A numerical analysis of fracture and high temperature creep characteristics of brittle matrix composites with discontinuous ductile reinforcements

Materials Science and Engineering, A 187 (1994) 125-138

125

A numerical analysis of fracture and high temperature creep characteristics of brittle matrix composites with discontinuous ductile reinforcements S. B. B i n e r

Ames Laboratory, Iowa State University, Ames, IA 50011 (USA) (Received May 10, 1993; in revised form December 3, 1993)

Abstract The role of the material parameters and interface characteristics in the fracture and creep behavior of discontinuous ductile fiber-reinforced brittle matrix composite systems was investigated numerically. To simulate the fracture behavior, the ductile fibers were modelled using a constitutive relationship that accounts for strength degradation resulting from the nucleation and growth of voids. The matrix was assumed to be elastic and fails according to requirements of a stress criterion. The debonding behavior at the fiber interfaces was simulated in terms of a cohesive zone model which describes the decohesion by both normal and tangential separation. Results indicate that for rigid interfaces between the ductile reinforcing phase and the matrix the contribution of the ductile reinforcement to the work-of-fracture value (toughness) of the composite increases with less exhaustion of its work-hardening capacity before the onset of matrix failure. Therefore the failure strength and elastic modulus values of the matrix become important material parameters. In the case of interfacial debonding the load transfer to the discontinuous reinforcements after matrix failure should be somehow maintained for utilization to full capacity of the reinforcement and interfacial behavior. In the creep regime, for rigidly bonded interfaces the creep rate of the composite is not significantly influenced by the material properties and geometric parameters of the ductile reinforcing phase owing to the development of triaxial stress state and constrained deformation in the reinforcement. For debonding interfaces the geometric parameters of the reinforcing phase become important; however, even with very weak interfacial behavior, low composite creep rates can be achieved by suitable selection of the geometric parameters of the ductile reinforcing phase. Significant increases in room temperature fracture toughness can be achieved without extensively sacrificing the creep strength by ductile discontinuous reinforcements.

1. Introduction T h e r e is ever-increasing interest in developing low density advanced material systems for service temperatures up to 1400°C. T h e s e materials should have reasonably good fracture toughness at low and intermediate temperatures combined with good creep resistance at elevated temperatures. Recent experimental studies have shown that brittle materials such as intermetallics and ceramics can be substantially toughened by incorporation of ductile reinforcements [1-8]. T h e c o m m o n feature in these experiments is that the matrix fails first by brittle fracture (i.e. transgranular or intergranular), while final failure of the composite occurs owing to damage formation in the form of nucleation and growth of voids within the ductile reinforcement. From these experimental studies it appears that the main contributions to the toughness stem from crack bridging and energy dissipation by plastic deformation of the ductile phase. It is also observed that the toughening contribution of the 0921-5093/94/$7.00 SSD10921-5093(94)09503-O

ductile phase can be further increased by modification of the interface characteristics to promote debonding between the reinforcement and the matrix [4]. During crack extension the ductile reinforcements bridge the crack and thus reduce the stress intensity factor at the crack tip (Fig. 1). T h e stress intensity factor due to the applied stress in this class of composites is usually expressed as [9-14] Kapplied = Kti p + Kbridge

(] )

where Kbridge is the reduction in stress intensity factor resulting from the pinning action of the crack surfaces by intact ductile reinforcements for some distance behind the crack tip. When the bridging zone length is comparable with either the crack length or the specimen dimensions, the applied stress intensity factor is also a function of the specimen geometry and size. At any given applied stress the magnitude of Kbridge is calculated from the work of fracture, Aw, of one ductile ligament (Fig. 1), which takes into account the volume fraction: © 1994 - Elsevier Sequoia. All rights reserved

126

S. B. Biner

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/

Fracture and creep analysis of brittle matrix composites

system through intelligent manipulation of the microstructure. In Section 2 the role of the matrix material parameters and interface characteristics in the evolution of the work of fracture, Ato (eqn. (2)), under simple uniaxial far-field loading is investigated numerically. The role of the material properties of the ductile reinforcing phase and interracial behavior in the creep deformation of the composite is presented in Section 3.

f

\

2. Fracture behavior

/ \

°

-/

/

j Fig. 1. A crack in a brittle matrix intersected by ductile reinforcements. The reinforcements stretch and fail as the crack opens. The work of fracture of the ductile reinforcements (inset) contributes to the toughness of the composite.

u*

Am = Vf J o(u) du

(2)

0

where o(u) is the nominal stress required to stretch a ductile ligament by u and u* is its failure stretch. Then the applied stress intensity factor can be rewritten as

[ E

/applied = gtip + / 1 - ~

A to J

(3)

where E is the Young modulus and v the Poisson ratio of the composite. The same result is obtained by considering the reduction in Kti p caused by the distribution of o(x) of closing stress inserted by ductile ligaments for a distance L (bridging length) behind the crack tip. The bridging stress intensity factor is given by [12] gbridge

=

o,x,ax

Vf j - ~

(4)

0

Although it has been demonstrated that significant improvements can be achieved in room temperature fracture toughness, the effects of ductile reinforcement on the creep behavior have not been investigated. Since these materials are developed primarily for applications at elevated temperatures, a good understanding of the factors influencing both the room temperature and high temperature behaviors could be most valuable in the design of these types of composite

