Master's Thesis

Department of Naval Architecture, Ocean & Marine Engineering

Implementation of Extended Finite Element Method

(X-FEM) in Fracture Mechanics

Author: Ioannis Tsichlis

Supervisor: Dr Erkan Oterkus

A thesis submitted in partial fulfilment for the requirement of the degree

Master of Science

Sustainable Engineering: Offshore Renewable Energy

2016

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Copyright Declaration

This thesis is the result of the author’s original research. It has been composed by the author and has

not been previously submitted for examination which has led to the award of a degree.

The copyright of this thesis belongs to the author under the terms of the United Kingdom Copyright

Acts as qualified by University of Strathclyde Regulation 3.50. Due acknowledgement must always be

made of the use of any material contained in, or derived from, this thesis.

Signed: Ioannis Tsichlis Date: 22nd August 2016

Acknowledgments

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Acknowledgments

First of all I would like to express my special thanks to my supervisor Dr Erkan Oterkus for his

continuous support and cooperation throughout this project. His outstanding guidance and valuable

comments led me to completion of this thesis.

I am very pleased and honored to have been selected among the recipients of the Offshore Renewable

Energy Britannia Scholarship. I feel gratitude to have been awarded such a remarkable assistance in

meeting the cost of tuition fees, materials, travel and living expenses. This generous financial aid

contributed to the successful fulfilment of my studies at the University of Strathclyde.

Many thanks to my parents for the financial and not only support, my sister and friends for standing by

me constantly during this year.

This work is devoted to Jina for her unremitting encouragement, faith in my skills and admiration.

Ioannis Tsichlis

August 2016, Glasgow, UK.

Acknowledgements

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Table of Contents

Acknowledgments .................................................................................................................................. iii

Table of Contents ................................................................................................................................... iv

Table of Figures ..................................................................................................................................... vi

Abstract ................................................................................................................................................. vii

1. Introduction ..................................................................................................................................... 1

1.1 Analysis of Structures ............................................................................................................. 1

1.2 Analysis of Discontinuities ..................................................................................................... 1

1.3 Modern Fracture Mechanics ................................................................................................... 2

1.4 Overview of the Report ........................................................................................................... 2

2. Fundamental Linear Elastic Fracture Mechanics (LEFM) ................................................. 3

2.1 Introduction to Fracture Mechanics ........................................................................................ 3

2.2 Conventional Failure Criteria .................................................................................................. 3

2.3 Griffith’s Energy Balance ....................................................................................................... 4

2.4 Modes of Fracture ................................................................................................................... 5

2.5 Singular Stress Fields .............................................................................................................. 6

2.5.1 Sliding Mode II ............................................................................................................... 7

2.5.2 Tearing Mode III ............................................................................................................. 7

2.6 Stress Intensity Factor ............................................................................................................. 7

2.7 Material Fracture Toughness .................................................................................................. 8

3. X-FEM Concept & Formulation ................................................................................................... 10

3.1 Introduction to X-FEM ......................................................................................................... 10

3.2 A review on X-FEM ............................................................................................................. 11

3.2.1 X-FEM Development .................................................................................................... 11

3.2.2 X-FEM Applications on other Fields ............................................................................ 12

3.3 The Partition of Unity as basis of the X-FEM approximation .............................................. 12

4. Numerical Study with X-FEM technique ..................................................................................... 15

Acknowledgements

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4.1 Static Crack Edge No Fracture (Steel) .................................................................................. 15

4.1.1 Problem Description ..................................................................................................... 15

4.1.2 Geometric and Material Properties ............................................................................... 15

4.1.3 Boundary and Initial Conditions ................................................................................... 15

4.1.4 Results ........................................................................................................................... 16

4.2 Static Crack Edge Fracture (Steel) ........................................................................................ 17




4.2.4 Results ........................................................................................................................... 18

4.3 Internal Crack No Fracture (Aluminium) ............................................................................. 20




4.3.4 Results ........................................................................................................................... 21

4.4 Internal Crack Fracture (Aluminium) ................................................................................... 23




4.4.4 Results ........................................................................................................................... 24

5. Conclusions & Recommendations ................................................................................................ 26

6. Bibliography ................................................................................................................................. 28

Appendix A Static Crack Edge No Fracture (Steel) .......................................................................... 29

Appendix B Static Crack Edge Fracture (Steel) ................................................................................ 35

Appendix C Internal Crack No Fracture (Aluminium) ..................................................................... 41

Appendix D Internal Crack Fracture (Aluminium) ........................................................................... 47

Acknowledgements

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Table of Figures

Figure 1: The fundamental modes of crack extension. (a) Opening mode I, (b) Sliding mode II, (c)

Tearing mode III ..................................................................................................................................... 6

Figure 2: Stress element in an infinite plate ........................................................................................... 6

Figure 3: 3D modelling with extended finite element method ............................................................ 11

Figure 4: Cracks and discontinuities modelling i) 2D static crack edge ii) crack propagation with

conventional FEM refinement technique iii) crack propagation with enrichment X-FEM technique . 13

Figure 5: Enriched X-FEM technique for strong and weak discontinuities modelling ........................ 14

Figure 6: Fracture Problem 1 Von Misses Stresses Distribution .......................................................... 16

Figure 7: Fracture Problem 1 S11 Principal Stresses Distribution ......................................................... 16

Figure 8: Fracture Problem 1 S22 Principal Stresses Distribution ......................................................... 17

Figure 9: Fracture Problem 1 Magnitude of Displacements ................................................................. 17

Figure 10: Fracture Problem 2 Von Misses Stresses Distribution ........................................................ 18

Figure 11: Fracture Problem 2 S11 Principal Stresses Distribution ....................................................... 19


Figure 13: Fracture Problem 2 Magnitude of Displacements ............................................................... 20







Figure 20: Fracture Problem 4 S22 Principal Stresses Distribution ..................................................... 25


Acknowledgements

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Abstract

The design of most engineering structures includes methods that evaluate the stress and displacement

distribution in conjunction with possible failure phenomena. Probably the most important issue at this

phase for all structures and shapes is the prediction of the event of failure itself. Also the history of the

humanity is a history of structures failures. There were catastrophic accidents that caused great loss to

people’s lives and financial losses as well. Accidents in various engineering fields involving failures of

bridges, steel structures, concrete buildings, tanker and cargo ships, pipelines, railway, geotechnical

failures. Great number of failures begun from cracks or ended at crack propagation.

A turn point in case of fracture mechanics was the massive failures that occurred in ships built in USA

before the Second World War. An approximation of 5000 merchant ships (tankers and cargo ships) with

an average age of 3 years cracked in a catastrophic sometimes way. A view on literature proves that it

is not so easy to predict crack propagation and fracture mechanism in advance for every structure. After

a range of brittle failures were occurred (the most dangerous and sudden failure phenomenon) can be

conclude that most brittle fractures occurred in low temperatures, in most cases the nominal stress was

well below the yield stress of the material and most failures begun from discontinuities in the structure

shape such as holes and cracks.

For many years conventional failure criteria try to give reliable results for fracture problems. However

conversional failure criteria cannot explain failures which occur with nominal stress well below the

yield or the ultimate stress of the material. They cannot take into account the size of the structure effects

as well. Thus modern techniques should developed in order to predict fracture mechanisms.

In the present project basic concepts and formulations of linear elastic fracture mechanics is presented.

Modern fracture mechanics is applied now with numerical and computational methods for complex 3-

dimensional structures and initial conditions has its origin to fundamental linear elastic fracture

mechanics and continuum mechanics. Thus references to those classic methods and theorems give the

reader a spherical approach. A state-of-the-art computational method in the field of fracture mechanics

is Extended Finite Element Method (X-FEM) which is presented in the present project. Few bench mark

problems have been selected and analyzed to investigate the implementation of X-FEM in that field.

For the scopes of the present work commercial software packages like Abaqus and Ansys have been

used.

