Caballero Dyskin 2008

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Mesh scalability in direct finite element simulation of brittle fracture

Antonio Caballero *, Arcady Dyskin

School 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 2007

Received in revised form 31 January 2008

Accepted 13 March 2008

Available online 31 March 2008

Keywords:

Singular stress

Non-singular stress

Scaling

Singularity exponent

a b s t r a c t

A new approach of dealing with mesh dependence in finite element modelling of fracture

processes 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 the

crack tip upon the element size; when the dependence approaches a power law with the

required accuracy, the mesh is called scalable. If the mesh is scalable and the exponent

and pre-factor are known, then the results of the computations can be scaled to the size

relevant to the scale of the physical microstructure of the material; the latter while not

being modelled directly ultimately controls the fracture propagation. To illustrate this

new approach, four 2D examples of a single straight crack loaded under tensile and shear

tractions applied either to the external boundary or to the crack faces are considered. It is

shown that combining the stresses at the crack tip computed using a set of similar meshes

of different densities with the crack tip asymptotic allows accurate recovery of the stress

intensity factors.

2008 Elsevier Ltd. All rights reserved.

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 of

sharp edges (cracks), produce infinite stress at the tip. There are two main approaches to deal with the unphysical unbounded

stress 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 a

wedge are expressed in terms of a power function [4] i.e. as self-similar distributions. Thus, the focus was on expressing

the criteria of the initiation and direction of crack growth through scale invariants such as the fracture energy [5] or the

stress intensity factors, or equivalent [1,6]. The direction of crack growth is determined by these invariants while the criteria

of crack growth include experimentally determined critical values of the invariants themselves or functions thereof. These

critical values incorporate (and conceal) the details of the mechanical behaviour of the material at the crack tip as well as the

fine geometry of the crack tip (these are the factors making the stress distributions bounded thus limiting the applicability of

the theory of elasticity).

0013-7944/$ - see front matter 2008 Elsevier Ltd. All rights reserved.doi:10.1016/j.engfracmech.2008.03.007

* Corresponding author. Tel.: +61 8 6488 3987; fax: +61 8 6488 1044.

E-mail addresses: [email protected] (A. Caballero), [email protected] (A. Dyskin).1 See Rossmanith [2] for a detailed overview of this unjustifiably forgotten work.

Engineering Fracture Mechanics 75 (2008) 40664084

Contents lists available at ScienceDirect

Engineering Fracture Mechanics

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / e n g f r a c m e c h
mailto:[email protected]:[email protected]://www.sciencedirect.com/science/journal/00137944http://www.elsevier.com/locate/engfracmechhttp://www.elsevier.com/locate/engfracmechhttp://www.sciencedirect.com/science/journal/00137944mailto:[email protected]:[email protected]


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Another approach is based on the introduction, in greater or lesser detail, of the fine aspects of the crack tip zone. The

simplest possibility is to average the stress singularity over a certain length [7,8]. Then the criterion of crack propagation

includes 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 [911].

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 the

self-similar stress distribution at the crack tip to the status of an intermediate asymptotic only valid for distances rfrom the

crack 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 crack

propagation and its path.

Despite the simplifications offered by the self-similarity of singular stress field near the crack tip, the actual computation

of the characteristics of the stress singularities becomes quite complex as soon as the crack geometry differs from simplest

straight and disc-like cracks. In the 1960s, Ngo and Scordelis [13] gave the starting shoot for one of the most fructiferous

research 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 other

works 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 reduction

of this size is restricted by the available computational resources and for that reason, in some situations the finite elements

may 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.

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 size

KI 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 meshS0y constant stressS1y constant stressS0xy constant stressS1xy constant stressb size coefficient (dy = b dx)r

