Cement Fracture

Fracture Mechanics of Concrete


P.K. Mehta and P.J.M. Monteiro, Concrete: Microstructure, Properties, and Materials

Fracture Mechanics

Fracture mechanics for concrete can be a useful tool for the designer because of the insight it provides on size effects, that is, how the size of a structural element will affect the ultimate load capacity.

Fracture mechanics also provides powerful criteria for the prediction of crack propagation.



Fracture Mechanics

Linear elastic fracture mechanics theory was developed in 1920, but not until 1961 was the first experimental research in concrete performed.

Fracture mechanics was used successfully in design for metallic and brittle materials early on; however comparatively few applications were found for concrete.

This trend continued up until the middle ‘70s when finally major advances were made.



Fracture Mechanics

In the ‘80s and ‘90s intensive research has been performed and applications of fracture mechanics in design of beams, anchorage and large dams are becoming more common.



Linear Elastic Fracture Mechanics

Griffith is often regarded as the founder of fracture mechanics.

He observed experimentally that small imperfections have a much less damaging effect on the material properties than the large imperfections.

Griffith suggested an energy balance approach based not only on the potential energy of the external loads and on the stored elastic strain energy but also on another energy term: the surface energy.



Crack in a plate



Crack in a plate

Griffith used a result obtained by Inglis that the change in strain energy due to an elliptical crack in an uniformly stressed plate is and therefore the change in potential energy of the external load is twice as much .

The change of energy of the plate due to the introduction of the crack is given by:

π a2 σ2 E

U cracked −Uuncracked = −

2 π a2 σ2

E+π a2 σ2

E+ 4 aγ



Crack in a plate

Minimizing the energy in relation to the crack length,

∂

∂a−π a2 σ2

E+ 4 aγ

⎛⎝⎜ ⎞

⎠= 0

gives the critical stress (for plane stress):

σ =2 E γ

π a



Importance of the equation

This equation is significant because it relates the size of the imperfection (2a) to the tensile strength of the material.

It predicts that small imperfections are less damaging than large imperfections, as observed experimentally.



Critical energy release rate

Irwin proposed that instead of using the thermodynamic surface energy, one should measure the characteristic surface energy of a material in a fracture test.

He introduced the quantity Gc as the work required to produce a unit increase in crack area. Gc is also referred to as the critical energy release rate.

Gc is determined experimentally, normally using simple specimen configuration.



Critical energy release rate

The energy release per unit increase crack area, G, is computed; if the energy release rate is lower than the critical energy release rate (G < Gc ) the crack is stable.

Conversely, if G> Gc, the crack propagates.

In the case when the energy release rate is equal to the critical energy release rate (G=Gc), a metastable equilibrium is obtained.



Computation of G

Considering a plate with thickness B, the energy released by crack growth

Δa

Δa

Δ = Δ −Δ eGB a P x U

Δ eU is the change in elastic energy due to crack growth

= − edUdxGB Pda da

Introducing the compliance = /c x P

the strain energy is given by

eU

2

2Δ =e

cPU

2( ) ( 2)/= −

d cP d cPGB Pda da

2

2=

P dcGB da



Example: Double cantilever beam with thickness B



Computation of G

3

2 3PaEI

δ=

The deflection of each cantilever

31 ( )12 2

=hI b

The compliance is given by

323

δ= =

acP EI

Therefore the energy release rate is given by

2 2 2

2= =

P dc P aGB da BEI



Stability criteria

A crack is stable if the derivative of the strain energy rate, with respect to crack length is negative.

1 0∂<

∂G

G a



For load control (double cantiliver specimen)

22∂=

∂G P aa BEI

(1 )( ) 2/ ∂ /∂ =G G a a

is a positive number, therefore the crack will propagate in an unstable way.



For displacement control

2

4

94

δ=

EIGa B

2

5

9 δ∂= −

∂G EIa a B

so

(1 )( ) 4/ ∂ /∂ = − /G G a a

is a negative number, therefore, the crack will propagate in a stable manner.



Modes

Mode I: opening or tensile mode,

Mode II: sliding or in-plane shear mode

Mode III: tearing or antiplane shear mode.



