spencer 1984
TRANSCRIPT
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1 INTRODUCTION
CONSTITUTIVE
THEORY
FOR
STRONGLY ANISOTROPIC SOLIDS
A.
J
M
SPENCER
Department of
heoretical
Mechanics
University of Nottingham
Nottingham NG RD
England
We
shal l discuss
a
number
of problems
concerned
with
s t ress and
deformation analysis of f ibre- reinforced composite,
and other s trongly
anisotropic, mater ials .
The kind of composite mater ia l in mind i s
one
in
which a matr ix mater i a l
i s
r e in fo rced by s t rong s t i f f f ib res which
are
systematical ly arranged in the matr ix. The
f ibres
are considered to be
long compared to thei r diameters and the f ibre spacings, and to be qui te
densely dis t r ibuted
so tha t
the f ibres form a substant ia l proportion
typical ly about
50
by
volume)
of the composite. There are
many
such
composite
materials now
in
use or under development;
examples
are carbon
f ibre reinforced epoxy res ins boron f ibre reinforced aluminium, and
nylon
or s teel
reinforced rubber which i s used in
tyres , hosepipes and
bel ts .
Since
we
assume
the f ibres to
be
systematical ly
arranged,
a
composite
of th is
kind
has strong direct ional
proper t ies ,
so tha t macroscopically
t
has to
be
regarded as
an
anisot ropic mater ial . In most cases th is
A. J. M. Spencer (ed.), Continuum Theory of the Mechanics of Fibre-Reinforced Composites
© Springer-Verlag Wien 1984
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2
A J M Spen
cer
ani
sotropy
i s
v
ery s t rong ,
so
th
a t m
echanical p ro p e r
t i es are
highly
depen
dent
o
n di re c t i on i n
the m
ater i a l; some exam
ples
of t
h is
w
il l be
given
l a t e r . I
f the mater i
a l i s
re inforce
d
with
a s ing le
f amily
of f ib re s ,
w
hich
are
ra
ndomly d is
t r ibu t e d i n
the cro
ss sec t ions
normal to t
he f ib re s
,
t
hen
t
he com
posite
m
ater ia l
has a s i
n g l e p re fe r r
ed
d i
rec t ion
w
hich
w
e
sha l l c a l l the fibr
e dire
ction an
d so i s t r ans
ver se ly i so t ro
p ic wi
th
resp
ec t to
t h i s d ire c t i on .
Th
e
f i
b r e d i r ec t ion
may be ch
arac ter ized
by a
un i t vec
tor a . Howeve
r, it i s
not necessary
t
h a t
the f
ib re d i r e c
t ion
be
th
e sa me
a t
each poin t ;
it i s
qu i t e p
oss ib le to a l ign
the f ib res
along a
fam ily
of
c
urves .
The
n
the
compos
ite i s ~ o c a ~ ~ y t r
an s v er se ly
a
n iso t rop ic
with
r espect to th
e local
f ib
re d i r ec t io
n , and
a i s a func t
ion of
pos i t ion
.
It
i s a l so poss ib le
to have
reinforce
men t by mor
e tha
n
one
family
of
f ib r
es . For examp
le, we may
cons ider
a la
minated p la te
b
u i l t up fro
m a
la r
g e
number of
th
in l
aminae,
each of which
i s
u
nid i r ec t io
n a l ly r
e info rced ,
bu t
which a
re s tack
ed a l t e rna t ely
with
the f ib res
a ligned
in tw o
d i f f e r e
n t
d i
rec t io ns .
n the
macrosco p
ic sca le such a l
aminate
wi l l
ha
ve
two
p re fe
r red d ire c t i on
s , and
so w i l l
h
ave o
r tho t rop ic
symmetry.
It i s
easy
to
envisage l aminates with
th re e
or
more
p re fe r red
d i rec t io ns .
Another
conf ig
u rat ion
of in
t e r e s t i s
t ha t of a c i r
c u l a r cyl inder
re
in fo r ced by
h
el ic a l f ib re
s ly in
g
in
conc
entr ic c i
r c u l a r c y l indr ic
a l su r
faces and wound
sym
m etrically
in
opposing d i r e c
t i ons .
This m ater ia l
i s lo c a l l y
or tho t rop i
c
but
the p re
fe r red di rec t ion
s vary wit
h pos i t ion .
n th
ese
cases we have
two o r more f ib
re d i r ec t io
n s ,
each of
whic
h may be charac
te ri zed
by a u n
i t
vec t
or a , b ,
e con
sider
t ha t the f ib res
are
d is
t r ibu te d
througho
ut
the
m ater ia l
the
p o s s i b
i l i ty o
f
var ia t i
ons
in f ib re densi t
y i s
not ex
c luded) . There
a r
e a l so in te re
s t i ng problems
in
whi
ch
t
he
f
ib res
l i e i n d i
s c r e t e su r
faces ,
bu t
these w
il l not be t r ea ted
here .
e are
concer
ned
with
the dev
elopment
of continuum t
h eor ies ,
s
o
we
assume
the f ib res
to be
con t inuo
us ly
d
i s t r ibu te d thro
ugh the m
ater ia l .
Then the f ib re
d
i r ec t ions
a ,
b,
may
be regarded
as
cont inuous
func t
ions o
f
pos i t ion
Thus on th
e ma
croscopic
sca
le we t r e
a t
th
e m ater ia l
as a s t rongly
an i so t r
op ic con
tinuum.
The
t h eo rie s
and
so lu t io ns
we
sha l l
develop
may
be
appl ied
no t
only
to
f ib re
re
in fo r ced
m a
ter ia ls , b
ut a l s
o to any s t ro
ng ly an i
so t rop ic
m a t e r ia l .
Howev
er, it i s con
venient t
o use term
inology assoc i
a ted with
f ib re
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Constitutive Theory
reinforced materials and
so
we shal l refer to the
direct ions character iz
ing
the
strong
anisotropy
as
f ibre direct ions
and to
their t ra jector ies
as f ibres .
This
continuum approach
excludes any consideration
of the
micro
mechanics
of the
composite which
involves in teract ions
between individual
f ibres
and
the
matrix.
There
are
many important problems on the
microscopic
scale but these wil l not be
considered. Another
important
problem
area i s tha t
of the relat ions between
the mechanical
properties of
the composite and those of the consti tuents
which form
the composite.
Such problems wil l also not
be
considered.
Clearly a f ibre-reinforced composite mater ia l may show a l l
kinds
of
mechanical
response. We
sha l l deal mainly
with e las t i c and
plas t ic
behaviour for
both large
and
small
deformations. There is also a
substant ia l body of theory w h i ~ h
does
not depend on material response
aspects of which
are dea l t
with
in Chapter I I .
2
LINEAR ELASTIC CONSTITUTIVE
EQUATIONS
FOR
FIBRE REINFORCED
MATERIAL
2 1
Linear elast ic i ty-
one
family of fibres
We
begin with the simplest case which
i s tha t
of a l inear ly elas t ic
sol id reinforced by a
s ingle family
of f ibres . The const i tu t ive equation
i s
therefore tha t of a t ransverse ly
i sotropic
l inear ly elas t ic sol id .
This const i tu t ive
equation
i s well
known; the
usual method
of
deriving t
i s
to
se lec t a coordinate system such tha t
one
of the coordinate axes
coincides
with the axis of
transverse isotropy and
examine
the
res t r ic t ions on the s t ra in-energy function which resul t
from
the
require
ments
of
invariance under
rotat ions about
th i s
axis .
We
proceed
in
a
ra ther
di f fe rent
though
equivalent
way. The main reason
for th i s i s
tha t because
the f ibre
direct ion
i s dependent on
posi t ion
t i s
convenient
to
have
a
formulation
which
does
not
depend
on
a
par t icular
choice of coordinate system. I t is qui te possible
to
transform the
standard resul ts so tha t
they do
not depend on the
choice
of coordinate
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4
A.J.M.
