A finite element model for asymmetric creep behavior of ceramics

MATERIALS SCIENCE & ENGINEERING

~:-~~,

j~

ELS EVI E R

Materials Science and Engineering A224 (1997) 125-130

A finite element model for asymmetric creep behavior of ceramics H.J. Lim a, J.W. Jung a, D.B. Han b, K.T. Kim a., a Department of Mechanical Engineering, Pohang Universzty of ScieJzce and Technology, Poha~zg 790-784, Sou~h Korea b Research Institute of Science and Technology, Pohang 790-784, South Korea

Received 27 February 1996: revised 11 September 1996

Abstract

Asymmetric creep responses of ceramics between tension and compression under high temperature were investigated. Multiaxial creep equations were proposed for ceramic materials under general loading conditions. The proposed constitutive equations were implemented into a finite element program (ABAQUS) to simulate asyrcm~etriccreep responses of ceramics under complex loading conditions. Finite element calculations were compared with experimental data in the literature for siliconized silicon carbide (Si-SiC) C-rings under compression and alumina and Si-SiC bending specimens under bending creep. Finite element calculations agreed well with experimental data when the principal stresses are smaller than the threshold stress for creep damage. A good agreement was also obtained for damage zone in a Si-SiC bending specimen compared with experimental data. © 1996 Elsevier Science S.A. Keywords: FEM; Model; Asymmetric creep; Alumina; Creep damage; C-rings; Bending

1. Introduction Structural ceramics have good corrosion and oxidation resistance, high strength and high thermal coefficient under high temperature. Combinations of these properties make ceramics very attractive for high temperature engineering applications [1,2]. Modern high performance ceramics are often used above half of their melting points, and they creep at this temperature range. Thus, it is very important to analyze stress re-distributions, damage and failure in ceramics during high temperature creep. Ceramics at high temperatures typically show creep rates in tension than in compression due to damages such as boundary cracks and cavities in tensile region. Asymmetric creep response of ceramics can often be observed in a bending specimen at high temperatures. For instance, monolithic SiC [3], sialon [4], alumina [5] and SiC fiber reinforced silicon nitride composites [6] show the asymmetric creep response. Thus, conventional symmetric creep responses are not appropriate to analyze creep responses of these ceramics. Ceramics in tension show a bilinear power-law creep response due to damage by cavitation at stresses be* Corresponding author. 0921-5093/97/$17.00 © 1997 Elsevier Science S.A. All rights reserved. P I I S 0 9 2 1 - 5 0 9 3 ( 9 6 ) 10557-8

yond a temperature dependent threshold level. In compression, ceramics also show a bilinear power-law creep response, although a larger applied stress is required to obtain the same rate of strain as in tension. Chnang and Wiederhorn [7] used a single power-law creep equation to simplify the mathematical analysis of the flexure problem by assuming that higher stress regimes of ceramic structures were not encountered in practice. Subsequently, Chuang and Wiederhorn [8] proposed an asymmetric creep response in tension and compression and compared its predictions with experimental data for damage zone sizes and shapes in Si-SiC C-rings under compression by using a simple curved beam theory. Recently, Chuang et al. [9] proposed multiaxial constitutive equations in a power-law creep form based on asymmetric creep responses under uniaxial creep. They also investigated creep responses of Si-SiC C-ring specimens under compression by using a finite element method. This paper reports on asymmetric creep responses of ceramics under high temperature. Multiaxial creep equations were proposed for ceramics under general loading conditions. These constitutive equations were implemented into a finite element program (ABAQUS) to simulate asymmetric creep responses of ceramics under complex loading conditions. Finite element cal-

126

H.J. Lira et aL /biaterials Science and Engineering A224 (1996) 125-I30

cuIations were compared with analytical solutions and experimental data for alumina and Si-SiC specimens under bending creep and Si-SiC C-rings under compression creep.

shear stresses does not exist. Hence, three dimensional creep equations in the principal directions may be written

(2)