In these simulations the ductile fibers are modelled using a constitutive relationship that accounts for strength degradation resulting from the nucleation and growth of voids. The matrix was assumed to be elastic and fails according to requirements of a stress criterion. The debonding behavior at the interfaces between the matrix and ductile reinforcements was simulated in terms of a cohesive zone model which describes the decohesion by both normal and tangential separation. In the analysis of a deforming two-phase material it is often necessary to make simplifying assumptions about the shape and distribution of the phases in order to make the problem tractable. A unit cell around the periodic array of aligned short fibers as shown in Fig. 2A was approximated by an axisymmetric model as shown in Fig. 2B. The initial cell and fiber geometries were specified by the cell half-length l~ and radius r~ and the fiber half-length If and radius re. The fiber volume fraction is

(5)

rf2lf Vf -- rc2l c

The initial fiber aspect ratio af and cell aspect ratio a c are defined by If

af = - , rf

lc

ac = -re

(6)

The deformation of the unit cell must be constrained to maintain compatibility and equilibrium with the adjacent material. This constraint requires that the cell boundaries remain straight and orthogonal and free of shear traction. Several methods for imposing these requirements on finite element modelling (FEM) have been suggested [15-17]; the procedure outlined in ref. 17 is utilized in this study.

2.1. Material model 2.1.1. Ductile reinforcing phase The discontinuous ductile fibers were modelled using a constitutive relationship that accounts for

S. B. Biner / Fractureand creep analysisof brittle matrix composites

127

I _o

Ic L

--

--

--T

.

.

.

.

.

.

.

.

.

x2

I I I

×2

ct

(a)

(b)

(c)

Fig. 2. (A) Unit cell. (B) Parameters of the unit cell. (C) FEM mesh used during the analysis.

strength degradation resulting from the nucleation and growth of microvoids. The basis for the constitutive model is a flow potential introduced by Gurson [18, 19] in which voids are represented in terms of a single internal variable f, the void volume fraction:

2f*qj cosh( q2ah] - 1 - q12f*2 = 0

¢=~+

at-

~ 2or ]

(7)

The growth rate is related to the macroscopic dilation rate by (f)growth = (1 -f)6i/Oi p

(11)

where 0i/p is the plastic part of the rate of deformation. The increase in void volume fraction due to the nucleation process is taken as

where

o~=~a':o ',

ah=l:a,

a'=o--lohI

(8)

and of is the flow strength of the fibers. The parameters ql and qz were introduced by Tvergaard [15, 20]; the case ql =q2 = 1 corresponds to Gurson's original formulation. The function f * was proposed by Tvergaard and Needleman [21] to account for the effect of rapid void coalescence at failure. Initially f * = f as originally proposed by Gurson, but at some critical void fraction f~ the dependence of f * on f is changed. This function is expressed by

f, f*=

fc

f
+f~*-f~ ( f f ) ,

f~_f,J-Jc,

f>--fc

(9)

The constant fu* is the value of f * at zero stress in eqn. (7) (i. e. fu* = 1 ~q l) and ff is the void fraction at fracture. As f--ff, f*~f~* and the material loses all stresscarrying capacity. The increase in void volume fraction f arises from the growth of existing voids and the nucleation of new voids. Thus / = (/)growth + (/)nucleation

(10)

The first term in eqn. (12) models the nucleation of voids due to the maximum normal stress as suggested in ref. 22, while the second term represents the plasticstrain-controlled nucleation process. As suggested by Chu and Needleman [23], void nucleation is assumed to follow a normal distribution. Thus plastic-strain-controlled nucleation is specified by s~(2:r)l/2 exp -

,

B=0

(13)

where fN is the volume fraction of void-nucleating particles, eN is the mean strain for nucleation and sN is the corresponding standard deviation. Similarly, for stresscontrolled nucleation with mean stress aN for nucleation

SN(27r)1/2

SN

(14)

D=0 Non-zero values of D and B are only used if Omax+Okk/3 and e f exceed their current maxima during the incremental solution.

128

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Fracture and creep analysis of brittle matrix composites

In the present investigation a rate-sensitive version of Gurson's model will be employed. In the ductile fibers the microscopic effective plastic strain rate ~fP is represented by the power-law relation

Of I 1/m

(15)

where m is the strain-rate-hardening exponent, go is a reference strain rate and efP is the current value of the effective plastic strain representing the actual microscopic strain state in the fibers. The function g(efP) represents the effective tensile flow stress in the fiber material in a tensile test carried out at a strain rate equal to the reference strain rate go. For a powerhardening fiber material the function g(ef p) is taken to be

