Crack propagation in concrete dams driven by internal water pressure

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Crack propagation in concrete dams driven by internal water pressure

José Sanchez Loarte & Maria Sohrabi

June 2017

TRITA-BKN. Master Thesis 522, Concrete Structures 2017 ISSN 1103-4297,

ISRN KTH/BKN/EX-522-SE

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Department of Civil and Architectural Engineering Division of Concrete Structures

Stockholm, Sweden, 2017

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Abstract

Concrete structures are in general expected to be subjected to cracking during its service life.

This is the reason why concrete is reinforced, where the reinforcement is only activated after cracks occur. However, cracks may be a concern in large concrete structures, such as dams, since it may result in reduced service life. The underlying mechanisms behind crack formations are well known at present day. On the other hand, information concerning the crack condition over time and its influence on the structure is limited, such as the influence of water pressure within the cracks.

The aim of this project is to study crack propagation influenced by water pressure and to define an experimental test setup that allows for crack propagation due to this load. Numerical analyses have been performed on an initial cracked specimen to study the pressure along the crack propagation. The finite element method has been used as the numerical analysis tool, through the use of the software ABAQUS. The finite element models included in these studies are based on linear or nonlinear material behavior to analyze the behavior during a successively increasing load.

The numerical results show that a crack propagates faster if the water is keeping up with the crack extension, i.e. lower water pressure is required to open up a new crack. When the water does not have time to develop within the crack propagation, more pressure is required to open up a new crack. The experimental results show that the connection between the water inlet and the specimen is heavily affected by the bonding material. In addition, concrete quality and crack geometry affects the propagation behavior.

Keywords: Concrete cracks, water pressure, concrete dams, crack propagation, finite element analysis, linear elastic fracture mechanics, instrumentation.

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Sammanfattning

Betongkonstruktioner förväntas i allmänhet att utsättas för sprickbildning under dess livslängd. Detta är anledningen till att betong armeras, där armeringen endast aktiveras efter sprickbildning. Sprickor kan orsaka problem även hos stora betongkonstruktioner, såsom i dammar exempelvis, eftersom dessa kan leda till minskad livslängd. De bakomliggande mekanismerna till sprickbildning är välkända idag. Emellertid är information om sprickförhållandet över tiden och dess inverkan på strukturen begränsad, såsom inverkan av vattentryck i sprickor.

Huvudsyftet med detta projekt är att studera spricktillväxten orsakad av vattentryck, samt att definiera en provningsmetod som tillåter spricktillväxt på grund av denna last. Numeriska analyser har utförts på en sprucken provkropp i syfte att studera vattentrycket längs sprick- propageringen. Finita element metoden har använts som verktyg för de numeriska analyserna, genom det kommersiellt använda programmet ABAQUS. De finita element modellerna inkluderade i dessa studier är baserade på linjär- eller icke-linjär material-beteende, som möjliggör analyser över beteendet under en successivt ökande last påfrestning.

De numeriska resultaten visar att en spricka växer fortare om vattnet hinner ikapp spricktillväxten, vilket innebär att ett lägre tryck krävs för att öppna upp en ny spricka. När vattnet inte har tid att utvecklas och avancera under spricktillväxten krävs det mer tryck för att öppna upp en ny spricka. De experimentella resultaten visar att kontakten mellan vatteninsläpp och provkropp är kraftigt påverkad av bindemedlet. Dessutom påverkas spricktillväxten av betong kvalitet och sprick geometri.

Nyckelord: Sprickor i betong, vattentryck, betongdammar, spricktillväxt, finita element analyser, linjär elastisk frakturmekanik, utrustning

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Preface

The research presented was carried out as a part of “Swedish Hydropower Centre – SVC”.

SVC has been established by the Swedish Energy Agency, Elforsk and Svenska Kraftnät in collaboration with Chalmers University of Technology, KTH Royal Institute of Technology, Luleå University of Technology and Uppsala University. www.svc.nu.

During the period of January to June 2017, the research was conducted at Sweco Energuide AB and the Department of Concrete Structures at the Royal Institute of Technology (KTH).

The project was initiated by Dr. Richard Malm, who also supervised the project, together with Dr. Lamis Ahmed and Adj. Prof. Manouchehr Hassanzadeh.

We would like to thank Dr. Richard Malm for his guidance, support and advice throughout this project. Furthermore, we would like to express our sincere gratitude to Dr. Lamis Ahmed for taking her invaluable time and support throughout the project at KTH.

We would also like to address our genuine appreciation to Adjunct Prof. Manouchehr Hassanzadeh at Sweco for his advice and for sharing his knowledge with us.

Alongside our supervisors, we would also like to express our sincere gratitude to Ph.D.

student Ali Nejad Ghafar at KTH and Research Assistant Patrick Rogers at CBI for their voluntary time and help throughout the experiment.

Finally, we would also like to thank Johan Blomdahl for giving us the opportunity to carry out the project at Sweco Energuide AB and KTH for giving us the opportunity to perform the experiment.

Stockholm, June 2017

José Sanchez Loarte & Maria Sohrabi

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1 Introduction

1.1 Background

Cracks in concrete are in general expected to form during the service life of the structure. Due to this issue, the concrete is reinforced. Cracking may be a concern in large concrete structures, such as concrete dams, since it may reduce the service life of the structure. Cracks in dams come in different shapes and propagation patterns which may differ based on the type of the dam. The mechanisms causing crack formations in dams are the same as in other concrete structures and easily recognized and identified. However, there is limited knowledge when evaluating the condition of cracks over time and its influence on the structure.

(Hassanzadeh and Westberg, 2016)

One of the mechanisms that potentially may have influence is water pressure within cracks.

The unilateral magnitude of the pressure in an existing crack is influenced in the same way as the uplift pressure on the interface between foundation and rock material. The magnitude of water pressure is important for existing cracks to propagate and the knowledge is limited and usually not taken into consideration in structural design. (Hassanzadeh, 2017)

Cracks in concrete structures give rise to several durability problems such as; leaching, corrosion of the reinforcement and reduced mechanical strength. These consequences may eventually lead to structural failure. (Hassanzadeh and Westberg, 2016) In order to prevent this, it is important to obtain more knowledge of the crack condition under the influence of water pressure.

Numerical models based on the finite element method (FEM), have been used to simulate crack propagation under water pressure loads. For the determination of the propagating behavior caused by water pressure, failure analyses have been carried out according to linear and nonlinear fracture mechanics. An experimental setup has been defined that allows for studying the behavior of crack propagation in concrete due to water pressure.

