Reference

NBS

PUBLICATIONS

NBSIR 85-2997

NATL INST. OF STAND & TECH

AlllDb ELDLE4

Estimation of Power-Law Creep

Parameters from Bend Test Data

T?.e-jer Chuang

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

Center for Materials Science

inorganic Materials Division

Gaithersburg, MD 20893

February 1985

NBS-COE -Contract No. DE-A1C5-3C0R23679

NBS-AFOSR-ISSA-84-OOOI

3

Prepared for Department of Energy

AR&TD Fossil Energy Materials Program

Washington, DC

a no _ Air Force

:s of Scientific Research

100 hington, DC

,1156

35-2997

1985

NBSIR 85-2997

ESTIMATION OF POWER-LAW CREEP

PARAMETERS FROM BEND TEST DATA

Tze-jer Chuang

NATIONAL BUBF.ATT OF STANDARD?

LIBRARY

a) 3s

r\ - (J-bfOb

m?r

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

Center for Materials Science

Inorganic Materials Division

Gaithersburg, MD 20899

February 1 985

NBS-DOE-Contract No. DE-A105-800R20679

NBS-AFOSR-ISSA-84-OOOI

Prepared for Department of Energy

AR&TD Fossil Energy Materials Program

Washington, DC

and

U.S. Air Force

Office of Scientific Research

Washington, DC

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

7444

ESTIMATION OF POWER-LAW CREEP PARAMETERS FROM BEND TEST DATA

by

Tze-jer Chuang

Center for Materials Science

National Bureau of Standards

Gaithersburg, MD 20899

ABSTRACT

Power-law creep parameters of brittle ceramic materials are commonly

deduced from load-point displacement data generated by four-point bend

experiments, under the assumption that tensile and compressive behaviors

obey the same constitutive law. However, because of microcracking and

cavitation, it is now well recognized that this premise may not always be

valid. The present paper presents an analysis which takes the differences

into account. Governing equations are first derived for the location of

the neutral axis of a beam under bending which does not in general pass

through the centroid of the cross section and for the creep response in

terms of both curvature rate and load-point displacement rate as functions

of the applied moment and power-law creep parameters. Numerical solutions

are obtained for any given set of material constants over a wide range of

applied moments. It is shown from the plots of creep response versus

applied moment on a logarithmic scale that even linear curves over a

narrow range of applied moment do not necessarily imply identical stress

exponents and that nonlinear curves concave upward signify a profound

difference in stress exponent between tension and compression. An example

of applying the present analysis to a set of load-point displacement data

on glass-alumina beam specimens crept at 1100°C is given. The results

show that the conventional method over/underestimates the creep rates in

compression/tension by two orders of magnitude, indicating a need of using

the more accurate analysis presented here. Several recommendations are

offered to improve the estimation of power-law creep parameters from bend

test data.
1 . Introduction

The present paper is concerned with steady-state creep deformation

behavior of structural ceramics which are candidate materials for hightemperature stress-bearing applications, and aims at developing a

mathematical scheme from which individual tensile and compressive powerlaw creep parameters can be estimated from conventional four-point bend

test measurements. Instead of directly relying on uniaxial testing,

flexural test methods are frequently adopted as an alternative to generate

data from which information on materials’ creep behavior may be extracted

(see for example [1-2]). This practice can be attributed to the fact that

a bending experiment is more stable and easier to perform without

involving problems of fixturing and alignment usually associated with

tension testing of brittle materials at high temperatures. A challenging

issue that must be resolved for a given set of data produced from crept

bend bars is: how can one, if possible, accurately estimate the uniaxial

creep behavior (both in tension and in compression) when the applied

stress is given? Since bending data contain both tensile and compressive

components, it is perhaps natural to expect that the resulting predictions

would be strongly influenced by the form of constitutive equations assumed

a prior i

Extensive literature review on the thermal creep of ceramics [3]

indicates that generally speaking the steady-state behavior relating creep

strain rate e s to applied stress o can be described by Norton's law in a

form £3 = A a n

, where A is pre-exponent constant depending only on test

temperature and material's properties; n is stress exponent which may or

2
may not depend on stress. Hollenberg, et al. [4] presented the first

analysis in which stresses and strains in the crept beam specimens can be

calculated from the bend test data provided tensile power-law creep

behaves identically to its compressive counterpart. This simplifies the

analytical work substantially and allows the solutions to be presented in

closed forms as the neutral axis location in this case always coincides

with the center line of the beam height regardless of the magnitude of the

applied loads.

However, it is now well recognized that tensile response might be

distinct from its compressive counterpart inside a beam for a given

material even tested under identical environments (see for example [5-6]).

