Fracture Mechanics of Cement Mortar and Plain Concrete Karihaloo

8/10/2019 Fracture Mechanics of Cement Mortar and Plain Concrete Karihaloo

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REVIEW ARTICLE

Fracture Mechanics of Cement Mortar

and Plain Concrete

B.L. Karihaloo, A. Carpinteri, and M. Elices

School of C ivil and M ining E ngineering, The U niversity of Sydney, A ustralia, D ipartim ento di

Ingegneria Strutturale, P olitechnico di Torino, Torino , Italy, and D epartam ento de C iencia de

M ateria les, U niversid ad P olitecnica d e M adrid , C iu dad U niversita ria, M adrid

Th e cla ss ic al G r if fit h theor y is m o di fie d to i nc lu de the prepeak non-

lin ea r a nd p os tp ea k te ns io n-s ofte nin g r es po nse o f c em en t m or ta r a nd

conc re te . T h is is done through the notion of an eq uivalent (effective)

b rittle m ateria l w hich p erm its the ca lculation o f the re al fra cture

energy CF of the maierial. T he l atter is used to ex pl ain th e d uc ti le -

br it tl e t rans it ion via the (e n ergy ) b r it tl en ess number. A simplified

p ro ce du re fo r s tu dy in g the cra ck grow th in cement m ortar and plain

concrete is [ormulated on the b asis of a n eq uiva le nt e la stic maierial,

and its lim itations are pointed out.

Aov

ANCEO CEMENT BASEO

MATERIALS

1993, l, 92-105

KEYWORDS: Elastic equivalences, Energy brittleness num-

ber, Equivalent (effective) brittle material, Modified Criffith

theory, Tension-softening

Il

e last decade has improved our under-

standing of the role of defects such as pores

and cracks in the tensile response of the ce-

ment based materials that were traditionalIy regarded

as being brittIe. It is now generally accepted that ali

cement based materials exhibit moderate strain hard-

ening prior to the attainment of their ultimate tensile

capacity (region AB in Figure I), reminiscent of the

response of high-strength metallic materials. How-

ever, unlike the latter, these materials are character-

ized by an increase in deformation with decreasing

tensile capacity past the ultimate strength (region BCO

in Figure 1). Such a response is called tension soften-

ing. The materials exhibiting moderate strain harden-

ing prior to the attainment of ultimate tensile strength

and tension softening thereafter may be called qua-

sibrittle. It was precisely because of such behavior that

the classical linear elastic fracture mechanics (LEFM)

was found to be inadequate for such rnaterials, al-

though attempts were made in the 1950s[1-4].A mod-

ification of the classical LEFMwas achieved through

the so-called cohesive crack model in which the real,

open macrocrack was assumed to terminate in a zone

Address correspondence lo: B.L. Karihaloo, School of Civil and Mining Engineer-

ing, The University of Sydney, NSW 2006, Australia.

Eisevier Science Publishing Co., Ine.

ISSN 1065-7355/93/ 6.00

with a residuai stress transfer capability (fracture pro-

cess zone) [5]and the corresponding toughness (frac-

ture energy) was designated

C F

The practical use of

cohesive crackmodel relies on sophisticated numerical

procedures that are often cumbersome. The existence

of a fracture process zone ahead of a macrocrack aiso

results in a size effectwhereby the larger the size of the

object the more brittle its observed response.

This articlewill first describe a direct modification of

the classicalGriffith theory

by

the inclusion ofprepeak

nonlinear and postpeak tension-softening response

and introduce the notion of an equivalent (effective)

brittle material, This will help in the direct calculation

of the rea fracture energy CF of cement mortar and

plain concrete. It will then explain the ductile-brittle

transition based on the notion of (energy) brittIeness

number

Se

= CAfP), where ft is the ultimate tensile

strength of the material and D a characteristic struc-

turaI dimension. Finally, it will present a simplified

procedure for evaluating crack growth in cement mor-

tar and plain concrete based on a generaI concept of an

equivalent elasticmaterial, pointing out the lirnitations

of this concept.

Prepeak and Postpeak

Nonl inear Response

All cement based materials contain pores and rnicro-

cracks even in their virgin, unstressed state. Under

applied tensile stresses, microcracks form at the mor-

tar/aggregate interfaces leading to the prepeak nonlin-

earity. The transition from linear to nonlinear response

(point A Figure 1) is primarily governed by the extent

of available interfaces.

For cement mortar and plain concrete, the threshold

and ultimate strains were accurately measured by Ka-

pIan [6] under several tensile loading combinations.

