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Elastic response of multi-directional coated-fiber composites
Composites Science and Technology 31 (1988) 273-293
Elastic Response of Multi-directional Coated-fiber Composites N.J. Pagano AFWAL/MLBM, Wright-Patterson AFB, OH 45433-6533 (USA)
& G. P. Tandon AdTech Systems Research Inc., 1342 North Fairfield Road, Dayton, OH 45432 (USA) (Received 27 July 1987; revised version received 30 September 1987; accepted 9 November 1987) ABSTRACT In this work a model is developed to approximate the elastic response of a composite body reinforced by coatedfibers oriented in various directions. The fundamental representative volume element is a three-phase concentric circular ~Tlinder under prescribed displacement components. The microstress distribution inside the.fiber, the coating, and the matrix has been determined under a un(/brm three-dimensional mechanical and/or hygrothermal loading. A parametrie study has also been conducted to illustrate how a coating applied to the fiber influences the effective thermoelastic properties and can alter the state of stress at the.fiber-matrix interface and thereby modify or control an observed mode of failure.
1 INTRODUCTION The mechanical behavior of a fiber-reinforced material, particularly its failure mode, is governed in part by the transfer of stress between fiber and matrix? This transfer occurs across the interface between the components, and the properties o f this interface, therefore, will affect the properties of the composite. For example, it has been shown by simplistic models that the strength of the interface in tension has a direct bearing on the composite transverse and compressive strengths and on delamination parallel to the 273 © 1988 u s Government
274
N. Ji Pagano, G. P. Tandon
fibers; whereas, the interface shear strength principally affects the k~ad transfer length, composite fracture under conditions of fiber pullout, and the deformation of the matrix. Recently, considerable research effort 2'3 has been directed towards the tailoring of the properties of the interface to achieve desirable composite properties. Coatings of different materials and of varying amounts applied to the fiber have been used to modify the overall composite behavior, such as elastic stiffness, damage tolerance and strength. For theoretical analysis, the interphase region between the fiber and the matrix can be modeled as a coating. 4 A single fiber test s has frequently been used to characterize the fiber-matrix interface (or rather 'interphase" since the region adjacent to the fiber has its own unique properties). Quite often~ the interphase region is a product of the processing conditions involved in the manufacture of the composite. In the case of u n i d i r e c t i o n a l continuous fiber composites, a number of papers 6- lo have addressed the problems of computation of the stress field in a composite subjected to thermo-mechanical loadings and prediction of its effective stiffness. The model used in the above papers is either a two-phase model consisting of an inner cylinder with fiber properties and an outer cylinder with the properties of the matrix, or a three-phase model in which one further cylinder is added to the outside of the two-phase model with the composite properties. Recently, Mikata and Taya 11 have studied the stress field in a coated continuous fiber composite which requires a four-phase model consisting of fiber, coating, matrix and surrounding composite body. All the thermoelastic properties of the surrounding body (compositet were obtained by the use of a rule of mixtures. The solution for the stress distribution was determined with the composite subjected to three independent axisymmetric boundary conditions, namely, uniform temperature change, uniaxial applied stress and equal biaxial applied stresses. The problem of a multidirectional fiber composite was treated by Rosen et al. 12 in their material model for spatially-oriented fiber composites. However, they have used a two-phase model which does not account for the effect of the interphase between the constituents. Also, the work was largely descriptive in nature in that the mathematical structure of the model was not reported. In this paper, a multidirectional coated continuous fiber composite has been analyzed by means of a three-phase concentric cylinder model. Effective thermoelastic properties have been determined and the solution for the stress distribution derived under a uniform three-dimensional mechanical and/or hygrothermal loading. We have also conducted a parametric study to examine the influence of coating properties for constituents typical of ceramic matrix composites.
Elastic response of multi-directional coated-fiber composites
275
2 THE COMPOSITE MODEL A coated, continuous fiber-reinforced composite is modeled by a representative volume element composed of N concentric, circular cylinder elements, in which the innermost cylinder is the fiber, the next ring is the coating and the outer ring is the matrix as shown in Fig. 1. Let us further denote the composite volume between the elements as the interstitial matrix region. Both the matrix in the composite cylinder and that in the interstitial region could be reinforced by particles. (J)
(j)
Each element orientation,j, is defined via the two cylindrical angles f~ and
~b with respect to a fixed
xl-Xa-X 3
coordinate system. The local element X
1
l I
ol X
1
i
•
I •
O
F - " eSS~
FIBER| 1 F'-'-
~
l
X2 <
\\ \(
\ ,
\
~P-2
Fig. 1.
N D S A N D S model.
//'-- COATING F FILLEDMATRIX
276
N. J. Pagano, G. P. Tandon (J)
(J)
(J)
Cartesian coordinate system is represented by X 1 - X 2 - X 3. It should be further noted that the local fiber axis, X~, coincides with the z-coordinate in (J)
(j)
the local cylindrical system, whereas X 2 - X 3 is the transverse r-O plane, with 0 being measured from the X 2 axis. (J)
(J)
If we denote Qkl = cos (Xk, Xl), then from the geometry in Fig. l, we have
' Qkl =
~
I' i cos
--
(J) (~>
(J) tJ))
sin ~ cos 05
sin fl sin 05
" COS~' COS 05
COS~ sin 4)
(.i)
(./)
- sin 05
cos 05
( 1)
3 MODEL ASSUMPTIONS The fiber, the coating and the matrix are assumed to be linearly elastic, homogeneous and perfectly bonded. In general, the constituent materials may have transversely-isotropic elastic and thermal expansion coefficients. Let the composite material volume be now subjected to a set of b o u n d a r y conditions of the form? ui(S) = e°xj
or
Ti(S) = o-°nj
(2a, b)
where nj is the unit outward normal vector on the b o u n d a r y surface S, -,j are the cartesian coordinates of that surface, e.° and a ° are constants and ui and Ti denote the components of the b o u n d a r y displacement and traction vector, respectively. F o r (2) prescribed, it can be shown that gij = eO =-
const,
or
#i.j = aO = const.
(3a, b)
respectively, where an overbar denotes the average value over the whole volume. When displacements are prescribed, the average strains are eg° , and 6ij have to be found and for prescribed traction, the average stresses are a ° and g~j need to be determined. To facilitate the analysis, we now introduce in Fig. 2 an equivalent homogeneous medium, having the effective composite properties, as a comparison material. We further assume that the displacements or traction acting on the b o u n d a r y of the composite cylinder elements, S¢, can be ? The use of (2a) and (2b) in conjunction with the approach described in this work leads to bounding solutions for the composite effective moduli.
Elastic response of multi-directional coated-fiber composites
277
8c
<--1
T (a) (b) Fig. 2. (a) ND composite, (b) 'Equivalent homogeneous body'.