Consequently, for application to plain concrete, Krajcinovic [7] developed

a damage theory for beams under pure bending in order to justify that

tensile stresses assume a parabolic distribution in terms of strain while

the compression behavior remains linearly elastic obeying Hook's law. More

recently, Rosenfield et al. [8] extended this time-independent analysis to

two more constitutive equations in tension, namely linear elastic with

lower effective Young's modulus and elastic-perf ectly plastic while once

again leaving the compressive portion unchanged. On the other hand,

within the arena of power-law creep, Finnie [9] was the first to recognize

the possible situation of pronounced differences between tensile and

compressive creep. His analysis permits A

t

* A

c and is capable of

predicting creep rates from data generated by creep bending of a

trapezoidal cross section beam provided n

t and n

Q are of unity; here

subscripts t and c refer to the cases of tension and compression

3
respectively. Talty and Dirks [10] extended the analysis of the same

trapezoidal beam to a more general case of N other than unity (i.e. n

t

=

n

c

= N).

The present paper extends the previous work to a completely general

case of unequal tension vs compression power-law creep behaviors wherein

not only the pre-exponent factors are permitted to be distinctive (A

t *

A

c

) but also the stress exponent constants may be unequal (n t * n c

). For

the sake of simplicity, only rectangular beams of uniform cross-sections

are considered. Further, the existence of a steady-state is assumed so

that time can be eliminated as a variable in the study. This assumption

requires that the transient stage be short-lived and thus can be ignored,

although it should be acknowledged that this phenomenon may sometimes

become important [11-12] when a well-defined steady state does not develop

[131.

In the next section, governing equations are first derived which

relate separately the position of neutral axis and the applied moment to

the curvature rate, K and (unknown, a_ priori) power-law creep parameters.

Computer programs were developed to solve these coupled non-linear

algebraic equations numerically. Solutions are obtained in graphic form

for an arbitrary set of power law constants. Graphic solutions are also

given in terms of the load-point displacement rate, Ap - a more measurable

quantity than K for collecting data on specimen’s response. Cases of n

t

=

n

c

= N as examined by Talty and Dirks [10] and A

t

= A c , = A, n t = n c= N

as analyzed by Hollenberg, et al. [4] are then presented as special cases

of the present investigation. From the point of view of an experimenter,
in order to apply the current theory, a parametric study method must be

used from which curves can be produced from the computer programs to match

the discrete data points which are dictated by creep bend tests. Once

accurate matching is achieved, the predicted power-law parameters become

available at once from the well-fitted curve. Fcr the sake of demonstration of how to use the theory, an example is given to estimate the four

power-law constants from a set of six bend test data on debased alumina

beams crept at 1100°C for a duration of more than 100 h. At conclusion, we

are able to make some recommendations from the present analysis for those

who prefer using the four-point bend test method to characterize power-law

creep behavior in structural ceramics. Those suggestions should lead to

more accurate characterization of creep properties.

2. Analysis

2.1 Derivation of the governing equations

In this section, we derive the control equations that relate material

response to external variables and material constants, for a rectangular

beam under four-point creep bending. (See Fig. 1)

A#s already discussed in the preceding section, under the action of

some constant external loads the material is assumed to respond in the

steady state according to a power law of the form:

n

e = A (a/a ) , a in compression (la) oC C O

and

5
, o in tension (1b) n

t

£ = A ( o/o ) St t o where e s j. s steady-state creep strain rate; A and n are materials

constant, o is the normal stress and o Q is a reference stress. The

subscript c and t refers to the case in compression and in tension

respectively. A schematic sketch of Eq. 1 applied to a beam is given in

Fig. 2.

The derivation that follows adopts the conventional simple beam theory

which entails a fundamental assumption, known as Bernoulli's hypothesis,

that planar sections remain plane during bending when creep is taking

place so that no warping will occur (because of the. need for geometric

compatibility this condition seems to hold in practice [14]). This

implies that the strain rate t of a fiber element is linearly dependent on

Y, the distance away from the neutral axis where t = 0 and the curvature

rate, K serves as a proportionality constant. Thus

e = K Y

and the stress distribution over the cross section of the beam is highly

nonlinear and has the following form, according to Eq. (1):

o(Y) o o

, Y K.l/n

1 Aj (2)

6
regardless of the sign of stress. Equilibrium requirements then dictate

that the total force acting on the compression side of the cross-sect ion

be counter-balanced by its tensile counterpart. This means that F

c = Ft

r c r z or B o dY = B

J 0

o dY where B is the beam width and H (=HC + Ht

) is

the beam height (see Fig. 1). By integrating a using Eqs. (2) and (1),

and after some mathematical manipulations, this force balancing equation

finally reduces to

n. - n

t c n

t n

t

( v1) n (n + 1

)

s. c t r n

c (nt V1 ) n

c (nt

+1)

1 k n

t (n

c

+ D h

c

* h

o - 1 (3)

^ • where R = A

t /Ac , k = KH/AC and hc

= Hc /H are dimensionless parameters.

Here we choose A

c as normalizing factor for R and K presumably because Ac

is much easier to measure than A

t

. Eq. (3) is a nonlinear algebraic

equation of the form Cxn + x = 1 for the unknown h

Q quantifying the

physical location of the neutral axis. Since both C and n are positive

definite, as parameters appeared in Eq. (3) are all positive quantities,

it can be proved from the form of this nonlinear algebraic equation that a

unique solution for x always exists in the range 0 < x < 1 beyond which no

physical meaning may be assigned to x.

