J
ournal o
f
A
pp
Published Onli
n
http://dx.doi.or
g
Open Access
De
v
El
e
ABSTRA
C
Most civil en
g
face-to-surfac
extremely va
r
p
erfect intera
c
tween two m
a
the process o
f
The novelty i
n
p
les prove th
e
Keywords: T
r
1. Introdu
c
Civil engine
e
that are syste
m
ious shapes a
n
and infrastru
c
p
erform the
m
For example,
p
ression is q
u
systematic as
s
[1]. Under la
t
to horizontal
compared to
t
though constr
u
computationa
l
ongoing rese
a
damentally s
o
analysis of s
t
encompassin
g
softening pro
p
gradient enha
n
ing damaged
i
N
umerical
p
er to illustr
clearly. The r
e
*
Corresponding
a
p
lied Mathemat
i
n
e November 2
0
g
/10.4236/jamp
.
v
elopm
e
e
ment
U
C
T
g
ineering stru
c
e interaction
p
r
iable; simplif
i
c
tion, on the
o
a
terials of diff
e
f
developing a
n
this paper i
s
e
stability of t
h
r
ansient Grad
i
c
tion
e
ring structure
m
atically and
r
n
d forms for a
e
c
ture. The as
s
m
odules und
e
although an
u
ite brittle,
m
s
embly of bri
c
t
eral loads wa
l
sliding are s
h
t
hose that fail
u
cted of same
l
ly efficient
m
a
rch with the
o
und, yet si
m
t
ructures, a r
e
g
two material
s
p
erties has be
e
n
ced non-loc
a
i
nterface inter
a
examples are
ate the proc
e
e
sults show th
e
a
utho
r
.
i
cs and Ph
y
sics
,
0
13 (http://ww
w
.2013.16009
e
nt of a
U
sing D
Ali
Science
c
tures are for
m
p
laying key ro
l
i
ed and extre
m
o
ther hand, pr
e
e
rent softenin
g
bi-material re
p
s
the develop
m
h
e approach fo
r
i
ent Non-Loca
e
s are formed
r
epetitively as
e
sthetically pl
e
s
embled struc
t
e
r similar loa
d
individual br
i
m
asonry arche
s
c
ks are relativ
e
l
l type structu
r
h
own to have
due to diago
n
materials [2].
m
odels are yet
objective of
d
m
plified mod
e
e
presentative
v
s
(Brick and
M
e
n developed
u
a
l homogenisa
a
ction of the t
w
provided thr
o
e
ss of formul
e
stability of t
h
,
2013, 1, 43-4
7
w
.scirp.org/jour
n
Bi-Ma
t
amage
d
Jelvehpour
*
and Engineerin
g
of Technolo
g
*
a.je
l
Receive
d
m
ed using a n
u
l
e on the over
a
m
ely detailed
m
e
dict unsafe b
e
g
properties is
d
p
resentative v
o
m
ent of non-lo
r
a simplified
R
l
; Homogenis
a
using modul
e
sembled in v
a
e
asing buildin
g
t
ures often o
u
d
ing conditio
n
i
ck under co
m
s
encompassi
n
e
ly more duct
i
r
es that fail d
u
higher ductili
t
n
al cracking
a
No reliable a
n
available. In
a
d
eveloping fu
n
e
ls for reliab
v
olume eleme
n
M
or
t
ar) of stra
i
u
sing a transie
n
tion incorpor
a
w
o materials.
o
ughout the p
ating the R
V
h
e approach.
7
n
al
/
jamp )
t
erial
R
d
Hom
o
*
, Manicka
D
g
Faculty, Que
e
g
y, Brisbane, A
u
l
e
d
September 20
u
mber of mat
e
a
ll response o
f
m
odels trialed
e
havior. In thi
s
developed usi
n
o
lume elemen
t
cal transient
d
R
VE and enc
o
a
tion; Damag
e
e
s
ar
-
g
s
ut
-
n
s.
m
-
n
g
i
le
u
e
t
y
a
l-
n
d
a
n
n
-
le
n
t
in
n
t,
at
-
a-
V
E
2. Co
n
Qu
a
2.1. E
l
In the
isotrop
i
stress-
s
where
and
i
C
Einstei
n
3). Th
e
damag
e
A d
a
enable
[4]:
where
current
ter wh
i
materi
a
damag
e
nearly
Kuhn-
T
R
eprese
o
genisa
D
hanasekar
e
nsland Univers
i
u
stralia
e
du.au
13
e
rials that are
b
f
the structure.
to date prove
s
paper a dam
n
g homogenis
a
t
(RVE) using
d
amage identi
f
o
urage applica
t
e
Mechanics
n
tinuum D
a
a
si-Brittle
M
l
asticity Bas
e
classical con
t
i
c damage qu
a
s
train relations
ij

