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Elastic moduli and stress field of plain-weave composites under tensile loading
Pll:
SO266-3538(97)00044-4
Composites Science and Technology Sl(l997) 787-800 0 1997 Elsevier Science Limited Printed in Northern Ireland. All rights reserved 0266-3538/97/$17.00
ELASTIC MODULI AND STRESS FIELD OF PLAIN-WEAVE COMPOSITES UNDER TENSILE LOADING
Makoto Ito” & Tsu-Wei Chad a 3rd Research Center, Japan Defense Agency, I-2-10 Sakae, Tachikawa, Tokyo 190, Japan b Center for Composite Materials and Department of Mechanical Engineering, Delaware 19716, USA
(Received 21 December
1995; revised 20 January 1997; accepted 21 January 1997)
warping allowed’ model. Suppressing the out-of-plane deformation owing to the tension&ending coupling effect, the analytical results approached those from experiments. Naik4 also studied the elastic properties of plain-weave lamina and laminated composites by the use of a three-dimensional geometrical model. It was shown that the elastic properties were almost the same for laminates with iso-phase and out-of-phase stacking. In Refs 14, the effect of tensiomtransverseshear coupling (ai6 of the compliance matrix) in the cross-section of a fabric composite was neglected. However, actually in plain-weave composites, the effect of tension/transverse-shear coupling on the elastic behavior is not insignificant because of yarn crimp. The second type of study, elastic and stress analyses using finite-element methods, can be represented by the works of Whitcomb and Foye.6 They conducted finite-element analyses of unit cells of laminates of out-of-phase stacking configuration. The elastic properties and stress distributions were investigated under tensile, shear and flexure loads. Although finiteelement analysis can adequately deal with threedimensional geometry and provide detailed stress analysis, the analysis is problem-specific and it is not suitable for parametric studies. Moreover, in plainweave laminated composites, the tension/bending coupling effect (b,, of the compliance matrix) can be suppressed by the neighboring layers. When the bending effects are negligible, simple models instead of finite-element analysis can provide stress distributions with sufficient accuracy. In this paper, iso-strain models and flexural models are developed to obtain the elastic properties and stress distributions of plain-weave composites. The results from these models are evaluated by finiteelement analyses. The tension/transverse-shear coupling and tension/bending coupling effects are investigated in terms of yarn undulation and laminar stacking configurations.
Abstract models have been developed for the investigation of the elastic properties and stress distributions of plain-weave composites under in-plane tensile loading. Analyses are performed for the configurations of single lamina, iso-phase laminate, outof-phase Iaminate and random-phase laminate. The analytical results are evaluated by finite-element analyses and good agreements have been achieved. The tension/bending coupling effect is significant for the single-lamina composite. The tension/transverse-shear coupling results in differences of Young’s modulus and stress distribution between the iso-phase laminate and out-of-phase laminate. This study has led to the conclusion that in the practical design of plain-weave composites, Young’s mod&i and stress distributions can be predicted by the iso-strain model. The Jlexural model is useful in assessing the effect of local bending of yarns. 0 1997 Elsevier Science Limited
Iso-strain
University of Delaware, Newark,
and flexural
Keywords: A. plain-weave fabrics; C. elastic properties;
stress analysis; micromechanics
1 INTRODUCTION Analyses of the elastic properties and stress distributions in plain-weave composites have been reported extensively in the literature. These previous studies can be categorized into two types. The first type of approach utilizes simplifying assumptions of iso-strain or iso-stress condition in the analytical models.14 The second type of approach involves more detailed analysis based upon finite-element methods.= The first type of study can be represented by the works of Chou and Ishikawa,‘” who obtained in-plane elastic properties both analytically and experimentally. In order to investigate the effect of out-of-plane constraint conditions, they developed two models: the ‘local warping constrained’ model and the ‘local 787
M. Ito, T.-W. Chou
788
a
I
hm
I
-b-C dx
Longitudinal yarn ,
Transverse yarn 2 Fig. 1. Two-dimensional unit cell of plain-weave composite. 2 TWO-DIMENSIONAL MODEL
GEOMETRICAL
of the transverse yarn, K,, are defined from the total fiber volume fraction of the composite, V, as
A two-dimensional unit cell as shown in Fig. 1 is adopted for plain-weave composites. The composite is assumed to have uniform xz cross-section. For simplicity, gap widths between transverse yarns are assumed to be zero. The center line of longitudinal yarn, zO, is defined as
z,(x)=-Tsin
4
2%-x
( 1 a
where the longitudinal yarn thickness, h, is assumed to be constant. The upper boundary of longitudinal yarn is z,= z0 - h,/2, and the lower boundary is z, = zO+ hJ2. Thicknesses for the transverse yarn, h,, and the matrix layer, h,, for (0
n-Vfh
V,h K’= 2h,’
3 ISO-STRAIN
Kt= 4h,
MODEL
3.1 Single-lamina model For a single-lamina composite, the whole composite structure can be represented by repeating the unit cell in Fig. 1. When a tensile force is applied in the longitudinal direction, the flexural deformation as represented by bll of the compliance matrix is significant and cannot be neglected. By assuming isostrain state in a vertical element with an infinitesimal length (dx) in the x direction, the following constitutive relationships are given for the element:
h,(x) = h, sin
i_i= h,(x)=
$- - $[sin($-)
+1 ]
[$j
;i
;i
i:i(?ij)
(3)
The local fiber direction is denoted as the 1 direction and the direction perpendicular to the fiber is the 2 direction. The angle 0 between the 1 axis and the x axis is
[ij=i,li;;
(@
In eqn (6) A,, B, and D, are stiffness constants and are expressed as functions of x (0
Since the longitudinal yarn and the transverse yarn should have the same total fiber areas in the unit cell, yarn packing fractions of the longitudinal yarn, K~,and
;j[;j
=
= h,&(x)
$ (ZI(X)’1
+fl
+ h,(x)Q;
Z"(X)2)Q~(X)
h c( --I - h,(x)
‘ L\L
+ h,(x)QT
j2 - ~,(x)~j
/
J
Qij
Plain-wave composites under tensile loading
r _------- . I
1 I
.
r I
--e-_-m. .
l
----___-
789
1 I
dx
I
L-2____,
Lo-phase
Out-of-phase
Random-phase 0
Macro-cell
Fig. 2. Laminar stacking configurations of iso-phase, out-of-phase and random-phase laminates. Macro-cells of iso-phase, outof-phase and random-phase laminates consist of one, two and an infinite number of the unit cells, respectively. For the iso-strain model, the iso-strain condition is assumed in the element with infinitesimal length (dx) through the thickness of the macrocell.
D,(X)
=
f
+
(Z*(x)3-
distributed parabolically across the thickness, &, is used instead of & with a weighting function given by9
Z,(x)3)Qs(x)
$[(G-h,(x))3-zI(X)Z]Qij
+ +[(+)‘-(+
(10)
-k(x))‘]Qr
(7)
The stresses of the transverse yarn and matrix are also obtained by using the following relationship:
where @j is the transformed reduced stiffness of a longitudinal yarn with respect to the fiber angle 0. a; and @’ are the reduced stiffnesses of transverse yam and matrix, respectively. aij, b, and d, in eqn (6) are compliances. K, denotes curvature. N, is the stress resultant. The effective Young’s modulus of the single-lamina model is obtained by averaging a,, over the yarn length:
Substituting the strains of eqn (6) into the stress/strain relationship of the longitudinal yarn, the stresses of the longitudinal yarn are obtained as
Here,
assuming
that the transverse
shear stress is
3.2 Laminate model In plain-weave laminated composites, three types of laminar stacking configurations are considered: isophase, out-of-phase and random-phase laminates. For each laminate, the repeating element of the whole composite structure, referred to as the macro-cell, is shown in Fig. 2. The macro-cells of the iso-phase, outof-phase and random-phase laminates consist of one, two and an infinite number of unit cells, respectively. For each macro-cell, the iso-strain condition is assumed in the element with infinitesimal length (dx) in the x direction. Each macro-cell satisfies the
M. Ito, T.-W. Chou
790
boundary condition in terms of deformation between macro-cells. We assume that flexural deformations can be neglected in the laminates because the flexural deformation of each layer is constrained by the neighboring layers. When the waviness ratio of the yarn, defined by h,la, is large, flexural deformations can no longer be neglected, and another stress analysis model is developed in Section 4. For the iso-phase and out-of-phase laminates, the relationship between stress resultant and strains in the infinitesimal element is
where A, for the iso-phase laminate are identical to those of eqn (7). For the out-of-phase laminate, A, are identical to those of eqn (7) except for Al6 and Az6, which are canceled in the macro-cell. The effective Young’s moduli of iso-phase and out-of-phase laminates are obtained by eqn (8). The stress distributions in the longitudinal yarn are obtained by substituting the strains of eqn (12) into
Stresses in the transverse yarn and matrix are also obtained by using the corresponding stiffness matrices. For the random-phase laminate, A, of an infinitesimal element in the macro-cell are independent of X, and the averaged stiffness constants, Aij, are defined by a&?