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

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

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

raverxy 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) 40664084 4067


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usually based upon the approximation of the displacement and/or stresses by polynomials, it is not possible to obtain an

accurate representation of the behaviour near to the singularity. To overcome this awkwardness some authors used to refine

more and more the finite element meshes near the crack tip [1416]. This method however proved to be too expensive in

terms 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 the

brittle crack concept). Some other works proposed minor changes to this approach, as for instance the method by Henshell

and Shaw [19], in which the mid-side node of a standard eight-node isoparametric element is displaced from its nominal

position. 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). Instead

of introducing special elements, the method proposed by Leung and Su [20,21] splits the domain into two regions: regular

and 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 resulting

large number of nodal variables is decreased by means of a global interpolating transformation in which the stress intensity

factors (SIFs) are the primary unknowns. The method has been successfully applied to compute the mode I SIF, the mixed

mode [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, the

displacement 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 other

than the regular polynomial functions used in the regular FEM. For instance, when dealing with crack problems, the method

could incorporate the self-similar near-tip asymptotic, which enables the domain to be modelled by finite elements without

remeshing [28]. In this method the sparsity and symmetry of the resulting stiffness matrix are retained. The model was

checked for the cases of strong discontinuities [29] as well as weak discontinuities [30]. The above methods may recover

the 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 view

to capitalise on the advantages of fixed density meshes. We illustrate the proposed approach using very simple examples for

which benchmark solutions are known: straight 2D brittle cracks under tensile or shear loading. We will use the proposed

methodology to recover the stress intensity factors from FE computations with different mesh densities. The simplicity of the

problem for this approach comes also from the fact that by the virtue of the assumption of brittleness, the problem does not

possesses a microscopic length which could otherwise be for instance related to the process zone; subsequently the chosen

finite element size represents the finest scale in the modelling. As we are only interested in the scaling properties, we keep

the 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 the

computations (in terms of strain and stresses) at the crack tip for successive refined meshes does not stabilise. The mesh

dependence is caused by the fact that the explicit modelling of fractures introduces a singularity in the stress and strain field

along any radial direction from the crack tip. However, it is also known that this singularity far of being random is propor-

tional to K=ffiffiffi

rp

where Kis the SIF and rthe radial distance from the crack tip (see Fig. 1). As mentioned in Introduction there

have been many methods which treat that singularity and/or mesh dependence of the solution near the crack tip on the size

of the finite element. Here another methodology is proposed to overcome difficulties associated with mesh dependence

which 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 a

self-similar manner), all mesh-dependent quantities should change as power functions of h, meaning that a fine enough

mesh becomes scale invariant (this type of scale invariance of the effective process zone length was confirmed by Pant

and Dyskin). This is a generalisation of the conventional notion of mesh independence in which case the scaling exponent

is 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 regression

line on a loglog plot), meaning that this mesh density is acceptable from the computational point of view. On the other

hand, 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 somewhat

representing 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.

4068 A. Caballero, A. Dyskin/ Engineering Fracture Mechanics 75 (2008) 40664084


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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 independent

of 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 element

methods 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 of

two 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 some

superficial resemblance since in our method the computations with different mesh densities are independent; rather than

linked in a process of iterative refinement. On top of that, our method relies upon linking the scales through a priori known

asymptotic 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 treating

the 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 are

characteristics 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 minimum

characteristic size of crack and/or load distribution. In numerical implementation, however, this possibility is limited by

the presence of another characteristic size which reflects the discrete nature of numerical approximation for instance the

mesh element size in the finite element method. In order to highlight the consequences of this restriction, we consider a

simple 2D example in which a straight crack of length 2a is uniformly loaded, Fig. 2. Let the element size be h and consider

the normal stress ry at the x-axis which is the line of crack continuation. We assume that the distance from the crack tip is

larger than the size of the element, rP h, and we focus on the interval h 6 r6 d ( a where one can expect the asymptotics

rasimpty KIffiffiffiffiffiffiffiffi2pr

p 1

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].

ryx pxffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 a2

p : 2

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

ravery x; h 1

h

Zxhx

ryxdx ph

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix h 2 a2

q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 a2

p : 3

In order to get an impression of the accuracy with which the square root asymptotic can be extracted from values (2) we

concentrate on the exponent (the singularity exponent) and see what element size is needed to recover the exponent value

of0.5 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.

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ax lnraver

y a r h; h lnravery a r; hlnr h lnr : 4

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 = 105a (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 the

asymptotics (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 ih KI

ffiffiffi2

pffiffiffiffiffiffiph

p : 5

Subsequently, after performing computations for different values of h, the singularity exponent that corresponds to scaling

the mesh can be recovered as

ascale lnravery a; 2h lnravery a; h

ln 2: 6

The result, as a function ofh, 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 = 0.01a

gives 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 stress

field. 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 instance

in order to reduce the computational time). In this case, the account of further asymptotic terms might be of benefit. We

investigate 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

-0.4

-0.3

r/a

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 the

element sizes in numerical computations).