(Ef,~f p )N g(efP) = o o k O'o +1 ,

g(O) = o o

(16)

with strain-hardening exponent N, elastic modulus Ef and yield stress o o. Using ~ = 0 as the plastic potential with the consistency condition, the values of f and of can be determined from known strain rates and macroscopic stress rates; the other details of F E M implementation can be found in refs. 15, 21 and 24. The material parameters for the ductile fibers appearing in eqns. (15) and (16) were chosen as Ef=500Oo, v=0.3, N = 0 . 2 , m = 0 . 0 1 and the reference strain rate go = 2 x 10 -3. Initially a zero volume fraction of voids is assumed in the fibers. The parameters appearing in eqns. (12)-(14) for void nucleation were taken as f y = 0 . 0 4 , SN=0.I, aN=2.2ao and eN =0.3. For accelerated void growth the parameters appearing in eqn. (9) were chosen as ff = 0.2, fc = 0.15 and fu*= 1. Also, ql =q2 = 1 was selected for eqn. (7) as in the original Gurson formulation. 2.1.2. Matrix material The matrix material is assumed to be linear elastic. During the course of the analysis various elastic moduli were used; these values will be indicated in the appropriate places. Throughout, a constant Poisson ratio v = 0.2 is assumed for the matrix. A critical value of the maximum principal tensile stress am* was chosen for the matrix material failure based on Griffith's fracture criterion. Upon attainment of this critical value of the maximum principal tensile stress at an integration point within an element, it is assumed that matrix failure has occurred at that element. The critical values of Om* will be given in the appropriate places.

2.2. Interface model To study the effects of fiber debonding and subsequent fiber pull-out during the deformation of fiberreinforced composites, it is necessary to simulate the interface failure by normal and tangential separation. A debonding model has been developed by Needleman [25] in terms of a potential that specifies the dependence of interface traction on the displacement differences at the interface. The potential used in ref. 25, which defines the non-linear variation in interface traction as a function of interface displacements, also contains three parameters, namely o'i, 0 i (ON, 0T) and a, where oi is the interfacial strength, complete separation is assumed to occur at uN = ON, and a specifies the ratio of shear to normal stiffness of the interface. These parameters are assumed to be intrinsic material properties. However, as discussed in ref. 26, the interface constitutive relationship given in ref. 25 describes the debonding only by normal separation. Therefore it is not suitable for tangential separation and fiber pull-out that occur under significant normal compression. An alternative model introduced in ref. 26 is utilized in this study. The normal and tangential tractions between the fiber and the matrix are given by

Ty = ~

F(2)

(17)

TT=a F(Z)

(18)

where F(2)is chosen as F(2)=~

o~(1- 2;t + 22)

~r0~2~1

and

(20) Equations (17) and (18) are valid as long as ). is monotonically increasing. Under normal compression, however, elastic springs with a high stiffness are used to approximately represent the contact instead of eqns. (17) and (18). The mesh used during the analysis is shown in Fig. 2C. The elements used are quadrilaterals, each built up from four triangular axisymmetric linear displacement elements. The ductile fibers constitute 20% of the total cell volume (i.e. Vf=0.2) and cell and fiber aspect ratios were chosen as a c = af = 5. The axial deformation rate of the unit cell was the same as the reference strain rate in eqn. (15). During the analysis the uniform stress and strain values were computed from the resulting

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Fracture and creep analysis of brittle matrix composites

reaction forces and axial displacements. The integration of the stress rate requires small time steps for stable numerical integration. The tangent modulus method of Peirce et al. [27] is used to increase the stable time step size. The tangent modulus provides a forward gradient estimate of the deformation rate based on a Taylor series expansion about the current deformation rate. The material failure both in the matrix and in the ductile fibers is implemented by the element vanish technique [21]. When the failure condition is met within an element, i.e. attainment of a critical value of either o,,* for the matrix or f = f f for the ductile fibers, that element no longer contributes to the virtual work. To avoid numerical instabilities, the nodal forces arising from the remaining stresses in failed elements are redistributed in several iterations. The role of the material parameters of the matrix in the deformation and fracture behavior of the composite system was studied first. During these analyses the interface between the discontinuous ductile fibers and the matrix was assumed to be rigid (i.e. no interface separation). The results of investigating the influence of matrix fracture strength am* on the composite deformation and fracture behavior are shown in Fig. 3. In these two simulations the matrix fracture strengths were chosen as Om*= 50o and 10ao and the elastic modulus of the matrix was kept constant at E m= 10Ef. Of course, this variation in the matrix strength yields two different work-of-fracture values for the matrix material. In both cases almost a linear deformation of

129

the composite system up to matrix failure was seen, even though the uniform stress level in the composite was about 6.25 times the yield stress a o of the ductile fibers. This is associated with the development of a triaxial stress state in the fibers which lowers the effective stress level. After failure of the matrix a large reduction in the stress carried by the composite can be seen and the deformation continues in a non-linear mode owing to the work hardening of the fibers. The subsequent load reduction is associated with damage development in the fibers resulting from the nucleation and growth of voids. In Fig. 3 the final failure of the composite occurs upon fulfilment of the ductile fracture in the fibers (Le. f-~ff). The distribution of maximum stress values in the unit cell just prior to failure of the matrix is shown in Fig. 4A. In all cases investigated, matrix cracking occurred first at the corner of the ductile fibers owing to the stress concentration effect, as can be seen from Fig. 4A. The contours of void volume fractions in the fiber just prior to final failure of the composite are shown in Fig. 4B. In this figure the cracked matrix elements are black. As can be seen, the damage accumulation in the fibers is very localized to the crack tip region. It appears that the deformation of the ductile fibers is still constrained by the matrix and limited to opening displacements of the crack in the matrix. The work-of-fracture values (i.e. area under the

/1/ 7.0 6.5

a.o

6.0

1.7.

