Computational Modelling of Brittle Fracture

Computational Modelling of Brittle FractureStructural engineering preliminary research project

Submitted by:Prabhu Gajawada2102867G

Supervised by:Dr. Chris Pearce

Summary

Advanced gas cooled reactors (AGRs) use graphite bricks as moderators. The graphite bricks, by virtue of function of AGRs, are exposed to internal stresses due to the dimensional changes caused by irradiation, thermal effects and radiolytic reactions. The stresses disturb the structural integrity of the bricks and may lead to fracture in the graphite bricks. Several studies are being carried out by many researchers both in academia and industry to understand and estimate the crack propagation in these bricks. Due to irradiation damage, bombardment of atoms and various chemical and mechanical actions, the mechanical properties of graphite change over the thickness of the brick making it heterogeneous. Heterogeneity can potentially influence the crack morphology and characteristics. The main aim of the project is to understand the influence of heterogeneity on the morphology of the cracks by analysing using finite element method. This report is a literature review for the main project and hence covers various technical concepts required. A brief introduction about nuclear reactors in general and AGRs in specific is presented in the first section. The second section introduces the concept of linear elastic fracture mechanics (LEFM) which serves as a foundation for this study. Various aspects related to LEFM are discussed. Limitations of LEFM are also presented.To study the morphology of cracks, it is important to know how to model cracks in a finite element. Various types of modelling cracks in a finite element mesh are presented. Their applicability, advantages and disadvantages are also explained. The last part of the report presents about the research performed by Dr. Chris Pearce and his team so far in this subject. This is followed by a brief explanation of my dissertation objectives.

Table of ContentsSummary11. Nuclear reactors32. Linear elastic fracture mechanics32.1. History and a brief introduction of fracture mechanics32.2. Modes of failure42.3. Crack tip field52.3.1. Longitudinal shear, mode III52.3.2. Plane strain and plane stress, mode I and mode II72.3.3. Mode-I crack-tip field82.4. K-Concept102.5. Crack interaction102.6. Stress concentration factor and stress intensity factors112.7. Energy Balance122.7.1. Constant load condition132.7.2. Constant displacement condition142.7.3. Relaxation analogy to calculate the strain energy152.8. J-Integral172.9. Limits of LEFM193. Modelling cracks203.1. Discrete crack models203.2. Smeared crack models213.3. X-FEM244. Computational modelling of brittle fracture in graphite28Summer Project28Aim28Objectives29Methodology29Project plan29References31

1. Nuclear reactorsNuclear plants generate electricity by boiling water into steam which is used to rotate the turbines. The heat required to boil water is produced by a physical process called as nuclear fission. Nuclear reactors are classified into different groups based on type of nuclear reaction, type of moderator or coolant. Nuclear reactors which use graphite as neutron moderator and carbon dioxide as coolant are classified as advanced gas cooled reactors (AGRs). These reactors are the II generation of British gas cooled reactors. There are about 14 AGRs, all in The Great Britain [1]. The most important part of an AGR is its reactor core. The nuclear fuel components are present in this section and all the nuclear reactions take place here. A nuclear reactor needs a neutron moderator to reduce the speed of fast neutrons. AGRs use graphite as the neutron moderator. The graphite bricks are assembled to create cylindrical spaces in which the reactions take place. A typical reactor core is shown in figure 1.1.