0 0.02 0.04 0.06 0.08-0.5

-0.49

-0.48

-0.47

h/a

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



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rasimpty KIffiffiffiffiffiffiffiffi2pr

p s0y s1yffiffiffi

rp Or3=2: 7

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

p.

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

hrasimpty ih KI

ffiffiffi2

pffiffiffiffiffiffiph

p s0y S1yffiffiffi

hp

; 8

where S1y 23 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 values

ofravery a; h determined for several values ofh. In fact this is a simple multivariate linear regression performed on the valuesof h1/2 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 ofKI reconstruction performed for different values ofh are shown in Fig. 5. It is seen that the accuracy is considerable

higher than the one achieved by using only the singular term. Thus for h = 0.1a, which is quite crude by all accounts, we have

the 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 a

finite 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 element

method the stresses in a standard element at the crack tip do not represent the stress singularity well enough, especially

with 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 these

values 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 elements

are two-dimensional, i.e. include points off the x-axis. It will also be taken into account that the elements do not have to be

symmetrical 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 the

asymptotic 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 shown

in Fig. 6 the asymptotics for the stress field can be expressed in the form [40]:

rasymptx KIffiffiffiffiffiffiffiffi2pr

p 1 sin h2

sin 3h2

cos

h

2 KIffiffiffiffiffiffiffiffi

2prp 2 cos h

2 cos 3h

2

sin

h

2 s0x s1x h

ffiffiffir

p Or3=2;

rasympty KIIffiffiffiffiffiffiffiffi2pr

p 1 sin h2

sin 3h2

cos

h

2 KIffiffiffiffiffiffiffiffi

2prp sin h

2cos

h

2 cos 3h

2b s0y s1yh

ffiffiffir

p Or3=2;

rasympt

xy K1ffiffiffiffiffiffiffiffi2prp

sinh

2 cos

h

2 cos

3h

2 KIIffiffiffiffiffiffiffiffi2prp

1

sinh

2 sin

3h

2 cos h

2 s0

xy s1

xyh ffiffiffirp

O

r3=2

:

9

0 0.02 0.04 0.06 0.081.76

1.77

1.78

1.79

1.8

1.81

h/a

ap

KI

Determ

ination

of SIF u

sing th

e sing

ular ex

pressio

n (5)

DeterminationofSIFusing threeterms

Exact value

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



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The average stresses then read

hrasymptx i 1

bd2

ZSd

rx dx dy; hrasympty i 1

bd2

ZSd

ry dx dy; hrasymptxy i 1

bd2

ZSd

rxy dx dy: 10

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 KIffiffiffi

dp CIb s0y S1

yffiffiffi

dp ;

hrasymptxy i KIIffiffiffi

dp CIIb s0xy S1xy

ffiffiffid

p;

11

where S1y ; S1

xy are coefficients and

CIb 1bffiffiffiffiffiffi

2pp

Zb0

dy

Z10

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

2r1 x

r

r1 1

21 x

r

y

r

2 dx;

CIIb 1bffiffiffiffiffiffi

2pp

Zb0

dy

Z10

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

2r1 x

r

r1 1

21 x

r

y

r

2 dx;

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 y2

p:

12

Fig. 7 shows the values of these factors. It is interesting that for all aspect ratios b, CI > CII, which means that the effect ofKII on

the 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 0; s0xy 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]:

x

y

r

d

dIII

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.

0 0.5 1 1.5 2 2.50.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

)(IC

)(IIC

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



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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 tangential

displacement is let free. However, for shear loading, both external and internal, tangential displacement is impeded on the

symmetry 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 x

and y direction also changes being for all cases b= 0.99701, 0.99797 and 0.99837.

Type 1 Type 4Type 3Type 2

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

Table 1

Characteristics 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-size

303 302 + 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.



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4.2. Finite element mesh

For the sake of simplicity, all meshes are structured and formed by identical 8-node quadratic continuum elements. The

level of refinement is controlled by the number of continuum elements used to discretise the crack. The crack length always

comprises odd number of elements, ensuring that in all cases there is a finite element in the centre of the crack. Due to the

symmetric simplification of the geometry, the central crack element is twice smaller in the x-direction, whereas the size in

the y-direction remains unchanged.