In addition, the requirement that the total summation of moments

produced by local tractions be equated to the external moment, M, forms

the second governing equation, viz.

7
1 result

H H

M = /

C

+ /oYBdY o o

Substitution of Eq. (2) for o = o(Y) and recognition of + hQ

=

in an equation relating the applied moment to the material’s response k:

2n +1

U 2n +1 c

m = 1/n. u

R 2n +1

V-/

(1-h

0

) 2n +1 c

(4)

where m = M/(BH^o Q

) is the normalized applied moment. The assignment of

the reference stress o Q is somewhat arbitrary. Since the unit of the

applied stress is generally expressed in MPa, it is convenient to set

o Q

= 1 MPa for simplicity.

Equations (3) and (4) constitute a system of algebraic equations for

the two unknowns h

Q and k, while the remaining parameters such as the

applied moment m and the materials parameters A's and n’s are being

treated as given. After examining the structure of these two equations,

we arrive at the unfortunate conclusion that analytical solutions in close

form cannot be obtained because they are highly nonlinear and coupled in

8
k. Accordingly, a numerical approach is the practical way to tackle this

problem. A computer program was developed which contains the following

primary tasks: first the development of a subroutine to solve h

Q from Eq.

(3) by a Newton-Raphson ' s iteration scheme. Input to this subroutine are

a R, n

t

, n

c and k, the latter being treated as an independent variable. Note

that initially an overshoot outside the range (0,1) for h

Q may arise

during numerical iterations, stability can then be restored if h

c be reset

to its boundary value. Once h

c is successfully solved from this

subroutine, it can then be used as input, together with the independent

a variable k, to Eq. (4) for the computation of m. At the end, a total of

a three one-dimensional arrays were generated, namely m, k and h

c

. This

concludes the computation phase of programming, and the plotting phase

a follows. The solutions were then displayed in graphic form for k versus

m as well as h

c versus m with any given A’s and n’s.

2.2 General Solutions

2.2.1 Location of neutral axis, h

c

The solutions for h

c are plotted as a function of m in Fig. 3 for

several values of n^. at n

c

= 4.0, R = 1000. Generally speaking for the

practical range of m > 0.5, as the applied stress increases, the

compressive zone keeps shrinking with increasing n

t

. Intuitively this

must be true since an increase in n

t implies that the material's ability

to support tensile stresses is reduced for a fixed creep rate. When n r -

n

c» hc

= °* 2 » independent of the applied stress. In fact, h c can be

9
expressed analytically in case of equal n. This special case will be

discussed later. Another interesting observation that can be extracted

from Fig. 3 is that when m is at a value of around one half, h

c is about

0.2 regardless of the value of n^. As a matter of fact, when examined in

a more broader base, we found that when m lies between 0.25 and 0.4 the

solutions are quite insensitive to both n

fc and n c suggesting that this

range of m should be avoided in a testing program that aims to

characterize the materials creep parameters. Fig. 4 presents another

solution for h

c for five values of R, fixing values of n^ and n c at 1 and

5 respectively. As expected, the higher the value of R, the lower the

size of compression zone as high R implies that the material's creep

resistance in tension is reduced at a given strain rate (or applied

moment). When m exceeds 10, however, a major portion of the beam is in

compression for R values up to 1000. Conversely, setting n

t = 5 and nc

=

1 demonstrated a reverse trend, as shown in Fig. 5 for the same five

values of R. Again, cases of higher R result in smaller compression zone

as expected.

2.2.2 Curvature rate of a beam element, k

The materials response in the form of curvature rate k under creep

bending for a given applied moment m is plotted in Fig. 6 on a log-log

scale for five values of n

fc and fixing n

c

= 4 and R = 1000. Two important

observations can be made here: (1) the curves appear linear with, of

course, always positive slopes when n

t

's are in the neighborhood of n

c but

when the gap between the values of n^ and nc widens, the curves become

nonlinear and concave upward; (2) a "blind point" in the vicinity of

m = 1/2 is also observed, similar to the solutions of h Fig. 7

10
demonstrates the solutions of k upon variation of R under fixed values of

n

t

= 1 and n c

= 5. Again, owing to big differences between the values of

n

t and n

c , concave upward curves are obtained. Another interesting

feature noteworthy to state is the solutions converge into one single

straight curve as the applied moment exceeds 15. This means that applying

a load in excess of m = 15 , would generate a straight line in k vs m plot,

regardless of the values of R and therefore is not particularly useful.

Lower loads (much less than m = 15) are thus recommended. If, on the

other hand, n

t

= 5 >> nc= 1 then the solutions are well-behaved in the

practical range of the applied load (.01-. £ m £ 100) as indicated in Fig.

8 , although most of the solutions appear to be linear.