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1.2 Aim and scope

The purpose of this project is to investigate and evaluate how crack propagation behaves under the influence of water pressure. This is performed by finite element analyses and an experimental test.

The first step is to design the experiment model with the numerical software ABAQUS.

Within this step, a parametric study is carried out to define allowable pressure magnitudes to propagate a crack. The crack propagation in the numerical model is approximated by using the contour integral method and the discrete crack approach. However, water pressure is the main factor to cause the propagating behavior at the crack plane and this can be approximated by adapting an iterative process in the analyses.

The second step is the performing of the experimental test, where different experimental designs can be performed to reach the optimal test setup. The last step is to evaluate and to compile the results.

The research questions to be answered within this project are the following:

 How should the fracture process be simulated with consideration of intruding water pressure?

 Following a crack propagation, how is the water pressure distributed along the crack?

 How could an experimental test setup be defined that allows for study of crack propagation due to water pressure?

1.3 Limitations

 In the experimental study, tests and simulations have been made on pre-defined specimens. In real applications, cracks in concrete structures can be affected by several factors such as corrosion residual products and efflorescence that may fill the cracks. Size effects may also have an influence on the results.

 Crack propagation has been simulated with one method, the discrete crack approach.

 The research has tended to focus on successively increasing loads.

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1.4 Outline

The content of this report is presented below to give an overview of the structure of this project.

Chapter 2, includes the general theory and background of concrete cracks associated with water pressure. Furthermore, it describes the theories behind fracture mechanics.

Chapter 3, contains general theory regarding the numerical analyses and nonlinear material behavior of concrete.

Chapter 4, presents the numerical model with descriptions of its attributes.

Chapter 5, presents the process of the experimental test. A brief description of the specimen and instrumentation is presented.

Chapter 6, compiles the results from the numerical modelling and the experimental test.

Chapter 7, presents the conclusions of this study, followed by suggestions for further research.

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2 Concrete cracks

2.1 Types of cracks

Concrete is a brittle material with low tensile strength and poor toughness, where different types of cracks can develop. Crack development affects the structural strength and lifetime of concrete structures. (Benarbia and Benguediab, 2015)

A crack can be defined as an interface and/or a gap within a structure or between two geometrical bodies in a structure. The geometrical bodies that surround the crack can spread apart or displace from each other without any subjected force.

Cracks in concrete occur in different stages and can be divided into two categories; Pre- hardening cracks and post-hardening cracks. There are two defined types of pre-hardened cracks: Plastic and constructional movement. Crack types of great importance before the hardening of the concrete is the plastic settlement and plastic shrinkage.

The post-hardening cracks occur through physical, chemical, thermal and structural processes.

In Table 2.1, a list of concrete cracks, and some of their possible causes are presented.

Table 2.1: Common crack types in concrete. Modified from Hassanzadeh and Westberg (2016).

Types of cracks Causes

Pre-hardening cracks

Plastic - Plastic settlement

- Plastic shrinkage Constructional

movement

- Movement in formwork - Sub-grade movement

Post-hardening cracks

Physical

- Shrinkable aggregates - Drying shrinkage - Crazing

Chemical

- Corrosion of reinforcement - Alkali-aggregate reactions - Cement carbonation

Thermal - Freeze thaw damage

- Weathering Structural

- Overload - Design loads

Some of the cracks mentioned in Table 2.1 are illustrated in the following figures. Figure 2.1 illustrates a crack caused by plastic settlement. The plastic settlement is caused by water separation which often occurs around the location of the reinforcement. Due to the water separation of freshly poured concrete, the solid parts of the concrete sinks and get blocked by

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the reinforcement which leads to crack formation. These cracks usually follow the direction of the reinforcement. (Hassanzadeh and Westberg, 2016)

Figure 2.1: Crack caused by plastic settlement, from Hassanzadeh and Westberg (2016).

Figure 2.2 illustrates cracks caused by plastic shrinkage. Plastic shrinkage occurs due to high water evaporation which causes the concrete surface to dry and shrink. These cracks appear on the surface of concrete while it is still fresh and plastic. They are usually appearing as several parallel cracks that are shallow and primarily occurring on horizontal surfaces.

(NRMCA, 2014)

Figure 2.2: Cracks caused by plastic shrinkage, from NRMCA (2014).

Cracks caused by crazing can be seen in Figure 2.3. Crazing often occurs due to shrinkage of the cement paste layer at the surface. Craze cracks can also occur due to poor concrete practices. These cracks are a growth of fine random cracks or fissures that appear on the surface of concrete, mortar or cement paste. (NRMCA, 2009)

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Figure 2.3: Crack caused by crazing, from ACI (2008).

2.2 Compressive behavior of concrete

The compressive strength of concrete is defined as the peak value of the nominal stress of a specimen subjected to a uniaxial compressive load test. The response of the concrete can be considered as linear elastic at early stages, i.e. 30-40% of 𝑓_𝑐𝑚, where 𝑓_𝑐𝑚is the mean value of the compressive strength. The formation of micro cracks initiates at this early stage in which energy is consumed and resulting in a decreased stiffness of the material. From this point, the material behaves nonlinear, i.e. the stress-strain curve gradually increases until reaching 70- 75% of the ultimate value of 𝑓_𝑐𝑚. This results in bond cracks between the aggregates and cement paste caused by the strains orthogonal to the applied load. From this point, further loading results in a significant reduction of the stiffness and the material response is defined as “softening” behavior. The crushing failure occurs at the ultimate strain. (Malm, 2016a) Figure 2.4 illustrates the typical behavior of concrete subjected to uniaxial compressive loading.

Figure 2.4: Stress-strain curve of uniaxial compressive loading, from Malm (2016).

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The non-linear stress-strain relationship can be described by the following equations, in accordance with Eurocode 2 (2004):

𝜎_𝑐

𝑓_𝑐𝑚 = 𝜅𝜂 − 𝜂²

1 + (𝜅 − 2)𝜂 (2.1)

𝜂 = 𝜀_𝑐

𝜀_𝑐1 (2.2)

𝜅 = 1.05𝐸_𝑐𝑚 𝜀_𝑐

𝑓_𝑐𝑚 (2.3)

where,

𝜀_𝑐 is the compressive strain [-]

𝜀_𝑐1 is the strain at peak compressive stress 𝑓_𝑐𝑚, [Pa]

𝜀_𝑐𝑢1 is the ultimate strain [-]

𝜎_𝑐 is the compressive stress in concrete [Pa]

𝑓_𝑐𝑚 is the mean value of concrete cylinder compressive strength [Pa]

𝐸_𝑐𝑚 is the mean elastic modulus [Pa]

𝜅 is a factor describing the actual stress compared to the compressive strength 𝑓_𝑐𝑚 𝜂 is a ratio of the compressive strain and the strain at peak compressive stress

2.3 Crack propagation

Crack initiation is a response of local damage in the previously un-cracked material.

