Archive of Applied Mechanics 65 (1995) 161 177 9 Springer-Verlag 1995
The formulation of constitutive equations for fibre-reinforced
composites in plane problems: Part II
O.4. Zheng, J. Betten
161
Summary From the continuum mechanics points ofview, most of commercial fibre-reinforced composites
(FRCs) can be considered to be anisotropic materials with one of the five material symmetries: transverse
isotropy, orthotropy, tetratropy, hexatropy and octotropy, as illustrated in the preceding paper (Part I)
[1]. No properly general formulation of constitutive equations for tetratropic, hexatropic and octoctropic
types of FRC has been found in the literature. In this paper, the restriction to the admissible deformation
of a FRC imposed by the failure strains of the fibres is investigated. The complete and irreducible
two-dimensional tensor function representations regarding tetratropy, hexatropy and octotropy derived
in Part I are applied to formulate constitutive equations for FRCs in plane problems of elasticity, yielding
and failure in the present work, and of heat conduction, continuum damage and asymmetric elasticity
in a continued work (Part III, forthcoming).
Key words Constitutive equations, composites, anisotropy, failure, tensor function
l
Introduction
An advanced fibre-reinforced composite (FRC) [2, 3] is comprised of continuous aligned highly stiff
and strong fibres (carbon or graphite, glass, boron, aramid) which are quite densely (typical volume
fraction about 50-60%) embedded in a relatively compliant matrix (plastics, metals, ceramics). Thus,
a unidirectional fibre composite occurs usually very strong directional properties; and for service in
engineering applications where multi-axial stress exists, fibres in FRCs should be arranged at angles. Most
of commercial FRCs is laminates that are stacked with unidirectional fibre plies at angles 0 ~ + 30 ~
_+ 45 ~ _ 60 ~ and/or 90~ and they possess one of the five macroscopic (or phenomenological, with
a homogenization technique) material symmetries: transverse isotropy, orthotropy, tetratropy, hexatropy
and octotropy, as illustrated in Figs. 1-4 of Part I. For instance, cross-ply laminates (e.g., [0;/90~ and
symmetric angle-ply laminates (e.g., [ + 0]s, [0~ 45~ are usually orthotropic, while laminates [0~176
[ + 45~ and [Off _+45~
are tetratropic; laminate [0~ 45~176 L is octotropic; and laminate [0~ 60~
is hexatropic. Unidirectional fibre composites are transversely isotropic in spatial problems and treated
as either othotropic or isotropic in plane problems as the fibre direction is either parallel or perpendicular
to the normal direction of the plane. In this paper, a unidirectional fibre composite in plane problems
being transverse to the fibre direction is called a T-isotropic FRC.
As is well known, the theoretical and experimental researches on thermomechanical behaviour of
FRCs have mainly focused on plane problems, involving plane stress, plane strain, plane heat flux, plane
continuum damage problems and so on, since the configurations of FRCs in engineering applications
are usually either plates or shells. In contrast to the fact that there are five types of material
Received 1 March 1993; accepted for publication 6 October 1994
Professor Q.-S. Zheng
Department of Engineering Mechanics, Tsinghua University,
Beijing 100084, P. R. China
Prof. Dr.-Ing. J. Betten
Lehr-u. Forschungsgebiet Math. Modelle in der Werkstoflkunde,
RWTH Aachen, Templergraben 55, D-52o56Aachen, Germany
The supports from the Alexander yon Humboldt Foundation,
Germany and the Research Foundation of Tsinghua University,
P. R. China are acknowledged by the first author.
162
symmetry with respect of FRCs, most authors, however, refer both hexatropic and octotropic FRCs to
as being quasi-isotropic ones, and do not distinguish between tetratropic and orthotropic FRCs. Such kind
of rough classification of FRCs does not reflect the nature of FRCs except for very limited and simple
mechanical models such as linear elasticity. Up to now no properly general modeling of complex
mechanical behaviour of tetratropic, hexatropic and octotropic FRCs has been found in the literature.
In this paper, the restriction to the admissible deformation of a FRC imposed by the failure strain, say
~ , of the fibres is investigated. It shows that for both hexatropic and octotropic FRCs the assumption
"inextensibility" [4] as a kinematics constrain is not available and the admissible strain can be
only of the oder 0 ( ~ ) . Study on the complex thermomechanical phenomena in plane problems of
T-isotropic, orthotropic, tetratropic, hexatropic and octotropic FRCs requires a rational fomulation of the
constitutive equations of these kinds of anisotropic advanced material. We perform such formulation
with respect to elasticity, yielding and failure in this paper, and to heat conduction, continuum damage
and asymmetric elasticity in a continued paper (Part III), by way of the complete and irreducible
two-dimensional tensor function representations presented in Part I. The readers may also refer to
another recent work [5] for the complete and irreducible two-dimensional tensor function representations
relative to all kinds of anisotropy [6].
Denote by a~ and a2 the two privileged orthogonal directions, as explained in Sect. 1 of Part I, of the
FRC considered and introduce the following two second-order symmetric tensors:
M=a~|174
N=al|174
r
(1)
According to the Remarks, i.e., Sect. 5 of Part I or the work of [5], we can redefine the basic tensors T,
H and C in the representations given in Tables 1 and 2 of Part I as:
T=M|174
H =M|174
(M|174
+N|174
N|174
(2)
C=M|174174
+N|174174
+M|174174
(M|174174
+N|174174
+ M|174174
+N|174174
+N|174174
For any plane second-order symmetric tensor A, introduce the abbreviations:
(TA)~:-- T~p~oA~,
IA1 = trA,
(HA)~p=H~7~.~A~Au~
, (CA)~: = C~:v~.~r162162
14 2 = 89
]A2
JA1 = tr(MA),
=
(3)
89
]A4 = ~tr(CaA)"
Ja3 = 89
In this paper, tr indicates trace, lower case Latin indices i, j, k . . . . . are running from 1 to 3 while lower
case Greece indices r fi, ~. . . . . from 1 to 2, and th e Summation convention is employed. Components
of vectors and tensors are referred to a rectangular Cartesian coordinate system {Xi} that the
X3-coordinate is either normal to the fibre plane in orthotropic, tetratropic, hexatropic and octotropic
cases or parallel to the fibre direction in T-isotropic case. The prefix term "plane" means the X, OX2 plane,
i.e., the X3-coodinate plane. From (3) one can easily obtain the following identities:
~A
~]~1
-1,
~A - M ,
--=A;
~A
~]A2 _ T~,
~A
~L3
~A
-- H a,
~L4
~A
(4)
= CA;
and
4Ia2 = -/il q- 2j2~ - 2Ja2"
(5)
Denote by e i the Xi-coordinate directions and introduce E~ = e~ | e~ - e2| e2 and E 2 = e, | e2 + e2 | el.
