Mesh scalability in direct finite element simulation of brittle fracture

$Page 1: Mesh scalability in direct finite element simulation of brittle fracture$
Engineering Fracture Mechanics 75 (2008) 4066–4084

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Engineering Fracture Mechanics

journal homepage: www.elsevier .com/locate /engfracmech

Mesh scalability in direct finite element simulation of brittle fracture

Antonio Caballero *, Arcady DyskinSchool of Civil and Resource Engineering, M051, The University of Western Australia, 35 Stirling Hwy, Crawley, WA 6009, Australia

a r t i c l e i n f o

Article history:Received 18 May 2007Received in revised form 31 January 2008Accepted 13 March 2008Available online 31 March 2008

Keywords:Singular stressNon-singular stressScalingSingularity exponent

0013-7944/$ - see front matter � 2008 Elsevier Ltddoi:10.1016/j.engfracmech.2008.03.007

* Corresponding author. Tel.: +61 8 6488 3987; faE-mail addresses: [email protected] (A

1 See Rossmanith [2] for a detailed overview of th

a b s t r a c t

A new approach of dealing with mesh dependence in finite element modelling of fractureprocesses is introduced. In particular, in brittle fracture modelling, the stress concentrationis mesh dependent as the results do not stabilise when refining the mesh. This paper pre-sents an approach based on the explicit incorporation of mesh dependence into the com-putations. The dependence of the relevant stress is quantified on the finite elements at thecrack tip upon the element size; when the dependence approaches a power law with therequired accuracy, the mesh is called scalable. If the mesh is scalable and the exponentand pre-factor are known, then the results of the computations can be scaled to the sizerelevant to the scale of the physical microstructure of the material; the latter while notbeing modelled directly ultimately controls the fracture propagation. To illustrate thisnew approach, four 2D examples of a single straight crack loaded under tensile and sheartractions applied either to the external boundary or to the crack faces are considered. It isshown that combining the stresses at the crack tip computed using a set of similar meshesof different densities with the crack tip asymptotic allows accurate recovery of the stressintensity factors.

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1. Introduction

1.1. Some historical approaches to fracture mechanics

Fracture Mechanics or Linear Elastic Fracture Mechanics as we know it, was originated by Wieghardt1 [1] and Inglis [3].Both independently showed that cavities and flaws in continuum materials act as stress concentrators which, in the limit ofsharp edges (cracks), produce infinite stress at the tip. There are two main approaches to deal with the unphysical unboundedstress obtained from elastic solutions.

The first approach is essentially based on the fact that the stress singularities at the crack tip, as well as at the tip of awedge are expressed in terms of a power function [4] i.e. as self-similar distributions. Thus, the focus was on expressingthe criteria of the initiation and direction of crack growth through scale invariants such as the fracture energy [5] or thestress intensity factors, or equivalent [1,6]. The direction of crack growth is determined by these invariants while the criteriaof crack growth include experimentally determined critical values of the invariants themselves or functions thereof. Thesecritical values incorporate (and conceal) the details of the mechanical behaviour of the material at the crack tip as well as thefine geometry of the crack tip (these are the factors making the stress distributions bounded thus limiting the applicability ofthe theory of elasticity).

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x: +61 8 6488 1044.. Caballero), [email protected] (A. Dyskin).

is unjustifiably forgotten work.

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Nomenclature

a semi-length of the crackCI normalized stress factor in mode ICII normalized stress factor in mode IId characteristic size of the averaging areah finite element sizeKI mode I stress intensity factorKII mode II stress intensity factorL size of the problem domainp applied normal loadr radial distance from the crack tipSd area of averaging in finite element meshS0

y constant stressS1

y constant stressS0

xy constant stressS1

xy constant stressb size coefficient (dy = b � dx)rasympt

y asymptotic stress along y direction (y-axis perpendicular to the crack)rasympt

xy asymptotic shear stress (y-axis perpendicular to the crack)raver

y averaging stress into area (d � bd) along y direction (y-axis perpendicular to the crack)raver

xy averaging shear stress into area (d � bd) (y-axis perpendicular to the crack)h polar angle

A. Caballero, A. Dyskin / Engineering Fracture Mechanics 75 (2008) 4066–4084 4067

Another approach is based on the introduction, in greater or lesser detail, of the fine aspects of the crack tip zone. Thesimplest possibility is to average the stress singularity over a certain length [7,8]. Then the criterion of crack propagationincludes two parameters: the length of the zone of averaging and the local strength of the material. More sophisticated mod-els involve specifying the details of non-linear constitutive behaviour of the material near the crack tip [9–11].

The introduction of the models of fine structure of the material near the crack tip zone puts the limitations to the appli-cability of the elastic solution: it is valid for distances much greater than the size d of the zone. This restriction demotes theself-similar stress distribution at the crack tip to the status of an intermediate asymptotic only valid for distances r from thecrack tip in a range d� r� L, Fig. 1, where L is the characteristic size of the crack or the load distribution on it [12].

If this range of validity exists (when the zone of non-elastic behaviour is sufficiently small) then the basic fracturemechanics associated with the self-similarity concept can be used to model, in the first approximation, the condition of crackpropagation and its path.

Despite the simplifications offered by the self-similarity of singular stress field near the crack tip, the actual computationof the characteristics of the stress singularities becomes quite complex as soon as the crack geometry differs from simpleststraight and disc-like cracks. In the 1960s, Ngo and Scordelis [13] gave the starting shoot for one of the most fructiferousresearch combination, the application of the finite element method to the fracture mechanics thus capitalizing on the con-siderable flexibility of the method in tackling various fracture mechanics problems. After that pioneering paper, many otherworks came up establishing the finite element method as a standard tool in the analysis of fracture mechanics problems.

The finite element method adds another characteristic size to the problem – the size of the finite element. The reductionof this size is restricted by the available computational resources and for that reason, in some situations the finite elementsmay turn out to be considerably larger than the physical size d. In addition, it was relatively soon noted that as the method is

d

Lr

Fig. 1. Zone of applicability of the elastic solution: d� r� L.

4068 A. Caballero, A. Dyskin / Engineering Fracture Mechanics 75 (2008) 4066–4084

usually based upon the approximation of the displacement and/or stresses by polynomials, it is not possible to obtain anaccurate representation of the behaviour near to the singularity. To overcome this awkwardness some authors used to refinemore and more the finite element meshes near the crack tip [14–16]. This method however proved to be too expensive interms of computing time and data management.

A solution to this problem was found in the development of special finite elements to be placed around the crack tip[17,18]. These new elements, so-called singular elements, incorporated the singularity and thus restored the local self-sim-ilarity of the solution at the crack tip (under the assumption that the non-elastic zone at the crack tip can be neglected – thebrittle crack concept). Some other works proposed minor changes to this approach, as for instance the method by Henshelland Shaw [19], in which the ‘mid-side’ node of a standard eight-node isoparametric element is displaced from its nominalposition. As a result, the singularity occurs exactly at the corner of the element.

