tension softening

7/31/2019 Tension Softening

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IntroductionFracture behaviour of concrete post crackingbehaviour Tension softening diagram

Knowledge of Stress displacement (w) inpost peak regime - determining fracture energy

2 and 3 dimensional micromechanical models

proposed link b`n microstructure of concreteand its tension softening response


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(w) relationDirect determination of (w) relationIndirect determination of (w) relation

Direct determination of (w) relationAppropriate method Displacement

controlled uniaxial tension testEccentricity should be avoided Reliableinformation Tension softening response


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(w) relation- uniaxial test on panelsStevin lab.-Closed loop servo hydraulic loading machine- double edgenotched specimen-LVDT

Envelope curves compared with static curves and both are found to becoincident(w) determination- detect elastic and prepeak inelastic deformation fromWt line from prepeak stress parallel to the initial loading curveNet inelastic deformation in post peak regime

f(w/wc)=[ 1+ C 1 (w/w c)3 ] e- (C2 w/wc )

f t`-Tensile strengthw-Crack opening,wc-Max.crack opening at end of softening diagram

1 / / ` / f wwww f f cct


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(w) relation- uniaxial test on tapered specimens

Wecharatana(1990)- specimen dog bone with edge notches

Stress displacement law

(/ft`) m +(w/wc) 2m = 1

Exponent m indirect measure of intrinsic brittleness of concrete-reflects slope of post peak curve

Smaller m-steeper the initial drop and more brittle the material


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Indirect determination of (w) relationRoelfstra and Wittmann(1986)

Proposed method - (w) relation-BilinearapproximationLoad deformation response of CT specimensreproduced in numerical simulation based FCMf 1 /f t`=1/6 -1/3

Gustafsson & Hillerborg(1985)Bilinear approximationCoordinates f 1=f t`/3

w1=0.8 G f /f t`wc=3.6 G f /f t`

Gf is determined by RILEM test method


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Indirect determination of (w) relationCEB FIP code (1993)-bilinear approx.

f 1= 0.15 f t`W c function of max. aggregate size

/ft` = 0.15(1- w/w 1) 0 0.15 ft` /ft` = 1-0.85 w/w 1 0.15 ft` ft`

w1 = (G f -22w c(G f /f )0.95

)/150(G f / f )0.95

f empirical coefficient depends on max.aggregate size

Bilinear (w) Matching of cracking opening profilesCrack opening profile by interferometry techniques matched withnumerical procedure based on FCMParameters assumed were optimized to achieve best fitG f and f t` were determined from RILEM method and uniaxialtension test


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Indirect determination of (w) relation(w) relation from J integral

(w) depends on microstructure of mix and not on size and geometry of the testspecimenJ=

DifferentiatingdJ / dw = (w)

(w) relation is gradient of J(w)Li et al.(1987) proposed a method for determining J(w)1. Load load point displacement (p- ) and load CMOD (p-w) plots are recorded

for notched specimens of any geometry

2. The p- and p-w curves averaged load is eliminated -diff. in potential energyis only because of diff. in notch sizes and is equal to shaded area

3. The shaded area is J- integral ,as post peak behavior is assumed to beindependent of size and geometry

4. Elimination of from w( ) and J( ) curves gives J(w) relation fromwhich (w) can be constructed


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In order to quantify ,it is necessary to describe the tension softeningbehaviour using Micromechanical modelling techniques

Several 2 and 3 dimensional tension softening models havebeen proposed

1. 2 dimensional model of Karihaloo et al. (1991)

2. 3 dimensional model of Karihaloo & Huang (1992)


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2 dimensional model Karihaloo et al (1991)Tension softening response of quasi brittle materials havelarge w c at rupture

Reasons for pronounced tail:Presence of poresFragmented microflaw doesn't lie in one plane

This model is appropriate for light weight and aerated concrete

At the beginning of tensioning softening discontinuousmacroflaw in model is assumed to consist of cracks of length2a0 interspersed by circular pores of diameter 2 r 0