Modes



Stresses at the tip of the crack for mode I

σy =

KI

2π rcos

ϕ

21+sin

ϕ

2sin

3ϕ2

⎡⎣⎢

⎤⎦⎥

σx =

KI

2 π rcos

ϕ

21 − sin

ϕ

2sin

3 ϕ2

⎡⎣⎢

⎤⎦⎥

τxy =

KI

2 π rsin

ϕ

2cos

ϕ

2cos

3 ϕ2

⎡⎣⎢

⎤⎦⎥



Stress-intensity factor

KI is called stress-intensity factor for Mode I.

Dimensional analysis of indicates that the stress-intensity factor must be linearly related to stress and to the square root of a characteristic length.

Assuming that this characteristic length is associated with the crack length:

K I = σ a f g( )



Fracture toughness

Suppose we measure the value of the stress at fracture in a given test.

Using the previous equations, we determine the critical stress intensity factor, Kc, or fracture toughness as it is usually called in the literature.



Concrete Fracture Mechanics

The first experimental research on fracture mechanics of concrete was performed by Kaplan in 1961.



Importance of Size Effect

( )π=

/c

cKp

a f a b

(1 )= − at t bp f



Brittleness number

Characterize the nature of the collapse; the lower the brittleness number the more brittle the behavior of the specimen. Fracture occurs in specimens with a small brittleness number, that is, for materials with a comparatively low fracture toughness, a high tensile strength, and in large specimens.

= /c ts K f b



Mesh sensitivity



Crack propagation in concrete

Courtesy from Prof. Nemati



Two simplifying assumptions for the strain in the fracture zone



Fictitious Crack Model

The fictitious crack model was created and expanded upon by Hillerborg, Petersson, and co-workers. One of the objectives of the model is to capture the complex nature of concrete in tension. The amount of microcracking in concrete, which is in tension, is small before the peak stress is reached, therefore, the deformation along the specimen can be assumed to be uniform, and the total elongation of the specimen can be expressed in terms of the length of the specimen

εΔ =l l




A localized fracture zone starts to develop just after the peak load is reached. In the model, this zone is assumed to form simultaneously across an entire cross section. As the total elongation increases, the stress decreases and the region outside the fracture zone experiences an unloading, while inside the fracture zone, there is softening.




The fracture zone remains localized and does not spread along the specimen, this is called strain localization, somewhat akin to that seen in plasticity. Beyond the peak stress, the total elongation of the specimen is the sum of the uniform deformation outside the fracture zone and the additional localized deformation

εΔ = +l l w




Two relationships are needed to characterize the mechanical behavior of concrete in tension: (1) a stress-strain relationship for the region outside the fracture zone, and (2) a stress-elongation relationship for the fracture zone.



Fracture energy

The fracture energy is equal to the area under the stress-elongation curve.

0( )σ

∞= ∫fG w dw



Experimental Results



CEB-FIP model code 1990

0 7( )α .= /f f cm cmoG f f

0.0532

0.0316

0.028

αf (Nmm / mm2)dmax



Slopes

The stress-strain and stress-elongation curves are related in the following manner: the slope of the stress-strain diagram is E, and the slope of the stress-deformation curve is proportional to

/( )/t f tf G f



Characteristic length

The ratio between the two slopes has units of length called the characteristic length of the material:

2= fch

t

EGl

f



Characteristic length

The characteristic length is often considered to be a material property, and it gives a measure of the brittleness of the material. Cement paste has a characteristic length in the range 5-15 mm, mortar in the range 100-200 mm, and concrete 200-400 mm. Compared to normal-strength concrete, high-strength concretes and light-weight aggregate concrete have lower characteristic lengths.



CEB-FIP model code 1990

for 0 9σ ε σ= ≤ .ct c ct ct ctmE f

0 1 (0 00015 ) for 0 90 00015 (0 9 )

σ ε σ.= − . − . ≤ ≤

. − . /ctm

ct ctm ct ctm ct ctmctm c

ff f ff E

For cracked section:

1

(1 0 85 ) for 0 15σ σ= − . . ≤ ≤ct ctm ctm ct ctmwf f fw

1

0 15 ( ) for 0 0 15σ σ.= − ≤ ≤ .

−ctm

ct c ct ctmc

f w w fw w

1

20 15 and β= − . =f f

c c Fctm ctm

G Gw w w

f f



Stress-strain and stress-elongation for concrete in uniaxial tension

Cement Fracture

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