Hereby, three levels of refinement have been considered: nelemy

= 168, 248 and 303. The characteristics of the considered

meshes 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 the

plate. The vertical displacement is prevented in all points of the lower boundary excluding the ones which belong to the

crack, the latter being free of load. The horizontal displacement is impeded on the right hand boundary. To avoid spurious

lateral displacements, the x-direction displacement is also impeded in a node located at the left-lower corner. Fig. 11 shows

the 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 2

provides a summary of the characteristics of the finite element meshes used in the computations. The last column of this

table contains the value ofry 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 2

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

Mesh h (cm) d (cm) ravery a; d (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 3

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

Stress

parameters

Reconstructed

values

Theoretical values computed for

infinite plane


finite plate

Absolute

error

Relative

error

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

s0y (MPa) 0.003 0 0 0.003

s1y (MPa m1/2) 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.

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The first column in Table 3 shows the results of the regression analysis for hrasympty i on variables CI(b)d1/2 andffiffiffi

dp

. The

second column of the table shows the theoretical value for KI obtained for the crack in infinite plane and for the constant

stress s0y which for cracks with free faces is always zero. The third column shows the corrected value of the stress intensity

factor 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 for

the 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 prevented

in right hand boundary, whereas horizontal displacement is impeded in lower boundary. Nodes located in the crack are left

free, 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 and

loading 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 similar

way 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 reasonable

close to the expected value (0.5) in the two first loading cases (external tension and shear). However we can see how the last

two loading cases show a significant difference with respect to the theoretical value, which corresponds to the results of the

simple 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 applied

load; 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 for

uniform 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 4

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

Mesh h (cm) d (cm) raverxy a; d (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


Table 5


Stress

parameters

Reconstructed

values


infinite plane

Theoretical values approximated for

finite plate

Absolute

error

Relative

error

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

s0xy (MPa) 0.003 0 0 0.003 s1xy (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 the

table 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.

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12/19

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 internal

opening pressure; (b) contour map of the module of the displacement plotted over the deformed shape.

Table 6


Mesh h (cm) d (cm) ravery a; d (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


Table 7


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 2.60

s0y (MPa) 1.013 0 0 0.013 1.30s1y (MPa m

1/2) 0.107

Table 8


Mesh h (cm) d (cm) raverxy a; d (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


Table 9


Stress

parameters

Reconstructed

values


infinite plane

Theoretical values approximated for

finite plate

Absolute

error

Relative

error (%)

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

s0xy (MPa) 1.002 0 0 0.002 0.20s1xy (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 the

table is obtained by using the corresponding stress intensity factor for the crack in the infinite plane (column 3 of the table) and applying the same

correction 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.



13/19

5. Discussion

We have seen that by interpolating the values of the stress concentration obtained from different mesh densities one can

achieve a reasonable accuracy in the determination of stress intensity factors. So, is this just a new method of computing

them? 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 would

be much more efficient. The message we are trying to convey is of a different nature. What we are striving to demonstrate is

that the computations conducted with meshes rougher than the final one still contain useful information. In theory, the

appropriate mesh density is determined by conducting series of computations with consecutive mesh refinement until

the results stabilise, even if in practice it might prove to be computationally demanding. After that, only the results obtained

with the finest mesh are utilised. All previous meshes are thrown in the rubbish bean or, at the best, used to demonstrate

that 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 proposed

methodology 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 the

crack tip where the stress given by the analytical solution coincided with the numerically calculated stress (the FLAC was

used for the computations and the crack was modelled as a slot of one element width). In the present examples we have

chosen the stress at the crack tip averaged over an area consisting of a fixed number of finite elements. By a sufficiently

refined mesh we understand a mesh on which the element size is much smaller than the minimum characteristic size of

the problem (i.e. for instance the crack length). Therefore, for fine enough meshes, in the abovementioned sense, we

expect the test characteristics to be power functions of the element size (for non-uniform meshes we presume that all

element sizes change proportionally with the mesh refinement) and thus be scale invariant in the general sense.

Fig. 14. Results of computations for crack under pure shear applied at the crack faces: (a) contour map ofrxy for the most refined mesh under internal shear

pressure; (b) contour map of the module of the displacement plotted over the deformed shape.