Ch.1 Introduction

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1. Introduction

1.1 Analysis of Structures

The finite element method (FEM) is nowadays the most popular computational tool for studying the

behavior of a wide range of engineering and physical problems. A lot of general purpose FEM software

packages are developed and are commercially available for those want to buy them. FEM applications

are widely thought in engineering universities in several undergraduate and postgraduate courses. The

crack propagation analysis is one of the fundamental FEM applications. These kind of applications

raised during the First World War in naval architecture laboratories trying to quantify linear elastic

fracture mechanics (LEFM) cases. LEFM has been applied to several crack and defect problems

however remained limited to simple geometries and loading conditions. The development of the FEM

increased much the range of LEFM applications. LEFM is now relying to a reliable and commercial

computational tool. Applications of FEM into LEFM expanded soon to elastic plastic fracture

mechanics (EPFM). During the time ago the synergy of them remained unchanged. LEFM basic

concepts are combined with classical FEM techniques based on continuum mechanics theory and

equations. The partition of unity theory and its application in the extended finite element method (X-

FEM) can be considered as a major breakthrough (Mohammadi, 2008).

1.2 Analysis of Discontinuities

Researches have been studied failure and fracture analysis of structures for many years. These studies

are addressed with the continuum mechanics theory, including plasticity theorems and damage

mechanics, or the discontinuous approach of fracture mechanics. On the one hand plasticity and damage

mechanics require the displacement field to be continuous everywhere (continuous mechanics). On the

other hand fracture mechanics is designed to solve problems with strong discontinuities where the stress

and displacement field behave in none linear nor smooth way across the discontinuous surface. Despite

fracture mechanics focuses and is designed for strong discontinuities it can be applied as well as to

weak discontinuities problems. Moreover plasticity theory and damage mechanics can be modified and

applied to fracture mechanics problems. Thus practically it is hard to distinguish fracture mechanics

from plasticity and damage mechanics applications. Crack problems can be simulated combining basic

fracture mechanics concepts with refining mesh techniques but with certain accuracy levels

(Mohammadi, 2008).

Ch1. Introduction

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1.3 Modern Fracture Mechanics

Fracture is very important as failure mode across a huge range of engineering fields such as civil

engineering, mechanical engineering, automobile, aerospace, marine engineering etc. A lot of accidents

started from cracks or ended at them in different engineering disciplines involving failures of bridges,

steel structures, concrete buildings, tanker and cargo ships, pipelines, railway and geotechnical failures.

Accidents caused enormous losses to people’s lives as well as financial losses (Zhuang, et al., 2014).

In most of the cases it is not that easy to quantitatively provide the causes of fracture phenomena

(Zhuang, et al., 2014). In addition numbers of structural and other failures cannot be explained by

traditional failure criteria in cases those failures occur at levels of stress more or even much lower than

the material’s ultimate strength (Gdoutos, 1993). As a result it is observed an upward trend of interest

to model discontinuities in an accurate way. Remarkable results can be noticed due to the development

of modern techniques the years before (Khoei, 2015).

Modern fracture mechanics expanded simultaneously with the high technology and large- scale

computers. That kind of development contributed to the enhancement of different numerical and

computational tools and experimental techniques. The basic traditional fracture mechanics guide the

growth of fracture mechanics applications in engineering. However fracture mechanics nowadays has

changed a lot due to the integration with science and high- technology methods (Zhuang, et al., 2014).

1.4 Overview of the Report

In the present report fundamental linear elastic fracture mechanics is presented in chapter 2. Contents

of this topic include conventional failure criteria, modes of fracture, singular stress fields, stress

intensity factor and material fracture toughness. In chapter 3 the basic concept, formulations and

governing equations of Extended Finite Element Method (X-FEM) are presented. Contents of this

chapter include X-FEM development and applications, a review on X-FEM as well as references to the

Partition of Unity method which is the basis of the enriched X-FEM approximation. Finally in chapter

4 cases of study of 2-Dimentional fracture problems are presented. The fracture problems have been

carried out in Abaqus software whereas only the meshing in Ansys. All fracture problems were fulfilled

with input files using Abaqus syntax code and import methodology. The full input files were imported

to Abaqus are shown on the Apprentices. Due to words limitation is not presented the full mesh of the

domains only just the first and last ten nodes and elements.

Ch.2 Fundamental LEFM

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2. Fundamental Linear Elastic Fracture Mechanics (LEFM)

2.1 Introduction to Fracture Mechanics

In fracture mechanics it is assumed defects are contained within the materials from which failures arise.

These discontinuities may already exist or appear under particular thermodynamic, microscopic and

loading conditions. For the estimation of a structure’s lifespan or a mechanical component’s the

knowledge of the stress and strain redistributed fields near to the crack tip is required. Generally

speaking cracks are synonym to non-linearity and non- smooth effects. They occur to remarkable

stresses near the crack tip with simultaneous release of energy; a point which requires special attention.

For hypothetically ideal brittle materials the linear theory of elasticity is adequate for the calculation of

the stress and strain distribution in the cracked domain. In some cases it is assumed linear behavior

without quite accurate results. (Gdoutos, 1993)

In this chapter an approach of the fundamental linear elastic fracture mechanics will be attempted with

references to the two dimensional fracture mechanics, the fundamental fracture modes, the stress

intensity factors and the fracture toughness and the energy release criterion (Gdoutos, 1993).

2.2 Conventional Failure Criteria

The prediction of the failure itself in a great range of engineering structures as well as the evaluation of

the stress and strain distribution is very important part of the mechanical design. The availability of the

related methods are huge nowadays. Theoretical analyses, numerical methods, computational methods

and simplified assumptions try to answer in a reasonable manner the above fundamental for the structure

questions. There are a lot of failure criteria in literature predict these kind of phenomena, however the

engineer should decide separately which criterion is more suitable and determine the most effective

method for each problem (Gdoutos, 1993) .

There are two main categories failure phenomena can be divided; brittle and ductile failures. In the first

scenario it is observed small deformation of the domain and it is most of the times sudden and

unexpected. In the second scenario the large deformation lasts significant long time with possible yield

and plastic flow. The mechanism of failure is affected noteworthy by defects that already existed or

developed in the structure shape. (Gdoutos, 1993)

In cases of ductile failure materials continue yielding up to the ultimate fracture. Criteria named yield

criteria developed for evaluating these kind of phenomena and are related to the magnitude of the

macroscopic stresses. (Gdoutos, 1993)


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The Tresca criterion is valid in multiaxial stress cases and quantifies the yielding effect

|𝜎1 − 𝜎3|

2= 𝑘, 𝜎1 > 𝜎2 > 𝜎3

(2.1)

where σ1, σ2, σ3 are the principal stresses and k is the yield stress in a pure shear stress.

The von Mises criterion is related to the distortional energy amount. According to this criterion the

material element yields when absorbs a certain distortional strain energy amount which is equal to the

distortional energy at the yield. (Gdoutos, 1993)

(𝜎1 − 𝜎2)2 + (𝜎2 − 𝜎3)2 + (𝜎3 − 𝜎1)2 = 2𝜎𝑌2 (2.2)

where σ1, σ2, σ3 are the principal stresses and σY is the yield stress in uniaxial tension

The Mohr- Coulomb criterion which is used basically in soil mechanics and rock mechanics problems

refers that fracture occurs when the shear stress becomes equal to a critical value with variables the

normal stresses. The Mohr Coulomb criterion in its simplest form is expressed by:

(

1 + sin 𝜔

2c cos 𝜔) 𝜎1 − (

1 − sin 𝜔

2c cos 𝜔) 𝜎3 = 1 (2.3)

where tan ω= μ and σ1>σ2>σ3.

2.3 Griffith’s Energy Balance

With main idea to model a crack system using reversible thermodynamic analysis Griffith proposed in

1920 a criterion relative to the energy amount the structure releases on the crack surface. It is considered

a theoretically infinite plate with tiny thickness related to its length. The plate is subjected to uniaxial

uniform tension. The length of the crack is raised is α. The strain energy is this case is (Zhuang, et al.,

2014):

𝑉𝜀 = (

−𝜎2

2Ε) πα2 (2.4)

where E is the Young’s modulus. The energy on the surface to arise a crack is

𝐸𝑠 = 2𝛾𝛼 (2.5)

where γ is the surface energy on a unit area.