Extending or growing of this initial crack due to exceeding of the materials failure strength is called crack propagation. (Shen et.al, 2014) Propagation of cracks in materials is described with the field of fracture mechanics.

2.3.1 Tensile behavior of concrete

The tensile behavior of a porous concrete material is brittle. The tensile failure is initiated by micro-cracks with increasing size and number and finally merging to a macro-crack, i.e.

creating a visible actual crack. Micro-cracks are the response to local damage in the material and are initiated in the weakened zones where the stress concentrations are high e.g. between aggregates and cement paste. The material response of a specimen subjected to a uniaxial tensile load is initially linear elastic up to a level just before reaching the tensile strength 𝑓_𝑡. When the applied load increases past this level i.e. at the level of 𝜎 = 𝑓_𝑡 the specimen reaches failure and is divided in two separate parts, see Figure 2.5. Before reaching the tensile strength, the extent of micro-cracks is small and distributed over the entire volume. Crack growth will stop if the load is maintained at this level. If the load increases past this level, crack propagation becomes unstable, i.e. uncontrolled propagation due to the amount of strain energy released to make the crack propagate by itself. At the maximum level of stress, micro- cracks propagate and are concentrated in a limited area, called the fracture process zone. All micro-cracking will occur within this area and the increased deformation will lead to merging of the micro-cracks. This behavior can be seen in Figure 2.5 where the stress-strain curve

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descends and the material is softening. When the micro-cracks finally are merged, a macro crack is created and visible. Macro-cracks are defined as traction free and visible; this step is the final stage in which the specimen is separated. (Malm, 2016a)

Figure 2.5: Formation of micro-cracks for a specimen under uniaxial tensile loading and the formation of the macro-crack within the fracture process zone, reconstructed from Hassanzadeh and

Westberg (2016).

To obtain a suitable descending curve due to uniaxial tensile loading, the material behavior has to be divided into two separate curves. This is due to the displacements formed by the elastic strain in un-cracked concrete and displacements due to the crack opening which can be seen in Figure 2.6. The total displacement can be written as ∆𝐿 = 𝜀𝐿 + 𝑤, where 𝐿 is the length of the specimen, 𝑤 is the crack opening displacement and 𝜀 is the elastic strain. (Malm, 2016a)

Figure 2.6: Two separate curves describing the linear and nonlinear behavior at uniaxial tensile loading, from Mier (1984).

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2.3.2 Fracture mechanics

Fracture mechanics describes the non-linear behavior of crack opening in concrete. The nonlinear behavior can be described as three different failure modes, see Figure 2.7. Mode I is a normal-opening mode in which concrete is subjected to tension. Mode II is caused by shear and mode III by tear. In concrete, mode I is the common type of failure which occurs in its pure form. The different failure modes can occur independently or ina combination of them.

Mode 2 can be initiated as mode I, i.e. as a crack subjected to tensile stress. (Malm, 2016a)

Figure 2.7: Different types of failure modes, from Malm (2016b).

Figure 2.8 illustrates the stress distribution along the fracture zone according to fracture mode 1. The crack is propagated by a macro-crack with an initial length 𝑎₀. Micro-cracks are successively formed along the fracture process zone 𝑙_𝑝. The width of the crack opening is denoted as 𝑤 and 𝑤_𝑐 is the width of the macro-crack. The stress at the transition zone between macro crack and fracture zone process is equal to zero. The stress increases in the fracture zone process reaching the maximum value equal to the tensile strength at the crack-tip.

(Malm, 2016b)

Figure 2.8: Illustration of stress distribution at crack tip according to fracture mode 1, reproduction from Hillerborg et al. (1976).

In order to determine the uniaxial tensile behavior of concrete, the fracture energy and shape of the unloading curve must be established. This information is not available in the Eurocode and must be determined using other sources such as the Model code (2010). These codes are based on experimental results and the expression used to estimate the fracture energy 𝐺_𝑓 in mode I is given as:

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𝐺_𝑓 = 73 ∙ 𝑓_𝑐𝑚^0.18 (2.4)

where,

𝐺_𝑓 is the fracture energy [^Nm_m₂]

𝑓_𝑐𝑚 is the mean compressive strength of concrete [MPa]

The fracture energy is defined as the amount of energy needed in order to obtain a stress free tensile crack of unit area, in which it can be illustrated in Figure 2.9. The area under the tensile behavior curve is denoted as 𝐺_𝑓, and varies along the fracture process zone to the crack tip. (Malm, 2016a)

Figure 2.9: Crack opening curves used for numerical analyses. Left to right, linear, bilinear and exponential, from Malm (2016b).

The linear and bilinear curves can be calculated according to the equations shown in Figure 2.9. The equation for calculating the exponential curve was proposed by Cornelissen et al.

(1986):

𝜎

𝑓_𝑡= 𝑓(𝑤) −_𝑤^𝑤

𝑐𝑓(𝑤_𝑐) (2.5)

in which:

𝑓(𝑤) = [1 + (^𝑐_𝑤¹^𝑤

𝑐)³] exp (−^𝑐_𝑤²^𝑤

𝑐) (2.6)

where,

𝑤 is the crack opening displacement [m]

𝑤_𝑐 is the crack opening displacement at which stress no longer can be transferred [m]

𝑐₁ is a material constant which 𝑐₁ = 3 for normal density concrete 𝑐₂ is a material constant which 𝑐₂ = 6.93 for normal density concrete 𝑓_𝑡 is the tensile strength [Pa]

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2.3.3 Linear elastic fracture mechanics (LEFM)

The non-linear fracture mechanics was introduced as a result of the fracture process zone ahead of the crack. Failure of concrete can however be estimated using only linear elastic fracture mechanics. Linear Elastic Fracture Mechanics (LEFM) assumes that the material is isotropic and linear elastic. In this theory, cracks are characterized by stress intensity approach and the energy balance approach for fracture. (Benarbia and Benguediab, 2015)

The stress intensity approach

In linear elastic fracture mechanics, a stress is applied perpendicular to the crack tip with linear elastic properties. An illustration of the stress at the tip can be seen in Figure 2.10. The stress concentration at the tip can be expressed approximately as:

𝜎_𝑦 = 𝐾

√2 ∙ 𝜋 ∙ 𝑥 (2.7)

where,

𝜎_𝑦 is the stress in the y-direction [Pa]

𝐾 is the stress intensity factor [Pa√m]

𝑥 is the distance from the crack tip [m]

Figure 2.10: Stress distribution at the crack tip based on LEFM, from Hillerborg (1988).

