2011-11-23 Topic 7 Fracture

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1.Introduction

2.Why do we need to consider fracture

mechanics in concrete?

3.Fracture mechanics for concrete

4.Computational Models for fracture

Fracture Mechanics forStructural Concrete

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What is fracture mechanics?

When a crack length reaches a certain criticallength, it can propagate, even though theaverage stress is much less than the tensilestrength of the test specimen.

Fracture mechanics tries to find the

quantitative relations between crack length,materials resistance to crack growth, and thestress at which cracks start to propagate.

Introduction

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Strength based approach

With classical linear elasticity, we cancalculate the stress concentration effect.

However, at the very sharp crack tip, thetheory generate an infinite stress.

Evidently, it is nonphysical.

If it is true, the strength will be near zero.Therefore, more rational method needed.

Introduction

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Energy-balance approach

0dE d dW dA dA dA

In this approach, the crack forms when totalenergy does not change or decrease, eventhough work is required for the crack to grow.

Introduction

dA: Incremental crack areaE : Total energy

: Potential energy supplied by release in internal strain energyW : Work required to create new surfaces

Critical condition

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d /dA: internal strainenergy decreasing rate

dW/dA: work required to

create new surface dA : surface energy

Energy-balance approach (contd)

2

an 2d s

d a dW dA E dA

Introduction

s

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Energy-balance approach (contd)

Internal strain energy is released by creatingfree surface.

Energy-balance concept can be expressed inenergy-release rate ( G ) concept.

Crack occurs when G reaches a critical value.

Introduction

d GdA

: Potential energy defined by U-FU : Strain energy stored in the body

F : Work done by external force

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Note that there is no compression crushing.

Crack modes

Introduction

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1.Introduction

2.Why fracture mechanics in concrete?3.Fracture mechanics for concrete



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Formation of cracks require a certain amountof energy, which agrees with the fracturemechanics concept.

Finite element analysis based on the strengthcriteria is unobjective around sharp crack,

that is, the analysis depends on the choice ofmesh size.

Why fracture mechanics in concrete?


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Limit analysis approach based on plasticitytheory cannot be justified in brittle types offailures, e.g., punching shear, shear failurewithout shear reinforcement, etc.Size effect can be addressed by fracturemechanics.

After all, tension softening type failure(Quasi-brittle failure) can be tackled byfracture mechanics.

Why fracture mechanics in concrete?(contd)


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Mesh size dependence

Element size along crack (a) 50 (b) 25 (c) 5


Loading

(b) (c)

(a)

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Mesh size dependence (contd)


Stress Distribution

0.0E+00

1.0E-04

2.0E-04

3.0E-04

4.0E-04

5.0E-04

6.0E-04

7.0E-04

0 50 100 150 200 250

Distance (mm)

S t r e s s

( M p a

)

Size 50

Size 25

Size 5

Model (a) is moresafe than model (c)?

Model (c) reflect thereality more accurately?

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Limit analysis can not explain crack propagation.

Limit of limit analysis


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In classical linear elasticity, critical stress doesnot depend on the structure size.

However, plain concrete shows strong sizeeffect, that is, the cracking stress depends on

the specimen or structure size.

Size effect


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: prediction for the nominal stress: tensile strength: maximum aggregate size: an empirical constant: structure or specimen size

Bazants size effect relation

'

1

t N

a

f d d


N

't f

ad

d

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Double punch test


Double Punch Test Setup Test Result

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Large Beam Failing in Shear

(Photo Courtesy of Shimizu Corporation)

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Influence of Member Depth and Maximum Aggregate Sizeon Shear Stress at Failure (Experimental Results, Shioya)

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Shear Design of Transfer Beams

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Construction and Loading of the Large Wide Beam, AT-1 UnderTesting Machine at the University of Toronto

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Crack Development for Beam AT-1

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Failure Surface of Beam AT-1

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Failure of Large Beam at 47% of ACI Shear Failure Load

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Observed Load-Deformation Response of Beam AT-1

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Influence of Minimum Shear Reinforcement onLoad-Deformation Response of Large Beams

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1.Introduction

2.Why fracture mechanics in concrete?




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Linear elastic fracture mechanics allows thestress to approach infinity at a crack tip.

However, infinite stress cannot develop in realmaterials.

A certain range of inelastic zone must exist

around crack tip.In metal, this zone is a yielding zone.

At the crack tip,

Fracture mechanics for concrete

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- Includes many microcracks.- Cohesive pressure still exists- Must be nonlinear.

- Finite tensile strength- Size is much larger than

steel.- In contrast to metals,

strain-softening, instead of strain-hardening, dominates.

