Dynamic Poisson's Ratio of Portland Cement

CEMENT and CONCRETE RESEARCH. Vol. I , pp. 559-583, 1971. Pergamon Press, Inc. Printed in the United States.

DYNAMIC POISSON'S RATIO OF PORTLAND

CEMENT PASTE, MORTAR AND CONCRETE

R. Narayan Swamy University of Sheffield, Sheffield, England

(Communicated by A. M. Neville)

ABSTRACT

Tests are reported to determine the Poisson's ratio of hardened paste, mortar and concrete from longitudinal and torsional resonant frequencies and from pulse velocity. It is shown that Poisson's ratio depends on the mix proportions, the type of aggregate and its Poisson's ratio, and the aggregate volume content. Pulse velocity Poisson's ratio is more consistent and greater than that obtained from longitudinal and torsional resonant frequencies. The considerable scatter found in Poisson's ratio values obtained from resonant frequency tests is shown to be due to anlsotropy and the imperfectly elastic behaviour of the paste, mortar and concrete.

On dEcrit des experiences pour determiner le rapport ~e Poisson pour la p~te durcie, le mortier et le bEten d'apr~s les fr~quences de r6sonance longtudinale et de torsion et d'apr~s la vitesse de vibration. On montre que le rapport de Poisson d~pend de proportions du melange, du genre d'agrSgat et de son rapport de Poisson et de la quantxte d agregat par volume. Le rapport de Poisson de la vitesse de vibration est plus uniforme et plus grand que celui obtenu d'apr~s les fr~quences de r~sonance longltudinale et de torsion. On montre que la dispersion considerable qu'on trouve dans les valeurs pour le rapport de Poisson obtenues par des examens de la fr~quence de r~sonance est due ~ l'anisotropie et au comportement Blastique in~gal de la p~te, du mortier ou du ciment.

559

560 Vol. I , No. 5 POISSON RATIO, DYNAMIC, CONCRETE

Introduction

Poisson's ratio is generally agreed to be the most difficult of the elas-

tic constants to measure. Its value is also normally not critical in engineer-

ing design and these two factors probably account for the limited amount of data

available on its experimental determination. A precise assessment of its value

is, however, necessary for multiaxial creep computations, in determining the

dynamic modulus from measurements of ultrasonic pulse velocity and in eval-

uating correction factors for the size and shape of the specimen vibrating in

longitudinal, flexural and torsional modes. Errors up to 30 per cent can be

introduced in the calculated values of dynamic modulus obtained from ultrasonic

pulse velocity by assuming a constant value of Poisson's ratio. A knowledge

of Poisson's ratio is also necessary to assess the spalling effects due to

thermal movements. Recent investigations have also shown that Poisson's ratio

has a significant role to play in studying the formation and propagation of

microcracks, and in studying the fracture mechanism of concrete (I, 2).

Experimental Program

The data reported here form part of an extensive investigation to study

the effects of the various constituents of concrete on the dynamic properties

of the composite material and, particularly, the effect of the presence of

relatively rigid aggregate inclusions in a cement paste matrix and in a mortar

matrix.

The experimental work consisted of three series of tests. In the first

series Poisson's ratio was determined both from the longitudinal and torsional

resonant frequencies, and from longitudinal wave velocity and the fundamental

longitudinal resonance frequency. The first series of tests was made on

twelve concrete mixes and their corresponding mortar mixes. Three aggregate-

cement ratios of 4.0, 6.0 and 7-5 were used, and for each aggregate-cement

ratio four water-cement ratios were used. The coarse aggregate for the concrete

specimens was a continuously graded crushed gravel with a maximum size of

19 mm (~ in.). The sand was washed and dried pit quartzite sand.

In the second series of tests Poisson's ratio was determined from the

longitudinal and torsional resonant frequencies. This series consisted of

tests on hardened pastes, mortars and concretes. The paste specimens had

water-cement ratios of 0.3, 0.4, 0.5 and 0.6. To control the shrinkage of

specimens with water-cement ratios of 0.5 and 0.6, additional specimens with

a plasticlsing grouting agent were also made. The tests on mortar consisted

of two water-cement ratios of 0.4 and 0.6, each water-cement ratio having

Vol. I , No. 5 561 POISSON RATIO, DYNAMIC, CONCRETE

sand contents of 20, ~0, 60 and 70 per cent by absolute volume. The concrete

specimens had a water-cement ratio of 0.6 with coarse aggregate contents of

20, ~0, 50, 60 and 70 per cent by absolute volume. Two types of aggregates

were used in this series - a crushed limestone and a crushed gravel, both

continuously graded with a maximum size of 19 mm (¼ in.).

In the third series Poisson's ratio was determined from longitudinal

wave velocity and fundamental longitudinal resonant frequency. T~lis series

of tests was carried out on nine concrete mixes and their corresponding

mortar mixes. Three aggregate-cement ratios of 4.0, 6.0 and 7.5 were used

and for each aggregate-cement ratio three water-cement ratios were used.

The coarse aggregate used was a continuously graded rounded river gravel

with 19 mm (¼ in.) maximum size. Washed and dried river sand was used in

all these tests.

Ordinary Portland cement (ASia[ type I) was used for all ~le three series

of tests. The cement for the first two series was obtained from one source,

while for the third series, it was obtained from a different source. No

attempt was made to blend the cement, but all the specimens in any one series

were made from a single delivery.

In the first and third series, and in the second series excepting the

paste specimens, the tests were conducted in the wet and dry condition. The

wet specimens were continuously cured under water underunoontrolled laboratory

conditions of temperature. For the dry conditions, the specimens cured under

water for fourteen days and hhen allowed to dry in a constant temperature and

humidity room at 68 ° and 65% Roll.

Testing.