As the crack length increases by dα the derivative of the strain energy released to the crack tip which

forces the crack to propagate and the derivative of the surface energy which resists to the crack

penetration should be balanced (Zhuang, et al., 2014). Thus,

𝜕(𝐸𝑠 + 𝑉𝑒)

𝜕𝛼= 0 (2.6)


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After the calculation of the above equations the external load for driving crack growth it is:

𝜎𝑓 = √2𝛦𝛾

𝜋𝛼 (2.7)

For ductile materials instead of the surface energy rate is used the critical energy release rate G. Thus

the above equation is equal to

𝜎𝑓 = √𝛦 𝐺

𝜋𝛼 (2.8)

There are cases that the happening of a crack cannot or it not cost effective to be prevented then the

crack propagation should be. Hence the crack should be detected before propagates to distance. The

stress and displacement field close to the crack tip is very important because those fields affect the

fracture process. Applying the popular method of stress intensity factor instead of the energy method

we shall go through the three fundamental modes of the crack extension investigating the stress field in

each occasion (Zhuang, et al., 2014) (Gdoutos, 1993).

2.4 Modes of Fracture

There are there fundamental modes of fracture subjected to crack extension. First of all is the opening

mode I. It is observed a symmetrically separation of the crack surfaces, displacements are opposite and

orientations are right- angled to each other. Mode I is very common fracture occurrence in engineering

fields. The second category is the sliding mode II. The crack surfaces slide relative to each other and

displacements remain opposite. The last category is the Tearing mode III. The antiplane displacements

occur between the two crack surfaces in an antiplane manner as presented in Figure 2.1c (Zhuang, et

al., 2014) (Gdoutos, 1990). It is noticeable that mode I and mode II corresponds to in-plane problems

and mode III to out of plane problems. However in real conditions a loading can have components of

more than one modes. This situation is named mixed applied loading and the fracture under such loading

mixed mode fracture. The justification of the suitable crack extension behavior mode is not always easy

to be achieved in real conditions (Kundu, 2008).


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2.5 Singular Stress Fields

Figure 2.2: Opening Mode I

The selected coordinate system has as original point the crack tip. In this case the x direction is the

direction of the crack’s propagation, the y direction is the normal of the crack surface direction and

finally the z direction is the out of the plane direction. It is considered also a polar coordinate system (r,

θ) (Zhuang, et al., 2014). The stress field given below is calculated applying the Westergaard method,

the presentation of this method is considered out of the scope of the present project. Thus,

Figure 2.1: Figure 1: The fundamental modes of crack extension. (a) Opening mode I, (b) Sliding mode II, (c) Tearing mode III (Zhuang,

et al., 2014)

Figure 2: Stress element in an infinite plate (Zhuang,

et al., 2014)


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The KI in the above system of equations is named stress intensity factor of the opening mode I crack.

Stress intensity factor is a very important quantity near the crack tip thus it will analytically be discussed

in a following section. The variables r and θ are dependent on the topology of the crack.

2.5.1 Sliding Mode II

In the same way KII is the stress intensity factors for the sliding mode II. Applying Westergaard method

the stress distribution in mode II cases is described by:

𝜎𝑥 = −

𝐾𝐼𝐼

√2𝜋𝑟sin

𝜃

2(2 + 𝑐𝑜𝑠

𝜃

2𝑐𝑜𝑠

3𝜃

2)

𝜎𝑦 =

𝐾𝐼𝐼

√2𝜋𝑟sin

𝜃

2𝑐𝑜𝑠

𝜃

2𝑐𝑜𝑠

3𝜃

2 (2.10)

𝜏𝑥𝑦 =

𝐾𝐼𝐼

√2𝜋𝑟cos

𝜃

2(1 − 𝑠𝑖𝑛

𝜃

2 𝑠𝑖𝑛

3𝜃

2)

2.5.2 Tearing Mode III

For the tearing (antiplane) mode III the in plane displacements are zero as it is considered only out of

plane deformation. For this crack extension William’s method is used for the analysis with results the

below stress field:

where w is the out of the plane displacement and KIII is the stress intensity factor for the tearing mode

III

2.6 Stress Intensity Factor

Equations 7 to 12 can be applied in two dimensional in-plane problems (plane stress and plane strain)

and calculate the stress distribution near the crack tip. The values of the stress intensity factors KI, KII

𝜎𝑥 =

𝐾𝐼

√2𝜋𝑟cos

𝜃

2(1 − 𝑠𝑖𝑛

𝜃

2𝑠𝑖𝑛

3𝜃

2)

𝜎𝑦 =

𝐾𝐼

√2𝜋𝑟cos

𝜃

2(1 + 𝑠𝑖𝑛

𝜃

2𝑠𝑖𝑛

3𝜃

2) (2.9)

𝜏𝑥𝑦 =

𝐾𝐼

√2𝜋𝑟cos

𝜃

2𝑠𝑖𝑛

𝜃

2 𝑐𝑜𝑠

3𝜃

2

𝜏𝑥𝑧 = −

𝐾𝐼𝐼𝐼

√2𝜋𝑟sin

𝜃

2

𝜏𝑦𝑧 =

𝐾𝐼𝐼𝐼

√2𝜋𝑟𝑐𝑜𝑠

𝜃

2 (2.11)

𝑤 =

2𝐾𝐼𝐼𝐼

𝜇√

𝑟

2𝜋𝑠𝑖𝑛

𝜃

2


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and KIII are a function of the geometry configuration of the problem and the loading conditions of the

body. The stress field near the crack tip is dependent in a linear manner by the constant of stress intensity

factor. If the stress intensity factor multiplied by a value A the same does and the stress field near to the

crack tip (Kundu, 2008). For all the crack extension modes it is remarkable that when r → 0, at the

original point of the coordinate system at the crack in this case the stress field tends towards infinity.

This phenomenon is known as stress singularity and it is a geometrically discontinuity of the crack

(Zhuang, et al., 2014).

Because of the extremely high stress values near the crack tip the prediction that a material failure

occurs for a stress levels bigger than the ultimate stress or yield stress is incorrect. Those criteria are

not applicable in this case. Despite the high stress values near the crack tip bigger than the critical value

of the material the plate does not fail a priori. A failure occurred after the applied loading exceeds a

critical value. The comparison of the constants of stress intensity factors with a critical value Kc is

applicable in this case. The constant Kc is known as critical stress intensity factor or material fracture

toughness. Kc is a material property and independent from the geometry and the loading whereas KI

and KII are. Under continuous increasing loading the stress intensity factors are increased proportionally

until the critical value of fracture toughness when a fracture occurs (Kundu, 2008).

A number of methods are considered related to the stress intensity factor determination. These methods

can be divided to:

1. Theoretical- Analytical (Westergaard semi-inverse method and method of complex potentials)

2. Numerical (Green’s function, alternating method, boundary collocation technique, integral

transforms, finite element methods)

3. Experimental (photoelasticity, holography) (Gdoutos, 1993)

In a small number of problems with simple geometrical configurations of cracks (line, circular and

elliptical cracks within two dimensional plates) and boundary conditions analytical methods of the

theory of elasticity can be applied. Otherwise for more complex conditions numerical tools (including

the extended finite element method) or experimental techniques are required (Kundu, 2008)

In addition for the stress intensity factor calculation references and manuals are available. These tools

applied for various geometries and loading conditions. The units of the stress intensity factor are

[𝑁 𝑚]−3

2 (Zhuang, et al., 2014).

2.7 Material Fracture Toughness

Particularly for opening mode I cracks the value of tension stress close to the crack tip with enough

energy to open the crack surface is,


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where ∝ is a geometrical parameter, Kc the fracture toughness and α is the length of the crack. The ∝

values can be estimated from a handbook for different conditions or calculated with analytic or

numerical solution.

With reference to the Griffith’s energy balance criterion the tension stress is equal to

𝜎𝑓 = √𝛦 𝐺𝐼𝐶

𝜋𝛼 (2.13)

where GIC is known as energy release rate which is compared to the total strain energy before and after

the crack propagation.