A,(X) dX
Thus, the effective Young’s modulus phase laminate is given by E,=
1 h$rl
(14) of a random-
(IS)
where tiij are the averaged compliances, which are obtained by inverting Ai> Though the Young’s modulus is an averaged property of the laminate, the stress distributions in a given layer of the random-phase laminate are strongly affected by the relative location
of the surrounding layers. Thus, the iso-strain condition in the macro-cell defined in Fig. 2 does not give a good approximation for the stress distributions of the random-phase laminate. Whitcomb et al.]’ have investigated the boundary effects in plain-weave composites. In their work, the boundary effect of a given layer propagates at most into the adjacent layers. Thus, as shown in Fig. 3, we assume that the stress field of a given layer (layer A) is affected by the upper adjacent layer with phase shift 4,” (layer B) and the lower adjacent layer with phase shift 4,” (layer C). Here, 4,” and 4,” are defined by using the shifts s,” and s,” of layers B and C, respectively, with respect to layer A as
The iso-strain condition is assumed in the element with infinitesimal length (dr) in the x direction through the thickness of the three layers. In eqn (12) A, is defined by Aij(x) + Aij[x - a( 4,8/2T)] + A,[x - a( +:/2 T)] based upon eqn (7). Using the same method as for the iso-phase and out-of-phase laminates, the stress distributions of layer A are obtained as functions of x, 4,” and $,“. In the same way, the stress distributions of layer B (or C) are defined by 4: (or 4:) and a phase shift with respect to the other adjacent layer. Thus, for the stress analysis of the random-phase laminate, each layer of the laminate can no longer satisfy the boundary condition between layers in terms of deformation. The stress distributions of layer A are equal to those of the iso-phase laminate when 4,” = 4,” = 0.
4 FLEXURAL MODELS When the waviness ratio of a composite laminate is large, flexural deformations must be considered in the analysis. Flexural deformations due to local fiber undulation in unidirectional composites have been studied by a number of authors. Lee and Harris” investigated the effective Young’s modulus and strength of unidirectional composites with iso-phase and out-of-phase curved fibers. In their work, the fibers were isotropic and the surrounding matrix carried no load. Here, the models of “for unidirectional composites are modified to deal with plainweave laminates. 4.1 Iso-phase As shown in consideration yarn part and
laminate model Fig. 4, the infinitesimal element under is divided into two parts: a longitudinal a combined part of matrix material and
791
Plain-wave composites under tensile loading
X
h
_)
c-
Layer B
c-
Layer A
c-
Layer C
I
Fig. 3. Iso-strain model for the stress distributions of a given layer A of a random-phase laminate. Iso-strain condition is assumed in the element with infinitesimal length (d_x)through the three layers (layers A, B and C). The shifts of layers B and C with respect to the layer A are s,” and s:, respectively.
a
c
Combined Part
quz Longitudinal yarn part
s, qlx
Fig. 4. Force and moment equilibrium of an infinitesimal element in the iso-phase laminate. The infinitesimal composed of a longitudinal yarn part and a combined part of matrix and transverse yarn.
element
is
792
M. Ito, T.-W. Chou
transverse yarn. Equations for force and moment equilibrium for (0
-dSY dx
to in-plane loading and is given by
Igo= 4112+ 412 =
dPY - dx + qlu
0
ASylPy+Sp,sy) h ( Y
(17)
The other term, +, termed as a pseudo-strain, is a contribution to axial strain by the bending rotation and is expressed as”
- qlx = 0
-dMY +py- dzo - + h&q,, + q,J - s, = 0 (19) dx dx
Eps=
$$)‘-(Z
+*y)2]_ (27)
for the longitudinal yarn part, and
d&,, - dx + 9”Z
- 412 =
dp,, ~dx + 91x dzo
P”, -
dx
4”X =
0
(20)
0
(21)
+ qux) - & = 0
(23)
where wy is the displacement of the yarn center line in the z direction, and I& and & are rotation angles caused by bending and shear deformation, respectively. Provided that & is uniform in the z direction and the normal stress in the z direction is neglected in the longitudinal yarn, & is given by
(&Py + S&S,)
(24)
Y
where 3; are the transformed reduced compliances of the longitudinal yarn and are obtained by inverting eqn (13). The axial strain of the longitudinal yarn at the center line is defined by y=
eYO+ x
-d@Y
Yh
_de,
%s
c;=
(22)
for the combined part. Here qu, qu,, qk and q,z are distributed forces per unit length in the x direction. Rotation of the center line of a longitudinal yarn is given by
Ex
&,U= Ej; + -hY
2
- $ (h - h&,x
yh = +
Hence, the axial strains at the upper (z = z0 - hy/2) and lower (z = z. + hy/2) boundaries of the longitudinal yarn are given, respectively, by
(25)
where E:’ is the strain along the yarn center line due
EY_ x
2
dx
dx
(28)
(29)
The bending equation of the longitudinal yarn is
M=-I,% Y
S:,
dx
(30)
where I, = h$12 is the bending stiffness of the longitudinal yarn. In the combined part, shear deformation caused by bending rotation of the longitudinal yarn is illustrated in Fig. 5.l’ Thus, the constitutive relationships of the combined part are given by P,, = (h &r = G0
hy)Eud
lc’y + (h - hy)&I
(31) (32)
where E,,, and G,, are the effective Young’s modulus and shear modulus of the combined part and are determined from the rule of mixtures: 2h,E, Eu, =
+ h,E,
h - h,
,
G,,=
(33)
Here, Et and G, are, respectively, Young’s modulus and shear modulus of the transverse yarn. E, and G, are the matrix Young’s modulus and shear modulus, respectively. With the above deformation and constitutive relationships, differential equations for force and
Plain-wave composites under tensile loading
793
After deformation
Before deformation
Longitudinal yarn part
Combined part
Longitudinal yarn part
Shear deformation by bending rotation 0, - 9, = Fig. 5. Shear deformation
moment equilibrium expressed as
Ah-hy ”
of the combined part due to bending rotation of the longitudinal yarn.
in the infinitesimal
element are
Py + (h - h,)E”l&; = P S, + G,[h +, + (h + hy) Cl
al2
L,(~,,P,,,S,)
cos( 2(2i;
l)‘r ) dx=O
(34)
=0
[i = 1,2,. . .]
(39)
[i = 1,2,...]
(40)
(35) al2 L,~$Y,PY,SY)cos(
-
hyG,,(&Cy) +Sy=O
(36)
2(2i;1)li)dx=0
and, at x=0 L2(rcIY,PYSY)
where P is an applied uniform force per thickness h. Since E: and & are functions of Py, S, and I&,, four differential equations, eqns (23) (34)-(36), are defined with respect to the four variables wY, P,,, S, and $,,. These differential equations can be solved approximately by means of the Galerkin method expressed by
c
CljCOS
a12
I
(41)
0
2(2i - 1)7r
n
i=1,2,...
cos(2(2i;1)T
p=
where the linear differential operators L,, L2, L3 and L4 denote eqns (23),(34)-(36) respectively. In order to satisfy the boundary conditions, wY I&, Py and S, are approximated by
WY=
L(wY~~Y~pYsY)
-
(
X
a
(42) )
)dXz()
0
[i = 1,2,. . .]
(37)
eyz
i
c2jps
i=1,2,...
2(2i-ljT
(
x
a
1
(43)
u/2 [LZ(~Y,PY,SY)
-
PI as
dx=O Py=C30+ [i = 1,2,. . .]
(38)
f: i=l,Z,...
