Analysis
o he
ic ional
slip
be ween
a
laye
and a
hal -space
R.
Abascal,
J.
Dommguez
Escuela
Supe io
de
Ingenie os
Indus ials,
Uni e sidad
de
Se illa,
A .
Reina
Me cedes,
s/n, 41012-Se illa,
Spain
ABSTRACT
The
nume ical
analysis
o a
boundless
elas ic
laye
on an
elas ic
hal -
space
wi h di e en ma e ial p ope ies unde
he
e ec s
o an
uni o m
su ace p essu e
and a
cyclic angen ial su ace o ce
is
p esen ed. F ic ional
con ac condi ions
a e
assumed.
The
s udy
is
ocussed
on he
e alua ion
o
he
maximum
ampli ude
o he
angen ial load
which
p oduces
localized
slip
be ween
he wo
egions du ing
he
i s
load cycle
bu no he
subsequen
ones.
The
mo e
simple
limi
o
which
no
slip
exis
e en
o he i s
cycle
is
also es ablished
INTRODUCTION
The
s udy
o he
slip
and
sepa a ion
ha
may
ake place
be ween
wo
su aces
in
con ac
when
hey
a e
unde
angen ial
cyclic
loading condi ions
is
e y impo an
o
p e en
he
ailu e
known
as
" e ing".
This
ailu e
mechanism
is
ini ia ed
by
localized
slip
be ween
wo
su aces
om
which
some
pa icles
a e
de ached.
These
pa icles
ac as an
ab asi e
in
subsequen
load cycles
and
de e io a e
he
ma e ial
apidly.
The
damage
mechanism
may
be
combined
wi h co osion
in he
case
o
me als[l].
The
slip
be ween
he wo
su aces
can be
a oided
by
in oducing
a
no mal
p essu e
be ween
hem.
I
his
p essu e
is
enough
he
p e en
slip
du ing
he i s
load cycle hen
he
sys em beha es
linea ly
and
slip
will
no
ake place du ing subsequen cycles
o he
same
ampli ude.
This
welded
con ac
p oblem
can be
sol ed easily.
The
limi
o
which
he
i s
slip
akes
place
can be
ob ained
om
he
welded
con ac
model.
I
co esponds
o he
angen ial
load
o
which
he
shea
ac ion
a a
poin
becomes
equal
o he
no mal
p essu e imes
he
ic ion
coe icien .
The e
a e
s ill
highe alues
o he
angen ial load
o
which
he e
is
pa ial
slip
du ing
he i s
load cycle
bu no in he
subsequen
ones.
The e o e
no
e ing
will
ake place
o
his
T ansac ions on Modelling and Simula ion ol 2, © 1993 WIT P ess, www.wi p ess.com, ISSN 1743-355X
210
Bounda y Elemen s
c
0
'
0
0
0
0
0
0
,
0
11111.
c,
,,
**
*:
1,1 1 1 1 ,
Q
^F kHoi
Su eca
Figu e
1.
Elas ic
laye
on
elas ic
hal -space
unde
angen ial
load
Q and
no mal
load
PO
load.
The
slip
o he i s
cycle lea es esidual s esses
which
p e en
he
wo
su aces
o
slip
du ing
he
nex cycles.
The
e alua ion
o he
highe
alue
o he
load
o
which
he
sys em
has
his
kind
o
beha io
is
impo an
since
i is he
ac ual
limi
o he
loads
which
do no
p oduce
e ing.
The
s udy equi es
a
mo e
complica ed
model
and a
nume ical
solu ion
app oach
as
shown
below.
The
p oblem
analyzed
in
his
wo k
e e s
o a
semi-in ini e
domain
consis ing
o a
boundless ho izon al
elas ic
laye wi h dep h
"a" on an
elas ic
hal -space.
The
su ace
o he
laye
is
subjec
o a
uni o m cons an
in
ime
p essu e
p<,
and o a
concen a ed angen ial load
o
ampli ude
Q
which
has
a
cyclic ime
dependence.
The
a ia ion
o he
load wi h ime
is
assumed
o
be
su icien ly
slow
so
ha
ine ial
e ec s
a e
negligible.
The e o e,
a
quasi-
s a ic
analysis
is
ca ied ou .
The ini e
egion wi h
he
bounda y
condi ions
shown
in
Figu e
1 is
used
o he
s udy.
The
p oblem
a
hand
emains
linea
o
small alues
o he
load
Q.
The
exp essions
o he
ac ions along
he
in e ace
in
such
case
can be
ound
o
ins ance
in
Re .[2].
The
s udy
o a
simila
p oblem
wi h
wo
ma e ials
o
iden ical
p ope ies
and a
single
loading p ocess
which
p oduces
i s ,
slip
a a
poin
hen,
one o wo
slip
zones
and inally,
sepa a ion,
was
done
analy ically
be
Schmuese ,
Comminou
and
Dundu s
[3].