The stress intensity factors, 𝐾 depends on the specimen geometry, loading conditions and crack length. The mode I stress intensity factor, 𝐾_𝐼, gives an overall intensity of the stress distribution. Stress intensity factor for mode I for some common geometries are given in Table 2.2 and illustrated in Figure 2.11.

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Table 2.2: Mode I stress intensity factors for some common geometries, from Fett (1998).

Type of Crack Stress intensity factor

Semi-infinite plate with centered crack

of length 𝟐𝒂 𝐾_𝐼 = 𝜎√𝜋 ∙ 𝑎

Finite width plate with centered crack

of length 𝟐𝒂 and width 𝑾 𝐾_𝐼 = 𝜎√𝜋𝑎 [sec (𝜋𝑎

2𝑊)

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] [1 − 0.025 (𝑎 𝑊)² + 0.06(𝑎

𝑊)⁴] Semi-infinite plate with edge crack of

length 𝒂 𝐾_𝐼 = 1.12𝜎√𝜋𝑎

Infinite body with a central penny-

shaped crack of radius 𝒂 𝐾_𝐼 = 2𝜎√𝑎

𝜋

Figure 2.11: Common geometries for determination of mode I stress intensity factors; a) Semi-infinite plate with centered crack b) Finite width plate with centered crack c) Semi-infinite plate with edge

crack d) Infinite body with a central penny shaped crack.

The conventional expression of the stress distribution within the distance 𝑥₁, in Figure 2.10, is not valid when the stresses exceed the tensile strength. When the stress approaches infinity, it is hard to draw conclusions of the crack stability and the propagating behavior in relation to the strength of the material. Therefore another criterion must be taken in to account which describes the crack propagation in terms of stress intensity factor. The crack starts to propagate as soon as the stress intensity factor 𝐾 reaches the critical stress intensity 𝐾_𝐶; this value is a measure of the material toughness. (Hillerborg, 1988) The failure stress 𝜎_𝑓 is thus given by:

𝜎_𝑓 = 𝐾_𝐼𝐶

𝛼√𝜋𝑎 (2.8)

where,

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𝐾_𝐼𝑐 is the critical stress intensity factor for mode 1 [Pa√m]

𝛼 is a geometrical parameter

The energy-balance approach

An alternative method to study the stress state near the crack tip is by the energy-balance approach. According to the first laws of thermodynamics, if a system undergoes changes from a non-equilibrium state to a state of equilibrium, the conclusion is that there will be a net decrease in energy. Based on this law, Griffith applied this into the formation of a crack in 1920 which became known as the Griffith energy criterion. The criterion can easily be described as an existing crack undergoing traction forces on the crack surface. At this point, the strain and potential energy remains constant, but this new state is not in an equilibrium state. The potential energy must instead reduce in order to achieve equilibrium. Griffith’s conclusion refers to the formation of a crack in which the process causes the total energy to decrease or remain constant. (Roylance, 2001) The mathematical expression can be

formulated as:

𝑑𝑈 𝑑𝐴= 𝑑𝛱

𝑑𝐴+𝑑𝑊

𝑑𝐴 = 0 (2.9)

where,

𝑑𝐴 is the incremental crack area 𝑈 is the total energy

𝛱 is the potential energy supplied by release in internal strain energy 𝑊 is the work required to create new crack surfaces

Figure 2.12 illustrates half of an infinite plate subjected to tensile stress. The triangular regions with the width 𝑎 and height 𝛽𝑎 is unloaded, while the remaining regions are subjected to the applied stress 𝜎.

Figure 2.12: Infinite plate subjected to tensile stress, from Roylance (2001).

The parameter 𝛽 in this case is selected as 𝛽 = 𝜋 with agreement with the Inglis’ solution.

Inglis’ work dealt with calculations of stress concentrations around elliptical holes. However, his work introduced a mathematical difficulty: when a perfectly sharp crack is subjected to

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tensile stresses, the stress reaches an infinite value at the crack tip. Instead of focusing on the crack tip stresses, Griffith developed an energy-balance approach for plain stress. (Roylance, 2001) The total strain energy 𝑈 is expressed as:

𝑈 = −𝜎²

2𝐸∙ 𝜋𝑎² (2.10)

where,

𝐸 is the young modulus [Pa]

𝑎 is the crack length [m]

The energy at the surface 𝑆 in relation to the crack length 𝑎 is:

𝑆 = 2𝛾𝑎 (2.11)

where,

𝛾 is the surface energy [J/m²] and the factor 2 in the equation above refers to the partition of the plate.

The terms mentioned by the expressions above can be written as:

𝑆 + 𝑈 = 2𝛾𝑎 −𝜎²

2𝐸∙ 𝜋𝑎² (2.12)

where the first expression on the right hand side represents the decrease in potential energy and the second term represents the increase in surface energy which is illustrated in Figure 2.13.

Figure 2.13: Fracture Energy balance, from Roylance (2001).

An increase of the stress level will give rise to growth of the crack 𝑎, and eventually reach a critical crack length 𝑎_𝑐. It can be shown from Figure 2.13 that the quadratic dependence of the strain energy will dominate in comparison to the surface energy passed the critical crack length. Beyond this critical length, crack growth is unstable.

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It can be shown that the first derivation of the total energy in relation to the critical crack is satisfied by the expression:

𝜕(𝑆 + 𝑈)

𝜕𝑎 = 2𝛾 −𝜎_𝑓²

𝐸 𝜋𝑎 = 0 (2.13)

This expression can be rewritten as:

𝜎_𝑓= √2𝐸𝛾

𝜋𝑎 (2.14)

which deals with brittle materials, since Griffith’s original work was based on specifically glass rods. The expression was however not accurate when dealing with ductility. This expression was therefore further developed with respect to energy dissipation due to plastic flow near the crack tip. It states that a significant fracture occurs when the strain energy is released at a sufficient rate, denoted as critical strain release rate 𝐺_𝑐 and introduced as:

𝜎_𝑓 = √𝐸𝐺_𝑐

𝜋𝑎 (2.15)

By comparing equation 2.8 and 2.15 with 𝑎 = 1, it can be seen that the energy and the stress intensity are interrelated:

𝜎_𝑓 = √𝐸𝐺_𝑐

𝜋𝑎 = 𝐾_𝐼𝑐

√𝜋𝑎→ 𝐾_𝐼𝑐² = 𝐸𝐺_𝑐 (2.16)

This interrelation is applicable for plane stress. For plain strain, the expression is given as:

(Roylance, 2001)

𝐾_𝐼𝑐² = 𝐸𝐺_𝑐(1 − 𝑣²) (2.17)

2.4 Fluid in cracks

Fluid driven crack propagation is a process where an existing crack is extended due to fluid pressure. Concrete structures such as gravity dams interact constantly with high water pressure. Existing cracks fills with a large amount of water which penetrates deeper into the dam and consequently reducing the bearing capacity and safety of the dam. (Sha and Zhang, 2017)

There are several earlier studies of fluid driven crack propagation in concrete structures.