0 2.2.3 Outer fibre stresses, o G and o t

As can be seen from the preceding section, during steady state creep

the neutral axis is displaced from the centroid and the stress

distributions are highly non-linear. As a result, it is to be expected

that the outer fibre stresses, both at the tensile side as well as at the

compression side must in general differ from the initial elastic stress

levels. In terms of the applied bending moment M, the outer fibre elastic

stresses both in tension and in compression have a value o e

= 6 M/(BH^)

from classical simple beam theory, and in terms of dimensionless

quantities, there results = 6 m, where o e = o e /o 0 is the

normalized outer fibre elastic stress.
The steady-state compresssive creep stress at the outer surface of

the beam is

o co (k h ) o c

1/n

c

as evidenced from Eq. (2). Normalizing against o e we have

o /o

c e

1_

6m

x 1 /n

(k h ) C. c (5)

Similarly the creep stress at the tensile edge, also normalized by o Q is

o e

i_

6m

[k (1-h )/R]

c

1/n,

( 6 )

Examination of these two equations indicates that the outer fibre creep

stresses, unlike their elastic counterpart, are not only a function of

applied moment, but also dependent on the neutral axis location and the

intrinsic power-law creep parameters. The maximum compressive stresses

for several n

t are plotted in Fig. 9, according to Eq. (5) for typical

values of n

c

= 4 and R = 1000. When m < 0.35 corresponding to an elastic

stress of about 2 MPa, higher values of n

t yield lower compressive

stresses; the reverse situation occurs if the applied stress exceeds 2

MPa. On the other hand, the maximum tensile stresses at the outer fibre,

as plotted in Fig. 10 from Eq. (6), show a unique feature, namely for a

fixed applied load, in the practical range (1 - 600 MPa) low values of n^.

yield higher tensile creep stresses. Note that in both cases, as shown in

12
Figs. 9 and 10, in the case of n the outer fibre stresses are Q

= n t

,

independent of the applied moment and as will be shown later in Sec. 2.3

they can be expressed in closed forms.

" •

2.2.4 Load-point displacement rate, Ap

As demonstrated in the preceding sections, R is a proper parameter to

measure the response of creep for a bend bar under the constant applied

moments. Unfortunately, the curvature time-rate of a beam at a fixed

location is difficult to measure in the laboratory and it is a general

practice to measure instead the load-point displacements continuously as a

function of time. Hence, it is desirable to present solutions in terms of

For a given material with a well defined k vs m relationship the

load-point displacement rate -A p

incurred from a four-point bend beam

A can be solved numerically by integration of k along the beam length x with

linear moment distribution in the outer span and a constant maximum moment

in the inner span. Here, it is assumed that shear effects on the beam

deflection y are negligible and the slopes of the deformed beam shape are

small (dy/dx << 1). The differential equation that needs to be solved is

then d

2y/dx2 = k(x) = f(m) with R = f(m) given in Sec. 2.2.2. Setting the

origin of the coordinate system at the mid span of the deformed beam, the

proper boundary conditions are y(0) = 0 and y’(0) = 0 due to symmetry.

After the deformed shapes y = y(x) are solved, the load-point displacement

rate is given by Ap

= y(L/2) - T(5,/2) where L and 2. are the lengths of

major and minor spans respectively. Typical solutions are plotted in Fig.

13
11 for nondimensional load-point displacement rate, defined as dp/HAc as a

function of the applied moment m. for several values of n

t with nc

= 4,

R = 1000, L = 42, and H/2, = 1/2. Similarities between Fig. 6 and 11 are

observed. Whenever n

c

= n t

= N, the solution can be described by a linear

• curve owing to the fact that the relationship between K

D and Apis linear

and has a form

K = 4(N+2)

p ( L-2, ) [L+ ( N+1 ) 2, ] p

2.3 The special cases

Having obtained the general solutions in the previous section, it is

easy to arrive at the results for the special cases as considered by

Hollenberg et al. [4] and Talty and Dirks [10].

2.3.1 The case n

t

= nc

= N ; R * 1

This is the case considered by Talty and Dirks [10] and Cohrt et al.

[12] . For this case, Eq. 3 describing the location of neutral axis takes

the following simple form

1

1 + R

1 /(N+1

)

(7)

independent of the applied moment m. This is in agreement with the work

of Cohrt et al. [12] (See Eq. 3 of Ref. 12 in which our R is equivalent to

their Sn

). In addition, it can be shown from Eq. 4 that the moment vs

curvature rate is linear when plotted on logarithmic scales, and therefore

can be described by
k * C m‘

J ( 3 )

where C is the intercept at m = 1 and is only a function of h

c , r and N:

C

( 2N + 1 )/N

2N+1 2N+1

N + (1-h ) N

c ,1/N

N

(9)

Fig. 12 presents the solutions of h

Q for several N, based on Eq. (7) as a

function of C defined in Eq. (9). Fig. 13 demonstrates the solutions for

a R. If nQ

= nt = N, the bend test data in terms of k vs m ought to be

fitted by a linear curve on logarithmic axis. A

c is automatically

•

* A | • obtainable from k and X (see Fig. 1 and definition of k). C as well as N

can then be extracted from this curve. Figs. 12 and 13 can then be used

to obtain h

Q and R (hence A and the complete creep parameters are

determined.