Comminou
and
Ba be
[4]
ex ended
his
analysis
o he
case
o
cyclic
loading
and
assuming
a
ic ion
coe icien
JJL
=
0.5.
These
au ho s
iden i ied
a
alue
X, o he
T ansac ions on Modelling and Simula ion ol 2, © 1993 WIT P ess, www.wi p ess.com, ISSN 1743-355X
Bounda y Elemen s
211
pa ame e
X =
Q/(po
a)
such
ha
o
loads
X < X, he
p oblem
emains
linea ,
and a
load ange
X, < X < X] o
which
he e
is
slip
bu
only
in he
i s
load cycle.
Loads
co esponding
o X > X?
p oduce
mul iple
slip
and/o
sepa a ion
which
may
become
gene al along
he
in e ace du ing
he
subsequen
load cycles.
The
pu pose
o
his
pape
is o
ca y
ou a
s udy
o
hese anges
o
he
load
o he
mo e
gene al case
when
he
laye
and he
hal -space ha e
di e en
ma e ial p ope ies.
The
s udy
is
done
nume ically
by
means
o he
Bounda y
Elemen
Me hod
(BEM)
which
is
e y well
sui ed
o
his
kind
o
p oblems.
A
pa ame ic s udy
is
done
using
as a
pa ame e
o
de ine
he
ela i e
s i ness
o he wo
ma e ials,
he
squa e oo
o he
a io
be ween
he
shea
modula
RCs =
(Gj/GJ^,
he
Poisson's
a ion
being
he
same
o
bo h ma e ials.
NUMERICAL
APPROACH
The
analysis
o he
p oblem
in
Figu e
1
s a s
by
disc e izing
he
bounda ies
in o
cons an elemen s
and
compu ing
he
usual
BEM
sys em
o
equa ions
o he
bounda y
nodes
o
each
one o he wo
sub egions
whe e
u*
and;/
a e he
displacemen
and
ac ion
ec o s espec i ely, along
he
bounda y
o he "z"
sub egion.
The
coupling
be ween
he wo
egions
is
done
by
means
o he
compa ibili y
and
equilib ium condi ions along
he
in e ace
co esponding
o one o he
ollowing
si ua ions:
bonding,
sepa a ion
o
sliding
wi h
ic ion.
A
Coulomb
ype
ic ion
is
assumed.
Sliding:
P = p u? + w»* = 0
Bonding:
P?* I < I M P? I .< . ui + «/ = 0 (3)
T ansac ions on Modelling and Simula ion ol 2, © 1993 WIT P ess, www.wi p ess.com, ISSN 1743-355X
212
Bounda y
Elemen s
Sepa a ion:
P?*
- 0 P?' = 0 5 > 0 W
whe e
he
supe index indica es
he
egion
o
which
he
a iable e e s
and he
subindexes
s and n
s and
o
angen ial
and
no mal
di ec ion
o he
in e ace,
espec i ely;
/x is he
ic ion coe icien
and 5 he
dis ance along
he
no mal
di ec ion
be ween
wo
poin s
which
could
be in
con ac .
The
sign
o he
shea ac ion du ing
he
sliding
is
de e mined
by he
condi ion
o
nega i e
wo k
(dissipa ion
o
ene gy)
du ing
he
p ocess.
In he
h ee
cases
abo e
ou equa ions
can be
w i en
o
each
couple
o
poin s
on
he
con ac su aces.
Those
equa ions plus
he wo BEM
equa ions
o
each
domain
de e mine
he
alues
o he wo
displacemen
componen s
and wo
ac ion
componen s
o
each
one o he wo
poin s
o a
couple
which
a e o
may be, in
con ac .
The
solu ion
o he
sys em
o
equa ions equi es
o an
i e a i e
p ocess
wi hin
each
load s ep.
This
p ocess
o a
load s ep
N
s a s
by
w i ing
he
B.E.
equa ions
(1) o
each
bounda y
node
as
whe e
x^ and /„ a e he
bounda y
unknown
ec o
and he
known
ec o ,
espec i ely.
The
la e
is
ob ained
om
he
p oduc
o he
known
bounda y
alues
and he
co esponding
columns
o
he
H and G
ma ix.
The
sys em
ma ix
A
con ains
mo e
columns
han
ows
as
bo h ac ions
and
displacemen s
a e
unknown
along
he
in e ace.
The
con ac equa ions
o he
nodes
on he
in e ace
a e
w i en
as
CN
*N = ° (6)
The
abo e
sys em
o
equa ions
(5)
plus
(6)
allows
o he
solu ion
o
he
p oblem
o he
load s ep
N
p o ided
ha
he
con ac condi ions
o all
he
poin s
on he
in e ace ( ep esen ed
by €„)
emain
cons an du ing
he
load s ep.
