcp. (3.25)

In all dynamic simulations, a fully explicit Newmark algorithm is used, withconstants β1 = 1/2 and β2 = 0.

3.7.2 Newton-Rhapson iterationIn the quasistatic simulations, the discretised, quasistatic equation

Ku = f, (3.26)

as well as Eq. (3.18) is solved in a staggered manner using alternating Newton-Rhapson iteration for u and d. Considering Eq. (3.26) (the procedure is iden-tical for Eq. (3.18)), the iteration scheme is given by considering (Bonet andWood, 2008)

Ku =−R

29

where initially R =−f. The residual force vector R is then updated by calcu-lating the equivalent nodal forces T =

∫Ω σ∇Ndxxx and setting R = T− f until

equilibrium, which is interpreted as the quotient R/f being within some toler-ance.

In the staggered algorithm, the coupled equations for u and d are solved byalternating between the two, first keeping d constant while solving for u andthen keeping the obtained u constant while solving for d etc. until tolerancesare met for both fields.

30

4. Contributions

4.1 Dynamic fracture (Paper I)In Paper I, a dynamic phase field model is compared with experiments per-formed on homogeneous 100% polylactic acid (PLA) and a wood fibre com-posite consisting of 20% pulp wood fibres and 80% PLA matrix material(Carlsson and Isaksson, 2018; Joffre et al., 2017).

Rectangular strip specimens with thickness T = 1 mm and dimensions H ×L = 50×100 mm2 are cut, and notches of varying lengths a0 are made usinga scalpel, Fig. 4.1. The specimens are clamped in a Shimadzu AGS-X tensiletesting machine with a load cell of 10 kN and cross head speed 5 mm/min. Thespecimens are then loaded slowly, until the onset of sudden, rapid fracture.Crack propagation is monitored using a high-speed camera (MotionPro Y8),recording at 70.000 frames per second.

The experimental results are compared to numerical results. The AT2 modelis used together with the spectral strain energy split (Miehe et al., 2010). Astate of plane stress is assumed. A geometry of 1×30× 100 mm3 is consid-ered, which is close to the free height of the specimens in the experiments.Material properties, including the phase field length parameter l, are presentedin Table 4.2. An initial notch is created by pre-defining a strain history field.Initial notch length a0 is varied. The model is loaded quasistatically using adisplacement increment of 5 μm until onset of fracture, after which crack prop-agation is simulated dynamically with a time step of 0.02 μs and a boundaryvelocity vy0 of 5 mm/min.

For both experiments and simulations, the average energy release rate iscalculated as (Nilsson, 1974, 1972)

Figure 4.1. Geometry and boundary conditions for simulations in Paper I. The upperboundary is subject to continuously increasing displacement until initiation of crackgrowth, after which a boundary velocity of 5 mm/min is applied.

31

Table 4.1. Material properties used for simulation in Paper I.

E [GPa] ν ρ [kg/m3] Gc [J/m2] cR [m/s] σc [MPa] l [mm]

Wood fibre composite 4.7 0.3 1000 800 1245 40–45 0.3PLA 3.8 0.3 1000 1500 1120 40–45 0.3

0 0.2 0.4 0.6 0.8 1

vtip

/cR

0

2

4

6

8

10

G/G

c

PLA

Simulation

Experiment

Branching

0 0.2 0.4 0.6 0.8 1

vtip

/ cR

0

2

4

6

8

10

G/G

c

Wood fibre composite

Simulation

Experiment

Branching

Figure 4.2. Crack speed vs energy release rate for pure PLA (left) and the wood fibrePLA composite (right).

G =u2

y0E

2H(1−ν2),

where uy0 is the vertical displacement on the boundary at the initiation of crackpropagation.

The average energy release rate versus average crack tip velocity is shownin Fig. 4.2. Samples for which branching has occurred are indicated with across. Both the pure PLA material and the wood fibre composites follow thenumerically predicted curve well. For simulations, as well as for PLA experi-ments, branching occurs for energy release rates just over 2Gc. For wood fibrecomposite experiments, energy release rates as high as 3− 4Gc occur with-out visible branching. Simulated fracture patterns for different initial cracklengths are shown in Fig. 4.3.