5.5

4.9

- " / c~ = 1 0 . 0 7 % / s ~ = 2.34. J

5.0

'4.3

4.5

I o.~o A ~

7 o.o~ ~

0 4.0 rO 3.5 (13 O3 i,i QE I.-03

0.1 7

1 :

3.0

I

2.5 2.0 1.5 1.0 0.5 B

0.0 0.000

i

I

i

i

0.00.4

0.008

0.012

0.016

0.020

STRAIN

Fig. 3. Stress-strain behavior of the composite for various fracture strength values of the matrix.

Fig. 4. (A) Distribution of maximum stress values in the unit cell just prior to failure of the matrix. The stress values were normalized by the fiber reference stress value o o. (B) Distribution of void volume fractions within the ductile fibers just prior to failure for the case of aM*= 5ao and a rigid interface.

S. B. Biner

130

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Fracture and creep analysis of brittle matrix composites

stress-strain curve) of the composite, ~Oc, normalized by the work of fracture of the matrix material, oM, for these cases are also included in Fig. 3. The effects of the elastic modulus of the matrix on the deformation and fracture behavior are shown in Fig. 5. During these simulations the fracture strength of the matrix was kept constant at Ore*= 500 and the elastic modulus ratio between the fibers and the matrix was taken as E m/Ef = 1, 3 and 10. However, owing to the changes in the elastic modulus value of the matrix, the work-of-fracture values for the matrix were also different for these cases. Figure 5 shares the same common features as in Fig. 3. However, for the case of the lowest elastic modulus ratio a more pronounced non-linear deformation behavior of the composite prior to the matrix failure can be seen. For the lowest modulus ratio the strain required for the failure of the fibers was also much smaller than for the other two cases in Fig. 5. This behavior is associated with the evolution of the damage within the fibers. The variation in the maximum void fraction values with axial strain of the unit cell is shown in Fig. 6. For the higher elastic modulus ratios the damage formation within the fibers began with matrix failure, whereas in the case of the lowest ratio the damage formation occurred before the onset of matrix cracking. For this case, after the initial void nucleation, very limited void growth took place until failure of the matrix due to the constrained deformation of the ductile fibers. The subsequent damage evolution with failure of the matrix was almost the same as in the other cases.

In the simulations presented in Fig. 7, the matrix work-of-fracture value was kept constant for the above modulus ratios by altering the matrix fracture strength am*. As can be seen, in this case the net gain in the work of fracture of the composite did not vary as significantly as in the other cases shown earlier. It is inter-

0.25

/

0.20

I

0 z

/

EM

EF

C) <

,,~ o.15 a

~

= 10.01 \\

"~ 0.10 M < 2~ 0.05

0.00

I

0.000

0.004

0.008

0.012

I

0.016

0.020

STRAIN

Fig. 6. Variation in the maximum void volume fraction within the ductile fibers with increasing uniform composite strain. Data are shown for various elastic modulus values of the matrix. 4.0

i

6.0 3.5

5.5

00

;E :,0ol

30

~'u 8.24J

~oM= 8.24

[E+/EF:3O] 0

b

E~ / E F = 1.o

o

EM

Er =

3.0

% ~u 8.89

2.0

z.0 2.5 03 Lt.I V--- 2.0 t/3 1.5 ~

L~ 1.5

03

k'1.0 0.5

1,0 0.5

0.0 0.0

0.000

0.000 0,004

O. 008

0.012

0.016

0.020

STRAIN Fig. 5. Stress-strain behavior of the composite for various elastic modulus values of the matrix.

0.004

I

I

I

0.008

0.012

0.016

0.020

STRAIN Fig. 7. Stress-strain behavior of the composite for constant

work-of-fracture value but various elastic modulus and fracture strength values of the matrix.

S. B. Biner

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Fracture and creep analysis of brittle matrix composites

esting to note that after matrix failure the work hardening of the ductile phase started at about the same stress level. In the next series of simulations the role of interface characteristics between the fibers and the matrix is investigated. For these simulations the matrix properties were kept constant at Em= 10El and am* = 5O,,. The critical separation distance was taken as 6 N= 6T = 0.01r f, which appears in the interface model. In these simulations non-dimensional interface elements were incorporated in both the top and side surfaces of the fibers. In Fig. 8 the stress-strain data obtained for two different interfacial strength values are compared with the data obtained for a rigid interface. The introduction of the interface elements did not have any effect on the stress level required for matrix failure, which indicates no interracial damage or insignificant separation before the onset of matrix failure with the chosen interface parameters. As can be seen, the net gain in the work of fracture of the composite is reduced with the introduction of the interface failure mechanism. After matrix failure, with decreasing interracial strength, there were reductions in the secondary increase in the composite strength levels resulting from the work hardening of the ductile fibers. The maximum value of damage accumulation in the ductile fibers with axial strain of the composite is shown in Fig. 9. It can be seen that the damage accumulation in the ductile

131

fibers is delayed owing to further relaxation of the build-up of hydrostatic stress by interface separation. However, the composite failure occurred before the full work-hardening capacity of the ductile fibers was reached. This is due to the premature failure of the top interface and the loss of the associated load transfer mechanism to the fibers. To further elucidate the role of the interface behavior and loading mechanism of short fibers, in the following simulations interface elements were only introduced in the side surfaces of the fibers; the top surface of the fibers was assumed to be rigidly connected to the matrix. For these cases the resulting stress-strain behavior was again compared with the rigid interface case (Fig. 10). When comparison is made with the previous analysis for the interface strength o~=5o o in Fig. 8, the importance of load transfer to the fibers during the debonding and fiber pull-out can be seen. Much higher overall composite ductility and a net increase in the work-of-fracture value were observed for the interfacial strength value of o~= 2.5Oo. The deformation pattern and the evolution of damage in the fibers are shown in Fig. 11 for oi=2.5Oo . The extent of the void nucleation and growth within the fibers and the extensive neck formation in the fibers can be clearly seen in this figure.

