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Effect of specimen width on the fracture of unidirectional metal matrix composites
Composites Science and Technology 45 (1992) 117-123
Effect of specimen width on the fracture of unidirectional metal matrix composites L. R. Dharani, S. Venkatakrishnaiah Department of Mechanical and Aerospace Engineering and Engineering Mechanics, University of Missouri-Rolla, Rolla, Missouri 65401-0249, USA &
R. A. D u p u y Department of Mechanical Engineering, University of Arkansas, Fayetteville, Arkansas 72701, USA (Received 28 January 1991; revised version received 1 August 1991; accepted 30 September 1991)
This paper presents the results from shear lag analysis of a finite-width, unidirectional metal matrix composite (MMC) containing a central crack and crack tip matrix yielding. The specimen is loaded in uniaxial tension parallel to the fiber axes while the crack is normal to the fibers. The emphasis has been on studying the effect of specimen width on the stress concentration and fracture behavior of a unidirectional MMC such as boron/aluminum. Since the stiffness of the fiber is very large compared to that of the matrix it is assumed that the fiber carries all the axial load while the matrix transfers load between and amongst fibers by shear. The governing equations consist of a system of coupled integral equations which are solved by a numerical quadrature technique. The parameters considered in obtaining the numerical results include the crack length, specimen width, plastic zone size, and the ultimate strength. Results show that the fiber stress concentrations at the notch, as well as at the specimen edges, are significantly higher than those obtained by an infinite plate solution.
Keywords: metal matrix composites, unidirectional composites, plasticity, finite width, shear-lag
1 INTRODUCTION
paper by Goree et al. 2 presents a literature review of experimental and analytical work on MMCs in particular. A significant part of the analytical work on modeling damage in unidirectional composites is based on the so-called shear-lag theory, originally proposed by Cox. 3 Early application of this theory to fracture of unidirectional composites is represented in the classic work of Hedgepeth, 4 and Van Dyke & Hedgepeth. 5 One very important feature of the shear lag theory is that it simplifies the equilibrium equations by removing the transverse displacement dependence from the longitudinal equilibrium equation. The fiber stress and the matrix shear stress can be determined without solving the transverse equilibrium equation. Goree & Gross 6 extended the
Unidirectional composites in which a matrix is reinforced by an arr~iy of parallel fibers form a useful class of materials, particularly with the advent of ceramic, intermetallic, and metal matrix composites for high temperature and other special applications. Of these, metal matrix composites (MMCs) have been studied extensively for their strength and fracture behavior. A paper by Awerbuch & Madhukar I gives an excellent review of work related to notched strength of composites in general while a recent * To whom correspondence should be addressed.
Composites Science and Technology 0266-3538/92/$05.00 © 1992 Elsevier Science Publishers Ltd. 117
118
L. R. Dharani, S. Venkatakrishnaiah, R. A. Dupuy
Hedgepeth 4 solution to include longitudinal matrix yielding and splitting for an arbitrary crack length. In addition to longitudinal yielding of the matrix, a certain amount of stable transverse extension of the initial notch under increasing applied load has been observed, 2.7,~ in tests on unidirectional MMC laminates. This transverse damage has been included in developing an analytical model for a unidirectional lamina of infinite width. 9 This paper is yet another extension of the above work, particularly that of Dharani et al.,~ to account for the effect of the finite width of a center notched unidirectional tension specimen. Since the concept of shear lag and the associated methodology for damage characterization in unidirectional composites is well documented, 2-6-9 the development of the analytical model is dealt with very briefly by using this available literature. An extensive parametric study is presented for a typical MMC.
metric in loading and geometry, only the upper quadrant of the specimen need be considered in the analysis as shown in Fig. 1. The fibers are taken to be of much greater extensional stiffness than the matrix so that all axial load may be assumed to be carried by the fibers while the matrix carries only shear. With this assumption the following classical shear lag" stress/displacement relations can be used: do,
Consider a center notched tension specimen of finite width with all fibers aligned parallel to the loading direction and normal to the notch. The initial damage consists of an arbitrary number of broken fibers which form the transverse notch (crack), longitudinal matrix damage in the form of yielding or splitting and an additional transverse damage zone in which the fibers are partially damaged. Since the problem is sym-
AFdO~ t
(2)
t- r , + , - r, = ( y - ll)
dy
(rN+, + ro)(b.N -
applied stress (
--
--
%0)
,k
14
i
GM r,+, = h (v,+, - v,,)
+ < y --
f
I ~
(l)
where o, is the axial stress in the fiber n; r,+~ is the shear stress in the matrix between fibers n and n + 1; EF is the Young's modulus of the fiber; GM is an equivalent shear modulus for the matrix; h is the shear transfer distance and v,, is the axial displacement of fiber n. By virtue of the shear lag assumption the longitudinal and transverse equilibrium equations become decoupled and the fiber axial displacements and stresses can be obtained without solving the transverse equilibrium equation." With reference to the free body diagram shown in Fig. 2, the longitudinal equilibrium equation, applicable to all fibers in Fig. 1, is given by
2 INTEGRAL EQUATION FORMULATION
,Jk, ~
O,, = E ~ - - dy
Jl
notch ~
transverse damage
r -I
"-- "¢b
Fig. 1. A unidirectional lamina with fiber and matrix damage.
