Materials Science Research International, Vol.1, No.1, pp. 17-22 (1995)

General paper

SIMULATION OF DEBONDING ON THE INTERFACE AND

SUBSEQUENT CREEP CRACK GROWTH IN THE

MATRIX FOR METAL MATRIX COMPOSITE

Mamoru MIZUNO, Koji TANAKA and Tatsuo INOUB Department of Energy Science and Engineering, Kyoto University,

Yoshida-Honmachi, Sakyo-ku, Kyoto 606-01, Japan

Abstract: Mode of debonding on the interface between Al matrix and SiC whisker aligned unidirection-

ally is simulated by using the finite element method, and the subsequent creep crack growth in the matrix

at 300•Ž is treated in the framework of the continuum damage mechanics. In order to estimate damage

and stress development in the vicinity of the whisker, a unit cell model which is based on a kind of pe-

riodical boundary condition is employed. Damage development associated with stress/strain for the unit

cell model is calculated as the axisymmetric and also plane strain conditions, and the difference between

both results are compared. The analyses of debonding and creep crack growth are carried out by using

a local approach, and macroscopic creep deformation of MMC as well as microscopic damage and stress

distribution is also evaluated, and the result is discussed in the comparison with the experimental data.

Key words: Metal matrix composite, Debonding on the interface, Creep crack growth in the matrix, Finite

element analysis, Unit cell model, Continuum damage mechanics, Aluminum matrix, SiC whisker

1 INTRODUCTION

Metal matrix composite, MMC is expected as one of candidated materials exposed to high temperature, and some trials have been carried out to evaluate the fundamental mechanical properties [1-3]. Since macroscopic properties of the composite materials are prescribed by the microscopic properties, the mech-anism of microscopic structural change affecting the macroscopic properties is required to be clarified.

Recent researches [4-6] based on the experimental observation in Al-SiC and other metal-ceramic com-posite systems indicate the ductile failure in the ma-trix of the composite due to the nucleation, growth and coalescence of voids and void formation on the whisker-matrix interface which leads to the debonding of the reinforcement from the matrix and subsequent crack growth in the matrix [6]. Mechanical behavior of the void in the metal matrix is, therefore, necessary to be evaluated in order to predict the debonding of the interface and failure of the matrix to correlate the deformation of a bulk material with the microscopic structural change.

In this paper, the nucleation, growth and coales-cence of voids are simulated in the framework of the continuum damage mechanics. Macroscopic deforma-tion of the composite material as well as microscopic damage and stress distribution is evaluated by the fi-nite element technique. Debonding of the interface and creep crack growth in the metal matrix are ana-lyzed by the local approach of the damage of scalar value incorporated into the finite element code. In order to simulate damage and stress development in the vicinity of the reinforcement, a unit cell model [7, 8] is introduced, and distribution of stress and dam-age for the unit cell model are calculated as both ax-

isymmetric and plane strain problems, and the calcu-

lated results for each problem are compared. In this

simulation, creep deformation at 300•Ž in aluminum

matrix composite reinforced unidirectionally by SiC

whisker is focused. Stress distribution and damage de-

velopment near the interface are elucidated, and the

mode of the macroscopic creep deformation is com-

pared with the experimental data.

2 METHOD OF SIMULATION

2.1 Unit Cell Model

Whiskers as the reinforcements of the same shape

are assumed to be aligned unidirectionally and period-

ically with the same interval. Under the assumptions,

two arrangements of whiskers, termed as pattern A

and B, are considered as shown in Figs. 1 (a) and (b).

(a) Pattern A

(b) Pattern BFig. 1. Alignment and the cross section of metal

matrix composite unidirectionally reinforced by

whiskers.

Received Sep. 29, 1994

17

Mamoru MIZUNO, Koji TANAKA and Tatsuo INOUE

Axisymmetric problem Case a1

Plane strain problem

Case a2

(a) Pattern A

(b) Pattern BFig. 2. Unit cell model.

Fig. 3. Boundary condition for unit cell model.

Then, a unit cell including whisker depicted by broken line in the figures is extracted from the bulk material. For the sake of simplicity, two series of simulations for pattern A are carried out either as axisymmetric or plane strain problem as indicated in Fig. 2 (a). The ef-fect of the stress distribution calculated by both ways on the damage development and creep curves are dis-cussed. The creep crack growth in the matrix on the other hand, is predicted by use of the model shown in Fig. 2 (b).

Taking into account the symmetricity and period-icity, a half of unit cell in the axisymmetric problem (Case a-1) and a quarter of the unit cell in the plane strain problem (Case a-2) are calculated under the boundary condition as illustrated in Fig. 3, as an ex-ample of case a-2. Upper end and right side of the unit cell are kept straight in order to retain consistency with the neighbor unit cells as depicted in Figs. 1 (a) and (b).

(a) Pattern A (b) Pattern BFig. 4. Finite element division.