approximated by the boundary conditions of the equivalent material, i.e., ui = e ° x j on Sc
or
Ti = a°nj on S c
(4a, b)
For boundary conditions (4a) then, eqns (3a) apply, while for (4b), eqns (3b) are applicable. An assumption of this nature has also been made by Rosen et al. 12 in their two-phase model. Thus, under boundary conditions leading to a constant strain or stress field within a homogeneous body, the stress and the displacement inside the composite cylinder element can now be evaluated as described in the following section. 4 THREE-PHASE CYLINDER MODEL The fiber (innermost cylinder), the coating, and the matrix (outermost cylinder) are denoted by the indexp = 1, 2, and 3, and their outer radii by r 1, r2, and r 3, respectively. For each constituent material, the equilibrium equations in the absence of body forces are given by 1
o" r - -
a~ , + 7 z,o,o + z,~,~ 4 - '
1
r
a o
- 0
2 r
~,o., + raO.o + ~0z.z + -~,0 = 0
1
1
Zrz ,r + r Z°z'° + az'z + -r zrz = 0
(5)
278
N. J. P a g a n o , G. P. T a n d o n
where differentiation is indicated by a comma. Letting the indices 1, 2, 3.4, 5 and 6 refer to z, r, O, rO, zO and rz, respectively, we get the stress-strain relations for a transversely isotropic constituent a: = C1 l(e= - e=) +
(;12(/2r - - •r) ~- (~12{[;0 - -
e0)
O-r ~--- C l 2 ( g , , z - - ez) + ('22(~;r - - er) -~ C23(~; O -
Co)
O"0 = C12(~;z - -
e=) +
~'23{~;r - - Cr) ~- ('22{~c;0 - -
e 0)
{6)
r~o = C447ro rOz = C 5 5 ) : 0 z ~'rz =
~ ' 5 5 / rz
where Cij are the elastic stiffness constants of the individual materials and e=, e, and e0 are the expansional (non-mechanical) strain c o m p o n e n t s along the longitudinal (z) and transverse directions ( r - 0 ) , respectively. In these relations we have used the effective properties of the filled matrix material, as discussed later, provided particle reinforcement is present. Further, for a transversely isotropic material Cr :
C0
C4. 4 = 0 - 5 ( C 2 2
-- C23 )
t7)
The engineering strain-displacement relations are given by ?;r :
Ur.r
1 ~o :
- -
i"
1 UO,O -+-
- -
Y
Ur
~;z = Uz.z
l
t
t8)
~rO = -t" ur'O -I- Oo, r - - r U o - -
~/rz :
Uz,r -1- Ur,z
"/oz :
n o , z - + - - uz'O
1 Y
Both engineering and mathematical strain c o m p o n e n t s will be employed in the remainder of this work depending on which representation is mathematically expedient. Substituting eqns (6) and (8) into {5), the
279
Elastic response of multi-directional coated-fiber composites
governing field equations for u,, Uo and u, can be expressed as
(
+1
C2 2 Ur,rr r ur'r
_iu,
r 2 ,]
1
+ c. 1
-~ Ur'O0 -{- C5 5Ur,zz 2I- (C2 3 "{- C44) r u°'r°
1 -- (C22 -{- C 4 4 ) ~ [!0,0 dr- (C12 --[- C55)Uz,rz = 0
(9) +C55 u z . . + r U z , , + ~- z,OO +C11u~.z~=0
C44
U O , r r + ~ U o , , - - - r- 2 Uo "-[- C22 ~Uo,oo + C55uo,zz -~-(C23 + C44 ) art 0
-~- (C22 + C 4 4 ) ~ U r , 0 "~ (C12 + C55 ) IIz,Oz ~- 0
In eqns (5)-(9) listed above, we have omitted the superscripts for the sake of brevity. However, in the remaining portion of this work they will be used for clarity. The nature of the boundary conditions (2a) or (2b) dictates the general form of the solution to the displacement and stress field within the constituents. In this study, however, we will focus our attention on the displacement boundary conditions since this has traditionally led to better agreement with experimental measurements of composite moduli. With the composite volume now subjected to prescribed displacements given by (2a), then, within each of the N elements, we have three displacement fields (in materials (1), (2) and (3) or fiber, coating and matrix, respectively) of the type (J,P)
(j,P)
(j,P)
(j,P)
(J,P)
(j,P)
(j,P)
(j,P)
(j,P)
(j,P)
ur (r, 0, z) = U l (r) cos 20 + U2(r ) sin 20 + U3(r ) + U4(r)z cos 0 + Us(r)z sin 0 (j,P)
Uo(r,O,z)= Vl(r)sin20+ Vz(r)cos20+ V4(r)zsinO+ Vs(r)zcosO
(J,P)
(j,P)
(j,P)
(j,p)
uz (r, 0, z) = z W3(r ) + W4(r) cos 0 + Ws(r) sin 0
where j = l , 2 ..... N
p=1,2,3
(10)
N..L Pagano. G. P. Tandon
280
(J,P)
(J.P)
[j,p)
and Ul(r), U g r ) . . . Ws(r ) are defined as (J,P)
(j,p~
(j,~
Ul(r)=Alr 3 f
{j,p)
+ A2
(J,p)
-~.T+Aar+-
(J'P)
A4 r
(J'P) ]
(J,P)
~3C2~ + C23)(i,,,~
o,p) Vl(F ) =
,
A:
f (J,P)
,J4,~
A1 v3 -~- 73 -- A 3 v +-?----'(]~}S->'_ A a
(j,p)
2C23 (J,m
~J,p)
Uz(r)= Blr 3
LJ,P) ]
~('2~- C,2) (i.,,~
2C22r
(J,P) + B2
(tip)
O.v~
~+
B3r+
B~ --
r
lJ,e)
V2(r) =
~3C22q- Cea (J,P)
/
2C23 (j,P)
(j,p)
U3r= D l r +
(j,p)
IJ,P)
~ C2 2 -- C232
t&p)
2C22r
(j,P) D 2 .....
r
(j,p)
(11)
W3(r ) = D 3 U,P
B2
B l r 3 . . . .V3 . . + B~r + . . . . . (jT~........ D4-
(J,p)
Ua(r ) = F 4 (J,P)
(J,P)
V4(r) = F3 (J'P)
(J,P) F5
W,~(r) -- ~
(J,P)
r
[J,P)
+ F6r
(J,P)
Us(r} = Ha (j,P)
(j,P)
Vs(r) = H 3 (j,p)
(J'P)
Hq
lj,p)
Ws(r } = - - ~ + H6r r
(J,p) (J,p)
tj,p)
where At, A2 . . . . H 6 are constants. A general solution similar to this form has been successfully employed by Pagano ~3 to determine the stress field in a cylindrically anisotropic body under two-dimensional surface tractions. Using the strain-displacement
281
Elastic response o f multi-directional coated-fiber composites
equations (8) and the stress-strain relationships (6), the stress field is expressed as (J,P)
(J,P}
(J,P)
(j,P)
a2 = cq (r) cos 20 + 0t2 (r) sin 20 + % (r) (J,P)
(J,P)
(j,P)
(J,P)
ao = fll (r) cos 20 + f12 (r) sin 20 + f13 (r) (J,P)
(J,P)
(j,P)
(j,P)
a, = (x (r) cos 20 + (2 (r) sin 20 + (3(r) (12) (J,P) (j,P) "grz = 64 (r) (J.P)
(j,P)
cos 0 + 65 (r) sin 0 (j,P)
(J,P)
Z~o = ~4 (r) sin 0 + ~5 (r) cos 0 (J,P)
(j,P)
(j,P)
r,o = 71 (r)sin20 + ?2 (r)cos20 (j,P) (j,P) (j,P) (j,P) where ~1 (r), ~tz (r)... ?x (r), ?2 (r) are defined as, r (J,P)
(J,P)
"~
(j.v,
(J.p, { ¿J.p) ~.v) ) J 3 A 1r 2 ' A, |
(j,p)
i/ (j.p)
~l(r) = C12 kC23 --
fl, (r)= [C22
t.i,v~
22/.1 (j---~-~ -I- (j-j~-~( [ C23 C22r2 J (J,P)
(j,P) xI -
-
C23)
--
(s,p) "~
{ is,v)
(1 (r) = ~C22 -- C23 )
(j,p) ~
(J,P)
3 k C22 + C23 J ~)r2
( -
-
(J'P)
-[- 7
C23
--
{ (J,P) (J,P)~ (J,P) (j,P) 'C22 + C23] A4 3A2 (J'P)} (J'P) r2 r* F A 3 C22
((J,P)
u,v)
71 (r) =
f
~
(j,p) ~
3 ~C22 "~- C23 )
(J,P) (J,p)
A1 r2
(j,p)
6 A2 r4
C23 f (J,P)
(J'P) 2A 3
(J,P)
(j,P) ~ (J,P)'~
C2 3 -+- C2 2 / A 4 [ (j,p)
F'2
C22 (j,P) "~
X 0"5 kC22 -- C23; r (J,p)
(j,p)
(j,p) f(j,p)
(j,p) X
[3Btr2
(J,p)
B, [
o~2(r)=C12 ~C23--C22 ) ~ t~.~ + ~ [ C23
C22r2J
(13)
IA ~c.
^,<
0
IA ~c.
IA ~c,
IA
~.
IA
IA
o
O
o-
,- - .
~9
~9
II
~
II
II
"-b
"~
~
~
+
I
+
+
II
II
~
o,
I
I
II
~
+
I
II
I
~c-.
I
~c:,
~c-. ~c.
+
s
I
4-
~
I
s
II ~
I
4-
II ~
~
i
I
II ~
s
mc
x
+
,"b.c.
~
II
~.c.
+
1",4
~c.
~c..
~c:.
II
.x~ c.
I
i~c..
"~
~c..
I'J
4-
i,,a
~c..
I
i',4
II
i.,a
b~
Elastic response of multi-directional coated-fiber composites (j,P)
(j,P)
(j,P)
283
(J,P)
Also A1, A2... Hs, H6 are constants to be evaluated by the following interface/boundary conditions: (i)
Displacements (eqn 4a) are prescribed at the outer boundary of the composite cylinder assemblage; (ii) Displacements and traction must be continuous across the fiber-coating and coating-matrix interfaces, which leads to the following equations: (j,l) ((j)'~ (j,2) ( (j)'~ ek \rl,] = Pk ~,rl}
(k = 1, 2,... 5)
(15)
(j,2) ((j)'~ (j, 3) ((j)'~ Pk ~,r21 = Pk \r2J
where P = U, V, ( and 7
ifk=l,
2
P = U, W and (
if k = 3
P = U, V, W and 6
if k = 4, 5
and (iii) Displacements and stresses must be bounded at the origin, r--0, which lead to the following identities: (j,l)
(j,1)
(j,1)
A 2 = A 4 = 0
B2 =
(j,1) B4 = 0
(j,1)
(j,l)
(j,1)
D2 = 0
F5 = 0
H5= 0
(16)
5 D E T E R M I N A T I O N OF STRESS AND DISPLACEMENT FIELDS (j)*
Let ~k~ represent the volume averaged strains in the local Cartesian coordinate system within the jth element. Thus, we have (j)*
(j} (J)
ek, = Qk,,Q,,e°.
(j = 1, 2... N)
(17) (j)
with (2a) prescribed. Here the components of the transformation matrix Qu are defined in (1) and the double-index strains represent the mathematical (j)*
strain components. Letting ui represent the displacement components on the boundary of the j t h element in local coordinates, we get (j)* (j)* (j) u, = ~i~X~
on Sc
(18)
284
N. J. Pagano, G. P. Tandon
where no summation on the superscripts is implied. With (18) prescribed, the displacement and the stress field inside the composite cylinder assemblage follow from eqns (10) and (12), respectively. Using (3), the stress and the strain field in the interstitial matrix region can be very easily evaluated using its effective properties. The composite sphere model introduced by Hashin t4 has been employed to determine the properties of the particle-reinforced matrix. In this model the upper and lower bounds for the effective bulk modulus are shown to coincide, whereas an approximate expression which gives values which lie between the bounds has been reported for the effective shear modulus. The composite stress, 6~j, can now be determined by volume averaging the stress field over the constituents, namely, the composite cylinder elements and the interstitial matrix. The stress-strain relation for the composite now takes the form ~ij = (~ijkt (gkl -- ~k~)
(19)
where Cijkl is the effective elastic stiffness, g~j = e° and ~j is the expansional strain of the composite.