Brühwiler and Saouma studied water pressure distributions’ in cracks. They have shown that the hydrostatic pressure inside a crack is a function of crack opening displacement given the stress continuity in the fracture process zone. This internal uplift pressure reduces from full hydrostatic pressure to zero along the fracture process zone. Slowik and Saouma (2000)

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examined the hydrostatic pressure distribution inside a crack with respect to time and the crack opening rate. They found that the crack opening rate plays an important role in controlling the internal water pressure distribution. Slow crack opening rate allow for the water to develop and propagate within the crack. The water behaves differently under fast crack opening rates and takes a longer time to develop. Barpi and Valente (2007) simulated water penetration inside a dam-foundation joint and analyzed the effect of crack propagation, resulting in the crest displacement being a monotonic function of the external load. It was also concluded that the crack initiation does not depend on dilatancy due to the fictitious process zone that moves from the upstream to the downstream side and creates a transition in the crack formation path. The load carrying capacity, however, depends on dilatance according to Barpi and Valente (2007).

2.4.1 Fluid flow in cracks

Crack propagation can occur due to static or dynamic loads. Static crack propagation occurs when the loading rate is low, i.e. no fluid flow. Higher loading rates correspond to dynamic propagation, i.e. the fluid actually flows in the crack. The flow can behave laminar or turbulent and depends comprehensively on the crack geometry. The fluid flow in the crack gives rise to a successively dropping pressure along the distance from the crack opening.

Besides the type of flow, the pressure drop depends also on the fluid properties. The pressure drop is largest for high viscous fluids due to large frictional losses along the crack boundaries.

(Rossmanith, 1992)

Longitudinal fluid flow, see Figure 2.14, in a crack can be determined with basic equations of fluid flow:

 Reynold’s lubrication theory defined by the continuity equation:

𝑔̇ +𝜕𝑞_𝑓

𝜕_𝑠 + 𝑣_𝑇+ 𝑣_𝐵 = 0 (2.18)

 Momentum equation for incompressible flow and Newtonian fluids through narrow parallel plates (Zielonka et al, 2014):

𝑞_𝑓= 𝑔³

12 ∙ 𝜇_𝑓∙𝜕𝑝_𝑓

𝜕_𝑠 (2.19)

where,

𝑔 is the fracture gap

𝑞_𝑓= 𝑣_𝑓∙ 𝑔 is the fracturing fluid flow

𝑣_𝑇 and 𝑣_𝐵 are the normal flow velocities of the fluid leaking into the surrounding porous medium

𝜇 is the fluid viscosity

𝑝_𝑓 is the fluid pressure along the fracture coordinate s

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Figure 2.14: Longitudinal fluid flow in a fracture, figure reproduced from Xielonka et al. (2014).

2.4.2 Fluid pressure in cracks

The hydrostatic pressure acting on concrete dams 𝜌𝑔ℎ, varies with depth ℎ on the upstream face. In a horizontal crack, the loads that drive the crack propagation are only the vertical tensile/compressive stress 𝜎_𝑦 and the water pressure 𝑃_𝑤 inside the crack; see Figure 2.15.

(Wang & Jia, 2016)

Figure 2.15: Stress condition around the crack in a gravity dam, from Wang & Jia (2016).

Load distributions in concrete dam cracks can be determined with two boreholes and a fracture linking them, see Figure 2.16. Figure 2.16a illustrates a concrete dam with a crack that has propagated far enough to reach the air on the downstream side. The fluid pressure will distribute linearly in such case. The pressure distribution will be even if the crack is completely sealed and can be expressed as, 𝑃 = 𝜌𝑔ℎ₁. However, this behavior is not realistic since concrete is a porous material and the water will thus spread out at the crack tip, see Figure 2.16c. The pressure at the crack tip can in such case be expressed as: 𝑃 = 𝜌𝑔ℎ₂. (Bergh, 2017)

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Figure 2.16: Fluid pressure distributions in a) crack in contact with air b) completely sealed crack c) realistic closed crack, from Bergh (2017).

Figure 2.16 illustrated fluid pressure distributions in fractures for steady state conditions.

Results from a study by (Shen et.al, 2014) shows that the pressure distribution varies with time, see Figure 2.17. The figure shows a dynamic process of fluid flow from the injection hole (green circle) to the extraction hole (white circle). Each color represents a fluid distribution at a specific time, where the distribution is nonlinear until the final stage. The black colored distribution illustrates the final fluid distribution, i.e. in steady state condition.

(Shen et.al, 2014)

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Figure 2.17: Fluid pressure distribution in a single fracture with time, figure from Shen et.al (2014).

2.5 Effects from cracking

The effects of cracks in concrete dams may be significant and could eventually reduce the bearing capacity of the structure or influence its durability. In this section, some of the predominate effects from cracking in concrete dam structures are described.

Cracking may cause deflection and deformation of the dam, i.e. movements and shape changes of the structure. It can also cause offset, i.e. one side of the cracked dam moves with respect to the other side of the dam, see Figure 2.18. It can occur at the edge of a crack or inside a crack and consequently cause horizontal or vertical movements. The Offset can also occur perpendicular to a crack and cause a movement upwards and outwards/inwards with respect to the bottom part of the cracked dam.

Figure 2.18: Offset due to cracking, from Tarbox and Charlwood (2014).

Delamination caused by cracks means splitting or separation of the top layer of the concrete and a plane parallel to the surface, see Figure 2.19. The cement used to bond the aggregate in

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concrete gets separated or delaminated. This phenomenon is typically caused by freeze/thaw damage or corrosion of the steel reinforcement.

Figure 2.19: Delamination of concrete, from Tarbox and Charlwood (2014).