4i7r
C3iCOS
-X (
a
1
(44)
794
M. Ito, T.-W. Chou 2(2i - 1)7r
(
cqicos
s, = i
i=l,Z,...
a
x 1
(45)
Substituting eqns (42)-(45) into eqns (37)-(41) Cji are obtained by solving the system of 4n + 1 equations. Calculations are conducted numerically by using software that can handle the symbolic computation.” In the calculation, h,, h,, E,, and GUI must be approximated by Fourier series representations to avoid discontinuities at x = 0 and a/2. The effective composite Young’s modulus is given bY
4.2 Out-of-phase laminate model The infinitesimal element for the out-of-phase laminated composite is composed of a longitudinal yarn part, an upper matrix part and a lower combined part of matrix and transverse yarn (Fig. 6). With respect to the longitudinal yarn part, equations for force and moment equilibrium are given by eqns (17)-(19). The force equilibrium equations for the other two parts are
--dpu dx
cl”,= 0
dP, + q,x = __ dx
The stresses in the longitudinal yarn are defined by
ux
Y=
P
Y
hY
-
_
2,
&I
d@
_,(T;=o,(TY
xz
dx
=
S
Y
(47)
P, = h,E,&
hY
#y + YP,
(48)
Y
(51)
PI = (h, + h,)E,&f,
(52)
4”Z = K”(W, - %)
(53)
4lz = - mw, = 0, c+:, = G, &
(50)
The deformation and constitutive relationship for the longitudinal yarn are identical to those of the isophase laminate (eqns (23)-(30)). Constitutive relationships for the other two parts are
where z, is the distance from the yarn center line. Stresses in the transverse yarn are defined by
a: = E;E~, u;
0
(49)
+ w0)
(54)
where E, is Young’s modulus of the lower combined part and calculated by using the rule of mixtures in the same way as eqn (33). KU and K, are equivalent spring
Upper matrix part
Longitudinal yarn pert
Lower combined hm+h part I
equilibrium of an infinitesimal element for out-of-phase laminated composite. The infinitesimal element is composed of a longitudinal yam, a upper matrix and a lower combined part of matrix and transverse yarn.
Fig. 6. Force and moment
Plain-wave composites under tensile loading
constants of the upper matrix and the lower combined part, respectively, and are given by Ku=
+,Kl= m
h,E,
(46) and (47) except for a: which is defined as #=
&E,
quz+412 +
z
2
(55)
+ h,E,
795
s,
sin
o
+
412- 4”Z
h,
%
2, (60)
Stresses in the transverse yam are w,, in eqns (53) and (54) is a constrained displacement and it is introduced to achieve zero sum of forces in the z direction at the upper and lower surfaces of the unit cell. This condition leads to 5 FINITE-ELEMENT
a/2 (quz + s,,) dx = 0 I0 Hence, the differential (23) and
-dSYdx
K&v,
(56)
equations to be solved are eqn
Py + h,E,e,”
+ (h, + h,)E,e;
(61)
= P
- wO) - K,( wy + w,,) = 0
(57)
MODELS
In order to evaluate the analytical models, finiteelement analyses are conducted by using ABAQUS Version 5.3-l. The single lamina, iso-phase laminate and out-of-phase laminate are modeled with the common unit cell (Fig. 7) under the corresponding boundary conditions. Plane stress elements (CPS8R and CPS6R) are used for the analyses. The boundary conditions are
(58) .(-+,z)=O,
+,z)=u,,
(62)
for the single-lamina model, +
h,E,
-&(h,.$)
+ + h, $$
+ s, = 0
(59)
These four differential equations and the constraint condition of eqn (56) can also be solved with the Galerkin method. Definitions of the effective Young’s modulus and stresses of the longitudinal yarn are identical to eqns
Fig. 7.
.(-+x)=0, +,+,, v(x,-
Unit cell used for finite-element analysis.
$)-u(,,t)=const.
(63)
M. Ito, r.-W. Chou
796
Table 1. Elastic properties of AS4 graphite and epoxy random
phase
Properties
AS413
Epoxy14
Err &, G2 G3
221 16.6 8.27 5.89
3.45 3.45 1.28 1.28
(GPa) @Pa) (GPa) @Pal
0.26
v12
20 1n ..Y
I
-
.
-
-
Flexural Model
---
Finite Element Modd ofcross-ply hminate is~ndkatede by an arrow
Ex 0
0
0.02
0.04
0.06
0.08
0.1
Waviness Ratio (by/a) Fig. 8.
lamina,
Effective Young’s moduli of cross-ply laminate, single iso-phase laminate, out-of-phase laminate and random-phase laminate.
for the iso-phase
laminate
model,
+g=o, .(x.-
5)
for the out-of-phase
to be 0.5 and h,lh = 0.48 was used. The elastic properties of the longitudinal and transverse yams were calculated by using micromechanical analysisi and the respective yarn packing fractions (eqn (5)). In the flexural models, the number of terms in the series of the Galerkin method is IZ=3 for adequate convergence. Table 2 summarizes the numerical analyses performed for the various mathematical models. assumed
Iso-strain Model -
and
.(~,z)=uo
=O,.(,,+) laminate
6.1 Effective Young’s moduli Figure 8 shows the variation of effective Young’s moduli with respect to the waviness ratio h,lu. Since the waviness ratios of plain-weave composites are about 0.02 to O-03 (Ref. 3) and O-05 to 0.08 (Ref. 16) from the sample measurements, the calculations were (64)
=const.
Table 2. Summary
Numerical analyses have been conducted with respect to the mathematical models developed in Sections 3-5. The material selected for calculations is AS4/epoxy; the elastic properties of the constituent materials are shown in Table 1. Total fiber volume fraction was
diagram
of stress components
of
the models analyses
Single lamina Iso-strain Fiexural Finite element a
and their locations
used
for
numerical
Laminate configuration
Models
model.
6 ANALYTICAL RESULTS AND DISCUSSION
Fig. 9. Schematic
0.35
X
X
Iso-phase
Out-ofphase
Randomphase
Xa X X
Xa X X
X” -
Bending effect not considered.
calculated
in the stress
respectively, the center line, upper boundary and lower boundary
analysis.
of the yarn.
CL, UB and LB denote,
797
Plain-wave composites under tensile loading performed at (0.01~ h,la < 0.1). The Young’s modulus of a cross-ply laminate is also shown in the figure. There is no significant difference among the iso-strain, flexural and finite-element models. With respect to laminate configuration, the magnitudes of the Young’s moduli are in the following order: cross-ply > randomThe lamina. phase > out-of-phase > iso-phase > single single-lamina composite shows the lowest Young’s moduli because of the large tension/bending coupling effect. When the waviness ratio increases. the iso-
phase laminate shows lower stiffness than those of the out-of-phase and random-phase laminates because the tension/transverse-shear coupling effect increases. The Young’s moduli of the out-of-phase laminate and the random-phase laminate show little difference and are insensitive to the waviness ratio. The predictions of Young’s moduli showed good agreement with experiments.“.”
5r
0 0.3
0.2
0.1
0.4
- - -
hestrain Model Flexural Model Finite Element Model
-_-
0
0.2
0.1
0.3
0.4
C 5
x/a
0.5
x/a
(4
(a) I
0.4,
-
- 0.3 -0.4
0
Lso-strain Model
--
‘%\\\
Finite Element Model
0.1
‘A_. 0.3
0.2
0.4
0
0.1
I
0.2
0.5
0.3
0.4
0.5
x/a
x/a
(4
@I
1
5 s
- - -_-
0.1~
ck I= 0
tso-strain Model Flexural Model Finite Element MC&l 0.1
0.2
0.3
0.4
0.5
x/a
Fig. 10. Stress distributions
of the single-lamina composite. Stresses are normalized by the far-field stress, (TV. (a) Longitudinai stress of longitudinal yarn, (b) transverse shear stress of longitudinal yarn, (c) transverse stress of transverse yarn.
11. Stress distributions of the iso-phase laminate. Stresses are normalized by the far-field stress, CT* (a) Longitudinal stress of longitudinal yarn, (b) transverse shear stress of longitudinal yarn, (c) transverse stress of transverse yarn. Fig.
798
h4. Ito, T.-w. Chou
-0.1.
-_-
i 0' 0
Finite Element Model 0.1
0.2
h-strain - -_-
I x./a
0.3
0.4
-
Finite Element
-0.2.