Since the tensors M, T, H and C are unaltered with respect to + a2, we may assume that {a~,a2} and
{e I, e2} are co-oriented and there is an angle 0(0 =< O < 27Q so that
Oaz;
e~ = cos 0a~ + sin 0a~,
e 2 = -- sin 0a~ +
E: = cos 20M + sin 20N,
E 2 = -- sin 20M + cos 20N.
cos
(6)
We adopt the new definitions (2) for T and C respectively rather than the original
of Part I, because we can write the following simple and unified expressions:
M = cos 20E~ -- sin 20E 2,
ones (2.1a)
and (z.lc)
]at = ACOS(20 + ~/a);
T A = A [cos(40 + ~A)E1 -- sin(40 + ~a)E2],
2]a2 = AZcos(40 + 2~a);
H a = A2[cos(60 + 2 ~ a ) E ~ -- sin(60 + 2~A)E~],
3]a3 = a 3 c o s ( 6 0 + 3~a);
CA = A 3 [cos (80 + 3~tA)E 1 -- sin (80 + 3~A)E2],
4]a 4 = A4cos(80 + 4~A);
163
(7)
where A > 0 and 0 < ~A < 2~Z defined by
trAE 1 = All
- - A22 =
A cos SA,
(8)
trAE z = 2A~2 -- A sin ~A"
2
Deformation and kinematics restriction
Suppose that a typical particle of the FRC that possesses initially the spatial point with the position
vector X~ moves currently to the spatial point with the position vector x~ = x t (X 1,X 2, X3). Denote by G~jthe
Green deformation tensor and by G = (G~p) its plane part. They relate to the deformation gradient
F 0 = ~xi/~Xj as follows:
Gij=
FmiF,,O,
(9)
G~a = Fm~Fm~.
Let 221,22~(with 2~, 22 > 0) be the two principal values of G and N~, N 2 the corresponding two principal
directions. The stretch, say 2~, in any plane direction N(q~) = cos q~N1 + sin cpN2 can be related with 21,
22 and ~o in the following way:
)~2 = &~2cos 2 cp + 2~ sin 2 qg.
(lo)
Conversely, 21 and 22 can be expressed in terms of stretches in any two plane non-collinear directions.
In particular, we have
2
2
22 = 2z~ + (2~2 - 22)tan
q~,
(11)
22{ = 22 + ,%+~J2
~2
~2 +=/2)sec 2~p.
+ (2~2 --/re
The shear strain, y~, of the two orthogonal directions N(~) and N(~, + hi2) can be expressed in the form:
-
]
?~ = arctan ~ 22122 sin 2 ~ .
(12)
From (11) and (12), it yields immediately the maximum and m i n i m u m plane stretches and shears as
follows:
2
2m~,=max(2v22)~
2mi~l= m i n (21,)o2)J '
~
7re,x;__ + arctan - Ymi~)
-221)'2
2
2'
= + arcsin
-]2~ + )~ I"
(13)
In a FRC with typical volume fraction about 50 ~ 60%, fibres in the FRC play a dominant role to
sustain applied loads and contribute to the strength of the FRC. Thus, the failure strain ey, (about 1.4 .-~ 2.0%
for carbon fibres, 1.8 ~ 5.2% for glass fibres) of the fibres of a FRC imposes restriction to the admissible
deformation of the FRC in the form:
[2~Ur~-- 1t =< e~,
(14)
where )'~b~edenotes the deformation stretch in a fibre direction. The difference between the tensile and
compressive failure strains of the fibres is assumed to be negligible here for the sake of simplicity.
Consider a hexatropic FRC which is reinforced with fibres in 0 and _+ 7rl3 directions. Since the failure
strain el, of the FRC imposes the restriction (14) for every fibre direction, in the light of the geometric
sense of the deformation ellipsis of G, it can easily be proved that the maximum and minimum
admissible plane stretches, denoted by Areax and Amin respectively, may occur as ~o= ~z/6.Inserting q~ = n/6,
2~ = 1 +_ el. and 2 2 = 1 -T-e~ in (11) and (13) yields
Ama*;=x/l+i(513)ef~+e}~'
Ami. j
164
/'max~ = + arcsin [
(1 + 5/3)ey~
]
F~i~J
1 + (1 - 5/3)el, + e~,
2 ,
(15a)
where Fm~~ and Fm~~ denote the maximum and minimum admissible plane shear strains.
In analogous, for an octotropic FRC reinforced with fibres in 0, re/2, and + rt/4 directions, the
maximum and minimum admissible plane stretches Am,~ and Ami~ may occur as (p = rt/8. Inserting
(p = re/8, 2~ = 1 _+ ef~ and 2~+~a = 1 -T-el. in (11) and (13) yields
Amax'~ = ~ 1 + 2x/2ef. + e~,
AminJ
F~.,~ = + arcsin[1 + (1 + ~ ) e f ~
FrninJ
(1 _ x/~) ef~ + e.~]"
(15b)
Two significant consequences of (15) are:
(i) admissible plane strains of both hexatropic and octotropic FRCs are of the order O (sf~), and
(ii) the kinematics constraint "inextensibility", i.e., )'~b~ = 1, in fibre directions is available for neither
hexatropic nor octotropic FRCs, although it is often assumed in theoretical researches of mechanical
behaviour of unidirectional fibre composites and cross-ply laminates [4].