ij

is the C
a
i
jkl
represents
n
’s summatio
n
e
Poisson’s ra
t
e
.
a
mage loadin
g
the model to

f
eq

~
, is the eq
u
state of strai
n
i
ch represents
a
l has experie
n
e
growth pre
s
elastic. Cha
n
T
ucker relatio
n
f
ntative
tion A
p
i
ty
b
onded to ea
c
Unfortunatel
y
quite comple
x
age mechanic
s
a
tion approac
h
damaged ho
m
f
ication algori
t
t
ion to other s
h
a
ma
g
e Mec
M
aterial
e
d Damage
M
t
inuum dama
g
a
ntity ω has
b
hip in the foll
o

1
ijkl
C 


a
uchy stress,

the elasticit
y
n
convention
a
t
io is assume
d
g
function is i
n
predict dama
g





eqeq
~
,
~
u
ivalent strain
n
tensor and

maximum eq
u
n
ced. For,
f
s
ent and mat
e
n
ge in para
m
n
s [4]:
,0,0 


 f
f
Volu
m
p
proac
h
c
h other with
t
y
these interac
x
. Models tha
t
s
based intera
c
h
. This paper
d
m
ogenisation a
p
t
hm. Numeric
a
h
apes of RVE
s
hanics for
M
echanics
g
e mechanics
b
een introduc
e
o
wing way [3]
kl

kl

is the lin
e
y
matrix co
m
a
pplies (i, j, k,
d
not to be af
f
n
t
r
oduced, in
g
e growth, as


which depen
d

, is a history
u
ivalent strai
n
0, there w
i
e
rial would b
e
m
eter

, foll
0

JAMP
m
e
h
t
heir su
r
-
tions are
t
assume
c
tion be-
d
escribes
p
proach.
a
l exam-
s
.
a scalar
e
d in the
:
(1)
e
ar strain
m
ponents.
l = 1, 2,
f
ected by
order to
follows
(2)
d
s on the
parame-
n
that the
i
ll be no
e
have li-
ows the
(3)

A. JELVEHPOUR, M. DHANASEKAR
Open Access JAMP
44
which implies that damage growth will happen when
0f, where history parameter satisfies eq


~
.
2.2. Definition of Equivalent Strain
Shape and size of the loading surface depends on the
definition of equivalent strain eq

~, which maps the
strain tensor into a scalar value by weighting its compo-
nents considering their different effects on cracking. In
this paper, a strain based modified von Mises definition
which is able to differentiate between tensile and com-
pressive strains. This differentiation is needed in order to
predict the behavior of quasi-brittle materials such as
concrete and brick. This definition reads as follows [5]:



 
1
2
2
12
22
1
212
1
112
212 1
eq
kI
k
kk
IJ
k








(4)
where 1
I, is the first invariant of strain tensor and 2 J,
is the second invariant of deviatoric strain tensor. Pa-
rameter k controls the sensitivity of the equivalent
strain to tension and compression which is usually set to
the ratio of compressive strength of material to its tensile
strength. In this equation, Poisson’s ratio has been repre-
sented by