Model
I
I
0.5
Model
Flexural Model
0
0.1
0.2
0.3
0.4
0.3
0.4
J
0.5
x/a (W
“A,,
g +m 2 c
0.1
0.1. - - -
ISO-strainModel Flexural Model Finite Element
-_-
- -_-
Model 0 0
0.1
0.2
x/a
0.3
0.4
0.5
(4
-0.1
0
-
Flexural Model Finite Element Model
0.1
0.2
x/a (4
Fig. 12. Stress distributions of out-of-phase laminate. Stresses are normalized by the far-field stress, gw (a) Longitudinal stress of longitudinal yarn, (b) transverse shear stress of longitudinal yarn, (c) transverse stress of transverse yarn, (d) transverse stress of longitudinal yarn. 6.2
Stress distributions
Figure 9 shows a schematic of stress components and their locations evaluated in this analysis. The stresses are transformed to the l-2 coordinate system. In plain-weave composites, the stress distributions relevant to the tensile strength are hte longitudinal stress, c+yl, and transverse shear stress, LT+J*,of the longistress of the tudinal yarn, and the transverse transverse yarn, a&. Thus, in this stress analysis, cry1 and ayZ are calculated at the center line (CL), the upper boundary (UB) and the lower boundary (LB) of the longitudinal yarn. g& is calculated at the center line (CL) of the transverse yarn, In addition, the transverse stress of the longitudinal yarn, I&, is calculated for the out-of-phase laminate because it is supposed to be high in this configuration. In the isostrain models, it is noted that stress distributions of iso-phase laminate and out-of-phase laminate are uniform through the thickness of each yarn because there is no flexural deformation. Thus, it is assumed that the calculated uniform stress occurs at the center line of each yarn. All stresses are calculated at the waviness ratio of 0.05 and normalized by the far-field
applied stress, CT@Because of symmetry, the stresses are calculated in the region 0
799
Plain-wave composites under tensile loading
Figure 12 shows stress distributions of the out-of-phase laminate. In order to compare the results of iso-phase and out-of-phase laminates, the same scales are used in Figs 11 and 12. All stress distributions of the three models are in good agreement, except for ah
1.8' 0
0.1
0.3
0.2
0.4
(
x/a Fig. 14. Longitudinal stress distribution in the longitudinal yarn of the random-phase laminate at phase shifts (#J:,+:) = ( f 7~,f n). Stresses are normalized by the farfield stress, (TV
(4
04
(4 Fig. 13. Maximum stress of random-phase laminate as
functions of phase shifts 4,” and 4:. Stresses are normalized by the far-field stress, U@ (a) Longitudinal stress of longitudinal yarn, (b) transverse shear stress of longitudinal yam, (c) transverse stress of transverse yarn.
predicted by the iso-strain model. Since (+& is caused by the local bending of yarns, the iso-strain model is less effective than the flexural model in approximating &. In the case of the random-phase laminate, the stress distributions are approximated by those of a layer with two adjacent layers of phase shifts 4,” and q5,” (see Section 3.2). Varying 4: and 4: from - 71 to r, the maximum values of &, (T:* and cz& are calculated (Fig. 13). gyl assumes its maximum at (@,+x”)=( fr, *r). When <+x”&x”)=( fr, *q), the maximum value occurs at x/a = 0 and 0.5 as shown in Fig. 14. a:;? and a& become maximum at ($~,+~) = (0,O). Thus, the maximum values of (+yZand a& of the random-phase model are identical with those of the iso-phase model. 6.3 Discussion The strength of plain-weave composites depends on the relationship between a:,, (T&, CT&,g& and their critical values.‘7,18 Catastrophic failure of the composite occurs when cry1 reaches its critical value. When u);~ or u&. reaches its critical value, interfacial debonding occurs at the boundaries of longitudinal yarns, and then each layer of the laminate behaves as a single lamina, separated from the neighboring ones. Transverse cracks initiated by ai2 have relatively little effect on the composite axial strength. The single-lamina composite shows high stress concentrations in every stress component, and thus possess the lowest strength among all the laminate configurations studied. The iso-phase laminate shows a high concentration of cryZ, while the out-of-phase laminate has the lowest stress concentration. This is due to the allowed or constrained transverse shear deformation of each layer. The out-of-phase laminate has a large concentration of u&, caused by the local bending of yarns. As for the AS4/epoxy unidirectional composite, the transverse shear strength is 1.3 times of the transverse tensile strength.13 Since ur2 of the iso-
800
M. Ito,
phase laminate is four times a& of the out-of-phase laminate, interfacial debonding occurs at a lower applied stress level in the iso-phase laminate than in the out-of-phase laminate. The random-phase laminate may have the same laminate configurations locally as those of the iso-phase and out-of-phase laminate. Thus, interfacial debonding in the random-phase laminate is caused by af. The strength predictions based upon these models showed good agreement with
7 CONCLUSIONS Iso-strain, flexural and finite-element models have been developed to investigate the elastic properties _ _. and stress distributions of plain-weave composites under in-plane tensile loading. Analyses have been performed for the configurations of a single lamina, iso-phase laminate, out-of-phase laminate and random-phase laminate. The tension/bending coupling effect is significant for the single-lamina composite. The tension/transverse-shear coupling effect gives rise to differences in the Young’s modulus and aY2 between the iso-phase laminate and the out-of-phase laminate. For all laminate configurations, Young’s moduli and stress distributions (except for uz2 of the out-of-phase laminate) can be predicted with the iso-strain model. The flexural model can predict the effect of local bending of yarns. In the practical design of plainweave composites, a single lamina and random-phase laminate are often used. As the number of layers increases, the overall composite behavior changes from that of a single lamina to that of a random-phase laminate. Moreover, the growth of interfacial debonding in a random-phase laminate renders each layer of the laminate as a single-lamina composite. In both configurations, the Young’s moduli and stress distributions of the laminates studied here can be adequately predicted by the iso-strain model, which is the simplest among the models developed. Thus, the iso-strain model is the most effective one for the design and characterization of plain-weave composites. REFERENCES 1. Chou, T. W., Microstructural Design of Fiber Composites. Cambridge University Press, Cambridge, UK, 1992.
T.-W. Chou
2. Ishikawa, T. and Chou, T. W., Stiffness and strength
properties of woven fabric composites. _I. Mater. Sci. 1982, 17,3211-3220.
3. Ishikawa, T., Matsushita, M., Hayashi, Y. and Chou, T.
W., Experimental confirmation of the theory of elastic moduli of fabric composites. J. Compos. Mater. 1985, 19, 443458.
4. Naik, N. K., Woven Fabric Composites. Technomic Lancaster, PA, 1994. 5. Whitcomb, J., Three-dimensional stress analysis of plain weave composites. In ASTM STP 1110. American Society for Testing and Materials, Philadelphia, PA, 1991, pp. 417-438. 6. Foye, R. L., Approximating the stress field within the unit cell of a fabric reinforced composite using replacement elements. NASA CR 191422, National Aeronautics and Space Administration, Washington, DC, 1993. 7. Dow, N. E and Rammath, V., Analysis of woven fabrics for reinforced composite materials. NASA-CR 178275, National Aeronautics and Space Administration, Washington, DC, 1987. 8. Paumelle, P., Hassim, A. and L&n&, F., Composites with woven reinforcements: calculation and parametric analysis of the properties of the homogeneous equivalent. La Recherche Aerospatiale 1990, 1,1-12. 9. Vinson, J. R. and Sierakowski, R. L., The Behavior of Structures Composed of Composite Materials. Kluwer Academic, Dordrecht, 1987, Ch. 2. 10. Whitcomb, J., Kondagunta, G. and Woo, K., Boundary effects in woven composites. J. Compos. Mater. 1995, 29, 507-524.
11. Lee, W. J. and Harris, C. E., A deformation-formulated micromechanics model of the effective Young’s modulus and strength of laminated composites containing local ply curvature. In ASTM STP 10.59. American Society for Testing and Materials, Philadelphia, PA, 1990, pp. 521-563. 12. Wolfram, S., Mathematics I/: 2.2. Wolfram Research, 1994. 13. Hercules data sheets. Hercules Inc., 1989. 14. Weeton, J. W., Peters, D. M. and Thomas, K. L., Engineers’ Guide to Composite Materials. ASM, Metals Park, OH, 1987. 1.5. Tsai, S. W. and Hahn, H. T., Introduction to Composite Materials. Technomic, Westport, CT, 1980, Ch. 9. 16. Yurgartis, S. W. and Maurer, J. P., Modeling weave and stacking configuration effects on interlaminar shear stresses in fabric laminates. Composites 1993, 24, 651-658.
17. Ito, M., Effects of yarn undulation on the stress and deformation of textile composites. PhD dissertation, University of Delaware, 1995. 18. Ito, M. and Chou, T. W., An analytical and experimental study of strength and failure behavior of plain weave composites. .I. Comp. Mater., 1977, 31.