The relation between the energy release rate GIC with the stress intensity factor is very important in

fracture mechanics. Using the Eq. (2.12) and (2.13), assuming unity ∝ and with ν considering the

Poisson’s ratio the energy release rate with the stress intensity factor are related as,

𝐸

1 − 𝜈2𝐺𝐼𝐶 = 𝐾𝐼𝐶

2 (2.14)

During the fracture process energy is released. This amount of energy is consumed at the crack tip

location. The critical strain energy release rate is known as Gcr is the strain energy release rate is

required during the energy consumption process. The Eq. (2.9) can be written as

𝜎𝑓 = √𝛦 𝐺𝑐𝑟

(1 − 𝜈2)𝜋𝛼 (2.15)

The above equation combines the material, the crack size and the stress value (Zhuang, et al., 2014).

𝜎𝑓 =

𝐾𝐼 𝐶

∝ √𝜋𝑎 (2.12)

Ch.3 X-FEM Concept & Formulation

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3. X-FEM Concept & Formulation

3.1 Introduction to X-FEM

As discussed in the previous chapter there was appeared a need for calculation the stress intensity factor

and the fracture mechanisms for complex problems, for those analytical methods are not applicable.

Numerical methods are more reliable and time cost- effective than the experimental in that case and a

lot of methods with the simultaneous technological growth appeared.

One of the most popular and commercial numerical tools is the finite element method (FEM). With

FEM stress and strain fields can be calculated in fast and not computational cost expensive way. The

FEM has significant applications in a wide range of engineering problems as well as scientific. The

domain is divided to regions named elements. The independent elements are connected together with a

topological map named mesh. With FEM complex geometries and structures can be handled easily

hence it became and it is up to now the most popular computational method (Khoei, 2015).

However the FEM suffers from certain disadvantages. For instance, FEM requires smooth linear fields

for accurate approximations. In cases there are non- smooth non-linear behaviors in the field of stresses

caused by strong geometrical discontinuities, such as cracks, FEM struggles to give reliable results

without huge computational costs (Khoei, 2015).

In fracture mechanics problems the stress distribution near the crack tip is considered to be infinity as

it is already proven. In the FEM the strong discontinuous fields near the crack tip are captured by local

refining of the mesh. In this case the number of degrees of freedom are increased. Moreover the crack

growth needs to be re-meshed frequently. This process causes a lot to computational cost and may affect

the accuracy of the approximate solution. It is seem that the conventional finite element techniques

achieved their limit for fracture mechanics problems (Khoei, 2015).

Trying to overcome these computational difficulties the first thought was the application of the

asymptotic stress field near the crack tip in the finite element analysis. Afterwards Dolbow (1999)

proposed an enrichment method for the crack modelling which was named “extended finite element

method” (X-FEM). In the X-FEM special functions such as the asymptotic displacement fields are

added for crack modelling. Applying this approach the location of the discontinuity can be arbitrary and

its propagation does not require local remesh. This technique is capable of providing more accurate

results within less time. X-FEM has proved a very powerful tool for cracks, discontinuities and fields

require enriching solutions. With X-FEM the power of the conventional FEM has enhanced with a great

range of applications and for complex domain configurations. In this chapter a review of the X-FEM

shall try with an emphasis on the field of linear elastic fracture mechanics (Khoei, 2015).


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3.2 A review on X-FEM

3.2.1 X-FEM Development

Due to its ability to manage the discontinuous displacement field without the need of remeshing the X-

FEM gained significant attention during the last decade. With X-FEM an accurate approximation can

be achieved in cases of jumps, singularities and non-smooth elements behavior. The adding enriched

terms in the conventional finite element basis are responsible for that. The non- smooth and non- linear

features are not related to the mesh and are captured independently (Khoei, 2015).

(Belytschko & Black, 1999) were the first they introduced an enriched finite element technique for

crack propagation without remeshing. The elements close to the crack tip and along the crack surface

are enriched with special functions which can be considered as an asymptotic displacement field using

the partition of unity theory. Afterwards a better and more straightforward approach was introduced by

(Moës, et al., 1999). They achieved with an elegant way to isolate the discontinuous displacement field

across the crack surface form crack tip itself. Next (Daux, et al., 2000) made one step more introducing

a concept is known as junction function concept. In this way they achieved to calculate multiple

branched cracks. The method was named extended finite element method (X-FEM) for the first time.

The geometric entities are no longer needed to be meshed. The technique was employed was very

promising as there is no need for conformation of cracks, voids or inhomogeneities to the initial mesh

configuration. The non-smooth displacement fields along the crack surface as well as the asymptotic

displacement fields at the crack tip are added to the finite element basis using the partition of unity

method. The additional degrees of freedom (DOF) within the enriched nodes are independent.

Additionally in the X-FEM the use of higher order elements or elements with some special functions is

reasonable (Khoei, 2015).

Figure 3: 3D modelling with extended finite element method (Areias & Belytschko, 2005)


Page | 12

The X-FEM very soon found a lot of computational applications in various engineering fields as well

as in physics disciplines because of the advantageous technique was introduced. Bordas and Legay

(2005) released the first open X-FEM codes while Dunant et al. (2007) were focused on the Method’s

efficiency aspects. Leading computational software companies such as LS-DYNA and ABAQUS

adapted X-FEM technique to their software packages. X-FEM is a robust technique based on previous

mesh free techniques. A lot of papers and articles have published as well as the first books summarizing

the development and the applications of the technique up to now (Khoei, 2015).

3.2.2 X-FEM Applications on other Fields

It is worth to be referred that X-FEM finds a wide range of applications in other fields apart from its

implementation in fracture mechanics and particularly in linear elastic fracture mechanics. The present

project focuses on the synergy of the X-FEM with fracture mechanics. Hence in-depth analysis and

examples of the implementation of the X-FEM in other fields will not shall try.

X-FEM is coupling with Level Set Method (LSM), is applying in cohesive fracture mechanics, in

composite materials and inhomogeneities, as well as in plasticity, damage and fatigue problems. Also

applications of the X-FEM can be found in shear band localization, fluid- structure interaction

(Pommier, et al., 2013).

3.3 The Partition of Unity as basis of the X-FEM approximation

The X-FEM is based on the partition of unity method. The main idea is the enrichment of the

conventional finite element basis to represent a given function ψ(χ) on a known domain Ω with the

below equation (Zhuang, et al., 2014):

𝜓(𝜒) = ∑ 𝑁𝐼(𝝌)𝛷(𝝌)

𝐼

(3.1)

where ΣI NI(χ)=1 the partition of unity to be satisfied.

Trying for the best approximation of the enriched function the constant cI was added with a result the

above Eq. (3.1) can be expressed as (Zhuang, et al., 2014):

𝜓(𝜒) = ∑ 𝑁𝐼(𝝌)cI𝛷(𝝌)

𝐼

(3.2)

Hence the conventional displacement field is enriched by those quantities to describe more sophisticated

displacement fields. In X-FEM the discontinuous displacement fields can be described with two parts

as bellow (Zhuang, et al., 2014):


Page | 13

𝑢ℎ = ∑ 𝑁𝐼(𝝌)𝑢𝐼 + 𝜓(𝝌)

𝐼

(3.3)

where NI indicates the conventional finite element polynomial basis, uI the conventional finite element

degrees of freedom whereas the ψ(χ) term expresses the enrichment. Applying the partition of unity Eq

(3.3) can be expressed as (Belytschko & Black, 1999):

𝑢ℎ = ∑ 𝑁𝐼(𝝌)𝑢𝐼

𝐼

+ ∑ 𝑁𝐽(𝝌)𝛷(𝝌)𝑞𝐽

𝐽

(3.4)

The first term indicates the conventional finite element approximation whereas the second term the

enriched approximation. The term qJ represents the enriched degrees of freedom added by the

enrichment function. It is remarkable that the term qJ does not have any physical meaning and it is

considered for more accurate approximations only. This enrichment finite element method technique

introduces the additional degrees of freedom at the element node in comparison with the conventional

finite element method. Considering the newly added function ψ(χ) equal to NJ(χ)Φ(χ) the Eq. (3.4) can

be written as below (Zhuang, et al., 2014):

𝑢ℎ = ∑ 𝑁𝐼(𝝌)𝑢𝐼

𝐼

+ ∑ 𝜓𝐽(𝝌)𝑞𝐽

𝐽

(3.5)

The shape function NJ on the right hand side of the Eq. (3.4) expresses the enrichment function may

differ from the function NI expresses the conventional finite element approximation. Moreover in Eq.