6 EFFECTIVE MODULI To evaluate the effective elastic moduli, we set the expansional strain components identically equal to zero, i.e., (J,P)
e~ = 0
(for all .LP)
(20)
With eqn (20), the stress-strain relation for the composite, eqn 19, therefore reduces to (using contracted notation) ~i = Cijgj
(i,j= 1,2 .... 6)
(21)
By setting each strain component equal to 1 individually, while all others are zero, we shall respectively obtain the j t h column of the Cij matrix: The composite engineering constants can now be defined in terms of the elastic compliances, 5'i~7 EXPANSIONAL STRAIN The expansional strains of a body subjected to thermomechanical loading can be computed in the following manner: (j,P)
Consider the case where ekt, the local material expansional strains, are
Elastic response of multi-directional coated-fiber composites
285
given their actual values according to some external stimulus, such as a temperature change. Suppose, we set ei~ = 0. Then, from eqn (19), we have eij =
-- Sijkl~kl
(22)
where the elastic compliance, Siik~, has already been determined and the composite stress, Okt, can be computed as explained earlier.
8 INITIAL F A I L U R E The problem of the analysis of failure of composite materials is much more difficult than the problem of effective elastic property prediction which has been discussed until now. When a composite specimen is subjected to increasing load and/or temperature, microfailures will develop at some stage. These may be in the form of matrix cracking with fibers bridging, fiber breaks, interface separation or debonding and local plastification. As loading continues, they will multiply and ultimately merge to produce catastrophic failure. Even the initial failure predictions are difficult to make because of the complexities involved, such as the material microstructure, existence of flaws and irregularities, material anisotropy, etc. The present model is considered as a first approximation to the addressing of certain important features of composite physical response, such as initiation of debonding or matrix cracking, or at least to the provision of material guidance to eliminate or minimize these failure mechanisms. The model contains quite drastic assumptions such as ideal elasticity, perfect bonding, homogeneity within phases, and solely orientation-dependent response of the cylindrical elements. Some of the classical theories of failure, such as the maximum stress, maximum shear stress, and maximum principal stress theories, in conjunction with extensive experimental observations, can be employed to gain an understanding of the factors controlling initial failure modes. These studies will also be used to direct further improvements in the model itself.
9 N U M E R I C A L RESULTS A N D DISCUSSION In order to illustrate the application of the present approach, we have selected a unidirectionally-oriented composite (aligned along xl) with a 0.60 fiber volume fraction for the analysis. It is assumed that no interstitial matrix exists. The material properties used in the calculations, representative of Nicalon
286
N. J. Pagano, G. P. Tandon
fiber and barium magnesium aluminosilicate (BMAS) matrix, both assumed to be isotropic, are listed as follows: Material
E (GPa)
G (GPa}
,z~(10-6/~Ct
Nicalon BMAS
200"0 106.0
77-0 43"0
3"2 2.7
The BMAS properties correspond to those of the bulk material. Although matrix and coating (possibly the fibers as well) in situ properties may be strongly influenced by processing conditions, this effect is neglected throughout this work owing to the lack of definitive data. As an approximation, the curing or the residual stresses can be estimated by subjecting the composite to a uniform temperature change. In this problem the only non-zero stress components predicted by the present model are ~r, ao and a=. The radial stress component near the fiber-matrix interface can be considered as a failure criterion for debonding, e.g., a negative value of at promotes contact between the fiber and the matrix at the interface, whereas a positive value suggests possible separation and initiation of debonding at this boundary. If 1000°C is chosen as a typical value of the cooling range, we calculate that a tensile stress of magnitude 15.27 MPa exists at the interface for this Nicalon/BMAS system. This tensile stress certainly is a possible explanation for the significant amount of debonding that has been reported by Kim 1~ on examining the SEM pictures of the composite. He also measured the transverse Young's modulus of the composite, E22, as approximately 20.7 GPa. For a 0"60 fiber volume fraction, the theoretical calculations, assuming perfect bonding, give a value of 152"84GPa. However, if we assume that the fibers are completely debonded, then we can approximate them as cylindrical holes in the matrix. This assumption leads to a theoretical estimate of 22.75 GPa, which is close to the experimentally measured modulus. We next conducted a parametric study to examine the influence of different hypothetical coating materials on the stress distribution when the composite was subjected to a uniform temperature change, AT = --I~C. This was motivated by the debonding observed during the curing of this composite, as earlier stated. The fiber volume fraction was set at 0.60 but the ratio of coating thickness to cylinder outer radius, defined as (r2 - r l)/r3, was treated as an independent variable. In Fig. 3, we have plotted the variation of radial stress, a,, at the interface. It is seen that a 'thick' coating of a 'soft' material with a 'low' coefficient of thermal expansion helps in reducing the stress concentration factor at the boundary, e.g., a coating thickness to cylinder outer radius ratio of 0.1 of a
287
Elastic response of multi-directional coated-fiber composites
O,(KPo)
Or(KPa) 17.5-
17-5-
~=3,~.~ ~'~ ~j" 12.5 -
"
~
12-5-
7-5-
5,~.~1"8
~
~
~3-45,1.8
2.5-
" ~ _ ~
-345¢01~
-2.5 0
I
I
.05
-1
('2" q)/'3 (a)
-2.5
I
0
.05
"345,1.8 ~'~-"-34 $ ,.018 ~3.45,.018 "34-$,.018 I
.1
(r2- rlgr3
(b) Fig. 3. Radial stress, o-,, at (a) fiber-coating interface; and (b) coating-matrix interface for Nicalon/BMAS composite under a uniform temperature change, AT = - I°C. (E¢: coating modulus (GPa); ~t¢:coefficient of thermal expansion of coating (10-6/°C).)
material having a Young's modulus, Ec, of 0.345 GPa and a coefficient of thermal expansion, ~c, of 0.018 x 10-6/°C changes the radial stress from tensile to compressive at both the fiber-coating and coating-matrix interfaces. This sign reversal can be helpful in controlling or minimizing interfacial debonding. The stress concentration is a function of both Young's modulus and the thermal expansion coefficient of the coating, besides its thickness. Figure 4 illustrates the variation of the longitudinal stress, a z, in the constituents. Under a uniform temperature change, the present model predicts a constant a= in each one of the individual materials. For E¢ < 3.45 GPa, it is observed that a z in the fiber decreases (compared to the uncoated value) while that in the coating increases, whereas for a higher coating modulus, the reverse behavior is seen to occur. The longitudinal stress, a=, in the matrix is seen to increase under all the simulated cases. The hoop stress, a o, is constant inside the fiber and discontinuous across the two interfaces. Within the coating, the algebraic maximum stress occurs at the coating-matrix interface. Inside the matrix the algebraic maximum stress occurs at the outer boundary if the stress is compressive, and at the inner boundary if it is tensile in nature. The variation of this algebraic maximum hoop stress is plotted in Fig. 5. As seen from the diagrams, it closely follows the trend depicted by the longitudinal stress component.
288
N. J. "Pagano, G. P. Tandon
az(Kp~) =
' 'a'c='018
0-
Oz(KPa)
EC='345,O-¢;= "OI8
-25~
50-0EC:=34"5,0-C="01B
3"45,'018
-31 -
375-
34'5,1"8
-373"45,'018
3-45,1"8
-43-
34s,.m8 34.5.18 "345.1"8
25"0-
-112"5-
-49-
12'5 '"
0
t
I
"05
-1
34.5,,018 I
-150.0 0
"05
('2- n)/'J (a)
3-45,1 e
-55-
J
0
"1
(r2- r ] y r 3
i
I
'05 (r2- rl)/r 3
'I
(c) (b) Fig. 4. Longitudinal stress, (r=,inside (a) fiber; (b) coating; and (c) matrix for Nicalon/BMAS composite under a uniform temperalure change, AT = - I"C. (E¢: coating modulus [GPa): 7c: coefficient of thermal expansion of coating (10 6/ C).)
GO (KPa)
% (KP.)
00 (KPa)
~
ECZ345,0-C: 01B
12"5t
.l" 8
. . ~ I ' 8
Ec=34.5 tlc=-OIB
]
/
3.45,-OI8
Ot
/~-34S.-OIB
3"45,018
1B'75-~
Ee=34"5'a'c=018 -37"5-
\
,518
750
6.25, ~-125
O-
-6.25 0
t 05
('2"'1)/'3 (a)
a -1
-150.0-.. / 0
t .05
(r2-'0/'3 (b)
-1
0
"05 ('2-',)/'3 (c)
.I
Fig. 5. Maximum hoop stress, a o, inside (a) fiber; (b) coating; and (c) matrix for Nicalon/' BMAS composite under a uniform temperature change, AT = - I°C. (Ec: coating modulus (GPa); ~¢: coefficient of thermal expansion of coating (10-6/°C).)