He found that they depend on only the volume frac-

tion of the coarse aggregate in the mix, but do not vary

with other microstructural parameters, such as the

type and texture of coarse aggregate or the water:ce-

ment ratio.

The interfacial microcracks first form between the

Received July 28, 1993

Accepled Septernber 7, 1993


2/14

Advn Cem Bas Mat

1993;1 :92-105

B

~~~w(a)

~o~

f

Deforrnation

FIGURE 1. Typical load-deformation response of a cement

based material in tension showing (1) the transition from

linear to nonlinear response (point A); (2) prepeak nonlin-

earity (AB);(3)anse t of localization of deformation (point B);

and (4) postpeak tension-softening response (BCD).

mortar and the Iargest size aggregates. With a further

increase in tensile stress, not onIy do the existing mi-

crocracks propagate stabIy, but also more microcracks

are progressively formed at interfaces between the

mortar and aggregates of smaller size. As cement

based materials usually contain a range of aggregate

size, the process of stable microcrack growth and for-

mation of new microcracks continues under increasing

tensile stress resulting in the prepeak nonlinearity

(stage ABin Figure 1).

This prepeak nonlinear response has been accurately

explained using damage mechanics modeis (see, e.g.,

ref 7). These models are applicable when the micro-

cracks are evenly distributed and are not too numer-

ous. However, even after the deformation has begun

to localize near the eventual fracture pIane (point Bin

Figure 1), fracture does not immediately follow. Point

Bcould even lie on the ascending branch. The decreas-

ing stress transfer capacity with increasing deforma-

tion in the tension-softening regime is a result of the

progressive rupture of the intact ligaments that break

the continuity of flaws. Because this progressive mi-

crodamage consumes energy, the intrinsic toughness

of quasibrittle materials is improved. Further improve-

ment oftoughness is achieved by external means, such

as fiber reinforcement, but this toughening mechanism

will not be pursued here.

Several two- and three-dimensional models have

been proposed to describe the tension-softening be-

havior of quasibrittle materiais. The two-dimensional

model considered first by Horii et al. [8] and later by

Ortiz [9] treats the discontinuous macroflaw as a row

of collinear cracks subjected to a normal tensile stress

well removed from the Iine of cracks. This modei pre-

dicts unstabie rupture when the ineiastic deformation

Fracture Mechanics 93

attains a criticaIvalue. Such unstable behavior, how-

ever, is not a material response but is a result of the

model assuming the cracks will coalesce, which may

not happen. In fact, it wouId seem (Melin [lO]) that

when the tips of adjacent straight cracks approach so

close to each other that the straight path would be

unstable, they avoid each other by curving and form

eyeIets in the processo Such crack curving and for-

mation of eyeIets have been repeatedIy observed in

tension tests on cement mortar and concrete speci-

mens [11].

The tension-softening response of most cement

based materials is characterized by a pronounced tail

[12],which has been predicted by the two-dimensional

model of Karihaloo et al.

[13].

In this model, the dis-

continuous macroflaw is treated as a row of collinear

cracks interspersed by circular pores and unbroken lig-

aments. The three-dimensional model ofLi and Huang

[14]regards the discontinuous macroflaw as consisting

of a row of penny-shaped cracks separated by the larg-

est second-phase particles (assumed as circular disks)

in the matrix, so that neighboring cracks do not inter-

act. However, as the tension-softening implies pro-

gressive approach of the neighboring broken segrnents

toward each other, their interaction cannot be ignored.

In the three-dimensional model of ref 15, the discon-

tinuous macroflaw is modeled by a singly or doubly

periodic array of penny-shaped cracks in the eventual

fracture pIane, allowing for full interactions between

the cracks.

In both the two- and three-dimensional modeis a

relationship between the decreasing stress transfer ca-

pacity, U , the average net inelastic opening across the

discontinuous fracture pIane, w, as a function of the

increasing damage parameter, ,/,is obtained

(1)

F} ( Y ) and F

2

( Y ) are determined from the solution of

the boundary value probIem corresponding to the cho-

sen model [8,9,13-15], and the microscopic ultimate

tensile strength t i is related to F}

( / 0 )

and the matrix

fracture toughness K J c / that is, the intrinsic crack

growth resistance of cement paste.

/0

is the (small)

accumulated damage at the onset of tension softening,

that is, it reflects the prepeak nonlinearity. Examples of

tension-softening diagrams obtained using the model

of ref 13 are shown in Figure 2.

Effective Gr iff ith Crack Concept

Asmentioned above, for cement mortar and plain con-

crete, the classicalGriffith approach to brittle materials


3/14

94

B.L. Karihaloo et al.

1.0

0.8

0.6

o/f,

0.4

0.2

o

l/

0.1

yo~

---0.02

0.04

----- 0.06

- - - 0.08

Edgemicrocracks

d , 1

d

r x

~ - o - y ; - < > - - < > --t>

2(~' \

Microvoid

o

OL J _ L_ L_~_~~_~_~_~~

~ o

2 3 4 5 6 7 8 9

wE'

1t(~o+ 30)f,

FIGURE 2. Tension-softening curves predicted by the model

shown in the inset for various values of initial damage (in-

cluding the circular pores).

has to be modified to account for the prepeak nonlin-