Figure 1.1: Reactor core showing graphite bricks [24]The graphite bricks are irreplaceable and the life of the nuclear reactors depends on the life of these bricks. Due to nuclear reactions, the neutrons bombard the graphite causing impacts, irradiation damages, heat and radiolytic oxidations. These change the chemistry and physics of the graphite which results in variation of mechanical properties through the cross section of the brick. Internal stresses are induced in the graphite bricks and the structural integrity of the bricks is disturbed due to this. This leads to cracking. To understand the cracking process, knowledge of linear elastic fracture mechanics is required. This is discussed in the next section.2. Linear elastic fracture mechanicsThis section explains various concepts of linear elastic fracture mechanics (LEFM). Various modes of cracking, concept of stress intensity factors, Griffiths theory and concepts related to energy balance are explained. Later, limitations of LEFM are also presented.2.1. History and a brief introduction of fracture mechanicsFracture mechanics is an enduring field of structural engineering. There were several disastrous incidents in the past due to the adverse effects of fracture. The failure of liberty ships during second world war, Tacoma Narrows bridge in the United States, the Kings Street bridge in Australia are a few spectacular failures which occurred in the past [2]. There has been a great amount of learning that has been absorbed by engineers by post incident investigations after every disaster. The driving force for fracture is the loads at the crack tip (which is expressed as stress intensity factor) and the energy available at the tip of the crack [3]. Resistance to fracture is known as toughness [2]. Fracture cannot be ignored. As seen in the examples of some spectacular failures which occurred in the past, ignoring fracture could cause damage of property and more importantly, damage of life. It is very important for the designer to know about fracture mechanisms in various materials subjected to different kinds of conditions. Same material subjected to different conditions and stresses may fracture in different ways. 2.2. Modes of failureFracture is usually classified into two types- brittle fracture and ductile fracture. This doesnt mean that ductile materials fail in ductile manner. There can be situations where a material as ductile as steel can fail in brittle manner. The failure mode depends on the type of loading and the conditions in which the body is stressed. In a few circumstances, there may be limited plasticity at the tip of the crack which leads to failure known as quasi-brittle failure [4]. Propagation of crack(s) when the stresses near the crack tip are smaller than the yield stress of the material is ideally referred to as brittle fracture. In situations where there is extensive plastic deformation ahead of the cracks, it leads to ductile fracture. Cracks formed due to ductile fracture are usually stable as long as the stresses remain unchanged [5]. In continuum mechanics, when seen at a macroscopic level, a crack is a cut in a body. Its boundaries are the crack surfaces or simply crack faces. The end of the crack is referred to as crack tip or crack front.

Figure 2.1: Cracked body showing crack faces and crack front [6]Crack opening, based on its deformation style, is classified into three modes. When a crack opens symmetrically about xz-plane as shown in the figure 2.2, it is termed as Mode-I cracking. Crack formation due to antisymmetric separation or sliding of crack faces due to relative displacements in the direction normal to the crack front (x-direction), is termed as Mode II failure. Mode III failure is separation caused when the relative displacement of the crack surfaces in the z-direction tangential to the crack front, see figure 2.2 [6].

Figure 2.2: Three modes of crack opening [6]During the process of cracking, microscopically complex processes like bond breaking take place in the region close to the crack front. This region is known as process zone. These processes cannot be usually defined in terms of classical continuum mechanics. The size of process zone is very small and localised, in fact microscopic in most materials thereby making it impossible to be dealt with classical mechanics. For instance, most metals have a much localised microscopic process zone. In few cases like granular materials or in concrete, the size of process zones may be larger.In linear elastic fracture mechanics, as the name suggests, the material behaviour is considered linearly elastic. Any inelastic processes are restricted to microscopic level; they are neglected and are assumed linear. Linear elastic fracture mechanics principally applies to brittle fractures.2.3. Crack tip fieldConsider a situation where a body has a straight crack and the problem is two dimensional (2D). A very small circular region of radius R is only focussed upon. For this, a local coordinate system with its origin at the tip of the crack is introduced, as shown in figure 2.3. 2.3.1. Longitudinal shear, mode IIIAnti-plane shear or longitudinal shear is the simplest 2D problem. Mode III crack opening is a consequence of the only non-vanishing displacement component w, which is perpendicular to x,y-plane. The crack tip field is given by a function (z), which is complex in nature [6]. (2.1)

Figure 2.3: Crack-tip and the coordinate axes chosen [6]Here, A is a complex constant and is a real exponent. For the non-singularity of the displacement at crack tips and for finite strain energy, is assumed to be positive.

Figure 2.4: Cracked body in complex plane showing coordinate axes [6][6] demonstrates that the solutions of stresses and displacements of a longitudinal shear problem as shown in figure 2.2, can be completely obtained from two equations, 2.1 & . (2.2)

Where ij is the shear stress in ij-plane for all values of i and j and w is the displacement in y direction. Using z = rei in the equations 2.1 and 2.2, we get, equation 2.3(2.3)

Applying the boundary conditions which say that the crack faces are traction free, i.e. we arrive at a set of homogeneous equations as presented below.(2.4)

(2.5)

If the determinant of the system equals zero, non-trivial solutions exist. So this eigenvalue problem is solved as follows [6]:(2.6)