Looking at the instantaneous energy release rate,

G =− dΠT da

=dΠp

T da− dΠk

T da,

it is possible to correlate variations in energy release to branching events, Fig.4.4. The number of simultaneous crack fronts suggest that each crack frontconsumes up to just above 2Gc, while for energies higher than this, the crackbranches.

So why do we not see branching in the composites? The answer most likelylies in the microstructure of the materials. As seen in the SEM micrographs,Fig. 4.5, the fracture surfaces of the composites are rugged and uneven com-pared to those of the pure PLA. It seems as if matrix cracking, fibre-matrixfracture and fibre pull-out might provide sufficient surface area to consume

32

Figure 4.3. Simulated damage patterns for different initial crack lengths. Top: shortcrack with G= 5.6Gc (a0/L = 0.006), and bottom: longer crack with G= 4.0Gc (a0/L= 0.01). The colourbar indicates the damage level.

Figure 4.4. Energy release rate versus crack tip position.

energy and to keep the velocities from reaching unusually high values, with-out observable macroscopic branching. In terms of fracture, there is thus adistinct difference between the pure PLA and the wood fibre composite – eventhough the wood fibre composite consists of 80% PLA and thus only a smallamount fibres – due to the heterogeneous microstructure. These small hetero-geneities on the microscale alter the crack behaviour, in spite of the fact thatthe wood fibre composite resembles a homogenous material in handling, andis definitely more similar to PLA than to typically heterogeneous materials.

Figure 4.5. Crack surfaces near the initial notch: pure PLA (left) and wood fibre PLAcomposite (right).

33

4.2 Influence of holes (Paper II)The influence of holes or pores in a material is studied experimentally as wellas numerically in Paper II (Carlsson and Isaksson, 2019). Simulations are per-formed on specimens of dimension 2×50×25 mm2 and a state of plane stressis assumed, Fig. 4.6. Material properties of PMMA are assumed, Table 4.2.The specimens have holes in various predefined configurations, Fig. 4.7. Tosimplify meshing around the hole boundaries, unstructured meshes are used,regardless of geometry, see the insert in Fig. 4.6. An initial notch of length a0is created by pre-defining a strain history field. The model is loaded quasistat-ically using a displacement increment of 0.5 μm until onset of fracture, afterwhich crack propagation is simulated dynamically using a time step of 0.02μs. The AT1 model together with the spectral strain energy split (Miehe et al.,2010) is used.

For comparison, samples of plexiglass (PMMA) are prepared, by cutting 2mm thick sheets of plexiglass to dimensions 50× 50 mm3 and drilling holesin the same pre-defined configurations. Initial notches of varying length arecut using a saw with a narrow blade. The samples are loaded slowly in aShimadsu AGS-X tensile tester with a cross-head speed of 5 mm/min, untilfracture occurs.

Table 4.2. Material properties of PMMA used for simulation in Paper II.

E [GPa] ν ρ [kg/m3] Gc [J/m2 ] cR [m/s] σc [MPa] l [mm]

3.24 0.35 1190 200 962 50 0.225

With the exception of geometry four, the crack paths are relatively consis-tent between the physical experiments, Fig. 4.7. The consistent crack paths arealso well predicted by the numerical simulations. In general, the experimentsand simulations with short initial notches are more affected by the presenceof holes in the geometry than the ones with longer notches. It is further notedthat the crack tends to speed up in the presence of a hole, but also that holes

Figure 4.6. Geometry and boundary conditions for simulations in Paper II. The upperand lower boundaries are subject to continuously increasing displacements until on-set of crack growth, after which crack propagation is simulated dynamically withoutfurther increase in prescribed displacement.

34

Figure 4.7. Experimental and numerical crack paths for the different geometries, fromtop to bottom: reference geometry, geometry 1, geometry 2, geometry 3 and geometry4. The left columns show experimental results, the right columns numerical results.Column 1 and 3 corresponds to long initial cracks (a0 = 2 mm), columns 2 and 4corresponds to short initial cracks (a0 = 0.75 mm). The length of the specimens is50 mm. For geometry 4, the experimental crack paths were not consistent betweendifferent experiments, hence no representative experimental sample is shown.