Table 10

Values of the exponents

External tension External shear Internal tension Internal shear

0.4825 0.4541 0.6175 0.7455

Table 11

Characteristics of the three different meshes used in the computations

nelemy nelemx Elements Nodes Elements crack





14/19

Subsequently, reaching the power law can be used as a quantifiable criterion to check whether the mesh is sufficiently

fine (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 in

the 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 own

right 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 the

latter 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, as

in 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 we

have considered is intentionally the hardest one as we used a uniform mesh. If the determination of stress intensity factors is

the 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 extra

dimension [42] on top of the conventional physical dimensions (two dimensions in the examples presented). We extended

this 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 stress

concentration, should scale with the element size according to certain asymptotic laws (e.g. as a power low for fine enough

meshes) 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 of

averaging 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 characteristics

when 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 accurate

numerical 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 main

feature that distinguishes this method from the multigrid finite element modeling.) This is especially important in the mesh-

L/2

a1

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.



15/19

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 Grant

DP0559737. 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 types

considered (Tables A1A16).

A.1. Type 1 loading

See Tables A1A4.

Table A1

Values ofry 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 4.366

2 0.494 2.802 2.774 2.686 3.203 3.392 3.258 3.473 4.263 4.465

3 0.309 2.787 2.818 2.718 3.116 3.378 3.233 3.217 3.937 4.934

4 0.494 2.996 2.955 2.862 3.533 3.784 3.313 3.639 5.025 6.405

5 0.790 2.953 2.971 2.860 3.347 3.674 3.651 3.381 4.544 5.542

6 0.494 2.917 2.997 2.844 3.174 3.578 3.887 3.124 4.099 5.051

7 0.309 3.179 3.228 3.126 3.742 4.128 3.673 3.422 4.662 9.535

8 0.494 3.114 3.227 3.077 3.471 3.935 4.272 3.163 4.062 7.712

9 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 ofry 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 1.628

2 0.494 1.654 1.636 1.583 1.900 2.013 1.932 2.068 2.543 2.664

3 0.309 1.028 1.039 1.001 1.155 1.253 1.198 1.197 1.467 1.842

4 0.494 1.772 1.746 1.690 2.100 2.251 1.967 2.171 3.004 3.833

5 0.790 2.794 2.810 2.702 3.182 3.496 3.473 3.225 4.344 5.304

6 0.494 1.725 1.772 1.680 1.885 2.127 2.312 1.861 2.447 3.020

7 0.309 1.177 1.196 1.157 1.393 1.538 1.366 1.277 1.744 3.574

8 0.494 1.845 1.912 1.822 2.066 2.345 2.546 1.887 2.429 4.622

9 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 A3

Values ofry 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 1.795

2 0.494 1.816 1.797 1.737 2.091 2.215 2.125 2.280 2.805 2.938

3 0.309 1.129 1.141 1.099 1.271 1.379 1.318 1.319 1.618 2.032

4 0.494 1.948 1.919 1.856 2.313 2.480 2.165 2.394 3.316 4.233

5 0.790 3.071 3.088 2.969 3.504 3.851 3.825 3.556 4.794 5.856

6 0.494 1.896 1.948 1.845 2.075 2.343 2.548 2.051 2.700 3.333

7 0.309 1.295 1.315 1.272 1.535 1.695 1.505 1.409 1.925 3.950

8 0.494 2.029 2.104 2.003 2.276 2.585 2.808 2.082 2.681 5.107

9 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



16/19

A.2. Type 2 loading

See Tables A5A8.

Table A4

Computation ofKI for plate under external tension

h 3 h ryffiffiffiffiffiffiffiffiffi

3 hp

1ffiffiffiffiffi3h

p KIffiffiffiffi2p

p B A

0.299 0.896 3.684 0.946 1.057

0.202 0.606 4.437 0.778 1.285

0.165 0.496 4.888 0.704 1.420

Computed coefficients 3.387 0.003 0.108

Computed KI 5.759

Theoretic KI 5.913

Error 2.602 %

Table A5

Values ofrxy 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. 91 0.309 0.551 0.706 0.799 0.489 0.744 0.950 0.374 0.628 1.331