Spencer
system, but
t
seems
more
sa t i s fac to ry to adopt
a formulation
which
i s
coordinate-free
from the outset . This
is
especial ly
the case
when we
proceed to consider more
complicated
const i tu t ive equations involving
f in i t e deformations and two
families
of f ibres .
All
vector and tensor components wil l be
refer red
to a system of
rectangular car tes ian coordinates xi ( i =
1,2,3) .
Components of the
inf in i tes imal
displacement
vector
u
are
denoted
by
u. , and components of
l.
the inf in i tes imal
s t ra in tensor e by e i j '
so tha t
e . .
l.J
[ au i aujJ
2
ax.
ax. J .
J
l.
The Cauchy s t ress tensor 0 has· components 0 .. and the f ibre d i rect ion
l.J
vector a has components a . .
- l.
In
l inear
elas t ic i ty ,
the
strain-energy function W
i s
a
quadrat ic
function of e
. .
, so that
l.J
W
= ~ c ijk.R. e i j ek R ,
(l)
(2)
where
the usual repeated
index
summation
convention
is
used,
and
cijk.R.
are
components of
the s t i f fness
tensor, which
possesses the symmetries
The
s t ress i s then
given
by
0 .
l.J
w
- \ - -
= c
. .
nekn.
ae
.
l.]k '
'
l.J
(3)
4)
The s t i f fness
components
cijk.R.
depend
on
the
f ibre
direct ion ,
and
so
may
vary with posi t ion.
To
determine
the
form of
the cijk.R. for a t ransverse ly i so t ropic
mater ia l we f i r s t
note
that , for a given deformation, W depends
on
e and
on the f ibre direct ion a. Thus
v = W(e,a) .
I f the
only anisotropic properties of
the material are
those which
ar ise
from
the
presence
of the
f ibres ,
then
W
i s
unchanged
i f both
the
deformation f ie ld
and
the
f ibres
undergo a rotat ion which
i s
described by
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Constitutive Theory
a
proper
orthogonal tensor
g. For this
new deformation, the s t ra in is
- T -
given
by
:
= 9:2
, and the f ibre direct ion
by
a g ~ . Thus
T
W(e,a)
=
W(QeQ
,Qa)
-
-
--
and
th is
holds for a l l proper orthogonal
tensors
g, tha t
is
for a l l
T T
tensors
such
tha t
= g g = I , det Q =
1,
where I is
the unit
tensor.
5
(5)
Equation
(5) i s a statement tha t
W
i s
an
isotropic invariant of : and
a.
Since
the
sense
of
is
not signif icant ,
Wmust be an even
function of
a,
and so it may
be
expressed as·
an
isotropic invariant
of
a n d ~ ~ ~ ,
where the
dyadic product
a®
a i s
the
second-order
tensor
with cartes ian
components
a .a
..
1 J
These
invariants
are tabulated
(see,
for
example,
[1]);
by reading off
from tables
we find
tha t
W can
be
expressed
as
a
function
of the
t races
of the following tensor
p r o d u c t s ~
e ,
a®a,
( a ~ a )
(a®al
3
,
e .a®a,
e . (a@a)
2
e
2
. a®a,
e
2
. (a®a)
2
•
-
-
-
- - -
- -
-
However,
since
a is a uni t
vector
a®a =
(a®al
2
a0a)
3
=
(6)
Also
t r a 0 a
=
1,
t r e .a®a = a .e .a , t r
e
2
. a®a (7)
and
so
the se t of
invar iants
reduces
to
t r
e ,
t r e
3
,
a .e .a ,
(8)
The
most
general quadratic
function in
e
which
can
be formed
from (8)
is
+2(\l -jl
) a . e . a + ~ S ( a . e . a )
L T - - - - - -
J.,A_ .
e . . ekk + l e . ke . + a.a . e . . a .ekk
11
T 1
1k
1
1J
J
+
2(\l
-jl
)a .e . .
e.kak+
~ S a . a . e . .
aka
0
ek ,
L
T
1
1J
J 1 J
1J
'
'
(9)
where A.
jlT, \lL a and
S
are elas t ic constants . Thus for
th is case
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6 A J M Spencer
..
1 ]
A.ekko . .
2ll
e . . +c:t(aka"ek
0
0 +a.a.ekk)
] T ] IV IV ]
1
J
where o . . denotes the Kronecker
del ta ,
so
tha t
1 ]
ci jk£ = A.oijoH +lJT(oikoj£+ojkoHl
+a aka£oi /a ia jokt l
An
al ternat ive derivation of th is resul t
i s
given
in
[2]. In
di rec t
notation, 10)
can
be writ ten as
(A.t re+aa.e .a) I+2l-
e+
a t r e+ S a . e . a ) a@ a
- - - - - - - - - - -
10)
+ 2 lJ
l 1
(a®a.e+e.a®al 12)
L T - - - - - -
The e las t i c constants lJL and l-IT represent shear moduli. The
other
e las t i c
constants A. a and S can
be
related
to
other
e las t i c
constants which have
more di rec t physical in terpretat ions , such as extension moduli and
Poisson's ra t ios . The admissible values of
the
e las t i c
constants are
res t r ic ted by the requirement
tha t
W
must
be posi t ive
def ini te .
suppose, for example, tha t the direct ion of the x
1
-axis
i s chosen
to
coincide
with
the f ibre d i rect ion , so tha t a has components (1,0,0) . Then
10) give
11
a22
a
a23
a31
a12
A.+2a+4lJL-2lJT+S
A.+a
A
a
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
13)
which
i s equivalent
to
the usual
form of the const i tu t ive equation
for
a
l inear ly
e las t i c material which i s transversely
isotropic with
respect to
t he x
1
- a x i s . From (13)
t
i s apparent t h a t ~ a n d ~ T a r e shea r moduli fo r
shear
on planes
para l le l
to
the f ibres ,
with direct ion of
shear in the
f ibre direct ion lJL) and normal
to
the f ibre d i rect ion lJT) respect ively .
Also
from
(13)
t
i s eas i ly
shown
tha t the extension moduli
EL
for
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Constitutive Theory
uniaxial tension in the f ibre direc t ion and
ET for uniaxial
tension in
direct ions normal to the f ibres
are
A + ~ T )
8 + 2 ~ L ) + ~ T A + 2 a ) - a
A + ~
T
4 ~ { A + ~ ) 8 + 2 ~ ) + ~ (A+2a)-a
}
T T L T
( A + 2 ~ ) < 8 + 2 ~ ) + 2 ~ (A+2a)-a
2
T L T
Expressions for the Poisson s ra t ios associated
with
extension in the
f ibre
and
t ransverse
direct ions are also readi ly derived from (13).
7
14)
Values
of the
elas t ic
constants
~ T ~ L
A
a
and
8
can be determined
from experimental
measurements.
For example, Harkham [3] obtained data
for a typical carbon f i b r e - epoxy resin composite
which
give (in uni ts of
10
9
Nm
2
)
=
5.66,
L
= 2.46,
and
hence,
in the
same
uni t s
239.35,
E
=
7.53
T
5.64,
a
-1.271
227.29
inematic
constraints.
We see from 15)
and
16) tha t the modulus
EL
15)
(16)
considerably exceeds the
other
extension and shear moduli.
This
of course
ref lec ts
the s t i f fness of
the material in the f ibre direc t ion, and
i s
a
feature
of many
f ibre-reinforced composites.
The
material i s res is tan t to
deformation by extension
in
the f ibre
direct ion,
and wil l prefer other
deformation mechanisms i f any are avai lable
t
i t .
This
suggests tha t
as
a
f i r s t approximation
we
might
consider the
l imit
in
which
EL ~
00
,
while
ET, ~ L
and ~ T
remain f in i te .
This
corresponds to the case
in
which the
material
i s
incapable
of
extension in
the
f ibre direc t ion, so tha t in
any
deformation
the s t ra in
component
a .a .e . . i s zero.
The
condition
a .a .e . .