(2.7)

(2.8)

This way, the stresses and displacements can be determined in this kind of failure mode. Here, stresses and displacements are a function of angle ; w0 describes a rigid body displacement.The value of r tends to 0 as the crack tip is approached and the field can be described by the dominant terms in the equations for stresses and displacements mentioned above, see equations 2.7 and 2.8. These stresses and displacements are expressed as below (Corresponding to smallest eigenvalue of =1/2):(2.9)

(2.10)

It also shows that a singularity of type r-1/2 is observed in the stresses at crack tip. The factor KIII which is used to determine the singular crack-tip field is known as Stress intensity factor or K-factor. Here, the subscript indicates its mode of failure, in this case being mode-III crack opening. KIII is a measure of strength of the crack-tip field. KIII can be calculated if the stresses and displacements at the crack-tip are known. From equation 2.9, KIII can be written as below: (2.11)

Similar to the stresses and displacements, the magnitude of the stress intensity factor also depends on the loading and geometry of the body. 2.3.2. Plane strain and plane stress, mode I and mode IIThe stresses and displacements for plane strain and plane stress conditions are calculated analytically similar to the longitudinal shear case. The complex mathematic equations involved in this analytical solution are not presented here. The final equations of stresses and displacements are given below [6]:Mode I:(2.12)

Mode II:(2.13)

where,plane strain:(2.14)

plane stress:

KI and KII are stress intensity factors corresponding to mode I and mode Ii crack opening types. (2.15)

The above equations are based on an analytical method of finding the stresses in terms of stress intensity factors, material properties and location coordinates of the point where these stresses are calculated. The equations presented in 2.12 are very widely used in calculation of stresses. Though the equations are robust, there is a drawback; the stresses at the crack tip tend to infinity as the value of r at crack tip is zero.

2.3.3. Mode-I crack-tip fieldThe mode-I crack tip field is defined by the set of equations, 2.12. As mentioned, the stresses and hence, as per Hookes the strains have r-1/2 type singularities. As this is an inverse relation, they increase infinitely as the value of r approaches zero. For instance, the plot of y against x,r is given in the figure 2.5.

Figure 2.5: Crack-tip field for a mode-I opening, showing variation of stresses and displacement [6]On the other hand, displacements show r1/2 type behaviour and for a positive value of stress intensity factor, the variation of displacement with respect to x,r is parabolic, see figure 2.5.(2.16)

For a negative value of stress intensity factor, the crack faces penetrate each other which is not possible physically. Essentially, the crack faces during the closure of the crack are in contact. If the previous equations of stresses mentioned for a mode-I type of opening are transformed from rectangular Cartesian coordinates to polar coordinates, they look as below; see figure 2.6 and equations 2.17. This transformation is done to make the calculation of principal stresses simpler. (2.17)

Figure 2.6: Variation of stresses with angle. Inset: The direction of various stresses acting [6]Based on the above stress equations, principal stresses characterised by an angle , can be calculated using the formulae below (plane stress and plane strain):(2.18)

For plane stress condition, the third principal stress is equal to zero but for plane strain condition, it is not zero, but is given in equation 2.19. (2.19)

As per basic laws of stresses, maximum shear stress is equal to the half of the difference between the maximum and minimum principal stresses. Hence,plane stress:(2.20)

plane strain:

2.4. K-ConceptThis topic specifically is limited to Mode-I failure mode which is the most common mode of crack opening. The crack-tip field in mode-I crack opening is fully characterized by its stress intensity factor KI. Figure 2.7 shows that the crack-field determined by KI is dominant in a region of radius R outside which, the higher order terms get considerable and hence cannot be ignored making the validity of K-determined field limited.