The tests were carried out on 100 mm x 100 mm x 500 mm (4 in. x A in x

20 in) prisms and the strength properties of the concrete in hbe three series

were determined from control specimens. The prisms were excited to resonance

in the fundamental modes of longitudinal and torsional vibration and their

respective resonance frequencies were recorded. The specimens were tested

at various ages up ~o 56 days in both the wet and dry conditions.

The apparatus for the vibration tests (Figs. 1 and 2) consisted of a

test bench for the appropriate mode of vibration, a suitable pich-up device

and an electromagnetic vibrator with their control system. A 5VA low-distor-

tion oscillator was used with the exciter, the combined system providing an

8.91~ (2 ibf) thrust wi~h a frequency range of 5 H~ to 20 kH~ and a maximum


FIG. I

Test set-up for longitudinal vibration

FIG. 2

Test set-up for torsional vibration


displacement of 2.5 mm (0.1 in.) peak to peak. The vibration exciter had a

low fundamental frequency and negligible mass. The signal from the pick-up

was amplified and displayed by a Type C amplifier Telex oscilloscope. The

frequency was directly read on a digital counter with a crystal stability

of + I part in 106 and a system accuracy of -+ 0.005 per cent.

A piezo-electric contact type crystal pick-up was used for longitudinal

measurements, and an electro-magnetic velocity sensitive (air-gap type)

vibration pick-up was used for torsional vibration. All the tests were

carried out in the fundamental mode. A preliminary series of tests, not

reported here, was carried out to investigate the support energy losses and

the repeatability of test results. Several specimens of the same mix were

vibrated in each mode (including flexural) and several readings were taken

for each specimen. With experience and better techniques of operation and

reduction of extraneous losses, it was found that consistent results could be

obtained with two specimens. All the data reported here are thus the average

of two specimens; for each specimen, the measurements were repeated until

three similar sets of results were obtained for a given mode of vibration.

Continuously water-cured specimens were returned to the water-tank immediately

after the measurements were taken, and no experimental evidence was found to

show ar~y change in the dynamic properties during the time of testing.

The longitudinal wave velocity through the test specimens was determined

with a well-known commercial ultrasonic pulse tester in which the individual

times of transmission could be measured to an accuracy of greater than one

per cent. A thin film of paraffin wax was used between the transducers and

the moulded faces of the specimens, and the reproducibility of the results

was found to be excellent. The time of propagation of the ultrasonic pulse

was determined across the centre of the section over the length of the prism,

and the mean of several observations was taken to compute the pulse velocity.

For laboratory specimens, the transmission times could be measured to within

-+ 0.2 microseconds.

Test Results

The dynamic moduli from the longitudinal and torsional modes of

vibration were determined from the well-established relationships between

elastic modulus and the mechanical resonance frequency (3,~,5). The

correction factor for lateral inertia for the fundamental longitud~-al

resonance is less than one half per cent and was neglected. The dy~Am4C


Poisson's ratio ~ was determined from the elastic relationship,

E 2--7- i = ~ (I)

where E and ~ are the dynamic Young's and torsional modulus respectively.

The ultrasonic pulse velocity V is related to the dynamic longitudinal

elastic modulus E by the equation

E = V20 (I + ~) (i - 2~) (2) (i - ~)

where p is the &ensity. Poisson's ratio was obtained from equation (2)

from the measured values of E and V.

Discussion of Test Results

The results of the test on paste, mortar and concrete are discussed

below. In this discussion, in order to identify the values of Poisson's

ratio determined by the two methods, the value obtained from longitu&inal

and torsional resonance tests is termed the dynamic Poisson's ratio. The

value obtained from pulse velocity and longitudinal resonance frequency is

termed the pulse velocity Poisson's ratio.

Although flexural measurements were taken in all the three series of

tests, to study damping characteristics (6), these measurements were not used

to evaluate Poisson's ratio from equation (I) for the following reasons.

For wet specimens, the dynamic modulus of the paste, mortar and concrete

obtained from longitudinal and flexural modes was nearly the same, although

minor differences existed at early ages. For dry specimens, the d~amic

modulus of mortar and concrete obtained from flexural resonant test was

generally less than that from longitudinal resonant test - at 56 days, the

differences were about 5 per cent (6).

The variation with age and water content of dynamic Poisson's ratio

of hardened paste without and ~th additive is shown in Fig.3, in which only

lines through the points for the paste without additive are shove. The

results for the paste without additive are generally consistent, but those

for the paste with additive show more scatter; nevertheless certain trends

are clearly evident. Poisson's ratio and differences due to age appear to

increase with water-cement ratio up to a value of 0.5. Poisson's ratio is

initially high, and decreases with increase in the strength and maturity of

the paste. The curve for 0.5 water-cement ratio without additive, however,

appears to rise at the beginning and then decrease more steeply than the rest;

with increase in age, the differences between curves for 0.5 and 0.6 for the

paste without additive become less and less.


0'36'

0"34'

~ 0.32.

a: 0 . 3 0

m

0-28

0 n 0 . 2 6

U $ < 0 2 4 z

0.22 '

0 '20 0

• 0.3 wlc • 0-4 w/c

• 05 W/C • 0.6 W/C

o 0 5 W/C WITH ADDITIVE

[] 0"6 W/C WITH ADDITIVE

• "r-------------._ 0 3

AGE IN DAYS

FIG. 3

Effect of water-cement ratio and age on the dynamic Poisson's ratio of saturated ordinary Portland cement paste without and with additive.

The slope of all the curves shows that Poisson's ratio is likely to

continue to change even after 28 days. The presence of an additive

increases the Poisson's ratio initially for the paste with 0.5 water-cement

ratio, but decreases it for the paste with 0.6 water-cement ratio. The

points for 0.6 with additive are well below those for 0.5 with additive.

These results of dynamic tests show quite a different trend to those

of static tests reported by other investigators (7). Tests on saturated

cement pastes at an age of 30 days for water-cement ratios of 0.3 to 0.5

showed that Poisson's ratio remained remarkably constant at 0.25 for

increasing water-cement ratio (7).