We note also that for a FRC reinforced in 0 and _+ 0 directions which is orthotropic (e.g., [0/___0]~),
the restriction in a similar form to (15) still holds if the coefficient 5/3 in (15a) or ~/2 in (15b) of the linear
term in ef~ is replaced with 1 + 2 cot2O. Furthermore, one can prove that if a FRC is reinforced regularly
in m( = integer > 3) directions, i.e., in 0, glm, 2rclm..... (m - 1)~lm directions, then the restriction is
of the form:
Amax~ = x/1 + 2km~f, + e}~, Fm~ = + arcsin [
Amin3
--
/~minJ
(l+km)%
]
(16)
i + (1 -- k~)sf~ + 8}. '
--
where {km:m = 3,4,5 .... } with
1 + 2 tan2(~/2m)
k~ =
I
(sec(n/m) = 1 + i
(as m is even)
(17)
2tan 2(n/2m)
-
(as rn is odd)
tan 2(n/2m)
is a constant series which approaches monotonously to 1 as m ~ oe. One may rough refer such FRCs
to as being quasi-isotropic, if the reinforcements are also equivalent in these rn directions.
Detailed classification of the material symmetries of these FRCs is investigated in [6].
In contrast, the failure strain el, imposes no restriction to the off-plane stretch and shears:
9
23= ~ 3 3 ,
f G,3
7~3= arcsm /
,
~/ Gll G33
723=arcsin
/G23
-- ,
~/G22G33
(18)
in orthotropic, tetratropic, hexatropic and octotropic cases. For instance, a fibre-reinforced plastics
allows normally quite large off-plane strains )'3, 7~3 and 723"In T-isotropic case, with the failure strain
~fu only we cannot anticipate similar restriction to (15) or (16), but have [).3 - 1 [ =<~f~. For cross-ply
laminates, the afu supples no restriction to the shear strain of the two orthogonal laminate directions and
the minimum stretch Am~, but imposes to the maximum stretch the restriction Am~X< x/2(1 + 8~,).
There are many kinds of fibres for which the difference between the tensile and compressive failure
strains 8f+ and ~f~ is significant. We can obtain the generalized form of (16) as follows:
--
/'a~]
~Fminj____ --+ a r c s i n
m
2
1
+
--
'
+2
(19)
--2
4 4 ( 1 _km)(2efu2 -+- - 2gfu
- - ~ - +efu
- 7 7 ~ -gfu )+- - ~ -(2 ~~;fu-28fu
:-- +g;2+gfuz5
"
In the sequel, we restrict our attention to the following general plane deformation:
X 1 = X 1 (Xl, X2) ,
x 2 = x2 (Xl,X2):~
x 3 = x3 (X3) ,
(20)
so that
/7++= &~/ax~, G =/7++= 0;
/7~, = a x j a x ~ ,
(2~)
G~t3 = Fe~F~#,
G33 = F~Z3, G3= = G~3 = O.
3
Elastic response
To describe the elastic response of a continuous material, it is most widely assumed the existence of
strain energy function W = W(eo) or complementary energy function W~= W~(to) (i.e., W + W ' = t~e o)
so that the stress-strain relation can be expressed in the forms:
W
to = - - ,
eo
~W ~
= --,
e~ ~ t o
(22)
where e~ is the Green strain tensor and t~ the second Piola-Kirchhoff stress tensor:
eij
(Gij
--
50)/2,
to= d ' - i / 7 ikFjlffkl ,
(23)
~..,jthe Kronecker's symbol, a 0 the Cauchy stress tensor, j = P/Po = det(~j), and Po and p the initial and
current mass densities. Denote by e, ~r and t the plane parts of eo, a 0 and t~j. In plane problem (20), it
is evident t h a t j = (F~F22 - Fa2F21)F33,
e=(FTF-1)/2,
% = ( F ; 32- - I ) 1 2 ,
e3==%=0;
t =j-1FTo'F,
e33 --- - j
0-~3~-- 0-3~ =
(24)
.--1
2
F330-33,
0;
and W(e, %), W ~= W~(t, t33).
According to Table a of Part I, as scalar-valued functions of a single symmetric tensor e or t and
a parameter % or t~3 the strain energy or complementary energy functions of FRCs in plane problem
(2o) can be expressed in the following most general forms:
Wr.i~ o = Wo(I~pI~ve33),
W~:_i,o = W~o(Itl, It2, t33);
wo.,~o= w]G~,G, LI, e . ) ,
W~o.~o --
~(I,,,I,+,]t~,. t.);
c
c
= G(I,.GL,
w,o.~ = w2G~,L,L, e33), %.~
t3~),,
(25)
wG~ = w~G. G].+. t.);
Wocto =
W4(/eI, Ie2'/e4' e3+),
c
W~o,o = W~(I., It2,s t33),
where the abbreviations/el . . . . . Jr4 are defined in (3). Thus, substituting (25) into (22) and using the
identities (4), we can immediately arrive at the following stress-strain relations:
~Wo
tr-i~~
~I~l
~W~
1
~W~e
~G1
~w~
aw~
er-i+~= ~/tl 1 +-~-t2 t;
r)Ie2
t~176
ttetra-= ~/el
a~
~Wo
+ 7e,
aW~
+~-1M'
aw~
q- ~-e2 e q- ~-~e2Te ,
aw~
.1
e~176 = 7 clo
aw~
aw~
aw~
+ ~ t 2 t +~-~-tl M;
, aw~.
aw~
etar, -- g~tl 1 t 7S--.