Figure 2.7: K- concept, figure shows the K-determined region and the plastic zone [6]Thus, formulation of stress intensity factor is now explained. The fracture process in the material initiates when the value of this stress intensity factor reaches a critical value known as critical stress intensity factor KIC. The value of this quantity depends on the material. Hence, it can be stated that there exists a critical state in the process zone at which the fracture starts or crack propagation gets initiated. The critical stress intensity factor is often termed as fracture toughness. It is determined experimentally for different materials. The term stress intensity factor was first used by Irwin in 1951[7]. Stress intensity factors are determined based on many methods. All the concepts related to linear elasticity i.e. linear elastic displacements and stresses can be applied in determining K-factors. Stress singularities must be identified and the concepts must be tailored accordingly. When there is a requirement of closed form of solutions, the problems are solved analytically. With increase in the complexity of the problem, use of numerical methods may also be required. Most common numerical method applied is the finite element method. Finite difference method and boundary element methods are also sometimes used. Few special problems demand physical experimentation to determine the stress intensity factors, which may include strain measurement at crack tip or even photoelasticity [8]. 2.5. Crack interactionIn most cases of fracture, one has to deal with a set of cracks may be two, three or many cracks. The effect of one crack on another depends on the distance between them. If the distance between two cracks is large, the effect of one on another is little, but if they are close, the effects are significant. As a basic assumption, each crack has been treated individually and its behaviour and propagation is discussed in the previous sections. But when two or more cracks are close, depending on geometry, the interaction between these different cracks may significantly change the stress intensity factor, which may sometimes increase and other times, decrease. This phenomenon is known as shielding-effect or amplification. For these types of problems, exact solution is possible only in certain cases. Even numerical methods may be limited. Solution is exact if the number of cracks interacting is small. For instance, consider a system of collinear cracks which are interacting, the solution can be exactly found in this case. Researchers like Kachanov have proposed methods to study the interaction of system of complex cracks [9] (This method is not discussed in this report as it is beyond its scope).2.6. Stress concentration factor and stress intensity factorsVery high stresses are observed at notches and these stresses get higher with decrease in the radius of notch. At the root of a notch, stresses are always finite or non-singular, which is not the case at crack-tips. The increase of stresses at notches is called as concentrations. The ratio of the maximum stress observed at the root of the notch to the nominal stress acting is known as stress concentration factor. This hence, has no unit but is just a ratio. Research shows that there is relationship between the stress fields at root a notch and at crack tips [3]. An elliptical notch with major and minor axes 2a& 2b, is considered, see figure 2.8. This specimen is subjected to a uniaxial tension of . The maximum stress occurs at the apexes where the value of x is . This maximum stress is given as below [6]:(2.21)

Here, is the radius of curvature at the apex of the ellipse, whose value is equal to b2/a. As the ellipse becomes narrower, the difference between a and b increases. For a very narrow ellipse, b is much smaller than a and hence the above equation 2.21 can be approximated as below:(2.22)

Figure 2.8: Uniaxial tension action on an elliptical crack [6]If the value of radius of curvature is further reduced, i.e. the elliptical notch turns into a crack of length 2a. The maximum stress at its tip tends to infinity. To compare stress field at an elliptic notch to the crack-tip field, a magnified look at the root of the notch is taken. The notch boundary seems like a parabola and if the origin of the coordinate system is shifted to its focus, the equation of the parabola can be written as below, see figure 2.9:(2.23)

By the method of superposition (of two fields), the stress-field in mode I type of notch is evaluated [6]. The stress field at the notch is given by the sum of stress field at crack-tip in a 2D (plane stress or plane strain) body which is discussed before and a second field which counters the boundary tractions along the parabola to zero is given as equations presented below[6]:(2.24)

Figure 2.9: Parabolic notch [6]The maximum stress at the root of the notch is given by (2.25)

The value of the stress intensity factor at the notch root hence equal to(2.26)

The value of stress intensity factor can be obtained when max() is known [6]. 2.7. Energy BalanceThe concept of energy balance is a very important one in linear elastic fracture mechanics. This section discusses about the energy balance and Griffiths theory which is a benchmark theory for brittle fractures. When a crack advances in a component, the following changes occur [10]:i. Stiffness of the component changesii. Strain energy in the component changes (may increase or decrease)iii. The points on the component where external loads are applied may or may not move; if these points move, external work is doneiv. Energy is being consumed to create two new surfaces (due to cracking)The key aspect of Griffiths approach to fracture mechanics is it was recognised that energy is absorbed in the formation of two new surfaces when a crack is formed [10]. There is inherent resistance persisting in material and formation of crack requires two new surfaces to be formed which consume energy. This energy has to come from some source. For the purpose of analysis, two extreme conditions are looked at- constant load and constant displacement conditions. 2.7.1. Constant load conditionConsider a situation of crack advancement in a component under a constant load. Say the initial crack length is a and it has advanced to a+da and the displacement due to P has changed from v to v+dv. The load displacement curve of this scenario is presented in figure 2.10.