~2 = klJ40-1 "JF ktJ50- 2 "]- Vii60- 3

2. Theory

el = K*70-1+ KJs0-2+ via90-3

To investigate creep responses of ceramic structures under general loading conditions, a constitutive equation for creep response under a general state of stress is necessary. Ceramics typically show much higher creep strain rate in tension than in compression. The constitutive equation for general asymmetric creep responses of ceramics, however, has not been thoroughly investigated so far. Assuming creep response of ceramics is governed by the sign of the largest principal stress in magnitude, Chuang et al. [9] proposed an asymmetric creep equation by generalizing a symmetric creep equation with different creep constants in tension and compression. Thus, the asymmetric creep equations by Chuang et al. [9] can be written 3 ~,=~A,6"'-l(0-,--G,,),

i = 1,2,3

A, = A t,

n, = n t

for 0-m~×> 0

A, = Ao,

n, = no

for O'max < 0

(1)

where k, is the principal strain rate, 0-, is the principal stress, O-.... is the largest principal stress in magnitude. 0-~= (0-~ + o-2 + 0-3)/3 is the mean of the principal stresses. A t and/7 t are the tensile creep coefficients and Ao and no are compressive creep coefficients. The approach by Chuang et al. [9] does not seem to have any problems when the difference in magnitudes of the principal stresses is distinct. However, it does not seem to be plausible when the difference in magnitudes of tensile and compressive principal stresses is very small. Fig. 1 shows two different states of stress interpreted by Chuang et al. [9] when (a) tensile principal stress is slightly larger than compressive principal stress and (b) compressive principal stress is slightly larger than tensile principal stress. Depending on the sign of the largest principal stress in magnitude, the model by Chuang et al. [9] may provide completely different states of stress as in Fig. 1. Thus, an incorrect solution may be obtained in the critical regime where the principal stresses have approximately similar magnitudes with different signs. In a numerical analysis for long term creep problems, total creep strain is obtained by accumulating the creep strain at each time interval. Hence, the numerical errors in the critical regime are also accumulated and these errors can not be ignored easily. To resolve this problem, we consider a general creep equation in the principal directions where the effect of

where Vla,.is a material constant. Assuming the incompressibility of a material and the material is subjected to uniaxial loading in the principal direction 1, we may write ~, = - (k2 + e3) = - 292 from the symmetry in the principal directions 2 and 3. Thus, we may write W1 = - 2~4 = - - 2~It7 from Eq. (2). Similarly, we consider the creep strains in the principal directions 2 and 3. Hence, Eq. (2) can be written 1 1 il : ~JIO-I --~ "t~20-2 --~ l"t/30"3 1 1 ~, = - ~ '-tq O-i + ~ 0 - ~ - ~ ~ 0 - 3

83 :

(3)

1 1 -- ~ I~'/10-I -- "~ kIJ2Z2 "~- kIJ30-3

Here ~, (i = 1, 2, 3) can be obtained from • in the symmetric creep response [10] depending on the sign of 0-, Thus, Eq. (3) can be written 8i ~_.

1 A2 ~ . . . . AIGnI-t0-1--~ -

10-2

_ _

1 ~A3 #'b-I0-3

(a) I~, I "-I';~ I IG[~ ~AeO~l)nmllGIAt IGI+AI~I

time = t

time = t +At

(b) I~ol>_l~,l

]o'l~

time = t

3 AC((GOnc-~'lO'JA't

time = t +At

Fig. I. Plane stress states interpreted by Chuang et al.'s model [9].

H.J, L#n et a l . / Materials Science and Engineering A224 (i996) I25-130

I27

3. Finite element analysis

I~2, i

(J~a= 0

O':':'=- ~

Finite element calculations by using the proposed multiaxial creep equations were compared with experimental data in the literature for alumina and Si-SiC specimens under bending creep and Si-SiC C-rings under compression creep. In this paper, we consider only elastic and creep deformations of ceramics. To include bilinear creep responses of ceramics, we used creep constants A, and n, from Table 1 for the corresponding stress range. The multiaxial creep Eqs. (4) and (5) were implemented into the user subroutine U M A T of ABAQUS [11] to compare with experimental data.

sin(280)

=-~280 =(G2-(~)80

3.1. Alumina

Fig. 2. Plane stress state by infinitesimal rotation cS0 relative to the principal axis direction. &=

1 1 - ~ AI~'q-10-1.-I- A 2 # " a - I 0 " 2 - ~ A3#n3-:0"3

(4)

1 1 ~3 = -- ~ AI# 'q - :0": -- ~ A2 #n2- 10"2-+-A3 #'3 - I0"3 A, = At,

G = nt

for 0-i > 0

As = At,

me= nt

for 0-, > 0

Numerical rotation of the coordinate axes into the principal direction may accompany small shear stresses as a numerical error. When these numerical errors are accumulated, it is hard to satisfy the equilibrium equations. Sometimes, it is also difficult to obtain solutions for complicated geometries and complex loading conditions if only Eq. (4) is considered by ignoring these shear stress terms. To include the effect of shear stress by numerical errors, we obtained the following small shear stress terms. The details for derivation of Eq. (5) can be found in Appendix A. Thus, ~1'2' =