2.2 Constitutive Equation

Aluminum matrix is assumed to obey Norton creep

law and Kachanov-Rabotnov creep damage hypothe-

sis. Constitutive equation of the creep strain ƒÃcij and

the evolution equation of damage variable D for alu-

minum matrix are assumed to be expressed as [9,10]:

(1)

(2)

where ƒÐ and sij represent equivalent stress and de-

viatoric stress, respectively, while A, n, B and k are

material constants. Damage variable D is regarded

as an internal state variable describing damage state

such as the density of voids, cavities and micro-crack

in the material, and is defined as:

D=0 at t=0 (initial undamaged state),D=1 at t=tR (final ruptured state).

(3)Whisker is, on the other hand, assumed to remain

elastic even in creep deformation process of the ma-trix.

2.3 Finite Element ModelFigures 4 (a) and (b) show the finite element divi-

sion with triangular simplex elements employed in this analysis, where shadowed parts represent the whisker and other elements correspond to the matrix under the assumption that the whisker is bonded perfectly with the matrix. In the vicinity of the interface, a fine mesh pattern is propounded due to severe stress gra-dients.

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SIMULATION OF DEBONDING AND CREEP CRACK GROWTH IN MMC

The initial strain method [11] is employed to incor-

porate the effect of the creep and creep damage into

the finite element calculation. The total strain vector

{} subtracted by the creep strain vector {} gives

the elastic strain vector, and stress vector {} in a

element reads

(4)

with the stress-strain matrix [De].

Since the creep strain varies with time, the creep

rate and creep damage rate are integrated step by step

for small time increment keeping the creep strain to

be constant in the interval. Thus, the effects of the

damage is introduced into the analysis only through

the creep strain ƒÃc in Eq. (4).

The analysis of debonding on the interface and the

subsequent creep crack growth within the matrix is

performed by the local approach of the fracture based

on the continuum damage mechanics and the finite el-

ement method [12]. Since the analysis is carried out

by use of triangular simplex elements, damage vari-

able is assumed to be constant in the element. When

damage variable D in an element attains to a criti-

cal value, the element is interpreted to failure, and

the rigidity of such element is reduced to vanish. The

aggregation of these failured elements is identified as

debonding on the interface or nucreation of a macro-

scopic crack in the matrix.

3 RESULTS OF ANALYSIS

3.1 Condition of Analysis

A composite with the matrix of 6061 aluminum al-

loy reinforced by SiC whisker with 15% volume frac-

tion are treated. Steady stress ƒÐ=70MPa in global

sense in axial direction is applied at 300•Ž. Material

constants in Eqs. (1) and (2) are identified by the ex-

perimental results of the tensile creep test of unrein-

forced aluminum of 6061 type at the temperature [13]. The aspect ratio (or the ratio of the length to the di-ameter) of the whisker is chosen as 7.5, while the ratio of the unit cell is 5.8, which is determined by the data of SEM observation [13] of SiCw/Al processed by the extrusion technique. The dimension of the unit cell shown in Fig. 4 is determined so that the volume frac-tion of the whisker in the cell for both patterns A and B as depicted by broken line in Fig. 1 is equal to 15%.

Material parameters employed in the simulation are listed in Table 1.

Table 1. Material parameters used for calculation.

(a) Axisymmetric problem

(b) Plane strain problemFig. 5. Equivalent stress distribution at t=6hr.

-Pattern A

3.2 Comparison of Simulated Strain and Dam-age between Axisymmetric and Plane Strain Problems

Figure 5 (a) shows the calculated equivalent stress distribution as the axisymmetric problem at t=6hr, and Fig. 5 (b) represents that as the plane strain con-dition for pattern A. Both results indicate the similar shape of stress distribution. Equivalent stress in the region of the whisker is high in both cases due to the increasing axial stress. Since the whisker is assumed to remain elastic during the creep deformation of matrix, stresses in the region of whisker develop with progress-ing creep deformation and elongating the unit cell in axial direction. Stress concentration is found at the upper end of whisker due to shear stress caused by the difference in Poisson's ratios between SiC whisker and aluminum matrix.

Development of damage variable around the whisker with several time steps is depicted in Fig. 6. In the

19

Mamoru MlIZUNO, Koji TANAKA and Tatsuo INOUE

(a) Axssymmetric problem -case a-1

(b) Plane strain problem -case a-2Fig. 6. Damage development with several time steps.

case a-1 as axisymmetric problem in Fig. 6 (a), damage nucleates at the center of the upper end of the whisker within the aluminum matrix approximately at t=4hr, and development of damage propagated to the cor-ner of the whisker on the upper end of the whisker. In the case a-2 as plane strain problem (see Fig. (b)), on the other hand, damage nucleates at the corner of the upper end of the whisker at about t=6hr and de-velopment of damage propagated to the center of the whisker on the upper end of the whisker. All dam-age variable on the upper end of the whisker attain to 1.0 approximately at t=8hr in the case of axisym-metric problem and at t=10hr in the plane strain condition. This is interpreted as the debonding be-tween aluminum matrix and SiC whisker. Thus, the tendency of damage development is revealed to dif-fer in axisymmetric and plane strain problems, which may be caused by the different mode of stress triaxi-ality. Flat distribution of the deformation under the axisymmetric assumption reduces the stress concen-tration appearing around the upper right corner of whisker, which is more remarkable under the plane strain assumption.