0.25

4.0

5.5 0.20 Z 0

3.0

L.) .< 0.15

2.5 b

o 2.0

FRG0~MNTER ACE = 8.24

/% / ....

Cb

1

....

%

/

~u

>

7.89

Z)

0.10

X .< ~E 0.05

I~

0,5 0.0 0.000

!]/% ~ 0.004

/ % = 5.0

s0 J 0.008

I 0.012

J 0.016

0.00 I 0.000

I 0.004

0.008

~ 0.012

i 0.016

0.020

0.020

STRAIN Fig. 8. Stress-strain behavior of the composite for various interfacial strength characteristics between the fibers and the matrix. In these simulations non-dimensional interface elements were incorporated in both the top and side surfaces of the fibers.

STRAIN

Fig. 9. Variationin the maximumvoid volumefraction within the ductile fibers with increasing uniform composite strain. Data are shown for various interracial strength characteristics. Nondimensional interface elements were incorporated in both the top and side surfaces of the fibers.

S. B. Biner I Fractureand creep analysis of brittle matrix composites

132 4.0

/ // 6

,

3.5

3.0

2.5

b0

2.0

IRIGIDINTERFACE1 ~ / ~u = 8.24

(/3

It%

0.5

O"i / 0 " °

~ 0.0

0.00

L

0.01

/

=

~M

/ ~u =

1

2.5 34.05 i

0.02

z/if

I

I

0.06 0.08 0.10

STRAIN

Fig. 10. Stress-strain behavior of the composite for various interracial strength characteristics between the fibers and the matrix. In these simulationsnon-dimensionalinterface elements were only incorporated in the side surfaces of the fibers; the top surface was rigidlyconnected to the matrix.

3. Creep behavior During the analyses both the ductile reinforcement and matrix creep behaviors were assumed to obey ~=go

(21)

where Oo is a reference stress (yield stress at test temperature), do is the reference strain rate, n is the creep exponent and o is the effective yon Mises stress. During the analyses the matrix properties were kept constant and the creep property of the ductile reinforcing phase was changed to give steady state creep ratios between the reinforcing phase and the matrix of gf/gm= 10, 100 and 500 at the applied nominal stress level. To simulate the constant-load creep test, normal traction was applied at the beginning of the solution to the end of the unit cell such that the area integral of the traction was equal to the applied load: f T~dA = o,4

(22)

A

where o was taken as a = 0.5 a o. The role of the material properties and the geometrical parameters of the ductile reinforcing phase in the creep behavior of the composite was investigated

first. In these simulations the interface between the ductile phase and the matrix was assumed to be rigid and analyses were carried out for a fiber aspect ratio af = 5. To elucidate the role of the creep rate of the ductile reinforcing phase, the gf/gm ratio was taken as 10, 100 and 500. The elastic modulus ratio between the ductile reinforcing phase and the matrix, Ef/Em, was assumed to be 0.2. From these simulations the resulting variations in the composite strain rates with the accumulated values of the strain are shown in Fig. 12. In this figure the composite creep rates were normalized by the matrix steady state creep rate, and the axial strain values obtained for a 20% volume fraction void-containing matrix (i.e. 20% void-containing unit cell instead of fiber) are also included for comparison. For all gf/gm ratios investigated, a true steady state condition for strain accumulations up to 6% was not achieved, as can be seen in Fig. 12. For /~f/gm ---- 10 the composite creep rate after a pronounced reduction in the initial stages of deformation (primary creep) steadily increased during subsequent deformation. For other values of gf/gm this primary stage of creep deformation was very short. As can be seen, a large variation in gf/gm did not significantly affect the composite creep rate and the values observed were about 20%-30% more than the matrix creep rate. On the other hand, the creep rate of the 20% void-containing matrix was 50%-60% more than the matrix creep rate in the same strain range. In the following simulations the role of the fiber geometry in the creep behavior was studied for fiber aspect ratios af = 2.5, 5 and 10. For these cases the material parameters were taken as Ef/Em=0.2 and ef/gm = 100. For various fiber aspect ratios the changes in the strain rates with accumulated values of strains are shown in Fig. 13. As in the previous case, the creep behavior was not significantly influenced by the fiber aspect ratio except in the early stages of creep deformation. The composite strain rates were again about 30% higher than the matrix strain rate. In the next set of analyses the role of debonding and fiber pull-out in the composite creep behavior was investigated. To evaluate the role of the interface strength, the ratio O ' i / O o w a s varied from 0.125 to 0.325 and the other interface parameters appearing in eqns. (17)-(20) were kept constant at 6N,T= 5 X 10-4q and a = 1. The analyses were carried out for a fiber aspect ratio af = 5 and the material parameters were chosen as Ef/E m = 0.2 and ef/gm = 100. The variation in the strain rate of the composite is plotted against the accumulated values of the axial strain in Fig. 14. During the analyses the interface failure always started from the center of the top surface of the fibers. The fiber pull-out events occurred in several stages in which observed strain rates varied significantly. As debonding

S. B. Biner

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Fracture and creep analysis of brittle matrix composites

133

and fiber pull-out progressed, there were several occurrences of acceleration and deceleration in the strain rates as can be seen in Fig. 14. The creep strain rates that are observed decrease with increasing inter2.00

o.o I

1.80 0.12 0.095

i

J

0.07~ 0.06~.