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2 dimensional model Karikaloo et al (1987)Numerical results for stress intensity factor K I at each crack tip in infiniterow and for COD for each macroflaw segment have been fitted bypolynomial approximation.

f(, )=(1+ r s C rs r s)[tan ( /2)/( /2)] 1/2 = r/a+r= 2(a+r)/d

Polynomial approximation differs from the numerical values by less than0.7%Crack growth criterion in the tension softening regime K I=K Icm

/ f t`= {(a0+r0)/(a+r)}f( 0, 0)/f(, ) (1)

Net inelastic deformation

w/w o = / f t`{g( , )/ g( 0, 0)-1 -(2)


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This model contains which is measure of relative fractions of micro voidsand micro cracks.

= 0 no micro voids, less accurate when is large.The tension softening diagrams for three values of a/r and several values of 0 are shown in fig.

As increases, not only does the critical value of crack tip openingdisplacement w c increase, but more importantly, the failure is less unstable.The equation 1 & 2 can be written as

wE`/ ft` d = (f( 0, 0)/2)[( o / )1/2[g( ,)/f( ,)] (g(0, 0)/f(0, 0))]

Tension softening models cannot predict the exact value of the cricticalcrack opening at rupture ( =1).A notational value may be used by limiting to 1- with governed bythe frictional pull-out characteristics of the hard phase from the matrixmaterial.


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For a/r =0.1 ,the unstable branch istotally absent, conforming the

stabilizing role of voids in thetension softening process


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Three dimensional model of Karihaloo & Huang (1992)Two dimensional model predict sudden drop in tensile carrying capacity

after the onset of tension softening

Provision of thickness direction result in a more gradual reduction in theinitial post peak tensile carrying capacity.

In three dimensional model, the discontinuous macroflaw is modelled by adoubly periodic array of penny shaped (circular) cracks (period=l) in theeventual failure plane

Degree of damage = = a2 / l 2

Assumption: diff. b`n the actual opening displacement of any one crack inthe array and the avg. opening displacement of all remaining cracks is sosmall as to be ignored

K I varies with polar angle


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Three dimensional model of Karihaloo & Huang (1992)

/ f t`= ( 0 / )1/4 f(0)/f()

3wE`/16a 0f t`= g( 0)/ 0( / f t` g(0)/g( )- 1)

where 0 = ao 2

/ l2

f t` = K ICm (2 ao) f(0)

Comparison of the 2 fig. shows that triaxiality produce a

more gradual loss in the tensile carrying capacity after thebeginning of tension softening .


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Calculation of fracture energy from (w) relation

Calculated from micromechanical models for tension softening

(w) is a function of intrinsic matrix fracture toughness K ICm and accumulated damage 0

0 depends only on volume fraction of coarse aggregates in the mix

The area under tension softening curve described by eqn 1 and 2 between0 1

-(3)

from eqn 1-(4)

0 1 ) / )(()( ft

dwwd w

)1(`2

1

3

00

f

f f

f f t


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)1 /()1sin

1(2 / 13

1

3

0

13

1

3

1

sr

r srs

sr

r srs C sC

f f

Sub the above eqn in eqn 4,we get

Where

then

00

002

`

) / *()(

f E

f gGK G f m IC f

]1 /(2sin

[2

*3

1

3

0

3

1

3

1

1

2

00

0

sr

r s rs

sr

r s rs f C C

f

g f G

00

00

2mIC

) / *(

)(K

`

f

f gGG E f f


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G*f includes energy expended on subcritical crack growth in prepeak from linear region

Fracture toughness fitted with polynomial approx.

Microvoids significant role in toughening process

The relative contribution of prepeak nonlinearity to toughening of mixwill be 10 -12%

Hence incremental toughness of matrix is mainly due to its post peak

tension softening

)1(][K

`00

1

0

3

10

1

0mIC

r s

r srs

r

r ro

f Y Y

G E


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tension softening

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