= 0
J
J
17)
i s an
example
of
a kinematic constraint and represents inextensibili ty
in
the
f ibre direc t ion
a. Kinematic
constra ints are not uncommon in
continuum
mechanics. The
cons t ra in t
which
we
encounter most often is tha t
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8
A.J.M. Spencer
of incompressibility which
i s
often used
in,
for example, f lu id
mechanics
and
f in i t e
e las t i c i ty theory. Although
no
mater ia l i s t ruly
incompressible,
there are
many
mater ials
and
applications
for
which
the
assumption
of
incompressibi l i ty
gives
sa t i s fac tory resul t s .
Because
it is
more famil iar ,
w shal l consider f i r s t the effect of considering the material to be
incompressible,
and
return
to the inextensibi l i ty const ra in t la ter .
ncompressible
material.
I f t r e = 0, then (for the transversely
i so t ropic l inear ly
e las t i c
mater ia l under consideration)
from
(8) W
becomes
a function of
t r
e
3
,
t r e
2
,
a . e . a ,
a .e
2
.a ,
but
w
may
add toW any multiple
of
t r
e .
Hence (9) i s replaced by
W = l l t r e
2
+2(l.l -l.l a .e
2
.a+l: S(a.e.a)
2
-ptre 18)
T LT -
where p may be
regarded
as a Lagrangian mult ip l ier . The number of
independent e las t i c
constants i s reduced
to three.
Then
(4)
gives ( in
direct notation)
r = -pi+2l.l e +S(a.e.a)a®a+2(l . l -l.l) (a®a.e+e.a®a)
- -
T-
- - - - - L T - - - - - -
19)
Here
p i s arbi t rary iP. the
sense
tha t
it
i s not given by a const i tu t ive
equation but has
to
be determined by equations of equilibr ium or
motion
and
boundary
conditions) and represents an
arbi t rary
hydrostat ic pressure.
This hydrostat ic pressure
i s
a reaction to the const ra in t of
incompressibi l i ty. The s t ress - p ~
does
no work in
any
deformation
which
conforms
to the const ra in t of
incompressibi l i ty, for
i f
ekk =
0 , then
From 19)
w
see tha t
w
may divide the s t ress
in to two
parts
(J
s
r or
(J
..
l J
s . . + r . .
l J l J
20)
where r represents the
reaction
stress and
i s
here of the form
-p i ,
and
s i s cal led the extra-stress. For a material subject to kinematic
const raints , the
extra-stress i s
given by const i tu t ive equations, and the
react ion
s t ress i s arbi t rary
in
the sense described above.
Since
p is
arbi t rary,
s
i s
arbi t rary
to
within
a
hydros ta t ic pressure,
so
without
loss
of general i ty w may
specify tha t
t r s
=
0.
Then s becomes the deviator ic
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Constituti
ve Theory
s t r e s
s
l
s
~ -
tr
21)
and from
19), and usin
g
tr
e = 0
and a
.
a =
l ,
w
e
have
s =
e
a . e . a ) a
+ 2 ~
- ~ ) a 2)a.
e+e.a 2)a)
~
L
T
- i3+4)J
-4)J
)
a .
e .a ) I
22)
L
T
Inexten
sible material
A s im i l a r p
rocedure can
be
f
o l lowed
if the
m a
ter ia l i s in
ex tens ib le in the
f ib
re
d ir
ec t io n bu t no
t incomp
ress ible .
Then
a . e . a
0 ,
and
9)
i s
rep laced
by
where T
i s
a
Lagrang
ian
m u l t i p lie r
, and again the num
ber o
f
indepe
ndent
e
la s t i c c
ons ta n ts
i s
re
duced
to
t
h re e .
Then
from
4)
o
T a 0 a + A i t
re + 2 ) J e+2 ) J
e + e . a ~ a )
L
T
23)
24)
The s t r e s s
Ta a
i s
an
a rb i t r a ry
tens ion
in the
f ib re d irec t ion
which
i s
a
r eac t i
o n
to
the i n e x t
e n s ib i l i ty
c o n
s t r a in t
and
does no work in
any
d
eform at ion
which
confo
rms
to
t h i s
c
o ns t r a in t , fo r
T
a.a .e . .
l J
l J
T a .e
.a 0 .
I f
w e
decompo
se
o
in to
a
r eac t i
on
s t r e s s r
a
nd an e x t r a - s t r e
s s s ,
as
in 20) , then
2
5)
and s
i s a rb i t r a r
y to
wi th in a
f ib re
te
n s ion .
Without
loss of
ge ne r a l
i ty
w e
m
ay
sp e
c ify a . s .
a 0, and then
s
A I-a 2
)a) t r e + 2 ) J
e + 2 ~
) a ~ a . e + e
. a 2 ) a )
L T
26)
ncom
pressible
nd inc
xtensible
material
I f the
mater i
a l i s
both
incom
pressible
and in
ex tens ib le in the
f ib re
d i r e c t io
n ,
the
n
W
t akes the
form
w
tre
2
+2 )J
)a .e
2
.a-ptr
e Ta.e.a
T
L T
27)
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10
A.J.M. Spencer
and hence
There
are
now
only
two independent
elas t ic
constants ~ T and ~ L · The
react ion s tress i s
and r
does
no work in any
deformation in which
a .e .a
0
and t r e 0.
- - - -
29)
The
extra-s tress
s is
indeterminate
to
within
an
arbi trary
pressure
and an
arbi trary tension the f ibre direc t ion. I f ,
without
loss of
generali ty,
w
specify
t r s 0
a .s .a 0 30)
then
it follows from
28)
that
s e
+ 2 ~
l a®a.e+e.a®al
- T- L T - - - - - -
31)
2.2 Linear elasticity - two families of fibres
Let
us now
consider
a material
which
has
l inear
elas t ic response
and
is reinforced by
two
families
of fibres,
with
fibre directions a and
b.
Suppose
tha t
the
only anisotropic
propert ies of
the material are those
which are due
to
the presence
of
the fibres,
so
that
there
are two
preferred directions a and b
a t
each point in
the
material .
I f the two families of f ibres are
orthogonal
as
in
a
cross-ply
laminated
material)
then
local ly
the material possesses material
symmetry
with respect
to
ref lect ions
in the
planes
normal to
the
f ibres
and so,
locally,
the material is orthotropi
with respect
to the
planes normal
to
the
f ibres and
the surfaces in
which
the f ibres l i e . I f
the
two families
are not necessari ly orthogonal but are mechanically equivalent i .e .
are
indist inguishable except
for
the i r
directions,
as
in
a
balanced
angle-ply
laminate) , then local ly the material
has material
symmetry with respect
to
ref lect ions
in
planes normal
to the bisectors of
the
two families of
f ibres and again
the material
i s local ly orthotropic,
but
now with respect
to
the planes
normal
to
the bisectors of the fibre
families
and
the
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8/16/2019 Spencer 1984
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Constitutive Theory
surfaces in
which
the f ibres
l ie .
However, in the f i r s t instance
we shal l
not r es t r i c t ourselves
to ei ther of
these
special cases,
and will consider
the general case
of
reinforcement
by
two
families
of
f ibres
which
are not
necessari ly
ei ther orthogonal or mechanically
equivalent.
By
arguments similar to those used for a
single fibre family,
W i s
quadratic in e, even in a
and
b,
and
such that
-
T
W Q.e.Q , Q.a, Q.b) = W e,a,b) ,
where
Q is any proper orthogonal tensor. I t follows that W i s an isotropic
invariant
of
e,
a0a
and
b®b.
