ORIGINAL ARTICLE

Toughness of fibre reinforced hydraulic lime mortar.Part-2: Dynamic response

Rachel Chan • Vivek Bindiganavile

Received: 20 February 2009 / Accepted: 5 April 2010 / Published online: 14 April 2010

� RILEM 2010

Abstract The seismic rehabilitation of historical

masonry buildings necessitates a quantitative under-

standing of the repointing mortar under variable

strain rates. In Part-1 of this paper, plain and fibre

reinforced hydraulic lime mortar specimens were

examined under compression, flexure and direct shear

to evaluate the post-crack response under quasi-static

loading. It was seen that although the fibres enhance

the flexural toughness of hydraulic lime mortar, the

material is weakest in Mode I fracture. In Part-2 of

this paper, the authors describe the strain rate

sensitivity of hydraulic lime mortar on the basis of

impact testing of notched beams. The mixes were

identical to those examined in Part-1, and the

dynamic response was evaluated using a drop-weight

impact machine for strain rates in the range of 10-6 to

10 s-1. The authors found that compared to fibre

reinforced Portland cement-based mortar and con-

crete, the flexural response of hydraulic lime mortar

is more sensitive to strain rate.

Keywords Hydraulic lime � Mortar �Fibre reinforcement � Impact � Strain rate sensitivity �Dynamic impact factor � Fracture toughness

1 Introduction

Many old stone masonry structures are located in

areas of seismic activity. The proper rehabilitation of

such buildings requires a quantitative knowledge of

the dynamic response of the masonry unit and its

components. In particular, the bond between the stone

blocks and the binding mortar is of concern [1]. As

explained in Part-1 of this report [2], in the restora-

tion of heritage stone masonry in Canada, hydraulic

lime mortar is chosen for it is intentionally weaker

than the stone blocks and also allows for their

movement over the first few months. While there is

considerable information on the strain rate sensitivity

of Portland cement based composites [3], very little is

known about fracture in hydraulic lime mortars

(HLM) under high strain rates. A civil engineering

structure may be subjected to a range of dynamic

loading from creep to blast loads resulting in strain

rates from 10-5 to 103 s-1 [4]. The first part to this

study [2] described the quasi-static response of these

mortars in compression, flexure and shear. It was

established that the principal mode of failure for both

plain and fibre reinforced HLM was in flexure. This

paper examines the flexural response at higher strain

rates to evaluate the critical strain rate sensitive

parameters for this material. In the absence of

available data on the dynamic response of HLM,

the authors drew lessons from existing literature on

the dynamic response of mortars and fibre reinforced

composites made with Portland cement.

R. Chan � V. Bindiganavile (&)

Department of Civil and Environmental Engineering,

University of Alberta, 9105-116th Street, Edmonton,

AB T6G2W2, Canada

e-mail: vivek@ualberta.ca

Materials and Structures (2010) 43:1445–1455

DOI 10.1617/s11527-010-9599-3

Schuler et al. [5] summarized the experimental

data over the past 30 years for cement based com-

posites in tension which was seen to obey the Comite

Euro-Beton equations as shown below [4]:

DIF ¼ _e_es

� �1:026d

for _e� 30 s�1 ð1aÞ

DIF ¼ c_e_es

� �1=3

for _e [ 30 s�1 ð1bÞ

where _e ¼ 10�6 s�1 and log c = 7.11d - 2.33, such

that:

d ¼ 1

10þ 6f 0cf 0co

ð1cÞ

However, Malvar and Ross [6] report that this

relation underestimates the dynamic impact factor

(DIF) for strain rates lower than 30 s-1 and suggest

the following modified expressions:

DIF ¼ _e_es

� �d

for _e� 1 s�1 ð2aÞ

DIF ¼ c_e_es

� �1=3

for _e [ 1 s�1 ð2bÞ

where _e ¼ 10�6 s�1 and log c = 6d - 2, such that:

d ¼ 1

1þ 8f 0cf 0co

ð2cÞ

The addition of discrete short fibres to Portland

cement mortar is seen to improve the fracture

toughness [7]. Studies under dynamic loading [8]

showed that when used at up to 1.4% of the mortar

volume, the steel fibres resulted in an increase of

toughness by 2.5 times at a stress rate of 103

MPa s-1. However, it must be noted that the data

available is for mortars with a tensile strength of

1.5 MPa, i.e. about 10 times more than that of HLM

investigated in this study. In such controlled low

strength materials, polymeric fibres may be better

suited due to their lower modulus of elasticity.