A different way to recover self-similar stress distribution is realised in the Fractal Finite Element Method (FFEM). Insteadof introducing special elements, the method proposed by Leung and Su [20,21] splits the domain into two regions: regularand singular. In the regular region, the conventional Finite Element Method (FEM) is used. The singular region is subdivided,following a self-similarity rule, into a large number of conventional regions where FEM is also used. However, the resultinglarge number of nodal variables is decreased by means of a global interpolating transformation in which the stress intensityfactors (SIFs) are the primary unknowns. The method has been successfully applied to compute the mode I SIF, the mixedmode [22] and mode III [23] SIFs as well as dynamic loading in mode I [24].

Another direction in the computational fracture mechanics – the eXtended Finite Element Method – was recently intro-duced by Belytschko and Black [25]. The XFEM is based on the local partition of unity (PUM) [26,27]. In this method, thedisplacement approximation is enriched by additional functions specific only to the regions near the crack tip (or disconti-nuities of other types such as material interface, etc.). As in the case of PUM, those specific functions can have the form otherthan the regular polynomial functions used in the regular FEM. For instance, when dealing with crack problems, the methodcould incorporate the self-similar near-tip asymptotic, which enables the domain to be modelled by finite elements withoutremeshing [28]. In this method the sparsity and symmetry of the resulting stiffness matrix are retained. The model waschecked for the cases of strong discontinuities [29] as well as weak discontinuities [30]. The above methods may recoverthe scale invariants, the stress intensity factors, throughout the known type of stress singularity.

In this paper we propose a methodology of dealing with mesh dependence in elastic brittle fracture mechanics modelling.The method is based on the determination of scaling properties of the model with respect to the element size with the viewto capitalise on the advantages of fixed density meshes. We illustrate the proposed approach using very simple examples forwhich benchmark solutions are known: straight 2D brittle cracks under tensile or shear loading. We will use the proposedmethodology to recover the stress intensity factors from FE computations with different mesh densities. The simplicity of theproblem for this approach comes also from the fact that by the virtue of the assumption of brittleness, the problem does notpossesses a microscopic length which could otherwise be for instance related to the process zone; subsequently the chosenfinite element size represents the finest scale in the modelling. As we are only interested in the scaling properties, we keepthe finite elements strictly of the same type such that the only parameter of the mesh is the element size.

2. Concept of scale-invariant mesh density

Attempts to model fractures explicitly in finite elements lead to mesh dependence in the way that the solution of thecomputations (in terms of strain and stresses) at the crack tip for successive refined meshes does not stabilise. The meshdependence is caused by the fact that the explicit modelling of fractures introduces a singularity in the stress and strain fieldalong any radial direction from the crack tip. However, it is also known that this singularity far of being random is propor-tional to K=

ffiffiffirp

where K is the SIF and r the radial distance from the crack tip (see Fig. 1). As mentioned in Introduction therehave been many methods which treat that singularity and/or mesh dependence of the solution near the crack tip on the sizeof the finite element. Here another methodology is proposed to overcome difficulties associated with mesh dependencewhich has been called: ‘The scale-invariant mesh density’. Instead of fighting the loosing battle of mesh-dependence we pro-pose to incorporate it in the computations. The method is based on the assumption that as the characteristic element size (ora similar characteristic length associated with the numerical method), h, tends to zero (the mesh is presumed to change in aself-similar manner), all mesh-dependent quantities should change as power functions of h, meaning that a fine enoughmesh becomes scale invariant (this type of scale invariance of the effective process zone length was confirmed by Pantand Dyskin). This is a generalisation of the conventional notion of mesh independence in which case the scaling exponentis zero. The proposed mesh scaling can be made to serve two purposes. On the one hand, the mesh can be considered suf-ficiently refined when the mesh-dependent quantities fit the power law (this is easy to check by fitting a linear regressionline on a log–log plot), meaning that this mesh density is acceptable from the computational point of view. On the otherhand, the obtained exponents can be used to extrapolate the mesh dependent quantities to the actual (physical) microstruc-tural sizes2 (if they are known) or to determine these sizes by back analysis.

2 The actual microstructure is not modelled directly, but the finite elements themselves can be viewed as artificial structural elements somewhatrepresenting the real ones. If the real microstructural element are too small and using that many finite elements is computationally prohibitive then the meshscalability can be used to extrapolate the mesh-dependent computational results down to the actual size of the physical microstructural elements.


It should be noted that our method differs from the methods that utilise the scale invariance of the energy release rate,which is the main term in the elastic energy variation associated with a small step of crack propagation as thus independentof the length of this step. This property was used to infer the stress intensity factors without resorting to special finite ele-ments. This was done by calculating the energy change due to virtual crack propagation, which, in terms of finite elementmethods corresponds to crack propagating at the length of one element [32,33], or by direct calculations of the energy re-lease rate [34]. The scale invariance of the energy makes this method mesh-independent. When the stress intensity factors oftwo modes need to be determined, the crack extensions in two different directions are needed [35,36].

The proposed approach cannot be considered as belonging to the class of multi grid methods [37] either, despite somesuperficial resemblance since in our method the computations with different mesh densities are independent; rather thanlinked in a process of iterative refinement. On top of that, our method relies upon linking the scales through a priori knownasymptotic relations.

In essence, the proposed method extracts the scale invariants through the use of meshes of different densities (i.e. by ana-lysing the change of the quantities of interest with the element size), which will be shown to be more efficient than treatingthe stress singularity as a function of the distance from the crack tip.

3. Finite element model of a crack without singular elements

3.1. On recovering parameters of stress concentration from direct numerical stress computations

As indicated in Section 1, the classical representations of crack growth criteria are based on the scale invariants which arecharacteristics of the main asymptotic terms of the elastic solutions at the crack tip. The determination of the asymptotic andthe pertinent invariants, require the possibility of approaching the crack tip close enough in comparison to the minimumcharacteristic size of crack and/or load distribution. In numerical implementation, however, this possibility is limited bythe presence of another characteristic size which reflects the discrete nature of numerical approximation for instance themesh element size in the finite element method. In order to highlight the consequences of this restriction, we consider asimple 2D example in which a straight crack of length 2a is uniformly loaded, Fig. 2. Let the element size be h and considerthe normal stress ry at the x-axis which is the line of crack continuation. We assume that the distance from the crack tip islarger than the size of the element, r P h, and we focus on the interval h 6 r 6 d� a where one can expect the asymptotics

rasimpty ¼ K Iffiffiffiffiffiffiffiffi

2prp ð1Þ

to represent the complete stress distribution with a sufficient accuracy.For the crack of Fig. 2 the stress field is well known. For instance, at the crack continuation the stress has the form [38].

ryðxÞ ¼pxffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

x2 � a2p : ð2Þ

As a model of numerical computations with a characteristic size h we assume an average of (2) over the length h:

ravery ðx;hÞ ¼ 1

h

Z xþh

xryðxÞdx ¼ p

h

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffixþ hð Þ2 � a2

q�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 � a2p� �

: ð3Þ

In order to get an impression of the accuracy with which the square root asymptotic can be extracted from values (2) weconcentrate on the exponent (the singularity exponent) and see what element size is needed to recover the exponent valueof �0.5 within a sufficient tolerance. We note that if at a point of the mesh, x = a + r, (3) is well approximated by a powerfunction, then the exponent should be

x

y

a

r

p

p

Fig. 2. The model: a 2D straight crack.