can shield a crack tip making the crack propagate slower, Fig. 4.8. Interest-ingly, the crack acceleration mechanism tends to dominate over the shieldingmechanism for short notches, i.e. for higher energy release rates. A transi-tion at around 2Gc was observed. On the other hand, in the presence of holes,energy release rate variations are slightly out of phase with the velocity. Thisbehaviour violates Eq. (3.3), which is valid only when the crack does notinteract with external boundaries (Fineberg and Bouchbinder, 2015), and im-plies that materials containing pores do not necessarily behave like continuaon a macroscopic scale. In relation to this, we observe that the holes in thegeometry affect the propagation of waves – generated by the crack– in thestructure, Fig. 4.9. The strain waves of the specimen with holes has a wavelength much larger than the general continuum, in the order of magnitude ofthe hole diameter.

In addition, the effect of the internal characteristic length l with respectto pore radius is studied. It is observed that since the elastic and kinematicenergies are not affected by any regularisation, small holes still affect crackdynamics by altering the strain and kinematic energy densities. Small holesdo not however localise damage.

35

Figure 4.8. Normalised crack tip velocity versus normalised crack tip position plottedover crack path (red area) at time t=52 μs for reference geometry (top row), geometry1 (middle row) and geometry 4 (bottom row). Left column: a0=2 mm, right column:a0 = 0.75 mm.

Figure 4.9. Strain energy distribution for numerical experiments with short initialnotches: Reference geometry (top row), and geometry 1 (bottom row) at 10, 30 and50 μs (from left to right). The colourbar refers to all figures.

36

4.3 Interaction between wood and tools (Paper III)In Paper III, a high-resolution elastic model of wood is developed and relatedto experiments. Interaction between grinding surface asperities and a woodstructure is studied numerically as well as experimentally with emphasis onthe groundwood pulping process (Carlsson et al., 2020). A simplified modelconsidering softwood clear-wood is used and modelled using the commercialsoftware Abaqus. The model consists of 4×7 cells, each with edge length 50μm and relative density 0.4, modelled as a linear elastic orthotropic continuumwith properties relative to the wood stem coordinates according to Table 4.3.The boundaries were constrained to mimic a repetitive behaviour.

Table 4.3. Material properties for simulations in Paper III. Moduli in GPa. Based ondata from Harrington et al. (1998), full data given in Table 2.3.

EL ER ET νLR νLT νRT μLR μLT μRT

S wall 60 9 9 0.1 0.1 0.4 3 3 3CML 20 5 5 0.1 0.1 0.3 2 2 2

Figure 4.10. Normal (top row) and shear (bottom row) strain for indentation withlarge (L), small (S) and truncated (T) indenters plotted against vertical distance (leftcolumn) and horizontal distance (right column). Red lines indicate plot paths.

Two sets of simulations are performed, with three different asperity geome-tries (pyramids with small radius (10 μm), pyramids with larger radius (30μm) and truncated pyramids (top side 40 μm)). In the first set of simulations,the tool asperity is indented a distance equal to half the cell height. Normaland shear stresses in the material underneath the indenter are shown in Fig.4.10. In general, strains are localised to the surface and extends only a fewcell diameters down. The truncated pyramids give strains deeper down in the

37

Figure 4.11. Experimental setup for CT. The small pyramids of the diamond grindingsurface are barely visible.

Figure 4.12. Strains from DVC: first principal strain (left) and maximum shear strain(middle). The volume used for calculation is shown on the right.

structure, and small pyramids give the most superficial strain. In a second setof simulations, the tool asperity is indented a distance equal to one tenth of thecell height, then slid horizontally a distance equal to one cell height.

Computational results are compared to experimental results using tools withsurface geometries similar to the modelled ones (small sharp pyramids, largepyramids with a truncated top of 5 μm width and truncated pyramids with atop side of 40 μm (Heldin, 2019)). In a first step, FE results are comparedto calculated strains from digital volume correlation (DVC) of image stacksobtained in computational X-ray tomography (CT) experiments. CT experi-ments are performed at Paul Scherrer Institute in Switzerland using a setupwhich simultaneously presses and slides a tool surface with small diamondpyramids against a piece of wood, Fig. 4.11. A total of 1201 radiographs arecaptured, each having a pixel size of 3.25 μm, while simultaneously rotatingthe sample 180◦. From the obtained CT stack, a DVC algorithm is used tocalculated displacements and strains (van Dijk et al., 2019), Fig. 4.12. Justlike in FE simulations, the strains are localised to the surface of the wood andextends only one or a few cell diameters down.