Elastic response of multi-directional coated-fiber composites
289
Besides the microstress distribution, we have also examined the effective elastic properties of the composite in the course of this investigation. In Fig. 6 we have plotted the variation in the transverse Young's modulus of the composite, E 22. What is particularly significant is the sensitivity of E2 2 to the elastic modulus of the coating, E c. As explained earlier, coatings/ interphases also occur as a result of processing conditions. This could therefore be an alternative explanation of the experimentally-measured low modulus value. The transverse Young's modulus of the composite decreases with increasing coating thickness, as expected, because we are replacing some matrix material with a softer material. Both the longitudinal shear m o d u l u s , G12 , and the in-plane shear modulus, G23 , follow a trend similar to E22. The longitudinal Young's modulus, E 11, on the other hand, is much less sensitive to the coating modulus as seen in Table 1. E22 (GPa)
180l lso--I~ .=34'5
60-
O[0
=
-05
=
-1
(r=-r,)/,3 Fig. 6. Transverse Young's modulus, E22, of Nicaion/BMAS composite. (Ec: coating modulus (GPa).)
N. J. Pagano, G, P. Tandon
290
TABLE 1 Unidirectional Composite Coating thickness
Coating modulus (GPa)
E11
E22
G12
G23
(GPa)
(GPa)
(GPa)
(GPa)
--
162"530
152'840
60"615
60,357
0"01
0'345 3'450 34"500
160"833 160'922 161-412
45-148 109"741 148"031
18"361 43.611 58-697
18-010 42'81t 58"263
0'05
0"345 3'450 34'500
153"986 154"296 156"799
22'891 58"332 131'901
10"149 23"057 52"043
9"232 22-850 51'464
0"10
0.345 3.450 34-500
145"002 145"560 150"704
14-624 37-123 115-995
6-778 14'852 45"307
5"826 14-500 44-914
Cylinder outer radius
To conclude, it is apparent that generally a reduction in the s t r e s s concentration can be made at the expense of the elastic moduli of the composite. Further, by a proper choice of coating thickness, modulus and coefficient of thermal expansion, the stress component of interest, which is instrumental in causing a specific mode of failure, can be controlled. We next consider a three-dimensional fibrous composite by arranging six fibers parallel to the six lines joining the opposite vertices of a regular icosahedron. These six axes can be oriented with reference to an orthogonal Cartesian coordinate system x 1 x 2 x 3 as follows: one pair in the .\-ix2 plane making angles of + 0' with the xl-axis, one pair in the X2X 3 plane making angles of + 0' with the x2-axis, and one pair in the x 3 x 1 plane making angles of + 0' with the x3-axis, where 0 ' = tan -1 (2 sin 18 °) = 31°4Y. As shown by TABLE 2 Three-dimensional 'Isotropic' Composite Coating thickness
Composite system
E (GPa)
v
Nicalon/BMAS
128-16
0.250
Nicalon/Nickel/BMAS
Cylinder outer radius
0.01
Nicalon/Carbon/BMAS
Nicalon/Polyimide/BMAS 0'10
Nicalon/Nickel/BMAS Nicalon/Carbon/BMAS
Nicalon/Polyimide/BMAS
G (GPa)
(10
('1
51"25
2.89
129.11 0 . 2 5 1 118-06 0 - 2 4 3 105-95 0 - 2 4 8
51.60 47-50 42.43
3.04 2-88 3-24
138-72 80.20 65.88
55.10 32.94 26.71
4.45 2-84 4,75
0.259 0.217 0"234
Elastic response of multi-directional coated-fiber composites
291
Rosen and Shu, 16 this type of arrangement gives rise to local isotropy. The isotropic relation. G =
E 2(1 + v) -
can be used as an independent check of the model. In general, this relation is not satisfied exactly in the present analysis, but the error is very small. The results for the Nicalon fiber and BMAS matrix system are shown in Table 2. Also shown in the table is the effect of different coating thicknesses and coating materials on the effective thermoelastic moduli. The material properties of the three coating materials used in the calculations were:
Coating material
E (GPa)
G (GPa)
o~(10-6/°C)
Nickel 11 ATJS, Carbon I 7 Polyimide i a
207.0 9-I 3"1
79"0 4.1 1"17
13-3 2.2 54"0
and the fiber volume fraction was set at 0.30. As seen from Table 2, both carbon and polyimide coatings, because of their lower elastic moduli compared to the BMAS matrix, reduce the effective elastic properties, whereas nickel, with a higher modulus, adds to the reinforcement. The thermal expansion coefficient of the composite, on the other hand, is influenced more by nickel and polyimide coatings because of the larger degree of mismatch between the constituents. The microstress distribution within the constituents, for a multidirectional fiber composite, in general, depends both on the type of loading and the fiber orientation. For the specific three-dimensional composite under consideration, the stress distribution remains identical for the six fiber orientations when the composite is subjected to a uniform temperature change. The effect of different coating materials and/or thicknesses on the stress concentrations is quite dramatic. As the trends are similar to those reported in the first example problem, they will not be addressed here. 10 THE NDSANDS P R O G R A M Micromechanical considerations in composite materials may require the use of a practical tool that can handle different constituent materials, arbitrary fiber orientations and multiaxial loading conditions. To address these requirements, the computer code called NDSANDS (N Directional Stiffness AND Strength) has been developed. It can be used either to analyze a composite or to conduct parametric study. By parametric study is meant that the user can change either a material property or the geometry of the
292
N. J. Pagano, G. P. Tandon
composite, one single variable at a time while the remainder are kept constant, and thereby examine the change in effective properties and stress distribution as a result of different input values of the parameter selected. When changing the material property, we must ensure that both the stiffness and compliance matrices remain positive definite at all times. ~9 The influence of constituent properties on the effective composite behavior has been examined, s° Composite response has been predicted with the composite subjected to different loading conditions, namely, uniaxial tension, longitudinal shear, in-plane shear, transverse tension and a uniform temperature change. 11 S U M M A R Y In summary, we have developed a first-order ideal material model to approximate the elastic response of a composite body reinforced by coated fibers oriented in various directions. The coating can either be applied intentionally to achieve the desirable composite properties or it can occur as a consequence of the processing conditions involved in composite manufacture. We have further demonstrated, through a parametric study, how an applied coating modifies the micro-stress distribution and the elastic properties of a Nicalon/BMAS system. The model can be used to provide material guidance for controlling the micromechanical failure modes. In conjunction with a disciplined experimental program, studies such as those conducted here, can be employed to direct further improvements in the model, such as the ability to handle discontinuity of some of the traction and/or displacement components at the boundary. This, formulation, along with the solution for the displacement and the stress field under prescribed traction boundary conditions, will be the subject of a subsequent paper.
REFERENCES 1. G.A. Cooper and A. Kelly, Role of the interface in the fracture offiber-composite materials, ASTM STP 452, (1969), pp. 90-106. 2. H. L. Hancox and H. Wells, The effects of fibre surface coatings on the mechanical properties of CFRP, Fibre Sci. & Tech., 10 (1977), 9-22. 3. J. H. Williams, Jr. and P. N. Kousiounelos, Thermoplastic fibre coatings enhance composite strength and toughness, Fibre Sci. & Tech., 11 (1978), 83-8. 4. D.F. Adams, A micromechanical analysis of the influence of the interface on the performance of polymer-matrix composites, Proc. Amer. Soc. for Comp., First Technical Conference (1986), pp. 207-26. 5. L. T. Drzal, Composite interphase characterization, S A M P E J., 19(5) (1983), 7-13.
Elastic response of multi-directional coated-fiber composites
293
6. T. Ishikawa, K. Koyama and S. Kobayashi, Thermal expansion coefficients of unidirectional composites, J. Comp. Mat., 12 (1978), 153-68. 7. D. Iesan, Thermal stresses in composite cylinders, J. Thermal Stress, 3 (1980), 495-508. 8. R. M. Christensen and H. Lo, Solutions for effective shear properties in threephase sphere and cylinder models, J. Mech. Phys. Solids, 27 (1979), 315-30. 9. Z. Hashin and B. W. Rosen, The elastic moduli of fiber-reinforced materials, J. Appl. Mech., 31 (1964), 223-32. 10. Y. Takeuchi, T. Furukawa and Y. Tanigawa, The effect of thermoelastic coupling for transient thermal stresses in a composite cylinder, ASME, WAM, DE-2 (1983). 11. Y. Mikata and M. Taya, Stress field in a coated continuous fiber composite subjected to thermo-mechanical loadings, J. Comp. Mat., 19 (1985), 554-79. 12. B. W. Rosen, S. N. Chatterjee and J. J. Kibler, An analysis model for spatially oriented fiber composites, ASTM STP 617 (1977) pp. 243-54. 13. N. J. Pagano, The stress field in a cylindrically anisotropic body under twodimensional surface tractions, J. Appl. Mech., 38 (1971), 1-6. 14. Z. Hashin, The elastic moduli of heterogeneous materials, J. Appl. Mech., 29 (1962), 143-50. 15. R. Y. Kim, UDRI (personal communication). 16. B.W. Rosen and L. S. Shu, On some symmetry conditions for three-dimensional fibrous composites, J. Comp. Mat., 5 (1971), 279-82. 17. D. E. Walrath and D. F. Adams, Finite element micromechanics and minimechanics modeling of a three-dimensional carbon--carbon composite material, Dept report UWME-DR-501-106-1 (1985), 57pp. 18. J.C. Haipin, Primer on composite materials: analysis, Technomic Publishing Co. Inc. Lancaster, PA (1984), 166pp. 19. R.M. Jones, Mechanics of composite materials, McGraw-Hill, New York (1975), pp. 37--45. 20. G. P. Tandon and N. J. Pagano, Effect of constituent material properties on the elastic behavior of composites, Proc. Amer. Soc. for Comp., 2nd Technical Conference, University of Delaware (1987), pp. 429-40.