From
tables
of such
invariants
[1], the
relat ions 6) and
7),
and
simi lar re la t ions for
b, it follows that
W
is
a
function
of
t r
e
t r e
2
, t r e
3
,
a . e . a
a .e
2
. a b.e .b
32)
b.e
2
.b
a.b)
2
= cos
2
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12
A J M Spencer
and the most general quadratic form for W is
where A, a
1
, a
2
, S
1
, S
2
and S
3
are
elas t ic constants . The
corresponding expression for the s t ress i s
a
1
t r e + S
1
a.e.a+f3
3
b.e .b)a®a
- - - - - - -
a
2
t r
e S
3
a.e .a S
2
b.e .b)b®b
-
- - - - -
I f
the coordinate axes are chosen so
that
has
components 1 ,0 ,0 )
and b has
components
0 , 1 , 0 ) , then 36) may be
writ ten
as
011
A 2a
1
S
1
A a
1
+ 2 ~ + 4 ~ 1
a2 S3
J.. a1
0
0 0
0
22
A a
1
A 2a
2
S
2
A a
2
0
0 0
a2 S3
+ 2 ~ + 4 ~ 2
0
3 3
A a
1
A a2
A + 2 ~
0
0 0
0
23
0 0
0
2 ( ~ + ~ 2 )
0
0
0
31
0
0 0
0
2 W ~
0
•
0
1 2
0 0
0 0
0
< ~ + ~ 1 + ~ 2 )
36)
e1 1
e22
e33
e23
e31
e12
37)
This
i s
of the usual form for an orthot ropic material , and enables the
e las t i c
constants A, a
1
, a
2
,
1
, S
2
and S
3
to
be related to
e las t i c
constants with
more
di rec t physical in terpretat ions;
for
example,
we see tha t ~ + ~ ~ + ~ ~ + ~ + ~ are shear moduli for shear on the planes
normal to and para l le l
to
the f ibres .
There
are nine
independent e las t i c
constants .
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Constitutive
Theory
3
I f
the m a t e r ia
l
i s
incompr
essible
then tr e
=
0
and t
he te rms
involvin
g
A
,
C l
and
a
2
a re om
it ted from
35)
and
36)
and a react io
n
s t r e s s
in
th e
form
of
a
h y d r o
s ta tic
pressure - p i
i s
added
on
the r ig h
t of
36) .
f
the
m at e
r ia l i s in
ex tensib le
in bo
th
f
i b r e d i r e c t i on
s ,
the
n
a .e .a
=
0
and b .
e .b = 0
Then W
takes
the form
38)
w
here
the La
grangian
m u l t
ip l ie r s
Ta and
Tb
repre s
en t a rb i tra ry
tens ions i
n
th e
tw o
f
ib re d i r e c t i o
n s .
Then s+ r
, where th
e
r eac
t io n
s t r e s s r i s
giv
en by
r
=
T
a 0 a
T b ® b .
a-
-
39)
The
e x t r a - s t r e s
s
s i s
giv
en by
40)
and has bee
n chosen
so th a t a . s . a
=
0 and
b .s
.b
= 0.
The number
of
independent
e l a s t i c
con s tan ts
i s
reduced
to
four .
I f the
m a t e r ia l
i s
both
incompres
s ible and inex ten
s ib le ,
then
r =
4
1)
42)
where s
has been
chosen
so
t
ha t t
r
s
= 0 , a . s .a
0
an
d b .
s . b
0
. There
a re
now
th r ee indep
endent e la
s t i c co n s
tan ts .
wo
mech nic
l ly equivalen
t f
amilies
of
fi
bres I f
the two fami l i
es of
f ib re
s
are mecha
nical ly eq u
iva len t ,
then W must
be
sym
metric
with re
spec t
to
in t
e r changes o f
a and
b
Hence
th
e se t 32) w
ith c o s 2 ¢ a .e
2
.b
om i
t ted)
i
s
rep laced by
tr
e
a . e . a + b .
e .b ,
a
.e .a )
b.e .b)
,
a .e
2
.a+ b
.e
2
. b ,
a .e
2
.a) b.e
2
.b) ,
cos 2¢ a .
e .
b ,
4
3)
The
m
ost gen
era l e
xpressio n
for W whi
ch
i s
quadra t ic
in e i s now
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4
A J M
Spe n
cer
w
44)
whe
re the
nine coeff ic ien
ts
A , ~ . y
1
, • • •
,y
7
are even
functions
of c
o s 2 ~ .
This
l
eads
to
the
cons t i tu ti
ve
equation
4
5)
In
m
any cases
t i s advantageo
us to express
the equations in
terms
of
the bisectors
of
the fibre
direct ions because
these bisectors a
re
mut
ually or
thogonal.
For th i s
we in troduce un
it ve
ctors c and d
,
w
here
c >
a+b) / c o s
~ , d
a - b ) / s
i n ~ ,
46)
a = cos J
+ s i n
~ ,
: c o s ~ -
~
s i n ~
.
On
s
ubst i tut in g
fo
r a and b from
46) in to
44), we
obta in a
n
expressi
on for
W of the
sa
m e form
as
35
),
w
ith
a an d b replac
ed by
c
a
nd
d
respe
ct ively,
and w
ith the coeff ic
ients be
coming fu
nctions
of
cos
2 ~ .
en
ce th i s
case
also
correspon
ds to orthotr
opic symmet
ry,
and
i f
c and
d
are chosen
to
l
ie
in
the
x
1
and
x
2
c
oordinate d ir
ect ions,
the
const i tutiv
e
e
quation as
sumes the form
37) .
There are again
nine independen
t e las t i
c
coefficie
nts wh
ich are functions
of c o s
2 ~ . T he
cons t i
tut ive
equatio
n
in
te
rms of
c
a
nd d is obt
ained by
substi tut
ing for a
and b from
46)
i
nto 45
).
The
expression
is
obviously complicated.
n
a lternat iv e
and
ra ther
si
mpler proce
dure i s to o
bserve from the
beginn
ing that
W can
be
express
ed
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Constitutive Theory 15
as
a function
of
e, c , d and
c o s 2 ~ ,
with c.d 0. Hence
t i s possible
to
proceed as
in
the case of
orthogonal
f ibres ,
with a
and
b
replaced by
c
and
d,
and
the
coeff ic ients
in
the
expression
for W
regarded as
functions
of
c o s 2 ~ . However, with
the
al ternat ive procedure
t i s
less
easy
to
proceed
to the case of f ibre
inextensibi l i ty.
I f
the
material is
incompressible
then
t r e
0 and
the terms
involving A
y
3
a n d y ~ are omitted from
44)
and 45), and
a
reaction
s t ress - p ~
i s
added.
I f
the material i s inextensible
in both f ibre
direct ions , then a.e .a = 0 and
b.e .b
=
O
and
45)
i s
replaced
by
47)
The number
of independent elas t ic constants
i s
reduced to f ive , and the
l a s t
two terms
represent the react ion s t ress . In
terms
of the vectors
c
and d, 47) i s
z
t r : +
211:
+ {
h cos 2 ~
t r : + 2y
2
: . : . : cos
2
P
+
~ . : . ~
s in
2
~ }
x
+ Ta-Tb)sin
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1
6
A J M Spenc
er
50)
I
n a l l
of
th
e cases di
scussed
above,
t
i s po ssib le t
o express
0 as
0 = r s ,
where r i s the
re act io
n s t r e s s , an
d w i
thout
lo
ss
of
ge ne r a l
i ty s
may
b
e chose
n
so
t ha t
a . s .a
=
0 b .s .b =
0
when t
he
m
ateria l i s
in ex tens ib le
in
the
d ire c t i ons
a
and
b ,
and
tr
s
=
0
when
the
m ater ia l
i s
incompress
ible .
3 FINITE
ELA
STIC CONST
ITUTIVE
EQU TIO
NS FOR
FIBR
E REINFORCE
D M TE
RI L
3 1
Kinematics of
finit def
ormations
So fa r we
have cons
idered only
small def
ormat ions
for which t
i s
no t
neces
sa ry
to dis t in g
u i s h
betw
een
the
f ib r e d i r e c
tions
in the
undeforme
d and deform
ed
conf igu ra t
ions o f a bod
y. e now tu rn
to
f i n i t e
d
eformations
.