.
2.3. Damage Evolution Law for Quasi-Brittle
Material
Fracture in quasi-brittle material is not a result of growth
of one dominant defect but a collective process of dam-
age growth and nucleation in its microstructure. Quasi-
brittle materials like concrete, brick and mortar, demon-
strate a gradual loss of strength, instead of a sudden loss
of deformation resistance like brittle fracture.
Several scalar damage evolution laws have been de-
veloped in the literature [6,7]. In engineering material
softening is nonlinear which has a relatively steep stress
drop when cracking starts and a moderate decrease af-
terwards. In this work, an exponential softening law for
concrete has been used in the form of



i
e
i 





 11 (5)
Due to crack bridging the experimentally obtained
load displacement data has a long tail. Using this expres-
sion when 

, stress approaches 
i
E


1 which
can represent this long tail. Parameter

controls the
damage growth rate which depends on the tensile frac-
ture energy of the material. When

is higher，model
will show a faster crack growth and a more brittle re-
sponse. This type of damage evolution law was used for
both constituents.
3. Non-Local Model for Strain Softening
Material
Damage growth is highly dependent on the microstruc-
ture of the material. In quasi-brittle material such as con-
crete, brick and mortar, cracks are bridged by aggregates.
Therefore, the fracture process is directly related to the
aggregates size and distribution. However, in classical
damage mechanics models the scale of microstructure
has not been included. This unrealistic shortcoming of
classical damage models results in the damage localisa-
tion [8]. To overcome this localisation problem in simu-
lation of strain softening material, we can introduce
non-locality to the constitutive relation so that the growth
of damage variable depends on the average deformation
of the material in a certain region. Addition of this non-
local concept to the damage model will result in a smooth
damage growth depending on the length scale [9].
3.1. Gradient Enhanced Non-Local Model
Non-local strain can be introduced as the solution of the
following partial differential equation
eqeqeq c

~
2 (6)
This means that the damage field variable should de-
pend on a non-local equivalent strain eq

, instead of
local equivalent strain eq

~
. Gradient parameter c, is a
constant related to the squared of the internal length pa-
rameter. eq

, can now be implicitly calculated in terms
of eq

~
, using a 0 C-continues finite element domain. To
solve this Helmholtz partial deferential equation a natural
boundary condition has been considered as proposed in
[9].
0.  n
eq
 (7)
in which n, is the unit normal to boundary .
3.2. The Transient-Gradient Damage Model
Using a constant parameter c, in the gradient model
leads to an increase of damage growth in and outside the
localisation zone. This issue can be resolved by consid-
ering a transient value instead of a constant for the gra-
dient parameter [10]. This modification transforms Equa-
tion (6) as follows
eqeqeq

~
2 (8)
in which

, represents the transient gradient parameter
and is defined as follows



























c
c
n
(9)

A. JELVEHPOUR, M. DHANASEKAR
Open Access JAMP
45
In order to solve this new partial differential equation,
an extra set of continuity equation needs to be added to
the original gradient enhanced model. To avoid adding
this extra continuity equation, Equation (9) can be di-
vided by 0

, which leads to the diffusion equation [11]





eq
eq
eq
~
2 (10)
which requires the same number of continuity equations
as the original gradient enhanced model. The transient
gradient parameter needs to be slightly changed to avoid
division by zero into [11]




























c
ccc
n
)( 00 (9)
in which 0 c, is considered to be an arbitrary positive
value so that non-local interaction is prevented at the
beginning of the analysis.
4. Computational Homogenisation
To derive an enhanced constitutive material model for a
complex composite like masonry computational homog-
enisation can be used so that we can derive the global
behaviour of the masonry from its constituents such as
concrete block and mortar. Uniform loading and periodic
geometry for masonry has been assumed and thus, ho-
mogenisation theory for periodic media which was
adopted in [12,13] seems suitable to use. Computations
have been performed on a single representative volume
element (RVE) which contains the information of the
entire mesostructure. A boundary value problem has
been solved on the RVE using finite element method.
Based on homogenisation theory for periodic media,
strains should be compatible and the stresses should be
anti-periodic on two opposite sides of the RVE. This will
ensure that two neighbouring RVEs fit together.
4.1. Strain-Periodic Displacement Field
The strain-periodic displacement field has the form
)(.)( xwxxu








(11)
where

, is the macroscopic strain tensor, x

, is the
position vector and )(xw


, is a mesoscopic displacement
fluctuation field which distinguishes the real meso-
structural displacement field from the linear x

.