2.3.2 The case of uniform property (n

t = = N ; R = 1

)

This case has already been considered by Hollenberg et al. [4]. The

neutral axis in this case always coincides with the centroid (i.e.

h

Q

= 1/2) as can be shown from Eq. 7 when R = 1 is substituted.

• .» a Constitutive equation 8 describing the relations between k and m is of

course still valid but the intercept C now takes the simple form:

15
C (N) = 2

2N + 1 N/2

N

( 10 )

Fig. 1 4 is a plot of N against C from Eq. (10). This can be used as a

first step to check whether the material has uniform properties in creep

by checking the observed value of C to see if it agrees with the predicted

value of C given by this plot.

2.4 Application

As an example of demonstrating the applicability of the present

analysis, three loads of different magnitude were applied to debased

alumina (commercially known as AD-86 V from Coors Corporation) beams at

1100°C in a four-point bend configuration with major and minor. spans set

at 40 and 10 mm respectively. Load-point displacements are continuously

monitored as a function of time. Apparent steady-state was observed in

all cases within 40 hour duration. Table I lists the pertinent data of

measurements. After data were taken, the first step is- to plot Ap versus

m on logarithmic scales. If the resulting curves are linear, the

solutions presented in Sec. 2.3 can be used to check whether special cases

apply. Since plotting of data in Table I indicated non-linear behavior

(suggesting n

t

* n

c

), the special cases discussed in Sec. 2.3 must be

ruled out. Solutions of different R and n were produced in order to fit

the data points. It was finally found that a special curve as shown in

Fig. 15 with the following constants n

Q

= 4, n

t

= 1/2, R = 50,000 fits the

]_/ Certain commercial equipment, instrument or materials are identified

in this paper in order to adequately specify the experimental procedure.

Such identification does not imply recommendation or endorsement by the

NBS, nor does it imply that the materials identified are necessarily the

best available for the purpose.

16
Thus the steady-state tension creep behavior

can be described by

e_ t = 4.25 x 10 d V

and the compression creep, on the ether hand, by

£ sc = 0.885 x 10-3 o 4

where e and o have the units of s

1 and MPa respectively.

The predictions strongly suggest a profound difference of creep

behavior between tension and compression. Uniaxial tension tests were

also performed which yield the same order of magnitude as predicted by the

current analysis. Additional tests in simple tension as well as in

compression are being performed in order to verify the predictions.

Detailed microstructures of these post crept beam specimens are also being

investigated in order to understand the rheology leading to the resulting

power-law stress exponents. One dominating factor obtained from the

preliminary studies is the observation of compositional changes due to

devitrification, resulting in drastic variation of viscosity in the grain

boundary liquid phases [15-16]. This strong stress-dependent viscosity

coupled with local recrystallization in the compression zone and dilatancy

and/or cavitation in the tension region may be responsible for the unequal

stress exponents.

17
In contrast, the conventional method assumes R = 1 and n^ = nc

= N so

that the outer fibre creep rate can be computed from the load-point

displacement rate data by the following equation

2(N+2)HA

' = E

ss ( L-S, ) [ L+ ( N+1 ) 2, ] and the outer fiber stresses by

Table II lists the resulting outer fiber creep rates and stresses computed

from the same set of load point displacement data tabulated in Table I. A

plot of £ ss versus o e on logarithmic scales by least squares method as

shown in Fig. 15 leads to a prediction of A = 1.71 x 1

-1

s~"' and N =

1.86. Hence we see there are substantial differences in the prediction of

power-law creep parameters between the present analysis and the

conventional method. Notice that A

fc < a < Ac and n t

< N < n G implying

that the conventional techniques, by imposing uniform properties,

effectively generate an average response in bending creep.

3. Discussion

We have presented a viable technique by which individual tension and

compression creep behaviors, being equal or not, can be predicted directly

from bend test data. Several main features in the analysis are

noteworthy: (1) the neutral axis of the beam cross-section is, in general

not located at the centroid, the location is not only a function of the

materials constants but also of the applied loads. This is generally

indicated by observations of significant densities of cavities developed

over 50 percent of the beam cross section, suggesting the neutral axis has

18
migrated towards the compressive side. (2) If a plot of data on R vs m

shows strong non-linearity which is concave upward, there will be a big

difference between n

c and n^. (3) There exists an applied load for all

materials, in the neighborhood of 1 - 2 MPa of initial outer fibre elastic

• V

stress, under which the material response in K or Ap will be insensitive

to n's suggesting that this load level is not useful in the test program.