The def
ormation
wi l l be r e fe r r ed
to a
f ix
ed
frame
of
re fe renc
e , and
to r
ec t angu la r
c a r t e s i a n
coord in
a tes
in
t h i s frame
. e cons ider a
body
which i s i n i t i a l l y
in
a
re ference configuration
in
which
a
typ ica l
p a r t ic le
has
p
os i t ion
vec
tor X w ith
componen
ts
XR
.
At
a subs
equent
t ime
the
body i s in
a efo rm
e configura
tion and th
e gener ic p
ar t i c le
has
pos i t ion vec to r
x , w i
th coord ina
tes
x
. . T
hus the deform
ation i s
descr ibed
l
by
eq
u ations of the form
x
x(X)
o r
(
51 )
wh
ich give
the
spa t i a l
coord i
na tes x i in
terms o
f
the
m ater ia l
coord ina
tes
XR.
Th
e
deforma
t ion grad ien t
ten
sor F has car tes ian
coord
ina tes
FiR
where
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8/16/2019 Spencer 1984
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C o
nstituti
ve The
ory
Cl
x./ClX
.
l
R
I f
ov and
ov are
volu
mes
of a
m a t e
r ia l v
olume
ele
m ent n t
h e
re fe
ren ce
a
nd de
formed
c
onf ig
u r a ti
ons re sp
e c t iv
e ly ,
and
p
0
a
nd p th
e
d e
n s i t ie
s o
f the
elem e
nt in
t
hese
tw o co n
f igu r
a t ion
s , then
ov
ov
p
det F .
We
sh a
l l a lso
employ
th
e d
eform
ation
t
en so r
s C and
B
, w it
h
c a r t
es ian
com
ponen
ts CRS a
nd
B ij
re
sp e c t
iv e ly ,
where
T T
c
F
.F
B
F.F
dX
dX
dX
dX
l
l
c
ClXR
ClXS
F
iRFiS
B . .
F.
F
.
RS
lJ
lX
R
C
lXR
lR
JR
7
5
2)
5
3)
54)
55
)
Supp
ose th a
t in
the
refe re
nce
conf ig
ura t i
o n a f ib re
d i r e
c t ion
i s
def in
ed by a
u n i t
v ec t
o r f ie ld
a
0
X
)
with car tes
i a n
co
mpone
nts a
o ) .
In
R
a
de
form a
tion the f i
b res ,
eing m a t
e r ia l
l in e
elem
ents ,
w il l be
conve
cted
with
the
p ar t i c le s
of the
body,
so
t h a t
in the
deformed
co n f igu ra t ion
the
f ib
re d i
rec t io
n
may be
descr i
b ed by a
u n
i t v ec
tor f i
e l d a
x) w
ith
c a r t
e s ian
compo
nents
a i .
In gener
a l
the
f
i b r e s w il
l a ls
o
s t re tc
h ; s
uppos
e
th a
t
a
f ib r
e e
lemen
t h
as le
ng th
o
L i
n t
h e re fe r
ence
co nf i
g u ra t
ion
and
len
g th o£
in t
he
d
eform
ed conf
igura
tion .
C
onside
r a f ib re
e
lem en
t who
se en
ds
X R a
~
in
the re
fe ren c
e con
f igu r
a t ion
def
ormed
conf i
gurat i
on .
Th
en,
from
51)
,
X
= X .
X )
l
l R
an
d it
fol lo
w s th a
t
Thus
a
. 6£
l
dX
)
l
0
6
dXR aR
L .
dX
.
l
x
. +a.o £
l
l
T
hen
the stret
h A
=
o
£/oL.
h
ave c
oord in
a tes
XR an
d
and
x.
and
x.+a .o
£ in
t
he
l
l
l
Aa.
l
ax
or
· ~ o
.
R
5
6)
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18
A J
M Sp
encer
T
his
r e l a t
e s
the f ib
re d
irec t i
on s
in the
re
feren c
e and
defo
rmed
c o n f
igu r a
t ions .
Also ,
s ince a
i s
a
u n i
t v
ec to r ,
dX dX
ax
ax
whic
h
d
eterm
ines
the
f ib re
s t re
tc h .
3 .
2 in
ite
el st
icity
fo
r one
fami
ly of
fibre
s
(5 7)
We cons id
er
a f in
i te e l
a s t i c
so l id
with
a
s t r a i
n ener
gy W
which i s
a
func ti
on
of th
e def
orm at
ion
g ra
d ien t
s
F.
. The
n b
y
s
t anda
rd
argum
ents
in
lR
the
devel
opm en
t of f i n
i t e
e la
s t i c i
ty theor
y
for ex
ample
[4 ) ) ,
which
are
in
no
w
ay af fec t
e d by the
prese
nce o f on
e
or
m
ore
fam i
l ie s
o f f ib
re s ,
W
can be
expr
essed a
s a fu
nc t ion
o f
th e
com
ponen
ts CRS
and t
he c o n s t
i tu t iv
e
equa t
ion
fo r th
e s
t r e s s
i s
(
58)
C onsi
der a
m a te
r ia l r
e info
rced
by a
s in
g le
f
ami ly
o
f f ib re
s
with
i n
i t i a l
f ib re d
i r ec t i
o n
The
n
by ar
gum en
ts s im
ila r
to th
ose used
in
th
e l i ne a
r e la
s t ic
case
, W
can be ex
press
e a
s a fu n
c t ion
of
C a
nd
o
w se l e
c t a
new
re fe r
ence c
on f ig
u ra t io
n w
hich i s o
bta in
ed
by a r i
g id
ro
ta t io
n of
th
e undefo
rmed
m
ater ia
l and the
f ib
re s , so
th a
t a t
y p ic a
l
p
a r t ic l
e
i
s
a t X
= Q.X
and the
f ib re
d i
re c t io
n i s
9 · ~ o
w
here Q
i s a
prope
r ort
hogon
al
t ens
o r .
Th
e d
eform
at ion
tenso
r from
th
e new
re fe r
ence
T
co n f igu ra t ion
i s
C =
Q.C .Q
.
However,
th i s
change
of refe rence
con
f ig u ra
t io n l
eaves
W
u na
l tered
, and so
59)
fo r a l
l pr
oper
orth
ogona
l tenso
rs Q.
He
nce W
i s
an
i s
o t r op
ic in
va r i a
n t of
~ a
n d
~
T here
fore,
by a
rgume
nts s
im i lar
to t
hose
used
in d e r
iv ing
th
e
s e t 8),
w it
h
e
a
nd a rep la
ced
by
and
re
sp e c t i
v e ly ,
t
fol lo
w s
th a t
W
can
be
expresse
as
a
fu nc t ion
of
the
in v a r ia n ts
-
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Cons
titutiv e Theor
y
t
r C
det C
wh
ere,
f
o r c
opvenience
,
tr
C, tr c
2
and tr C
3
have
been
rep laced
b
y the
equ i
va len t s
e t
I
1
I
2
and
I
3
•
From 5
8)
it
then
fol lows
t ha t
lJ
5
- ~
\
I
3
F.F.
L
lR
JS
a=
l
[
3I
3
I
a
w
a RS C
SR
whe
re
W
denotes
3w/3I . Fro
m 60) we
ob ta in
a a
3I
2
= Io -c
3c RS
R
S
RS
l i
3
I
0 I C
C C
3CRS
=
2
RS
1
RS RP P
S I
Hence
61)
can
be w
ri t t en as
0
Using
54)
,
56
) and
60), t h i s become
s
19
60)
61)
62)
63)
Fu r
ther
s im p
l i f ica t io n
ol lows by
us ing the
Cayley-H
amilton theor
em fo r
namely
o
64)
a
nd,
s
ince
d
e t B
0 th e
r e la
t i on
0
65)
-
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20
A.J.M. Spencer
to
eliminate B
3
and B
2
from 63)
ir. favour
of
B-
1
which gives
J
This
is equivalent to
resul ts
given
by Ericksen
and Riviin [5] for
t ransversely
isotropic e las t i c
materia ls .