, field
[13]. The fluctuation field is assumed to be periodic. The
volume average of the mesoscopic strain field resulting
from equation (11) is given by
 
 



RVERVE
dwxdu
RVERVE
 . 1
)(
1 (12)
which shows the volume average of mesoscopic strain
field is equal to macroscopic strain

. By using the
Hill-Mandel work equivalence the total macro-stress can
be determined as





RVE
d
m
RVE

1 (13)
in which

, and m

, represent macro and meso stress,
respectively.
4.2. Mesoscopic Representative Volume Element
(RVE)
In order to minimize the computational cost at the
mesoscopic scale and also capture all possible failure
mechanisms, the RVE should be chosen carefully. It is
important to note that if the average behaviour remains
unique any periodic RVE predicts equivalent results, i.e.
in an infinitesimal strain setting and before localisation
happens. For masonry, due to periodicity of the initial
mesostructure, the RVE is chosen as the smallest peri-
odic element. Based on the assumption that arrangement
of the constituent materials is the main cause of average
stiffness degradation, the initial and damage induced
anisotropy will be captured correctly using this RVE
[13].
4.3. RVE’s Boundary Conditions
Periodic boundary conditions have been applied on three
controlling nodes (see Figure 1) of the RVE as indicated
in [13] and justified in [12].
The periodicity conditions for edges can then be for-
mulated in terms of the controlling nodes as
,,
,,
,,
32
31
21
eq Feq F
eq Eeq B
eq Deq A
FC
EB
DA
uuuu
uuuu
uuuu



















(14)
Figure 1. Controlling nodes and periodicity conditions on a
typical masonry RVE.

Open Access
46
This mean
s
riodic in the
p
eriodic stres
is applied on
p
oints as illu
s
and L, are wi
tively. More
o
allel to bed j
shear stress, r
e
5. Numeri
c
After imple
m
sections, a
R
length, 60 m
m
110 mm. The
5 mm mortar
critisation w
h
lustrated in
F
have been c
o
experimental
obtained in [1
Mesoscopi
c
can be found
chanics has b
e
Quadratic
i
ered for the
d
functions ha
v
Parameters f
o
model are sh
o
method is e
m
b
een obtaine
d
Two loadi
n
compression
p
p
endicular to
have been ap
p
Evolution of
dicular and p
a
and 5 , respec
t
to bed joint,
d
brick-mortar
i
In the second
allel to bed j
o
spot and cont
i
Figure
s
both displac
e
RVE. These
p
ses and strain
the RVE by
m
s
trated in Fi
gu
dth, height a
n
o
ver,
xx
 ,
y

oint, stress p
e
e
spectively.
c
al Exampl
e
m
enting the p
r
R
VE has bee
n
m
height and
RVE consist
s
around all its
h
ich was used
F
i
g
ure 3 . Th
e
o
nsidered, in
o
results on c
4].
c
material pr
o
in Table 1 .
e
en considere
d
i
nterpolation
f
d
isplacement
f
v
e been consi
d
o
r the transien
t
o
wn in Table
m
ployed since
t
d
under load c
o
n
g cases have
p
arallel to be
d
bed joint.
T
p
lied on the R
V
damage und
e
a
rallel to bed j
t
ively. In case
d
amage growt
h
i
nterface and
c
case which i
s
o
int, damage g
i
nues to grow
i
2. Loading mo
d
e
ment and da
m
p
eriodicity co
n
s mesoscopic
m
eans of the t
h
u
re 2 . In thes
e
n
d length of t
h
y
y
, and,
xy