(3)
The fracture of unidirectional metal matrix composites 'l ~F(Y+ ~)1 n1~(y+,fy)l n+l
T-(y+g~y)1n
~(Y)ln+l
(:YM(Y)I n+l
GM(Y)In d/~
~(Y)ln ,9
,9
'I;(y)ln
(~F(y) In "~(Y)I~+~
Fig. 2. Free body diagram of a typical fiber/matrix element.
where Av is the area of the fiber, t is lamina thickness, ( y - l ) = l for y - l and ( y - l ) = 0 for y > l, 6~ is the Kronecker symbol (6~j = 1, for i = j and 6~j = 0 for i :/:j), and ro is the matrix yield stress in shear. Substituting the shear lag stress/displacement relationships (1), (2) into eqn (3) and then using certain change of variables, 9 a non-dimensional equilibrium equation is obtained as d2V.
--+ dr/z
119
reduced to a single ordinary differential equation by using Fourier transforms and the orthogonality properties of the circular functions. 9 The solution to the problem of vanishing stresses and displacements at infinity and uniform pressure on the ends of the broken fibers is first sought. The total solution is obtained by adding the results corresponding to uniform remote axial stress and no broken fibers to the above solution. The appropriate boundary conditions for the former problem are V, = 0
as r/--->oo for all fibers
(6a)
V, = 0
at r/= 0 for all unbroken fibers
(6b)
o,=dV~ - - 1 dr/ O~ =
at r/ = 0 for broken fibers
(6c)
dVo
= - 1 + yOM+~ dr/ at r/= 0 for constrained fibers
(6d)
where 7 is a constant representing the constraint. The solution to eqn (4) satisfying the boundary conditions (6a) and (6b) is given by v.(r/) = -
e -~" ~ B,, cos(m0) cos(n0) dO
V . + , - 2I,'. + V._, = f~(r/)(6.(,V+l)- 6.N)
+ ½ ~(t)D,(n, r/, t) dt +
(4)
+ ira2 2jo f2(t)D2(n, r/, t) dt
where for r/ c~
f,(r/) = V~ - V ~ + , - L ( r / - fl,) f,(r/) = 0 f~(r/) = V~ - v~+, + ~ ( r / -
for r / > fl:
~)
The non-dimensional variables (r/, o:~, 1,1,, ~'o, i:b) are given by y = ~Pr/, li = ~ i L = ~pa"
i = 1, 2
On = 0 ~ 0 n
GM
~o
~:b
h
2 f= F~(0) Di(n' r/' t)=-~ Jo 6 x {e -~l'-"l - e-~"+")}cos(n0) dO F~(0) = cos(N0) - cos[(N + 1)0] 6 = 2 sin(0/2)
Un = (~Vn "~b
where
F2(0) = cos(P0) - cos[(P + 1)0]
(5)
"~o
(7)
where A~vh ~P= ~I GMt .~ / A F h
(P = V EvGMt o® The differential difference equation (4) can be
In the above formulation, if 132--->oo and i:~ = 0 we obtain the solution of a finite width region containing a total of (2P + 1 = N W ) fibers. The unknown Fourier constants are obtained by satisfying the stress boundary conditions, eqns (6c), (6d), as -
6B,. cos(m0) + F~(0)
e-'tf~(t) dt
=
+ F2(0
)fo
}
e-~f2(t) dt cos(n0) dO = - 1 forn=0,1,...,N
(8)
120
2
L. R. Dharani, S. Venkatakrishnaiah, R. A. Dupuy
=
6
B,. cos(m0) + F,(0)
e-~'f,(t) dt
m=O
X { c o s ( n 0 ) - y cos[(M + 1)0]} dO = -1 + y
for n = N + 1 , . . . ,
M
(9)
The other unknown functions f,(t) and f2(t) are given by fl(t) = gl(t) -- 7o f2(t) = g2(t) where M
e -6~ ~ m=0
+ ½
B,,,cos(m0)F~(0) dO
(a) transverse crack only, tr = fl, = 0; (b) transverse crack with longitudinal matrix yielding, a~ = 5, ,61 = 0; and (c) transverse crack with longitudinal matrix splitting as in the thermoset polymer matrix composites, a = fl~ = 5.