The results from FE and DVC of initial contact are also compared to grind-ing experiments in which the wood is submerged in 90◦C water and scratched20 times with different asperity geometries pressed against the wood with anormal load of 10 N, Fig. 4.13. The surface of the wood after scratching looks

38

Figure 4.13. Surfaces scratched with, from left to right: small pyramids, large pyra-mids and truncated pyramids.

very different depending on which tool surface was used. For the sharper pyra-mids, the surface shows parallel grooves where the fibres have been cut. Forthe truncated pyramids, fibres are starting to be pulled out.

In light of the differences between the methods utilised in the study, thecorrelation between the results of the different methods is surprising. Both FEand DVC show very local deformation and shallow strain fields for the smallerpyramid radius. Such extreme local deformation is likely to promote localdamage and fibre fracture rather than fibre detachment. Affirmatively, in thegrinding experiments, we find that the sharper pyramids tend to cut the fibresinto short segments and the worn wood surface shows distinct parallel scratchmarks, whereas the truncated pyramids actually grind out fibres as opposed tosegments.

39

4.4 Fracture in a wood microstructure (Paper IV)In Paper IV, fracture in a wood microstructure is simulated and the importanceof the middle lamella is investigated. Fracture modes at the cell scale, such asintercellular and trans-wall fracture, are accounted for by introducing a mid-dle lamella interface zone between cells, similar to the approach adopted byHansen-Dörr et al. (2019).

The middle lamella is not directly visible in the original image, but since themiddle lamella (or CML) is reported to influence crack behaviour (Thuvanderand Berglund, 2000; Boatright and Garrett, 1983; Stanzl-Tschegg et al., 2011;Ashby et al., 1985; Dill-Langer et al., 2002), a fictional middle lamella is cre-ated using a watershed transform. The middle lamella has a width of 1.3 μm.The width of the middle lamella is not sufficient for damage to localise com-pletely. In order to have proper energy release as a crack propagates in theCML, the toughness of the CML is reduced from Gc to Gc. Using numericalintegration of the crack profile, a value of Gc = 200 J/m has been found togive a dissipated energy close to Gc for the CML, which is around 350 J/m(Gibson and Ashby, 1997). All other material properties are reported in Table4.4. The numerical model is an AT1 model with hydrostatic-deviatoric split(Amor et al., 2009), and plane strain conditions are assumed.

Table 4.4. Material properties of the different cell wall layers of wood used for sim-ulation in Paper IV (Harrington et al., 1998; Gibson and Ashby, 1997). The fractureenergy for the CML has been reduced to 200 J/m.

E [GPa] ν ρ [kg/m3] Gc [J/m2]

S wall 9 0.35 1500 1650CML 5 0.35 1500 350 (200)

First, a simple two-cell model is studied quasistatically, Fig. 4.14. Each cellhas an outer length of 50 μm. The mesh consists of first-order elements withedge length 0.33 μm. The size of the cell lumina (hole) is varied. It is notedthat small cell lumina lead to abrupt failure. For an intermediate size hole, theload drops suddenly, but the middle of the interface is shielded from stress andcontinues to carry load for some additional displacement. For a large hole,damage does not localise to the middle lamella but to the edges of the holes.

Simulations are then performed on a 20×10 cells wood microstructure withas well as without CML. The geometry is created by image processing of a mi-crograph, and discretisation is performed directly from the processed image,so that each pixel becomes one element. This results in a mesh with around1.5 million first-order elements, each having element edge length 0.33 μm.Simulations are performed either using quasistatic simulation with a displace-ment increment of 0.05 μm or dynamically using a time step of 0.03 ns with aboundary velocity v0 of 10 or 20 m/s, Fig. 4.15. It is observed that the exis-tence of the CML alters the crack propagation significantly, Fig. 4.16. Due to