Elastic Response of Multi-directional Coated-fiber Composites N.J. Pagano AFWAL/MLBM, Wright-Patterson AFB, OH 45433-6533 (USA)
& G. P. Tandon AdTech Systems Research Inc., 1342 North Fairfield Road, Dayton, OH 45432 (USA) (Received 27 July 1987; revised version received 30 September 1987; accepted 9 November 1987) ABSTRACT In this work a model is developed to approximate the elastic response of a composite body reinforced by coatedfibers oriented in various directions. The fundamental representative volume element is a three-phase concentric circular ~Tlinder under prescribed displacement components. The microstress distribution inside the.fiber, the coating, and the matrix has been determined under a un(/brm three-dimensional mechanical and/or hygrothermal loading. A parametrie study has also been conducted to illustrate how a coating applied to the fiber influences the effective thermoelastic properties and can alter the state of stress at the.fiber-matrix interface and thereby modify or control an observed mode of failure.
1 INTRODUCTION The mechanical behavior of a fiber-reinforced material, particularly its failure mode, is governed in part by the transfer of stress between fiber and matrix? This transfer occurs across the interface between the components, and the properties o f this interface, therefore, will affect the properties of the composite. For example, it has been shown by simplistic models that the strength of the interface in tension has a direct bearing on the composite transverse and compressive strengths and on delamination parallel to the 273 © 1988 u s Government
274
N. Ji Pagano, G. P. Tandon
fibers; whereas, the interface shear strength principally affects the k~ad transfer length, composite fracture under conditions of fiber pullout, and the deformation of the matrix. Recently, considerable research effort 2'3 has been directed towards the tailoring of the properties of the interface to achieve desirable composite properties. Coatings of different materials and of varying amounts applied to the fiber have been used to modify the overall composite behavior, such as elastic stiffness, damage tolerance and strength. For theoretical analysis, the interphase region between the fiber and the matrix can be modeled as a coating. 4 A single fiber test s has frequently been used to characterize the fiber-matrix interface (or rather 'interphase" since the region adjacent to the fiber has its own unique properties). Quite often~ the interphase region is a product of the processing conditions involved in the manufacture of the composite. In the case of u n i d i r e c t i o n a l continuous fiber composites, a number of papers 6- lo have addressed the problems of computation of the stress field in a composite subjected to thermo-mechanical loadings and prediction of its effective stiffness. The model used in the above papers is either a two-phase model consisting of an inner cylinder with fiber properties and an outer cylinder with the properties of the matrix, or a three-phase model in which one further cylinder is added to the outside of the two-phase model with the composite properties. Recently, Mikata and Taya 11 have studied the stress field in a coated continuous fiber composite which requires a four-phase model consisting of fiber, coating, matrix and surrounding composite body. All the thermoelastic properties of the surrounding body (compositet were obtained by the use of a rule of mixtures. The solution for the stress distribution was determined with the composite subjected to three independent axisymmetric boundary conditions, namely, uniform temperature change, uniaxial applied stress and equal biaxial applied stresses. The problem of a multidirectional fiber composite was treated by Rosen et al. 12 in their material model for spatially-oriented fiber composites. However, they have used a two-phase model which does not account for the effect of the interphase between the constituents. Also, the work was largely descriptive in nature in that the mathematical structure of the model was not reported. In this paper, a multidirectional coated continuous fiber composite has been analyzed by means of a three-phase concentric cylinder model. Effective thermoelastic properties have been determined and the solution for the stress distribution derived under a uniform three-dimensional mechanical and/or hygrothermal loading. We have also conducted a parametric study to examine the influence of coating properties for constituents typical of ceramic matrix composites.
Elastic response of multi-directional coated-fiber composites
275
2 THE COMPOSITE MODEL A coated, continuous fiber-reinforced composite is modeled by a representative volume element composed of N concentric, circular cylinder elements, in which the innermost cylinder is the fiber, the next ring is the coating and the outer ring is the matrix as shown in Fig. 1. Let us further denote the composite volume between the elements as the interstitial matrix region. Both the matrix in the composite cylinder and that in the interstitial region could be reinforced by particles. (J)
(j)
Each element orientation,j, is defined via the two cylindrical angles f~ and
~b with respect to a fixed
xl-Xa-X 3
coordinate system. The local element X
1
l I
ol X
1
i
•
I •
O
F - " eSS~
FIBER| 1 F'-'-
~
l
X2 <
\\ \(
\ ,
\
~P-2
Fig. 1.
N D S A N D S model.
//'-- COATING F FILLEDMATRIX
276
N. J. Pagano, G. P. Tandon (J)
(J)
(J)
Cartesian coordinate system is represented by X 1 - X 2 - X 3. It should be further noted that the local fiber axis, X~, coincides with the z-coordinate in (J)
(j)
the local cylindrical system, whereas X 2 - X 3 is the transverse r-O plane, with 0 being measured from the X 2 axis. (J)
(J)
If we denote Qkl = cos (Xk, Xl), then from the geometry in Fig. l, we have
' Qkl =
~
I' i cos
--
(J) (~>
(J) tJ))
sin ~ cos 05
sin fl sin 05
" COS~' COS 05
COS~ sin 4)
(.i)
(./)
- sin 05
cos 05
( 1)
3 MODEL ASSUMPTIONS The fiber, the coating and the matrix are assumed to be linearly elastic, homogeneous and perfectly bonded. In general, the constituent materials may have transversely-isotropic elastic and thermal expansion coefficients. Let the composite material volume be now subjected to a set of b o u n d a r y conditions of the form? ui(S) = e°xj
or
Ti(S) = o-°nj
(2a, b)
where nj is the unit outward normal vector on the b o u n d a r y surface S, -,j are the cartesian coordinates of that surface, e.° and a ° are constants and ui and Ti denote the components of the b o u n d a r y displacement and traction vector, respectively. F o r (2) prescribed, it can be shown that gij = eO =-
const,
or
#i.j = aO = const.
(3a, b)
respectively, where an overbar denotes the average value over the whole volume. When displacements are prescribed, the average strains are eg° , and 6ij have to be found and for prescribed traction, the average stresses are a ° and g~j need to be determined. To facilitate the analysis, we now introduce in Fig. 2 an equivalent homogeneous medium, having the effective composite properties, as a comparison material. We further assume that the displacements or traction acting on the b o u n d a r y of the composite cylinder elements, S¢, can be ? The use of (2a) and (2b) in conjunction with the approach described in this work leads to bounding solutions for the composite effective moduli.
Elastic response of multi-directional coated-fiber composites
277
8c
<--1
T (a) (b) Fig. 2. (a) ND composite, (b) 'Equivalent homogeneous body'.