0 0.02 0.04 0.06 0.08-0.5

-0.4

-0.3

r/a

α

h=0.01a

h=0.001ah=0.0001a

h=0.00001a

Fig. 3. Dependence of the singularity exponent (4) for the average stress vs. the distance from the crack tip for different averaging sizes (modelling theelement sizes in numerical computations).


aðxÞ ¼ln raver

y ðaþ r þ h;hÞ � ln ravery ðaþ r;hÞ

lnðr þ hÞ � lnðrÞ : ð4Þ

The results for different values of h are plotted in Fig. 3. It is seen that in order to get close to the correct value of singu-larity exponent (�0.5) one needs to choose a very small mesh size of h = 10�5a (100,000 elements per half crack length). Itwas actually this property that made it impractical to perform the explicit computations and the direct extracting of theasymptotics (1) and brought about the introduction of special elements of various kinds.

In the spirit of the discussed methodology of mesh scalability we can suggest another method of recovering the singular-ity exponent. Indeed, according to (1), for different averaging lengths h the average stress at the crack tip should scale as

hrasympty iðhÞ ¼ K I

ffiffiffi2pffiffiffiffiffiffiphp : ð5Þ

Subsequently, after performing computations for different values of h, the singularity exponent that corresponds to scalingthe mesh can be recovered as

ascale ¼ln raver

y ða;2hÞ � ln ravery ða;hÞ

ln 2: ð6Þ

The result, as a function of h, is shown in Fig. 4. Already for h = 0.01a we have ascale = �0.496 which is much better than thevalue a = 0.4914 obtained by choosing h = 0.0001a in the method (4) and close to a = 0.4972 obtained with h = 0.00001a.

Fig. 5 shows the reconstruction of KI by fitting the singular term (5) to the values (3). It is seen that choosing h = 0.01agives an error in KI determination of 0.25%.

3.2. The inclusion of further terms of asymptotic expansion

The asymptotics (1) is based on the notion that as r ? 0 the singular term supersedes the non-singular part of the stressfield. However, as soon as a finite characteristic length, h, is involved, the influence of the non-singular part can be essential[39]. This is especially important in numerical modelling given the natural desire to keep h as large as possible (for instancein order to reduce the computational time). In this case, the account of further asymptotic terms might be of benefit. Weinvestigate the possible benefit by considering a three term asymptotic expansion of the Williams’ series:

0 0.02 0.04 0.06 0.08-0.5

-0.49

-0.48

-0.47

h/a

α

Fig. 4. Dependence of the singularity exponent (6) for the average stress vs. the element size.

0 0.02 0.04 0.06 0.081.76

1.77

1.78

1.79

1.8

1.81

h/a

ap

KI

Determination of SIF using the singular expression (5)

Determination of SIF using three terms

Exact value π

Fig. 5. Reconstruction of KI using the expressions containing only the singular term (5) and three asymptotic terms (8).


rasimpty ¼ K Iffiffiffiffiffiffiffiffi

2prp þ s0

y þ s1y

ffiffiffirpþ Oðr3=2Þ: ð7Þ

The presence of the constant stress s0y is controlled by the type of loading: for instance, for the loading shown in Fig. 2 this

stress is zero, while if the uniform load p is applied directly to the crack faces, s0y ¼ �p.

After the averaging, we arrive at the following dependence of the average stress on h:

hrasimpty iðhÞ ¼ K I

ffiffiffi2pffiffiffiffiffiffiphp þ s0

y þ S1y

ffiffiffihp

; ð8Þ

where S1y ¼ 2

3 s1y . We will however not dwell onto the interpretation of this coefficient.

When the representation (8) is adopted, the coefficients can be found by performing the regression analysis of the valuesof raver

y ða;hÞ determined for several values of h. In fact this is a simple multivariate linear regression performed on the valuesof h�1/2 and h1/2, such that it is sufficient to use only three values of h. We will use averaging lengths of h, 2h and 4h. Theresults of KI reconstruction performed for different values of h are shown in Fig. 5. It is seen that the accuracy is considerablehigher than the one achieved by using only the singular term. Thus for h = 0.1a, which is quite crude by all accounts, we havethe accuracy of 0.33% as opposite to 2.5% when only the singular term is used.

These observations suggest the following algorithm for recovering the exponent and the stress intensity factors from afinite element/finite difference model:

1. Choose a mesh size, h and determine the average stress at the crack tip. Given that, for instance in the finite elementmethod the stresses in a standard element at the crack tip do not represent the stress singularity well enough, especiallywith the low order elements, the averaging should be performed over a region consisting of a number of elements,d = n � h where n = 1,2, . . .

2. Repeat the computations with few smaller sizes, say h/2 and h/4, and fit the asymptotics of the type shown in (8) to thesevalues thus determining the stress intensity factor. The curve fitting over several points also serves to mitigate the influ-ence of the usual numerical errors.

In the following section this procedure will be upgraded to take into account the fact that in 2D problems the elementsare two-dimensional, i.e. include points off the x-axis. It will also be taken into account that the elements do not have to besymmetrical with respect to the x-axis: we will consider the elements that are offset.

3.3. Stress at the crack tip averaged over a rectangular region

The above recovery algorithm is based on the averaging over an area near the crack tip which, as we assume, scales pro-portionally to the chosen element size. In order to match this averaging we need to perform a similar averaging for theasymptotic stress field. In view of the finite element analysis and taking into account that the averaging region can span sev-eral finite elements, we will denote its size by d, reserving h for the size of the finite element. In the coordinate frame shownin Fig. 6 the asymptotics for the stress field can be expressed in the form [40]:

rasymptx ¼ K Iffiffiffiffiffiffiffiffi

2prp 1� sin

h2� sin

3h2

� �cos

h2� K Iffiffiffiffiffiffiffiffi

2prp 2þ cos

h2� cos

3h2

� �sin

h2þ s0

x þ s1x ðhÞ

ffiffiffirpþ Oðr3=2Þ;

rasympty ¼ K IIffiffiffiffiffiffiffiffi

2prp 1þ sin

h2� sin

3h2

� �cos

h2þ K Iffiffiffiffiffiffiffiffi

2prp sin

h2

cosh2� cos

3h2

bþ s0y þ s1

yðhÞffiffiffirpþ Oðr3=2Þ;

rasymptxy ¼ K1ffiffiffiffiffiffiffiffi

2prp sin

h2� cos

h2� cos

3h2þ K IIffiffiffiffiffiffiffiffi

2prp 1� sin

h2� sin

3h2

� �cos

h2þ s0

xy þ s1xyðhÞ

ffiffiffirpþ Oðr3=2Þ:

ð9Þ

x

y

rθ

d

βdIII

KK ,

dS

Fig. 6. Rectangular region Sd of averaging the singular stress field at the crack tip loaded in such a way that it produces the stress intensity factors KI and KII.In principle, the geometry of the region does not have to coincide with the geometry of finite element mesh, though in some cases it might be convenient.