Yig.$ is typical of the results of the first series of resonance tests

in which the variation of dynamic Poisson's ratio with age for wet and dry

concrete an& its corresponding mortar matrix was also studied. For both

mortar and concrete, the poisson's ratio is initially high and decreases with

age and continued hydration (8,9,10). Pickett reported that Poisson's ratio

increased as hydration proceeded, and that values greater t~an 0.25 were

obtained for specimens moulded under pressure (3); it is not known why

Poisson's ratio increased with hydration and no simS1 ar phenomenon has been


030 -

0.28"

" 0.26"

lI 0.24-

~ 0.22- 0 n

_0 O.2O-

< Z ~ oJg '

0.16.

0,14

SERIES I

CONCRETE •

AGG./CEMENT 750

WATER/CEMENT O,51

MORTAR MATRIX •

WET

DRY

.

\ • \

" -

0 ,~ 24 3~ 4b 48 AGE IN DAYS

FIG. 4

Typical variation of dynamic Poisson's ratio with age of wet and dry concrete and its corresponding mortar matrix.

--A

5~

reported. The greatest decrease in Poisson's ratio occurs between I and 7

days, being about 10 to 20 per cent, and is greater for mortar matrix than

for concrete. After 7 days, the decrease is more gradual, and at the end

of two months, the values were still slowly decreasing. Similar results

have also been obtained by others from longitudinal wave velocity and

longitudinal resonance frequency measurements (8, 9).

The effect of drying is to reduce the Poisson's ratio further (Fig.4).

During the rapid drying stage - which corresponds to the removal of moisture

from capillary channels - the decrease in Poisson's ratio is rapid. After

about 10 days drying, the rate of decrease of Poisson's ratio becomes the

same for both concrete, and the mortar matrix. Again, the effect of drying

is much more pronounced with mortar th~n with concrete, althoug~h the

differences tend to decrease with continued drying up to about four weeks.

After about five weeks of dry, the Poisson's ratio remains sensibly

constant for both the concrete and its mortar matrix. Ultimately, Poisson's

ratio of dry mortar remains greater than that of dry concreteo

The results shown in Fig. 4 are representative of the behaviour of

other mixes in the first series. In addition to drying and wetting, curing


conditions are also known to affect the static Poisson's ratio (II), and the

nature of the environment, such as corrosive atmosphere, similarly influences

the value of Poisson' s ratio (I0).

Influence of A~Eregate Content on the Poisson's Ratio of the ~trix.

Comparing Figs. 3 and 4, it is seen that the addition of fine aggregate

to paste lowers the Poisson's ratio of the composite material. The results

of the second series of tests in which the sand content of the mortar matri~

was varied further showed that for a given water-cement ratio, Poisson's

ratio decreased as the volume fraction of sand increased. For each water-

cement ratio only four different sand volume contents were used; the results

are thus too few to define a precise relationship. However, with a water-

cement ratio of 0.4, for example, the Poisson's ratio at 28 days decreased

from about 0.28 for the cement paste (wet) to about 0.17 for mortar (dry) with

70 per cent of absolute volume content of sand. Similar results have been

reported by Ishai (12) who found that pulse velocity Poisson's ratio decreased

from about 0.27 to about 0.16 with 80 per cent sand content. In both cases,

the decrease was gradual up to about 50 per cent, beyond which the rate of

decrease increased.

Similarly the addition of coarse aggregate decreased the Poisson's

ratio of the composite material, and, for a given water-cement ratio,

increasing the quantity of coarse aggregate caused a decrease in the Poisson's

ratio of the composite material. The results are again too few to state

definite relationships; nevertheless trer~s similar to those diSCUSSed above

were observed. Thus, for limestone aggregate with a water-cement ratio of

0.6, the Poisson's ratio at 28 days decreased from about 0.22 for the mortar

matrix (dry) to about 0.19 for concrete (dry) with 70 per cent absolute

volume of aggregate. Similar results have been reported on static tests by

A son (7).

Sand and coarse aggregates are generally stiffer than cement paste and,

as stiff inclusions in the paste, they restrain the lateral ~pansion of the

softer matrix. In fact both Young's modulus and Poisson's ratio of the

aggregate influence the Poisson's ratio of the resulting concrete (13). The

results of series two tests showed that with crushed gravel aggregates having

an elastic modulus of about 55 kN/m m2 (8.0 x 106 psi) and a Poisson's ratio

of about 0.22, their addition reduced the overall Poisson's ratio of the

matrix to values comparable to that of ~he aggregate. Crushed limestone with

an elastic modulus of about 69 kN/mm 2 (i0.O x 106 psi) and a Poisson's ratio


of about 0.33 (13), on the other hand, was found to produce concretes with a

higher Poisson's ratio than concrete with crushed gravel.

Anson (7) has shown that the static Poisson's ratio of concrete v C

is related to that of the paste v by the equation P

Vc = Vp(l - Va)n = Vp Vnp (3)

where V and V represent the volume fraction of the total aggregate and of a p cement pas~respectively. The constant n in equation (3) can be considered

as a measure of the degree of restraint imposed by the aggregate inclusion on

the lateral expansion of the matrix, and its value would depend on the stiff-

ness of the aggregate and its Poisson's ratio• For river gravel aggregates

with a Young's modulus of about 69kN/mm 2 (10.0 x 106 psi) and a Poisson's

ratio of 0.22, n is about 0.42 for Vp = 0.25 (14).

An examination of all test data obtained in this investigation showed

that, for a given aggregate-cement ratio, Poisson's ratio generally increased

with increase in water-cement ratio, and that, for a given water-cement ratio,

Poisson's ratio decreased with increase in aggregate-cement ratio. Similar

results were also obtained by Anson (7). Fig. 5 shows the influm~ce of water

023-

O 22-

0.21"

020

_o o l 9

0-18'

o m o.17- m o Q.