Figure 2.10: Crack advancement under constant load- experimental setup [11]

Figure 2.11: Load-displacement curve Constant loadFrom the load displacement curve, Change in strain energy ( ) due to change in displacement from v to v+dv is:(2.27)

External work done (W) is given by:(2.28)

Hence, in the case of constant load,(2.29)

2.7.2. Constant displacement conditionConsider a situation in which a component undergoes crack advancement under constant displacement. The crack advancement takes place due to change in load from P1=P to P2=P+dP. The load displacement curve of this scenario is given in figure 2.11. When the crack length changes from a to a+da, the stiffness decreases, see load-displacement curve.

Figure 2.12: Load displacement curve- Constant displacementFrom the load-displacement curve for constant displacement condition, figure 2.12, the change in strain energy ( and external work done (W) are written below:(2.30)

(2.31)

To calculate the strain energy in the presence of a crack, the following methods are used usually [11]:i. Dimensional analysisii. Relaxation analogyiii. Actual calculations based on crack face displacement; this requires knowledge of stress fields and displacement fields.For a central crack (crack with two tips) in an infinite panel with unit thickness, the strain energy is given by Ua:(2.32)

where, is the nominal stress acting, E is the Youngs modulus and 2a is the length of the crack. This formula is derived based on relaxation anology.2.7.3. Relaxation analogy to calculate the strain energyConsider a plate of thickness B which is stretched between fixed grips as shown in figure 2.13. Since the grips are fixed, there is no external work done on the system. A central crack of length 2a is introduced in this plate. It is assumed that the strain energy is released from two triangular regions in the neighbourhood of the crack as shown in the figure 2.13. This strain energy released is used in the formation of two new surfaces.

Figure 2.13: Central crack and the triangular regions which release strain energyThe strain energy released is equal to:

(2.33)

Here, is a proportionality constant which is equal to /2 for thin plates idealising it as a plane stress problems [11]. Hence for a thin plate of thickness B, the strain energy in the presence of a crack is given by(2.34)

This shows that the strain energy is proportional to a2. If s is the surface energy per unit area of one surface, the surface energy required (Us) to create two new surfaces in a model of thickness B in this case of fixed grips or constant displacement is hence given as below:(2.35)

From equation 2.35, it is clear that the surface energy is a linear function of the crack length. Applying energy balance, change in strain energy is equated to change in surface energy per crack length. This is represented below:(2.36)

The expressions for Ua and Us are already given. So plugging these values from 2.34 and 2.35 in 2.36, we get

Hence,

(2.37)

Or, it can also be written as

(2.38)

The above is the kernel of Griffiths approach to fracture mechanics which states that the formation of two new surfaces requires energy and this energy comes from either the external work done, or the strain energy released or both. In this case of fixed grips, the external work done being zero, the strain energy released is used in the formation of two new surfaces. Griffiths theory which is based on energy balance is widely used in analysing crack-tip field problems.2.8. J-IntegralJ- Integral is a very powerful parameter in LEFM. J-integral facilitates in calculation of energy release rate or stress intensity factors in a material. This section explains what a J-integral is and how it is useful in solving an LEFM problem. J-integral is a parameter which is equivalent to the stress intensity factor or the energy release rate for linear elastic materials under quasi-static loading. The theory of J-integrals can be applied to inelastic materials also [6]. An elastic body whose strain energy density is U(ij) is considered. It is assumed that there are no volume forces acting on this. For the sake of simplicity, deformations are considered to be very small which means strains are infinitesimally small. The J-integral vector for this system is given as below [6]:(2.39)

In the above equation, u is the displacement, is a closed surface which has nj as its unit normal vector pointing outwards, this is demonstrated in figure 2.14 a. The energy momentum tensor or configurational stress tensor, bkj is given as below [6]:(2.40)

Figure 2.14: J- Integral [3]In accordance with the theory of divergence, the value of J-integral vector is zero for a closed surface dV if there are no defects in the material and bkj has no discontinuities [6]. A discontinuity AD is introduced in the surface V as shown in figure 2.15 and traction forces ti=ijnj are acting on the boundary of this elastic body. Now, this boundary which is along AD is moved by dsk keeping the tractions unaltered. Due to this displacement, the change in total internal energy of the system is given by:(2.41)