+It2 t + ~ ' ~ Tt;
(26)
165
th'~ = ~
1 + - eaI~ + -----H~
eL3
ehe~a-- ~t~ 1 + 7~/~
t + -~]t~
-Ht;
e octo
-
t
-
c'
Linear elastic response of FRCs is of particular interest. Suppose that both W and t..,jvanish as e~ being
zero, and both W~ and % vanish as t~j being zero. From (25) and (26), for all %isotropic, hexatropic and
octotropic FRCs we have the following formulation:
166
W=~Ie] + 2#I~2 + 2'Iae33 + ( ~ +
#')e]s;
/
t = (2I~ + 2'e33) 1 + 2#e,
t
t33----- /el + (2 + 2#')e33 ,
(27)
t33/ / 2'
2'
2'+2#'
t~zJ L 0
0
0
00 /e33/~e22t;
2g ~e~2j
and
EIcW = -~Is
v)It2-- v'lt~t~3 + ~t~3;
+ (1 +
Ee = - (vI, t + v't33) 1 + (1 + v)t, Ee33 =- - v'Ita + ~t33,
(28)
ken)
0
0
1 + v.] It@
with 4 independent material constants, i.e., the Lam6 moduli A, A', ~, t~', or equivalently, the Young's
moduli E, E' = E/~ and Poisson's ratios v, v'. The relations between these two sets of material constants are:
22=
2# -
~E
(1--V)r
E
1 + V'
E
'2
2//' =
l+v'
2'-
v'E
( 1 - - V) ~ -- 2V'2'
(1 - v - v')E
(1 -- V)~ -- 2V'2'
(29)
E' = (2 + #)(2' + 2#') -- 2 ,2
E = 4/z [(2 + #) (2' + 2#') -- 2 ,2
2+#
(2 + 2#)(,;[' + 2#') -- ;t'z '
V --
2(2' + 2p') - 2,2
(2 + 2/2)(2' + 2#') -- 2 '2'
V~ ----
22'#
(2 + 2#)(2' + 2 # ' ) - - 2 '2"
Noting the relation (5) so that 1e21and 1~ can be replace by]~z and ]t2 respectively, for orthotropic
FRCs we have
W=~t~l+2iale2+2Ie~e33+
+I1' ~3+#(aJez+(~lJej+o(J~2%);
t = (2Ie~ + )Je33 + ~#]el) 1 + 2#e + ~#T~ + #(cs
t~3 = 2'Ie~ + ('~' + 2#')e33 + ~'#Jel,
+ ~'e33)M,
{
}['2+(2+c~+2~)#
3. - ~12
01
2' + ~'#
2 + (2 + a - 2 ~ ) #
2 ' - - a'#
t33
2' - a'#
2' + 2#'
t12
0
0
(30)
le3'
2 (1 -- o:) I2 (e~2 j
and
1
~
-
!
~
167
E e = -- (vlta + v't33 + fi].)l + (1 + v)t - a T t - (filt~ + a't3~)M,
a2,
0]I,11
Ee33 -= _ v'It~ + ~t33 - a'Jt p
Ede22~ = I
(e~a)
L
--v+a
+
1--a+2fi--v'+a'
0
0
0
0
t~2
o
q2
l + v + 2a
(3~)
l
with 7 independent material constants (2, 2', #, #', ~, 4, ~', or equivalently, E, E' = EI{, v, v', a, e/, a'). The
relations between these two sets of material constants are:
[(1 + 3 v - - 4 a ) ~ - - 4 a ' 2
42-
+ 2v'2]E
z[(1 -- v)~ -- 2v'2]E
4# -
_~.
[(1 -- v ) ~ - - 2v'Z]E
E
1 + v+2a'
E
-t- 1 + v + 2 a '
2'=
[(1 + v -- 2a)v' + 2fia']E
[(l+v--2a)(1--v--v')--2fia'--4fi']
2#' =
_,~
,
E
1 +v +2a'
[2t/v + a'(1 -- v)]E
t
(a'v' + fi~)E
$2E = 4(1 -- ~2)[()v _}_]~)(~/ _}_2#') -- .~,2] __ 4(1 -- C~)12[~,2 (2 + #) + 2a'~2' + ~2 (;f, + 2#')],
E'=
(1+~)[(2+12)(2'+212')--2'2]--[~'2(2+12)+2~'~2'+~z(2'+212')]#
(1 + ~ ) ( 2 + # ) _ a 2 1 2
$2v = (2 + c~2# -- :~212/2) (2' + 2#') -- 2 '2 -- [~'z(2 -- 12 + 2cr
,
(32a)
+ ~'~2'] 12,
$2v' = 2(1 - cr [(1 + ~ ) 2 ' - - :(~#] 12,
.Qa = c~[(2 + #) (2' + 2p') -- 2 '2] -- [c( 2 (2 + #) + 2a'~2' -4- ~2(2' + 212')] #/2,
.f2fi = (1 -- ~) [~(2' + 212') -- a'2'] #,
t2a' = 2(1 - ~)[a'(,~ + 12) - ~2']/~,
in which we have used the abbreviations
~ = (1 + v - - 2 a ) [(1 - v) ~ - - 2v'21 - - 2 [ a a ( 1 - v) + 2~2~ + 4c/a'v],
(32b)
.(2-=--(2 + 2# -- azfi -- d2~12) (2 ' if- 2#') -- 2 ,2 -- [~'2(2 + 3# -- 2~#) + c(~2'] #/2.