(3.4) the displacement field uh is a vector as a result uI and qJ are vectors as well with same order. The

enriched finite element technique is used in the X-FEM is similar to the p-order refinement of the

conventional finite element method technique. With the p-order refinement of the FEM the number of

elements are increased locally whereas with the X-FEM approach the feature of the shape function is

improved in general. In the p-order refinement technique the location of the added DOF is the inside

part of the element domain whereas in X-FEM they are located at the initial element nodes.

Figure 4: Cracks and discontinuities modelling i) 2D static crack edge ii) crack propagation with conventional FEM

refinement technique iii) crack propagation with enrichment X-FEM technique (Khoei, 2015)


Page | 14

With X-FEM technique strong and weak discontinuities can be modelled without considering their

geometries. In the X-FEM the mesh generation does not include the discontinuities. They are considered

separately and depended on their nature with enriched special functions in the finite element basis. The

technique targets on the simulation of the discontinuities (strong and weak) with the minimum possible

enrichment. The external boundary conditions in the X-FEM are considered at the mesh generation

whereas internal boundary conditions such as cracks are not considered at this phase. Applications of

the X-FEM’s enrichment technique include crack propagation, punching etc. (Khoei, 2015) (Pais,

2013).

At this point an explanation of the enrichment concept shall try. It is considered discontinuity Γd in a

domain Ω as shown in the following figure 3.2. The technique focuses on the finite element

approximation of the displacement field that can be discontinuous along the discontinuity Γd. The

conventional finite element technique focuses on the conformation of the line of the discontinuity with

appropriate mesh generation. The element edges in that case are aligned to the discontinuity. Though

this technique creates the discontinuity, troublesome phenomena appear if the discontinuity propagates

and evolves in time or its initial configuration should be changed or modified. In comparison in the X-

FEM the discontinuity is modelled with enriched functions and the circled nodes additional DOF as

presented in figure 3.2 (Khoei, 2015).

Figure 5: Enriched X-FEM technique for strong and weak discontinuities modelling (Khoei, 2015)

Ch.4 Numerical Study with X-FEM technique

Page | 15

4. Numerical Study with X-FEM technique

4.1 Static Crack Edge No Fracture (Steel)

4.1.1 Problem Description

In a 2-dimentional steel plate with thickness (plane stress condition) the same uniformly distributed

load is applied to the top and the bottom edge. The top edge is not allowed to translate in the horizontal

direction while the bottom edge in both horizontal and vertical. The static crack is not aligned with any

element edge. The problem has been analyzed with Abaqus whereas the mesh discretization of the

domain has been done with Ansys. In this case the maximum principal stress criterion (MAXPS) as

damage initiation criterion has been applied. The DOF of the elements close to the crack tip were

enriched with X-FEM technique.

4.1.2 Geometric and Material Properties

Width of Plate = w = 1m

Height of Plate = h = 1m

Young’s Modulus of Material = E = 210 GPa

Poisson’s Ratio = 𝜈 = 0.30

Ultimate strength of steel = 500 MPa

Length of crack = 0.25m

Elements per edge = 21

4.1.3 Boundary and Initial Conditions

Boundary Condition at top edge: Ux = 0; URz = 0

Boundary Condition at bottom edge: Ux = 0; Uy = 0; URz = 0

Uniformly distributed load at top and bottom edge: σ = 10 N/m


Page | 16

4.1.4 Results

Figure 6: Fracture Problem 1 Von Misses Stresses Distribution

Figure 7: Fracture Problem 1 S11 Principal Stresses Distribution

The Von Misses stresses, S11 and S22 stresses are maximum on the right hand side of the crack edge as

presented in the figures 6, 7 and 8. The magnitude of the displacements field is maximum at the above

left corner of the domain as shown in figure 9.


Page | 17


Figure 9: Fracture Problem 1 Magnitude of Displacements

4.2 Static Crack Edge Fracture (Steel)


This problem is described in the same way, methodology and failure criteria with the first case study.

The difference in this problem is the applied pressure which is much bigger (σ = 10 kN/m) than the


Page | 18

pressure applied in the first problem. The DOF of the elements close to the crack were enriched with

X-FEM technique.






Ultimate tensile strength of aluminium = 500 MPa

Length of crack = 0.25m



Boundary Condition at top edge: Ux = 0; URz = 0

Boundary Condition at bottom edge: Ux = 0; Uy = 0; URz = 0

Uniformly distributed load at top and bottom edge: σ = 10 kN/m

4.2.4 Results



Page | 19



The crack tip opens due to the distributed loading condition The Von Misses stresses, S11 and S22

stresses are maximum on the right hand side of the crack edge as presented in the figures 10, 11 and 12.

The magnitude of the displacements field is maximum at the above left corner of the domain as shown

on figure 13.


Page | 20


4.3 Internal Crack No Fracture (Aluminium)


In a 2-dimentional aluminium plate with thickness (plane stress condition) uniformly distributed load

is applied at the right edge. The left edge is not allowed to translate in both horizontal and vertical

direction. The internal of the domain crack has 0.4 m length. The problem has been analyzed with

Abaqus whereas the mesh discretization of the domain has been done with Ansys. In this case the

maximum principal stress criterion (MAXPS) as damage initiation criterion has been applied. The DOF

of the elements close to the crack were enriched with X-FEM technique.







Length of crack = 0.8 m



Page | 21


Boundary Condition at left edge: Ux = 0; Uy = 0; URz = 0

Uniformly distributed load at right edge: σ = 10 N/m

4.3.4 Results




Page | 22

The Von Misses stresses and the S11 principal stresses are maximum at the areas above and below the

internal crack as presented in figures 14 and 15. The S22 principal stresses are maximum at the right of

the domain edge centrally as shown on figure 16.



The magnitude of the displacements field is maximum on the right hand side of the internal crack to

the right of the domain edge as shown on figure 17.


Page | 23

4.4 Internal Crack Fracture (Aluminium)


This problem is described in the same way, methodology and failure criteria with the third case study.

The difference in this problem is the applied pressure is much bigger (σ = 10 kN/m) than the pressure

applied in the third problem. The DOF of the elements close to the crack were enriched with X-FEM

technique.







Length of crack = 0.8 m



Boundary Condition at left edge: Ux = 0; Uy = 0; URz = 0

Uniformly distributed load at right edge: σ = 10 kN/m


Page | 24

4.4.4 Results


The Von Misses stresses and the S11 principal stresses are maximum at the areas above and below the

internal crack as presented in figures 4.4.1 and 4.4.2.



Page | 25


The S22 principal stresses are maximum at the right of the domain edge centrally as shown on figure

4.4.3. , whereas The magnitude of the displacements field is maximum on the right hand side of the

internal crack to the right of the domain edge as presented in figure 4.4.4.


Ch.5 Conclusions & Recommendations

Page | 26

5. Conclusions & Recommendations

No doubt the conventional Finite Element Method (FEM) is the most popular and commercial Finite

Element tool nowadays. As it is proven already the conventional FEM achieved its limits in cases of

fracture problems. This occurs because the classical FEM requires smooth and linear fields of stresses

and displacements. In fracture problems where strong discontinuous fields are observed especially near

the crack tip where the stress field is theoretically infinite conventional finite element method could not

give reasonable results without huge computational cost and time consuming attempts.

In the present project the approach of the Extended Finite Element Method (X-FEM) has been analyzed

as part of better and more accurate fracture modelling. X-FEM does not require any refinement of the

mesh locally even near the crack tip. In X-FEM there are additional degrees of freedom (DOF) for the

elements are enriched close to the non-smooth discontinuous field.