I f
the material is incompressible, then I
3
= 1, W is a function
of
I
1
, I
2
, I
4
and Is but a term - ~ p ( I
- l ) , where
p s a Lagrangian
mult ip l ier may
be
added to W This leads
to the
const i tut ive equation
J
and p s
a
reaction pressure.
66)
67)
I f in
addit ion,
the
material
is
inextensible
in the f ibre direc t ion
a,
then
I
4
= A
2
= 1, W depends
on
I
1
I2
and Is and a term ~ T I - l ) may
be
added to W where T
i s
another Lagrangian mult ip l ier . Then
J 68)
and
so
again
T
i s ident i f ied
as
an
arbi t rary
f ibre
tension
which i s
a
react ion to the
inextensibi l i ty
constra int .
3 3 Finite el sticity for
two
families of fibres
I f
an e las t i c body i s reinforced by two
families of
f ibres whose
direct ions
in
the
reference configurat ion are
defined
by uni t
vector
f ie lds
and respect ively , and in the deformed configurat ion
by
a and
b
respect ively , then,
by
arguments
similar to those used above,
the
strain-energy function w
i s an
isotropic invar iant
of C,
~ 0 ~
and
® ~ o ·
I t follows by
analogy
with
32)
and
33) that W can
be
expressed
as
a function
of
II
I2
,
I3
,
I4
,
Is
I6
· ~ · ~ o
I7
=
2
bo · ~ o
I a
= cos 4> · ~ · ~ o
and
cos
2
4>
,
69)
where
cos 4>
~ o · ~ o
is the cosine of the angle between the
two
families
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Constitutive Theory
2
of f ibres
in
the reference
configurat ion.
The angle between the
famil ies of f ibres in the deformed configuration is
given
by
cos
a .b
I f the famil ies
of
f ibres are orthogonal in the reference
configurat ion, then the material i s orthotropic in th is
configurat ion;
and W i s a function of I
1
, ••• I
7
• Then, by arguments
s imilar to
those
which
lead
to
66), the
const i tu t ive equation may be writ ten as
J
70)
This
is in agreement with resul ts
for
orthotropic
elas t ic
materials
given
by Smith and Rivlin [6] and Green and Adkins [7]. I f in addit ion, the
material i s
incompressible, then
I
3
= l a n d the term W W in 71)
i s replaced by - ~ p ~ , where p s
a
reaction pressure. I f furthermore,
the
material i s
inextensible
in the
two f ibre d i rect ions, then I
4
= 1,
I
6
= l and 71) is replaced by
:: = 2 { W ~ - ? J ~ - l + W ( ~ 0 ~ . ~ + ~ . ~ 0 ~ ) + W ( ~ 0 ~ · ~ + ~ . ~ 0 ~ J }
where the
l a s t three terms represent
the react ion s t ress and Ta
and
Tb
are arbi t rary f ibre tensions.
I f
the two
famil ies
of
f ibres
are mechanically
equivalent ,
then
the
mater ia l
i s
local ly orthot ropic
in the reference
configuration
with
respect
to
the planes which
bisect
and and the planes containing ~
and ~ o · Then W
i s
a
function
of I
1
, •••
I
8
and
s y w ~ e t r i c
with respect
to
interchanges of and ~ o · Hence W can be expressed as a function of
and
73)
However, t
can
be shown
tha t
I
12
can
be
expressed in terms
of the
other
invar iants , and
soW
can
be
expressed
as
a function of the seven
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A J M Spencer
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - ~ - - - -
22
invariants
a
and
(74)
Alternative::_y,
can
J: e
E..X:t-H . . . > ~ G .Ll •
-·'·
~ · ~ . 12
1:.ul..ually
orthogonal
uni t
vectors which bisect
and ~
but
because mater ia l l ine elements
which bisec t f ibre
direct ions
in the reference configurat ion do not, in
general , bisec t f ibre
direct ions
in the deformed configurat ion,
it
i s not
usual ly a d v ~ n t a g e o u s to do th is .
When
W
is expressed as
a
function of the
se t (74) ,
we
obtain from
(58)
a
(75)
I f the material is incompressible and inextensible in the
two
f ibre
di rect ions,
then
I
3
=
I
4
=
I
6
=
1,
and
hence
1
9
=
2,
I
10
=
1,
and
(75)
i s
replaced
by
(76)
where again the l a s t
three
terms represent the react ion s t ress .
4 PLASTICITY
THEORY
FOR FIBRE REINFORCED t· ATERIAL
4.1
ield functions for
one
family of fibres
We now consider
tha t
the
material
has plas t ic response. We follow a
standard formulation of
plas t i c i ty
theory, beginning with the yield
function.
The
appl ica t ions .we have in
mind
are
to f ibre-reinforced
composites with metal matrices, but the theory i s
not
l imited
to
any
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Constitutive Theory
23
par t icular type
of
material .
e
postulate a yield function f
O
.. ) such
tha t in
admissible
s t ress
l]
s ta tes
f
0
with
f
=
0
when
plas t ic
deformation
i s
taking
place.
For
strain-hardening material some s l igh t modifications are needed, which we
take
up
l a te r .
I f the plas t ic material i s i so t ropic
then
it i s well known tha t f can
be expressed as a function of
the s t ress
invar iants t r o,
t r 0
2
and
t r o
3
•
In i so t ropic metal plas t i c i ty it
i s
observed experimentally
tha t
for many
mater ials yie lding i s effect ively independent of a superposed hydros ta t ic
pressure.
This
i s incorporated in the
theory
by res t r ic t ing f to
depend
on
the
deviator ic
s t ress
s
s
. .
l J
s = o-
. .I
t r
o .
-
3_
-
77)
Then t r s 0 and f can be expressed as a function of t r s
2
and t r
s
3
• We
note
tha t
s
i s
the
extra-stress
for an incompressible mater ial .
For anisotropic
materials f
i s
a function of
0 . .
or s . . ) which
i s
l J l J
invar iant
under
the
appropriate transformation group.
For
a
f ibre- reinforced material
the
yield
properties
wil l
depend
on
the
orientat ion of the f ibres so
we
propose f to be a function of and
l J
or, since
the
sense o f ~ has no signif icance, of
o
. .
and a.a
. .
l J l J
For a f ibre- reinforced metal
we
ex.pect yielding to remain independent
of superposed hydrostatic s t ress . Also
for
a metal
reinforced
by
inextensible
f ibres
it i s reasonable
to
expect tha t yie lding i s not
affected
by
a superposed
tension
in
the f ibre
direct ion ,
since such
a
tension produces no
s t ress in the
matrix. These conditions can be
incorporated by assuming f to depend on
0
only
through the
extra-stress
s
where
0
r
s
r = -pi T a ~ a •
Here r
i s
the react ion s t ress
and
the indeterminacy in s
i s
removed by
imposing
the
conditions
t r
s = 0
a .s .a
o.
Then it follows from 78) and 79)
tha t
78)
79)
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24
A J M Spencer
t r
= -3p T,
a.O.a
-p+
T
and t follows by eliminating
p
and T from
(78)
that
s =
o - ~ t r o - a . o . a ) I + l : i t r o - 3 a . o . a ) a ~ a .
80)
Now f must be invariant under rotat ions of the
s t ress f ie ld ,
with the
f ibres
moving with
the
s t ress f ie ld .
Hence,
for any
orthogonal
tensor Q
i
T
s = Q.s.Q
then
we require
f s , a0a )
a = Q.a
Hence f i s an i sotropic invariant
of
s a n d
a®a. By standard resul ts in
invariant theory
[1],
and taking in to
account
79) and (6), t follows
\
that
f
can be
expressed
as
a function
of
81)
In
the solut ion
of
problems,
even in isotropic plas t ic i ty theory,
t
is
usually necessary to
assume some
special form
for
the
yield
function,
the most commonly
adopted
yie ld functions being those of von Mises and
Tresca. The
natural approach for
a
f ibre-re inforced
mater ia l i s
to
t ry
to
general ise these.