The average stresses then read

hrasymptx i ¼ 1

bd2

ZSd

rx dxdy; hrasympty i ¼ 1

bd2

ZSd

ry dxdy; hrasymptxy i ¼ 1

bd2

ZSd

rxy dxdy: ð10Þ

For the simple configuration shown in Fig. 6 we shall use only the components ry for KI determination and rxy for KII deter-mination. The average values for these components are

hrasympty i ¼ K Iffiffiffi

dp CIðbÞ þ s0

y þ S1y

ffiffiffidp

;

hrasymptxy i ¼ K IIffiffiffi

dp CIIðbÞ þ s0

xy þ S1xy

ffiffiffidp

;

ð11Þ

where S1y ; S

1xy are coefficients and

CIðbÞ ¼1

bffiffiffiffiffiffi2pp

Z b

0dyZ 1

0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi12r

1þ xr

� �r1� 1

21� x

r

� �þ y

r

� �2� �

dx;

CIIðbÞ ¼1

bffiffiffiffiffiffi2pp

Z b

0dyZ 1

0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi12r

1þ xr

� �r1þ 1

21� x

r

� �� y

r

� �2� �

dx;

r ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ y2

p:

ð12Þ

Fig. 7 shows the values of these factors. It is interesting that for all aspect ratios b, CI > CII, which means that the effect of KII onthe average shear stress is smaller than the effect of KI on the normal stress contrary to what is observed on just the x-axiswhere the effects are equal.

We will now use (11) and (12) to recover the stress intensity factors for the following examples of crack loadings.

3.4. Uniform loading at crack faces

Remote loading leaves the crack faces unloaded and subsequently the constant terms in (11) are s0y ¼ 0; s0

xy ¼ 0. We shallnow consider the difference created by the case when these terms are present. The simplest loading for this case is the uni-form loading of crack faces. Let, for the sake of simplicity, the crack faces be loaded by uniform normal load p. The exact solu-tion for this case has the form [38]:

0 0.5 1 1.5 2 2.50.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

β

)(βIC

)(βIIC

Fig. 7. The values of factors CI and CII as functions of the aspect ratio b of the averaging area.

0 0.02 0.04 0.06 0.08 0.1-0.65

-0.6

-0.55

-0.5

h/a

α

0 0.02 0.04 0.06 0.08 0.1-0.65

-0.6

-0.55

-0.5

h/a

α

Fig. 8. Dependence of the singularity exponent (6) for the average stress vs. the element size.

0 0.02 0.04 0.06 0.081.4

1.5

1.6

1.7

h/a

ap

KI

Determination of SIF using the singular expression (5)

Determination of SIF using three terms

Exact value π

Fig. 9. Reconstruction of KI using the expressions containing only the singular term (5) and three asymptotic terms (8) for the crack with uniformly loadedfaces.


ryðxÞ ¼pxffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

x2 � a2p � p: ð13Þ

In this case the model of numerical computations with a characteristic size h – an average of (13) over the length h is

ravery ðx;hÞ ¼ 1

h

Z xþh

xryðxÞdx ¼ p

h

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðxþ hÞ2 � a2

q�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 � a2p� �

� p: ð14Þ

The result, as a function of h, is shown in Fig. 8. For h = 0.01a we have ascale = �0.54 which is much worse than for the case ofremote loading for which the value a = 0.4914 was obtained. Only by choosing the element size two orders of magnitudesmaller, h = 0.0001a we can obtain ascale = �0.504 which achieves accuracy close to the case of remote loading.

The obvious reason for such a dramatic decrease of accuracy is the presence of the constant term in (11). Indeed, in thiscase the constant is the first non-singular term thus the relative error of neglecting the non-singular part is of the order ofO

ffiffiffihp� �

, while in the case of the remote loading, when the first non-singular term is itself of the order of Offiffiffihp� �

, the relativeerror of neglecting the non-singular part is of the order of O(h). It is not surprising then that the reconstruction of KI by usingonly the singular term (5) has a larger error (see the bottom line in Fig. 9). However, when three asymptotic terms (8) areused for the reconstruction the accuracy is quite high.

4. Numerical examples

4.1. Description of geometry and loading

In line with the remarks made in the previous section according to the difference between the remote loading and theinternal crack loading, in our examples we consider both cases separately for normal and shear loadings. The summary ofthe loading types is given in Fig. 10.

To test the proposed scalability mesh method, an example of a 2D squared plate with a crack in the middle is chosen. Thedimensions of the plate are L � L and the length of the crack is L/5. The crack which is located at the centre of the plate ismodelled by a nodal disconnection of those continuum elements which have one of their lateral faces lying on one of thetwo faces of the crack. Hereby, the nodes located at the crack faces behave as nodes located at any free boundary, being freeto move depending only on the external load and the problem boundary conditions. Due to the symmetry of the geometryand loading conditions, the problem is reduced and only a quarter of the plate is modelled. In that simplification new bound-

Type 1 Type 4Type 3Type 2

Fig. 10. Schematics of the types of loading used in the numerical examples.


ary conditions according to the symmetry assumption and to the type of loading have to be established. In particular for uni-axial tension, either internal or external, normal displacements are impeded along the two symmetry axis while tangentialdisplacement is let free. However, for shear loading, both external and internal, tangential displacement is impeded on thesymmetry axis at the same time that normal displacement is let free.

The element size in the x-direction is denoted by h while in the y-direction by b � h. The averaging area, in these compu-tations, consists of 3 � 3 elements such that the size in the x-direction is d = 3h. The method proposed requires differentcomputations in which the element size is the only changing variable. Hereby, the relation between element sizes along xand y direction also changes being for all cases b= 0.99701, 0.99797 and 0.99837.

Table 1Characteristics of the three different meshes used in the computations

nelemy nelemx Elements Nodes Elements crack

168 167 + 1 of half x-size 28.224 85.345 33 + 1 of half x-size248 247 + 1 of half x-size 61.504 185.505 49 + 1 of half x-size303 302 + 1 of half x-size 91.809 276.640 60 + 1 of half x-size

Fig. 11. Results of computations for crack under uniaxial tension applied at external boundaries: (a) contour map of ry for the most refined mesh; (b)contour map of the module of the displacement plotted over the deformed shape.


4.2. Finite element mesh

For the sake of simplicity, all meshes are structured and formed by identical 8-node quadratic continuum elements. Thelevel of refinement is controlled by the number of continuum elements used to discretise the crack. The crack length alwayscomprises odd number of elements, ensuring that in all cases there is a finite element in the centre of the crack. Due to thesymmetric simplification of the geometry, the central crack element is twice smaller in the x-direction, whereas the size inthe y-direction remains unchanged.

Hereby, three levels of refinement have been considered: nelemy = 168, 248 and 303. The characteristics of the consideredmeshes are shown in Table 1.

4.3. Type 1 loading: plate under external tension

The plate is loaded under uniaxial tension of magnitude p = 1 MPa directly applied to the upper external boundary of theplate. The vertical displacement is prevented in all points of the lower boundary excluding the ones which belong to thecrack, the latter being free of load. The horizontal displacement is impeded on the right hand boundary. To avoid spuriouslateral displacements, the x-direction displacement is also impeded in a node located at the left-lower corner. Fig. 11 showsthe contour plots of stress ry (Fig. 11a) and displacement uy (Fig. 11b).