Fracture process zone of concrete

't f


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Comparison of crack tips


Linear Fracture Metal Concrete

F : Fracture process zone

N : Nonlinear hardening zone

L : Linear zone

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In ductile metal, fracture process zone, whichshows strain-softening, is usually small.

In concrete, fracture process zone is largecompared to metal.

Therefore, in concrete, crack tip is not

defined clearly.

Comparison of crack tips (Contd)


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Fracture energy is the energy required toopen unit area of crack surface.

Fracture energy is a material constant.

It is independent of structure size orgeometry of structure.

Fracture energy


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Gq : Fracture energy

GIc : Energy to create new surface

: Energy to separate the

surfaces completely

Fracture energy (Contd)


0

t w

q IcG G w dw

0

t w

w dw

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In general, G Ic is ignored.

To determine this curve, f t, G f ,and the shape of curve haveto be known.

The shape can be linear, bi-linear, quadratic, etc.

Note that the energy is not a

function of strain, butdisplacement.

Fracture energy (Contd)


Typical shape of fracture energy

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1.Introduction

2.Why fracture mechanics in concrete?




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Fictitious Crack Model by Hillerborg (1976)- First proposed model applying fracture energy

- Discrete model approach

Crack Band Model by Bazant and Oh (1983)

- Smeared crack model approach- Proposed minimum element size

Two models

Computational Models for Fracture

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Before the Hillerborg model


In the Dugdale model, plasticzone near the crack tip.

In that model, the stressequals to the yield stressacross the crack.

Barenblatt model is similar to

the Dugdale model, but thestress is assumed to vary withthe deformation.

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Fictitious crack model -assumption


The crack is assumed to propagatewhen the stress reaches the tensilestrength,

When the crack opens, the stress isnot assumed to fall to zero at once.

After cracking, the stress at thenode is calculated by the fractureenergy curve.

When implemented in finite elementmethod, the node is separated intotwo nodes. (discrete model)

't f

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Fictitious crack model - Example


Three points bend beam Finite element mesh

Result

F1 is computed based onfracture energy, not by thelinear elastic fracturemechanics

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Characteristic length


Hillerborg introduced the concept ofcharacteristic length,

Characteristic length is material property also.

Characteristic length is index for thebrittleness.

If it is large, the material is ductile, whereas ifit is small, brittle.

2/ 'c c t l EG f

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Fictitious crack mode -Shortcomings


When the crack extends through a certainnode, the node must be split into two nodes.

The optimized band should be calculated ineach step.

If the direction of crack is not known inadvance, identifying the direction in which theenergy release rate is maximum should becarried out.

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Fictitious crack mode Shortcomings(contd)


Even if the calculation is possible, thedirection may not propagate through nodes. Inthat case, the calculation may be misleading.

C i l M d l f F

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Crack band model - Introduction


In the theory of randomly inhomogeneousmaterials, equivalent continuum stresses andstrains are defined over a certain

representative volume.The volume size must be at least several

times the maximum aggregate size.

The distribution of stress or strain overdistance less than several aggregate sizeshas no physical meaning.

C t ti l M d l f F t

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Crack band model Introduction(Contd)


Bazant and Oh modeled thefracture process zone by aband of smeared crack band

with a fixed width hc.To remedy the

unobjectivity of finite elementanalysis, cracking criterion offracture energy wasintroduced.

Comp tational Models for Fract re

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Crack width of crack band model


In the smeared crack band, the crack opening, w,is calculated as the product of average strain andband width.

Note that the fracture energy relation is changedinto one between stress and strain.


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Crack band width


In those paper, Bazant and Oh assumed stress strainrelation in the fracture process zone as linear.In this case, the crack band width (w c) is calculated as

1

2

2 1 1

'

f

ct t

G

w f E E


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Characteristics of cracked element


If a certain element is in fracture process zone, thestiffness in the direction normal to crack is decreasedgradually according to fracture energy relationship.


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Characteristics of cracked element(Contd)

1 01

1 0

1

x x

y y

z z f

E


Where x, y and z are principaldirection.

is the additional strain dueto opening of the microcracks.

f

1

' f t z

f

f C

z z

f z

't f 't f

0 0 p

f C

1 E

1

t E

1


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Characteristics of cracked element(Contd)

0

1 011 0

x x

y y

z t z

E E E


0

1 1 1

where 0

'

t f

t

f

E E C

f C

Finally, for plane stress case, orthotropic stiffness matrix is

2

0

' '

' '

x x t t

z z t t

E E E

E E

' 1 1

2

1where '

'1

t f

E E C

E E


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Element size of crack band model


In finite element analysis, the crack band width is theelement size of fracture process path.Since the size of fracture process zone is w c, the

smaller element than w c makes no sense.However, larger element than w c can be used with

proper correction.The crack width will be the product of crack band

width and element strain.

2011-11-23 Topic 7 Fracture

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