{ D , ( N , rh t) - D , ( N + 1, rh t)}
x (g~(t) - ~:o) dt
+ ½
{D2(N, rh t) - D2(N + 1. r/, t)}
X
g2(t) dt
2
foxe -6~ ~M Bm cos(mO)cos(PO)
(10) dO
m = O
+½£~D , ( P ,
+lfo
rI, t)(g,(t) - 70) dt
D2(P, rl, t)gz(t) dt
always occur in a self-similar fashion along the plane of the initial notch. These fibers break on either side of the initial crack plane. This phenomenon has generally been attributed to the statistical nature of the fiber strength and size distribution along the fiber axis. In the absence of crack tip longitudinal damage, matrix yielding and/or splitting, maximum fiber stress occurs along the initial crack plane. However, with the introduction of longitudinal damage the fiber stress distribution changes in such a way that the maximum stress occurs in the region of longitudinal damage as shown in Fig. 3. The width of the specimen is N W , the total number of fibers in the specimen. Fiber stress concentration along the fiber is plotted for the following three cases:
(11)
Along with g l ( a 0 = 7 o , the final governing equations (8)-(11) contain the unknown Fourier constants B,,, and the unknown functions g,(r/) and g2(r/). The governing equations are solved numerically. The integrals over the semi-infinite domain are evaluated by using Gauss-Laguerre quadrature and the integrals over a finite domain are evaluated with Gauss-Legendre quadrature. With this, the governing equations reduce to a set of linear algebraic equations with the unknowns being the Fourier constants B,, and the functional values of g,(r/) and gz(r/) at quadrature points.
3. RESULTS
Stress concentration along the crack plane is in fact reduced with the introduction of any longitudinal damage. However, the maximum stress concentration for the two later cases is much higher than that of a transverse crack alone and occurs away from the crack plane, validating the non self-similar fiber breaking observed in experiments. 2'~ For each fiber ahead of the
6 NTBF = 7 NW = 17
z o 5
q=~-
Y
g}
e,ILl
3
II.
1
i
i
i
1
2
3
i
4
,
5
q
It has been observed experimentally 2'8 that the fiber fracture in unidirectional MMCs does not
Fig. 3. E f f e c t o f m a t r i x splitting a n d y i e l d i n g o n t h e fiber stress concentration.
The fracture of unidirectional metal matrix composites
z o
NTBF = 7 N W = 17
0
or,
a=5.0
,< IE
121
Z NTBF = 7 ~( = 5.0
i-
zt l l
z
~3
(j
\,
Z
o t9 ID
z
o ~J (n u)
'.,, M A X I M U M
Ill n,
k
e~ .1
w
X W
2
13
E
L
/
15 17
/
21
29 i
5
i
i
i
i
6
7
8
4
~
i
6
B
FIBER INDEX
i
i
i
10
1.2
14
FIBER INDEX
Fig. 4. Maximum and crack plane fiber stress concentration distribution ahead of the crack tip.
Fig. 6. Maximum stress concentration ahead of crack tip, matrix yielding.
damage zone the stress concentration along the crack plane (x axis) and the maximum stress concentration, if it occurs at points other than along the crack plane, are shown in Fig. 4. Interestingly, the maximum stress concentration occurs along the crack plane for most fibers with the exception of the first unbroken fiber at the crack tip and the last fiber at the free edge. Next, the effect of finite width on fiber stress distribution for three different damage situations is considered. All three damage cases have the same crack length, seven broken fibers, while
differing in the nature of the crack tip damage. Figures 5 and 6 show the stress concentration along the crack plane and the maximum stress concentration in a fiber, respectively, for the case of a transverse crack with longitudinal matrix yielding (tr = 5). The trend is identical for all finite specimen widths. Figure 7 corresponds to matrix splitting (o: = fl~ = 5) and shows a similar response albeit with higher maximum stresses. The case of additional fiber damage (constrained fibers, tr = fl] = 0, )' = 0-8) at the crack tip is also investigated and results are shown in Fig. 8 for
3
Z
O if-. .<
NTBF-- 7
~Z
~=5.0
ae IZ "'
.i (J
(J
z
o (J (n u) •n,
6
NTBF = 7
/11 = 5 . 0 b
Z
O tO
(n Iu) ll n" I" (n
2
4
II 1 1/ 1'
e~
u.
M.
2
J
21 29
1 __
i
i
6
8
i
i
i
10
12
14
FIBER INDEX
Fig. 5. Fiber stress concentration along the crack plane ahead of the crack tip.
4
6
8
10
12
14
FIBER INDEX
Fig. 7. Maximum stress concentration ahead of crack tip, matrix splitting.
122
L. R. Dharani, S. Venkatakrishnaiah, R. A. Dupuy 2.5
+0! .