earity and the postpeak tension softening, that is, for

the energy expended on microdamage rather than on

the creation of critical fracture surfaces (criticalmacro-

crack).This diffuse zone ofmicrodamage is commonly

called the process zone.

A modification based on the concept of an equiva-

lent (oreffective)crackwas proposed by[enq and Shah

[16]and Nallathambi and Karihaloo [17],who used the

classical Griffith theory but with the size of any pre-

crack augmented to account for the energy expended

in the process zone. [Here it isworth recalling that test

specimens used to determine fracture toughness

K

1c

and/or critical energy release rate C

Ic

are usually pro-

vided with a well-defined sharp starter crack (length

ao ) from which it is easier to study the instability phe-

nomenon.] It was reasoned that the energy expended

in the process zone can be equated to that required to

create a fictitious (traction-free) crack (Figure 3). In

practice, though, the length of the latter was deter-

mined not from the indicated energy balance but

rather by comparing either the compliances [16]or the

stiffnesses [17]. The size of the supplementary ficti-

tious crack so calculated is designated by f : .a or f : .a

e

and

the corresponding fracture toughness

I G c

or K rc to dis-

tinguish it from the fracture toughness

K

1c

of an ideally

brittle material. It will be noted that in this modified

Griffith approach the instability condition in quasibrit-

tle materials is governed by two parameters [K ic and a

a

o

+ f:.a(or the criticalcrack opening corresponding

to it) in the two-parameter model of Shah and Jenq

[16] ; K Jc

and

a

e

= a

o

+

Sa ,

in the effective crackmodel

ofKarihaloo and Nallathambi [18]),as opposed to just

one parameter (KIc) in brittle materials.

The procedures for determining

K~c

and

K

are sum-

Advn Cem Bas Ma

1993;1 :92-105

1 .. ~ Cr~~~~g 1

'Iraction-free crack,

ao Bridging-zone zone

I

Fracture process-zone,

lp I

~

Main crack

I

ffective traction-free crack, a

e

FIGURE 3. Process zone ahead of real traction-free crack an

the concept of effective traction-free crack.

marized in ref 18 and will not be repeated here. W

note en passant that for normal age concrete, K

an

I G c

increase with increasing I l whereas S a , and Sa de

pict the opposite trend. Thus, it is Sa or

f : .a

rather tha

K L or K~c which defines the brittleness of a cemen

based material; the higher the tensile strength, I l th

lower Sa or f : .a for structures of identical geometry

This situation is unlike that in metals for which th

fradure toughness decreases with increasing tensil

strength.

From the previous rnicromechanical models for ten

sion softening it is possible to calculate the energy re

quired for complete fracture of the material. This en

ergy per unit area, denoted

C F

is given by the are

under the curve shown, for example, in Figure 2

C F = [o w a da = = [ W c a w dw

J f J o

2

where

a w

defines the tension softening and is a fune

tion of the accumulated damage at the onset of defor

mation localization (eq1). In practice, CF is determined

from tests conducted in various loading geometrie

(e.g., three-point bending [19]), but this value ma

grossly overestimate the real fracture energy because

includes energy expended on processes other than th


4/14

Advn Cem Bas Mat

1993;1 :92-105

formation of fracture surfaces, for example on cracking

in the bulk of the specimen, etc.

The area under the tension-softening curve (eq 1)

between the limits - V a ~

-V ~

1 gives the real fracture

energy

l

a

fl

a a

CF = w a ) d a = = w - v)

a

d- v.

/. '(o-V

The energy

CF

(per unit area) expended on the creation

of a traction-free crack of relative length (1 - - V a ) at

complete rupture can now be used to put the effective

fracture toughness K~e actually the effective toughness

C/ c

corresponding to it

(C~e

=

(K le)2/E ),

on a firm phys-

ical footing simply by equating the total energies (per

unit thickness)

CF =C fJl - -V )

=

(K fei(1 - -Va) E .

The toughening ratio

K f K : ;

can be shown to be [13-

15,20]

where F 3 - V ; - V a is again determined from the solution

of the corresponding boundary value problem.

Figure 4 shows the ratio of the effective fracture

toughness K fe to the matrix fracture toughness

K k

(eq

5) for severallevels of the accumulated damage at peak

load -V a as a function of the parameter [3.The latter in

2.0

...... . .. .. . oooor . ....

8

Y o =

6

Kfc

K~

4

1.2

1.0 L.. __ . . L . . . _ _ _ _ _ _ _ _ . . . .

O 0.2 0.6 0.8

.4

FIGURE

4. Variation of the toughening ratio with 13for sev-

erallevels of the accumulated damage at peak load, lo.

Fracture Mechanics

95

(3)

this model [13] is a measure of the relative fractions of

microvoids and microcracks. Thus, [3

=

Odenotes the

absence of microvoids, whereas [3

=

1 implies micro-

voids with negligibly small edge microcracks. The

curves in Figure 4 are reminiscent of the classical creep

curves, with an initial primary region of increasing

K U

K k

with increasing [3

0.6. The variation of

K f d K k

with - V a for a fixed

[3is fairlymoderate (Figure 5). It is evident that micro-

cracks alone cannot make a large contribution to the

toughening of quasibrittle materials. For further illus-

tration, we write explicitly the toughening ratio from

eq 5 for the two extreme cases of [3

=

O(no microvoids)

and [3=1 (microvoids with negligibly small edge mi-