(4) If R vs. m data demonstrates a linear response on logarithmic axes,

then the results given in Sec. 2.3 can be first used to ascertain whether

the material poses uniform creep properties. Otherwise, the general

solution scheme as detailed in Sec. 2.1 has to be adopted since a linear

curve within a short range of m does not necessarily mean that n

t

= n„ as

evidenced from Fig. 7. An -example was given in Sec. 2 to show how to

apply the current analysis to a realistic case.

However, the analysis does have some restrictions and limitations

built-in which ought to be borne in mind. First, the constitutive law is

assumed to take a power-law equation form with the distinction being made

through the variations in n and A. Microscopic observations including SEM

and STEM have showed that in a general ceramic system, cracks and cavities

are developed in different patterns both in terms of density and

orientations inside the tensile and compressive zones of a crept bend

specimen. Hence, cracking and cavitation play an important role in the

contribution to creep resulting in distinct behaviors. Secondly, no

deformation mechanism changes are assumed to take place under uniaxial

loading so that a single power-law equation completely character izes the

creep behavior. Consequently, the current analysis is not applicable to

materials exhibiting bi-linear law, although in principle, a numerical

19
scheme may be developed to handle this case. Finally the problem of creep

fracture is not addressed in this paper. As a result, the solutions

presented here are assumed to be valid irrespective of how high the loads

are applied. Realistically, however, as the loads become higher and

higher, the solutions will always be termed invalid somewhere by premature

failure due to void growth and flaw linkage. The prediction of rupture

time as a function of applied stress is thus an important area which

warrants further studies.

Rosenfield et al. [8] recently presented a similar time-independent

analysis on a beam with three different tensile laws. By fixing the

compression behavior as linearly elastic, they found an interesting result

that at 1/5 of the beam depth from the tension edge, the stress there is

fairly fixed regardless of the form of the tensile deformation law.

However, with the removal of the assumption of linear behavior in

compression, the results given by the present analysis did not show this

feature. Accordingly, we must conclude that their results possess a

strong limitation induced by the imposed linear elastic behavior in

compression.

4. Summary and Recommendations

In lieu of summary of the analysis presented in the paper, the

following recommendations are offered to experimenters who intend to use

bend tests for characterizing the materials creep behavior:

(1) The range of the applied loads should be as wide as possible; at

least two orders of magnitude in loads (e.g. 5 - 600 MPa) are recommended.

20
(2) In case that higher moments cannot be achieved owing to premature

fracture a supplemental test program either in tension or in compression

should be performed for the purpose of reducing computer work, thus

improving the accuracy of the results.

(3) Direct measurement, of uniform curvature rate in the inner span

are preferred to load-point displacements measurements as the latter

induce complications such as shearing effects, although the former may be

more difficult to do.

Acknowledgments

Financial support by AR&TD Fossil Energy Materials program of the U.S.

Department of Energy under interagency agreement DE-A1 05-800R20679 and by

the U.S. Air Force Office of Scientific Research under the agreement

AF0SR-ISSA-84-0001 3 with NBS are hereby acknowledged. The author is

grateful to Professor Karl Jakus of the University of Massachusetts,

Amherst, MA for carrying out the bending experiment on glass-bonded

alumina, and to Drs. Richard J. Fields and Sheldon M. Wiederhorn of NBS

for critically reviewing the manuscript.

References

1. S.U. Din and P.S. Nicholson, J. Mater. Sci. 10 (1975) 1375.

2. S.U. Din and P.S. Nicholson, J. Am. Ceram. Soc. 58 (1975) 500.

3. W.R. Cannon and T.G. Langdon, J. Mater. Sci. 18 (1983) 1.

4. G.W. Hollenberg, G.R. Terwilliger and R.S. Gordon, J. Am. Ceram. Soc.

54 (1 971 ) 1 96.

5. M.S. Seltzer, Bui. Am. Ceram. Soc. 56 (1977) 4 1 8

6. R. Morrell and K.H.G. Ashbee, J. Mater. Sci. 8 (1973) 1253.

7. D. Krajcinovic, J. Appl. Mech. 46 (1979) 592.

21
8. A. R . Rosenfield, D.K. Shetty and W.H. Duckworth, to appear in

J. Mater. Sci.

9. I. Finnie, J. Am. Ceram. Soc. 49 (1966) 218.

10. P.K. Talty and R. A. Dirks, J. Mater. Sci. 13 (1978) 580.

11. T. Fett and D. Munz, ibid. 19 (1984) 1791.

12. H. Cohrt, G. Grathwohl and F. Thgummler, Res. Mechanics, 10 (1984) 55

13* R.M. Arons and J.K. Tien, J. Mater Sci. 15 (1980) 2046.

14. G.H. MacCullough, Trans. ASME (1933) 55.

15. S.M. Wiederhorn, B.J. Hockey, R.F. Krause and K. Jakus, submitted to

J. Mater. Sci.

16. W.D. Kingery, H.K. Bowen and D.R. Uhlmann, "Introduction to Ceramics"

John Wiley & Sons, Inc. (1976) p. 757.