Setting ~' = c~= 0 and a' = fi = 0 in (30)-(32) yields immediately the corresponding results for
tetratropic FRCs as follows:
W=}
r162
e~3+:~,ha;
+#'
t = (2Ie~ + 2'e33)1 q- 2/ze + or
t33 = J['lel q- ()/ q- 2#')e33 ,
(33)
168
t221= / /~--~#
t33j [
,~t
t12
2 + ( 2 +c~)#
2'
%;
2"
2 ' + 2#'
[e33j
0
0
0
2(1-c~)#
et2
and
Ewes_ _~2 _1_(1_}_v)ii2_~,itlt33.q_~3_a]t2;
Ee =
-- (Fit1 -4-
,e11,
V't33)1 + (1 + v)t -- aT t,
E~e=t =
--v+a
1--a
-v'
~e33[
-v'
-v'
~e12)
0
0
Ee33 = -- v'Ita + ~t33,
0 ]{ttl
0
t22 ;
~
0
[33
0
1 + v + 2a
t12
(34)
with 5 independent material constants (2, 2', #, p', c~, or equivalently, E, E' = E/i, v, v', a). The relations
between these two sets of material constants are:
22=
{E
( 1 - - v ) ~ --21/2
(1 + v)E
2#--(1+v)2
4a 2'
(1 + v)E
(1 + v ) 2 - - 4 a 2'
2'-
v'E
( 1 - - v ) ~ - - 2 v '2'
(1 -- v - - v')E
2/~'=(l_v)~_2v,2,
2a
= 1 + v'
E=
4(1 -- 0r [(2 + #) 0[' + 2#') --,~'2] #
(2 + 2# -- ~2/z)(2' + 2#') -- 2 ,2 '
(35)
E'=
(2 + ~t)(2' + 2#') -- 2 '2
2+#
)
(2 + o~2#) (2 ' + 2#') -- 2 '2
- a~);~'u
V= (2 + 2# -- 2a2#) (2' + 2/f) -- 2 '2, Vr= (2 + 2it --2(12~2p)
(2' + 2/f) -- 2 ,2,
a=
c~[(2 + #)(2' + 2#') -- 2'21
(2 + 2/~ -- 2or (2 ' + 2/~') -- 2 ,2.
We emphasize that in (30), (31), (33) and (34), the X 1- and X2-coordinates are specially taken in the
two priviledged directions al and a2.
The discussion given above shows that from the observation of linear elastic responses of FRCs one
cannot distinguish among T-isotropic, hexatropic and octotropic cases. To characterize hexatropy and
octotropy in terms of stress-strain relations, at least the second-order and third-order effects respectively
have to be taken into account, as explained below by way of the complementary energy functions.
Expand W~ = W;(I,I, It2, t33) in T-isotropic case as a polynomial:
W~ = 1~o0}+ D~(2) + 1~o{3) + . . . ,
(36)
where
E~(1) --
~It 2 ~-(1-[-,')Jt2--~'ItxtB3-[-~t~3,
(37)
EWg(2 ) = b,I~ + b21,1t(, 4, (b3t~ 4, b41a)t33 + bs~3 4. b~t~3,
and so on. We summarize from (25)-(37) that the lowest order characteristic complementary energy
functions in the plane problem (2o) of T-isotropic, orthotropic, tetratropic, hexatropic and octotropic
FRCs are of the forms:
w;;,o = wLI,
WCortho= W~(1) -- (aJt2 4, a/tllu 4,
a'Jt~t33),
WCtetra= W0(1) -- aJ~2,
(38)
W~he~ = Woo >+ W;(23- bit3,
%,~ =
+ W;o(2>+ w & -
dL.
The material constants both d and a', a, b and d are the basic characteristic ones of orthotropic, tetratropic,
hexatropic, and octotropic FRCs, respectively.
4
Determination of material elastic constants
For simplicity, we expalin the determination of elastic constants of FRCs in plane stress and small
deformation problems. Thus, we may replace (t, t33) and (e, %)in (25)-(38) with the Cauchy stress
(O', 0"33) with ~33 = 0 and the infinitesimal strain (~, e33) respectively, where
eli = (Fij 4- Fji)/2 -- ~Sij, or
(39)
g.l~ = (Fo~ + Fa~)/2 -- c~.~,
g33 = F33 -- 1,
83~ = e~3 = O.
From (37)-(38) we can write the characteristic stress-strain relations for FRCs as follows:
Egr_is o
=
(1 + v ) o ' - - vI,r
Egorth o = (1 + v)(r-- vI~al -- a t e - - die, M ,
Egrets.,, = (1 + v ) t r - - v I # a l -- aT,,,
(4o1
E~h,= = (1 + v ) t r - - v I ~ : l + (3b~I2: + b 2 / j 1 + b 2 t , ~ t r - b H , ,
Ego, o = (1 + v) a - - v/#~l + ( 3 b l I ~ + b2I#2 + 4d~I3~ + 2d2I~I~2) 1 + (b2I,~ + d212~ + 2d3I~2) ( r - dC,.
Let the X~-coordinate directions e~ be related to a~ in accordance with (6). Denote by go = q~, el = %
and 7o = 2q2 the plane tensile strains and shear in response to the FRC being sustained a simple tension
stress tr = 0"0el@ el, as illustrated in Fig. 1. From (4o) and with the aid of (7), we can relate these strains
and shear to 0"o in detailed forms, i.e.,
Eg~ -- l,
(TO
--
= -- v,
0-0
E];~ = O,
(41)
GO
in T-isotropic case;
Eeo
- - = l -- d c o s 20 -- acos 40,
fro
Eg~
- -
GO
- -- v + a cos 40,
ETa
O-o
- 2 a sin 40,
(42)
169
(1+~~/~o
(~0 /
//
X2
/
//
X2,
.
/ ...........
--/" ' 1
o
17o
X~
X1
Fig. la,b. A FRC specimen in directional simple tension: a the specimen made of a FRC, b simple tension
in orthotropic case;
Ee__s
Eel_
Go
v+acos40,
(70
EY~
(43)
(Y0
in tetratropic case;
Ee--~~
3b1+
-bcos60
O"0
ao,
--=-v+
O"0
bcos60 ao,
3b1+
2- +
O-~
(44)
in hexatropic case; and
Ee--~~
1~o= + 3(b' + ~) a~
(4dl+2dz+d3-dc~176176
(45)
Ee_~=tyo_v+(3bl..~)~7o+(4d1Jvd2+dcos80)(72o ' E~O(To= 2d(sin80)a2o
in octotropic case. Shear strain accompaniment, as shown in Fig. l(b), or analytically, in (42), (43), (44)
and (45) for off-fibre directional simple tension tests of FRCs is a typical phenomenon differing from
that of isotropic one, as shown in (41).