approximated by the boundary conditions of the equivalent material, i.e., ui = e ° x j on Sc
or
Ti = a°nj on S c
(4a, b)
For boundary conditions (4a) then, eqns (3a) apply, while for (4b), eqns (3b) are applicable. An assumption of this nature has also been made by Rosen et al. 12 in their two-phase model. Thus, under boundary conditions leading to a constant strain or stress field within a homogeneous body, the stress and the displacement inside the composite cylinder element can now be evaluated as described in the following section. 4 THREE-PHASE CYLINDER MODEL The fiber (innermost cylinder), the coating, and the matrix (outermost cylinder) are denoted by the indexp = 1, 2, and 3, and their outer radii by r 1, r2, and r 3, respectively. For each constituent material, the equilibrium equations in the absence of body forces are given by 1
o" r - -
a~ , + 7 z,o,o + z,~,~ 4 - '
1
r
a o
- 0
2 r
~,o., + raO.o + ~0z.z + -~,0 = 0
1
1
Zrz ,r + r Z°z'° + az'z + -r zrz = 0
(5)
278
N. J. P a g a n o , G. P. T a n d o n
where differentiation is indicated by a comma. Letting the indices 1, 2, 3.4, 5 and 6 refer to z, r, O, rO, zO and rz, respectively, we get the stress-strain relations for a transversely isotropic constituent a: = C1 l(e= - e=) +
(;12(/2r - - •r) ~- (~12{[;0 - -
e0)
O-r ~--- C l 2 ( g , , z - - ez) + ('22(~;r - - er) -~ C23(~; O -
Co)
O"0 = C12(~;z - -
e=) +
~'23{~;r - - Cr) ~- ('22{~c;0 - -
e 0)
{6)
r~o = C447ro rOz = C 5 5 ) : 0 z ~'rz =
~ ' 5 5 / rz
where Cij are the elastic stiffness constants of the individual materials and e=, e, and e0 are the expansional (non-mechanical) strain c o m p o n e n t s along the longitudinal (z) and transverse directions ( r - 0 ) , respectively. In these relations we have used the effective properties of the filled matrix material, as discussed later, provided particle reinforcement is present. Further, for a transversely isotropic material Cr :
C0
C4. 4 = 0 - 5 ( C 2 2
-- C23 )
t7)
The engineering strain-displacement relations are given by ?;r :
Ur.r
1 ~o :
- -
i"
1 UO,O -+-
- -
Y
Ur
~;z = Uz.z
l
t
t8)
~rO = -t" ur'O -I- Oo, r - - r U o - -
~/rz :
Uz,r -1- Ur,z
"/oz :
n o , z - + - - uz'O
1 Y
Both engineering and mathematical strain c o m p o n e n t s will be employed in the remainder of this work depending on which representation is mathematically expedient. Substituting eqns (6) and (8) into {5), the
279
Elastic response of multi-directional coated-fiber composites
governing field equations for u,, Uo and u, can be expressed as
(
+1
C2 2 Ur,rr r ur'r
_iu,
r 2 ,]
1
+ c. 1
-~ Ur'O0 -{- C5 5Ur,zz 2I- (C2 3 "{- C44) r u°'r°
1 -- (C22 -{- C 4 4 ) ~ [!0,0 dr- (C12 --[- C55)Uz,rz = 0
(9) +C55 u z . . + r U z , , + ~- z,OO +C11u~.z~=0
C44
U O , r r + ~ U o , , - - - r- 2 Uo "-[- C22 ~Uo,oo + C55uo,zz -~-(C23 + C44 ) art 0
-~- (C22 + C 4 4 ) ~ U r , 0 "~ (C12 + C55 ) IIz,Oz ~- 0
In eqns (5)-(9) listed above, we have omitted the superscripts for the sake of brevity. However, in the remaining portion of this work they will be used for clarity. The nature of the boundary conditions (2a) or (2b) dictates the general form of the solution to the displacement and stress field within the constituents. In this study, however, we will focus our attention on the displacement boundary conditions since this has traditionally led to better agreement with experimental measurements of composite moduli. With the composite volume now subjected to prescribed displacements given by (2a), then, within each of the N elements, we have three displacement fields (in materials (1), (2) and (3) or fiber, coating and matrix, respectively) of the type (J,P)
(j,P)
(j,P)
(j,P)
(J,P)
(j,P)
(j,P)
(j,P)
(j,P)
(j,P)
ur (r, 0, z) = U l (r) cos 20 + U2(r ) sin 20 + U3(r ) + U4(r)z cos 0 + Us(r)z sin 0 (j,P)
Uo(r,O,z)= Vl(r)sin20+ Vz(r)cos20+ V4(r)zsinO+ Vs(r)zcosO
(J,P)
(j,P)
(j,P)
(j,p)
uz (r, 0, z) = z W3(r ) + W4(r) cos 0 + Ws(r) sin 0
where j = l , 2 ..... N
p=1,2,3
(10)
N..L Pagano. G. P. Tandon
280
(J,P)
(J.P)
[j,p)
and Ul(r), U g r ) . . . Ws(r ) are defined as (J,P)
(j,p~
(j,~
Ul(r)=Alr 3 f
{j,p)
+ A2
(J,p)
-~.T+Aar+-
(J'P)
A4 r
(J'P) ]
(J,P)
~3C2~ + C23)(i,,,~
o,p) Vl(F ) =
,
A:
f (J,P)
,J4,~
A1 v3 -~- 73 -- A 3 v +-?----'(]~}S->'_ A a
(j,p)
2C23 (J,m
~J,p)
Uz(r)= Blr 3
LJ,P) ]
~('2~- C,2) (i.,,~
2C22r
(J,P) + B2
(tip)
O.v~
~+
B3r+
B~ --
r
lJ,e)
V2(r) =
~3C22q- Cea (J,P)
/
2C23 (j,P)
(j,p)
U3r= D l r +
(j,p)
IJ,P)
~ C2 2 -- C232
t&p)
2C22r
(j,P) D 2 .....
r
(j,p)
(11)
W3(r ) = D 3 U,P
B2
B l r 3 . . . .V3 . . + B~r + . . . . . (jT~........ D4-
(J,p)
Ua(r ) = F 4 (J,P)
(J,P)
V4(r) = F3 (J'P)
(J,P) F5
W,~(r) -- ~
(J,P)
r
[J,P)
+ F6r
(J,P)
Us(r} = Ha (j,P)
(j,P)
Vs(r) = H 3 (j,p)
(J'P)
Hq
lj,p)
Ws(r } = - - ~ + H6r r
(J,p) (J,p)
tj,p)
where At, A2 . . . . H 6 are constants. A general solution similar to this form has been successfully employed by Pagano ~3 to determine the stress field in a cylindrically anisotropic body under two-dimensional surface tractions. Using the strain-displacement
281
Elastic response o f multi-directional coated-fiber composites
equations (8) and the stress-strain relationships (6), the stress field is expressed as (J,P)
(J,P}
(J,P)
(j,P)
a2 = cq (r) cos 20 + 0t2 (r) sin 20 + % (r) (J,P)
(J,P)
(j,P)
(J,P)
ao = fll (r) cos 20 + f12 (r) sin 20 + f13 (r) (J,P)
(J,P)
(j,P)
(j,P)
a, = (x (r) cos 20 + (2 (r) sin 20 + (3(r) (12) (J,P) (j,P) "grz = 64 (r) (J.P)
(j,P)
cos 0 + 65 (r) sin 0 (j,P)
(J,P)
Z~o = ~4 (r) sin 0 + ~5 (r) cos 0 (J,P)
(j,P)
(j,P)
r,o = 71 (r)sin20 + ?2 (r)cos20 (j,P) (j,P) (j,P) (j,P) where ~1 (r), ~tz (r)... ?x (r), ?2 (r) are defined as, r (J,P)
(J,P)
"~
(j.v,
(J.p, { ¿J.p) ~.v) ) J 3 A 1r 2 ' A, |
(j,p)
i/ (j.p)
~l(r) = C12 kC23 --
fl, (r)= [C22
t.i,v~
22/.1 (j---~-~ -I- (j-j~-~( [ C23 C22r2 J (J,P)
(j,P) xI -
-
C23)
--
(s,p) "~
{ is,v)
(1 (r) = ~C22 -- C23 )
(j,p) ~
(J,P)
3 k C22 + C23 J ~)r2
( -
-
(J'P)
-[- 7
C23
--
{ (J,P) (J,P)~ (J,P) (j,P) 'C22 + C23] A4 3A2 (J'P)} (J'P) r2 r* F A 3 C22
((J,P)
u,v)
71 (r) =
f
~
(j,p) ~
3 ~C22 "~- C23 )
(J,P) (J,p)
A1 r2
(j,p)
6 A2 r4
C23 f (J,P)
(J'P) 2A 3
(J,P)
(j,P) ~ (J,P)'~
C2 3 -+- C2 2 / A 4 [ (j,p)
F'2
C22 (j,P) "~
X 0"5 kC22 -- C23; r (J,p)
(j,p)
(j,p) f(j,p)
(j,p) X
[3Btr2
(J,p)
B, [
o~2(r)=C12 ~C23--C22 ) ~ t~.~ + ~ [ C23
C22r2J
(13)
IA ~c.
^,<
0
IA ~c.
IA ~c,
IA
~.
IA
IA
o
O
o-
,- - .
~9
~9
II
~
II
II
"-b
"~
~
~
+
I
+
+
II
II
~
o,
I
I
II
~
+
I
II
I
~c-.
I
~c:,
~c-. ~c.
+
s
I
4-
~
I
s
II ~
I
4-
II ~
~
i
I
II ~
s
mc
x
+
,"b.c.
~
II
~.c.
+
1",4
~c.
~c..
~c:.
II
.x~ c.
I
i~c..
"~
~c..
I'J
4-
i,,a
~c..
I
i',4
II
i.,a
b~
Elastic response of multi-directional coated-fiber composites (j,P)
(j,P)
(j,P)
283
(J,P)
Also A1, A2... Hs, H6 are constants to be evaluated by the following interface/boundary conditions: (i)
Displacements (eqn 4a) are prescribed at the outer boundary of the composite cylinder assemblage; (ii) Displacements and traction must be continuous across the fiber-coating and coating-matrix interfaces, which leads to the following equations: (j,l) ((j)'~ (j,2) ( (j)'~ ek \rl,] = Pk ~,rl}
(k = 1, 2,... 5)
(15)
(j,2) ((j)'~ (j, 3) ((j)'~ Pk ~,r21 = Pk \r2J
where P = U, V, ( and 7
ifk=l,
2
P = U, W and (
if k = 3
P = U, V, W and 6
if k = 4, 5
and (iii) Displacements and stresses must be bounded at the origin, r--0, which lead to the following identities: (j,l)
(j,1)
(j,1)
A 2 = A 4 = 0
B2 =
(j,1) B4 = 0
(j,1)
(j,l)
(j,1)
D2 = 0
F5 = 0
H5= 0
(16)
5 D E T E R M I N A T I O N OF STRESS AND DISPLACEMENT FIELDS (j)*
Let ~k~ represent the volume averaged strains in the local Cartesian coordinate system within the jth element. Thus, we have (j)*
(j} (J)
ek, = Qk,,Q,,e°.
(j = 1, 2... N)
(17) (j)
with (2a) prescribed. Here the components of the transformation matrix Qu are defined in (1) and the double-index strains represent the mathematical (j)*
strain components. Letting ui represent the displacement components on the boundary of the j t h element in local coordinates, we get (j)* (j)* (j) u, = ~i~X~
on Sc
(18)
284
N. J. Pagano, G. P. Tandon
where no summation on the superscripts is implied. With (18) prescribed, the displacement and the stress field inside the composite cylinder assemblage follow from eqns (10) and (12), respectively. Using (3), the stress and the strain field in the interstitial matrix region can be very easily evaluated using its effective properties. The composite sphere model introduced by Hashin t4 has been employed to determine the properties of the particle-reinforced matrix. In this model the upper and lower bounds for the effective bulk modulus are shown to coincide, whereas an approximate expression which gives values which lie between the bounds has been reported for the effective shear modulus. The composite stress, 6~j, can now be determined by volume averaging the stress field over the constituents, namely, the composite cylinder elements and the interstitial matrix. The stress-strain relation for the composite now takes the form ~ij = (~ijkt (gkl -- ~k~)
(19)
where Cijkl is the effective elastic stiffness, g~j = e° and ~j is the expansional strain of the composite.