40

0 0.05 0.1 0.15

U/H

0

0.2

0.4

0.6

0.8

1

1.2

F/F

0

rc=H/3

rc=2H/5

rc=4H/9

Figure 4.14. A preliminary test using only two cells, top left: geometry, top right:normalised force F/F0, bottom left: damage profile for the geometries with smallerholes and bottom right: damage profile for the geometry with large hole. Note that thedamage profiles are not plotted against the full length of the sample but a short sectionclose to the interface; dashed vertical lines indicate the width of the interface (CML).

its lower fracture toughness, damage is localised to the CML. Moreover, thereis not one single crack but rather a coalition of several microcracks, and thecracks only rarely seek out the holes (lumen). When comparing damage pat-terns obtained in simulations with different loading velocities, it appears thatan increased loading velocity v0 results in more damage, and that this damagebecomes scattered over a larger area. This applies to all models with a dis-cretised microstructure, whether the CML is modelled or not. For sufficientlyslow loading velocities the quasistatic results should be recovered, but onlyif the crack propagation itself is stable. Simulations using boundary veloci-ties as slow as 5 m/s have been performed (not reported here) but truly stablecrack propagation has not been observed, indicating that crack growth in theseporous materials is typically unstable in nature, and that fracture cannot beproperly accounted for with quasistatic models.

The study indicates that in the cell scale, fracture in wood can be seen asa combination of alternately stable and unstable cracking. It appears as ifintercellular (middle lamella) damage evolves ahead of the crack tip, however,actual crack propagation – it seems – must to some extent also consist of trans-wall fracture.

41

Figure 4.15. Geometry and boundary conditions in wood microstructure simulations.The upper and lower boundaries are subject to continually increasing prescribed ver-tical displacements; in the dynamic simulations the time step is controlled so that thedisplacement rate amounts to a pre-defined velocity, v0. The insert shows the compu-tational grid (element edges not visible) with the CML visible in dark grey.

No CML CML Continuum

Qua

si-s

tatic

v 0=

10m

/sv 0

=20

m/s

Figure 4.16. Dynamic crack patterns when the global load carrying capacity hasdropped by 50%, influence of velocity and CML. Left column: no CML, middlecolumn: CML included, right column: Continuum. Top row: quasi-static loading,middle row: v0 = 10 m/s, bottom row: v0 = 20 m/s. Colourbar refers to all images.

42

4.5 Length scales and continuum modelling (Paper V)Paper V addresses the issue of how to treat the length parameter l in phase fieldmodelling of notch sensitivity, strength and crack path in pre-notched samplesof varying notch length in a material with a microstructure resembling that ofwood in the tangential direction. The paper is primarily related to the issue ofmoving between different scales in a multi-scale material.

Table 4.5. Material properties used for simulation in Paper V (Harrington et al.,1998; Gibson and Ashby, 1997).

Eμ [GPa] νμ ρμ [kg/m3] Gcμ [J/m2]

9 0.35 1500 1650

The numerical model used is an AT2 model with hydrostatic-deviatoricstrain energy split (Amor et al., 2009), and plane strain conditions are as-sumed. Microstructured as well as homogeneous continuum simulations areconsidered, Fig. 4.17. Microscale material properties are given in Table 4.5.Macroscale stiffness scales with the square of the relative density, Fig. 4.18, somacroscale material properties are given in Table 2.4. In order for the forcesat fracture of the microstructured and the continuum models to coincide, l isalways chosen to be the same in the two models, Fig. 4.18.

Pre-loading is simulated by continually increasing the prescribed displace-ment (Δu = 0.5 μm) at the horizontal boundaries until initiation of crack prop-agation, Fig. 4.17. Crack propagation is then simulated either quasistaticallyor dynamically without further increase in prescribed displacement. In thecase of dynamic simulations, the time step Δt = 0.1 ns.

It is found that the difference between the continuum and the microstruc-tured models in terms of fracture force is large for l smaller than around 2dc(where dc is the cell diameter), Fig. 4.19. For l � 2dc the microstructuredmodel also recovers the homogeneous continuum reference toughness, Fig.4.20. If the toughness is plotted versus critical crack length according to Eq.(2.2), the dependence on relative density disappears.

Figure 4.17. Geometry and boundary conditions for simulations in Paper V. The sizeof the model is 1.14×0.82 mm2.