crocracks), using the appropriate coefficients

(4)

~ I~

=

1.0039

+

0. 3265-V o

[ 3

=

O

le le 1.7360 + 0.6732 -V o [ 3 =1)

(6)

(5)

Thus, for a material such as high-strength concrete,

even when the accumulated damage (including any

preexisting damage) at peak load is large

- V a =

0.25,

say), the toughening provided by this mechanism ([3

=

Oin eq 6) would be just 8.5%. Concretes of normal

strength and cement mortars, though, can be expected

to have a [3value in the range of 0.15 ~ [3~ 0.6, so that

the toughening provided by the tension softening

would range between 15%and 20% (Figure 5) depend-

ing on the damage present at peak load.

2.0

r

8

6

K~c

K~ 4

0.75

0.15 - O.

1.2

1.0

O

1.0 L====:::r:==:::L:=~:J

O 0.05 0.10

0.15

0.20

0.25

Y o

FIGURE 5. Variation of the toughening ratio with lo for var-

ious relative fractions of microvoids and microcracks (13).


5/14

96

B.L. Karihaloo et al.

To complete this section, it may be pointed out that

the damage accumulated in concrete at peak load Y o )

depends only on the volume fraction of the coarse ag-

gregate V i [6]. It follows therefore from eq 5that

K ic

can

be expressed in terms of only

K f: :

and

V i

(in the range

0.15 ,,;;[ 3 , , ; ; 0.6), that is, a similar dependence to that

predicted by the tension-softening model for concrete

proposed in ref 14.

A comparison of the effective fracture toughness

K i c

calculated above from a micromechanical point of view

with the fracture toughness obtained using the effec-

tive crack model [18]clearly demonstrates that

supple-

mentary fictitious traction-free crack Sa is a measure

of the brittleness of the cement based material, The

smaller its value, the more brittle the material. In fact,

the microscopic parameters

t i

and Y o can be expressed

in terms of

K i c

and

Sa

[21], so that we can define

brittleness using the effective crack concept.

u ctility S ize ffects

From a practical point of view, an approach based on

the concept of brittleness number has been found to be

more useful, Brittleness is often confused with low ma-

terial strength. The two properties are in fact entirely

distinct. Materials can be strong but brittle (glass, ce-

ramics, cast iron, etc.) and they can be weak but tough

(concrete, rock, wood, etc.). Brittleness is the tendency

to break suddenly and noisily but not necessarlly at

low stresses.

One of the most significant discoveries in recent

years in the field of Structural Engineering was that the

brittleness of structural elements depends on their

size. Small elements fai in a ductile or plastic manner,

while large elements of the same material fail in a brit-

tle or catastr ophic m anner. T hs can be explained by

the fact that the two intrinsic properties of the material,

namely the strength and toughness, have different

physical dimensions: strength is force per unit area,

that is, energy per unit volume, while toughness is

force per unit of length, that is, energy per unit area. It

is possible to demonstrate that the brittleness of struc-

tures may be measured through the ratio of the tough-

ness of the material

CF

to the product of its strength t i

and of the maximum size of the structure D. This ratio

is a pure number, with no physical dimension, and

depends on the dimensional scale of the body. Based

on this ratio, a small glass body and a large steel ship

of same geometry may appear to be equally brittle, just

as a thin steel body and a microscopic glass filament

may appear equally ductile.

The variation in structural response as the size of the

structure changes is called d uctile-brittle transition,

and in recent years has led to a flourishing crop of

Advn Cem Bas M

1993;1:92-105

theories and models capable of describing and repro

ducing this surprising phenomenon. As has alway

occurred in the history of science, there has been muc

discussion, at times heated, among researchers b

cause of the fact that the new and generaI models mu

only simulate the transition, but must not contradic

the most classic and by now firmly accepted facts. M

terial behavior laws have therefore been generalize

to include, as limit cases, theory ofplasticity on the on

hand, and linear elastic fracture mechanics on th

other. New mathematical theories concerning catastro

phes [22] and fractals are also providing a useful co

tribution to the description and understanding of th

phenomenon, which is irregular, discontinuous, an

without any physical similitude as the scale changes

The classic elastc-perfectly plastic constitutive la

(Figure 6a) is obviously not appropriate for describin

the localization of damage into a crack, nor brittleness

in the sense of a sudden structural unloading. Qui

recently it has been pointed out how even the strain

softening constitutive laws (Figure 6b), which repr

sent tension (J as decreasing with strain E, are not sui

able for simulating localization phenomena, especial

for reproducing dimensional effects. Only a doub

constitutive law which uses stress and strain up to th

maximum load, and stress and crack opening in th

softening stage (Figure 6c) is useful for this purpos

(cf. eq 1).

Let us consider three bars of the same material but

different lengths under tension (Figure 7a). While th

stiffness of the bars decreases with length, peak-stres

t i and dissipated energy W

d

= CFA, where CF = fra

ture energy and

A

=

area of the cross-section, mu

o

o