SO266-3538(97)00044-4
Composites Science and Technology Sl(l997) 787-800 0 1997 Elsevier Science Limited Printed in Northern Ireland. All rights reserved 0266-3538/97/$17.00
ELASTIC MODULI AND STRESS FIELD OF PLAIN-WEAVE COMPOSITES UNDER TENSILE LOADING
Makoto Ito” & Tsu-Wei Chad a 3rd Research Center, Japan Defense Agency, I-2-10 Sakae, Tachikawa, Tokyo 190, Japan b Center for Composite Materials and Department of Mechanical Engineering, Delaware 19716, USA
(Received 21 December
1995; revised 20 January 1997; accepted 21 January 1997)
warping allowed’ model. Suppressing the out-of-plane deformation owing to the tension&ending coupling effect, the analytical results approached those from experiments. Naik4 also studied the elastic properties of plain-weave lamina and laminated composites by the use of a three-dimensional geometrical model. It was shown that the elastic properties were almost the same for laminates with iso-phase and out-of-phase stacking. In Refs 14, the effect of tensiomtransverseshear coupling (ai6 of the compliance matrix) in the cross-section of a fabric composite was neglected. However, actually in plain-weave composites, the effect of tension/transverse-shear coupling on the elastic behavior is not insignificant because of yarn crimp. The second type of study, elastic and stress analyses using finite-element methods, can be represented by the works of Whitcomb and Foye.6 They conducted finite-element analyses of unit cells of laminates of out-of-phase stacking configuration. The elastic properties and stress distributions were investigated under tensile, shear and flexure loads. Although finiteelement analysis can adequately deal with threedimensional geometry and provide detailed stress analysis, the analysis is problem-specific and it is not suitable for parametric studies. Moreover, in plainweave laminated composites, the tension/bending coupling effect (b,, of the compliance matrix) can be suppressed by the neighboring layers. When the bending effects are negligible, simple models instead of finite-element analysis can provide stress distributions with sufficient accuracy. In this paper, iso-strain models and flexural models are developed to obtain the elastic properties and stress distributions of plain-weave composites. The results from these models are evaluated by finiteelement analyses. The tension/transverse-shear coupling and tension/bending coupling effects are investigated in terms of yarn undulation and laminar stacking configurations.
Abstract models have been developed for the investigation of the elastic properties and stress distributions of plain-weave composites under in-plane tensile loading. Analyses are performed for the configurations of single lamina, iso-phase laminate, outof-phase Iaminate and random-phase laminate. The analytical results are evaluated by finite-element analyses and good agreements have been achieved. The tension/bending coupling effect is significant for the single-lamina composite. The tension/transverse-shear coupling results in differences of Young’s modulus and stress distribution between the iso-phase laminate and out-of-phase laminate. This study has led to the conclusion that in the practical design of plain-weave composites, Young’s mod&i and stress distributions can be predicted by the iso-strain model. The Jlexural model is useful in assessing the effect of local bending of yarns. 0 1997 Elsevier Science Limited
Iso-strain
University of Delaware, Newark,
and flexural
Keywords: A. plain-weave fabrics; C. elastic properties;
stress analysis; micromechanics
1 INTRODUCTION Analyses of the elastic properties and stress distributions in plain-weave composites have been reported extensively in the literature. These previous studies can be categorized into two types. The first type of approach utilizes simplifying assumptions of iso-strain or iso-stress condition in the analytical models.14 The second type of approach involves more detailed analysis based upon finite-element methods.= The first type of study can be represented by the works of Chou and Ishikawa,‘” who obtained in-plane elastic properties both analytically and experimentally. In order to investigate the effect of out-of-plane constraint conditions, they developed two models: the ‘local warping constrained’ model and the ‘local 787
M. Ito, T.-W. Chou
788
a
I
hm
I
-b-C dx
Longitudinal yarn ,
Transverse yarn 2 Fig. 1. Two-dimensional unit cell of plain-weave composite. 2 TWO-DIMENSIONAL MODEL
GEOMETRICAL
of the transverse yarn, K,, are defined from the total fiber volume fraction of the composite, V, as
A two-dimensional unit cell as shown in Fig. 1 is adopted for plain-weave composites. The composite is assumed to have uniform xz cross-section. For simplicity, gap widths between transverse yarns are assumed to be zero. The center line of longitudinal yarn, zO, is defined as
z,(x)=-Tsin
4
2%-x
( 1 a
where the longitudinal yarn thickness, h, is assumed to be constant. The upper boundary of longitudinal yarn is z,= z0 - h,/2, and the lower boundary is z, = zO+ hJ2. Thicknesses for the transverse yarn, h,, and the matrix layer, h,, for (0
n-Vfh
V,h K’= 2h,’
3 ISO-STRAIN
Kt= 4h,
MODEL
3.1 Single-lamina model For a single-lamina composite, the whole composite structure can be represented by repeating the unit cell in Fig. 1. When a tensile force is applied in the longitudinal direction, the flexural deformation as represented by bll of the compliance matrix is significant and cannot be neglected. By assuming isostrain state in a vertical element with an infinitesimal length (dx) in the x direction, the following constitutive relationships are given for the element:
h,(x) = h, sin
i_i= h,(x)=
$- - $[sin($-)
+1 ]
[$j
;i
;i
i:i(?ij)
(3)
The local fiber direction is denoted as the 1 direction and the direction perpendicular to the fiber is the 2 direction. The angle 0 between the 1 axis and the x axis is
[ij=i,li;;
(@
In eqn (6) A,, B, and D, are stiffness constants and are expressed as functions of x (0
Since the longitudinal yarn and the transverse yarn should have the same total fiber areas in the unit cell, yarn packing fractions of the longitudinal yarn, K~,and
;j[;j
=
= h,&(x)
$ (ZI(X)’1
+fl
+ h,(x)Q;
Z"(X)2)Q~(X)
h c( --I - h,(x)
‘ L\L
+ h,(x)QT
j2 - ~,(x)~j
/
J
Qij
Plain-wave composites under tensile loading
r _------- . I
1 I
.
r I
--e-_-m. .
l
----___-
789
1 I
dx
I
L-2____,
Lo-phase
Out-of-phase
Random-phase 0
Macro-cell
Fig. 2. Laminar stacking configurations of iso-phase, out-of-phase and random-phase laminates. Macro-cells of iso-phase, outof-phase and random-phase laminates consist of one, two and an infinite number of the unit cells, respectively. For the iso-strain model, the iso-strain condition is assumed in the element with infinitesimal length (dx) through the thickness of the macrocell.
D,(X)
=
f
+
(Z*(x)3-
distributed parabolically across the thickness, &, is used instead of & with a weighting function given by9
Z,(x)3)Qs(x)
$[(G-h,(x))3-zI(X)Z]Qij
+ +[(+)‘-(+
(10)
-k(x))‘]Qr
(7)
The stresses of the transverse yarn and matrix are also obtained by using the following relationship:
where @j is the transformed reduced stiffness of a longitudinal yarn with respect to the fiber angle 0. a; and @’ are the reduced stiffnesses of transverse yam and matrix, respectively. aij, b, and d, in eqn (6) are compliances. K, denotes curvature. N, is the stress resultant. The effective Young’s modulus of the single-lamina model is obtained by averaging a,, over the yarn length:
Substituting the strains of eqn (6) into the stress/strain relationship of the longitudinal yarn, the stresses of the longitudinal yarn are obtained as
Here,
assuming
that the transverse
shear stress is
3.2 Laminate model In plain-weave laminated composites, three types of laminar stacking configurations are considered: isophase, out-of-phase and random-phase laminates. For each laminate, the repeating element of the whole composite structure, referred to as the macro-cell, is shown in Fig. 2. The macro-cells of the iso-phase, outof-phase and random-phase laminates consist of one, two and an infinite number of unit cells, respectively. For each macro-cell, the iso-strain condition is assumed in the element with infinitesimal length (dx) in the x direction. Each macro-cell satisfies the
M. Ito, T.-W. Chou
790
boundary condition in terms of deformation between macro-cells. We assume that flexural deformations can be neglected in the laminates because the flexural deformation of each layer is constrained by the neighboring layers. When the waviness ratio of the yarn, defined by h,la, is large, flexural deformations can no longer be neglected, and another stress analysis model is developed in Section 4. For the iso-phase and out-of-phase laminates, the relationship between stress resultant and strains in the infinitesimal element is
where A, for the iso-phase laminate are identical to those of eqn (7). For the out-of-phase laminate, A, are identical to those of eqn (7) except for Al6 and Az6, which are canceled in the macro-cell. The effective Young’s moduli of iso-phase and out-of-phase laminates are obtained by eqn (8). The stress distributions in the longitudinal yarn are obtained by substituting the strains of eqn (12) into
Stresses in the transverse yarn and matrix are also obtained by using the corresponding stiffness matrices. For the random-phase laminate, A, of an infinitesimal element in the macro-cell are independent of X, and the averaged stiffness constants, Aij, are defined by a&?