81~
NTBF=7 NC=4 ~=5.0 )' = 0.8
Z
_o I-
< /Z
m
t7
.... ~
0.8 i
/"
n.,,,
NW=15
~ 2.0 Z o
\\
m
LLI I'--
o
17
~\\\ /i//
ua
~~5'
.i
m 1.5
0,6
JY9
0,2
i
1.0
r
7
i
6
8
10
/ /
- 7 - - - - - - v
12
14
0
3
Pig. 8. Maximum stress concentration ahead of crack tip, matrix yielding and transverse damage.
four constrained fibers, two at each crack tip. In all of these cases the stresses for finite width configuration are always higher than those of the infinite plate having the same crack length. We define a finite width correction factor, f ( a / w ) , as Kv = KFuf(a/w)
(12)
where Kv is the fiber stress concentration for the finite width plate and Kvr~ is the corresponding fiber stress concentration for an infinite plate containing a crack of the same size. Following Hedgepeth, + the crack tip fiber stress concentration for a crack in an infinite unidirectional
,Y
o
I-
~( = 0.0
,< u.
O
NW=93
4
29/
O .i n. nO (.J I--
£
tu I-
1 i
.0
0.2
"r
0.4
q
0.6
- - l ~
r
0.8
1.0
2alw
9. Finite
width correction concentration.
factor
for
stress
6
9
12
15
18
21
24
27
NUMBER OF BROKEN FIBERS
FIBER INDEX
z
" ~
Fig. 10, Residual strength curves.
composite can be given as 4.6.8 .......... = 3.5.7
...........
(2NTBF+2) (2 N T B F + 1)
(t3)
where NTBF is the total number of broken fibers. The finite width correction factor thus calculated, from eqns (12) and (13), is shown in Fig. 9 as a function of normalized crack length for various specimen widths. The corresponding isotropic plate solution is also shown on the same figure for comparison. For a/w in the range of practical interest (a/w < 0.75) the isotropic plate solution gives conservative correction factors. There is some dependence on actual number of fibers comprising the width of the plate, NW. The final results deal with the effect of specimen width on the residual strength and the size of longitudinal yielding. For a given crack size (number of broken fibers) the remote applied stress is increased until the maximum stress in the first unbroken fiber at the crack tip reaches the fiber ultimate strength and the corresponding size of the yield zone, c~, is recorded. These two parameters are shown in Fig. 10 as functions of the crack size (number of broken fibers) for various specimen widths, including that of an infinite plate, with the following material and geometric properties: 2 fiber Young's modulus, E v = 475 GPa; fiber area, AF=0.0159mmZ; lamina thickness, t = 0.165 mm; fiber ultimate strength o,~t = 3.98 GPa. The effect of finite width of the specimen on both the longitudinal damage size and the residual
The fracture of unidirectional metal matrix composites strength is significant. A n infinite plate solution, however, gives higher residual strength and a larger yield zone. 3. 4 CONCLUSIONS 4. The crack extends in a non-self-similar fashion (fiber breaks) in the presence of crack tip matrix damage either in the form of matrix yielding or matrix splitting. T h e fiber stress concentrations at the notch, as well as at the specimen edges, are significantly higher than those o b t a i n e d by an infinite plate solution. The effect of specimen width on the interpretation of residual strength data is significant and must be a c c o u n t e d for.
5. 6.
7.
REFERENCES
8.
1. Awerbuch, J. & Madhukar, M. S., Notched Strength of Composite Laminates: Predictions and Experiments. J. Reinforced Plastics and Composites, 4 (1985) 1-159. 2. Goree, J. G., Dharani, L. R. & Jones, W. F., Crack
9.
123
Growth and Fracture of Continuous Fiber Metal Matrix Composites: Analysis and Experiments. Metal Matrix Composites: Testing, Analysis, and Failure Modes, ASTM STP 1032, ed. W. S. Johnson. American Society of Testing and Materials, Philadelphia, 1989, 251-69. Cox, H. L., The Elasticity and Strength of Paper and Other Fibrous Materials. British J. Appl. Phys., 3 (1952) 72-9. Hedgepeth, J. M., Stress Concentrations in Filamentary Structures, NASA TN D-882, 1961. Van Dyke, P. & Hedgepeth, J. M., Stress Concentrations from Single Filament Failures in Composite Materials. Textile Res., 39 (1969) 613-26. Goree, J. G. & Gross, R. S., Analysis of a Unidirectional Composite Containing Broken Fibers and Matrix Damage. Engng Fracture Mech., 13 (1979) 563-78. Awerbuch, J. & Bakuckas, J. G., On the Applicability of Acoustic Emission for Monitoring Damage Progression in Metal Matrix Composites. Metal Matrix Composites: Testing, Analysis, and Failure Modes, ASTM STP 1032, ed. W. S. Johnson. American Society of Testing and Materials, Philadelphia, 1989, 68-89. Jones, W. F. & Goree, J. G., Fracture Behavior of Unidirectional Boron/Aluminum Composite Laminates. Mechanics of Composite Materials--1983, ASME AMD, Vol. 58, 1983, pp. 171-8. Dharani, L. R., Jones, W. F. & Goree, J. G., Mathematical Modeling of Damage in Unidirectional Composites. Engng Fracture Mech., 17 (1983) 555-73.