6 EFFECTIVE MODULI To evaluate the effective elastic moduli, we set the expansional strain components identically equal to zero, i.e., (J,P)
e~ = 0
(for all .LP)
(20)
With eqn (20), the stress-strain relation for the composite, eqn 19, therefore reduces to (using contracted notation) ~i = Cijgj
(i,j= 1,2 .... 6)
(21)
By setting each strain component equal to 1 individually, while all others are zero, we shall respectively obtain the j t h column of the Cij matrix: The composite engineering constants can now be defined in terms of the elastic compliances, 5'i~7 EXPANSIONAL STRAIN The expansional strains of a body subjected to thermomechanical loading can be computed in the following manner: (j,P)
Consider the case where ekt, the local material expansional strains, are
Elastic response of multi-directional coated-fiber composites
285
given their actual values according to some external stimulus, such as a temperature change. Suppose, we set ei~ = 0. Then, from eqn (19), we have eij =
-- Sijkl~kl
(22)
where the elastic compliance, Siik~, has already been determined and the composite stress, Okt, can be computed as explained earlier.
8 INITIAL F A I L U R E The problem of the analysis of failure of composite materials is much more difficult than the problem of effective elastic property prediction which has been discussed until now. When a composite specimen is subjected to increasing load and/or temperature, microfailures will develop at some stage. These may be in the form of matrix cracking with fibers bridging, fiber breaks, interface separation or debonding and local plastification. As loading continues, they will multiply and ultimately merge to produce catastrophic failure. Even the initial failure predictions are difficult to make because of the complexities involved, such as the material microstructure, existence of flaws and irregularities, material anisotropy, etc. The present model is considered as a first approximation to the addressing of certain important features of composite physical response, such as initiation of debonding or matrix cracking, or at least to the provision of material guidance to eliminate or minimize these failure mechanisms. The model contains quite drastic assumptions such as ideal elasticity, perfect bonding, homogeneity within phases, and solely orientation-dependent response of the cylindrical elements. Some of the classical theories of failure, such as the maximum stress, maximum shear stress, and maximum principal stress theories, in conjunction with extensive experimental observations, can be employed to gain an understanding of the factors controlling initial failure modes. These studies will also be used to direct further improvements in the model itself.
9 N U M E R I C A L RESULTS A N D DISCUSSION In order to illustrate the application of the present approach, we have selected a unidirectionally-oriented composite (aligned along xl) with a 0.60 fiber volume fraction for the analysis. It is assumed that no interstitial matrix exists. The material properties used in the calculations, representative of Nicalon
286
N. J. Pagano, G. P. Tandon
fiber and barium magnesium aluminosilicate (BMAS) matrix, both assumed to be isotropic, are listed as follows: Material
E (GPa)
G (GPa}
,z~(10-6/~Ct
Nicalon BMAS
200"0 106.0
77-0 43"0
3"2 2.7
The BMAS properties correspond to those of the bulk material. Although matrix and coating (possibly the fibers as well) in situ properties may be strongly influenced by processing conditions, this effect is neglected throughout this work owing to the lack of definitive data. As an approximation, the curing or the residual stresses can be estimated by subjecting the composite to a uniform temperature change. In this problem the only non-zero stress components predicted by the present model are ~r, ao and a=. The radial stress component near the fiber-matrix interface can be considered as a failure criterion for debonding, e.g., a negative value of at promotes contact between the fiber and the matrix at the interface, whereas a positive value suggests possible separation and initiation of debonding at this boundary. If 1000°C is chosen as a typical value of the cooling range, we calculate that a tensile stress of magnitude 15.27 MPa exists at the interface for this Nicalon/BMAS system. This tensile stress certainly is a possible explanation for the significant amount of debonding that has been reported by Kim 1~ on examining the SEM pictures of the composite. He also measured the transverse Young's modulus of the composite, E22, as approximately 20.7 GPa. For a 0"60 fiber volume fraction, the theoretical calculations, assuming perfect bonding, give a value of 152"84GPa. However, if we assume that the fibers are completely debonded, then we can approximate them as cylindrical holes in the matrix. This assumption leads to a theoretical estimate of 22.75 GPa, which is close to the experimentally measured modulus. We next conducted a parametric study to examine the influence of different hypothetical coating materials on the stress distribution when the composite was subjected to a uniform temperature change, AT = --I~C. This was motivated by the debonding observed during the curing of this composite, as earlier stated. The fiber volume fraction was set at 0.60 but the ratio of coating thickness to cylinder outer radius, defined as (r2 - r l)/r3, was treated as an independent variable. In Fig. 3, we have plotted the variation of radial stress, a,, at the interface. It is seen that a 'thick' coating of a 'soft' material with a 'low' coefficient of thermal expansion helps in reducing the stress concentration factor at the boundary, e.g., a coating thickness to cylinder outer radius ratio of 0.1 of a
287
Elastic response of multi-directional coated-fiber composites
O,(KPo)
Or(KPa) 17.5-
17-5-
~=3,~.~ ~'~ ~j" 12.5 -
"
~
12-5-
7-5-
5,~.~1"8
~
~
~3-45,1.8
2.5-
" ~ _ ~
-345¢01~
-2.5 0
I
I
.05
-1
('2" q)/'3 (a)
-2.5
I
0
.05
"345,1.8 ~'~-"-34 $ ,.018 ~3.45,.018 "34-$,.018 I
.1
(r2- rlgr3
(b) Fig. 3. Radial stress, o-,, at (a) fiber-coating interface; and (b) coating-matrix interface for Nicalon/BMAS composite under a uniform temperature change, AT = - I°C. (E¢: coating modulus (GPa); ~t¢:coefficient of thermal expansion of coating (10-6/°C).)
material having a Young's modulus, Ec, of 0.345 GPa and a coefficient of thermal expansion, ~c, of 0.018 x 10-6/°C changes the radial stress from tensile to compressive at both the fiber-coating and coating-matrix interfaces. This sign reversal can be helpful in controlling or minimizing interfacial debonding. The stress concentration is a function of both Young's modulus and the thermal expansion coefficient of the coating, besides its thickness. Figure 4 illustrates the variation of the longitudinal stress, a z, in the constituents. Under a uniform temperature change, the present model predicts a constant a= in each one of the individual materials. For E¢ < 3.45 GPa, it is observed that a z in the fiber decreases (compared to the uncoated value) while that in the coating increases, whereas for a higher coating modulus, the reverse behavior is seen to occur. The longitudinal stress, a=, in the matrix is seen to increase under all the simulated cases. The hoop stress, a o, is constant inside the fiber and discontinuous across the two interfaces. Within the coating, the algebraic maximum stress occurs at the coating-matrix interface. Inside the matrix the algebraic maximum stress occurs at the outer boundary if the stress is compressive, and at the inner boundary if it is tensile in nature. The variation of this algebraic maximum hoop stress is plotted in Fig. 5. As seen from the diagrams, it closely follows the trend depicted by the longitudinal stress component.
288
N. J. "Pagano, G. P. Tandon
az(Kp~) =
' 'a'c='018
0-
Oz(KPa)
EC='345,O-¢;= "OI8
-25~
50-0EC:=34"5,0-C="01B
3"45,'018
-31 -
375-
34'5,1"8
-373"45,'018
3-45,1"8
-43-
34s,.m8 34.5.18 "345.1"8
25"0-
-112"5-
-49-
12'5 '"
0
t
I
"05
-1
34.5,,018 I
-150.0 0
"05
('2- n)/'J (a)
3-45,1 e
-55-
J
0
"1
(r2- r ] y r 3
i
I
'05 (r2- rl)/r 3
'I
(c) (b) Fig. 4. Longitudinal stress, (r=,inside (a) fiber; (b) coating; and (c) matrix for Nicalon/BMAS composite under a uniform temperalure change, AT = - I"C. (E¢: coating modulus [GPa): 7c: coefficient of thermal expansion of coating (10 6/ C).)
GO (KPa)
% (KP.)
00 (KPa)
~
ECZ345,0-C: 01B
12"5t
.l" 8
. . ~ I ' 8
Ec=34.5 tlc=-OIB
]
/
3.45,-OI8
Ot
/~-34S.-OIB
3"45,018
1B'75-~
Ee=34"5'a'c=018 -37"5-
\
,518
750
6.25, ~-125
O-
-6.25 0
t 05
('2"'1)/'3 (a)
a -1
-150.0-.. / 0
t .05
(r2-'0/'3 (b)
-1
0
"05 ('2-',)/'3 (c)
.I
Fig. 5. Maximum hoop stress, a o, inside (a) fiber; (b) coating; and (c) matrix for Nicalon/' BMAS composite under a uniform temperature change, AT = - I°C. (Ec: coating modulus (GPa); ~¢: coefficient of thermal expansion of coating (10-6/°C).)