Another predominate effect of cracking is efflorescence which appears as a white substance on the surface of the dam; this phenomenon is caused by a chemical reaction within the concrete and is transported to the surface by moisture transport, see Figure 2.20.

Figure 2.20: Efflorescence stains at the concrete surface due to cracking, from Tarbox and Charlwood (2014).

Water leakage through concrete cracks is a major problem for underground structures, such as dams. Leakage is a process of discharge of any material (liquid or gas) trough a crack, see Figure 2.21. (Tarbox and Charlwood, 2014)

Figure 2.21: Leakage stains from flowing water on the concrete surface, from Tarbox and Charlwood (2014).

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3 Finite element method

The finite element method (FEM) is a sophisticated numerical method used in many engineering fields to obtain approximate solutions to continuum problems. The FEM was first applied to stress analysis and has later become applicable in many other fields. FEM can be described as the piecewise polynomial interpolation at the nodes of an element, in other words the field quantity such as displacements are interpolated at the nodes within an element connected to adjacent elements and so on. The main advantage of FEM is its versatility compared to classical methods. An example of its versatility is in terms of no restrictions regarding the geometry, boundary conditions and loading conditions which give the ability to combine components of different mechanical behavior. (Cook, 1995)

In this chapter, the FE- models for cracks in concrete are introduced. There are several different material models which describe the structural behavior of concrete. However, the material models that will be presented in this report are those provided in ABAQUS.

3.1 Material models

The nonlinear behavior of concrete has a significant influence on the structure. This section presents a brief description of common types of material models that describe the nonlinear behavior of concrete.

Crack propagation in concrete can be described by two main approaches, discrete crack approach and smeared crack approach. In a smeared crack approach, the cracks are distributed over the elements while in a discrete crack approach; there is a physical separation of two crack surfaces. (Malm, 2016b)

3.1.1 Discrete crack approach

The discrete crack approach is initiated by defining a crack opening at the intersection of two elements. This will introduce an early separation at the element edges and give rise to the geometry of an existing crack, see Figure 3.1. The numerical model introduced by Ngo and Ingraffea (1967) defines the crack propagation caused by the nodal force which is transferred from the crack tip into the adjacent node and creates a propagating pattern. In this crack approach, the separating parts are given linear properties of concrete, while the interface is defined with nonlinear properties. Hence, the propagation will only occur at the interface.

(Malm, 2016b)

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Figure 3.1: Discrete crack model, from Malm (2016b).

The discrete cracks method is however limited since it is only applicable at the interface between concrete elements which means that the crack locations needs to be predefined before selecting a failure surface. This causes bias when selecting the appropriate mesh to the model. Adaptive methods can be used to reduce bias by refining the mesh into finer element sizes. Another method used to reduce the bias effects is the extended FEM (XFEM) developed by Belytschko and Black (1999). This method initiates a crack within the element where the enriched nodes are located and splits them into two separate elements given the condition that the tensile strength is reached. (Malm, 2016b)

3.1.2 Smeared crack approach

In this approach, cracks are formed in the integration points within the element, the effects are later transferred to the whole element, see Figure 3.2. The strain in this approach will consist of both elastic and non-linear strain. The elastic strain is formed from the uncracked concrete material and the non-linear from the crack opening. The total strain can be expressed as:

𝜀_𝑡𝑜𝑡 = 𝜀_{𝑒𝑙𝑎𝑠𝑡𝑖𝑐}+ 𝜀_{𝑐𝑟𝑎𝑐𝑘} (3.1)

The crack opening strain can be defined as the relationship between the crack opening displacement and the crack band length:

𝜀_{𝑐𝑟𝑎𝑐𝑘} = 𝑤

ℎ (3.2)

Figure 3.2: Smeared crack model, from Malm (2016b).

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Within the smeared crack method, there are two different approaches which describe the crack propagation, the fixed crack model and the rotated crack model. (Malm, 2016b) The main difference between the two approaches is illustrated in Figure 3.3.

Figure 3.3: Fixed and rotated the crack model, from Malm (2016b).

3.2 Crack modelling

Different techniques can be used to study delamination behavior of layered composites, such as virtual crack close technique (VCCT) and cohesive elements. VCCT is based on fracture mechanics where delamination grows when the energy release rate exceeds a critical value.

The cohesive element is a technique based on damage mechanics where the delamination interface is modeled by a damageable material. Delamination grows when the damage variable reaches its maximum value. (Burlayenko et al, 2008) Delamination can also be modeled by using a surface-based cohesive interaction that is similar to cohesive elements. A brief description of the different techniques is presented below.

3.2.1 Debond using VCCT

VCCT (Virtual crack close technique) is a technique based on linear elastic mechanics to model delamination growth. The delamination is considered as a crack in the debonding of two layers where its growth is based on the fracture toughness of the bond and the strain energy release rate at the crack tip. The VCCT approach is based on two assumptions:

 Irwin’s assumptions; the released energy in crack growth is equal to the work needed to close the crack to its initial length.

 The crack growth has a constant state at the crack tip.

The energy to close and open the crack, assumed that the crack closure behaves linear elastic, can be calculated from the following equations:

−1 2

𝐹_𝑗∆𝑈_𝑖

∆𝐴 = 𝐺_𝐼 (3.3)

∆𝐴 = 𝛿𝑎𝑏 (3.4)

where,

a j

x y

m m e

e 1

2

s

t t

x, y = global coordinate system m , m = material coordinate system₁ ₂ e , e = principal strain direction₁ ₂

1

2

c1 c2

a

x y

e, m e , m

1

2

x, y = global coordinate system m , m = material coordinate system₁ ₂ e , e = principal strain direction₁ ₂

1

2

sc2 sc1

Fixed crack approach Rotated crack approach

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26 𝐹_𝑗 is the node reaction force j [N]

∆𝑈_𝑖 is the displacement between released nodes at I [m]

𝛿𝑎 is the crack extension [m]

𝑏 is the width of the crack [m]

𝐺_𝐼 is the energy release rate [J/m²]

Cracks start to grow when the energy release rate exceeds a critical value: G_I≥ G_IC where G_IC is the Mode I fracture toughness parameter. (Burlayenko et al, 2008)

3.2.2 Cohesive behavior

The adhesive interaction can be simulated in two ways: using cohesive elements or surface- based cohesive behavior. The surface-based cohesive behavior is similar to the behavior of cohesive elements and is defined using a traction-separation law. Surface-based behavior is easier to use since no additional elements are needed and can be used in a wider range of interaction. (Dassault Systèmes, 2014)

Cohesive Elements

Cohesive elements can be useful when difficulties of the implementation of VCCT into finite element codes occur. This technique includes both initiation and propagation of delamination.