Besides, the strength of the matrix is also known to

affect the material’s strain rate sensitivity: At similar

densities, higher strength concrete typically has a

lower sensitivity to the loading rate [9]. Polymeric

fibres depict a significant rate sensitivity and the

fracture energy and impact strength of plain concrete

are improved even at a dosage of 0.5% volume

fraction [10]. In fact the resulting flexural toughness

may even exceed that of a corresponding mix

containing an equal amount of steel fibres [11].

Additionally, with polymeric fibres, the mode of

failure may switch from one of pullout to that of fibre

fracture, indicating embrittlement at higher loading

rates [12].

2 Experimental details

An instrumented drop-weight impact-testing machine

as shown in Fig. 1 was used to impart dynamic

loading. The machine has a capacity of 1000 J and is

capable of dropping a 60 kg mass from heights of up

to 2.5 m. A plain hydraulic lime mortar and two fibre

reinforced mixes with proportions as described in

Part-1 [2] were examined in three-point flexure. Note

that the water content was kept constant across all

three mixes (equal to 7.7%), although the addition of

fibres affected the workability as determined by the

spread on a flow table. The specimens were subjected

to impact loading from two drop heights of 250 mm

and 500 mm, respectively. This produced a corre-

sponding impact velocity of 2.20 and 3.10 m s-1,

respectively. The dimension of specimens was iden-

tical to those used for the flexural tests under quasi-

static loading (100 mm 9 100 mm 9 350 mm), so

that together with the two dynamic rates, a range of

strain rates between 10-6 to 10 s-1 was examined in

this program. These strain rates capture up to the

lower end of seismic activity.

Fig. 1 Instrumented drop-weight impact machine

1446 Materials and Structures (2010) 43:1445–1455

The loading history was captured with a load cell,

equipped with a Wheatstone bridge mounted on a tup

(Fig. 2). The acceleration history was captured using

a piezoelectric accelerometer that was attached to the

bottom of the specimen at midspan. The data from

the accelerometer was collected by a data acquisition

system at a rate of 100,000 Hz and was integrated

twice with respect to time so as to yield the deflection

history at mid-span of the beam. Since a suddenly

applied load creates an inertial response from the

specimen, it is required that this inertia be accounted

for to derive the flexural stresses on the beam as

experienced by the material [13]. The equivalent

static response has been derived based on the single-

degree-of-freedom approach. For a beam subjected to

3-point bending under impact, the generalized inertial

load on the specimen during the impact Pi(t) is

represented by [14]:

PiðtÞ ¼ qAaoðtÞ1

3þ 8ðovÞ3

3l2

" #ð3aÞ

where ao(t) = acceleration at midspan of the beam at

time, t; q = mass density for the beam material d

A = cross-sectional area of the beam; l = clear span

of the beam; ov = length of overhanging portion of

the Beam

Thus, the actual stressing load, Pb(t) is derived

after subtracting the inertial load from the total load

recorded by the load cell, Pt(t).