Von
Mises yie ld function
i s
the most
general i sotropic
yie ld function which i s quadratic
in
the
s t ress
components. The most
general function of the invariants (81) which
is
quadratic in the s t ress
i s
(82)
The coeff icients
of
J
1
and S
2
are writ ten
in
th is
way
because
t
can
then
eas i ly be
shown
tha t
kT
and
kL
can be ident i f ied with the shear yield
s t ress
on
the
planes containing the
f ibres
for shear in
the
direct ions
transverse to and para l le l
to
the
f ibres
respect ively.
I f
kT
=
kL
(or,
more
generally ,
i f
f
depends
on
J
1
and
J
3
only) ,
the
f ibre orientat ion does not enter in to the yie ld condition, and the
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Constitutive Theory 25
material
behaves
as a
constrained isotropic material In such
a
material
the
extra-stress
response
i s i sot ropic .
However,
the
experimental
evidence
is
tha t
kT
kL,
except
perhaps
for
low f ibre
densi t ies .
A
yie ld function analogous to Tresca 's , which
i s
essent ia l ly
due
to
Lance and Robinson [8] i s
( ~ J 1 - J 2 l
- 1
for
J2 <
k2
kT
L ,
f
(83)
J2
-
1
for
( :;Jl-J2)
<
k2
k
T
L
This effect ively s ta tes that yie ld occurs when the component
in ei ther
the
t ransverse
direct ion
or the
longitudinal
direct ion of the shear s t ress
on
planes
containing
the f ibres reaches
a
c r i t i ca l value.
s an example, consider uniaxial
tension
in the
x
1
-direc t ion
of
a
rectangular block of material reinforced
by
s t ra ight para l le l
f ibres
in
the
(x
1
,x
2
) planes
and incl ined
a t an
angle to the
x
1
-axis .
Thus
0
a
=
( c o s ~
s i n ~
0)
which
gives
J
2
. 2 , ~ , 2 , ~ ,
2
=
1
1
s ln
't'
cos
t •
On subst i tu t ing these
in to (82)
we
find
that
f =
0
when
±2k k
L T
: :;
•
s i n ~
c o s ~
k ~
tan
2
4 k ~ )
Alternat ively , the
yie ld
condition
(83)
gives
in
th is case
_ {
±kL/sin > c o s ~
,
al l -
±2kT/sin
2
~
,
Both
(84)
and
(85)
give very good f i t s to experimental data with
appropriate
choices
of
kT
and
kL.
(84)
(85)
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6
A.J.M. Spencer
4.2 Yield functions for two families of fibres
Similar arguments can be
used in
the case of two
famil ies
of f ibres .
In th is case the
extra-stress
s sa t i s f i e s
the
conditions
t r s
= 0 , a .s .a 0 b.s .b = 0 , 86)
and it follows, af ter some manipulation,
tha t
{ t r o -
2 cosec
2
2¢ a .o .a - cosec
2
2 ¢ - 3 co t
2
2¢)b.o.b}a®a
.... .... .... -
87)
where
cos
2¢
= ~ · ~ · Then it i s assumed that
f
i s
a
function of s , a®a
and b®b, and by arguments s imi lar
to those
used
above
it follows tha t
f
i s
an i so t ropic
invar iant
of these tensors. I t then follows by standard
resul t s in
invar iant theory
tha t
f
can
be expressed as a
function
of
a .s
2
. a ,
J
4
= b .s
2
. b ,
cos cp ~ . : • ,
and 88)
I f the
two
famil ies
of
f ibres are
mechanically
equivalent ,
then f
must be symmetrical with respect to
interchanges
of a and b ,
and
then
dependence on J
2
and
J
4
can be replaced
by dependence on
J
2
+J
4
and
J
2
J
4
•
However
J
2
J
4
can be expressed in terms of J
1
, J
3
, J
5
, J
6
and cos
2
2¢, and
so it
may
be
omitted.
Thus
in
th is case
f becomes a
function of J
1
,
J
2
+J
4
, J
3
, J
5
,
J
6
and
cos
2
2¢.
The
most general quadrat ic
yield
function
i s
then
can
be expressed in terms of otl1er invar iants and omitted)
f
where
c
1
,
c
2
and
c
3
have the dimensions of s t ress and
are
functions of
cos
2
2¢.
89)
As an example consider yie lding of
a
rectangular block reinforced by
two
famil ies of s t ra ight para l le l
f ibres
so tha t
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Constitutive Theory
27
a cos
sin, 0) ,
b (cos< >, -sin< >, 0) .
(90)
By inser t ing
(90) in 88), and expressing s . . in
terms
of
a . using 87),
lJ lJ
t can
be shown that n
this
configuration J
2
+J
4
,
J
1
and J
6
are each
l inear
combinations
of
a ~
o;
3
and { a
11
-a
33
)sin
2
- (a
22
-a
33
)cos
2
}
2
•
Hence for
this
configuration the
yield
function (89)
can
be expressed in
the form
f
(91)
where
Y
k
1
and
k
2
are functions
of
< >which,
with
some
manipulation,
can
be related
to c
1
, c
2
and c
3
• The parameters k
1
and k
2
can
be in terpreted
as shear
yield
s t resses
for
shear on planes x
3
= constant in the x
1
and
x
2
direct ions
respectively. I f
Y
1
,
Y
2
and
Y
3
are defined
by
then Y
1
, Y
2
and Y
3
are yield s t resses in
uniaxial
tension in the x
1
, x
2
and x
3
direct ions ,
for th is
yield
condition.
Hence in principle k
1
,
k
2
and Y may be determined experimentally.
Suppose
the block
i s subjected to simple tension
P
along
an axis
defined by the
uni t
vector cos8,
s in8 0).
Then
~ P ( l
+cos 28) , 0
22
~ P l -
cos 28) ,
s in 28 ,
By
subst i tu t ing
these in (91) we find that
p
±
cos 28 - cos 2¢
2Y
(92)
93)
The same yield s t resses Y k
1
and
k
2
occur
in
problems
of deformations of
hel ical ly reinforced cylinders .
4.3
low rul s
To
complete a
theory
of plas t ic i ty
const i tu t ive
equations are
required. One common procedure for
formulating
these is to assume
that
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28
A J M Sper,cer
the
yield
function i s
also a plas t ic
potent ia l
function,
so tha t
the
components d ~ , of the plas t ic s t ra in- ra te dp are
given
by
1 ]
dP = AClf/Clcr
, ,
i j 1 ]
94)
where i s
a
scalar mult ip l ier (not
a
mater ia l
constant) .
In
a
r ig id
plas t ic
theory
dp
i s the
to ta l s t r a in - r a t ed with components d, . , so
tha t
1 ]
d ~ .
]
1
av.
av .
di j
= 2
d
d
J 1
95)
where v. are
components
of veloci ty . There are various jus t i f ica t ions for
1
(94); for
example, it may be deduced as a consequence of Drucker s
s t ab i l i ty postulate . The usual arguments leading to 94)
remain
val id in
the presence of kinematic const raints . I t is straightforward to
calculate
dp
from
94)
for
the yield functions
discussed in th i s chapter.
When f has
the
form (82),
the flow
rule 94) becomes
and th i s can be expressed in terms
of
cr by using
80) .
When
f has
the
form (89), the flow
rule becomes
where
:
i s now
given
by
(87).
The dp
obtained
as above automatically sa t i s fy
the
const ra in ts of
plas t ic incompressibi l i ty and f ibre inextensibi l i ty.
4 4
Hardening
rules
For
a perfect ly plas t ic i . e . non-hardening) mater ia l
the
yield
(96)
(97)
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Constitutive Theory
29
stresses, such
as
kL and kT in
82),
are constants . We now consider the
poss ibi l i ty tha t
the
material is work-
or
strain-hardening.
In
general the
hardening
proper t ies
of
a
highly
anisotropic
plas t ic
sol id wi l l be
complicated,
as even in the simplest cases e.g. 82))
several
parameters
are required to describe the current yield surface.