22
Table

I.

Calculations

of

normalized

load-point

displacement

rates

from

^-point

bend

data

of

glass

alumina

tested

at

1100°C.

Power-law

creep

parameters

were

then

obtained

based

on

the

present

analysis. o <o < 1 X o N <— Q. • < X

Cl >- • < O • <

CO Ns oo X

bO —i -—

<v s X s s

03X

<X) cqc

<us • O O 2

<Da

cn

CO

O C— in -

cn CTv CO CM CO

CTv vo CO co Ov !

CM CM CM .=r iiIi

CM -=T CO on CM iiiii cVO av f- o in 0-

CO VO CM -=T i

r— • • • • • • CM CM CM 111|

in cn VO o iI c j CO r

r— in n n

r— 0- on vO • • • • • r— n •=T vO vO 11ij

on on o o vO vO on on o o vO VO on on o o VO on on in in vO vO o o o o O 111 o

CM ou on on

in in o o o o

CO CO c-- c— c— 0-

t- 0- c

t- t- • • • • • • CM CM CM CM CM CM 1i ^ CM on -=T n !1

VO
Table

II.

Computation

of

strain

rates

for

the

same

set

of

data

listed

in

Table

Different

power

law

creep

parameters

were

then

predicted

based

on

conventional

method. i 0

jj

S- oa • < o

CO 0 0

t. CO -—

n co 0 0 Cl, C- 5:

4-3 ^

s. 00

CD

Cx,

Q.

CD0

i- o 0 CO 0 o- VO 0 X 7— on in .— in CM O' -=T O' in O' O

CL

O • . . • • • 0 r— in •=r VO VO *— C\J 0 r—

CO0

CO 0 CO S- 0 CL'' •rH 4-3 CO Cl c/} ClS

CO 0 -rH

4-3 4-3 0 3 CO o O 01—

Cxi

4-3 X

ho —H ^-s 0 E X E E W 0 X 0Xc0E

•rH o0

Cl

CO

in O'

r- vO

CO ,

r— in

O c~~ E

r— 0- on vO in in E • • • • • • in ^=r 10 vO t— 1— O

r—

on on

O

O to- c

J

O' O' -=r •=r co CO • . - • • • . VO vO in in on on vO

r— r— CM CM on on CO o

CM

O

CM

O

on oon o•=r

in in o

co co c—

t- c~- >-

CM CM CM CM CM CM

CM on in VO

11 EEo

.<X

CM+X

CM I -3 • 0X

on 0

4-> o
P/2 L14P/2 2

Fig.

1.

Schematic

sketches

ofa

four-point

bend

beam.

(a)

loading

configuration

(b)

representati

ve

cross-section

(c)

typical

element

of

deformed

shape

(d)

strain-rate

distribution

and

(e)

stress

distribution.
b° N

b

I

Fig.

2.

Schematic

of

power-law

constitutive

equations

in

creep

showing

distinct

behaviors

between

tension

and

compression.
OUTERFIBER

ELASTIC

STRESS,

<xe

(MPa)

10

100

CMo

O

o

I o

0)

o

CO Is" <0 LO 't CO CM c ^ OO• • ft ft I I ft • ft o OOO OOO

(h/°h=)Vnoiivooi sixv ivainaN

LO

4-. o o

C II o

4-> U C o

C T3 3 C

j-> re 3

r-> 4) tTl o sz • 00 *-> —

m

ao

bu

NORMALIZED

APPLIED

MOMENT
CM

t. • O >,

<*-. r-t <D

4-> >

C

<D 4-> S O

o a? s a

<n •o <u

a> t.

•3-

to •H

Cju

NORMALIZED

APPLIED

MOMENT
CM

Fig.

5

.

Compressive

zone

size

solutions

as

a

function

of

the

applied

moment

at

nt

«

5,n

c

=1

and

R

=

0.1,

1,

10,

100

and

1000

respectively.
S’

H

Z

LLi 2

o

Q

LU

_!

GL

CL

<

Q

LUN

<

<ro

(ov/hm• =) * ‘3ivd aamvAano aaznvwuoN V

CL LO

Cl

CO o

c

C-i cO O

-=T c

o

•-H in

4-> • o CM c

3

LTi

• cO r—

<n «

co LC

• o O

jj

CO n i. jj

cu c

s-3 "O

j-> c

CO cO >

t- o

3 o

O o

•a

CD n NM cc iH

cOE -=T t.O II c

o .