{AIB12 # n : - 1 -t- A 2 B 2 1 ~ n 2 - 1}o"i, 2,

~2'3' =

{A2B23 t~n2-

1 -t-

3.2. Sitioonized SiC

A3B32 ~ 3 - i} 0-2'3'

(5)

@'v = {A3B3:~3 - : + AIB:36""~ - 1}0.3':' where B12

0-i

B23 ~ _ _ 0 " 2

0"1 - - 0"2 7

B21 =

0"2 0-2-

, 0"1

B3: =

0"2 - - 0"3 ~

832 =

0"3 0"3 - - 0"2

,

Fig. 3(a) shows a schematic drawing for a ceramic material under a four point bending creep test. Fig, 3(b) shows finite element meshes for an alumina bending specimen. By using the plane stress condition, the righthalf of the specimen was analyzed due to symmetry about y-axis. Fifty elements in the x-direction and 14 in the y-direction were used. Two dimensional, plane stress, solid 8-node biquadratic element (CPS2DS) was used in the finite element analysis. The maximum bending stress was in the range of 30-60 MPa during four point creep tests to compare with experimental data in the literature. The symmetric condition about x-plane (ux = @.,.-= (/: = 0) was applied on the nodal points at x = 0 and the condition z5, = 0 was applied at the point x = 20, y = 0. The creep constants obtained by Ferber et al. [5] were used and shown in Table 1. We did not consider a bilinear creep response for alumina bending specimens under bending creep since the bilinear creep data for alumina were not available in the literature.

0"3 0"3 - - 0"1

Bt s =

Fig. 3(c) shows finite dement meshes for a Si-SiC bending specimen. Fig. 4(a) shows a schematic drawing of C-ring compression test. We considered a C-ring Table 1 Creep parameters used in the finite element analysis

0":

n

.4 (s-:)

0"1 - - 0"3

where the coordinate axes x, (i = 1, 2, 3) are the principal directions in theory and x,, ( i = 1, 2, 3) are the principal directions obtained in numerical analysis. When the state of stress is rotated by d0 infinitesimally in the principal direction as in Fig. 2, the rotation angle cs0 can be represented by the ratio of shear stress and the difference in two principal stresses. Thus, shear stress can be expressed by imposing the weight factor i50 on each principal stress.

Alumina (AD94) [5] Tension Compression Si-SiC (KX01) [9] Tension (MPa) 0 100 Compression (MPa) --200<(r<0 or< - 2 0 0

5.6 1.7

2.972 x 10 - : 7 5.555 x 10- :~

4 I0

5 x 1 0 -:7 5 × I0 -29

4 10

5 x 10-:s 5 × 10 -30

H.J. Lira et al./Materials Science and Engineering A224 (1996) 125-130

128

L

i ]-2

!

L,

~'

r\

/ Specimen F/2

To c o m p a r e with experimental data for a cavitation zone obtained by C h u a n g and Wiederhorn [7], we analyzed a S i - S i C bending specimen under bending creep. F o r t y elements in the x-direction and 14 elements in the ),-direction were used as shown in Fig. 3(c). The load was applied at x = 5, y = 3 so that the initial m a x i m u m bending stress was 250 MPa. The symmetric condition a b o u t the x-plane (u~ = @ = ¢,~ = 0) was applied on the nodal points at x = 0 and the condition uy = 0 was applied at the point x = 20, y = 0. The creep constants used here are shown in Table 1. F o r both tensile and compressive creep deformations, bilinear creep responses were considered.

F/2

Fig. 3. (a) A schematic drawing of four point bending test. (b) Finite element meshes for an alumina bending specimen. (c) Finite element meshes for a Si-SiC bending specimen.

specimen that has the same size and loading condition as Ct specimen in C h u a n g et al. [9]. The C-ring specimen is 6.44 m m in width, 15.575 m m in inner radius and 19.015 m m in outer radius. The C-ring specimen was subjected to a compressive load of 86.24 N. Fig. 4(b) shows finite element meshes for the S i - S i C C-ring compression specimen. Only the upper half o f the specimen is considered due to symmetry. The portion right o f the loading point is not shown since it is not subjected to the load. Due to symmetry in thickness, only half o f the thickness was considered for 3-D analysis. Six elements in radial direction, 40 elements in tangential direction and two elements in thickness direction were used. Three dimensional solid 20-node quadratic, reduced integration elements (C3D20R) were used for a C-ring specimen under compression creep. The load P / 2 was applied on the nodal point x = 0, y = R~ in Fig. 4(b). F r o m the symmetry conditions described earlier, a symmetric condition about the z-plane (u_ = p.,. = ¢,~.= 0) was applied on the nodal points at z = 0 and a symmetric condition about the );-plane 0%. = P.~ = ¢'.- = 0) was applied on the nodal points at y = 0 . The condition u~ = 0 was also applied on the nodal point at x = - R 2 , y = 0, z = 0 to prevent rigid rotation. The creep constants by C h u a n g et al. [9] were used and shown in Table 1. The bilinear creep response was included in finite element calculations for a S i - S i C C-ring under compression creep.