For more spherically understanding a holistic approach has been attempted for the scope of the present

project. Thus, fundamental linear elastic fracture mechanics aspects have been provided in chapter 2 as

well as remarkable references to the X-FEM technique, concepts and formulations.

As part of this project few work bench fracture problems have been selected, modelled and presented

in chapter 4. For each fracture problem a unique file has been created in notepad. The mesh

discretization has been carried out beforehand in Ansys. Afterwards the input file imported to Abaqus

where the analysis run. The results of the analysis are presented in figures are reasonable. For instance

the maximum value of the Von misses and principal stress field is near to the crack tip. This is something

it is known beforehand from the analytic solutions.

For the analysis of the fracture problems are presented in this project 4- node solid elements have been

selected. For further study could be very interesting and promising the modelling of more complex 3-

Dimentional fracture problems. By the research has been made shell elements are applicable with the

Extended Finite Element Method (X-FEM) as well as Abaqus software. In the near future more complex

geometries and initial conditions could be modelled using shell elements. Complex 3- Dimensional

geometries and shell elements were considered out of the scope of the present project due to the time

limitation.

In addition, a comparison of the results of the work bench problems are presented in this project could

be attempt using LS-DYNA the other software has adapted the X-FEM technique in its package.

Extended Finite Element Method (X-FEM) is considered a very strong computational tool for over a

decade with solid implementation in the field of fracture mechanics. Fracture problems can be modelled

Ch5. Conclusions & Recommendations

Page | 27

with X-FEM technique in a reliable way with accurate results whereas huge difficulties of the

conventional FEM gaps could overcome.

Bibliography

Page | 28

6. Bibliography

Areias, P. & Belytschko, T., 2005. Analysis of three-dimensional crack initiation and propagation using

the extended finite element method. International Journal forNumerical Methods in Engineering, Issue

63, pp. 760-788.

Belytschko, T. & Black, T., 1999. Elastic crack growth in finite elements with minimal remeshing.

Internation Journal for Numerical Methods in Engineering, Issue 45, pp. 601-620.

Daux, C., Moës, N. & Dolbow, J., 2000. Arbitrary branched and intersecting cracks with the extended

finite element method. International Journal for Numerical Methods in Engineering, Issue 48, p. 1741–

1760.

Gdoutos, E. E., 1990. Fracture Mechanics Criteria and Applications. Dordrecht: Kluwe academic

Publishers.

Gdoutos, E. E., 1993. Fracture Mechanics: An introduction. Dordrecht: Kluwer academic Publishers.

Khoei, A. R., 2015. Extended finite element method: theory and applications. Chichester, West Sussex,

United Kingdom: Wiley.

Kundu, T., 2008. Fundamentals of fracture mechanics. Boca Raton: CRC Press.

Moës, N., Dolbow, J. & Belytschko, T., 1999. A finite element method for crack growth without

remeshing. International Journal for Numerical Methods in Engineering, Issue 46, pp. 131-150.

Mohammadi, S., 2008. Extended finite element method for fracture analysis of structures. Tehran:

Blackwell Publishing Ltd.

Pais, M., 2013. [Online].

Pommier, S., Gravouil, A., Combescure, A. & Moes, N., 2013. Extended Finite Element Method for

Crack Propagation. London: Iste.

Zhuang, Z., Liu, Z., Cheng, B. & Liao, J., 2014. Extended Finite Element Method. 1st ed. Oxford:

Elsevier.

Appendix A

Page | 29

Appendix A Static Crack Edge No Fracture (Steel)

*Heading

** Job name: StaticCrackEdgeNoFracture Model name: Model-1

** Generated by: Abaqus/CAE 6.13

*Preprint, echo=NO, model=NO, history=NO, contact=NO

**

** PARTS

**

*Part, name=Crack

*End Part

**

*Part, name=Plate

*Node

1, -1., -1.

2, 1., -1.

3, -0.90476, -1.

4, -0.80952, -1.

5, -0.71429, -1.

6, -0.61905, -1.

7, -0.52381, -1.

8, -0.42857, -1.

9, -0.33333, -1.

10, -0.23810, -1.

--------------------------------------------------------------

475, 0.90476, 0.04762

476, 0.90476, 0.14286

477, 0.90476, 0.23810

478, 0.90476, 0.33333

479, 0.90476, 0.42857

480, 0.90476, 0.52381

481, 0.90476, 0.61905

Appendix A

Page | 30

482, 0.90476, 0.71429

483, 0.90476, 0.80952

484, 0.90476, 0.90476

*Element, type=CPS4R

1, 1, 3, 85, 84

2, 3, 4, 105, 85

3, 4, 5, 125, 105

4, 5, 6, 145, 125

5, 6, 7, 165, 145

6, 7, 8, 185, 165

7, 8, 9, 205, 185

8, 9, 10, 225, 205

9, 10, 11, 245, 225

10, 11, 12, 265, 245

--------------------------------------------------------------

432, 304, 324, 53, 54

433, 324, 344, 52, 53

434, 344, 364, 51, 52

435, 364, 384, 50, 51

436, 384, 404, 49, 50

437, 404, 424, 48, 49

438, 424, 444, 47, 48

439, 444, 464, 46, 47

440, 464, 484, 45, 46

441, 484, 43, 23, 45

*Nset, nset=_PickedSet2, internal, generate

1, 484, 1

*Elset, elset=_PickedSet2, internal, generate

1, 441, 1

** Section: Main

*Solid Section, elset=_PickedSet2, material=Steel

1.,

*End Part

**

Appendix A

Page | 31

**

** ASSEMBLY

**

*Assembly, name=Assembly

**

*Instance, name=Part-1-1, part=Plate

*End Instance

**

*Instance, name=Part-2-1, part=Crack

*End Instance

**

*Nset, nset=_PickedSet6, internal, instance=Part-1-1, generate

1, 484, 1

*Elset, elset=_PickedSet6, internal, instance=Part-1-1, generate

1, 441, 1

*Nset, nset=_PickedSet10, internal, instance=Part-1-1

22,


484,

*Elset, elset=__PickedSurf8_S3, internal, instance=Part-1-1, generate

420, 441, 1

*Surface, type=ELEMENT, name=_PickedSurf8, internal

__PickedSurf8_S3, S3


1, 21, 1



*Enrichment, name=Crack-1, type=PROPAGATION CRACK, elset=_PickedSet6

*End Assembly

**

** MATERIALS

**

*Material, name=Steel

*Damage Initiation, criterion=MAXPS

Appendix A

Page | 32

5e+08,

*Damage Evolution, type=DISPLACEMENT

1.,

*Elastic

21e+10, 0.30

*Initial condition, type=ENRICHMENT

Part-1-1.215, 1, Crack-1, -0.01

Part-1-1.215, 2, Crack-1, -0.01

Part-1-1.215, 3, Crack-1, 0.01

Part-1-1.215, 4, Crack-1, 0.01

Part-1-1.214, 1, Crack-1, -0.01

Part-1-1.214, 2, Crack-1, -0.01

Part-1-1.214, 3, Crack-1, 0.01

Part-1-1.214, 4, Crack-1, 0.01

Part-1-1.213, 1, Crack-1, -0.01

Part-1-1.213, 2, Crack-1, -0.01

Part-1-1.213, 3, Crack-1, 0.01

Part-1-1.213, 4, Crack-1, 0.01

Part-1-1.212, 1, Crack-1, -0.01

Part-1-1.212, 2, Crack-1, -0.01

Part-1-1.212, 3, Crack-1, 0.01

Part-1-1.212, 4, Crack-1, 0.01

Part-1-1.211, 1, Crack-1, -0.01

Part-1-1.211, 2, Crack-1, -0.01

Part-1-1.211, 3, Crack-1, 0.01

Part-1-1.211, 4, Crack-1, 0.01

** --------------------------------------------------------------

** STEP: Loading

**

*Step, name=Loading

*Static

1., 1., 1e-05, 1.