The damage occurs when the stresses exceed a strength criterion, and final separation of the material is modeled by using fracture mechanics parameters. (Burlayenko et al, 2008)

Cohesive elements are used to model the behavior of adhesives joints, fracture at bonded interfaces, gaskets and rock fracture. The constitutive behavior of the cohesive elements depends on the specific application and can be defined with a:

 Continuum-based constitutive model – Suitable when modelling the actual thickness of the interface.

 Traction-separation constitutive model – Suitable when the thickness of the interface can for practical purposes be considered zero, e.g. cracks in concrete.

 Uniaxial stress-based constitutive model – Useful in modelling gaskets and/or unconstrained adhesive patches – Suitable when using only macroscopic material properties such as stiffness and strength using conventional material models.

Cohesive elements can be constrained to surrounding components in different ways. It can either be constrained on both its surfaces or free at one, see Figure 3.4-3.6. (Dassault Systèmes, 2014)

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Figure 3.4: Cohesive elements a) sharing nodes with surrounding elements, from Dassault Systèmes (2014).

Figure 3.5: Cohesive elements connected to surrounding components with surface-based tie constraints, from Dassault Systèmes (2014).

Figure 3.6: Cohesive elements connected with contact interaction on one side and tie constraints on the other, from Dassault Systèmes (2014).

Damage of the traction-separation constitutive model is defined with a framework that is used for conventional materials. A combination of multiple damage mechanisms acting on the

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same material at the same time is allowable in this framework. The damage mechanisms consist each of a:

 damage initiation criterion

 damage evolution law

 choice of element deletion when it reaches a fully damaged state

The initial traction-separation response of the cohesive element is assumed to be linear, see Figure 3.7. Material damage will occur when the damage initiation criterion is specified with a corresponding damage evolution law. Damage on the cohesive layer will not occur under pure compression. (Dassault Systèmes, 2014)

Figure 3.7: Traction-separation response, from Dassault Systèmes (2014).

Surface-based cohesive behavior

Surface-based cohesive behavior is mainly used when the interface thickness of the adhesive material is negligibly small. Cohesive elements are suitable for interfaces with finite thickness if properties such as stiffness and strength of the material are available.

The cohesive surface behavior defines an interaction between two surfaces with given cohesive property. In order to prevent over-constraints, a pure master-slave formulation is enforced for these surfaces. This master-slave formulation is further described in Section 3.4.

Damage modeling for cohesive surfaces with traction-separation is defined within the same general framework as for the cohesive elements. The difference in interpretation for traction and separation for cohesive elements and the cohesive surface can be seen in Table 3.1.

However, it’s important to note that damage in cohesive surface behavior is an interaction property, not a material property. (Dassault Systémes, 2014).

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Table 3.1: Traction and separation for cohesive elements and cohesive surfaces, from Dassault Systèmes (2014).

Cohesive elements Cohesive surfaces

Separation

𝑁𝑜𝑚𝑖𝑛𝑎𝑙 𝑠𝑡𝑟𝑎𝑖𝑛 (𝜀) =

𝑅𝑒𝑙𝑎𝑡𝑖𝑣𝑒 𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡 (𝛿) 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑡ℎ𝑒 𝑡𝑜𝑝

𝑎𝑛𝑑 𝑏𝑜𝑡𝑡𝑜𝑚 𝑜𝑓 𝑡ℎ𝑒 𝑐𝑜ℎ𝑒𝑠𝑖𝑣𝑒 𝑙𝑎𝑦𝑒𝑟

𝐼𝑛𝑖𝑡𝑖𝑎𝑙 𝑡ℎ𝑖𝑐𝑘𝑛𝑒𝑠𝑠 (𝑇0)

𝐶𝑜𝑛𝑡𝑎𝑐𝑡 𝑠𝑒𝑝𝑎𝑟𝑎𝑡𝑖𝑜𝑛 (𝛿)

Traction 𝑁𝑜𝑚𝑖𝑛𝑎𝑙 𝑠𝑡𝑟𝑒𝑠𝑠 (𝜎) 𝐶𝑜𝑛𝑡𝑎𝑐𝑡 𝑠𝑡𝑟𝑒𝑠𝑠 (𝑡) = 𝐶𝑜𝑛𝑡𝑎𝑐𝑡 𝑓𝑜𝑟𝑐𝑒 (𝐹)

𝐶𝑢𝑟𝑟𝑒𝑛𝑡 𝑎𝑟𝑒𝑎(𝐴) 𝑎𝑡 𝑒𝑎𝑐ℎ 𝑐𝑜𝑛𝑡𝑎𝑐𝑡 𝑝𝑜𝑖𝑛𝑡

3.3 Pre-defined crack modelling

Another method for modelling crack propagation is the contour integral evaluation. This method is an iterative process based on pre-defined cracks in which important factors, such as the J-integral, in which the stress intensity factor is extracted from.

3.3.1 Contour integral evaluation

The contour integral can be evaluated using two different approaches. The first approach is based on conventional FEM, which requires the user to adapt the mesh to the crack geometry in order to obtain accurate results. Mesh adaptation involves specifying the crack front and the direction in which the crack will extend. The second approach is based on the extended finite element method (XFEM) and does not require mesh matching into the crack geometry. The crack front and virtual crack extension are determined automatically by defining an enrichment zone. Figure 3.8 illustrates the virtual crack extension for evaluation of the contour integral.

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Figure 3.8: Virtual crack extension, from Dassault Systèmes (2014).

The basic principle of the contour integral in two dimensional cases can be thought of as a virtual motion of a block of material around the crack tip, for three dimensional cases the virtual motion occurs on the surrounding of each node along the crack line. Blocks are defined as contours in which each contour is a ring of elements surrounding the crack tip or nodes along the crack line from a crack face to another in the opposite direction. ABAQUS provides the evaluation for the type of contour integral to be calculated, these include the evaluation of the J-integral, Ct-integral, stress intensity factors and T-stresses. The J-integral describes the energy release rate associated with crack propagation. The Ct-integral characterizes the rate of growth of the crack-tip creep zone, which is time dependent. For small-scale creep, i.e. the elastic strains dominate in the material, crack growth is governed by the stress intensity factor in failure mode I. T-stresses represent the stresses parallel to the crack faces and is associated with the crack stability. (Dassault Systèmes, 2014)

In order to avoid convergence problems around the crack tip, cracks can be modeled with a desired singularity. Singularity arises if the geometry of the crack region is sharp and the strain fields become “singular” at the crack tip. This can be included in ABAQUS by editing the crack tip with collapsed quadrilateral elements, also called the quarter point technique, see Figure 3.9. Certain conditions must be fulfilled in order to obtain singularity; elements around the crack tip must collapse i.e. resulting in a zero length of the edge located near the crack tip.