0.05"

, . i

Ef / E m ---- 0 . 2 Ef // Em---- 1 0 0

O.O4-

r,,

2 0 Z Void ConfainTng ]

1.60

Z <[ ty

O.O4~

0.02",=

t:3 L=J N ..J .<( ]E

0.008. 0.OO4"

t~ z

1.40

1t /

rf = 2 . 5 }

1.20

I,,/ r,=I0,01 1.00

B

A

C

Fig. 11. Distribution of void volume fractions within the ductile fibers at various stages of deformation for the case of o~= 2.5o o. Uniform strain values for the unit cell were (A) 0.01, (B) 0.04 and (C) 0.094 respectively.

0.80 0,001

i

i

i

i

i = = =I 0.010

i

,

,

, , , , 0.100

STRAIN

Fig. 13. Variation in axial strain rates with accumulated values of axial strain for various fiber aspect ratios. 2.00

.....

i

2.00

T

If / rf = 5 Ef / E m = 0 . 2

Et / Ef /

1.80

Em = 0 . 2 =

if /

100

rf = 5 . 0

t~1 = 5 x l 0 - 4 r f

1.80

2 0 Z Void } Ld n-

2 0 Z Void C o n t a i n i n g ]

1.60

,~ P~

z

r'~ bJ N .-J ~E t~ O Z

r

1.60

z

[ ~f /

1.40

i

= 5oo

u~ lad N .J ~E n," O Z

1.20

/

I',, < ,oo]

1.20

~

OCe I~, / °° -- ° ' ~

1.00

0.001

1.40

I

1.00

0.010

0.100

STRAIN

Fig. 12. Variation in axial strain rates with accumulated values of axial strain for various ~r/~m ratios. The composite strain rates were normalized by the steady state creep rate of the matrix.

0.80 ~ 0.001

-

0.010

0.100

STRAIN

Fig. 14. Variation in axial strain rates with accumulated values of axial strain for various interracial strengths between the ductile reinforcements and the matrix.

134

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Fracture and creep analysis of brittle matrix composites

face strength. Both the time and strain required for the debonding and fiber pull-out also increase with interface strength, but it appears that this does not occur in a linear fashion. The results in Fig. 14 also indicate that complete interface separation did not take place for strain values up to 6%, otherwise the composite strain rate should have been equal to the creep rate of the matrix containing 20% voids as observed in an earlier study [28]. The combined effects of interface separation and fiber aspect ratio were studied in the next set of analyses. In these analyses the material parameters were again chosen a s E f / E m = 0.2 and gf/gm = 100 and the interface properties were taken as oi/Oo = 0.25 and a = 1, while the fiber aspect ratio was varied as af = 2 . 5 , 5 and 10. Since the fiber diameter varies with the aspect ratio, the work of separation per unit interface area is also affected owing to changes in the separation distance via the 6N.x = 5 × 10 -4rf relation. Therefore an increase in the fiber aspect ratio yields smaller separations and hence a lower value of the work of separation for constant values of 0i/Oo=0.25 and a = 1. The results observed from these simulations are presented in Fig. 15. In contrast with the rigid interface (Fig. 13), the same variation in the fiber aspect ratio exhibited a significant difference in strain rates. Although the fiber aspect ratio of 10 has the lowest value of the work of separation per unit interface area (i.e. the weakest

2.00

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.

.

.

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.

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.

.

.

.

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1.80

1.60 z

1.40 N

,0 z

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1.00

0.80 I 0.001

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STRAIN

Fig. 15. Variation in axial strain rates with accumulated values of strain for various aspect ratios with debonding interfaces.

interface), it gave the most delayed debonding and fiber pull-out as well as the lowest secondary creep rates. For these cases, except for the sudden elevations in the strain rates during debonding, the composite strain rates were about 30%-80% more than the matrix strain rate.