Furthermore, considering the simple shear strain 70 = 2212 in response to the simple shear stress
--=z0E 2 for 0 = 0, we have
EY----~~
= 2(1 + v)
in T-isotropic and hexatropic cases,
"CO
EYe~= 2 (1 + v + 2a)
in orthotropic and tetratropic cases,
(46)
170
E,__~
0v = 2(1 + v) + 4(d 3 - 4d)z 0 in octotropic case.
"~0
Finally, based on (41)-(46) we can experimentally determine, as shown in Table 1, all the elastic constants
appeared in (40).
For a given value of a o with respect to tetratropic, hexatropic and octotropic FRCs, the mechanical
evidence that the tangent Young's moduli E o = daoldeo should have greater values in fibre directions than
that in off-fibre directions indicates that the symbolic plots of (Eo, 0) in a polar coordinate system should
be regularly convex closed curves, as delineated in Figs. 2(c), 2(d) and 2(e). This observation yields
that the material constants a, b, and d should be positive. For comparison we provide also Figsi z(a)
and 2(b) in T-isotropic and orthotropic cases, respectively. In the latter case g, irE/2 < E0, is positive.
The readers may consider more constraints to the elastic material constants imposed, for instance, by the
first and second thermomechanical laws.
Table 1: Determination of elastic constants of FRCs by way of simple tensile stress and simple shear
stress tests
Type of FRC
Experimental relations
Determined material constants
T-isotropic
Eg o = O"0
ET0 = 2 ( 1 + v)%
Eg o = (1 -- a -- 2a)a 0
EG/2 = (1 -- a + 2a) Gl~
EGu = (1 + a ) G i 4
E
v
(1 - a - 2 d ) l E
(1 -- a ) / E
Ez o=2(1
v
orthotropic
tetratropic
E
+ v + 2a)%
Eg o = (1 -- a ) a o
EGt4 = (l + a ) G l ,
E'c0 = 2(1 + v + 2a)'c 0
Eg o = a 0 + (3b I + 3 b J 2 - b)r
Ez o =2(1 + v)z o
EGn = a.n + (3bl + 3b2/2 + b) o'2=/z
E e I = -- v a o + (3b, + b J 2 + b)a20
hexatropic
octotropic
v
E , 3 b 1 + 3b212 - b
v
b
b,
E e o = a o + 3 (b~ + b212)a2o + (4d~ + 2 d 2 + d 3 - d)a~
X2
I
171
(1 -- a ) l E
E
E , b I + b212,4d ~ + 2 d 2 + d 3 - d
E z o = 2(1 + v)z o + 4(d 3 -- 4d)zo2
v, d 3
EG/s = a~s + 3 (b~ + b212)a2o+ (4dr + 2d 2 + d 3 - d) r 8
EgTo = -- v a o + (3b~ + bz/2)o- ~ + (4d L+ d 2 - d)a3o
d
d~
X2 ,
EOor
Eeor
Eoor
)
1*
J
1
X~
b
X2'
@.
X ~ ~
Eoor
O)
I
X1
Eoor
( /'~ c,o(o)
Xl
d
Fig. 2a-e. Directional tensile tangent Young's modttli and strengths (thick diagonals indicate the fibre directions):
a T-isotropy, b orthotropy, c tetratropy, d hexatropy, e octotropy
5
Failure and yield criteria
It is m o s t w i d e l y a s s u m e d ( s e e , for e x a m p l e , [ 7 - 9 ] ) that t h e failure a n d / o r yield criteria o f a c o n t i n u o u s
m a t e r i a l c a n b e d e s c r i b e d in t e r m s o f a s c a l a r - v a l u e d f u n c t i o n F o f t h e C a u c h y stress t e n s o r ~rUin t h e form:
F(r
= 1.
In t h e p l a n e p r o b l e m (20), t h e o f f - d i a g o n a l s t r e s s c o m p o n e n t s v a n i s h a n d a 3 = a33 is a p r i n c i p a l
(47)
stress. Introducing the deviatoric stress tensor
sij = aq -- ~ - 6 5 q ,
(48)
denoting by s the plane part of s~j, and noting that
F in the form:
F = F(s,/si,
172
$33 =
--
trs
=
-~sl
and s3~ = s3 = 0, we can express
o-ram)"
(49)
Thus, according to Table 1 of Part I, the most general invariant forms of the failure or yield functions
for FRCs in the plane problem (2o) are:
Fr_i,o = F0(I~, IIs, a~m),
F ortho = El (Is' I!s' Jsl, 0-rnm)'
(50)
F tetra ~" F2 ( Is, IIs, Js2, o-ram)'
G.x~ = F3(I.,ns, L3, 0-m~),
F octo = F4(Is, IIs, ls4, 0-mm),
where the abbreviations l~ and II~:
II s = I a - I~1/4 = [2trs 2 - (trs) 2]
I s = Is~ -- trs,
(51 )
are used instead of/~ a n d / ~ . With the aid of (7), we can give two useful sets of expressions, i.e., (i)
when a = a~ e~ | e 1 + a 2e 2 | e 2, then:
S-- 0-1 + 0-2 -- 20-3 1 + al -- 0-2E1'
6
2
Is-
0-I -~ 0-2 -- 20"3
,
3
(O"1 -- 0-2)2
2
I I s - - - ,
Jsn -(0-1 --0-2)ncos(2nO),
n
(52)
(17= 1,2,3,4);
and (ii) when o = z (el | e2 + e2 | e~), then:
0-~
s = --~1
3
~--
20-3,
3
+
r
IIs = z 2 ,
Is1 = -- 2 r s i n 2 0 ,
Js2 =
(53)
-
2zZc~
8-F3
Is3 = - - s i n 6 0 ,
3
Js4 =
4@co880.