Q-t c >. o i—1 c_ a;

CD o > c C_ T-C o o

4-> c <u 3 c a

rH B cn o O <u CO 5 s_

vO

bO H

Cl4
o

CVI

(ov/h*=)>i ‘3iva aanivAano aaznvwaoN • V

NORMALIZED

APPLIED

MOMENT,
CM

(°v/h>i =) m *3iva aunivAuno aaznvwaoN

NORMALIZED

APPLIED

MOMENT,
‘ss3His awoo'xvw aaznvwuoN

NORMALIZED

APPLIED

MOMENT
®x>/^‘ss3ais ansNai'xvw aaznvwaoNI I

NORMALIZED

APPLIED

MOMENT
INITIAL

OUTERFIBER

ELASTIC

STRESS,

cre

(MPa)

1

10

100

°VH/ d

y '3ivu"idsia lNiod-avoi aaznvwaoN

NORMALIZED

APPLIED

MOMENT
LOo

(h/°h=) 'Nouvooi sixv ivainaN

MAGNITUDE

OF

INTERCEPT
MAGNITUDE

OF

INTERCEPT
10.0

co

N ‘!N3NOdX3 SS3U1S

MAGNITUDE

OF

INTERCEPT,.
(s/o ss?‘3iva Nivais assuasino

ovH/ dv ‘3ivandSiaiNiod-avo"i aaznvwaoN

c\j r—i

4-3 CO \ 03

03 c «- c

—4 n O

4-3 M> II £. •M

CL 0) o o 4-3 <0 v o<t Lz_> c

S- —' c 03 o a 03 >

C T3 03 • c

05 "-I c CM o

c 03 a \ C3 •M E 03 r— E •=r 4-3 03 3 CO a II JC

r

O n •M 4-3 05 »-4 a 1 O 03 c c

03 03 C o

03 3 a.

05 03 - v— a

f—( 1_ O c J 03 CO 03 O 03 0) 03 > O £0 03 c •> 4-3 J J3 o • o o

orn 03 T— 4-3 03 •X S- X O

4-3 x n 03 lT> f-H 05 s CM Cl o dec 03 •

• < c. L.

4-i —- s_ 03 03 W o 4-3 0 03

05 CO Cm 03 c

4-3 O c= -U>—

rH C 03 <£ 1—

•c O t_ c Cm -m 03 • • 03

03 O 4-3 a. .sr • JD 3 - c

03 r-4 CL i 03 0)

4-5 C3 O 03 03 >

c 03 03 03 O 03 a. £_ C o CO o 03 03 C3 Cl

c. » S. o

1 03 4-3 5- 3 03

£- JC 03 (S J CL rH 3 4-3 f—i 03 03 O JC l on c

Cm c o S- i o 03 -M 4-3 03 1 03 Cm 03 3 O —4 O O E O -— £_ o o CL X 0) o

4-3 o *o in Cl — O O O 03 ao E H — o H • O o

0- «— CO H O 03 E

in

co •H

Lx-
'
NBS-1 1 JA qev. 2-^c U.S. DEPT. O OF COMM.

BIBLIOGRAPHIC DATA

1. PUBLICATION OR

REPORT NO.

2. Performing Organ. Report No. 3. Publication Date

SHEET (See instructions) NBSIR 85-2997 February 1985

4. TITLE AND SUBTITLE

Estimation of Power-law Creep Parameters from Bend Test Data

5. AUTHOR(S)

Tze-jer Chuang

6. PE RFORMING ORGAN 1 ZATI ON (If joint or other than N BS, see instructions)

fc

c-TO^6t5T?06799

NATIONAL BUREAU OF STANDARDS AFOSR-ISSA-84-OOOl

3

DEPARTMENT OF COMMERCE 8. Type of Report & Period Covered

WASHINGTON, D.C. 20234

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City, State, ZIP)

U.S. Department of Energy AR&TD Fossil Energy Materials Program, Washington, DC 20545

U.S. Air Force Office of Scientific Research, Bldg. 410, Bolling AFB , Washington, DC

20332

10. SUPPLEMENTARY NOTES

Document describes a computer program; SF-185, FIPS Software Summary, is attached.

11. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey mention it herel . . . , Power-law creep parameters or brittle ceramic materials are commonly deduced from

load-point displacement data generated by four-point bend experiment, under the

assumption that tensile and compressive behaviors obey the same constitutive law.

However, thanks to different roles played by microcracking and cavitation, it is now well recognized that this premise on occasions may not be valid. The present

paper undertakes an analysis which takes the differences into account. Governing

equations are first derived for the locations of neutral axis of a beam under

bending and for the creep responses in terms of both curvature rate and load point

displacement rate as functions of the applied moment and power-law creep parameters. Numerical solutions are obtained for any given set of materials constants

over a wide range of applied moments. It was shown from the plots of creep

response versus applied moment on logarithmic scales that nonlinear curves concave

upward signify profound difference in stress exponent between tension and compression. An example of applying of the present analysis to a set of load-point

displacement data on glass-alumina beam specimens crept at 1100°C is given. The

results show that the conventional method over/underestimates the creep rates ir. compression/tension by two orders of magnitude, indicating a need of undertaking - more accurate analysis. Several recommendations are offered to experimental:.

as to enhance the reliability in the estimation of power-law creep parameter::

solely on bend test data.

12. KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key wore-. . creep analysis; creep of glass alumina; creep of ceramics; creep bend test; four-:

bend test; power-law creep

13. availability

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