Pb tð Þ ¼ Pt tð Þ � Pi tð Þ ð3bÞAlso, the velocity and displacement histories, vo(t),

and, do(t), at the load-point were obtained by

integrating the acceleration history with respect to

time:

voðtÞ ¼Z

aoðtÞdt ð4Þ

doðtÞ ¼Z

voðtÞdt ð5Þ

The displacement history as derived in Eq. 5 was

used to represent the strain rate. In addition, two

high-speed cameras were employed to obtain a

stereoscopic record of the dynamic event. Both

cameras captured images at a frequency of

10,000 frames s-1 and were later analyzed with an

image processing software to derive displacement

histories. The load cell, the accelerometer and the two

cameras were all connected to an optical sensor that

served as the trigger device. This sensor was placed at

a distance of 2 mm above the top contact surface of

the specimen, Fig. 3. The devices were triggered

immediately once the hammer passed the sensor

causing a disconnection in the optical circuit. This

caused a previous voltage of 5 V to drop to 0 V,

effecting the start of data collection by the acquisition

system (EDAQ). By using this trigger mechanism,

the raw data from all four sources were synchronized

to the same time stamp. Six beams were tested per

mix so that the results shown in the following

sections are the representative data averaged over

each series of specimens.

3 Results and discussion

3.1 Analysis of impact test data

In this study, the acceleration under impact loading

was measured through three different sources

namely, a piezoelectric accelerometer, a blade load

cell based on a Wheatstone bridge and through high

speed cameras. Where as the accelerometer and the

load cell directly measured accelerations, the cameras

measured actual displacement. While the cameras

were set to record data at only 10,000 Hz, the load

cell and the accelerometer registered data at

100,000 Hz. The high speed images were analyzedFig. 2 600 blade load cell: a location of strain gauges in tup and

b Wheatstone bridge circuit

Materials and Structures (2010) 43:1445–1455 1447

to derive an acceleration-time response at midspan of

the beam and the acceleration histories from all three

sources are shown in Fig. 4. It is clear that the actual

displacements as recorded by the cameras are in

keeping with the accelerometer data, while the load

cell records lower accelerations. The strain rates

corresponding to the two drop heights of impact

are shown in Table 1. It was approximately 5 s-1 for

the drop height of 250 mm, while about 10 s-1 for

the higher drop height of 500 mm.

Fig. 3 Trigger mechanism

for activating the high speed

data collection

0

500

1000

1500

2000

2500

3000

3500

4000

0

1000

2000

3000

4000

5000

6000

7000

8000

Acc

eler

atio

n (m

/s)

0

1000

2000

3000

4000

5000

6000

7000

Acc

eler

atio

n (m

/s)

0 2 4 6 8 10 12 14 160 2 4 6 8 10 12 14 160

500

1000

1500

2000

2500

3000

3500

0 2 4 6 8 10 12 14 16

0

500

1000

1500

2000

2500

3000

3500

4000

0 2 4 6 8 10 12 14 160 2 4 6 8 10 12 14 160

500

1000

1500

2000

2500

3000

3500

0 2 4 6 8 10 12 14 16

Time (ms) Time (ms)

Time (ms) Time (ms)

Time (ms)

Time(ms)

accelerometer

High speed camera

Load-cell

accelerometer

High speed camera

Load-cell

accelerometer

High-speed camera

Load-cell

accelerometer accelerometer

Load-cell

Load-cell

High speed camera High speed camera

accelerometer

High speed camera

Load-cell

(a) (b) (c)

(d) (e) (f)

Fig. 4 Acceleration histories for fibre reinforced HLM obtained from various measuring devices. a 0.25 m drop; Vf = 0%, b 0.25 m

drop; Vf = 0.25%, c 0.25 m drop; Vf = 0.50%, d 0.50 m drop; Vf = 0%, e 0.50 m drop; Vf = 0.25% and f 0.50 m drop; Vf = 0.50%

1448 Materials and Structures (2010) 43:1445–1455

3.2 Fracture toughness

Figure 5a–c shows the load–displacement response

of plain and fibre reinforced hydraulic lime mortar at

three rates of loading. As described in Part-1 of this

paper, the plain mix had higher compressive strength

as it was better compacted than the fibre reinforced

mixes. Note that the workability in these mortars was

kept deliberately low (compared to typical Portland

cement based mortars), as the former are intended for

masonry-repointing applications. As stated in Sect. 2,

the water content was kept constant, at 7.7%.