Figure 2.15: Generalised forces [6]The change in internal energy is the amount of strain energy stored in the layer of thickness equal to the dsknk. The value of change in external potential energy is given as below [6]:(2.42)

The above equation 2.42 is the external energy which is external force multiplied by displacement caused due to that fore. Hence, from above equations, we deduce the total change in potential energy as below:(2.43)

As the value of the above integral over a closed surface is zero, it can be written as (Book):(2.44)

Comparing equations 2.39 and 2.44, dJk can be written as:(2.45)

This result is not just valid for this particular case but for any arbitrary discontinuity surfaces. The major aspect of J- integral is that its value is path-independent; independent of the path of integral taken around the crack. Jk is often termed as generalised force or configurational force [6]. The equation 2.45 also means that the value of J-integral is equal to the derivative of total potential energy with respect to crack length. The energy release rate during cracking is precisely this and hence, J is equivalent to energy release rate G during cracking. J integral can also be related to stress intensity factors by applying the laws of LEFM. (2.46)

The concept of J-integral hence craves out a way to calculate the strain energy release rate or work done per unit fracture area in a material. This can be used to solve various kinds of cracking problems. Path-independence and applicability to inelastic materials make J-integrals very important.

Proof: J- integral is path independentConsider a crack in a body and a closed path 1, +, 2 and - around it as shown in the figure 2.16. For paths + and -, the value of J-integral is zero as d(x2)=0 and since crack surfaces are traction free, ij=0. Since the contour considered is closed and the J-integral is calculated around the contour, its value is 0. If J1, J+, J2 and J- are J-integrals corresponding to the paths 1, +, 2 and - , J+ and J- are equal to 0. (2.47)

This shows that the value of J-integral calculated on the paths 1 and 2 are equal and hence J-integral is path-independent.

Figure 2.16: J-integral- path independence 2.9. Limits of LEFMLEFM though a very robust and powerful method to analyse and solve crack propagation problems, has a few limitations. LEFM assumes that the size of the plastic zone is very minute compared to K-dominant region. This means that the yielding s limited to a small zone. This is known as small scale yielding. The deviations of the material response from linear elasticity are observed only in the plastic zone, which is shown in figure 2.7. Estimation of the size of this plastic zone is complex. An approximate estimate of the plastic zone is obtained by solving the crack-tip field problem for mode-I by assuming ideal plastic material behaviour. LEFM fails when the material behaviour is not elastic, most often in materials which exhibit significant plastic behaviour (Ductile or stable crack growth) [6]. LEFM is very limited in cases where there is fatigue crack growth. Fatigue crack problems usually arise due to thermal-mechanical or purely thermal stresses. The materials used in these problems typically have high fracture toughness and lower strength. These materials undergo varying stresses which may exceed the yield strength. Fatigue problems may potentially arise in thick walled pipes, nozzles, turbine casings, boiler vessels, nuclear reactors and many more. LEFM is extremely limited in these applications. LEFM is also limited when sustained loads are applied and in conditions where creep is significant [12].Size-effect is another key issue which limits the applicability of LEFM. Consider a huge concrete block, the fracture phenomenon in this block takes place in the cement-aggregate interface. In this case, the size of process zone depends on size of aggregate. So, in a huge concrete block, the size of the process zone is so small compared to the overall dimensions of the structure making LEFM applicable. But in contrast to it, the same concrete block with much smaller dimensions would limit the applicability of LEFM. This phenomenon is known as size effect. 3. Modelling cracksCracks are modelled in several ways in a finite element. The type of modelling is decided based on the type of problem. In general, to model a single dominant crack or a few dominant cracks, discrete crack models are used; to model diffused cracks, smeared crack models are used. There are other modelling methods like X-FEM and cohesive zone models. X-FEM is a method to model cracks independent of the mesh. Various methods of modelling cracks are discussed in this section. Their applicability, advantages and disadvantages are also presented. 3.1. Discrete crack modelsIn discrete crack approach, a crack is introduced in the finite element mesh as a geometric entity. The crack grows when the nodal force at the tip of the crack exceeds the set strength of the material. The node splits into two and the crack propagates. The crack tip shifts to the next node. When the strength criterion is violated at this node, the crack propagates further. Discrete crack approach is demonstrated in figure 4.1. Discrete crack modelling is very useful to model single or few dominant cracks. Advantages of this approach are its simplicity and ease of use. On the flip side, discrete crack approach has several drawbacks. The cracks have to propagate only through element boundaries which in turn introduce mesh bias [14]. Ingraffea and co-workers have developed various remeshing codes to diminish, if not to remove the mesh bias [15]. Discrete crack approach by the virtue of its process has a computational difficulty which involves continuously changing the topology [14]. Other difficulties with this method of modelling cracks include requirement of large computational resources with finite element methods, difficulties in 3D crack modelling and the way in which the support of a node is changed during crack propagation (Support node of a crack is explained in the section 3.3.)[16]. In addition to this, there has to be a background mesh to ensure correct calculations of forces and stiffness matrices. All these disadvantages together made the applicability of this method limited.