A
major simplicat ion resul ts
i f
t i s assumed tha t
the
current s ta te
of
hardening of the
material can be described
by a single parameter, which
we take
to
be the plas t ic work W , defined by
p
w
p
0 . d ~ 0
1 1
S ,
. d ~
. •
1 1
98)
This point of
view is doubtless
an
oversimplificat ion
of the
real
si tuat ion,
but
is
plausible
in
some circumstances -
for
example, i f
we
consider a f ibre- reinforced
composite
with an
isotropical ly-hardening
metal matrix. Then
t
is reasonable to assume tha t the s ta te of hardening
of the
composite i s
controlled by the s ta te of hardening of
the
matrix,
which in
turn,
for i so t ropic hardening, depends on a single
parameter
such
as
the
plas t ic work.
We therefore
assume
that the
yield
condition can be expressed in the
form
g O, .) = k W ) I
1
p
99)
where k has the dimensions
of
s t ress . e
further assume
that g O . . ) k
1
i s
a convex
surface in
o
. . space and
that
{g O
. . ) }
2
is
a homogeneous
1 1
function of
degree
two in 0 . . ( th is i s
not qui te
the same as assuming g
1
i s
homogeneous
of
degree one;
th is
formulation
avoids
ambiguities
which
ar ise when taking
square
roots) .
By the flow rule 94)
p .
d , , E:dg/30, , lQO)
1 1
and
E has dimensions t ime)-
1
• Then
w
p
EO. ,dg/30. 0
Eg O . . )
Ek W) I
1 1 1
p
101)
by
Euler s theorem for
homogeneous
functions.
Equation
101) es tabl ishes
a correspondence between Wp and E; i f k is an
increasing
function of Wp
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30 A J M
Spencer
th i s correspondence i s one-to-one.
Hence
k may be regarded as a funct ion
of
E rather
than of
W . Also E
has the dimensions of
a s t ra in- ra te ,
and
p
so
may
be
regarded
as
an
equivalent
s t ra in- ra te .
To
obta in
an exp l i c i t
expression
for
E
it i s necessary
to
solve 100)
for a
. in terms of d ~ .
_
J
_
J
and E.
Then subs t i tu t ing for J
in
the yield condit ion gives E as a
l.J
funct ion
of d ~ . .
l.J
In prac t ice it may be
d i f f i cu l t
to express the yield condit ion
in
the
form
99) .
For
example th i s
i s the case with the
yield
funct ion
82) when
k and k are regarded as functions of W or E. I f we simplify fur ther
T L · p
and
suppose tha t
the r a t io kT/kL = a remains cons tant th is
assumption
can
also
be made
plaus ib le) , then
the yield condit ion
corresponding
to 82)
takes
the form
102)
which i s of the
form 99) . Then E
can
be related to
the
s t ress- ra te by
k
T .
E
d
Also from
100)
and
102)
and
equivalent ly to
96) when
kT akL)
d ~ .
l.J
and from
102)
and
104) it
follows
af te r
a l i t t l e
manipulation
tha t
E
2 { d ~ . d ~ . + 2 a -
- l ) a . a . d ~ k d ~ k } .
l.J l.J
_
J
_
J
103)
104)
105)
Some simpler problems for
strain-hardening
materials wil l be considered
in
Chapter
IX
inconnection with
dynamic
problems for beams
and
pla tes .
4 5 Small
elastic plastic deformations
So fa r we have considered
only
r ig id-plas t ic materia ls .
In
construct ing
an
e la s t i c -p la s t i c
theory
we
r e s t r i c t
at tent ion to
problems
of small
deformations.
Then
the
usual procedure, which we follow, i s to
assume
tha t
the
s t ra in- ra te tensor d
i s
the sum
of an
e las t i c s t ra in- ra te
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onstitutive Theory
de and
a
plas t ic s t ra in- ra te dP, so
that
d
d
1
d ~ .
1
1
3
106)
We consider tha t the e las t i c s t ra in- ra te
depends
l inear ly on the
s t ress - ra te and so
the
const i tut ive equations for
the
elas t ic s t ra in- ra te
are
analogous
to the l inear
e las t i c
s t ress -s t ra in relat ions
descr ibed
ear l i e r
and require no
fur ther
discussion.
For a composite comprising a duct i le matrix reinforced with e las t i c
f ibres
t i s
plausib le
tha t
only
the
e las t i c s t r a in
contributes to
the
volume
change and
to
the
extensions
in
the
f ibre direc t ions.
Consequently
the plas t ic
s t r a in
involves
no volume
change or
f ibre
extensions. I f the
flow
ru le 94)
i s
adopted,
th i s
means that the yie ld function f must
be
a
function
of the
invariants 81)
for
a
single
family
of
f ibres
and
88)
for
two
famil ies of f ibres .
Thus the yie ld function takes the same form as
in
the r ig id-plas t ic theory, and dp i s
given by
the flow ru le
94),
so tha t
the resul ts given
above
for
the
r ig id-plas t ic theory
apply provided
that
dp i s interpre ted
as
the plas t ic par t of
the
s t ra in- ra te .
REFEREN ES
[1] SPENCER
A.J.M.,
Theory
of
invar iants ,
in Continuum
Physics
Vol.l
Eringen, A.C.,
Ed.,
Academic Press, New York 1971) 239-
253
[2]
SPENCER A.J.M.,
The formulation
of
const i tut ive
equations for
anisotropic sol ids in Comportement mecanique des solides
anisotropes Colloques
internationaux
u C.N.R.S. No.295),
Boehler, J-P. and Sawczuk, A., Eds., Editions
Scient i f iques
du
C.N.R.S., Paris
1982)
3-26
[3] MARKHA.t-l l4 .F . Composites
1 1970)
145-149
[4] SPENCER
A.J.M.,
Cont-inuum Mechanics,
Longman, London,
1980
[5]
ERICKSEN,
J.E.
and RIVLIN,
R.S., J. Rat.
Mech
Anal.
3 1954)
281-
301
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3 A J M Spencer
[6]
SMITH
G . ~ . and RIVLIN,
R.S.
Trans.
Amer Math
Sao.
88
1958)
175-193
[7]
GREEN
A.E.
and
ADKINS
J .E.
Large
E ~ s t i o
Deformations
Clarendon
Press
Oxford 1960
[8] LANCE R.H. and
ROBINSON
D.N.
J. Meoh
Phys. SoLids
19 1971) 49-60
dditional bibliography
PIPKIN, A.C. Fini te deformations
of
ideal f iber- re inforced composites
in
Composite MateriaLs VoL.2
Sendeckyj G.P. Ed.
Academic Press New York 1973)
251-308
PIPKIN,
A.C.
Advances
in
AppLied Mechanics
19 1979) 1-51
ROGERS T.G. Anisotropic elas t ic and
plas t ic mater ia ls in
Continuum
Mechanics
Aspects of
Geodynamics and Rook Fracture
Mechanics Thoft-Christensen P. Ed. Reidel
Dordrecht
1975) 177-200
ROGERS
T.G.
Fini te
deformations
of strongly anisotropic materials
in
TheoreticaL RheoLogy
Hutton
J .F . Pearson
J.R.A. and
Walters
K.
Eds. Applied
Science
Publishers London
1975) 141-168
SPENCER
A.J.M.
Deformations
of
Fibre reinforced
MateriaLs
Clarendon
Press
Oxford
1972
SPENCER A.J.M. The formulation
of
const i tu t ive
equations
in continuum
models
of
f ibre-re inforced composites
in
Proceedings
of
the Third Symposium
on
Continuum ModeLs
of Discrete
Systems
Kroner
E. and Anthony K-H. Eds. University
of
Waterloo
Press
1980) 479-489
SPENCER
A.J.M.
Continuum models
of
f ibre-reinforced mater ia ls
in
Proceedings
of
the InternationaL
Symposium on the
MechanicaL
Behaviour
of Structured
Media Selvadurai
A.P.S. Ed.
Elsevier Amsterdam 1981) 1-26