The computed values of stresses at the Gauss points belonging to the averaging area are presented in Appendix. Table 2provides a summary of the characteristics of the finite element meshes used in the computations. The last column of thistable contains the value of ry obtained in the finite element analysis and averaged on a region of 3 � 3 finite elements whilethe second and third columns refer to the element size and averaging region size along the x-direction.

Table 2Average stress at the crack tip obtained for three mesh sizes for loading type 1

Mesh h (cm) d (cm) ravery ða; dÞ (MPa)

168 � 167.5 0.299 0.896 3.684248 � 247.5 0.202 0.606 4.437303 � 302.5 0.165 0.496 4.888

The first number in the mesh size refers to the y-direction; the second number refers to the x-direction in which one element is of half length.

Table 3Results of the determination of the coefficients of first Eq. (11) for type 1 loading

Stressparameters

Reconstructedvalues

Theoretical values computed forinfinite plane

Theoretical values computed forfinite plate

Absoluteerror

Relativeerror

KI (MPa m1/2) 5.759 5.605 5.913 �0.154 �2.6%s0

y (MPa) 0.003 0 0 0.003 –s1

y (MPa m�1/2) 0.108 – – – –

Fig. 12. Results of computations for crack under pure shear applied at external boundaries: (a) contour map of rxy for the most refined mesh under; (b)contour map of the module of the displacement plotted over the deformed shape.


The first column in Table 3 shows the results of the regression analysis for hrasympty i on variables CI(b)d�1/2 and

ffiffiffidp

. Thesecond column of the table shows the theoretical value for KI obtained for the crack in infinite plane and for the constantstress s0

y which for cracks with free faces is always zero. The third column shows the corrected value of the stress intensityfactor which takes into account the influence of external boundaries taken from [41]. The last two columns show the abso-lute and relative errors of the proposed computations with respect to the theoretical values of the stress intensity factor forthe crack in finite plate.

4.4. Type 2 loading: plate under external shear

The plate is loaded under pure shear directly applied on the left and upper boundary. Vertical displacement is preventedin right hand boundary, whereas horizontal displacement is impeded in lower boundary. Nodes located in the crack are leftfree, see Fig. 12.

Tables 4 and 5 provide similar information as Tables 2 and 3, respectively, but for the external shear loading case.

4.5. Type 3 loading: crack pressurised from inside

The plate is loaded under opening pressure directly applied to one of the sides of the crack. According to the mesh andloading symmetries, same boundary conditions as uniaxial tension case are specified, see Fig. 13.

Tables 6 and 7 show the corresponding results for the case of internal opening pressure. Results are organized in a similarway as presented in loading types 1 and 2 (see Tables 8 and 9).

4.6. Type 4 loading: Crack with shear loading applied to its faces

Shear tractions are applied directly to one of the crack sides. Boundary conditions are the same as in the pure shear test,see Fig. 14.

4.7. Singularity exponent

In Table 10 the results for the exponents computed from the two finest meshes are given. The exponents are reasonableclose to the expected value (0.5) in the two first loading cases (external tension and shear). However we can see how the lasttwo loading cases show a significant difference with respect to the theoretical value, which corresponds to the results of thesimple example considered in Section 3.4. This difference reflects the difference in scaling for the cases of internal and exter-nal loading since in the formal the asymptotic of stresses at the crack tip includes the constant term equal to the appliedload; in the case of external loading this term is absent.

To check the reasoning presented in Section 3.4, two new meshes have been prepared and computations performed foruniform tensile loading at the crack faces (type 3 loading). The characteristics of those meshes are given in Table 11. As ex-pected, the recovery of the singularity exponent using these meshes gives a slightly better result of 0.5916.

Table 4Average stress at the crack tip obtained for three mesh sizes for type 2 loading

Mesh h (cm) d (cm) raverxy ða; dÞ (MPa)

168 � 167.5 0.299 0.896 2.067248 � 247.5 0.202 0.606 2.453303 � 302.5 0.165 0.496 2.687



Stressparameters

Reconstructedvalues


Theoretical values approximated forfinite plate

Absoluteerror

Relativeerror

KII (MPa m1/2) 5.761 5.605 5.913a �0.152 �2.58%s0

xy (MPa) �0.003 0 0 0.003 –s1

xy (MPa m�1/2) 0.161 – – – –

a The exact value of the mode II stress intensity factor for a shear crack in a finite plate of this configuration was not available. The value presented in thetable is obtained by using the corresponding stress intensity factor for the crack in the infinite plane (column 3 of the table) and applying the samecorrection factor as for the case of tension (loading type 1). By performing such an approximation we obtain an estimate for the error of our method.

Fig. 13. Results of computations for crack under uniaxial tension applied at the crack faces: (a) contour map of ry for the most refined mesh under internalopening pressure; (b) contour map of the module of the displacement plotted over the deformed shape.


Mesh h (cm) d (cm) ravery ða; dÞ (MPa)

168 � 167.5 0.299 0.896 2.667248 � 247.5 0.202 0.606 3.420303 � 302.5 0.165 0.496 3.871



Stressparameters

Reconstructedvalues


Theoretical values computed forfinite plate

Absoluteerror

Relativeerror (%)

KI (MPa m1/2) 5.759 5.605 5.913 �0.154 �2.60s0

y (MPa) �1.013 0 0 �0.013 �1.30s1

y (MPa m�1/2) 0.107 – – – –


Mesh h (cm) d (cm) raverxy ða; dÞ (MPa)

168 � 167.5 0.299 0.896 1.065248 � 247.5 0.202 0.606 1.452303 � 302.5 0.165 0.496 1.686



Stressparameters

Reconstructedvalues


Theoretical values approximated forfinite plate

Absoluteerror

Relativeerror (%)

KII (MPa m1/2) 5.761 5.605 5.913a �0.152 �2.57s0

xy (MPa) �1.002 0 0 �0.002 �0.20s1

xy (MPa m�1/2) 0.158 – – – –

a The exact value of the mode II stress intensity factor for a shear crack in a finite plate of this configuration was not available. The value presented in thetable is obtained by using the corresponding stress intensity factor for the crack in the infinite plane (column 3 of the table) and applying the samecorrection factor as for the case of tension (loading type 1). By performing such an approximation we obtain an estimate for the error of our method.


Fig. 14. Results of computations for crack under pure shear applied at the crack faces: (a) contour map of rxy for the most refined mesh under internal shearpressure; (b) contour map of the module of the displacement plotted over the deformed shape.