However, while the plain mortar had a slump flow

of 90–105%, the fibre reinforced mortars dropped in

flow to between 35 and 45% [2]. Therefore, even with

a relatively low fibre content (i.e. volume fraction

B0.5%), there was a significant drop in the work-

ability in the fibre reinforced mortars which in turn

led to lower compact-ability. Note that there was a

significant increase in the apparent strength under

impact loading. As described in Eqs. 3a and 3b, the

loads shown in Fig. 5 are the actual stressing load

where the inertial component has already been

accounted for. It is also clear from the pre-peak

portion of the plot that the material became stiffer

with increasing rate of loading. Figure 6 describes the

post peak flexural toughness factor for the three

mixes across all rates of loading. There was a

significant increase in this parameter at impact rates.

Table 1 Test parameters at variable strain rates

Loading rate Quasi-static (Q-S) Impact (I-250) Impact (I-500)

Drop-height 0 250 mm 500 mm

Strain rate (s-1) 10-6 5 9

Fibre volume fraction (%) 0 0.25 0.50 0 0.25 0.50 0 0.25 0.50

0

1000

2000

3000

4000

5000

6000

0

1000

2000

3000

load

(N

)

4000

5000

60007000

0 0.2 0.4 0.6 0.8 1 1.2 1.40 0.2 0.4 0.6 0.8 1 1.2 1.4

1.60

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Quasi-static Quasi-staticQuasi-static

Impact-250 mm

Impact-250 mm

Impact-250 mm

Impact-500 mmImpact-500 mm

Impact-500 mm

Deflection (mm) Deflection (mm)Deflection (mm)

(a) (b) (c)

Fig. 5 Flexural response of plain and fibre reinforced hydraulic lime mortar under quasi-static and dynamic loading. a Fibre

content = 0%, b Fibre content = 0.25% and c Fibre content = 0.50%

Fig. 6 Post-peak flexural toughness factors (FTF) under quasi-

static and dynamic loading for plain and fibre reinforced

hydraulic lime mortar

Materials and Structures (2010) 43:1445–1455 1449

Similar improvement in the post-crack response was

noted previously in studies with fibre reinforced

concrete at high strain rates [14, 15]. However, there

was a subsequent drop in the post-peak FTF for all

the mixes when the drop height was raised from 250

to 500 mm. Thus, while the modulus of rupture

registered an increase with an increase in the strain

rate, the drop in the post-peak FTF at higher rates

indicates an embrittlement in the composite and a

limit to its energy absorbing capacity.

The fracture toughness of plain and fibre rein-

forced mortars were evaluated at three rates of

loading based on the flexural load–displacement plots

shown in Fig. 5a–c. The crack mouth opening

displacement (CMOD) is known to vary between

2D/3 to 4D/3, where D is the displacement of the

neutral axis at mid-span [16]. Thus, in order to

evaluate the fracture properties, the mid-span beam

deflection in Fig. 5 was assumed to be equal to the

crack mouth opening displacement at the notch.

These plots were analyzed to generate crack growth

resistance curves (Fig. 7a–c), where the crack growth

resistance is expressed as the stress intensity factor,

K, at the tip of the crack as it extends into the matrix

leading to an effective crack length, aeff. The

derivation is outlined in Appendix 1.

The peak value on the plots in Fig. 7, is noted as the

fracture toughness, KIC, which is a material property

(Table 2). It is clear from Fig. 8, that the fracture

toughness of plain hydraulic lime mortar increases

with an increase in the rate of loading. Similarly, note

that for the fibre reinforced mix containing 0.25%

fibres by volume, there was an increase in KIC at

higher rates of loading. However, for the mix with

0.50% Vf, there was a drop in the fracture toughness at

500 mm drop height. A comparison of R-curves from

the impact tests in Fig. 7c suggests that while the

stress intensity factor was higher initially from a

500 mm drop, it fell below that for the 250 mm drop

at increasing effective crack length, especially after

80 mm. Also, the addition of fibres did not necessarily

lead to an increase in the fracture toughness especially

Table 2 Mechanical properties of HLM under variable strain rates

Mix designation fr (kPa) FTF (kPa) KIC (MPa m-0.5)