4. Discussion

The analyses presented in this study are not based on detailed quantitative data for the parameters that appear in the constitutive models of the ductile fibers and matrix or the parameters in the cohesive zone model of a particular composite system. Rather, the results give mainly qualitative information about the influence of various parameters on the evolution of the work of fracture, Ato (eqn. (2)), under simple uniaxial far-field loading. For rigid interfaces between the ductile reinforcement and the matrix the role of the matrix material parameters in the fracture characteristics was summarized in Figs. 3, 5 and 7. The changes in the various matrix properties resulted in different work-of-fracture values for the matrix in Figs. 3 and 5. In contrast, in Fig. 7 for different elastic modulus values of the matrix the work-of-fracture values for the matrix were kept constant by altering the matrix fracture strength. In all cases except for the lowest elastic modulus ratio the stress-strain behavior of the composite was almost linear until the failure of the matrix, even though the uniform stress level in the composite was as high as 6.25 times the yield strength of the ductile reinforcing phase. As will be shown later, this behavior is associated with the development of a triaxial stress state in the ductile reinforcing phase which lowers the equivalent stress and mitigates the plastic deformation of the ductile phase. From the simulations it appears that the work of fracture of the composite is composed of two regimes. The first regime is controlled by the matrix behavior and terminates with matrix failure. In the second regime the constrained deformation of the ductile phase terminates with damage accumulation within the ductile phase, leading to failure. For a given ductile reinforcement its contribution to the composite work of fracture appears to depend upon the magnitude of the exhaustion of the work-hardening capacity of the ductile phase in the first regime, i.e. before the onset of matrix failure. For rigidly bonded interfaces, even if the nucleation of voids occurs before matrix failure and in the presence of a large hydrostatic stress, the growth of these voids, and hence the damage accumulation, is extremely small owing to the constraint deformation of the ductile phase as seen in Fig. 6. Therefore the minimum exhaustion of the work-hardening capacity

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of the ductile phase in this first regime will yield a larger contribution to the composite work of fracture during subsequent deformation. This can be seen in Figs. 3, 5 and 7, where the increase in the fracture strength (Fig. 3) and reductions in the matrix elastic modulus values (Fig. 5) caused the consumption of most of the work-hardening capacity of the ductile phase before the onset of matrix failure, resulting in smaller improvements in the toughness values. The opposite can be seen in Fig. 7: when failure of the matrix occurred, the level of work hardening in the ductile phase was about the same in all three cases, yielding similar toughness values for three matrices with different elastic modulus and fracture strength values. The weakening interfacial strength resulted in a change in the deformation mode of the ductile fibers from localized deformation to more diffuse neck formation. However, to achieve the full work-hardening capacity of the ductile reinforcing phase, the load transfer to the reinforcement should be maintained during the debonding and pull-out process (Figs. 8 and 9). One of the reasons for the early loss of load transfer with decreasing interfacial strength in the simulations presented in Fig. 8 is assumed to the location of matrix cracking. As seen in Fig. 4A, the ductile phase induced a large stress concentration in the matrix and the matrix cracking always started at the corner of the fibers. As mentioned in refs. 29 and 30, rounding of the fiber corners may considerably alter the stress state in the matrix and hence may move the incidence of matrix cracking to a more favorable location. Otherwise, in order for this type of discontinuous fiber reinforcement to maintain the load transfer, the interface modification for selective areas of the fibers must be achieved by other processing means. The benefits of such reinforcement with selectively modified interfaces can be seen in Figs. 10 and 11, where continuous debonding along the side surface of the reinforcing phase resulted in delayed damage formation within the ductile phase owing to a shift in the stress state. For rigid or weak interfacial behavior, another important aspect is the large loss in the stress-carrying capacity of the composite after matrix cracking. This should be balanced with the work-hardening and damage accumulation behavior of the suitably selected ductile reinforcing phase for enhanced composite behavior. From the results presented in this study, it appears that the contribution of ductile reinforcements to the composite toughness is not only a function of the volume fraction, strength and ductility characteristics of the ductile phase, as assumed in eqn. (2) or (4), but is also strongly influenced by the fracture strength and Young modulus of the matrix and by the interfacial behavior.

135

In the case of rigid interfaces between the matrix and ductile reinforcements, the material properties and geometric parameters of the ductile reinforcement did not show any significant influence on the creep deformation of the composites. The creep rates for the composites were about 20%-30% more than that for the matrix; also, the presence of true steady state creep rates up to strain accumulations of 6% was not observed. To elucidate this behavior, for rigidly bonded interfaces the variation in the equivalent stress, hydrostatic stress and resulting equivalent strain distributions within the unit cell at various stages of creep deformation are shown in Fig. 16 for a fiber aspect ratio af = 5. As can be seen, large values of the equivalent stress in the matrix were located between the reinforcements during the early stages of deformation. A stress redistribution takes place with increasing deformation: while the equivalent stress value increases in the regions at the top of the fibers, a reduction occurs in the areas between the fibers. As a result of this stress distribution, the stresses carried by the fibers were also increased; however, this increase was not as large as the changes that were occurring within the matrix. When a comparison is made between the equivalent stress and hydrostatic stress values, the relatively large development of a triaxial stress state within the ductile reinforcement can also be seen in Fig. 16. Initially the large strain accumulation within the fibers occurred at the top region of the fibers, resulting from the stress concentration effect of the fiber corners. The magnitude of the strains within the fibers was much smaller than that of the strains in the matrix owing to this stress triaxiality in the fibers. In subsequent deformation a more uniform strain accumulation within the matrix can be seen owing to stress redistribution. Again, because of the slight elevation in the stresses within the fibers as a result of this stress redistribution, the deformation rate in the fibers increases and becomes comparable with the strain accumulation in the matrix owing to the constraint created by the matrix and also to the triaxial stress state. In the case of the debonding interface between the fibers and the matrix a strong dependence of the fiber aspect ratio on the composite creep behavior was observed in Fig. 15. This behavior is associated with the differences in the magnitude of the stress triaxiality and their relaxation rates when the fiber-debonding process occurs in composites with different fiber aspect ratios, which has been discussed in detail in an earlier study [28]. The results obtained in this study indicate that significant (several-fold) improvements in the room temperature fracture properties of brittle materials can be achieved without extensively sacrificing the creep strength by the use of ductile reinforcements. For