A,(X) dX
Thus, the effective Young’s modulus phase laminate is given by E,=
1 h$rl
(14) of a random-
(IS)
where tiij are the averaged compliances, which are obtained by inverting Ai> Though the Young’s modulus is an averaged property of the laminate, the stress distributions in a given layer of the random-phase laminate are strongly affected by the relative location
of the surrounding layers. Thus, the iso-strain condition in the macro-cell defined in Fig. 2 does not give a good approximation for the stress distributions of the random-phase laminate. Whitcomb et al.]’ have investigated the boundary effects in plain-weave composites. In their work, the boundary effect of a given layer propagates at most into the adjacent layers. Thus, as shown in Fig. 3, we assume that the stress field of a given layer (layer A) is affected by the upper adjacent layer with phase shift 4,” (layer B) and the lower adjacent layer with phase shift 4,” (layer C). Here, 4,” and 4,” are defined by using the shifts s,” and s,” of layers B and C, respectively, with respect to layer A as
The iso-strain condition is assumed in the element with infinitesimal length (dr) in the x direction through the thickness of the three layers. In eqn (12) A, is defined by Aij(x) + Aij[x - a( 4,8/2T)] + A,[x - a( +:/2 T)] based upon eqn (7). Using the same method as for the iso-phase and out-of-phase laminates, the stress distributions of layer A are obtained as functions of x, 4,” and $,“. In the same way, the stress distributions of layer B (or C) are defined by 4: (or 4:) and a phase shift with respect to the other adjacent layer. Thus, for the stress analysis of the random-phase laminate, each layer of the laminate can no longer satisfy the boundary condition between layers in terms of deformation. The stress distributions of layer A are equal to those of the iso-phase laminate when 4,” = 4,” = 0.
4 FLEXURAL MODELS When the waviness ratio of a composite laminate is large, flexural deformations must be considered in the analysis. Flexural deformations due to local fiber undulation in unidirectional composites have been studied by a number of authors. Lee and Harris” investigated the effective Young’s modulus and strength of unidirectional composites with iso-phase and out-of-phase curved fibers. In their work, the fibers were isotropic and the surrounding matrix carried no load. Here, the models of “for unidirectional composites are modified to deal with plainweave laminates. 4.1 Iso-phase As shown in consideration yarn part and
laminate model Fig. 4, the infinitesimal element under is divided into two parts: a longitudinal a combined part of matrix material and
791
Plain-wave composites under tensile loading
X
h
_)
c-
Layer B
c-
Layer A
c-
Layer C
I
Fig. 3. Iso-strain model for the stress distributions of a given layer A of a random-phase laminate. Iso-strain condition is assumed in the element with infinitesimal length (d_x)through the three layers (layers A, B and C). The shifts of layers B and C with respect to the layer A are s,” and s:, respectively.
a
c
Combined Part
quz Longitudinal yarn part
s, qlx
Fig. 4. Force and moment equilibrium of an infinitesimal element in the iso-phase laminate. The infinitesimal composed of a longitudinal yarn part and a combined part of matrix and transverse yarn.
element
is
792
M. Ito, T.-W. Chou
transverse yarn. Equations for force and moment equilibrium for (0
-dSY dx
to in-plane loading and is given by
Igo= 4112+ 412 =
dPY - dx + qlu
0
ASylPy+Sp,sy) h ( Y
(17)
The other term, +, termed as a pseudo-strain, is a contribution to axial strain by the bending rotation and is expressed as”
- qlx = 0
-dMY +py- dzo - + h&q,, + q,J - s, = 0 (19) dx dx
Eps=
$$)‘-(Z
+*y)2]_ (27)
for the longitudinal yarn part, and
d&,, - dx + 9”Z
- 412 =
dp,, ~dx + 91x dzo
P”, -
dx
4”X =
0
(20)
0
(21)
+ qux) - & = 0
(23)
where wy is the displacement of the yarn center line in the z direction, and I& and & are rotation angles caused by bending and shear deformation, respectively. Provided that & is uniform in the z direction and the normal stress in the z direction is neglected in the longitudinal yarn, & is given by
(&Py + S&S,)
(24)
Y
where 3; are the transformed reduced compliances of the longitudinal yarn and are obtained by inverting eqn (13). The axial strain of the longitudinal yarn at the center line is defined by y=
eYO+ x
-d@Y
Yh
_de,
%s
c;=
(22)
for the combined part. Here qu, qu,, qk and q,z are distributed forces per unit length in the x direction. Rotation of the center line of a longitudinal yarn is given by
Ex
&,U= Ej; + -hY
2
- $ (h - h&,x
yh = +
Hence, the axial strains at the upper (z = z0 - hy/2) and lower (z = z. + hy/2) boundaries of the longitudinal yarn are given, respectively, by
(25)
where E:’ is the strain along the yarn center line due
EY_ x
2
dx
dx
(28)
(29)
The bending equation of the longitudinal yarn is
M=-I,% Y
S:,
dx
(30)
where I, = h$12 is the bending stiffness of the longitudinal yarn. In the combined part, shear deformation caused by bending rotation of the longitudinal yarn is illustrated in Fig. 5.l’ Thus, the constitutive relationships of the combined part are given by P,, = (h &r = G0
hy)Eud
lc’y + (h - hy)&I
(31) (32)
where E,,, and G,, are the effective Young’s modulus and shear modulus of the combined part and are determined from the rule of mixtures: 2h,E, Eu, =
+ h,E,
h - h,
,
G,,=
(33)
Here, Et and G, are, respectively, Young’s modulus and shear modulus of the transverse yarn. E, and G, are the matrix Young’s modulus and shear modulus, respectively. With the above deformation and constitutive relationships, differential equations for force and
Plain-wave composites under tensile loading
793
After deformation
Before deformation
Longitudinal yarn part
Combined part
Longitudinal yarn part
Shear deformation by bending rotation 0, - 9, = Fig. 5. Shear deformation
moment equilibrium expressed as
Ah-hy ”
of the combined part due to bending rotation of the longitudinal yarn.
in the infinitesimal
element are
Py + (h - h,)E”l&; = P S, + G,[h +, + (h + hy) Cl
al2
L,(~,,P,,,S,)
cos( 2(2i;
l)‘r ) dx=O
(34)
=0
[i = 1,2,. . .]
(39)
[i = 1,2,...]
(40)
(35) al2 L,~$Y,PY,SY)cos(
-
hyG,,(&Cy) +Sy=O
(36)
2(2i;1)li)dx=0
and, at x=0 L2(rcIY,PYSY)
where P is an applied uniform force per thickness h. Since E: and & are functions of Py, S, and I&,, four differential equations, eqns (23) (34)-(36), are defined with respect to the four variables wY, P,,, S, and $,,. These differential equations can be solved approximately by means of the Galerkin method expressed by
c
CljCOS
a12
I
(41)
0
2(2i - 1)7r
n
i=1,2,...
cos(2(2i;1)T
p=
where the linear differential operators L,, L2, L3 and L4 denote eqns (23),(34)-(36) respectively. In order to satisfy the boundary conditions, wY I&, Py and S, are approximated by
WY=
L(wY~~Y~pYsY)
-
(
X
a
(42) )
)dXz()
0
[i = 1,2,. . .]
(37)
eyz
i
c2jps
i=1,2,...
2(2i-ljT
(
x
a
1
(43)
u/2 [LZ(~Y,PY,SY)
-
PI as
dx=O Py=C30+ [i = 1,2,. . .]
(38)
f: i=l,Z,...