Q-S I-250 I-500 Q-S I-250 I-500 Q-S I-250 I-500

Plain HLM 193 2123 2239 6 2040 1300 0.097 3.33 5.1

FR-HLM (0.25% Vf) 205 1435 2091 95 1600 1250 0.21 3.71 10

FR-HLM (0.50% Vf) 225 1620 2273 165 1720 960 0.25 5.83 4.9

0

1

2

3

4

5

6

7

8

9

10

0

1

2

3

4

5

6

0 20 40 60 80 100 0 20 40 60 80 100

aeff (mm) aeff (mm) aeff (mm)

Impact-500 mm

Impact-250 mm

Q-static

Impact-500 mm

Impact-250 mm

Q-Static

Q-Static

Impact-500 mm

Impact-250 mm

(a) (b) (c)

Fig. 7 Crack growth resistance in hydraulic lime mortar at different strain rates: a 0% Fibre content, b 0.25% Fibre content and c0.50% Fibre content

1450 Materials and Structures (2010) 43:1445–1455

at the higher drop height of 500 mm, as shown in

Fig. 8. This is likely due to a lack of proper dispersion

at higher fibre dosage, which led to a reduction in the

compressive strength of the fibre reinforced mixes.

While the polymer fibre-matrix bond is known to

increase significantly under higher loading rates, this

improvement may have been offset by the lower

compressive strength of the fibre reinforced mixes.

The post-peak FTF is therefore a better indicator of

the fibre efficiency.

3.3 Strain rate sensitivity

The modulus of rupture and fracture toughness are

plotted as a function of strain rate in Fig. 9a and b,

respectively. It is evident that hydraulic lime mortar

is a strain rate sensitive material such that both its

modulus of rupture and its fracture toughness depict a

time dependent response. The dynamic impact factors

as predicted by Eqs. 2a–2c from Malvar and Ross [6]

are shown alongside. Note that the firm lines in

Fig. 9a depict the response from regular concrete

(f 0c = 30 MPa) while the dotted lines were drawn for

hydraulic lime mortar to account for its much lower

compressive strength [Malvar–Ross (m)]. It is readily

seen that hydraulic lime mortar is more sensitive to

high strain rates than conventional concrete. More

significantly, when the compressive strength of the

material was taken into account via the correction

factor, d (Eq. 2c), the predicted dynamic response

was far greater than the actual experimental value.

Therefore, it is clear that while the constitutive laws

formulated for regular concrete underestimate the

strain rate effects for HLM, adopting the correction

factor for its low strength will lead to non-conserva-

tive estimates of its dynamic response. The Malvar–

Ross adaptation of the CEB formulation addresses

cement based composites as low as 10 MPa in

compression. Clearly, as more data is generated on

hydraulic lime mortars, the available models such as

that in Eqs. 1a–1c and 2a–2c may be extended to

describe rate effects in controlled low strength

building materials. In Fig. 9b, the fracture toughness,

KIC, was plotted alongside the expression for the DIF

of fracture energy of conventional concrete, as

suggested by Schuler et al. [5]. Assuming a propor-

tional relation between fracture toughness and frac-

ture energy for a given material, this expression

underestimates the dynamic fracture response of the

HLM mixes. As is evident from their respective

values of DIF, the fracture toughness of HLM was far

more sensitive to strain rates than the modulus of

rupture. This is in keeping with previous reports for

plain concrete [17, 18]. A comparison with fibre

reinforced Portland cement mortar and concrete is

Fig. 8 Mode I fracture toughness of plain and fibre reinforced

hydraulic lime mortar (Fibre volume fraction = 0%, 0.25%

and 0.50%)