Extensive experimental evidence has shown that materials, isotropic and anisotropic can withstand
very high hydrostatic pressures. This had given rise to the widely accepted assumption of the independence
o f f on atom (see, for example, [9]), namely (50) takes the following forms:
Fr_is o = F0(Is, IIs),
For,ho = F1 (Is, Hs, 1,1),
Ftetra =
F2 (Is, IIs, Is2),
Fhexa = F3 (Is, IIs, L3),
Foc,o =/:4 (/s'/Is' Is4)'
(54)
Polynomial failure criteria as introduced in [lO, 11] are widely used in engineering. The most general
failure or yield criteria in quadratic forms of FRCs can be expressed as:
II~
I:
Is
Fo=~5+
y2
~--
F1 --
l,
2II~-L2
2IIsq-Js 2 I:
k2
+
k2
-} y2
Is
(5
2II~-L2
I~
F2 --
k2
2G+L2
{-
/2
k2
+ y2
Isis1 ]sl
M
(5 -- 1,
(55a)
(5 -- l,
173
for T-isotropic as well as hexatropic and octotropic, orthotropic and tetratropic FRCs respectively. The
linear terms coincide with the strength or yield differential effect which is present in all materials, isotropic
and anisotropic and appears in all loading modes [12]. The strength-differential or S-D effect is described
in more detail in [13-15], where suitable yield criteria are proposed for polymers and sintered powder
materials.
To distinguish hexatropic and octotropic cases from T-isotropic one, the simplest characteristic failure
or yield criteria may be propose as:
F3 = k-2 q y2
(5
H
(55b)
iio ~ i~ L , - 4 n 2
F~=Fq y~ a
2c =1.
The aim of involving a T-isotropic invariant term II~ in (55b) is to simplify the relation between the sets
of material constants and failure or yield stresses, as shown below.
Relative to (52), the criteria (55) can be referred to as being the failure or yield surface in the principal
stress space, i.e.,
(0-I -- 0-2) 2
F~ --
4k 2
Fsin220
(0-1 -}- 0-2 -- 20-3) 2
0-1 "~- 0"2 -- 20-3
9Y 2
3(5
+
(0-1 -- 0-2)2
C~
F~=LT+TJ
4
q
- 1,
(0-1 -}- 0-2 -- 20-3)2
9Y 2
0-1 q- 0-2 -- 20-3
3c5
0-1 + 0-z -- 20-3 1 \
3--k/[
+ ~)(0-1 -- ~ cOs20 = 1,
(56)
I-sin220 . c0s220]
(0-1
-j
/73 -- (0-1 -- 0-2) 2
4k 2
-
-
0-2) 2
4
(0-1 ~- 0-2 -- 20"3)2
+
9Y 2
0-1 -~ 0-2 -- 20-3
--1,
3(5
(O-1 + 0-2 -- 20"3) 2
0-1 -}- 0-2 -- 20-3
(0-1 -- 0"2) 3
9Y 2
3(5
3H
F
F4 = (0-1 - 0-2)2 (0-1 q- 0-2 -- 20-3)2
4k 2
}9Y 2
0-1 + G2 -- 20-3
3~
.n
cosot~ = 1,
(or, -- G2)4 . 2,
2C
sm 4t~ = 1.
In the stress state (53), the failure or yield criteria become:
v2
40-~
20-3
Fo= F + g # +T~ =1,
Fsin220 cos220q 2 40-~
2(73
(20-3
Fl=L~q--~T--J'c
+~-y-5+-~--2\3 M
l~z
gj sin20=1,
I-sin220
z2
cos220~
4a~
F~=~+:~+
40.~
"172
4o~ 263
+ ~-5 + ~-
2%
8z3 sin60 = 1,
36
3H
20- 3
8"~ 3
1,
(57)
F4 = ~ + 9 - ~ + ~ - + - C sin240 = 1.
174
In (55) for all T-isotropic, orthotropic, tetratropic, hexatropic and octotropic FRCs, the material constant
k corresponds to the failure or yield stress L for plane simple shear of the a1 and a 2 directions, Y and
6 can be determined from the failure or yield stresses 0-c3
+ and 0-c3
- for simple tension and
compression in X3-direction, i.e.,
2
k=~,
+
§
2
g=3 ~ ,
@3 -- @3
3a-
0.+ -
(58)
c3 O-c3
In T-isotropic case (55), denoting by 0.~+ and 0.- the failure or yield stresses for simple tension and
compression respectively in a plane direction, from (56) we can further arrive at the following
additional relations:
I
1
4k5 + 9 y2
1
0.c+ 0.<_,
O'c3+
2
-36
-=
2 0.+
- =
- - 0.C3
~c~+ 0.~-
-- 0.Z
(59)
0.~+0.7
In orthotropic case (55), the extra material constants k, M and ~ can be related to failure or yield
stresses in the forms:
=_-7_--7-_ -q
0.el 0.cl
"~+
2
--
3M
2f+g~
2
f+
a
f~-
"+
--
0.c3
2
-
--
"-
0. c3
C'U
,
0.c3 0.c3)
1
--
1
+
(60)
acl 0.cl- '
0.c2 0.c2
+ %3
0.c3
+
0.cl - - 0.cl
+ 0.cl 0.cl
- - '1~c- _
4--
0.c+ 0.c2
-
+
+
~
- - 0.c2
+ - "
0.c2 0.c2
Here, we denote by a + and a + the failure or yield stresses for simple tension and compression in a 1
and a2 directions respectively, by a~3 (f) the failure or yield stresses ~r3 in the deviatoric stress state
s = - (0.3/3) 1 + zE 2 for 0 = n/4 with given z = f, as shown in (53), by f+ (a3) the failure or yield stresses
z for 0 = ___n/4 with given 0.3 in the deviatoric stress state s = -(0.J3)1 + zE 2, and use the following
abbreviations:
d 0.+
0.+ = 0"I (0),
O~ = ~-~ 3 (0);
.
' d
"of ----"~+(0), f f = d0.3 "c+(0).