016- _u

0"14

013 0 '3

! o

• • & | O

4 & &

El " ~l A O

| • •

, • 28 DAYS

• ,, 56 DAYS o •

• 9 f • i o V D

o

SERIES I. CONCRETE MORTAR MIX

WET DRY WET DRy • O • O

0:4 o:s & o:~ 0:8 o~

FIG. 5

Influence of water-cement ratio on the dynamic Poisson's ratio of wet and dry concrete and its mortar matrix.


content on the Poisson's ratio of wet and dry concrete and its mortar matrix.

There is considerable scatter to define a single relationship; the results,

however, show a trend. Leslie and Cheesman (15) found from tests on water-

saturated prisms that Poisson's ratio obtained from flexural and torsional

moduli decreased with increase in density - the values ranging from about

0.30 to 0.24 for densities of about ll20 to 2@20 kg/m 3 (70 to 150 lb/ft3).

The data given in Figs. 3, ~ and 5 show that the dynamic values of

Poisson's ratio are generally greater than the generally known static values

~ch, for most concretes, fall between 0.15 and 0°20 although other values

have also been reported. At high stresses or under conditions of rapidly

alternating loads, however, changes in the static value do occur. Probst

(16) has shown that, with repeated loading, Poisson's ratio shows a

consistent decrease. On the other hand, a marked increase in Poisson's

ratio at very high stresses has been observed by several investigators (1,

17, 18, 19, 20). Similar results have also been reported in triaxial tests

on concrete cylinders. Gardner (21) has shown that Poisson's ratio changes

from about 0.10 at low stresses to about 0.25 for low values of triaxial

stresses. When Polsson's ratio is less than 0.50, there is a decrease in

the volume of the specimen subjected to compressive loading. At about 70

to 95 per cent of the failure load, increases in Poisson's ratio to over 0.50

have been measured (17, 18, 21). At this stage, the volume of the specimen

begins to increase, and this has been suggested as the long-term sustained

strength of concrete, and represents the beginning of hhe process of

disintegration and internal discontinuity (17, 19).

Relation Between Poisson's Ratio and Dynamic Mo&,lus

The variation of d~namic Poisson's ratio with longitudinal dynamic

modulus of concrete and its corresponding mortar matrix is shown in Figs. 6

and 7. The plotted points in these two figures refer to all the tests on

concrete and mortar specimens of series oneand two explained in the

experimental programme. For concrete, the data shown in Fig. 6 comprise two

types of aggregates (limestone and crushed gravel), three aggregate-cement

ratios (with crushed gravel) with a wide range of water-cement ratios, and wet

and dry curing conditions up to an age of 56 days. The data in Fig. 7 refer

to the mortar matrix of the concrete shown in Fig. 6; they cover a wide range

of sand contents and water-cement ratios under wet and dry conditions up to

an age of 56 days.


0"32

0-30

(>28,

_ '0.26'

0-: )4

0 m 0':~:~.

n

IE < Z

Ot8.

o.1@

0"14 2"0

w #

!

v

| ,

%

÷ ÷

x

AGG./CEI~ENT 4-0 8"O "7.5

L~ESTONE GRAVEL

AGE

• f3 & ~ v e •

; , t , , : : ' , • ' : t ~ 4 , , . ,

• ~ ' 6 , I j

I 3'0 4:0 5 b 8~0 7.C) LONGITUDIB~L DYNAMIC MOOULUS X I0 6 p.$.i.

FIG. 6

Variation of dynamic Poisson's ratio with longitudinal dynamic modulus of wet and dry' concrete.

W E T D R Y

• 6

w •

+ x

• D

I - - 5 6 DAYS

S E R I E S I . ICRUSI,'IED GRAVEL I

SERIES 2.

It is well-known that the dynamic modulus increases with age and it

has been shown that Polsson's ratio decreases with age. For both mortar and

concrete, Poissen's ratio decreases ~th increase in d~/namic modulus, but the

results are too scattered to define any one relationship. The results show

t.hat there is perhaps no unique relations~hip between Poisson's ratio and

dynamic modulus, but that such relationships depend upon the composition of

concrete. For a wide range of mix proportions and age, Simmons (22) also

found considerable sca~ter of values between Poisson's ratio and dynamic

modulus. However, for one mix proportion only, it was found that a linear

relationship existed between Poisson's ratio and dynamic modulus. It is

noteworthy from Fig. 6 that higher values of Poisson's ratio were generally

obtained for limestone concrete than for crushed gravel concrete. Similar

results have also been reported by Jones (8). Although limited, these

results show the influence of the nature of the aggregate and confirm that

it is unlikely that a single relationship will exist for all types of

aggregates (8).

Figs. 6 and 7 further show that there is no significant difference in

range of Poisson's ratio values for concrete (0.15 - 0.32) and for mortar

matrix (0.17 - 0.30). It is further confirmed that in general Poisson's


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0 2 8 .

• 026- o

rr 024 "

A/1

z 0 022-

n

0 2 0 "

018-

016"

OI4 0

• 0

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I o

o W •

0 l 0 •

W • v • ° °

' i • ~ , , g % + , o. , . , • o i •

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i'o 2:o 3:o 4:o s b LONGITUDINAL DYNAMIC MOOULUS X IO ~ p+s+i

MORTAR MATRIX

WET DRY

• °t A A SERIES I V •

• o ( 0 4 W/~ SERIES2)

+ x ( O 6 W/~ SERIES 2 )

AGE I - 56 DAYS

riG. 7

Variation of dynamic Poisson's ratio with longi~idinal d~namic modulus of wet and dry mortar matrix.

ratio is less for both mortar and concrete in the dry condition than in their

wet condition. McCoy and Mather (23) also found a wide range of Poisscn's

ratio values for moist-cured limestone concrete tested under flexural and

torsional vibrations - from 0.11 to 0.33 at 14 days and from 0.13 to 0.39 at

180 days. Similar variations have been reported for static tests - showing

a slight increase with applied load and the strength of the concrete.