Table 10Values of the exponents

External tension External shear Internal tension Internal shear

0.4825 0.4541 0.6175 0.7455

Table 11Characteristics of the three different meshes used in the computations

nelemy nelemx Elements Nodes Elements crack

388 387 + 1 of half x-size 154.449 464.920 77 + 1 of half x-size493 492 + 1 of half x-size 243.049 731.120 98 + 1 of half x-size


5. Discussion

We have seen that by interpolating the values of the stress concentration obtained from different mesh densities one canachieve a reasonable accuracy in the determination of stress intensity factors. So, is this just a new method of computingthem? – By no means. Clearly, in conventional situations the usage of conventional computational methods of the linear frac-ture mechanics such as special types of the finite element method or the other methods, like the collocation method wouldbe much more efficient. The message we are trying to convey is of a different nature. What we are striving to demonstrate isthat the computations conducted with meshes rougher than the final one still contain useful information. In theory, theappropriate mesh density is determined by conducting series of computations with consecutive mesh refinement untilthe results stabilise, even if in practice it might prove to be computationally demanding. After that, only the results obtainedwith the finest mesh are utilised. All previous meshes are thrown in the rubbish bean or, at the best, used to demonstratethat the final mesh is chosen appropriately. The methodology we are developing utilises the additional information con-tained in the computations with the coarser meshes to determine the characteristics of the fracture process. The proposedmethodology consists of the following elements:

1. Identification of the characteristics that are hypothesised to be scale-invariant with respect to the size of the finite ele-ment, at least for sufficiently refined meshes. In it was the effective length of the process zone, i.e. the distance from thecrack tip where the stress given by the analytical solution coincided with the numerically calculated stress (the FLAC wasused for the computations and the crack was modelled as a slot of one element width). In the present examples we havechosen the stress at the crack tip averaged over an area consisting of a fixed number of finite elements. By a sufficientlyrefined mesh we understand a mesh on which the element size is much smaller than the minimum characteristic size ofthe problem (i.e. for instance the crack length). Therefore, for fine enough meshes, in the abovementioned sense, weexpect the test characteristics to be power functions of the element size (for non-uniform meshes we presume that allelement sizes change proportionally with the mesh refinement) and thus be scale invariant in the general sense.


Subsequently, reaching the power law can be used as a quantifiable criterion to check whether the mesh is sufficientlyfine (in the conventional mesh-independent situations the exponent is simply zero) though in practice this can be com-putationally demanding. The situation is of course simplified in the case when the exponent is known in advance, as inthe case considered in the present paper where the exponent was �0.5.

2. Determination of the pre-factors of the power law of the characteristics. These pre-factors can either be used in their ownright as mesh-independent characteristics of the process under consideration, as in the case presented where the pre-fac-tors were proportional to the stress intensity factors. Alternatively, the pre-factors can be utilised to extrapolate the val-ues of the characteristics for the element sizes that correspond to the real (physical) microstructural size in the case thelatter is much smaller than the element size permitted by the existing computational capacity.

3. When the asymptotic behaviour of the characteristics is known in more detail, for instance from theoretical reasoning, asin the case considered here case of the average stress near the crack tip, this knowledge can be used to increase the accu-racy of reconstruction of the parameters of this asymptotic. This possibility was demonstrated in the present paper.

As far as the accuracy of the reconstruction of the stress intensity factors is concerned, we should note that the case wehave considered is intentionally the hardest one as we used a uniform mesh. If the determination of stress intensity factors isthe main aim then non-uniform meshes condensing at the crack tips are called upon.

In conclusion, it can be said that the proposed method follows the methodology that treats the (actual) scale as an extradimension [42] on top of the conventional physical dimensions (two dimensions in the examples presented). We extendedthis notion to the virtual reality where the mesh characteristic size plays the role of the scale.

6. Conclusions

The paper introduces the concept of mesh scalability which implies that since the mesh-dependent quantities, like stressconcentration, should scale with the element size according to certain asymptotic laws (e.g. as a power low for fine enoughmeshes) important additional information about the modelled object can be obtained by collating the results of the compu-tations performed using similar meshes of different densities. We demonstrate that the straightforward determination ofaveraging stresses at the crack tip using the finite element method over different densities of uniform meshes and the sub-sequent fitting of the theoretical dependence allows the recovery of the stress intensity factor. The meshes up to 300 � 300elements allow the determination of the stress intensity factors with relative error below 3%.

This result is significant in two ways. First, it presents a method of determination of the fracture mechanics characteristicswhen special singular elements are either not available in the finite element code at hand or not desirable. Most important,the method does not even need non-uniform meshes, which might be handy when the crack propagation is being modelled.The only thing required is the computation with meshes of different densities, which is a requirement of any accuratenumerical modelling anyway. Secondly, it introduces a new philosophy of modelling when independent simulations withmeshes of various densities are used simultaneously (the independent simulation with different density meshes is the mainfeature that distinguishes this method from the multigrid finite element modeling.) This is especially important in the mesh-

L/2

a 1

5

93

2 8

4 7

6

3

2

1

6

5

4

9

8

7

Fig. A1. The numbering and location of the continuum elements and the Gauss points.


dependent situations, as the mesh dependence, albeit unavoidable, is placed in the frame of the power law. This allows scal-ing the mesh dependence to the sizes where a certain physical since can be assigned to the element size.

Acknowledgements

The authors acknowledge the financial support from the Australian Research Council through the Discovery GrantDP0559737. A.V.D. acknowledges the financial support from the Australian Computational Earth Systems Simulator (ACcESS) –a Major National Research Facility.

Appendix A. Below we present the details of the mesh (Fig. A1) and the results of computations for all four loading typesconsidered (Tables A1–A16).

A.1. Type 1 loading

See Tables A1–A4.

Table A1Values of ry at the Gauss points for the coarse mesh under external tension

Gauss point Weight ry Elem. 1 ry Elem. 2 ry Elem. 3 ry Elem. 4 ry Elem. 5 ry Elem. 6 ry Elem. 7 ry Elem. 8 ry Elem. 9

1 0.309 2.824 2.741 2.641 3.304 3.419 3.181 3.730 4.625 4.3662 0.494 2.802 2.774 2.686 3.203 3.392 3.258 3.473 4.263 4.4653 0.309 2.787 2.818 2.718 3.116 3.378 3.233 3.217 3.937 4.9344 0.494 2.996 2.955 2.862 3.533 3.784 3.313 3.639 5.025 6.4055 0.790 2.953 2.971 2.860 3.347 3.674 3.651 3.381 4.544 5.5426 0.494 2.917 2.997 2.844 3.174 3.578 3.887 3.124 4.099 5.0517 0.309 3.179 3.228 3.126 3.742 4.128 3.673 3.422 4.662 9.5358 0.494 3.114 3.227 3.077 3.471 3.935 4.272 3.163 4.062 7.7129 0.309 3.056 3.235 3.014 3.213 3.756 4.769 2.906 3.498 6.260

Sy average 2.958 2.990 2.868 3.345 3.672 3.686 3.346 4.342 5.949

Table A2Values of ry at the Gauss points for the medium mesh under external tension


1 0.309 1.042 1.010 0.972 1.225 1.268 1.178 1.389 1.725 1.6282 0.494 1.654 1.636 1.583 1.900 2.013 1.932 2.068 2.543 2.6643 0.309 1.028 1.039 1.001 1.155 1.253 1.198 1.197 1.467 1.8424 0.494 1.772 1.746 1.690 2.100 2.251 1.967 2.171 3.004 3.8335 0.790 2.794 2.810 2.702 3.182 3.496 3.473 3.225 4.344 5.3046 0.494 1.725 1.772 1.680 1.885 2.127 2.312 1.861 2.447 3.0207 0.309 1.177 1.196 1.157 1.393 1.538 1.366 1.277 1.744 3.5748 0.494 1.845 1.912 1.822 2.066 2.345 2.546 1.887 2.429 4.6229 0.309 1.132 1.199 1.115 1.194 1.398 1.778 1.083 1.306 2.343