Before Creep

After Creep

z

X

Fig. 4. (a) A schematic drawing of C-ring compression test. (b) Finite element meshes for a Si-SiC C-ring compression specimen.

H.J. Lira et a l . / Materials Scielzce and Engineering A224 (i996) 125-I30

I29

4

1.0

/

; x~'~,..,

Alumina

" ~ - \

U= co

1000°C, 30MPa

"u 0.5 co

--------

E

0 5 100 200

.....

o Z

.....

0

h h h h

-20

/

3 /

"~:,..

"-r

FEM-50MPa ..... Experiment-50 MPa --[3--- FEM-60 MPa ..... Experiment-60MPa

/

.~ 2

"~.'-,, \\',, ~.~\

0

2o

40

0

20

Stress, MPa

4. Results and discussion

4.1. Alumina

Fig. 5 shows the variation of stress distribution with time in the cross-section of an alumina bending specimen under initial flexural stress of 30 MPa at 1000°C. The initial elastic stress is gradually redistributed nonlinearly as time elapses and the neutral axis moves toward the compressive region. The stress distribution attains the steady state after about 200 h. Fig. 6 shows a comparison between finite element calculation and the analytical result by Chuang and Wiederhorn [7] for stress distribution in the cross-section of an alumina bending specimen at 1000°C after 200 h under initial flexural stress of 60 MPa. The

b••

U=

-~ 0 . 5 0 ._N "@

o

1000°C, 60 MPa

Analytical Result

0.00

,

-150

i

,

I

i

-100

,

I

,

I

-50

,

~ ,

~ I

~

,

60

80

100

120

I

0 50 Stress, MPa

~ ,

~ ~ I

100

Fig. 7. Comparisons between finite element calculations and experimental data [5] for the variation of creep strain with time for alumina bending specimens at 1000°C under initial fiexural stress of 50 and 60 MPa.

agreement between the analytical result and the finite element calculation is very good. Fig. 7 shows comparison between finite element calculations and experimental data [5] for the variation of creep strain with time for alumina bending specimens at 1000°C under initial flexural stresses of 50 and 60 MPa. The agreement in flexural stress of 50 MPa is very good. In the case of 60 MPa, however, the experimental data shows about 3 - 4 times higher strain rate than the finite element calculation. This can be attributed to damage in the tensile region of the specimen under flexural stress of 60 MPa [5]. Finite element calculations from the present model are applicable only when the principal stresses are smaller than the threshold stress for creep damage, since the creep damage model in the tensile region is not incorporated in the present analysis.

4.2. Siliconized SiC

Alumina

FEM

o Z

40

Time, h

Fig. 5. Variation of stress distribution with time in the cross-section of an alumina bending specimen under initial flexural stress of 30 MPa at 1000°C.

1.00

Alumina 1000°C

/

,

,

t

~

150

Fig. 6. Comparison between finite element calculation and the analytical result [7] for stress distribution in the cross-section of an aIumina bending specimen at 1000°C under initial flexural stress of 60 MPa.

Fig. 8 shows finite element calcuIations for the variation of principal stress distribution with time at the cross-section y = 0, z = 0 of a Si-SiC C-ring compression specimen at 1300°C. The steady state was attained after 100 h under compression creep. The tensile region beyond 100 MPa was not observed throughout the creep time. The deflection rate at the load point of the C-ring specimen was predicted as 1.06 gm h - 1 which is very close to the experimental data [8] of the deflection rate 1.08 ~m h - L Fig. 9 shows a comparison among the finite element calculation, the analytical result [7] and experimental data [7] for a cavitation zone of a Si-SiC bending specimen during four point bending under initial flexural stress of 250 MPa. It is understood that only the

130

H.J. Lira eta/./"Materials Science and Engineering ,4224 (I996) I25-130 Appendix

1.20

/I,'" / i

1.16

,•,I

Si-SiC

Shear stress 5vz generated during mtmerical rotation in the principal direction can be expressed