Krenchel (24) has reported values of Poisson's ratio of 0.15 to 0.18 for weak

concrete and of 0.17 to 0.25 for strong concrete of about 48 N/ram 2 (7000 psi).

Pulse Veloci~ Poisson' s Ratio.

Poisson's ratio obtained from pulse velocity and longitudinal resonant

frequency is shown in Fig. 8. The data shown cover the first aria third

series of tests on concrete and the corresponding matrix and comprise a wide

r~unge of aggregate-cement and water-cement ratios, two different types of

agcregate (coarse and fine) and 28 day wet and dry curing conditions. If all

the concrete and mortar matrix results ax'e considered, the data show

considerable scatter. The rest~ts show that for both concrete and mortar

matrix different relationships exist for different aggregates. Both the

coarse and fine aggregate were different for the two series of" tests, and


_o

Z O

O r l

03(

02'

O 21

02"

O 2~

O25

O.2,~

022

022

O21

02O 30

SERIES I : CRUSHED GRAVEL CONCRETE • o

MORTAR MIX • A

28 DAY WET DRY

28 DAY WET (CRUSHED GRAVEL)

8 DAY WET (CRUSHED GRAVEL)

A @ • OX + + O O •

O • A X + O

+ + • X X •

+ o

+

I I 1 @

SERIES 3 : ROUNDED GRAVEL + •

x ®

WET DRY

, A J

4.0 5"0 6 0 7 0 - 6

LONGITUDINAL DYNAMIC -MODULUS X IO PS I

FIGo 8

V a r i a t i o n o f p u l s e v e l o c i t y P o i s s o n ' s r a t i o ~rl'~h longitudinal d~u~amic modulus of wet and dry concrete and i t s m o r t a r m a t r i x .

the results sho~ the dependence of Poisson's ratio on the type of coarse and

fine aggregate.

If, however, the 28 day wet results alone are considered for a given

type of aggregate, the results show that an approximately linear variation

exists for both concrete and its mortar matrix for a wide range of

aggregate-cement and water-cement ratios. Simmons (22) also four~l slm~lar

linear relationships between pulse velocity Poisscn's ratio and the

longitudinal dynamic modulus.

The influence of water-cement ratio on the Poisson's ratio of

concrete and its mortar matrix obtained from measurements of longitudinal

wave velocity and fundamental longitudinal resonance frequency is shown

in Fig. 9.

The data shown comgrise, for concrete, three aggregate-cement ratios tested

at 28 days in the wet and dry state and their corresponding mortar matrices.

The results show the following:-

1 ) Poisscn's ratio increases with water-cement ratio for

both mortar and concrete.

2) The process of drying decreases the Poisson's ratio of

Vol I , No 5 573 POISSON RATIO, DYNAMIC, CONCRETE

_o n,,

v1 Z 0

0 n

o I..U >

n

028"

027:

026-

025"

0.24"

023-

022

021

020

019 0

SERIES 3

WET DRY WET DRY WET DRY

MORTAR MATRIX , a x ® + •

COI,K:RETE • o j J • ,

AGG./CEMENT 40 60 7.5 • ~ AGE 28 DAYS

• l

o.4 ols o~ o~ o8 WATER-- CEMENT PATIO

FIG. 9

Influence of water-cement ratio on pulse velocity Poisson's ratio of concrete and its mortar matrix for various aggregate contents.

the wet specimens of concrete and mortar, the decrease

for mortar being generally less than that for the

concrete, although this depen&s on the aggregate content

and water-cement ratio.

3) The Poisson' s ratio of both wet an~ dry mortar matrix is

greater than that of the corresponding wet and dry

concrete •

4) Withim the limitations of the test variables, Poisson's

ratio obtained from wave velocity and longitudinal

resonance frequency appears to be more consistent and

shows less variation than that obtained from longitudinal

and torsional resonance frequencies (Fig. 5); and the

scatter is much less for wet specimens than for dry

specimens.

5) Poisson's ratio depends on the aggregate content and the

type of aggregate and its Poisson's ratio (13). It is

therefore unlikely that there will be one unique

relationship between Poisson's ratio and either water-


cement ratio, compressive strength or dynamic modulus

(Figs. 6 and 7).

6) The data of Fig. 9 show that significant errors are li/cely

to occur if dynamic modulus is determined from longitudinal

wave velocity equations when a single constant value of

Poisson's ratio is assumed. Leslie and Cheesman (15),

however, found that good results could be obtained by

using a value of Poisson's ratio = 0.24 for concretes with

a density in excess of 2240 k~m 3 (140 lb/ft3).

7) Comparing Figs° 5 and 8, it is seen that Poisson's ratio

obtained from wave velocity and longitudinal resonance

frequency is generally higher than the corresponding value

obtained from longitudinal and torsional resonance

frequencies although equality of these two values has also

been reported (25).

The variation of compressive strength with the Poisson's ratio of

concrete and its mortar matrix obtained from wave velocity and fundamental

longitudinal resonance frequency is shown in Fig. 10. The data generally

confirm the results shown in Fig. 9 and show that Poisson's ratio tends to

0 3(>

0 ,~ 0 2 9 iX

i l l 0 2 8

U3 cO

0 2 7 CL

>- I'--

) 0,26"

d w

> 025 - uJ ,,.n _1 :D a. 0.24-

021

AGG./CEMENT 4 0 6 0 75

CONCRETE • A ,

M O I ~ T A R M A T R I X o A v

o

4 0 0 0 6 0 0 0 8 0 0 0 K3000

C U B E C R U S H I N G S T R E N G T H - - P.S.I.

FIG. I0

Influence of compressive strength on the pulse velocity Poisson's ratio of wet concrete and its mortar matrix.