**

** BOUNDARY CONDITIONS

Appendix A

Page | 33

**

** Name: FixedBRC Type: Displacement/Rotation

*Boundary

_PickedSet10, 1, 1

_PickedSet10, 2, 2

_PickedSet10, 6, 6

** Name: RollerTRC Type: Displacement/Rotation

*Boundary

_PickedSet11, 1, 1

_PickedSet11, 6, 6

**

** LOADS

**

** Name: BottomPressure Type: Pressure

*Dsload

_PickedSurf9, P, -10.

** Name: Top Pressure Type: Pressure

*Dsload


**

** INTERACTIONS

**

** Interaction: Growth

*Enrichment Activation, name=Crack-1, activate=ON

**

** OUTPUT REQUESTS

**

*Restart, write, frequency=0

**

** FIELD OUTPUT: F-Output-1

**

*Output, field

*Node Output

CF, PHILSM, RF, U

Appendix A

Page | 34

*Element Output, directions=YES

LE, PE, PEEQ, PEMAG, S

*Contact Output

CDISP, CSTRESS

**

** HISTORY OUTPUT: H-Output-1

**

*Output, history, variable=PRESELECT

*End Step

Appendix B

Page | 35

Appendix B Static Crack Edge Fracture (Steel)

*Heading

** Job name: StaticCrackEdgeFracture Model name: Model-1



**

** PARTS

**

*Part, name=Crack

*End Part

**

*Part, name=Plate

*Node

1, -1., -1.

2, 1., -1.

3, -0.90476, -1.

4, -0.80952, -1.

5, -0.71429, -1.

6, -0.61905, -1.

7, -0.52381, -1.

8, -0.42857, -1.

9, -0.33333, -1.

10, -0.23810, -1.

--------------------------------------------------------------

475, 0.90476, 0.04762

476, 0.90476, 0.14286

477, 0.90476, 0.23810

478, 0.90476, 0.33333

479, 0.90476, 0.42857

480, 0.90476, 0.52381

481, 0.90476, 0.61905

Appendix B

Page | 36

482, 0.90476, 0.71429

483, 0.90476, 0.80952

484, 0.90476, 0.90476


1, 1, 3, 85, 84

2, 3, 4, 105, 85

3, 4, 5, 125, 105

4, 5, 6, 145, 125

5, 6, 7, 165, 145

6, 7, 8, 185, 165

7, 8, 9, 205, 185

8, 9, 10, 225, 205

9, 10, 11, 245, 225

10, 11, 12, 265, 245

--------------------------------------------------------------

432, 304, 324, 53, 54

433, 324, 344, 52, 53

434, 344, 364, 51, 52

435, 364, 384, 50, 51

436, 384, 404, 49, 50

437, 404, 424, 48, 49

438, 424, 444, 47, 48

439, 444, 464, 46, 47

440, 464, 484, 45, 46

441, 484, 43, 23, 45


1, 484, 1


1, 441, 1

** Section: Main

*Solid Section, elset=_PickedSet2, material=Steel

1.,

*End Part

**

Appendix B

Page | 37

**

** ASSEMBLY

**


**

*Instance, name=Part-1-1, part=Plate

*End Instance

**

*Instance, name=Part-2-1, part=Crack

*End Instance

**

*Nset, nset=_PickedSet6, internal, instance=Part-1-1, generate

1, 484, 1

*Elset, elset=_PickedSet6, internal, instance=Part-1-1, generate

1, 441, 1


22,


484,


420, 441, 1




1, 21, 1



*Enrichment, name=Crack-1, type=PROPAGATION CRACK, elset=_PickedSet6

*End Assembly

**

** MATERIALS

**

*Material, name=Steel


Appendix B

Page | 38

5e+08,


1.,

*Elastic

21e+10, 0.30


Part-1-1.215, 1, Crack-1, -0.01

Part-1-1.215, 2, Crack-1, -0.01

Part-1-1.215, 3, Crack-1, 0.01

Part-1-1.215, 4, Crack-1, 0.01

Part-1-1.214, 1, Crack-1, -0.01

Part-1-1.214, 2, Crack-1, -0.01

Part-1-1.214, 3, Crack-1, 0.01

Part-1-1.214, 4, Crack-1, 0.01

Part-1-1.213, 1, Crack-1, -0.01

Part-1-1.213, 2, Crack-1, -0.01

Part-1-1.213, 3, Crack-1, 0.01

Part-1-1.213, 4, Crack-1, 0.01

Part-1-1.212, 1, Crack-1, -0.01

Part-1-1.212, 2, Crack-1, -0.01

Part-1-1.212, 3, Crack-1, 0.01

Part-1-1.212, 4, Crack-1, 0.01

Part-1-1.211, 1, Crack-1, -0.01

Part-1-1.211, 2, Crack-1, -0.01

Part-1-1.211, 3, Crack-1, 0.01

Part-1-1.211, 4, Crack-1, 0.01

** --------------------------------------------------------------

** STEP: Loading

**

*Step, name=Loading

*Static

1., 1., 1e-05, 1.

**


Appendix B

Page | 39

**

** Name: FixedBRC Type: Displacement/Rotation

*Boundary

_PickedSet10, 1, 1

_PickedSet10, 2, 2

_PickedSet10, 6, 6

** Name: RollerTRC Type: Displacement/Rotation

*Boundary

_PickedSet11, 1, 1

_PickedSet11, 6, 6

**

** LOADS

**

** Name: BottomPressure Type: Pressure

*Dsload


** Name: Top Pressure Type: Pressure

*Dsload


**

** INTERACTIONS

**

** Interaction: Growth

*Enrichment Activation, name=Crack-1, activate=ON

**

** OUTPUT REQUESTS

**


**


**

*Output, field

*Node Output

CF, PHILSM, RF, U

Appendix B

Page | 40



*Contact Output

CDISP, CSTRESS

**


**


*End Step

Appendix C

Page | 41

Appendix C Internal Crack No Fracture (Aluminium)