Figure 3.1: Discrete crack modelling3.2. Smeared crack modelsOne of the most common ways of modelling fracture is by smeared crack models. This method comes very handy when the cracks are diffused. In smeared crack modelling, the nucleation of cracks occurs in a small volume that is certain to an integration point. This is transformed into a deterioration of existing strength and stiffness of that integration point. Crack is initiated when the stresses satisfy particular conditions (e.g. crack initiates when the maximum principal stress exceeds the tensile strength of the material).This demonstrates that the isotropic stress-strain variation is replaced by elastic orthotropic relation [14]. The methodical explanation of smeared crack modelling is presented in this section. Due to localised deformations in the cracked regions, the total strain ( is separated into continuum part () and inelastic part () [17]. (3.1)

As per Hookes law, the continuum part of strain is related to the stress by elasticity matrix (Dco); shown in equation 3.2.(3.2)

The relationship between stress traction acting on the fractured plane () and the crack strain ) is given as below:(3.3)

is the elasticity matrix corresponding to the cracked part. These parameters are expressed in local coordinate system of the crack. The localised deformation w is transformed to a cracking strain. To do this, w is divided by ws, the width over which the crack is smeared out [17]. Local cracking strain discussed is transformed to global coordinate system by multiplying it with a transformation matrix. The traction is also expressed in terms of the continuum stress; equations 3.4 and 3.5. (3.4)

(3.5)

In cases where there are more than one crack occurring concurrently, the strains and tractions of each of these cracks are assembled in single vectors- traction vector and strain vector. Similarly, one transformation matrix is compiled for all the cracks. The constitutive relation mentioned in equation 3.2 hence remains unaltered for multi-crack scenario. These kinds of models are generally considered to have fixed crack directions and positive incremental shear stiffness. The principal stresses rotate after the fracture causing large shear stresses in a crack. This is one of the major disadvantages of smeared cracked models. One remedy for this problem is to rotate the crack direction along with that of principal stress. This approach may not seem practical as real cracks dont rotate this way but the mean principal direction of microcracks can rotate during their initiation. There is another shortcoming of smeared crack approach which is strain localisation. This is discussed in the following section.

Problem of strain localisationStrain localisation is a major setback to the smeared crack modelling. To explain the problem of strain localisation, a bar under uniaxial tension as shown in figure 3.2 is considered. The bar has a constant area of cross-section of A and a length of L fixed between two supports. The material behaviour is assumed to be initially linearly elastic followed by linear softening. Hence, the stress strain curve for this bar when it is tensioned uniaxially is plotted as in figure 3.3a. The stress increases uniformly till a strain of , which is equal to , where ft is the tensile strength of the material and E is its Youngs modulus. After this peak, the material softens linearly. The response of the beam is linearly elastic up to a strain of which means a displacement. If the method is displacement controlled, the resistance of the beam decreases with any further increase in displacement after. Due to constant cross-section, the stress distribution is constant. This doesnt mean that the strain is continuous along the bar. For every stress value, there are two strains possible, one on linear elastic part and other on the linear softening part, see figure 3.3b. Any strain distribution which varies as a step function with values between u and s is valid; shown in figure 3.4. Length of the bar along which strain is equal to s is Ls and the length where strain is u is Lu. Hence, total length L = Ls+Lu. When the bar reaches complete failure (separated into two parts), the stress in it is zero and the strain equals final strain f. Hence, the total elongation in the bar when stress is zero is given in equation 2.6. The strain in the unloading region, u = 0. (3.6)