Fig. 9 Strain rate

sensitivity of plain and fibre

reinforced hydraulic lime

mortar showing a Modulus

of rupture and b Mode I

fracture toughness

Materials and Structures (2010) 43:1445–1455 1451

made in Fig. 10. Note that the data on Portland

cement mortar represents a direct tensile response

while the rest were taken from flexural tests. The

fracture energy of Portland cement mortar is known

to be less strain rate sensitive than that of concrete

according to Lu and Xu [19], who examined the

compressive response. They attributed it to the

presence of the interfacial transition zone around

the coarse aggregates in concrete. It is clear from

Fig. 10 that fibre reinforced HLM is more rate

sensitive than either Portland cement based mortar

[8] or fibre reinforced concrete [17]. This was true for

both its flexural strength and its fracture toughness,

KIC. However, an increase in the fibre volume

fraction appears to decrease the DIF, when measured

for fracture toughness. It is documented in the

literature that among Portland cement based

materials, those that possess lower strength tend to

exhibit higher strain rate sensitivity [9, 20, 21]. As

hydraulic lime mortar is a softer material, which

gradually accepts load due to its longer curing time

unlike Portland cement, the fracture process within

HLM is less linear elastic than that of mortar or even

concrete made with Portland cement. This may

account for its higher strain rate sensitivity.

4 Conclusions

The following conclusions may be drawn from this

study:

1. When subjected to impact loading during a drop-

weight test, the accelerations as measured by the

hydraulic lime mortar are always higher than that

measured by the loading tup. The resulting

deflections may be satisfactorily estimated by

means of a high speed camera.

2. Hydraulic lime mortars exhibit strain rate sensi-

tivity in flexure, with the tensile response as

determined by its modulus of rupture being more

sensitive than that for Portland cement based

mortars or concrete.

3. Crack growth resistance curves indicate that the

Mode I fracture toughness of hydraulic lime

mortar is strain rate sensitive. The fracture

toughness of fibre reinforced HLM was found

to be more sensitive to strain rate than its

modulus of rupture.

4. Polymeric micro fibres impart post-peak energy

dissipation to hydraulic lime mortar. However,

the post-peak flexural toughness factors

Fig. 10 Stress rate

sensitivity of various fibre

reinforced composites

showing a Tensile strength

and b Mode I fracture

toughness. FR-HLM fibre

reinforced hydraulic lime

mortar (this study),

FR-PCM fibre reinforced

Portland cement mortar [8],

FRC fibre reinforced

concrete [17]

Fig. 11 Compliance found experimentally for any load and

corresponding CMOD

1452 Materials and Structures (2010) 43:1445–1455

decreased with an increase in the rate of loading.

This implies embrittlement of the fibre reinforced

mortar at high strain rates.

5. Modified ‘Comite Euro-Beton’ (CEB) expres-

sions that describe the dynamic impact factors

for conventional concrete may not be directly

applied to hydraulic lime mortars. In particular,

accounting for the low strength of this material

leads to a significant overestimation of the

dynamic impact factors.

Acknowledgements The authors wish to thank the Network

of Centres of Excellence on Intelligent Systems for Innovative

Structures (ISIS-Canada) and the Natural Sciences and

Engineering Research Council (NSERC) Canada, for

financial support to this study. In addition, the authors are

grateful to the Masonry Contractors Association of Alberta

(Northern Region) and Public Works Canada, for their in-kind

contributions.

Appendix 1: Evaluation of effective crack length

and the stress intensity factor

Let:

P = applied load

S = span

B = beam width

D = beam depth

D = midspan deflection

CMOD = crack mouth opening displacement

r = flexural stress

ao, aeff = initial crack length and effective crack

length (Note: aeff C ao)

C = compliance = CMOD/P

E = modulus of elasticity

We thus have, C(ao), as the initial compliance and

C(aeff), the compliance at a crack length aeff ([ao).

The compliance at any instant of loading may be

experimentally derived from the load, P, and the

crack mouth opening displacement, CMOD, as

shown in Fig. 11.

The compliance corresponding to Point A0, which

represents a general loading condition on Fig. 11 is

given by:

C aeff

� �¼ CMOD að Þ

Pð6Þ

The initial compliance may be found as:

C aoð Þ ¼CMOD

Pð7Þ

where CMOD and P correspond to the bending over

point, A, in Fig. 11.