(61)
In tetratropic case (55), denoting by a f the failure or yield stress for simple tension and compression
in a fibre direction (e.g., 0 = 0), by ~ the failure or yield stress for simple shear of two orthogonal maximum
off-fibre directions (e.g., 0 = _ n/4), we have
4
1
)--1/2
(62)
+~-In hexatropic and octotropic cases (55), we have
3H
1
2C
2"c7
i
1
-
=
\ z2c
+
ffc30.c3
+
_
_--g-7~7,
=3 ,
)k.<+~,)
(63)
=2 2
(z~/~<)
16"Pc
0"c3 0.c3
(64)
Here, rY~and ~c correspond to the failure or yield stresses for simple tension in a maximum off-fibre
direction (e.g., 0 = n/2 in hexatropic and 0 = rd8 in octotropic cases) and for simple shear of two
orthogonal maximum off-fibre directions (e.g., 0 = ~112, 7r/2 + ~/12 in hexatropic and 0 = 7z/8, 7t12 + ~/8
in octotropic cases).
6
The convexity of failure or yield surfaces
It is evident that the failure or yield surfaces (56) in the principal stress space {a~, cr2, a~} are cylinders
because of the independence of F on % . We consider in plane stress state the convexity of the curves:
<G:GI~=0 = 1,
(n = 0, 1 , 2 , 3 , 4 ) ,
(65)
175
as intersections of these cylinders with the {or1, a2} plane. Introducing the coordinate transformation:
X = a 1 - o - 2, y=-o" 1 + o - 2,
(66)
from (56)-(64) we can express c~ in the forms:
x~
~0:~-~
(y + %-}-
-- O'er) 2
(CTC~ "Jr- 0 " ~ ) 2
+ -
+ -
O'c3 ~ c3
(sin220
G c3 (7c3
cosZ20~x 2 + y 2
+
-
+
,
+ 2(%+ _ cr~)y
-
+
-[(Y-aa+G2)%G~-(7-%
--
+
--
+G1)GzG2]
2xcos20
+ - +
=4,
0"el O-el O-c2 O'c2
(67)
[sin220
COS220"~
2
(3' + G - ~7~)~ (~? + ~g)~
c~2:[-~-2 + ~
x
+ + ~ c3 0"c3
G c3 ~ c3
"rc )
qx2
2+2(a~+--@~)y
qr 7{ + y
X2
~4:~+
Tc
+ -
O'c3 O'c3
2 .q_ 2 ( G O ~ - - G c 3 ) y
y
+ -
Gc3 Gc3
( 1 - - "c /r~)
2 3
cos60 = 4,
2.?3
1
-
=2 2 4
rJz~)x
4-~4
sin2 40 = 4,
The curves cg,, n = 0, 1. . . . . 4, are symbolic plotted in Fig. 3. Both ~0 and cg2 for any angle 0 are ellipses
on the {x,y} plane; cg4 for any angle O is a convex closed ellipse-like curve; cg~ is an ellipse for any angle
0 if and only if
-
;
(68)
and cd3 is an ellipse if gc = zc, consists of a convex closed pear-like curve and a open curve if
x/3/2 < fc/z c < 1,
(69)
and consists of three open curves if 0 < f~/z c _-<x/3/2. Based on (67), one an easily calculate the
characteristic sizes a o, a 1. . . . . a 4 in Fig. 3. Higher yield stress level corresponds to the open curve in Fig. 3(d)
than that to the closed curve in the same Figure. The former should be unstable and thus not real.
Consider the failure or yield stress G(0) in simple tension of the 0 direction. Suppose that the differential
effect is negligible. From (67) by setting x = y -- G(0), we have
ego: ( ~ + ~2c3/0-2c(0) = 4 ,
/sin220
cos220
vv
1
cos20
-2-~c2
0-.2 + 2 ~ -0-c,
cos20\
)r
2
y,
i
~r
"" ,,...
..................
/;:~.+........
!\ ..~
oc,/
.....
A +~
%3- %3
!\1/
a
,..."
I- a0 "F ao
176
01i::
o
b
~
yJ
a2
I
a2
y~
.......................IL ........
:
(YC /'"W
/5"
~c3
d
e
/
~.t+x
I,~c3" (~c3
..I
a3
a4
Fig. 3a-e. Failure or yield surfaces: a T-isotropy, b orthotropy, c tetratropy, d hexatropy, e octotropy
c~2: \
sin220 c0s220
z ~ - - + ,[+.[~
~3:
+
1
G(O)
1\ z0
1) 2
~3 @ ( 0 ) = 4 ,
2"7~
(7o)
cosov = 4,
(1--~zJ~2)o-~(O) . 2.~
s,n
=4
One may anticipate that G(O) in fibre directions are greater than that in off-fibre directions. This yields
in (7o) ~c > zc for tetratrppic FRCs and z~ > ~o H > 0 and C > 0 for hexatropic and octotropic FRCs. In
(7o) for orthotropic FRCs, from G(0) > ac0Z/2) it can yield G~ > G2 and m > 0.
For simplicity, Fig. 2 may serve also as symbolic plots from above observation for (G(O), O) in a polar
coordinate system. However, we emphasize that it is impractical that yield stress and the elastic constant
show same tendency with respect to the fibre directions. For example, we can read the following
experimental data from [16]: E o -= 30.0 MPa, Eo/E~5= 0.95, Eo/E90 = 1.14, G(0) = 500 MPa, ac(O)/ac(15) =
1.49, G(O)/G(90) = 1.3, for a triaxial woven fabrics (as a hexagonal composite). The Young's modulus
E o in the fibre direction 0 ~ is weaker than the Young's modulus E~5 in the off-fibre direction 15~ while
the tensile strength G(0) in the fibre direction 0 ~ is stronger than the tensile length G(15) in the off-fibre
direction 15 ~.
7
Conclusions
Most commercial fibre-reinforced composites (FRCs) are appropriately classified in terms of the five
macroscopic material symmetries: transverse isotropy, orthotropy, tetratropy, hexatropy and octotropy.
The formulation of constitutive equations for FRCs in plane problems of elasticity, yielding and failure
is performed, based on the complete and irreducible two-dimensional tensor function representations.
Hexatropic and octotropic FRCs possess significant different kinematic and physical properties from
transversely isotropic, orthotropic and tetratropic ones. The theoretical determination of material elastic,
yielding, and failure constants for FRCs is demonstrated. This calls for further experimental tests and
verifications for the practical determination of these constants.
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177