Effect of specimen width on the fracture of unidirectional metal matrix composites L. R. Dharani, S. Venkatakrishnaiah Department of Mechanical and Aerospace Engineering and Engineering Mechanics, University of Missouri-Rolla, Rolla, Missouri 65401-0249, USA &
R. A. D u p u y Department of Mechanical Engineering, University of Arkansas, Fayetteville, Arkansas 72701, USA (Received 28 January 1991; revised version received 1 August 1991; accepted 30 September 1991)
This paper presents the results from shear lag analysis of a finite-width, unidirectional metal matrix composite (MMC) containing a central crack and crack tip matrix yielding. The specimen is loaded in uniaxial tension parallel to the fiber axes while the crack is normal to the fibers. The emphasis has been on studying the effect of specimen width on the stress concentration and fracture behavior of a unidirectional MMC such as boron/aluminum. Since the stiffness of the fiber is very large compared to that of the matrix it is assumed that the fiber carries all the axial load while the matrix transfers load between and amongst fibers by shear. The governing equations consist of a system of coupled integral equations which are solved by a numerical quadrature technique. The parameters considered in obtaining the numerical results include the crack length, specimen width, plastic zone size, and the ultimate strength. Results show that the fiber stress concentrations at the notch, as well as at the specimen edges, are significantly higher than those obtained by an infinite plate solution.
Keywords: metal matrix composites, unidirectional composites, plasticity, finite width, shear-lag
1 INTRODUCTION
paper by Goree et al. 2 presents a literature review of experimental and analytical work on MMCs in particular. A significant part of the analytical work on modeling damage in unidirectional composites is based on the so-called shear-lag theory, originally proposed by Cox. 3 Early application of this theory to fracture of unidirectional composites is represented in the classic work of Hedgepeth, 4 and Van Dyke & Hedgepeth. 5 One very important feature of the shear lag theory is that it simplifies the equilibrium equations by removing the transverse displacement dependence from the longitudinal equilibrium equation. The fiber stress and the matrix shear stress can be determined without solving the transverse equilibrium equation. Goree & Gross 6 extended the
Unidirectional composites in which a matrix is reinforced by an arr~iy of parallel fibers form a useful class of materials, particularly with the advent of ceramic, intermetallic, and metal matrix composites for high temperature and other special applications. Of these, metal matrix composites (MMCs) have been studied extensively for their strength and fracture behavior. A paper by Awerbuch & Madhukar I gives an excellent review of work related to notched strength of composites in general while a recent * To whom correspondence should be addressed.
Composites Science and Technology 0266-3538/92/$05.00 © 1992 Elsevier Science Publishers Ltd. 117
118
L. R. Dharani, S. Venkatakrishnaiah, R. A. Dupuy
Hedgepeth 4 solution to include longitudinal matrix yielding and splitting for an arbitrary crack length. In addition to longitudinal yielding of the matrix, a certain amount of stable transverse extension of the initial notch under increasing applied load has been observed, 2.7,~ in tests on unidirectional MMC laminates. This transverse damage has been included in developing an analytical model for a unidirectional lamina of infinite width. 9 This paper is yet another extension of the above work, particularly that of Dharani et al.,~ to account for the effect of the finite width of a center notched unidirectional tension specimen. Since the concept of shear lag and the associated methodology for damage characterization in unidirectional composites is well documented, 2-6-9 the development of the analytical model is dealt with very briefly by using this available literature. An extensive parametric study is presented for a typical MMC.
metric in loading and geometry, only the upper quadrant of the specimen need be considered in the analysis as shown in Fig. 1. The fibers are taken to be of much greater extensional stiffness than the matrix so that all axial load may be assumed to be carried by the fibers while the matrix carries only shear. With this assumption the following classical shear lag" stress/displacement relations can be used: do,
Consider a center notched tension specimen of finite width with all fibers aligned parallel to the loading direction and normal to the notch. The initial damage consists of an arbitrary number of broken fibers which form the transverse notch (crack), longitudinal matrix damage in the form of yielding or splitting and an additional transverse damage zone in which the fibers are partially damaged. Since the problem is sym-
AFdO~ t
(2)
t- r , + , - r, = ( y - ll)
dy
(rN+, + ro)(b.N -
applied stress (
--
--
%0)
,k
14
i
GM r,+, = h (v,+, - v,,)
+ < y --
f
I ~
(l)
where o, is the axial stress in the fiber n; r,+~ is the shear stress in the matrix between fibers n and n + 1; EF is the Young's modulus of the fiber; GM is an equivalent shear modulus for the matrix; h is the shear transfer distance and v,, is the axial displacement of fiber n. By virtue of the shear lag assumption the longitudinal and transverse equilibrium equations become decoupled and the fiber axial displacements and stresses can be obtained without solving the transverse equilibrium equation." With reference to the free body diagram shown in Fig. 2, the longitudinal equilibrium equation, applicable to all fibers in Fig. 1, is given by
2 INTEGRAL EQUATION FORMULATION
,Jk, ~
O,, = E ~ - - dy
Jl
notch ~
transverse damage
r -I
"-- "¢b
Fig. 1. A unidirectional lamina with fiber and matrix damage.