Since
hese condi ions
will,
in
gene al,
change
du ing
he
load
s ep,
each
load s ep mus include se e al
i e a ions
wi h
di e en
con ac
T ansac ions on Modelling and Simula ion ol 2, © 1993 WIT P ess, www.wi p ess.com, ISSN 1743-355X
Bounda y
Elemen s
213
condi ions CN*.
Fo
each
load s ep
one
wo ks
wi h inc emen s
o he
a iables wi h espec
o he
p e ious s ep
»
-/«-
0
= A .
(7)
The
inc emen s
o he
a iables
a e
subdi ided
in o
M
possible sub-
inc emen s
and
w i en
as
A
*„ = £ A ^*
(8)
whe e
each
sub-inc emen
co esponds
o
di e en
con ac condi ions
Gj.
The
i e a ion
p ocess
o
load s ep
N
begins
by
sol ing
he
sys em
(7)
assuming
he
same
con ac condi ions
o he end o he
p e ious s ep
and
applying
he
comple e
load inc emen A/^.
A
scale ac o /?„'
is
compu ed
om
he
solu ion
o
his
sys em
such
ha
i
ep esen s
he
pa
o he
load
which
can be
applied wi hou
change
in he
con ac condi ions.
This
load
can
be
w i en
as
and
A
4
(9)
whe e
A * is he
solu ion
o (7)
wi h A/*.
Nex
he
con ac condi ions
a e
modi ied
and he
load
(1 - #/)
A/%, applied.
The
p ocess
con inues
up o he
poin
when
/?/ > 1.
Then
he
compu a ion
o
load s ep
N is
inished.
A
simila
i e a i e
p ocess
was
applied
by he
au ho s
o
dynamic
p oblems
in
Re s.
[6] and
[7].
LAYER
ON A
HALF-SPACE
The
ollowing alues
o he
pa ame e s
ha e
been
used
o he
model
in
Figu e
1: a = 1 m; L = 10 m ; H = 10 m;
Poisson's
a io
u, = ^ =
T ansac ions on Modelling and Simula ion ol 2, © 1993 WIT P ess, www.wi p ess.com, ISSN 1743-355X

214
Bounda y Elemen s
1/3
;
po=l
Nw/n
;
ic ion
coe icien
p = 0.5 and a
shea
modulus
o he
hal -space
62 = 1
Nw/n .
The
pa ame ic s udy
is
done
by
changing
he
shea
modulus
o he
laye ma e ial
and he
load ampli ude
Q. The
load
is
assumed
o be
applied
in 100
s eps
ollowing
he
a ia ion
shown
in
Figu e
2.
• I 32 41 64
LaW Sup
Figu e
2.
Tangen ial
load
a ia ion
Cons an
Bounda y
Elemen s
a e
used
o he
s udy.
The
disc e iza ion
consis s
o 50
equal elemen s
on
each side
o he
con ac
in e ace
and on he
laye ee su ace,
3
equal elemen s
on
each
la e al
bounda y
o he
laye ,
5 on
each
la e al
bounda y
o he
hal -space
and 10
equal elemen s
o he
bo om
o he
model.
In
o de
o
es
his
model
which ex ends
o a
ini e
egion
a ound
he
load
o
ep esen
an
in ini e
domain
and
includes
a
cons an
Bounda y
Elemen
app oxima ion,
a
p oblem
wi h
known
analy ical solu ion
is
sol ed.
Such
solu ion
was
ob ained
by
Comninou
and
Ba be
[5] o he
case
o wo
iden ical
ma e ial p ope ies
o he
laye
and he
hal -space. Figu e
3
shows
a
compa ison
be ween
he
analy ical
and he
p esen nume ical
esul s
o he
maximum
load
o he
second
load cycle (poin
C o
Figu e
2)
wi h
a
load
ampli ude
X =
2.353.
Slip
be ween
he
laye
and he
hal -space akes place
o
his
alue
o
X
du ing
he
i s
cycle
bu no in he
subsequen ones.
The
no mal
and
shea
ac ions
along
he
in e ace
e sus
he
ho izon al
dis ance
x o he
poin load
a e
shown
in
Figu e
3. The
no mal
ac ions
ha e
a
smoo h
a ia ion
and a e
almos
he
same
as in he
linea
case.
On he
o he
hand,
he
shea
ac ions
show
a
clea di e ence wi h hose
o he
linea
case
because
o he
esidual
ac ions
exis ing
in he
zones
whe e
slip
T ansac ions on Modelling and Simula ion ol 2, © 1993 WIT P ess, www.wi p ess.com, ISSN 1743-355X
Bounda y Elemen s
215
ook
place
in he
p e ious load cycle.