,
e
rpendicular t
o
e
s
r
ocess explai
n
n
constructed
an ou
t
-of-
p
l
a
s
of a 230*50
c
sides. The fi
n
for the anal
y
e
se geometri
c
o
rder to sim
u
onventional
m
o
perties used
f
Elasticity bas
e
d
for both mat
e
functions ha
v
f
ields and lin
e
d
ered for the
t
gradient enh
a
2 . Conventio
n
t
he experime
n
o
ntrol
t
ests.
been conside
r
d
joint and c
o
T
hese loading
V
E as illustra
t
e
r compressi
o
oint can be s
e
of compressi
o
h
initiates on t
h
c
ontinues into
m
s
under compr
e
g
rowth initiate
s
i
nto the perpe
n
d
es applied on
A. JELVEHPO
U
m
age will be p
n
ditions lead
t
fields. Loadi
n
h
ree controlli
n
e
figures W,
H
h
e RVE, respe
,
are stress p
a
o
bed joint a
n
n
ed in previo
u
with 240 m
m
ne thickness
o
c
lay brick an
d
n
ite element d
i
sis has been
i
c
configuratio
n
u
late the biax
i
m
asonry pan
e
f
or the analy
s
e
d damage m
e
e
rials.
v
e been consi
d
e
ar interpolati
o
non-local fiel
a
nced non-loc
n
al load contr
o
n
tal results ha
v
r
ed for analys
i
o
mpression p
e
configuratio
n
t
ed in Fi
g
ure
o
n load perpe
n
e
en in Fi
g
ures
o
n perpendicul
h
e corner of t
h
m
ortar bed joi
n
e
ssion load p
a
s
from the sa
m
n
d joints.
the RVE.
U
R, M. DHAN
A
e-
t
o
n
g
n
g
H
,
c-
ar
-
n
d
u
s
m
o
f
d
a
i
s-
i
l-
n
s
i
al
e
ls
s
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-
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al
ol
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i
s;
er
-
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2 .
n
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ar
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Figure
Tab
Mate
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c
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t
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a
Mate
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Figure
perpen
d
Da
m
have a
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erime
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sient g
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6. Co
n
In this
als wit
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ous da
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n
the mo
A
SEKAR
3. Typical fini
t
le 1. Material
p
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ial E
ck
14000
t
a
r
3000
a
ble 2. Parame
t
r
ial
c
ck
2
t
a
r
2
4. Evolution o
f
d
icular to bed
j
m
age evolution
good agreem
e
n
tal investiga
t
r
adient model
h
n
clusion
contribution,
h
different so
u
sing a homo
g
m
age growth
a
n
hanced non-
l
del. Influenc
e
t
e element mes
h
p
arameters for


0.2 1.0
0.2 1.0
t
ers for tran sie
n
0
c
0.01
0.01
f
damage in th
e
j
oint.
patterns obta
i
e
nt with failu
r
t
ions. Mesh s
h
as been inve
s
the interactio
n
ftening prope
r
g
enisation app
r
a
nd mesh sens
l
ocal model
h
e
of the choic
e
h
used in comp
RVE comput
a


800 10
100 10
n
t gradient m
o
n

1.0
1.0
e
RVE for co
m
i
ned from the
r
e patterns se
e
ensitivity of
t
s
tigated in [10
,
n
between tw
o
r
ties has bee
n
r
oach. To avo
itivity, a trans
h
as been intro
d
e
of this mod
e
JAMP
utations.
a
tions.
i

0.0002
0.0003
o
del.
e

0.002
0.003
m
pression
analysis
e
n in ex-
t
he tran-
,
11].
o
materi-
n
investi-
id spuri-
ient gra-
d
uced to
e
l on the