,,';/

1300°C

,'/A

0"i

, ' - ,/

1.12

A

,' /

0"I'2' =

G1'2' 0"1--0"2

'-

1.08

0

<,-"

55

/ "/""

1.04

,5

--

~'

-- " -

h

=

0"I'2'

0"1

--

0"2

h h

--

0"2 --0"1'2' 0"[--0"2 "Jr-

(A1)

0"1'2' 0

"

0"1

-

-

0-2

2

--

0"1

F r o m Fig. 2, 50 can be written

100 h

/

1 .000

~

I

-200

,

-1 O0

I

,

0

I

30

,

100

200

Stress, M P a

=

0"1'2' 0-2 - - 0.I

Thus, Eq. (A1) can be written

Fig. 8. Finite element calculations for the ,~ariations of principal stress distribution with time at the cross-section y = 0, z= 0 of a Si-SiC C-ring compression specimen at 1300°C. right half of the bending specimen is shown due to symmetry in the ?'-axis. The finite element calculation agreed well with experimental data for the cavitation zone o f a S i - S i C bending specimen. The present model predicts the experimental data for cavitation zone better than the analytical result by C h u a n g and Wiederhorn [7]. Here, the cavitation zone represents a region with tensile stress beyond 100 MPa, where cavities in S i - S i C specimens grow rapidly at about 100 M P a [7].

0.r2' = 0.1( - 50) + 0.z50

By using Eq. (A2), we m a y assume the shear strain rate Sl,2, = st( - 50) + @50

(A3)

since the strain rate ~ is proportional to 0" under steady state creep with constant temperature. Substituting a power-law creep equation in Eq. (A3) and arranging, we can write @2' = AI ~'q - l ° h ( -- c50) + A2~"-'-10"2g0 -

= A 10"''~ 5. C o n c l u s i o n

i

F/2 Y 3

Unit: mm Experfment ........ AnalyticalResult .... FEN

2 _---_---i--~-5-21........ 1

0

0

2

4,

6

8

10

12 14

1'6 1'8

X

Fig. 9. Comparisons among the finite element calculation, the analytical result [7] and experimental data [7] for cavitation zone of a Si-SiC bending speclmen, during four point bending under initial flexural stress of 250 MPa.

0"2

0"1

- 0"1 - - 0"2

o..i,2, + A z # , , z - i

--= (A1B12gni -1 q_ AzB2t #n2- 1)0"1,2 ' This paper reports on asymmetric creep responses of ceramics under high temperature. The proposed multiaxial creep equations were implemented into a finite element p r o g r a m to simulate creep responses of alumina and S i - S i C specimens under bending creep and a S i - S i C C-ring under compression creep. The finite element calculations from the present model agreed well with analytical results and experimental data. The present model also showed a g o o d agreement with the experimental data for the cavitation zone of a S i - S i C bending specimen under bending creep.

(A2)

-

-

0" i'2'

0"2 - - 0"1

(A4)

References

[1] E. Dorre and H. Hubner, Ahtmina, Processing, Properties and Applications, Springer-Verlag, New York, I984. [2] R.N. Katz, Mater. Sci. Eng., 71 (1985) 227. [3] R.F. Krause and T.-J. Chuang, in P. Vincenzine (ed.), Ceramic Today-Tomorrow's Ceramics, Elsevier Science, Amsterdam, 199I, p. 1865. [4] C.F. Chen and T.-J. Chuang, d. Am. Ceram. Soc., 73 (1990) 2366. [5] M.K. Ferber, M.G. Jenkins and V.J. Tennery, Ceram. Eng. Sci. Proc., 11 (1990) 1028. [6] B.J. Hockey, S.M. Wiederhorn, W. Liu, J.G. Baldon and S.-T. Buljan, J. Mater. Sci., 26 (I991) 393I. [7] T.-J. Chuang and S.M. Wiederhorn, J. Am. Ceram. Soc., 71 (i988) 595. [8] T.-J. Chuang and S.M. Wiederhorn, d. Am. Ceram. Soc., 74 (199I) 2531. [9] T.-J. Chuang, Z.-D. Wang and D. Wu, J. Eng. Mater. Tech., 114 (1992) 31I. [10] J.T. Boyle and J. Spence, Stress Analysis for Creep, Butterworths, London, 1983. [1I] ABAQUS User's I and II Manual, Habbitt, Karlsson and Sorensen, 1992.