Vol . 1, No. 5 575 POISSON RATIO, DYNAMIC, CONCRETE

decrease with an increase in compressive strength or a decrease in water-

cement ratio, but there appear to be different relations for different mix

proportions and different types of aggregate (26). In general, any factor

that increases the strength of concrete or its elastic modulus tens to

decrease the value of Poisson's ratio obtained from pulse velocity or

resonance frequencies.

Comparison of Poisson's Ratio Obtained by Different Methods.

A comparison of Poisson's ratio obtained from longitudinal wave

velocity and fundamental longitudinal resonance frequency and that from

longitudinal and torsional resonance frequencies is shown in Fig. 11.

The results cover concrete and its corresponding mortar matrix over a wide

0"30"

o ~ O,29.

"~ 0.28. ul

a. 0-27.

>,. t.- U 028- q w

0.25. i,1

J ~ 0-24'

CONCRETE •

MORTAR MIX o

28 DAY WET

/ @

f O O

0.is o;m o.',7 o: ls 0:~ o:2o 0:21 o:22 o-'23 o.'24 DYNAMIC POISSON'S RATIO

FIG. 11

Comparison between dynamic Poisson's ratio and pulse velocity Poisson's ratio.

range of aggregate-cement ratios of 4.0 to 7.5, and a wide range of

water-cement ratios of 0.37 to 0.87. There is some scatter for both

concrete and mortar. For both mortar and concrete, the Poisson's ratio

obtained from pulse veloci~ covers the same range of about 0.25 to 0.28;

for concrete, the pulse velocity Poisson's ratio is about ~0 to 50 per cent,

and for mortar 20 to 25 per cent, greater than the value obtained from

longitudinal and torsional moduli. Values of Poisson's ratio obtained

from longitudinal and torsional resonance frequencies are thus intermediate

between ~hose from direct static measurements and those from pulse velocity

an~ longitudinal frequency (9, 22).

576 Vol . I , No. 5 POISSON RATIO, DYNAMIC, CONCRETE

%

× 4c

~c ,~ 2c Z_

k-

=7

'_O 2C x

8 IC

_ . - - e .e

_ _ ,:,,

8 I 2

AGE IN DAYS

~a 8 . /"r ." _..~ . . . . ~ - .~ . , .

j 2c~- / , ~ . - - - .A •

O i , A ~ J i a O B 16 24

AGE IN DAYS

• .

I I - -II

' ~ ' ,~ ' 2 ~ '

AGE IN DAYS

- - e 0 3 W/C

- - m 0 4 W / C

- - v OS W/C

- - ~ o 6 w / c

- - - v O5 W/C PASTE WITH ADDITIVE

- - - a O 6 W/C PASTE WITH ADOITIVE

FIG. 12

Typical dynamic Young' s modulus, flexural modulus and torsional modulus for hardened paste.

Scatter of Te st Re sults.

The considerable scatter of results observed in Figs. 3, 5, 6, 7 and

8 casts doubt upon either the reliability of the experimental technique or

the validity of equation (1). The reliability of the experimental methods

employed in this investigation is established by the consistent values of

Young's and torsional modulus obtained for all cementitious materiels,

typical values of which are shovm in Fig. 12 (6). It is therefore

considered that the scatter of data found is real and that it is primarily

due to the poorly-conditioned nature of equation (S) and the anistropy of

concrete. Of course, the lateral inertia corrections which were ignored,

although small in relation to dynamic modulus, may also have some important

influence on the derived value of the Poisson's ratio.

Equation (I) is basically a poorly-conditioned equation to calculate

Poisson's ratio - small variations in E, and especially in G, produce large

changes in v . The error in ~ in terms of E and G may be assessed by

differentiating equation (I) and dividing it by equation (I) which leads to

dv E dE dG E-2G ('-E --G) (l~)


Equation (~) shows that errors of like sign in measuring both E and G tend

to cancel out each other, whereas errors of unlike sign are additive° In

the latter case, significant errors in ~ might result when multiplie& by

E/E-2@ which is of the order of @.0 to 5.0 for concrete. Assuming mean

values of E and @ of ~1.5 kN/mm 2 (6.0 x iO 6 psi) and 16.5 kN/mm 2 (2.4 x iO 6

psi) respectively, a variation of + 2 per cent from the mean values of E or

G would cause Poisson's ratio to vary by + i0 per cent.

Anisotropy of Concrete.

The secQnd important factor causing discrepancies of results is the

anisctropy of concrete. Both equations (i) and (2) relating Foisson's ratio

to dynamic moduli and pulse velocity are based on the assumption of concrete

being a continuous, homogeneous, isotropio and elastic material. The hetero-

geneity of cementitious materials, however, creates anisotropic conditions

through the presence of a gradient in the elastic modulus. In good quality

uniform concrete such variations tend to be negligible, but the presence of

small degrees of anisotropy an& of an elastic modulus gradient can introduce

serious discrepancies in the values obtained from dynamic tests.

Such anisotropic conditions are known to exist in metals and rocks.

In polycrystalline metals, such as copper, brass an~ silver, an appreciable

degree of anisotropy can exist due to preferred orientation of the grains,

and this can produce anomalous results in Pcisson's ratio (27, 28)@ If the

grain size is small compared to the specimen size, and the grains are randomly

oriented, isotropic conditions are generally satisfied. However, preferred

crystal orientations occur commonly in casting, working and annealing. Then

the elastic properties are functions of direction. If the material becomes

completely asymmetric, elastic behaviour can only be satisfied by 21

independent elastic constants as opposed to 2 for an isotropic material.

Determination of Poisson's ratio from vibration tests on bars of copper

and silver have shown values for copper varying between 0.271 and 0.781 an~

for silver between 0.185 and 0.802, the actual value depending on the type of

heat treatment given to the specimen (27). Bradfield and Pursey (27) found

that values of this apparent Poisson's ratio agreed with values determined

by other methods only when the specimens had been thoroughly annealed.