Elastic response of multi-directional coated-fiber composites
289
Besides the microstress distribution, we have also examined the effective elastic properties of the composite in the course of this investigation. In Fig. 6 we have plotted the variation in the transverse Young's modulus of the composite, E 22. What is particularly significant is the sensitivity of E2 2 to the elastic modulus of the coating, E c. As explained earlier, coatings/ interphases also occur as a result of processing conditions. This could therefore be an alternative explanation of the experimentally-measured low modulus value. The transverse Young's modulus of the composite decreases with increasing coating thickness, as expected, because we are replacing some matrix material with a softer material. Both the longitudinal shear m o d u l u s , G12 , and the in-plane shear modulus, G23 , follow a trend similar to E22. The longitudinal Young's modulus, E 11, on the other hand, is much less sensitive to the coating modulus as seen in Table 1. E22 (GPa)
180l lso--I~ .=34'5
60-
O[0
=
-05
=
-1
(r=-r,)/,3 Fig. 6. Transverse Young's modulus, E22, of Nicaion/BMAS composite. (Ec: coating modulus (GPa).)
N. J. Pagano, G, P. Tandon
290
TABLE 1 Unidirectional Composite Coating thickness
Coating modulus (GPa)
E11
E22
G12
G23
(GPa)
(GPa)
(GPa)
(GPa)
--
162"530
152'840
60"615
60,357
0"01
0'345 3'450 34"500
160"833 160'922 161-412
45-148 109"741 148"031
18"361 43.611 58-697
18-010 42'81t 58"263
0'05
0"345 3'450 34'500
153"986 154"296 156"799
22'891 58"332 131'901
10"149 23"057 52"043
9"232 22-850 51'464
0"10
0.345 3.450 34-500
145"002 145"560 150"704
14-624 37-123 115-995
6-778 14'852 45"307
5"826 14-500 44-914
Cylinder outer radius
To conclude, it is apparent that generally a reduction in the s t r e s s concentration can be made at the expense of the elastic moduli of the composite. Further, by a proper choice of coating thickness, modulus and coefficient of thermal expansion, the stress component of interest, which is instrumental in causing a specific mode of failure, can be controlled. We next consider a three-dimensional fibrous composite by arranging six fibers parallel to the six lines joining the opposite vertices of a regular icosahedron. These six axes can be oriented with reference to an orthogonal Cartesian coordinate system x 1 x 2 x 3 as follows: one pair in the .\-ix2 plane making angles of + 0' with the xl-axis, one pair in the X2X 3 plane making angles of + 0' with the x2-axis, and one pair in the x 3 x 1 plane making angles of + 0' with the x3-axis, where 0 ' = tan -1 (2 sin 18 °) = 31°4Y. As shown by TABLE 2 Three-dimensional 'Isotropic' Composite Coating thickness
Composite system
E (GPa)
v
Nicalon/BMAS
128-16
0.250
Nicalon/Nickel/BMAS
Cylinder outer radius
0.01
Nicalon/Carbon/BMAS
Nicalon/Polyimide/BMAS 0'10
Nicalon/Nickel/BMAS Nicalon/Carbon/BMAS
Nicalon/Polyimide/BMAS
G (GPa)
(10
('1
51"25
2.89
129.11 0 . 2 5 1 118-06 0 - 2 4 3 105-95 0 - 2 4 8
51.60 47-50 42.43
3.04 2-88 3-24
138-72 80.20 65.88
55.10 32.94 26.71
4.45 2-84 4,75
0.259 0.217 0"234
Elastic response of multi-directional coated-fiber composites
291
Rosen and Shu, 16 this type of arrangement gives rise to local isotropy. The isotropic relation. G =
E 2(1 + v) -
can be used as an independent check of the model. In general, this relation is not satisfied exactly in the present analysis, but the error is very small. The results for the Nicalon fiber and BMAS matrix system are shown in Table 2. Also shown in the table is the effect of different coating thicknesses and coating materials on the effective thermoelastic moduli. The material properties of the three coating materials used in the calculations were:
Coating material
E (GPa)
G (GPa)
o~(10-6/°C)
Nickel 11 ATJS, Carbon I 7 Polyimide i a
207.0 9-I 3"1
79"0 4.1 1"17
13-3 2.2 54"0
and the fiber volume fraction was set at 0.30. As seen from Table 2, both carbon and polyimide coatings, because of their lower elastic moduli compared to the BMAS matrix, reduce the effective elastic properties, whereas nickel, with a higher modulus, adds to the reinforcement. The thermal expansion coefficient of the composite, on the other hand, is influenced more by nickel and polyimide coatings because of the larger degree of mismatch between the constituents. The microstress distribution within the constituents, for a multidirectional fiber composite, in general, depends both on the type of loading and the fiber orientation. For the specific three-dimensional composite under consideration, the stress distribution remains identical for the six fiber orientations when the composite is subjected to a uniform temperature change. The effect of different coating materials and/or thicknesses on the stress concentrations is quite dramatic. As the trends are similar to those reported in the first example problem, they will not be addressed here. 10 THE NDSANDS P R O G R A M Micromechanical considerations in composite materials may require the use of a practical tool that can handle different constituent materials, arbitrary fiber orientations and multiaxial loading conditions. To address these requirements, the computer code called NDSANDS (N Directional Stiffness AND Strength) has been developed. It can be used either to analyze a composite or to conduct parametric study. By parametric study is meant that the user can change either a material property or the geometry of the
292
N. J. Pagano, G. P. Tandon
composite, one single variable at a time while the remainder are kept constant, and thereby examine the change in effective properties and stress distribution as a result of different input values of the parameter selected. When changing the material property, we must ensure that both the stiffness and compliance matrices remain positive definite at all times. ~9 The influence of constituent properties on the effective composite behavior has been examined, s° Composite response has been predicted with the composite subjected to different loading conditions, namely, uniaxial tension, longitudinal shear, in-plane shear, transverse tension and a uniform temperature change. 11 S U M M A R Y In summary, we have developed a first-order ideal material model to approximate the elastic response of a composite body reinforced by coated fibers oriented in various directions. The coating can either be applied intentionally to achieve the desirable composite properties or it can occur as a consequence of the processing conditions involved in composite manufacture. We have further demonstrated, through a parametric study, how an applied coating modifies the micro-stress distribution and the elastic properties of a Nicalon/BMAS system. The model can be used to provide material guidance for controlling the micromechanical failure modes. In conjunction with a disciplined experimental program, studies such as those conducted here, can be employed to direct further improvements in the model, such as the ability to handle discontinuity of some of the traction and/or displacement components at the boundary. This, formulation, along with the solution for the displacement and the stress field under prescribed traction boundary conditions, will be the subject of a subsequent paper.
REFERENCES 1. G.A. Cooper and A. Kelly, Role of the interface in the fracture offiber-composite materials, ASTM STP 452, (1969), pp. 90-106. 2. H. L. Hancox and H. Wells, The effects of fibre surface coatings on the mechanical properties of CFRP, Fibre Sci. & Tech., 10 (1977), 9-22. 3. J. H. Williams, Jr. and P. N. Kousiounelos, Thermoplastic fibre coatings enhance composite strength and toughness, Fibre Sci. & Tech., 11 (1978), 83-8. 4. D.F. Adams, A micromechanical analysis of the influence of the interface on the performance of polymer-matrix composites, Proc. Amer. Soc. for Comp., First Technical Conference (1986), pp. 207-26. 5. L. T. Drzal, Composite interphase characterization, S A M P E J., 19(5) (1983), 7-13.
Elastic response of multi-directional coated-fiber composites
293
6. T. Ishikawa, K. Koyama and S. Kobayashi, Thermal expansion coefficients of unidirectional composites, J. Comp. Mat., 12 (1978), 153-68. 7. D. Iesan, Thermal stresses in composite cylinders, J. Thermal Stress, 3 (1980), 495-508. 8. R. M. Christensen and H. Lo, Solutions for effective shear properties in threephase sphere and cylinder models, J. Mech. Phys. Solids, 27 (1979), 315-30. 9. Z. Hashin and B. W. Rosen, The elastic moduli of fiber-reinforced materials, J. Appl. Mech., 31 (1964), 223-32. 10. Y. Takeuchi, T. Furukawa and Y. Tanigawa, The effect of thermoelastic coupling for transient thermal stresses in a composite cylinder, ASME, WAM, DE-2 (1983). 11. Y. Mikata and M. Taya, Stress field in a coated continuous fiber composite subjected to thermo-mechanical loadings, J. Comp. Mat., 19 (1985), 554-79. 12. B. W. Rosen, S. N. Chatterjee and J. J. Kibler, An analysis model for spatially oriented fiber composites, ASTM STP 617 (1977) pp. 243-54. 13. N. J. Pagano, The stress field in a cylindrically anisotropic body under twodimensional surface tractions, J. Appl. Mech., 38 (1971), 1-6. 14. Z. Hashin, The elastic moduli of heterogeneous materials, J. Appl. Mech., 29 (1962), 143-50. 15. R. Y. Kim, UDRI (personal communication). 16. B.W. Rosen and L. S. Shu, On some symmetry conditions for three-dimensional fibrous composites, J. Comp. Mat., 5 (1971), 279-82. 17. D. E. Walrath and D. F. Adams, Finite element micromechanics and minimechanics modeling of a three-dimensional carbon--carbon composite material, Dept report UWME-DR-501-106-1 (1985), 57pp. 18. J.C. Haipin, Primer on composite materials: analysis, Technomic Publishing Co. Inc. Lancaster, PA (1984), 166pp. 19. R.M. Jones, Mechanics of composite materials, McGraw-Hill, New York (1975), pp. 37--45. 20. G. P. Tandon and N. J. Pagano, Effect of constituent material properties on the elastic behavior of composites, Proc. Amer. Soc. for Comp., 2nd Technical Conference, University of Delaware (1987), pp. 429-40.