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Fracture and creep analysis of brittle matrix composites

W

T I M E : 2.38

T I M E : 25.91

TIME : 4 1 0 . 5 0

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Fig. 16. Variation in equivalent stress (top row), hydrostatic stress (middle row) and equivalent strain (bottom row) distributions within the unit cell with time.

rigidly bonded interfaces the material properties and the geometrical parameters of the ductile reinforcing phase do not play a significant role in the creep deformation behavior of the composites. In this case the selection of the ductile reinforcement should be based

on its room temperature deformation behavior and its physical properties (e.g. chemical compatibility with the matrix, oxidation resistance, thermal compatibility, etc.). For debonding interfaces at elevated temperatures the geometrical parameters of the ductile re-

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Fracture and creep analysis of.brittle matrix composites

inforcing phase were seen as an important factor. However, even for very weak interfacial behavior, low creep rates for the composite can be achieved by suitable selection of the aspect ratio of the reinforcements. In this study the room temperature fracture characteristics and high temperature creep behavior of ductile discontinuous reinforced composites were investigated by using unit cell analysis under simple uniaxial farfield loading. In the unit cell analysis the assumptions of fully aligned fibers and a completely periodic pattern of distribution of reinforcing phase are an idealization. Since the geometric parameters of the reinforcing phase and their distribution in the matrix are nonuniform, the failure of the matrix, the interface failure and pull-out mechanisms do not occur simultaneously in all reinforcement locations but may occur sequentially like the strain jumps observed experimentally [3]. In the room temperature fracture analyses the possible thermal stresses due to the differences in the thermal expansion coefficients of the phases, and in the high temperature creep simulations the possible stress relaxation resulting from diffusion mechanisms [31, 32], were also neglected during the analyses. In spite of these simplifications, it is expected that the present analyses give reasonable indications of the role of the material parameters, the geometric parameters of the reinforcement and the interfacial behavior in the room temperature fracture characteristics and high temperature creep deformation behavior of composites consisting of a brittle matrix and ductile discontinuous reinforcements.

5. Conclusions The role of the matrix material parameters and interface characteristics in the fracture and creep behavior of discontinuous ductile fiber-reinforced brittle matrix composite systems was studied numerically. The results indicate the following. ( 1 ) For rigid interfaces between the ductile reinforcing phase and the matrix the contribution of the ductile reinforcement to the work-of-fracture value (toughness) of the composite increases with less exhaustion of its work-hardening capacity before the onset of matrix failure. Therefore the failure strength and elastic modulus values of the matrix become important material parameters. (2) In the case of interfacial debonding, the load transfer to the discontinuous reinforcements after matrix failure should be somehow maintained for utilization to full capacity of the reinforcement and interfacial behavior.

137

(3) For rigidly bonded interfaces the creep rate of the composite is not significantly influenced by the material properties and geometric parameters of the ductile reinforcing phase owing to the development of a triaxial stress state and constrained deformation in the reinforcement. (4) For debonding interfaces the geometric parameters of the reinforcing phase become important; however, even for very weak interracial behavior, low composite creep rates can be achieved by suitable selection of the geometric parameters of the ductile reinforcing phase. (5) Significant increases in room temperature fracture toughness can be achieved without extensively sacrificing the creep strength by ductile discontinuous reinforcements.

Acknowledgments This work was performed for the United States Department of Energy by Iowa State University under contract W-7405-Eng-82. This research was supported by the Director of Energy Research, Office of Basic Energy Sciences. The author wishes to thank R. Winther for useful discussions during the computer implementation of FEM.

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18 A. L. Gurson, Plastic flow and fracture behavior of ductile materials incorporating void nucleation, growth and interaction, Ph.D. Thesis, Brown University, 1975. 19 A.L. Gurson, J. Eng. Mater. Technol., 99 (1977) 2. 20 V. Tvergaard, Int. J. Fract., 18 (1982) 237. 21 V. Tvergaard and A. Needleman, Acta Metall. Mater., 32 (1988) 157. 22 A. Needleman and J. R. Rice, in D. P. Koistinen and N. M. Wang (eds.), Mechanics of Sheet Metal Forming, Plenum, New York, 1978, p. 237. 23 C. C. Chu and A. Needleman, J. Eng. Mater. Technol., 102 (1980) 249. 24 V. Tvergaard, J. Mech. Phys. Solids, 30 (1982) 399.

25 A. Needleman, J. Appl. Mech., 4 (1987) 525. 26 V. Tvergaard, Mater. Sci. Eng. A, 125 (1990) 203. 27 D. Peirce, C. E Shih and A. Needleman, Comput. Struct., 18 (1984) 875. 28 S.B. Biner, Mater. Sci. Eng. A, 156 (1992) 21. 29 A. Needleman and S. Suresh, Acta Metall. Mater., 39 (1991) 2317. 30 S.R. Nutt and A. Needleman, Scr. Metall., 21 (1987) 705. 31 K. Wakashima, B. H. Choi and T. Mori, Mater. Sci. Eng. A, 127(1990) 57. 32 J. Rosier, B. Bao and A. G. Evans, Acta Metall. Mater., 39 (1991)2733.