A. JELVEHPOUR, M. DHANASEKAR
Open Access JAMP
47
Figure 5. Evolution of damage in the RVE for compression
parallel to bed joint.
evolution of damage under two loading cases has been
investigated. The overall RVE behaviour using the tran-
sient gradient model indicates that calibration of the tran-
sient parameter under different loading conditions needs
to be investigated.
REFERENCES
[1] A. Cavicchi and L. Gambarotta, “Collapse Analysis of
Masonry Bridges Taking into Account Arch-Fill Interac-
tion,” Engineering Structures, Vol. 27, No. 4, 2005, pp.
605-615.
http://dx.doi.org/10.1016/j.engstruct.2004.12.002
[2] M. Dhanasekar, “Review of Modelling of Masonry Shear,”
International Journal of Advanced Engineering Science
Applied Mathematics, Vol. 2, No. 3, 2010, pp. 106-118.
http://dx.doi.org/10.1007/s12572-011-0022-2
[3] J. L. Chaboche, “Continuum Damage Mechanics. Present
State and Future Trends,” Nuclear Engineering and De-
sign, Vol. 105, No. 1, 1987, pp. 19-33.
http://dx.doi.org/10.1016/0029-5493(87)90225-1
[4] R. H. J. Peerlings, “Enhanced Damage Modelling for
Fracture and Fatigue,” Ph.D. Thesis, Technische Univer-
siteit Eindhoven, 1999.
[5] J. H. P. De Vree, W. A. M. Brekelmans and M. A. J. Van
Gils, “Comparison of Nonlocal Approaches in Continuum
Damage Mechanics,” Computers and Structures, Vol. 55,
No. 4, 1995 pp. 581-588.
http://dx.doi.org/10.1016/0045-7949(94)00501-S
[6] G. Pijaudier-Cabot, Z. P. Bažant and M. Tabbara, “Com-
parison of Various Models for Strain-Softening,” Engi-
neering Computations, Vol. 5, No. 2, 1988, pp. 141-150.
http://dx.doi.org/10.1108/eb023732
[7] R. H. J. Peerlings, R. de Borst, W. A. M. Brekelmans and
M. G. D. Geers, “Gradient Enhanced Damage Modelling
of Concrete Fracture,” Mechanics of Cohesive-Frictional
Material, Vol. 3, No. 4, 1998, pp. 323-342.
http://dx.doi.org/10.1002/(SICI)1097-0207(19961015)39:
19<3391::AID-NME7>3.0.CO;2-D
[8] Z. P. Bažant, T. Belytschko and T. Chang, “Continuum
Theory for Strain-Softening,” Journal of Engineering
Mechanics, Vol. 110, No. 12, 1984, pp. 1666-1692.
http://dx.doi.org/10.1061/(ASCE)0733-9399(1984)110:12
(1666)
[9] R. H. J. Peerlings, R. de Borst, W. A. M. Brekelmans and
J. H. P. de Vree, “Gradient Enhanced Damage for Quasi-
Brittle Materials,” International Journal for Numerical
Methods in Engineering, Vol. 39, No. 19, 1996, pp. 3391-
3403.
http://dx.doi.org/10.1002/(SICI)1097-0207(19961015)39:
19<3391::AID-NME7>3.0.CO;2-D
[10] M. G. D. Geers, R. de Borst, W. A. M. Brekelmans and R.
H. J. Peerlings, “Strain-Based Transient-Gradient Dam-
age Model for Failure Analyses,” Computer Methods in
Applied Mechanics and Engineering, Vol. 160, No. 1-2,
1998, pp. 133-153.
http://dx.doi.org/10.1016/S0045-7825(98)80011-X
[11] S. Saroukhani, R. Vafadari and A. Simone, “A Simplified
Implementation of a Gradient-Enhanced Damage Model
with Transient Length Scale Effects,” Computational Me-
chanics, Vol. 51, No. 6, 2013, pp. 899-909.
http://dx.doi.org/10.1007/s00466-012-0769-8
[12] A. Anthoine, “Derivation of the In-Plane Elastic Charac-
teristics of Masonry through Homogenization Theory,”
International Journal of Solids and Structures, Vol. 32,
No. 2, 1995, pp. 137-163.
http://dx.doi.org/10.1016/0020-7683(94)00140-R
[13] T. J. Massart, “Multi-Scale Modelling of Damage in Ma-
sonry Structures,” Ph.D. Thesis, Technische Universiteit
Eindhoven, 2003.
[14] M. Dhanasekar, “Comparison of Nonlocal Approaches in
Continuum Damage Mechanics,” Ph.D. Thesis, Univer-
sity of Newcastle, 1985.