~~~ ~

~ ~ ~

(a)

(b)

o o

c)

FIGURE 6. Material constitutive laws: (a) elastic perfectl

plastic; (b) strain-softening; and (c) cohesive.


6/14


7/14

98 B.L.Karihalooet al.

FIGURE 10 (a) Crack in the flexural concrete element; (b)

cohesive crack model (cf. Figure 3).

uniaxial tension test. As the initial depth of the crack,

aiD

increases, a decrease in stiffness and loading ca-

pacity is observed, aswell as an unexpected increase in

ductility (Figure lla). As C F decreases to a value more

characteristicfor mortar than for plain concrete (Figure

llb), the various tendencies remain, but the system

appears tobe much more brittle-so much indeed that

the snap-back phenomenon occurs for

aiD ~

0.25.For

aiD =

O that is, for an initialIy uncracked beam, the

cusp is especialIy sharp and implies an explosive re-

lease of kinetic energy.

Ifinstead the material is kept unchanged, say a con-

crete with C F

=

50 N/m, but only the depth scale is

changed (with C /O

=

4), one notices a considerable

increase in brittleness as the beam depth D increases

(Figure 12).As mentioned previously, all of these ten-

dencies may be captured by changing only the nondi-

mensional parameter s e (eq 7). Figure 13 shows the

structural responses considered earlier in a nondimen-

sional form as

s e

changes over four orders of magni-

tude. At the lower extremity of the range (curve A),

one can find both especialIybrittle materials (lowCF/ t

ratio) and especially large beams (high value of D).

For the purpose of building safety, it is therefore

very important to control the evolution ofthe mechan-

icaldamage processes, which, as we have seen, can be

intrinsicalIy stable, unstable, or catastrophic. The de-

formation of a loaded structure increases first in pro-

portion to the load and then more rapidly as the stiff-

ness reduces because of damage. Thus, as deforma-

tions increase beyond a certain level of damage, it is

necessary to decrease the applied force to avoid sud-

den failure. This unstable damage stage may be kept

under checkby controlling the monotonicalIy growing

deformation but not by controlling the force, which

must decrease. Often the unstable stage is folIowedby

a third catastrophic stage when a macrocrack forms

A d v n

Cem Bas Mat

1993;1 :92-105

18000

16000

14000

12000

, . . . . . . ,

6

10000

~

d


8/14

40000

ao/D =0.0

D =0.4 m

ao/D =0.0

32000

0.1

0.2

g

24000

-

Documents

Fracture Mechanics of Cement Mortar and Plain Concrete Karihaloo