*Heading

** Job name: InternalCrackNoFracture Model name: Model-1



**

** PARTS

**

*Part, name=Crack

*End Part

**

*Part, name=Plate

*Node

1, -0.50000, -0.50000

2, 0.50000, -0.50000

3, -0.45000, -0.50000

4, -0.40000, -0.50000

5, -0.35000, -0.50000

6, -0.30000, -0.50000

7, -0.25000, -0.50000

8, -0.20000, -0.50000

9, -0.15000, -0.50000

10, -0.10000, -0.50000

--------------------------------------------------------------

432, 0.45000, -0.11102E-15

433, 0.45000, 0.50000E-01

434, 0.45000, 0.10000

435, 0.45000, 0.15000

436, 0.45000, 0.20000

437, 0.45000, 0.25000

438, 0.45000, 0.30000

Appendix C

Page | 42

439, 0.45000, 0.35000

440, 0.45000, 0.40000

441, 0.45000, 0.45000


1, 1, 3, 81, 80

2, 3, 4, 100, 81

3, 4, 5, 119, 100

4, 5, 6, 138, 119

5, 6, 7, 157, 138

6, 7, 8, 176, 157

7, 8, 9, 195, 176

8, 9, 10, 214, 195

9, 10, 11, 233, 214

10, 11, 12, 252, 233

--------------------------------------------------------------

391, 270, 289, 51, 52

392, 289, 308, 50, 51

393, 308, 327, 49, 50

394, 327, 346, 48, 49

395, 346, 365, 47, 48

396, 365, 384, 46, 47

397, 384, 403, 45, 46

398, 403, 422, 44, 45

399, 422, 441, 43, 44

400, 441, 41, 22, 43


1, 441, 1


1, 400, 1

** Section: Aluminium

*Solid Section, elset=_PickedSet2, material=Aluminium

1.,

*End Part

**

Appendix C

Page | 43

**

** ASSEMBLY

**


**

*Instance, name=Plate-1, part=Plate

*End Instance

**

*Instance, name=Crack-1, part=Crack

*End Instance

**

*Nset, nset=_PickedSet4, internal, instance=Plate-1, generate

62, 80, 1

*Elset, elset=_PickedSet4, internal, instance=Plate-1, generate

1, 381, 20


1, 441, 1


1, 400, 1


1, 441, 1


1, 400, 1

*Elset, elset=__PickedSurf5_S2, internal, instance=Plate-1, generate

20, 400, 20



*Enrichment, name=Crack, type=PROPAGATION CRACK, elset=_PickedSet10

*End Assembly

**

** MATERIALS

**

*Material, name=Aluminium


Appendix C

Page | 44

5e+08,


1.,

*Elastic

7e+10, 0.33

*Expansion

2.52e-05,


Plate-1.115, 1, Crack, -0.1

Plate-1.115, 2, Crack, 0.1

Plate-1.115, 3, Crack, 0.1

Plate-1.115, 4, Crack, -0.1

Plate-1.135, 1, Crack, -0.1

Plate-1.135, 2, Crack, 0.1

Plate-1.135, 3, Crack, 0.1

Plate-1.135, 4, Crack, -0.1

Plate-1.155, 1, Crack, -0.1

Plate-1.155, 2, Crack, 0.1

Plate-1.155, 3, Crack, 0.1

Plate-1.155, 4, Crack, -0.1

Plate-1.175, 1, Crack, -0.1

Plate-1.175, 2, Crack, 0.1

Plate-1.175, 3, Crack, 0.1

Plate-1.175, 4, Crack, -0.1

Plate-1.195, 1, Crack, -0.1

Plate-1.195, 2, Crack, 0.1

Plate-1.195, 3, Crack, 0.1

Plate-1.195, 4, Crack, -0.1

Plate-1.215, 1, Crack, -0.1

Plate-1.215, 2, Crack, 0.1

Plate-1.215, 3, Crack, 0.1

Plate-1.215, 4, Crack, -0.1

Plate-1.235, 1, Crack, -0.1

Plate-1.235, 2, Crack, 0.1

Appendix C

Page | 45

Plate-1.235, 3, Crack, 0.1

Plate-1.235, 4, Crack, -0.1

Plate-1.255, 1, Crack, -0.1

Plate-1.255, 2, Crack, 0.1

Plate-1.255, 3, Crack, 0.1

Plate-1.255, 4, Crack, -0.1

Plate-1.275, 1, Crack, -0.1

Plate-1.275, 2, Crack, 0.1

Plate-1.275, 3, Crack, 0.1

Plate-1.275, 4, Crack, -0.1

**

** ----------------------------------------------------------------

**

** STEP: Loading

**

*Step, name=Loading

*Static

0.1, 1., 1e-05, 0.1

**


**

** Name: BC-1 Type: Displacement/Rotation

*Boundary

_PickedSet4, 1, 1

_PickedSet4, 2, 2

_PickedSet4, 6, 6

**

** LOADS

**

** Name: Pressure Type: Pressure

*Dsload


**

**

Appendix C

Page | 46

** OUTPUT REQUESTS

**


**


**

*Output, field

*Node Output

CF, NT, PHILSM, RF, U



*Contact Output

CDISP, CSTRESS

**


**


*End Step

Appendix D

Page | 47

Appendix D Internal Crack Fracture (Aluminium)

*Heading

** Job name: InternalCrackFracture Model name: Model-1



**

** PARTS

**

*Part, name=Crack

*End Part

**

*Part, name=Plate

*Node

1, -0.50000, -0.50000

2, 0.50000, -0.50000

3, -0.45000, -0.50000

4, -0.40000, -0.50000

5, -0.35000, -0.50000

6, -0.30000, -0.50000

7, -0.25000, -0.50000

8, -0.20000, -0.50000

9, -0.15000, -0.50000

10, -0.10000, -0.50000

----------------------------------------------------------------

432, 0.45000, -0.11102E-15

433, 0.45000, 0.50000E-01

434, 0.45000, 0.10000

435, 0.45000, 0.15000

436, 0.45000, 0.20000

437, 0.45000, 0.25000

438, 0.45000, 0.30000

Appendix D

Page | 48

439, 0.45000, 0.35000

440, 0.45000, 0.40000

441, 0.45000, 0.45000


1, 1, 3, 81, 80

2, 3, 4, 100, 81

3, 4, 5, 119, 100

4, 5, 6, 138, 119

5, 6, 7, 157, 138

6, 7, 8, 176, 157

7, 8, 9, 195, 176

8, 9, 10, 214, 195

9, 10, 11, 233, 214

10, 11, 12, 252, 233

----------------------------------------------------------------

391, 270, 289, 51, 52

392, 289, 308, 50, 51

393, 308, 327, 49, 50

394, 327, 346, 48, 49

395, 346, 365, 47, 48

396, 365, 384, 46, 47

397, 384, 403, 45, 46

398, 403, 422, 44, 45

399, 422, 441, 43, 44

400, 441, 41, 22, 43


1, 441, 1


1, 400, 1

** Section: Aluminium

*Solid Section, elset=_PickedSet2, material=Aluminium

1.,

*End Part

**

Appendix D

Page | 49

**

** ASSEMBLY

**


**

*Instance, name=Plate-1, part=Plate

*End Instance

**

*Instance, name=Crack-1, part=Crack

*End Instance

**


62, 80, 1


1, 381, 20


1, 441, 1


1, 400, 1


1, 441, 1


1, 400, 1

*Elset, elset=__PickedSurf5_S2, internal, instance=Plate-1, generate

20, 400, 20



*Enrichment, name=Crack, type=PROPAGATION CRACK, elset=_PickedSet10

*End Assembly

**

** MATERIALS

**

*Material, name=Aluminium


Appendix D

Page | 50

5e+08,


1.,

*Elastic

7e+10, 0.33

*Expansion

2.52e-05,


Plate-1.115, 1, Crack, -0.1

Plate-1.115, 2, Crack, 0.1

Plate-1.115, 3, Crack, 0.1

Plate-1.115, 4, Crack, -0.1

Plate-1.135, 1, Crack, -0.1

Plate-1.135, 2, Crack, 0.1

Plate-1.135, 3, Crack, 0.1

Plate-1.135, 4, Crack, -0.1

Plate-1.155, 1, Crack, -0.1

Plate-1.155, 2, Crack, 0.1

Plate-1.155, 3, Crack, 0.1

Plate-1.155, 4, Crack, -0.1

Plate-1.175, 1, Crack, -0.1

Plate-1.175, 2, Crack, 0.1

Plate-1.175, 3, Crack, 0.1

Plate-1.175, 4, Crack, -0.1

Plate-1.195, 1, Crack, -0.1

Plate-1.195, 2, Crack, 0.1

Plate-1.195, 3, Crack, 0.1

Plate-1.195, 4, Crack, -0.1

Plate-1.215, 1, Crack, -0.1

Plate-1.215, 2, Crack, 0.1

Plate-1.215, 3, Crack, 0.1

Plate-1.215, 4, Crack, -0.1

Plate-1.235, 1, Crack, -0.1

Plate-1.235, 2, Crack, 0.1

Appendix D

Page | 51

Plate-1.235, 3, Crack, 0.1

Plate-1.235, 4, Crack, -0.1

Plate-1.255, 1, Crack, -0.1

Plate-1.255, 2, Crack, 0.1

Plate-1.255, 3, Crack, 0.1

Plate-1.255, 4, Crack, -0.1

Plate-1.275, 1, Crack, -0.1

Plate-1.275, 2, Crack, 0.1

Plate-1.275, 3, Crack, 0.1

Plate-1.275, 4, Crack, -0.1

**

** ----------------------------------------------------------------

**

** STEP: Loading

**

*Step, name=Loading

*Static

0.1, 1., 1e-05, 0.1

**


**

** Name: BC-1 Type: Displacement/Rotation

*Boundary

_PickedSet4, 1, 1

_PickedSet4, 2, 2

_PickedSet4, 6, 6

**

** LOADS

**

** Name: Pressure Type: Pressure

*Dsload


**

**

Appendix D

Page | 52

** OUTPUT REQUESTS

**


**


**

*Output, field

*Node Output

CF, NT, PHILSM, RF, U



*Contact Output

CDISP, CSTRESS

**


**


*End Step

Documents

Master's Thesis