(3)
The fracture of unidirectional metal matrix composites 'l ~F(Y+ ~)1 n1~(y+,fy)l n+l
T-(y+g~y)1n
~(Y)ln+l
(:YM(Y)I n+l
GM(Y)In d/~
~(Y)ln ,9
,9
'I;(y)ln
(~F(y) In "~(Y)I~+~
Fig. 2. Free body diagram of a typical fiber/matrix element.
where Av is the area of the fiber, t is lamina thickness, ( y - l ) = l for y - l and ( y - l ) = 0 for y > l, 6~ is the Kronecker symbol (6~j = 1, for i = j and 6~j = 0 for i :/:j), and ro is the matrix yield stress in shear. Substituting the shear lag stress/displacement relationships (1), (2) into eqn (3) and then using certain change of variables, 9 a non-dimensional equilibrium equation is obtained as d2V.
--+ dr/z
119
reduced to a single ordinary differential equation by using Fourier transforms and the orthogonality properties of the circular functions. 9 The solution to the problem of vanishing stresses and displacements at infinity and uniform pressure on the ends of the broken fibers is first sought. The total solution is obtained by adding the results corresponding to uniform remote axial stress and no broken fibers to the above solution. The appropriate boundary conditions for the former problem are V, = 0
as r/--->oo for all fibers
(6a)
V, = 0
at r/= 0 for all unbroken fibers
(6b)
o,=dV~ - - 1 dr/ O~ =
at r/ = 0 for broken fibers
(6c)
dVo
= - 1 + yOM+~ dr/ at r/= 0 for constrained fibers
(6d)
where 7 is a constant representing the constraint. The solution to eqn (4) satisfying the boundary conditions (6a) and (6b) is given by v.(r/) = -
e -~" ~ B,, cos(m0) cos(n0) dO
V . + , - 2I,'. + V._, = f~(r/)(6.(,V+l)- 6.N)
+ ½ ~(t)D,(n, r/, t) dt +
(4)
+ ira2 2jo f2(t)D2(n, r/, t) dt
where for r/
f,(r/) = V~ - V ~ + , - L ( r / - fl,) f,(r/) = 0 f~(r/) = V~ - v~+, + ~ ( r / -
for r / > fl:
~)
The non-dimensional variables (r/, o:~, 1,1,, ~'o, i:b) are given by y = ~Pr/, li = ~ i L = ~pa"
i = 1, 2
On = 0 ~ 0 n
GM
~o
~:b
h
2 f= F~(0) Di(n' r/' t)=-~ Jo 6 x {e -~l'-"l - e-~"+")}cos(n0) dO F~(0) = cos(N0) - cos[(N + 1)0] 6 = 2 sin(0/2)
Un = (~Vn "~b
where
F2(0) = cos(P0) - cos[(P + 1)0]
(5)
"~o
(7)
where A~vh ~P= ~I GMt .~ / A F h
(P = V EvGMt o® The differential difference equation (4) can be
In the above formulation, if 132--->oo and i:~ = 0 we obtain the solution of a finite width region containing a total of (2P + 1 = N W ) fibers. The unknown Fourier constants are obtained by satisfying the stress boundary conditions, eqns (6c), (6d), as -
6B,. cos(m0) + F~(0)
e-'tf~(t) dt
=
+ F2(0
)fo
}
e-~f2(t) dt cos(n0) dO = - 1 forn=0,1,...,N
(8)
120
2
L. R. Dharani, S. Venkatakrishnaiah, R. A. Dupuy
=
6
B,. cos(m0) + F,(0)
e-~'f,(t) dt
m=O
X { c o s ( n 0 ) - y cos[(M + 1)0]} dO = -1 + y
for n = N + 1 , . . . ,
M
(9)
The other unknown functions f,(t) and f2(t) are given by fl(t) = gl(t) -- 7o f2(t) = g2(t) where M
e -6~ ~ m=0
+ ½
B,,,cos(m0)F~(0) dO
(a) transverse crack only, tr = fl, = 0; (b) transverse crack with longitudinal matrix yielding, a~ = 5, ,61 = 0; and (c) transverse crack with longitudinal matrix splitting as in the thermoset polymer matrix composites, a = fl~ = 5.