Creep curves calculated by the unit cell model are presented in Fig. 7. Solid line is the results under plane strain condition, while broken line indicates the

result by axisymmetric problem. Open circles repre-sent the experimental data [13]. Macroscopic creep strain is evaluated by the displacement of the end of unit cell divided by the initial length of the cell. In the case of plane strain problem, steady state creep and tertiary creep can be simulated. Although little dif-ference in the onset of tertiary creep, strain and time for rupture between simulation and experimental data are found, steady state creep is identified well with ex-perimental data. In the case of plane strain problem, however, in spite of the steady state creep coincided with experimental data, tertiary creep starts more ear-lier, and tertiary creep is not found clearly in compari-son with experimental data. Though rupture strain is different in both cases, rupture time in axisymmetric problem is the same as that in plane strain problem.

3.3 Simulation of Creep Crack Growth in the Matrix

Figure 8 shows the equivalent stress distribution simulated as the plane strain problem for pattern B at t=13hr. Equivalent stress presents the similar distri-bution as both results discussed above. Dependence of the equivalent stress distribution on the arrangement of the whisker can not be found.

Development of damage variable around the whisker

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SIMULATION OF DEBONDING AND CREEP CRACK GROWTH IN MMC

Fig. 7. Creep curves under steady stress in axial

direction ƒÐ=70MPa at 300•Ž. -Pattern A

Fig. 8. Equivalent stress distribution at t=13hr.-Pattern B

with several time steps is depicted in Fig. 9 (a) through (c). Damage nucleates at the end corner of the whisker at about t=13hr (see, Fig. 9 (a)) and development of damage propagated to the center of the whisker on the interface (Fig. 9 (b)). Damage variable on the end of the whisker attains to 1.0 approximately at t=17hr as shown in Fig. 9 (c). Though the same tendency as the case of pattern A (Fig. 6 (b)) is indicated quali-tatively, time of the initiation of damage development and completion of the debonding on the interface is different quantitatively.

The contour of damage variable at t=18hr cal-culated by the unit cell model shown in Fig. 2 (b) is illustrated in Fig. 10. Development of damage vari-able extends to the flank of the whisker within the matrix. It is explicated that creep crack grows in the matrix toward the side of the neighbor whisker after the debonding on the interface is completed.

(a) t=13hr

(b) t=15hr

(c) t=17hr

Fig. 9. Mode of damage development -Pattern B.

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Mamoru MIZUNO, Koji TANAKA and Tatsuo INOUE

Fig. 10. Contour of damage variable at t=18hr.- Pattern B

Simulated creep curve is presented by solid line in Fig. 11 while open circles indicate the experimental data [13]. Difference in the steady state creep rate, rupture strain and rupture time between the simula-tion and experimental data is found, the steady state creep and tertiary creep can be simulated well. The prediction close to the experimental data is provided in comparison with the result by using the unit cell as shown in Fig. 2 (a-2).

4 CONCLUDING REMARKS

Creep deformation of aluminum matrix composite

reinforced by SiC whisker aligned unidirectionally at

300•Ž is simulated by the finite element technique

based on the continuum damage mechanics in order

to evaluate macroscopic creep deformation as well as

the stress distribution and damage development in the

vicinity of the whisker. For the sake of simplifica-

tion, the unit cell model for both axisymmetric and

plane strain problems is employed and the difference between both cases are discussed.

As a result of the simulation, microscopic stress and

damage development and macroscopic creep deforma-

tion are elucidated as follows:

1. Equivalent stress distribution near the whisker in-

dicates that stress in the region of the whisker is

high, and stress concentration is found on the end

of the whisker.

2. Damage occurs on the end of the whisker in both

problems, and debonding on the interface is pre-

dicted. However, nucleation site and direction of

the propagating of the damage is different in ax-

isymmetric problem and plane strain problem.

3. It is elucidated that creep crack grows in the ma-

Fig. 11. Creep curve under steady stress in axial

direction ƒÐ=70MPa at 300•Ž. -Pattern B

trix toward the flank of the neighbor whisker after

the debonding on the end of the whisker is com-

pleted.

4. The simulated results under plane strain assump-

tion (both in pattern A and B) indicate a possi-

bility in describing tertiary creep region in creep

curves, and, especially, steady state region in pat-

tern A is coincided well with experimental data,

while the result of axisymmetric problem gives

discrepancy with experiment.

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