The value of Ls may be anything between 0 and L, which gives countless load-displacement curves for this case, each being a valid solution. This problem mathematically is ill-posed. One solution for this problem is that the whole bar is unloading elastically, and second solution is that the whole bar is softening [17]. This problem can be overcome by considering imperfections. It is assumed that the tensile strength of a part of the bar is slightly lesser that the rest over a region of length Ls. When the applied load reaches the reduced tensile strength, the material in this weaker region starts to soften and the stresses begin to decrease. Elastic unloading is observed in the material which is outside this weak region. This means that the size of softening region is related to the size of weak zone. This region might be so small that the softening part of the equilibrium path can be close to linear elastic one.

Figure 3.2: Bar of length L under uniaxial tension [17]

Figure 3.3: a) Stress-strain curve b) Strain values corresponding to a particular stress [17]

Figure 3.4: Uniaxial damage represented as a set of parallel fibres which are failing at different strain values [17]This leads to another problem known as mesh dependence, in finite element modelling. Assuming that the localised solution is properly captured, the softening region is confined to one element which corresponds to the weak zone in the bar. The post peak response of member depends on the total number of finite elements considered. As the number increases, the post peak response tends towards the initial elastic response as shown in figure 2.9. Here, the load-displacement curve depends on the level of discretization. To overcome this problem, the stress-strain curve is made dependent on the discretization making load-displacement curve independent of it [17].

Figure 3.5: Effect of discretization on the results: a) Load- displacement curve b) Strain profiles [17]

3.3. X-FEMX-FEM or extended finite element method is a numerical method that is tailored to treat discontinuities. Discontinuities are usually classified into strong and weak. The discontinuities in the solution variables of a problem are known as strong discontinuities, e.g. displacements. The discontinuities in their derivatives are known as weak, e.g. strains. X-FEM is a powerful technique used to model cracks or displacement discontinuities. In an XFEM model, the crack is not represented by the mesh boundaries, it rather crosses the elements. This section explains how X-FEM is governed and how the discontinuity is made mesh-free.In order to represent a crack in a finite element mesh, the nodes have to be placed along the crack and on the crack-tip. This ensures explicit delineation of the geometry, see figure 3.6.

Figure 3.6: Explicit representation of crack in a finite element mesh [20]Crack-tips are usually associated with asymptotic fields and hence mesh needs to be refined near cracks. With the propagation of crack, the mesh has to be modified accordingly. In many cases, the results of the problem need to be correlated from one mesh to other when mesh changes as a result of crack advancement. This increases both computational time and cost. X-FEM is a remedy to this problem. X-FEM is based on the concept of partition of unity. Partition of unity is a set of functions that satisfies the following:(3.7)

Interpolation functions in finite element analysis satisfy this condition. Say N1, N2,.. are interpolation functions in a finite element, . The displacement at a point in the mesh is given by:(3.8)

Here, ui is the displacement at ith node and Ni is the interpolation function corresponding to that node. X-FEM uses certain special functions called enrichment functions in order to introduce the notion of discontinuity. By incorporating the enrichment functions in the equation 3.8, it gets modified to:(3.9)

Here, pj represents the enrichment functions, ui are the finite element degrees of freedom and aij are the extra degrees of freedom [18]. The type of enrichment function and the parts of the approximations that have to be enriched have to be determined. To demonstrate the use of enrichment functions and overall idea of XFEM, a simple example is considered. Mesh 1 is a finite element mesh which has a crack or a discontinuity as shown in figure 3.7. Mesh 2 is a crack-free continuous finite element mesh. The objective is to represent the former in terms of the latter plus some enrichment terms.

Figure 3.7: Mesh 1- with discontinuity, Mesh 2- without discontinuity [21]

The finite element approximation of the first mesh is(3.10)

The displacements terms, u9 and u10 can be expressed in terms of a and b where and . Hence u9= a+b and u10=a-b. By replacing the 9th and 10th displacement terms in the equation 3.10, we get(3.11)

The function H is to represent the discontinuity or jump. It takes the value of 1 for all values of y>0 and -1 for the values of y