{ D , ( N , rh t) - D , ( N + 1, rh t)}
x (g~(t) - ~:o) dt
+ ½
{D2(N, rh t) - D2(N + 1. r/, t)}
X
g2(t) dt
2
foxe -6~ ~M Bm cos(mO)cos(PO)
(10) dO
m = O
+½£~D , ( P ,
+lfo
rI, t)(g,(t) - 70) dt
D2(P, rl, t)gz(t) dt
always occur in a self-similar fashion along the plane of the initial notch. These fibers break on either side of the initial crack plane. This phenomenon has generally been attributed to the statistical nature of the fiber strength and size distribution along the fiber axis. In the absence of crack tip longitudinal damage, matrix yielding and/or splitting, maximum fiber stress occurs along the initial crack plane. However, with the introduction of longitudinal damage the fiber stress distribution changes in such a way that the maximum stress occurs in the region of longitudinal damage as shown in Fig. 3. The width of the specimen is N W , the total number of fibers in the specimen. Fiber stress concentration along the fiber is plotted for the following three cases:
(11)
Along with g l ( a 0 = 7 o , the final governing equations (8)-(11) contain the unknown Fourier constants B,,, and the unknown functions g,(r/) and g2(r/). The governing equations are solved numerically. The integrals over the semi-infinite domain are evaluated by using Gauss-Laguerre quadrature and the integrals over a finite domain are evaluated with Gauss-Legendre quadrature. With this, the governing equations reduce to a set of linear algebraic equations with the unknowns being the Fourier constants B,, and the functional values of g,(r/) and gz(r/) at quadrature points.
3. RESULTS
Stress concentration along the crack plane is in fact reduced with the introduction of any longitudinal damage. However, the maximum stress concentration for the two later cases is much higher than that of a transverse crack alone and occurs away from the crack plane, validating the non self-similar fiber breaking observed in experiments. 2'~ For each fiber ahead of the
6 NTBF = 7 NW = 17
z o 5
q=~-
Y
g}
e,ILl
3
II.
1
i
i
i
1
2
3
i
4
,
5
q
It has been observed experimentally 2'8 that the fiber fracture in unidirectional MMCs does not
Fig. 3. E f f e c t o f m a t r i x splitting a n d y i e l d i n g o n t h e fiber stress concentration.
The fracture of unidirectional metal matrix composites
z o
NTBF = 7 N W = 17
0
or,
a=5.0
,< IE
121
Z NTBF = 7 ~( = 5.0
i-
zt l l
z
~3
(j
\,
Z
o t9 ID
z
o ~J (n u)
'.,, M A X I M U M
Ill n,
k
e~ .1
w
X W
2
13
E
L
/
15 17
/
21
29 i
5
i
i
i
i
6
7
8
4
~
i
6
B
FIBER INDEX
i
i
i
10
1.2
14
FIBER INDEX
Fig. 4. Maximum and crack plane fiber stress concentration distribution ahead of the crack tip.
Fig. 6. Maximum stress concentration ahead of crack tip, matrix yielding.
damage zone the stress concentration along the crack plane (x axis) and the maximum stress concentration, if it occurs at points other than along the crack plane, are shown in Fig. 4. Interestingly, the maximum stress concentration occurs along the crack plane for most fibers with the exception of the first unbroken fiber at the crack tip and the last fiber at the free edge. Next, the effect of finite width on fiber stress distribution for three different damage situations is considered. All three damage cases have the same crack length, seven broken fibers, while
differing in the nature of the crack tip damage. Figures 5 and 6 show the stress concentration along the crack plane and the maximum stress concentration in a fiber, respectively, for the case of a transverse crack with longitudinal matrix yielding (tr = 5). The trend is identical for all finite specimen widths. Figure 7 corresponds to matrix splitting (o: = fl~ = 5) and shows a similar response albeit with higher maximum stresses. The case of additional fiber damage (constrained fibers, tr = fl] = 0, )' = 0-8) at the crack tip is also investigated and results are shown in Fig. 8 for
3
Z
O if-. .<
NTBF-- 7
~Z
~=5.0
ae IZ "'
.i (J
(J
z
o (J (n u) •n,
6
NTBF = 7
/11 = 5 . 0 b
Z
O tO
(n Iu) ll n" I" (n
2
4
II 1 1/ 1'
e~
u.
M.
2
J
21 29
1 __
i
i
6
8
i
i
i
10
12
14
FIBER INDEX
Fig. 5. Fiber stress concentration along the crack plane ahead of the crack tip.
4
6
8
10
12
14
FIBER INDEX
Fig. 7. Maximum stress concentration ahead of crack tip, matrix splitting.
122
L. R. Dharani, S. Venkatakrishnaiah, R. A. Dupuy 2.5
+0! .