4i7r
C3iCOS
-X (
a
1
(44)
794
M. Ito, T.-W. Chou 2(2i - 1)7r
(
cqicos
s, = i
i=l,Z,...
a
x 1
(45)
Substituting eqns (42)-(45) into eqns (37)-(41) Cji are obtained by solving the system of 4n + 1 equations. Calculations are conducted numerically by using software that can handle the symbolic computation.” In the calculation, h,, h,, E,, and GUI must be approximated by Fourier series representations to avoid discontinuities at x = 0 and a/2. The effective composite Young’s modulus is given bY
4.2 Out-of-phase laminate model The infinitesimal element for the out-of-phase laminated composite is composed of a longitudinal yarn part, an upper matrix part and a lower combined part of matrix and transverse yarn (Fig. 6). With respect to the longitudinal yarn part, equations for force and moment equilibrium are given by eqns (17)-(19). The force equilibrium equations for the other two parts are
--dpu dx
cl”,= 0
dP, + q,x = __ dx
The stresses in the longitudinal yarn are defined by
ux
Y=
P
Y
hY
-
_
2,
&I
d@
_,(T;=o,(TY
xz
dx
=
S
Y
(47)
P, = h,E,&
hY
#y + YP,
(48)
Y
(51)
PI = (h, + h,)E,&f,
(52)
4”Z = K”(W, - %)
(53)
4lz = - mw, = 0, c+:, = G, &
(50)
The deformation and constitutive relationship for the longitudinal yarn are identical to those of the isophase laminate (eqns (23)-(30)). Constitutive relationships for the other two parts are
where z, is the distance from the yarn center line. Stresses in the transverse yarn are defined by
a: = E;E~, u;
0
(49)
+ w0)
(54)
where E, is Young’s modulus of the lower combined part and calculated by using the rule of mixtures in the same way as eqn (33). KU and K, are equivalent spring
Upper matrix part
Longitudinal yarn pert
Lower combined hm+h part I
equilibrium of an infinitesimal element for out-of-phase laminated composite. The infinitesimal element is composed of a longitudinal yam, a upper matrix and a lower combined part of matrix and transverse yarn.
Fig. 6. Force and moment
Plain-wave composites under tensile loading
constants of the upper matrix and the lower combined part, respectively, and are given by Ku=
+,Kl= m
h,E,
(46) and (47) except for a: which is defined as #=
&E,
quz+412 +
z
2
(55)
+ h,E,
795
s,
sin
o
+
412- 4”Z
h,
%
2, (60)
Stresses in the transverse yam are w,, in eqns (53) and (54) is a constrained displacement and it is introduced to achieve zero sum of forces in the z direction at the upper and lower surfaces of the unit cell. This condition leads to 5 FINITE-ELEMENT
a/2 (quz + s,,) dx = 0 I0 Hence, the differential (23) and
-dSYdx
K&v,
(56)
equations to be solved are eqn
Py + h,E,e,”
+ (h, + h,)E,e;
(61)
= P
- wO) - K,( wy + w,,) = 0
(57)
MODELS
In order to evaluate the analytical models, finiteelement analyses are conducted by using ABAQUS Version 5.3-l. The single lamina, iso-phase laminate and out-of-phase laminate are modeled with the common unit cell (Fig. 7) under the corresponding boundary conditions. Plane stress elements (CPS8R and CPS6R) are used for the analyses. The boundary conditions are
(58) .(-+,z)=O,
+,z)=u,,
(62)
for the single-lamina model, +
h,E,
-&(h,.$)
+ + h, $$
+ s, = 0
(59)
These four differential equations and the constraint condition of eqn (56) can also be solved with the Galerkin method. Definitions of the effective Young’s modulus and stresses of the longitudinal yarn are identical to eqns
Fig. 7.
.(-+x)=0, +,+,, v(x,-
Unit cell used for finite-element analysis.
$)-u(,,t)=const.
(63)
M. Ito, r.-W. Chou
796
Table 1. Elastic properties of AS4 graphite and epoxy random
phase
Properties
AS413
Epoxy14
Err &, G2 G3
221 16.6 8.27 5.89
3.45 3.45 1.28 1.28
(GPa) @Pa) (GPa) @Pal
0.26
v12
20 1n ..Y
I
-
.
-
-
Flexural Model
---
Finite Element Modd ofcross-ply hminate is~ndkatede by an arrow
Ex 0
0
0.02
0.04
0.06
0.08
0.1
Waviness Ratio (by/a) Fig. 8.
lamina,
Effective Young’s moduli of cross-ply laminate, single iso-phase laminate, out-of-phase laminate and random-phase laminate.
for the iso-phase
laminate
model,
+g=o, .(x.-
5)
for the out-of-phase
to be 0.5 and h,lh = 0.48 was used. The elastic properties of the longitudinal and transverse yams were calculated by using micromechanical analysisi and the respective yarn packing fractions (eqn (5)). In the flexural models, the number of terms in the series of the Galerkin method is IZ=3 for adequate convergence. Table 2 summarizes the numerical analyses performed for the various mathematical models. assumed
Iso-strain Model -
and
.(~,z)=uo
=O,.(,,+) laminate
6.1 Effective Young’s moduli Figure 8 shows the variation of effective Young’s moduli with respect to the waviness ratio h,lu. Since the waviness ratios of plain-weave composites are about 0.02 to O-03 (Ref. 3) and O-05 to 0.08 (Ref. 16) from the sample measurements, the calculations were (64)
=const.
Table 2. Summary
Numerical analyses have been conducted with respect to the mathematical models developed in Sections 3-5. The material selected for calculations is AS4/epoxy; the elastic properties of the constituent materials are shown in Table 1. Total fiber volume fraction was
diagram
of stress components
of
the models analyses
Single lamina Iso-strain Fiexural Finite element a
and their locations
used
for
numerical
Laminate configuration
Models
model.
6 ANALYTICAL RESULTS AND DISCUSSION
Fig. 9. Schematic
0.35
X
X
Iso-phase
Out-ofphase
Randomphase
Xa X X
Xa X X
X” -
Bending effect not considered.
calculated
in the stress
respectively, the center line, upper boundary and lower boundary
analysis.
of the yarn.
CL, UB and LB denote,
797
Plain-wave composites under tensile loading performed at (0.01~ h,la < 0.1). The Young’s modulus of a cross-ply laminate is also shown in the figure. There is no significant difference among the iso-strain, flexural and finite-element models. With respect to laminate configuration, the magnitudes of the Young’s moduli are in the following order: cross-ply > randomThe lamina. phase > out-of-phase > iso-phase > single single-lamina composite shows the lowest Young’s moduli because of the large tension/bending coupling effect. When the waviness ratio increases. the iso-
phase laminate shows lower stiffness than those of the out-of-phase and random-phase laminates because the tension/transverse-shear coupling effect increases. The Young’s moduli of the out-of-phase laminate and the random-phase laminate show little difference and are insensitive to the waviness ratio. The predictions of Young’s moduli showed good agreement with experiments.“.”
5r
0 0.3
0.2
0.1
0.4
- - -
hestrain Model Flexural Model Finite Element Model
-_-
0
0.2
0.1
0.3
0.4
C 5
x/a
0.5
x/a
(4
(a) I
0.4,
-
- 0.3 -0.4
0
Lso-strain Model
--
‘%\\\
Finite Element Model
0.1
‘A_. 0.3
0.2
0.4
0
0.1
I
0.2
0.5
0.3
0.4
0.5
x/a
x/a
(4
@I
1
5 s
- - -_-
0.1~
ck I= 0
tso-strain Model Flexural Model Finite Element MC&l 0.1
0.2
0.3
0.4
0.5
x/a
Fig. 10. Stress distributions
of the single-lamina composite. Stresses are normalized by the far-field stress, (TV. (a) Longitudinai stress of longitudinal yarn, (b) transverse shear stress of longitudinal yarn, (c) transverse stress of transverse yarn.
11. Stress distributions of the iso-phase laminate. Stresses are normalized by the far-field stress, CT* (a) Longitudinal stress of longitudinal yarn, (b) transverse shear stress of longitudinal yarn, (c) transverse stress of transverse yarn. Fig.
798
h4. Ito, T.-w. Chou
-0.1.
-_-
i 0' 0
Finite Element Model 0.1
0.2
h-strain - -_-
I x./a
0.3
0.4
-
Finite Element
-0.2.