2 0.494 0.801 1.031 1.234 0.703 1.009 1.435 0.562 0.833 1.650

3 0.309 0.454 0.587 0.741 0.392 0.531 0.822 0.321 0.400 0.813

4 0.494 0.856 1.159 1.358 0.733 1.133 1.655 0.575 0.882 2.496

5 0.790 1.236 1.652 2.065 1.050 1.528 2.424 0.869 1.226 2.598

6 0.494 0.695 0.913 1.218 0.584 0.797 1.341 0.500 0.629 0.881

7 0.309 0.514 0.739 0.912 0.424 0.647 1.150 0.352 0.421 1.836

8 0.494 0.735 1.030 1.368 0.606 0.860 1.646 0.536 0.614 1.673

9 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 ofrxy 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 1.608

2 0.494 0.936 1.220 1.470 0.818 1.197 1.718 0.650 0.986 1.986

3 0.309 0.528 0.691 0.881 0.453 0.625 0.983 0.368 0.468 0.974

4 0.494 1.005 1.377 1.623 0.858 1.351 1.989 0.668 1.049 3.020

5 0.790 1.443 1.957 2.463 1.221 1.811 2.909 1.003 1.448 3.129

6 0.494 0.805 1.076 1.451 0.673 0.938 1.605 0.572 0.736 1.049

7 0.309 0.603 0.880 1.093 0.496 0.771 1.386 0.409 0.498 2.226

8 0.494 0.857 1.221 1.636 0.702 1.017 1.980 0.620 0.721 2.018

9 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 A7

Values ofrxy 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 1.773

2 0.494 1.019 1.334 1.612 0.890 1.310 1.889 0.704 1.078 2.188

3 0.309 0.573 0.755 0.966 0.491 0.682 1.079 0.397 0.510 1.071

4 0.494 1.096 1.510 1.782 0.935 1.482 2.190 0.726 1.150 3.333

5 0.790 1.570 2.141 2.704 1.326 1.982 3.200 1.087 1.584 3.448

6 0.494 0.874 1.175 1.592 0.729 1.024 1.763 0.617 0.802 1.151

7 0.309 0.658 0.965 1.201 0.539 0.845 1.528 0.445 0.545 2.458

8 0.494 0.933 1.337 1.797 0.762 1.113 2.180 0.671 0.787 2.225

9 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



17/19

A.3. Type 3 loading

See Tables A9A12.