43

10-1

100

Relative density *

10-3

10-2

10-1

100

Rela

tiv

e s

tiff

ness E

/E

(*)2

(*)3

*

10-1

100

Relative density *

10-1

100

101

Rela

tiv

e f

orc

e F

/F

l =l

l =l*

l =l/*

(*)-0.5

(*)0.5

Figure 4.18. Stiffness of a continuum model scales with the square of the relativedensity (left). Force at failure for the microstructured model coincides with that of thecontinuum model if the length parameter is chosen to be the same (right).

0 1 2 3 4 5

l/dc

0

0.1

0.2

0.3

0.4

0.5

Err

or

*=0.25

*=0.5

*=0.75

Figure 4.19. Mean relative errors between continuum and microstructured models,considering all crack lengths for different relative densities and different quotients ofl/dc. Error bars represent standard deviation.

In terms of crack propagation, for simulations with small l there is a markeddifference between the microstructured and the continuum models, Figs. 4.21–4.22. This distinction diffuses for larger l. The microstructured models arealso sensitive to instability in crack propagation. Even though both the contin-uum models and the microstructured models are experiencing unstable crackgrowth, it is only for the microstructured model that we see a difference be-tween the dynamic and quasistatic simulations in terms of crack paths.

It is concluded that if the quotient l/ac (or l/dc, but this does not take rel-ative density into account) for a porous material is sufficiently large, then amicrostructured model can be replaced by a continuum model. It is also sug-gested that the localisation behaviour should determine the length parameterl, and rather that Gc be chosen to obtain the correct stress at failure.

44

0 1 2 3 4 5

l/dc

0.2

0.4

0.6

0.8

1

1.2

G/G

c

*=0.25

*=0.5

*=0.75

0 5 10 15 20

l/ac

0.2

0.4

0.6

0.8

1

1.2

G/G

c

*=0.25

*=0.5

*=0.75

Figure 4.20. The quotient G/Gc for un-notched micro-structured models of differentrelative densities and different quotients of l/dc (left) and different quotients of l/ac(right).

l = 0.5dc l = dc l = 3dc

Qua

si-s

tatic

Dyn

amic

Figure 4.21. Crack paths for continuum models with ρ∗ = 0.5, visualised when theload-bearing capacity has dropped by 50%, with l = 0.5dc (left column), l = dc (mid-dle column) and l = 3dc (right column), all with initial crack length a0 ≈ 10dc. Toprow: quasi-static simulations, bottom row: dynamic simulations.

l = 0.5dc l = dc l = 3dc

Qua

si-s

tatic

Dyn

amic

Figure 4.22. Crack paths for micro-structured models with ρ∗ = 0.5, visualised whenthe load-bearing capacity has dropped by 50%, with l = 0.5dc (left column), l = dc(middle column) and l = 3dc (right column), all with initial crack length a0 ≈ 10dc.Top row: quasi-static simulations, bottom row: dynamic simulations.

45

5. Conclusions

Natural variation, sub-structural features, heterogeneity and porosity makefracture modelling of wood and many other heterogeneous and cellular ma-terials challenging. The phase field method is a promising approach, as it hasbeen shown to overcome a number of obstacles such as complicated geometry,heterogeneity, anisotropy, dynamics etc.

In this thesis, fracture is simulated in heterogeneous and cellular materialsusing a phase field method. In Papers I and II, the dynamic phase field modelis compared to experiments. Paper I studies crack paths and crack velocity inpure PLA and wood fibre/PLA composites. Crack propagation velocities andenergy release rates are well predicted for both materials, but the inclusion ofa small amount of wood fibre has an unexpected effect on the crack paths, effi-ciently limiting the amount of visible branching. Paper II studies the effect ofholes or pores in a material. It is found that a propagating crack is attracted toholes, especially if the energy release rate is high. In other cases, the presenceof holes tends to shield the crack tip.

In Paper III, a high-resolution wood model is created and related to experi-ments. No fracture model is included, instead numerically obtained strains arecorrelated to deformations from in-situ CT scans and scratching experiments.

Papers IV and V are focussed on high-resolution numerical analyses of frac-ture in cellular microstructures. In Paper IV we see that the middle lamella isnecessary to accurately simulate fracture in wood, as damage is localised tothe weaker interface material. In Paper V we find that localisation in a cellu-lar material is governed by the geometry rather than the properties of the cellwalls, indicating that material inherent lengths of the microstructural geom-etry play a role similar to that of the phase field characteristic length. BothPapers IV and V show that crack propagation in cellular microstructures, alsounder slow loading, is often unstable, meaning that inertial effects must beconsidered.