These
esidual ac ions a oid
he
sliding
in he
second
and
subsequen load cycles.
The
esul s
p esen ed
in
Figu e
3
show
a
e y
good
ag eemen
be ween
he
analy ical
and he
p esen
nume ical
solu ion.
-
No mal Theo .
* *
Lo.d
•
No mal
B.E.M.
«
{?Sl£
—Shea Theo .
^ « Rd - l
•
Shea
B.EJ*.
* .
,.oj
-4-3-2'
-1
Figu e
3.
Resul s
o wo
iden ical
ma e ials
20
18
16
14
I*"
O 10
Zone
1
0 Oj 1 lj 2 2J 3 3J 4 4j
Ra«
Figu e
4.
F e ing
zones
T ansac ions on Modelling and Simula ion ol 2, © 1993 WIT P ess, www.wi p ess.com, ISSN 1743-355X
216
0.4
0
3
02
4 «
•0.4
Bounda y Elemen s
-5
-4 3 -2
-1012345
«/*
0.4
035
O
"
&
025
g
02
I
0.15
3 0.1
*
0.05
0
4X05
LoadC
#-03
X- X
-
RC -0.25
-RCs-0.50
RCm-l
-RCs-2
-
RC«-4
-5-4-3-2-1012345
0.4
035
0.15
0.1
0.05
0
•0.05
Lo dA
P-O5
X
• X
-
RCs-025
RCs-OJO
-RCs-2
-RCm-4
5 -4 3 -2
-1012345
%/*
OJ5
03
025
02
0.15
0.1
0.05
0
-0.05
5 -4 -3 -2
-1012345
%/a
Figu e
5.
T ac ioo5
dis ibu ion
along
he
in e ace
Once
he
B.E. model and'
he
i e a i e
app oach ha e
been
alida ed
he
pa ame ic s udy
o a
ange
o
he
s i ness
a io
RCs =
(GJG^
going
om
0 o 4.5 is
ca ied
ou .
Fo
each
s i ness
a io
wo
limi ing
alues
o
he
load pa ame e s
X =
Q/PO
a a e
de e mined.
The
i s
one,
X, is he
limi
o he
loads
which
do no
p oduce non-linea
e ec s
on he
con ac in e ace.
This
limi
de ine
he
linea
zone (zone
1) in
Figu e
4. The
second
limi
X?
de ines
he
load
zone
(zone
2) o
which he e
is
slip
along
he
in e ace
T ansac ions on Modelling and Simula ion ol 2, © 1993 WIT P ess, www.wi p ess.com, ISSN 1743-355X
Bounda y Elemen s
217
du ing
he i s
load cycle
bu he
esidual
ac ions
a oid
slip
in he
subsequen load cycles. Values
o X > X%
(zone
3)
co espond
o
loads
which
p oduce
slip
in all he
load cycles
and
he e o e
he
e ing
p ocess
can no be
con olled.
The
alues
o X, and X%
ob ained analy ically
by
Comninou
and
Ba be
[5] o he
case
o RCs = 1 a e
shown
as
do s
in he
igu e.
Figu e
5
shows
some
o he
ac ion
dis ibu ions
which
appea
along
he
in e lace du ing
he
loading
p ocess.
Only
in one
case
a e he
no mal
ac ions
ep esen ed (Figu e
5a)
since hose
ac ions
show
e y
li le
di e ence wi h
he
linea
ones
o
zone
2.
The
shea ac ions ep esen ed
in
Figu e
5b
co espond
o he
second
cycle (Poin
C) and X = X,.
The e o e,
no
slip
has
aken place
and he
cu es
in
his
igu e
ep esen
he
shea
ac ion
dis ibu ion
o a
linea
p oblem.
Figu es
5c and 5d
show
he
shea
ac ion
dis ibu ion
o X = X2
du ing
he
i s
and
second
load cycles (Poin s
A and C,
espec i ely).
In he
o me
case,
he e ha e
been
slip
only
in he
nega i e pa
o he
x-axis
and
esidual
ac ions ha e
appea ed.
I can be
seen
in
Figu e
5d
ha
he
esidual
ac ions
co esponding
o
poin
A
emain
and
some
mo e
appea
along
he
posi i e
pa
o he
x-axis
which
ha e
been
p oduced
du ing
he
nega i e pa
o
he
i s
load cycle (poin
B).
2.0
1.5
1.0
7«
3- O'O
&
-OJ
-1.0
-1J
-zo
-
RCs-0.25
RCs-0.50
-RCs-1
-RCs-2
-•RCs
-4
LoadC
-0.5
A
-
-5-4-3-2-1012345
x/a
Figu e
6.
Rela i e angen ial
displacemen s
along
he
in e ace
T ansac ions on Modelling and Simula ion ol 2, © 1993 WIT P ess, www.wi p ess.com, ISSN 1743-355X