In a flexural test, the theoretical compliance is

influenced by the span-to-depth (S/D) ratio as well as

the extent of the crack. If a = (a/D) and b = (S/D).

Thus, the general expression for the compliance for

b C 2.5 is given by Guinea et al. [22] as follows:

C að Þ¼ 72

432EBa �7:8224þ6:3904b�0:6327b2� �

þ 72

864EBa2 11�0:88bð Þ 1:24�0:14bð Þ

� 72

432EBa3 �1:24þ0:14bð Þ2

þ 72

EB

2:65ab512 1�að Þ �4:64þ0:13bð Þ� �

þ 72

EB

a 2:65bð Þ2 2�að Þ512 1�að Þ2

( )

� 72

EB

a 16:48þ45:19bð Þ2

82944 1þ3að Þ

( )

þ 72

EB

�21:5296þ63:1104b�10:339b2

4096

� �ln 1�að Þ

þ 72

EB

5350:0672þ14129:7728b�1482:6839b2

995328

� �

ln 1þ3að Þ ð8Þ

Substituting for b = 3, Eq. 8 reduces to the

following:

C að Þ ¼ 1

EB

0:942aþ 0:571a2 � 0:112a3 � 1:584a1�a

þ8:888a 2�að Þ1�að Þ2 � 20:07a

1þ3aþ

1:362 ln 1� að Þ þ 2:448 ln 1þ 3að Þ

8>>><>>>:

9>>>=>>>;ð9Þ

Now, the crack mouth opening displacement corre-

sponding to any applied load, P, is given by CMOD

(a,b) and is expressed as a function of both the crack

length and the span-to-depth ratio [22]. For the present

case of b = 3, the CMOD may be expressed as

follows:

Materials and Structures (2010) 43:1445–1455 1453

CMODðaÞ ¼ 24P

EBffiffiffipp 0:0052� 1:12aþ 0:2456a2

� 0:176a3 þ 0:086a4 � 0:01148a5

� 0:227 ln(1 + aÞþ0:616 ln(1 + 3aÞ

� 0:879

1� aþ 0:8745

ð1� aÞ2

)ð10Þ

As mentioned earlier, for the impact tests, the

CMOD was measured using high speed cameras. On

the other hand, for the quasi-static tests, the mid-span

deflection, D, was taken to be equal to the CMOD.

This was justified on the basis of the observation by

Armelin and Banthia [16]. Upon relating the theo-

retical result of CMOD(a) for a given value of the

applied load, P, from Eq. 10 with the experimental

value, one obtains the corresponding value for the

modulus of elasticity, E. Since the elastic modulus

remains constant throughout the extension of the

crack, it is found from the bending over point,

corresponding to the initial crack length, ao. Substi-

tuting this value of E in Eq. 9, results in the

theoretical evaluation of the compliance correspond-

ing to P, at any instant. Equating the experimentally

found compliance at any point, A0, as given by Eq. 6,

with the corresponding theoretical value from Eq. 9,

we may determine the corresponding effective crack

length, aeff.

Finally, the compliance is related to the specific

energy release rate, G, and in turn to the stress

intensity factor, KI, as given by the well-known

relationships in Broek [23]:

G ¼ P2

2BD

dC að Þda

ð11Þ

K1 ¼ffiffiffiffiffiffiffiGEp

ð12ÞBased on the effective crack length, aeff, derived in

Eq. 10, the stress intensity factor for the three point

bend test was determined as follows [21]:

KIðaÞ ¼6M

D5=2

ffiffiffiap

ð1� aÞ3=2ð1þ 3aÞ

" #ð13Þ

p1ðaÞ þ4

3p4ðaÞ � p1ðaÞ½ �

� �ð14Þ

where

M ¼ PD

2; ð15Þ

p4ðaÞ ¼ 1:9þ 0:41aþ 0:51a2 � 0:17a3 ð16aÞ

and

p1ðaÞ ¼ 1:99þ 0:83a� 0:31a2 þ 0:14a3 ð16bÞ

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