In sum, the thesis demonstrates the potential of the proposed phase fieldmethod in terms of its ability to simulate complicated geometries as well ascrack dynamics, but also limitations in our current understanding of both qua-sistatic and dynamic fracture of heterogeneous and cellular materials. In orderto make full use of phase field models for fracture, it is necessary to properlychoose the characteristic length, which remains a topic for further study. Thefindings of this thesis improve our understanding of fracture in heterogeneousand cellular materials, knowledge that may be applied in a wide range of fieldssuch as construction, pulping, and medicine.

46

Sammanfattning på svenska

De modeller och metoder som används inom hållfasthetsläran bygger ofta påantaganden om att ett material är kontinuerligt och homogent, d.v.s. att varjeenskild liten bit av materialet har samma egenskaper som varje annan liten bitoch att dessa bitar i sin tur har samma egenskaper som den större biten. Förmaterial som trä, ben, kompositer, papper m.m. stämmer inte detta. I mångafall kan man bortse från inhomogeniteten och använda sig av materialets ge-nomsnittliga egenskaper. Detta gäller dock inte för brott, eftersom det är enlokal process, vilket gör att lokala egenskaper spelar stor roll.

Som en förenklad modell kan trä beskrivas som bestående av axiellt orien-terade, tubformade celler, vilka hålls samman av ett svagare matrismaterial,den så kallade mittlamellen, Fig. 1. I den här modellen är trä både heterogent(då det består av olika material) och poröst (eftersom de tubformade cellernaär ihåliga). Andra heterogena material är t.ex. kompositer, där fibrer inneslutsi ett matrismaterial. Exempel på porösa material är t.ex. fasta skum och trabe-kulärt ben.

Brott är en process i vilken ett föremål separeras i två eller flera delar. Menbrott kan också ses som en process i vilken elastisk eller kinetisk energi om-vandlas till ytenergi genom att nya ytor skapas. Energin som krävs för att skapadessa nya ytor ges av materialets seghet Gc. Brottprocessen kan därför uttryc-kas som ett variationsproblem och vedertagna variationsprinciper som mini-mering av potentiella energin eller minsta verkans princip tillämpas. Metodenär särskilt användbar i kombination med en regularisering av sprickan, i vil-ken sprickan beskrivs av ett kontinuerligt varierande skadefält (fasfält), Fig.

Figur 1. En enkel modell av trä: axiellt orienterade, tubformade celler hålls sammanav en mittlamell.

47

2. Till skillnad från en diskret spricka är skadefältet definierat i hela kroppen,och den regulariserade sprickans bredd styrs av regulariseringslängden l. I av-handlingen presenteras en sådan fasfältsmodell. Denna används sedan i fyra avde bifogade artiklarna för att lösa kvasistatiska och dynamiska brottproblem ikomplicerade geometrier och heterogena mikrostrukturer.

I Artikel I jämförs fasfältsmodellen med dynamiska sprickpropagerings-experiment på träfiberkomposit respektive det rena matrismaterialet (PLA).Det finns en god överensstämmelse mellan simulerad och experimentell sprick-spetshastighet vspets givet en viss randförskjutning, Fig. 3. Randförskjutningenkorrelerar med energifrigörelsehastigheten G, vilken i experiment kan styrasmed, och i praktiken varierar beroende på, initialsprickans längd a0. Vi serockså en god överrensstämmelse mellan simulering och experiment med av-seende på sprickvägar, men bara för det rena matrismaterialet. En förklaringantas vara att sprickan påverkas av mikrostrukturen i träfiberkompositen.

I Artikel II studeras hur närvaron av hål påverkar sprickans beteende genomatt numeriskt och experimentellt studera sprickpropagering i bitar av plexiglasmed förborrade hålstrukturer. Överrensstämmelsen mellan experiment och si-muleringar är generellt god, Fig. 4. I artikeln konstateras att sprickor sökersig till hål, vilket leder till förhöjda sprickspetshastigheter, men bara om myc-ket töjningsenergi finns tillgängligt i materialet, d.v.s. för korta initialsprickora0. I annat fall tenderar hålen att avskärma sprickspetsen så att sprickspets-hastigheten sjunker.