Similarly rocks which possess distinct bedding planes an~ minute

fissures are known to exhibit anisotropic elastic characteristics owing

to orientation of structural and textural features. Irrational values of

Poisson's ratio from static and dynamic tests on rock specimens have been


reported (23, 29). ~ith rock cores, instantaneous elastic relief of stress

and the resulting creep would certainly affect the measured elastic

properties; further, the drilling operation itself might set up an

asymmetrical orientation of structural elements that could lead to the

measurement of negative and time-dependent Poisson's ratio. The behaviour

of such cores may be considered analogous to the behaviour of a stopper

forced into the neck of a bottle (23).

Concrete can exhibit non-homogeneous and anisotropic characteristics,

the extent of which depends on the quality of the concrete mix and tha care

with which the specimens are made. In heterogeneous materials made up of

various constituents, the components can be anisotropic in themselves.

As the total volume of the specimen increases in relation to the size ofthe

constituents, the material may be regarded statistically homogeneous and

isotropic.

Anisotropy in concrete can occur in a number of ways. It is well-

known that water-gain can trap water or cause voids under aggregate particles

and can result in lower aggregate-matrix bond strength at the bottom of the

aggregates than at the top (30, 31). This can occur in all mixes and can

cause anisotropic conditions in the direction of casting and differential

strength characteristics in different directions (32). In such cases, the

velocity and propagation of a longitudinal wave will be variable. The

presence of anisotropy has been shown by the differing ultrasonic pulse

velocities through the depth of a specimen in the direction of casting, the

extent of variation depending upon the distance travelled by the pulse in

relation to the size of the inhomageneities (8). In addition, reflection

and refraction will also occur at the interfacial boundaries of the various

elements (33), and the wave-front of the pulse will be complex ~th varying

velocity and direction of propagation. These effects are minimised as the

size of specimen increases in relation to aggregate size, and the velocity of

propagation in different directions becomes "almost equal. It is probable

that anisotropy occurs during compaction due to segregation of the mixing

water, and is diminished by reducing t~ water content or by using an air-

entraining agent (3A). The viscoelastic nature of concrete also affects

the dispersion and attenuation of the pulse, but its effects are less

important than the heterogeneity of concrete (35).

The use of equation (1) to obtain Poisson'~ ratio from longitudinal

and torsional resonance will therefore almost certainly lead to a different


value to that obtained from equation (2) from measurements of longitudinal

resonance and pulse velocity. Examples illustrating this difference in

values are shown in Table 1.

Z~LE !l

Anisotropic Corre0tions__for D~fnamic ~oduli and Poisson's Ratio

EXPERIMENTAL

m !

m~/mm 2

l

&6o5

k~mm 2

C0~&K;TED

!

F2~ ,~/~_i i~I. T TAL

t

CORRECTED

~mm 2

[-

G kN/mm 2

&8.2

v=E-1 2G

19.8

elastic constants) may be obtained and a value

the equation

m !

' = (2G' i) (5)

A value of Poisson's ratio ~ can also be obtained from measurements of

longituainal wave velocity (equation 2).

In good quality uniform concrete, the results of these two different

testing methods would nearly be the same and the anom~!ies due to anisotropy

of concrete would almost be entirely absent. If, however, an elastic modulus

gradient exists, serious errors can arise in interpreting the results of the

longitudinal and resonance tests by equation (5). The propagation of high-

frequency pulses has the advantage over the resonance tests in that they

Pursey and Cox (28) have outlined a me~hod of estimating the degree of

anisotropy whereby the true elastic properties of a material in its isotropic

form can be obtained from longitudinal and torsional tests on slightly

anisotropic specimens. Provided certain assumptions are made, the

corrections procedure applied to concrete can highlight the influence of

anisotropy on the results of elasticity measurements.

From the longitudinal and torsional resonance tests, the experimental

values of the dynamic moduli E'and G' (i.e. the apparent values of the

' of Poisson's ratio from

19.9 0.168 0.229 47.7 19.5 0.221

48.9 20.9 0.167 0.224 h9.9 20.5 0.220

!20.1 0.176 0.225 0.219


propagate in a path normal to the direction of maximum gradient of elastic

modulus (36), and in that short wave-lengths are affected to a lesser extent

by anisotropy (35). The value of Poisson's ratio obtained from

ultrasonic pulse velocity measurements is therefore more likely to be nearer

to the true Poisson's ratio. In any case, a useful measure of the elastic

anisotropy of the material can be obtained from the fractional difference ! !

between ~ and ~ and is given by

(~v --

: ( 7 l ) (6 )

The real isotropic constants may then be obtained from the following

E = E' (I - 0.40a~') (7)

G G' (i + 0.40c~' -- ) i + ,~' (8)

where E a na G represent the true (corrected) values of Young's modulus and

~.~odulus of Rigidity.

Table I gives three examples to illustrate the order of magnitude of

the correction. The experimental values were obtained from three specimens

made as nearly identical as possible from the same mix. It is clearly seen

that anisotropy has considerable influence on both the elastic moduli and

that the corrected values of Poisson's ratio agree closely with the value

obtained from equation (2).

The results shown in Table I --nd the method given for dealing with

,n~ sotropy might appear to discredit the data given in this paper, and

inAeed, the dynamic method of determining Poisson's ratio. This is, of

course, not the case. The implication of Table i is that the value of

Pcisson's ratio determined from longitudinal and torsional resonant

frequency tests is inherently subject to error - firstly, due to the

ill-oon~itione~ nature of the mathematical relationship, and secondly, due

to the anisotropy of concrete. It is important to appreciate the

limitations of dynamic methods in evaluatimg Poisson's ratio, and to know

that anisotropio corrections are necessary to give a more realistic

assessment of the elastic constants. The results further show that errors

are bound to occur in computing one elastic constant from the experimental

determination of another elastic constant.

equations (26)


Conclusions

Extensive test &ata are presented on the value of Poisson's ratio of

hardened paste, mortar and concrete obtained from longitudinal and torsional

resonant frequencies, and from pulse velocity and longitudinal resonant

frequency. The mortar and paste specimens were made of ~he same matrix

as ~hat of the corresponding concrete and mortar. The test variables

included different water-cement ratios, different aggregate-cement ratios,

different ages, wet and dry curing conditions and three different types of

coarse aggregate. Each test was performed on two specimens, and for each

specimen, the measurements were repeated until three similar sets of results

were obtained. One type of cement and two types of sand were used. Within

~he limitations of these tests and variables, the following conclusions can

be drawn; the results are compared, where appropriate, to earlier findings.