81~
NTBF=7 NC=4 ~=5.0 )' = 0.8
Z
_o I-
< /Z
m
t7
.... ~
0.8 i
/"
n.,,,
NW=15
~ 2.0 Z o
\\
m
LLI I'--
o
17
~\\\ /i//
ua
~~5'
.i
m 1.5
0,6
JY9
0,2
i
1.0
r
7
i
6
8
10
/ /
- 7 - - - - - - v
12
14
0
3
Pig. 8. Maximum stress concentration ahead of crack tip, matrix yielding and transverse damage.
four constrained fibers, two at each crack tip. In all of these cases the stresses for finite width configuration are always higher than those of the infinite plate having the same crack length. We define a finite width correction factor, f ( a / w ) , as Kv = KFuf(a/w)
(12)
where Kv is the fiber stress concentration for the finite width plate and Kvr~ is the corresponding fiber stress concentration for an infinite plate containing a crack of the same size. Following Hedgepeth, + the crack tip fiber stress concentration for a crack in an infinite unidirectional
,Y
o
I-
~( = 0.0
,< u.
O
NW=93
4
29/
O .i n. nO (.J I--
£
tu I-
1 i
.0
0.2
"r
0.4
q
0.6
- - l ~
r
0.8
1.0
2alw
9. Finite
width correction concentration.
factor
for
stress
6
9
12
15
18
21
24
27
NUMBER OF BROKEN FIBERS
FIBER INDEX
z
" ~
Fig. 10, Residual strength curves.
composite can be given as 4.6.8 .......... = 3.5.7
...........
(2NTBF+2) (2 N T B F + 1)
(t3)
where NTBF is the total number of broken fibers. The finite width correction factor thus calculated, from eqns (12) and (13), is shown in Fig. 9 as a function of normalized crack length for various specimen widths. The corresponding isotropic plate solution is also shown on the same figure for comparison. For a/w in the range of practical interest (a/w < 0.75) the isotropic plate solution gives conservative correction factors. There is some dependence on actual number of fibers comprising the width of the plate, NW. The final results deal with the effect of specimen width on the residual strength and the size of longitudinal yielding. For a given crack size (number of broken fibers) the remote applied stress is increased until the maximum stress in the first unbroken fiber at the crack tip reaches the fiber ultimate strength and the corresponding size of the yield zone, c~, is recorded. These two parameters are shown in Fig. 10 as functions of the crack size (number of broken fibers) for various specimen widths, including that of an infinite plate, with the following material and geometric properties: 2 fiber Young's modulus, E v = 475 GPa; fiber area, AF=0.0159mmZ; lamina thickness, t = 0.165 mm; fiber ultimate strength o,~t = 3.98 GPa. The effect of finite width of the specimen on both the longitudinal damage size and the residual
The fracture of unidirectional metal matrix composites strength is significant. A n infinite plate solution, however, gives higher residual strength and a larger yield zone. 3. 4 CONCLUSIONS 4. The crack extends in a non-self-similar fashion (fiber breaks) in the presence of crack tip matrix damage either in the form of matrix yielding or matrix splitting. T h e fiber stress concentrations at the notch, as well as at the specimen edges, are significantly higher than those o b t a i n e d by an infinite plate solution. The effect of specimen width on the interpretation of residual strength data is significant and must be a c c o u n t e d for.
5. 6.
7.
REFERENCES
8.
1. Awerbuch, J. & Madhukar, M. S., Notched Strength of Composite Laminates: Predictions and Experiments. J. Reinforced Plastics and Composites, 4 (1985) 1-159. 2. Goree, J. G., Dharani, L. R. & Jones, W. F., Crack
9.
123
Growth and Fracture of Continuous Fiber Metal Matrix Composites: Analysis and Experiments. Metal Matrix Composites: Testing, Analysis, and Failure Modes, ASTM STP 1032, ed. W. S. Johnson. American Society of Testing and Materials, Philadelphia, 1989, 251-69. Cox, H. L., The Elasticity and Strength of Paper and Other Fibrous Materials. British J. Appl. Phys., 3 (1952) 72-9. Hedgepeth, J. M., Stress Concentrations in Filamentary Structures, NASA TN D-882, 1961. Van Dyke, P. & Hedgepeth, J. M., Stress Concentrations from Single Filament Failures in Composite Materials. Textile Res., 39 (1969) 613-26. Goree, J. G. & Gross, R. S., Analysis of a Unidirectional Composite Containing Broken Fibers and Matrix Damage. Engng Fracture Mech., 13 (1979) 563-78. Awerbuch, J. & Bakuckas, J. G., On the Applicability of Acoustic Emission for Monitoring Damage Progression in Metal Matrix Composites. Metal Matrix Composites: Testing, Analysis, and Failure Modes, ASTM STP 1032, ed. W. S. Johnson. American Society of Testing and Materials, Philadelphia, 1989, 68-89. Jones, W. F. & Goree, J. G., Fracture Behavior of Unidirectional Boron/Aluminum Composite Laminates. Mechanics of Composite Materials--1983, ASME AMD, Vol. 58, 1983, pp. 171-8. Dharani, L. R., Jones, W. F. & Goree, J. G., Mathematical Modeling of Damage in Unidirectional Composites. Engng Fracture Mech., 17 (1983) 555-73.