Sy average 3.542 3.580 3.430 4.025 4.422 4.437 4.040 5.252 7.207

Table A3Values of ry at the Gauss points for the finest mesh under external tension


1 0.309 1.144 1.108 1.066 1.349 1.396 1.296 1.532 1.903 1.7952 0.494 1.816 1.797 1.737 2.091 2.215 2.125 2.280 2.805 2.9383 0.309 1.129 1.141 1.099 1.271 1.379 1.318 1.319 1.618 2.0324 0.494 1.948 1.919 1.856 2.313 2.480 2.165 2.394 3.316 4.2335 0.790 3.071 3.088 2.969 3.504 3.851 3.825 3.556 4.794 5.8566 0.494 1.896 1.948 1.845 2.075 2.343 2.548 2.051 2.700 3.3337 0.309 1.295 1.315 1.272 1.535 1.695 1.505 1.409 1.925 3.9508 0.494 2.029 2.104 2.003 2.276 2.585 2.808 2.082 2.681 5.1079 0.309 1.245 1.319 1.226 1.316 1.541 1.961 1.194 1.441 2.588

Sy average 3.893 3.935 3.768 4.432 4.871 4.888 4.454 5.796 7.958

Table A4Computation of KI for plate under external tension

h 3 � h ry

ffiffiffiffiffiffiffiffiffi3 � hp

1ffiffiffiffiffi3�hp K Iffiffiffiffi

2pp B A

0.299 0.896 3.684 0.946 1.0570.202 0.606 4.437 0.778 1.2850.165 0.496 4.888 0.704 1.420Computed coefficients 3.387 0.003 0.108Computed KI 5.759Theoretic KI 5.913

Error 2.602 %


A.2. Type 2 loading

See Tables A5–A8.

Table A5Values of rxy at the Gauss points for the coarse mesh under external shear

Gauss Point Weight rxy Elem. 1 rxy Elem. 2 rxy Elem. 3 rxy Elem. 4 rxy Elem. 5 rxy Elem. 6 rxy Elem. 7 rxy Elem. 8 rxy Elem. 9

1 0.309 0.551 0.706 0.799 0.489 0.744 0.950 0.374 0.628 1.3312 0.494 0.801 1.031 1.234 0.703 1.009 1.435 0.562 0.833 1.6503 0.309 0.454 0.587 0.741 0.392 0.531 0.822 0.321 0.400 0.8134 0.494 0.856 1.159 1.358 0.733 1.133 1.655 0.575 0.882 2.4965 0.790 1.236 1.652 2.065 1.050 1.528 2.424 0.869 1.226 2.5986 0.494 0.695 0.913 1.218 0.584 0.797 1.341 0.500 0.629 0.8817 0.309 0.514 0.739 0.912 0.424 0.647 1.150 0.352 0.421 1.8368 0.494 0.735 1.030 1.368 0.606 0.860 1.646 0.536 0.614 1.6739 0.309 0.409 0.552 0.795 0.336 0.441 0.886 0.311 0.333 0.336

Sy average 1.563 2.092 2.623 1.329 1.923 3.077 1.100 1.491 3.404

Table A6Values of rxy at the Gauss points for the medium mesh under external shear

Gauss point Weight rxy Elem. 1 rxy Elem. 2 rxy Elem. 3 rxy Elem. 4 rxy Elem. 5 rxy Elem. 6 rxy Elem. 7 rxy Elem. 8 rxy Elem. 9

1 0.309 0.647 0.838 0.952 0.574 0.887 1.139 0.435 0.748 1.6082 0.494 0.936 1.220 1.470 0.818 1.197 1.718 0.650 0.986 1.9863 0.309 0.528 0.691 0.881 0.453 0.625 0.983 0.368 0.468 0.9744 0.494 1.005 1.377 1.623 0.858 1.351 1.989 0.668 1.049 3.0205 0.790 1.443 1.957 2.463 1.221 1.811 2.909 1.003 1.448 3.1296 0.494 0.805 1.076 1.451 0.673 0.938 1.605 0.572 0.736 1.0497 0.309 0.603 0.880 1.093 0.496 0.771 1.386 0.409 0.498 2.2268 0.494 0.857 1.221 1.636 0.702 1.017 1.980 0.620 0.721 2.0189 0.309 0.473 0.650 0.949 0.385 0.517 1.062 0.356 0.387 0.396

Sy average 1.824 2.478 3.130 1.545 2.278 3.693 1.270 1.760 4.102

Table A7Values of rxy at the Gauss points for the finest mesh under external shear


1 0.309 0.706 0.918 1.045 0.625 0.973 1.253 0.473 0.820 1.7732 0.494 1.019 1.334 1.612 0.890 1.310 1.889 0.704 1.078 2.1883 0.309 0.573 0.755 0.966 0.491 0.682 1.079 0.397 0.510 1.0714 0.494 1.096 1.510 1.782 0.935 1.482 2.190 0.726 1.150 3.3335 0.790 1.570 2.141 2.704 1.326 1.982 3.200 1.087 1.584 3.4486 0.494 0.874 1.175 1.592 0.729 1.024 1.763 0.617 0.802 1.1517 0.309 0.658 0.965 1.201 0.539 0.845 1.528 0.445 0.545 2.4588 0.494 0.933 1.337 1.797 0.762 1.113 2.180 0.671 0.787 2.2259 0.309 0.513 0.710 1.042 0.416 0.563 1.168 0.385 0.421 0.432

Sy average 1.985 2.711 3.435 1.678 2.494 4.062 1.376 1.924 4.520

Table A8Computation of KII for plate under external shear

h 3 � h rxy


1ffiffiffiffiffi3�hp KIIffiffiffiffi

2pp B A

0.299 0.896 2.067 0.946 1.0570.202 0.606 2.453 0.778 1.2850.165 0.496 2.687 0.704 1.420Computed coefficients 1.815 �0.003 0.161Computed KII 5.761Theoretic KII 5.605

Error �2.779 %


A.3. Type 3 loading

See Tables A9–A12.