I Artikel III utvecklas en beräkningsmodell av trä på cellnivå. Därefter re-lateras numeriskt beräknade deformationer i en trämikrostruktur till experi-mentella observationer, utan hänsyn till brott. Simulering av deformationsfältunder en spets jämförs med deformationsfält från datortomografi av en fysiskspets i kontakt med trä och slutligen med resultat av slipexperiment. Med tankepå att metoderna är sinsemellan mycket olika är korrelationen förvånansvärtgod.

I Artikel IV kombineras fasfältsmodellen från Artikel I–II med cellskale-modellen av trä från Artikel III. Här simuleras brott i en trämikrostruktur, därmodellen skapas utifrån en mikroskopibild av verkligt trä. Inverkan av mittla-mellen visar sig vara betydande både vid låga och höga belastningshastigheter,

Figur 2. En skarp spricka Γ i en kropp Ω (vänster) representeras med en sprickdensi-tetsfunktion γ(d,∇d) som ett diffust skadefasfält d (höger).

48

0 0.2 0.4 0.6 0.8 1

vspets

/cR

0

2

4

6

8

10

G/G

c

PLA

Simulering

Experiment

Förgrening

0 0.2 0.4 0.6 0.8 1

vspets

/ cR

0

2

4

6

8

10

G/G

c

Träfiberkomposit

Simulering

Experiment

Förgrening

Figur 3. Normerad sprickspetshastighet som funktion av normerad energifrigörelse-hastighet för ren PLA (vänster) respektive träfiberkomposit (höger). Sprickspetshas-tigheten har normerats med Rayleigh-hastigheten för de respektive materialen.

Figur 4. Sprickvägar i experiment (vänster) och skadefältet d i beräkning (höger) förolika geometrier, uppifrån och ned: referensgeometrin, geometri 1, geometri 2 ochgeometri 3. Provbitarna i kolumn 1 och 3 har långa initialsprickor (a0 = 2 mm), de ikolumn 2 och 4 har korta initialsprickor (a0 = 0.75 mm).

49

Utan mittlamell Med mittlamell Kontinuum

Kva

sist

atis

ktv 0

=10

m/s

v 0=

20m

/s

Figur 5. Sprickvägar i kvasistatiska och dynamiska simuleringar. Här visas skadefältetd när den lastbärande förmågan har reducerats med 50 %. Vänster: ingen mittlamell(CML), mitten: med mittlamell och höger: kontinuumreferens. Övre raden: kvasisita-tisk simulering, mittenraden: v0 = 10 m/s och nedre raden: v0 = 20 m/s. Färgskalangäller alla bilder.

då skada lokaliseras till det svagare mittlamellmaterialet, Fig. 5. Även om detännu saknas mycket kunskap om mittlamellens egenskaper dras slutsatsen attmittlamellen är av central betydelse för modellering av brott i trä.

Slutligen i Artikel V studeras ett rent cellulärt material, med mikrostruk-tur liknande den i Artikel III–IV, men utan mittlamell. Syftet är att förstå hurregulariseringslängden l i fasfältsmodellen samspelar med mikrostrukturgeo-metrin, specifikt cellstorleken dc och en kritisk defektlängd ac. Det konstaterasatt lokalisering vid korta sprickor styrs av mikrostrukturens geometri, vilkendärför spelar en roll liknande regulariseringslängdens.

Genomgående visar avhandlingen på dynamikens betydelse. Dynamiskt (in-stabilt) brott skiljer sig ofta väsentligt från kvasistatiskt brott. Instabilt brott ärvanligt i heterogena material, och hänsyn måste därför tas till dynamiska ef-fekter.

Sammantaget visar avhandlingen på möjligheter med fasfältsmodeller förbrott för simulering av komplicerade geometrier, heterogenitet och dynamik,samtidigt som den också visar på begränsningar i nuvarande förståelse av t.ex.regulariseringslängden. Avhandlingen bidrar också till att öka förståelsen förbrott i heterogena material och visar att brott i heterogena material typisktskiljer sig från brott i homogena, kontinuerliga material.

50

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