Table A8

Computation ofKII for plate under external shear

h 3 h rxyffiffiffiffiffiffiffiffiffi

3 hp

1ffiffiffiffiffi3h

p KIIffiffiffiffi2p

p B A

0.299 0.896 2.067 0.946 1.057

0.202 0.606 2.453 0.778 1.285

0.165 0.496 2.687 0.704 1.420

Computed coefficients 1.815 0.003 0.161Computed KII 5.761

Theoretic KII 5.605

Error 2.779 %

Table A9

Values ofry 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 1.025

2 0.494 0.886 0.872 0.828 1.084 1.173 1.107 1.221 1.599 1.694

3 0.309 0.549 0.558 0.527 0.651 0.729 0.684 0.685 0.905 1.201

4 0.494 0.981 0.960 0.914 1.245 1.361 1.132 1.307 1.982 2.635

5 0.790 1.536 1.548 1.461 1.848 2.098 2.077 1.889 2.791 3.535

6 0.494 0.943 0.980 0.905 1.071 1.268 1.415 1.054 1.526 1.958

7 0.309 0.669 0.683 0.652 0.843 0.954 0.815 0.750 1.155 2.590

8 0.494 1.039 1.092 1.019 1.216 1.441 1.598 1.072 1.542 3.240

9 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 A10

Values ofry 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 1.305

2 0.494 1.156 1.138 1.085 1.402 1.511 1.429 1.574 2.037 2.154

3 0.309 0.717 0.728 0.690 0.844 0.939 0.884 0.888 1.157 1.520

4 0.494 1.273 1.247 1.190 1.601 1.743 1.463 1.681 2.505 3.305

5 0.790 1.997 2.011 1.904 2.385 2.691 2.664 2.442 3.545 4.460

6 0.494 1.227 1.272 1.180 1.389 1.629 1.808 1.372 1.949 2.483

7 0.309 0.866 0.883 0.844 1.081 1.218 1.048 0.971 1.460 3.222

8 0.494 1.347 1.411 1.321 1.568 1.842 2.035 1.398 1.965 4.054

9 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 A11

Values ofry 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 1.473

2 0.494 1.318 1.298 1.239 1.593 1.714 1.623 1.786 2.298 2.428

3 0.309 0.818 0.830 0.787 0.960 1.066 1.005 1.010 1.307 1.710

4 0.494 1.449 1.419 1.357 1.814 1.972 1.661 1.905 2.817 3.705

5 0.790 2.274 2.289 2.170 2.707 3.046 3.017 2.773 3.995 5.012

6 0.494 1.398 1.448 1.346 1.579 1.845 2.043 1.563 2.202 2.797

7 0.309 0.983 1.002 0.959 1.223 1.376 1.187 1.103 1.642 3.598

8 0.494 1.531 1.603 1.502 1.778 2.082 2.296 1.592 2.218 4.539

9 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



18/19

A.4. Type 4 loading

See Tables A13A16.

Table A12

Computation ofKI for plate under internal crack opening pressure

h 3 h ryffiffiffiffiffiffiffiffiffi

3 hp

1ffiffiffiffiffi3h

p KIffiffiffiffi2p

p B A

0.299 0.896 2.667 0.946 1.057

0.202 0.606 3.420 0.778 1.285

0.165 0.496 3.871 0.704 1.420

Computed coefficients 3.386 1.013 0.107Computed KI 5.759

Theoretic KI 5.605

Error 2.752 %

Table A13

Values ofrxy 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 1.020

2 0.494 0.307 0.536 0.738 0.209 0.515 0.938 0.069 0.339 1.154

3 0.309 0.146 0.277 0.431 0.083 0.222 0.512 0.012 0.093 0.502

4 0.494 0.362 0.663 0.862 0.239 0.638 1.157 0.080 0.391 2.000

5 0.790 0.445 0.860 1.271 0.260 0.738 1.629 0.078 0.438 1.804

6 0.494 0.201 0.418 0.723 0.090 0.304 0.845 0.005 0.137 0.382

7 0.309 0.205 0.429 0.602 0.116 0.338 0.839 0.041 0.118 1.527

8 0.494 0.241 0.535 0.871 0.112 0.367 1.149 0.040 0.124 1.176

9 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 A14

Values ofrxy 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 1.297

2 0.494 0.442 0.725 0.974 0.325 0.703 1.221 0.157 0.492 1.491

3 0.309 0.219 0.382 0.572 0.144 0.316 0.673 0.059 0.161 0.664

4 0.494 0.511 0.882 1.127 0.365 0.856 1.492 0.174 0.559 2.525

5 0.790 0.653 1.165 1.670 0.431 1.021 2.114 0.213 0.661 2.336

6 0.494 0.312 0.581 0.956 0.179 0.445 1.109 0.078 0.244 0.551

7 0.309 0.295 0.571 0.782 0.187 0.461 1.075 0.098 0.196 1.917

8 0.494 0.364 0.726 1.140 0.209 0.524 1.483 0.124 0.232 1.522

9 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 A15

Values ofrxy 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 1.462

2 0.494 0.525 0.839 1.116 0.396 0.816 1.392 0.211 0.584 1.692

3 0.309 0.264 0.446 0.656 0.182 0.374 0.769 0.089 0.203 0.760

4 0.494 0.602 1.015 1.286 0.442 0.987 1.693 0.231 0.660 2.838

5 0.790 0.780 1.349 1.910 0.537 1.192 2.405 0.297 0.797 2.655

6 0.494 0.380 0.681 1.096 0.235 0.531 1.268 0.123 0.309 0.652

7 0.309 0.349 0.656 0.890 0.231 0.536 1.217 0.134 0.243 2.150

8 0.494 0.439 0.842 1.301 0.268 0.620 1.683 0.176 0.298 1.729

9 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



19/19

References

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Table A16

Computation ofKII for plate under internal crack shear pressure

h 3 h rxyffiffiffiffiffiffiffiffiffi

3 hp

KIIffiffiffiffi2p

p B A

0.299 0.896 1.065 0.946 1.057

0.202 0.606 1.452 0.778 1.285

0.165 0.496 1.686 0.704 1.420

Computed coefficients 1.815 1.002 0.158Computed KII 5.761

Theoretic KII 5.605

Error 2.775 %


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Caballero Dyskin 2008