Model
I
I
0.5
Model
Flexural Model
0
0.1
0.2
0.3
0.4
0.3
0.4
J
0.5
x/a (W
“A,,
g +m 2 c
0.1
0.1. - - -
ISO-strainModel Flexural Model Finite Element
-_-
- -_-
Model 0 0
0.1
0.2
x/a
0.3
0.4
0.5
(4
-0.1
0
-
Flexural Model Finite Element Model
0.1
0.2
x/a (4
Fig. 12. Stress distributions of out-of-phase laminate. Stresses are normalized by the far-field stress, gw (a) Longitudinal stress of longitudinal yarn, (b) transverse shear stress of longitudinal yarn, (c) transverse stress of transverse yarn, (d) transverse stress of longitudinal yarn. 6.2
Stress distributions
Figure 9 shows a schematic of stress components and their locations evaluated in this analysis. The stresses are transformed to the l-2 coordinate system. In plain-weave composites, the stress distributions relevant to the tensile strength are hte longitudinal stress, c+yl, and transverse shear stress, LT+J*,of the longistress of the tudinal yarn, and the transverse transverse yarn, a&. Thus, in this stress analysis, cry1 and ayZ are calculated at the center line (CL), the upper boundary (UB) and the lower boundary (LB) of the longitudinal yarn. g& is calculated at the center line (CL) of the transverse yarn, In addition, the transverse stress of the longitudinal yarn, I&, is calculated for the out-of-phase laminate because it is supposed to be high in this configuration. In the isostrain models, it is noted that stress distributions of iso-phase laminate and out-of-phase laminate are uniform through the thickness of each yarn because there is no flexural deformation. Thus, it is assumed that the calculated uniform stress occurs at the center line of each yarn. All stresses are calculated at the waviness ratio of 0.05 and normalized by the far-field
applied stress, CT@Because of symmetry, the stresses are calculated in the region 0
799
Plain-wave composites under tensile loading
Figure 12 shows stress distributions of the out-of-phase laminate. In order to compare the results of iso-phase and out-of-phase laminates, the same scales are used in Figs 11 and 12. All stress distributions of the three models are in good agreement, except for ah
1.8' 0
0.1
0.3
0.2
0.4
(
x/a Fig. 14. Longitudinal stress distribution in the longitudinal yarn of the random-phase laminate at phase shifts (#J:,+:) = ( f 7~,f n). Stresses are normalized by the farfield stress, (TV
(4
04
(4 Fig. 13. Maximum stress of random-phase laminate as
functions of phase shifts 4,” and 4:. Stresses are normalized by the far-field stress, U@ (a) Longitudinal stress of longitudinal yarn, (b) transverse shear stress of longitudinal yam, (c) transverse stress of transverse yarn.
predicted by the iso-strain model. Since (+& is caused by the local bending of yarns, the iso-strain model is less effective than the flexural model in approximating &. In the case of the random-phase laminate, the stress distributions are approximated by those of a layer with two adjacent layers of phase shifts 4,” and q5,” (see Section 3.2). Varying 4: and 4: from - 71 to r, the maximum values of &, (T:* and cz& are calculated (Fig. 13). gyl assumes its maximum at (@,+x”)=( fr, *r). When <+x”&x”)=( fr, *q), the maximum value occurs at x/a = 0 and 0.5 as shown in Fig. 14. a:;? and a& become maximum at ($~,+~) = (0,O). Thus, the maximum values of (+yZand a& of the random-phase model are identical with those of the iso-phase model. 6.3 Discussion The strength of plain-weave composites depends on the relationship between a:,, (T&, CT&,g& and their critical values.‘7,18 Catastrophic failure of the composite occurs when cry1 reaches its critical value. When u);~ or u&. reaches its critical value, interfacial debonding occurs at the boundaries of longitudinal yarns, and then each layer of the laminate behaves as a single lamina, separated from the neighboring ones. Transverse cracks initiated by ai2 have relatively little effect on the composite axial strength. The single-lamina composite shows high stress concentrations in every stress component, and thus possess the lowest strength among all the laminate configurations studied. The iso-phase laminate shows a high concentration of cryZ, while the out-of-phase laminate has the lowest stress concentration. This is due to the allowed or constrained transverse shear deformation of each layer. The out-of-phase laminate has a large concentration of u&, caused by the local bending of yarns. As for the AS4/epoxy unidirectional composite, the transverse shear strength is 1.3 times of the transverse tensile strength.13 Since ur2 of the iso-
800
M. Ito,
phase laminate is four times a& of the out-of-phase laminate, interfacial debonding occurs at a lower applied stress level in the iso-phase laminate than in the out-of-phase laminate. The random-phase laminate may have the same laminate configurations locally as those of the iso-phase and out-of-phase laminate. Thus, interfacial debonding in the random-phase laminate is caused by af. The strength predictions based upon these models showed good agreement with
7 CONCLUSIONS Iso-strain, flexural and finite-element models have been developed to investigate the elastic properties _ _. and stress distributions of plain-weave composites under in-plane tensile loading. Analyses have been performed for the configurations of a single lamina, iso-phase laminate, out-of-phase laminate and random-phase laminate. The tension/bending coupling effect is significant for the single-lamina composite. The tension/transverse-shear coupling effect gives rise to differences in the Young’s modulus and aY2 between the iso-phase laminate and the out-of-phase laminate. For all laminate configurations, Young’s moduli and stress distributions (except for uz2 of the out-of-phase laminate) can be predicted with the iso-strain model. The flexural model can predict the effect of local bending of yarns. In the practical design of plainweave composites, a single lamina and random-phase laminate are often used. As the number of layers increases, the overall composite behavior changes from that of a single lamina to that of a random-phase laminate. Moreover, the growth of interfacial debonding in a random-phase laminate renders each layer of the laminate as a single-lamina composite. In both configurations, the Young’s moduli and stress distributions of the laminates studied here can be adequately predicted by the iso-strain model, which is the simplest among the models developed. Thus, the iso-strain model is the most effective one for the design and characterization of plain-weave composites. REFERENCES 1. Chou, T. W., Microstructural Design of Fiber Composites. Cambridge University Press, Cambridge, UK, 1992.
T.-W. Chou
2. Ishikawa, T. and Chou, T. W., Stiffness and strength
properties of woven fabric composites. _I. Mater. Sci. 1982, 17,3211-3220.
3. Ishikawa, T., Matsushita, M., Hayashi, Y. and Chou, T.
W., Experimental confirmation of the theory of elastic moduli of fabric composites. J. Compos. Mater. 1985, 19, 443458.
4. Naik, N. K., Woven Fabric Composites. Technomic Lancaster, PA, 1994. 5. Whitcomb, J., Three-dimensional stress analysis of plain weave composites. In ASTM STP 1110. American Society for Testing and Materials, Philadelphia, PA, 1991, pp. 417-438. 6. Foye, R. L., Approximating the stress field within the unit cell of a fabric reinforced composite using replacement elements. NASA CR 191422, National Aeronautics and Space Administration, Washington, DC, 1993. 7. Dow, N. E and Rammath, V., Analysis of woven fabrics for reinforced composite materials. NASA-CR 178275, National Aeronautics and Space Administration, Washington, DC, 1987. 8. Paumelle, P., Hassim, A. and L&n&, F., Composites with woven reinforcements: calculation and parametric analysis of the properties of the homogeneous equivalent. La Recherche Aerospatiale 1990, 1,1-12. 9. Vinson, J. R. and Sierakowski, R. L., The Behavior of Structures Composed of Composite Materials. Kluwer Academic, Dordrecht, 1987, Ch. 2. 10. Whitcomb, J., Kondagunta, G. and Woo, K., Boundary effects in woven composites. J. Compos. Mater. 1995, 29, 507-524.
11. Lee, W. J. and Harris, C. E., A deformation-formulated micromechanics model of the effective Young’s modulus and strength of laminated composites containing local ply curvature. In ASTM STP 10.59. American Society for Testing and Materials, Philadelphia, PA, 1990, pp. 521-563. 12. Wolfram, S., Mathematics I/: 2.2. Wolfram Research, 1994. 13. Hercules data sheets. Hercules Inc., 1989. 14. Weeton, J. W., Peters, D. M. and Thomas, K. L., Engineers’ Guide to Composite Materials. ASM, Metals Park, OH, 1987. 1.5. Tsai, S. W. and Hahn, H. T., Introduction to Composite Materials. Technomic, Westport, CT, 1980, Ch. 9. 16. Yurgartis, S. W. and Maurer, J. P., Modeling weave and stacking configuration effects on interlaminar shear stresses in fabric laminates. Composites 1993, 24, 651-658.
17. Ito, M., Effects of yarn undulation on the stress and deformation of textile composites. PhD dissertation, University of Delaware, 1995. 18. Ito, M. and Chou, T. W., An analytical and experimental study of strength and failure behavior of plain weave composites. .I. Comp. Mater., 1977, 31.