Table A9Values of ry at the Gauss points for the coarse mesh under internal crack opening pressure


1 0.309 0.560 0.534 0.504 0.708 0.741 0.668 0.842 1.105 1.0252 0.494 0.886 0.872 0.828 1.084 1.173 1.107 1.221 1.599 1.6943 0.309 0.549 0.558 0.527 0.651 0.729 0.684 0.685 0.905 1.2014 0.494 0.981 0.960 0.914 1.245 1.361 1.132 1.307 1.982 2.6355 0.790 1.536 1.548 1.461 1.848 2.098 2.077 1.889 2.791 3.5356 0.494 0.943 0.980 0.905 1.071 1.268 1.415 1.054 1.526 1.9587 0.309 0.669 0.683 0.652 0.843 0.954 0.815 0.750 1.155 2.5908 0.494 1.039 1.092 1.019 1.216 1.441 1.598 1.072 1.542 3.2409 0.309 0.632 0.685 0.617 0.682 0.851 1.153 0.591 0.785 1.568

Sy average 1.949 1.978 1.857 2.337 2.654 2.662 2.353 3.348 4.862

Table A10Values of ry at the Gauss points for the medium mesh under internal crack opening pressure

Gauss Point Weight ry Elem. 1 ry Elem. 2 ry Elem. 3 ry Elem. 4 ry Elem. 5 ry Elem. 6 ry Elem. 7 ry Elem. 8 ry Elem. 9

1 0.309 0.730 0.698 0.661 0.913 0.954 0.865 1.080 1.403 1.3052 0.494 1.156 1.138 1.085 1.402 1.511 1.429 1.574 2.037 2.1543 0.309 0.717 0.728 0.690 0.844 0.939 0.884 0.888 1.157 1.5204 0.494 1.273 1.247 1.190 1.601 1.743 1.463 1.681 2.505 3.3055 0.790 1.997 2.011 1.904 2.385 2.691 2.664 2.442 3.545 4.4606 0.494 1.227 1.272 1.180 1.389 1.629 1.808 1.372 1.949 2.4837 0.309 0.866 0.883 0.844 1.081 1.218 1.048 0.971 1.460 3.2228 0.494 1.347 1.411 1.321 1.568 1.842 2.035 1.398 1.965 4.0549 0.309 0.821 0.885 0.802 0.885 1.090 1.459 0.776 1.011 1.980

Sy average 2.533 2.568 2.419 3.017 3.404 3.414 3.046 4.258 6.121

Table A11Values of ry at the Gauss points for the finest mesh under internal crack opening pressure

Gauss point Weight ry Elem. 1 ry Elem. 2 ry Elem. 3 ry Elem. 4 ry Elem. 5 ry Elem. 6 ry Elem. 7 ry Elem. 8 ry Elem. 9z

1 0.309 0.832 0.797 0.755 1.037 1.082 0.983 1.223 1.581 1.4732 0.494 1.318 1.298 1.239 1.593 1.714 1.623 1.786 2.298 2.4283 0.309 0.818 0.830 0.787 0.960 1.066 1.005 1.010 1.307 1.7104 0.494 1.449 1.419 1.357 1.814 1.972 1.661 1.905 2.817 3.7055 0.790 2.274 2.289 2.170 2.707 3.046 3.017 2.773 3.995 5.0126 0.494 1.398 1.448 1.346 1.579 1.845 2.043 1.563 2.202 2.7977 0.309 0.983 1.002 0.959 1.223 1.376 1.187 1.103 1.642 3.5988 0.494 1.531 1.603 1.502 1.778 2.082 2.296 1.592 2.218 4.5399 0.309 0.934 1.005 0.913 1.006 1.233 1.642 0.888 1.146 2.225

Sy average 2.885 2.923 2.757 3.424 3.853 3.864 3.460 4.802 6.872

Table A12Computation of KI for plate under internal crack opening pressure

h 3 � h ry


1ffiffiffiffiffi3�hp K Iffiffiffiffi

2pp B A

0.299 0.896 2.667 0.946 1.0570.202 0.606 3.420 0.778 1.2850.165 0.496 3.871 0.704 1.420Computed coefficients 3.386 �1.013 0.107Computed KI 5.759Theoretic KI 5.605

Error �2.752 %


A.4. Type 4 loading

See Tables A13–A16.

Table A13Values of rxy at the Gauss points for the coarse mesh under internal crack shear pressure


1 0.309 0.242 0.397 0.489 0.181 0.435 0.639 0.066 0.318 1.0202 0.494 0.307 0.536 0.738 0.209 0.515 0.938 0.069 0.339 1.1543 0.309 0.146 0.277 0.431 0.083 0.222 0.512 0.012 0.093 0.5024 0.494 0.362 0.663 0.862 0.239 0.638 1.157 0.080 0.391 2.0005 0.790 0.445 0.860 1.271 0.260 0.738 1.629 0.078 0.438 1.8046 0.494 0.201 0.418 0.723 0.090 0.304 0.845 0.005 0.137 0.3827 0.309 0.205 0.429 0.602 0.116 0.338 0.839 0.041 0.118 1.5278 0.494 0.241 0.535 0.871 0.112 0.367 1.149 0.040 0.124 1.1769 0.309 0.100 0.243 0.485 0.027 0.133 0.576 0.002 0.025 0.023

Sy average 0.562 1.090 1.618 0.329 0.922 2.071 0.099 0.496 2.397

Table A14Values of rxy at the Gauss points for the medium mesh under internal crack shear pressure


1 0.309 0.338 0.529 0.642 0.265 0.578 0.828 0.128 0.438 1.2972 0.494 0.442 0.725 0.974 0.325 0.703 1.221 0.157 0.492 1.4913 0.309 0.219 0.382 0.572 0.144 0.316 0.673 0.059 0.161 0.6644 0.494 0.511 0.882 1.127 0.365 0.856 1.492 0.174 0.559 2.5255 0.790 0.653 1.165 1.670 0.431 1.021 2.114 0.213 0.661 2.3366 0.494 0.312 0.581 0.956 0.179 0.445 1.109 0.078 0.244 0.5517 0.309 0.295 0.571 0.782 0.187 0.461 1.075 0.098 0.196 1.9178 0.494 0.364 0.726 1.140 0.209 0.524 1.483 0.124 0.232 1.5229 0.309 0.164 0.342 0.639 0.077 0.209 0.753 0.047 0.079 0.083

Sy average 0.824 1.476 2.125 0.545 1.278 2.687 0.269 0.765 3.096

Table A15Values of rxy at the Gauss points for the finest mesh under internal crack shear pressure


1 0.309 0.397 0.609 0.735 0.317 0.664 0.942 0.165 0.511 1.4622 0.494 0.525 0.839 1.116 0.396 0.816 1.392 0.211 0.584 1.6923 0.309 0.264 0.446 0.656 0.182 0.374 0.769 0.089 0.203 0.7604 0.494 0.602 1.015 1.286 0.442 0.987 1.693 0.231 0.660 2.8385 0.790 0.780 1.349 1.910 0.537 1.192 2.405 0.297 0.797 2.6556 0.494 0.380 0.681 1.096 0.235 0.531 1.268 0.123 0.309 0.6527 0.309 0.349 0.656 0.890 0.231 0.536 1.217 0.134 0.243 2.1508 0.494 0.439 0.842 1.301 0.268 0.620 1.683 0.176 0.298 1.7299 0.309 0.204 0.402 0.732 0.108 0.256 0.859 0.075 0.113 0.119

Sy average 0.985 1.709 2.431 0.679 1.494 3.057 0.375 0.929 3.514

Table A16Computation of KII for plate under internal crack shear pressure

h 3 � h rxy


K IIffiffiffiffi2pp B A

0.299 0.896 1.065 0.946 1.0570.202 0.606 1.452 0.778 1.2850.165 0.496 1.686 0.704 1.420Computed coefficients 1.815 �1.002 0.158Computed KII 5.761Theoretic KII 5.605

Error �2.775 %


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Documents

Mesh scalability in direct finite element simulation of brittle fracture