The results confirm that Poisson's ratio is initially high and

decreases with increase in the strength and maturity of the material. It

also increases with increase in water-cement ratio. The process of drying

decreases the Poisson's ratio of the specimens, ~he decrease for mortar being

generally less than that for concrete, although ~his depends on the aggregate

content and water-cement ratio.

The addition of ag~reo~ate inclusions is shown to reduce the Poisson's

ratio of ~he paste and mortar matrix. Aggregate inclusions not only stiffen

the composite material but restrain the lateral expansion of the softer matrix.

The Poisson's ratio of both wet and dry mortar matrix is ~hus generally

greater than that of the corresponding wet and dry concrete. For a given

water-cement ratio, increasing the volume fraction of the aggregate content

decreases the Poisson's ratio of the composite material, but the rate of

decrease depends on the aggregate content.

Poisson's ratio of c~crete thus depends on the aggregate volume and

the type of aggregate and its Poisson's ratio. It is shown that, in general,

any factor that increases the strength of concrete or its elastic modulus

tends to decrease its Poisson's ratio, but there appears %0 be d/fferent

relations for different mix proportions. It is therefore unlikely that there

is one unique relationship between Poisson's ratio and either water-cement

ratio, compressive strength or dynamic modnlus for all types of concrete.

Poisson's ratio obtained from wave velocity measurements is higher

and generally more consistent and shows less scatter than that obtained from

longitudinal and torsional resonance frequencies. The latter values lie


between those from static measurements and pulse velocity measurements.

The value of Poisson's ratio thus depends on the method of its determination.

The considerable scatter and discrepancies found in the value of

Poisson's ratio obtained from longitudinal and torsional resonant frequency

tests are considered to be real and primarily due to the poorly-conditioned

nature of the equation and the anisotropy of concrete. These tests show

that paste, mortar and concrete are basically imperfect elastic media.

The large discrepancies arise from the use of formulae which are inadequate

to describe the actual elastic behaviour of these materials. The

discrepancies found in computing one elastic constant from the experimental

determination of another are thus real, and cannot be made without running

me risk of making large errors.

Reference s

1. G.S. Robinson, Proc.Inter.Conf. The structure of concrete and its Behavior Under Load, London, Sept. 1965, 131 (1968).

2. A.L.L. Baker, Proe.Inst.of Civ.Eng. 45, 269 (1970).

3. G. Pickett, Proc.Am.Soc. Test.~at. 45, 8&7 (1945).

4. S. Spinner and 17.E. Teft, Proc.Am.Soc.Test.Mat. 61, 1221 (1961).

5. Am.Soc.Test.Mat. 1968 Book of AST}~ Standards, Part 10, 155 (1968).

6. R.N. Swamy and G. Rigby, RILE~! Mat. and Struct. 4,13 (1971).

7. M. Anson, Mag.Conc.Res. 16, 73 (1964).

8. R. Jones, Mag.Conc.Res. 2, 67 (1949).

9. R.H. Elvery, RILE~[ Inter.Symp. Non-destructive Test, Mat. and Struct., 1, 111 (1953).

10. M.J. Chefdeville, RILE~! Bull. 15, 61 (1953).

iS. J.R. Keeton, Proc. High Res.Bd. 39, 310 (1960).

12. 0. Ishai, J.Am.Conc.Inst. 58, 611 (1961).

13. R. Jones, Non-destructive Testing of Concrete, p.103, Cambridge University Press (1962).

14. K. Newman, Proc.Int.Conf. The Structure of Conc. and its Behavior under Load, London, Sept. 1965, 13 (1968).

15. J.R. Leslie and W.J. Cheesman, J.Am.Conc.Inst. 46, 17 (1949).

16. E. Probst, Struct.Eng. 9, &lO (1931).

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18. E. Hognestad, N.W. Hanson and D. McHenry, J.Am.Conc.lnst. 52, 455 ( 955).

It

19. H. Rusch, Zement-Kalk-Gips, 12, I (1959).

20. T.T.C. Hsu, F.O. Slate, F.0. Sturman and G. Winter, J.Am.Conc.Inst. 60, 209 (1963).


21.

22.

23.

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33-

34.

35.

36.

N.J. Gardner, J.Am.Conc.lnst. 66, 136 (1969).

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H.E. Krenchel, Proc.Inter.Conf.Structure Conc. and its Behavior under Load, London, Sept. 1965, 160 (1968).

J.W.H. King, Mag.Conc.Res. 8, 39 (1956).

E.W. Bennett and Z.M. Khilji, J.Br. Granite and V~instone Fed. 3, 17 (1963).

Go Bradfield and H. Pursey, Phil.~ag. 44, 437 (1953).

H. Pursey and H.L. Cox, Phil.~lag. 45, 295 (1954).

L.J. Nitchell, Proc. High.Res.Bd. 33, 242 (1954).

H.H. Bache and P. Nepper-Christensen, Proc.Int.Conf. The Structure of Concrete and its Behavior under Load, London, Sept. 1965, 93 (1968).

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Acknowledgement

The author records his deep gratitude to Mr. @. Rigby who carried

out the tests and to Mr. K.L. Anand who helped with the computations.

The investigation is part of the project on fracture mechanism of

cementitious materials supported by the Science Research Council, England.

Dynamic Poisson's Ratio of Portland Cement

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