For the two dimensional cases, singularity is modeled as ¹

√𝑟 and ¹

𝑟 at the crack tip. (Dassault Systèmes, 2014)

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Figure 3.9: Illustration of contour mesh, from Dassault Systèmes (2014).

Domain integral method

ABAQUS uses the domain integral method to evaluate contour integrals. This method is quite robust in terms of evaluating contour integrals accurately with no mesh refinement. This is due to the integral taking over a domain of elements that surrounds the crack, decreasing the effect of errors in local solution parameters. In ABAQUS, the J-integral is calculated first from which the stress intensity factors, 𝐾 can be extracted from. (Dassault Systémes, 2014) The J-integral

The J-integral is typically used in rate-independent quasi-static fracture analysis to evaluate the energy release associated with the crack propagation. The energy release rate in relation to the crack propagation for a two dimensional case is given by:

𝐽 = lim

Γ→0∫𝒏 ⋅ 𝑯 ⋅ 𝒒𝑑Γ

Γ (3.5)

where,

Γ is the contour starting at the bottom crack surface and ending at the top surface q is a unit vector of the direction of the crack extension

n is the outward normal to Γ

𝑯 is given by the following expression,

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32 𝑯 = 𝑊𝑰 − 𝜎 ⋅𝜕𝑢

𝜕𝑥 (3.6)

where,

𝑊 is the elastic strain energy in elastic material behavior And for the three dimensional case the expression is written as:

𝐽̅ = ∫𝜆(𝑠)𝒏 ∙ 𝑯 ∙ 𝒒𝑑𝐴

𝐴

(3.7) where,

𝜆(𝑠) is the virtual crack advancement

𝑑𝐴 is a surface element along a vanishing small tubular surface enclosing the crack tip or crack-line

𝒏 is the normal to 𝑑𝐴 and

𝒒 is a vector direction in which crack propagates H is related to the elastic strain energy of the material

Stress intensity factors

The stress intensity factors 𝐾_𝐼, 𝐾_𝐼𝐼, 𝐾_𝐼𝐼𝐼 are of great importance in linear elastic fracture mechanics (LEFM). These factors characterize the influence of load or deformation which is transferred to the local crack tip in terms of stress and strain. The parameters measure also the propensity for crack propagation or the driving forces in which propagation is initiated. For linear elastic materials the stress intensity factor can be related to the energy release rate according to:

𝐽 = 1

8𝜋𝑲^𝑇⋅ 𝑩⁻¹⋅ 𝑲 (3.8)

where 𝑲 = [𝐾_𝐼, 𝐾_𝐼𝐼, 𝐾_𝐼𝐼𝐼]^𝑇 and 𝑩 is the pre-logarithmic energy factor matrix. The above formulation can be simplified with respect to homogeneous isotropic materials where the matrix 𝑩 is diagonal as,

𝐽 = 1

𝐸^′(𝐾_𝐼² + 𝐾_𝐼𝐼²) + 1

2𝐺𝐾_𝐼𝐼𝐼² , (3.9)

where,

𝐺 is the shear modulus

𝐸^′ is the young modulus for plane stress and 𝐸^′= 𝐸/(1 − 𝑣²) for plane strain For cracks located at the interface of two different isotropic materials, the following parameters are introduced as, 𝐺₁ = 𝐸₁⁄2(1 + 𝑣₁) and 𝐺₂ = 𝐸₂⁄2(1 + 𝑣₂) which corresponds to the shear modulus of the interacting materials. The J-integral can be rewritten with respect to the interacting materials as,

𝐽 =(1 − 𝛽²) 2 ⋅ (1

𝐸₁^′+ 1

𝐸₂^′) ⋅ (𝐾_𝐼²+ 𝐾_𝐼𝐼²) +1 4(1

𝐺₁+ 1

𝐺₂) ⋅ 𝐾_𝐼𝐼𝐼² (3.10)

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𝛽 =𝐺₁(𝜅₂− 1) − 𝐺₂(𝜅₁− 1)

𝐺₁(𝜅₂+ 1) + 𝐺₂(𝜅₁+ 1) (3.11)

where,

𝜅 = 3 − 4𝑣 for plain strain

𝜅 = (3 − 𝑣)/(1 + 𝑣) for plain stress

The interfacial crack does however not behave as Mode I and Mode II in its pure form and introduces complexity to the K parameters. (Dassault Systémes, 2014)

3.4 Interactions

In this section, a description of interactions and interaction properties that may be needed in a typical crack modelling is presented.

Surface to surface contact

The surface to surface contact describes the contact between two deformable geometrical parts or the contact between a deformable and a rigid geometrical part. Within the contact, definitions can be assigned to the interaction independently i.e. a set of data may contain different properties. An important part when choosing the right contact interaction is the definition of the master- and slave surface. The main characteristics attached for the selection of the master surface is that it must be applied to analytical rigid surfaces and rigid-element- based surfaces, the slave surface, on the other hand, is attached to deformable bodies.

A surface-to-surface based interaction provides a more accurate stress and pressure results compared to a node-to-surface interaction. A node-to-surface based interaction constrains the slave nodes to penetrate into the master surface but this does not apply to the master surface itself. The surface-to- surface base interaction resists penetration to occur and can be seen as a smoothing effect. (Dassault Systèmes, 2014)

Pressure penetration

The pressure penetration interaction is used to simulate the stress distribution caused by the fluid penetrating two surfaces. This interaction is only appropriated in surface-to-surface based interaction. The fluid pressure is applied orthogonally to the crack plane causing the bodies to separate from each other. In Abaqus, a fluid penetration can be used by the standard analysis, i.e. with implicit solver. The definition is initially made by identifying the surfaces in contact that will undergo exposure to fluid pressure. Within this interaction the magnitude of pressure in respect to critical contact pressure must be defined, this setup is mainly defined at the nodes of the contact surface. The fluid can penetrate into one or multiple regions of the surface which does not consider the actual status of the contact until a critical contact pressure is reached. The fluid penetrates easier at higher critical contact pressures. (Dassault Systèmes, 2014)

Crack propagation in concrete dams driven by internal water pressure