A11100 T^Sfl73 NBS

PUBLICATIONS

NBSIR 82-2504

Effects of Crack Growth on the

Load-Displacement Characteristics

of Precracked Specimens Under

Bending

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

National Measurement Laboratory

Center for Materials Science

Fracture and Deformation Division

Washington, DC 20234

May 1982

Prepared for Department of Energy

Advanced Research & Technology Development

Energy Materials Program

jjU ngton, DC 20545

,U56

82-2504

1982

ATIOMAX UWR1AW

Of STANDARD*

MBRaRT

NBSIR 82-2504

jyj_ 7 1982

EFFECTS OF CRACK GROWTH ON THE

LOAD-DISPLACEMENT CHARACTERISTICS

OF PRECRACKED SPECIMENS UNDER

BENDING

T.-J. Chuang, E.R. Fuller, Jr., R. J. Fields, and L. Chuck

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

National Measurement Laboratory

Center for Materials Science

Fracture and Deformation Division

Washington, DC 20234

May 1982

Prepared for Department of Energy

Advanced Research & Technology Development

Fossil Energy Materials Program

Washington, DC 20545

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

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director
I
Effects of Crack Growth on the

Load-displacement Characteristics of Precracked

Specimens under Bending

T.-J. Chuang, E. R. Fuller, Jr., R. J. Fields

and L. Chuck

Fracture and Deformation Division

National Bureau of Standards

Washington, DC 20234

ABSTRACT

A critical evaluation of the feasibility of obtaining crack growth

parameters from bend tests is presented. First derived are the

governing differential equations which characterize the time-history of

bend test parameters for a given elastic material exhibiting power law

crack growth behavior. A numerical solution scheme is then developed

which is capable of solving the initial value problem, thus

quantitatively assessing the influence of crack growth on the

load-displacement output. The results of this analysis indicate that

for high N materials (where N is the exponent in the power law crack

growth equation) the flexural test method gives a broad error band in N

prediction and hence is not a reliable technique. However, it can be

used by a designer to quickly screen the new materials with high N

values which are potential candidates for structural application.

1

I . INTRODUCTION

Due to practical limitations to the technique of uniaxial tension

test (such as difficulty in setting up the test fixture and gripping the

specimen), flexural tests (e.g. three and four point bend tests) are

widely adopted as alternative methods for obtaining the fracture

parameters of brittle materials at elevated temperatures . 1-3 In a

typical bending test, a specimen is first precracked at the bottom edge

of the midspan, enclosed in a furnace and then loaded by a mechanical

testing machine in a pure flexural manner. Generally, the crosshead of

the loading machine is kept at a slow constant displacement rate. Total

load applied to the specimen is recorded continuously on a chart as a

function of time. Under certain conditions, particularly those at high

temperatures, nominally elastic specimens exhibit nonlinear

load-displacement curves. This nonlinearity has been attributed to

stable crack growth in the specimen . 1-4 Since time is directly related

to loadpoint displacement at a given crosshead speed, these nonlinear

load vs. displacement curves contain information from which stable crack

growth parameters can be deduced to give lifetime prediction. Assuming

that the non-linearity of the load-displacement curve is due wholly to

the subcritical crack growth without plastic deformation, several

researchers 1 ’ 3 ’ 4 have developed general procedures by which the crack

growth kinetic law of the specimen can be extracted.

A question arises as to the accuracy of this resulting kinetic law

for slight variations in the load-displacement curve unavoidably

introduced by the chart recording system or other sources of error.
2

This is an important issue as it may critically determine the

applicability of this testing methodology. We believe, in order to

resolve this issue, that a sensitivity analysis of the relation which

maps curves of load-displacement (P vs 5) to crack growth law (u vs K^)

is required. In what follows, we carry out such an analysis by

numerically generating load-displacement curves for various crack growth

laws. By varying the crack growth law, we can determine how sensitively

the load-displacement curve reflects changes in crack growth behavior.

To perform the above sensitivity analysis, we have deduced a

general method of solving the time-dependent problem of predicting the

load-displacement curve for a given crack growth law. In Section II, a

system of coupled ordinary differential equations is derived which

characterizes the whole problem. Then a computer program which employs

the Runge-Kutta numerical scheme is developed to solve for all relevant

parameters as functions of displacement (or time). Using this program,

the load-displacement solution as well as solutions for other relevant

parameters are presented in Section III for a typical case. Further,

with the availability of this program, it becomes possible to evaluate

the effects of variation in crack growth parameters on load-displacement

characteristics. Section IV discusses several important aspects of

those influences.

Also, as a byproduct of the current analysis, the accuracy of

fracture toughness (K^) determinations can be assessed. In general,

maximum load and maximum stress intensity do not coincide in time, as
3

will be shown later. Consequently, the conventional method of computing

KTr in terms of peak load may lead to an erroneous result.

II . Analysis

In this section, a detailed derivation of the control equations

leading to solutions of the initial value problem is presented for a

class of materials exhibiting a power law relationship between crack

velocity (u) and stress intensity factor (K^). In this regard, the

following basic assumptions are made in order to simplify the analysis:

(1) the displacement rate, 6, is kept constant at all times. This

assumption is consistent with the conditions under which a typical

dynamic fatigue test is carried out; (2) the plastic deformation of the

bulk is less significant than its elastic counterpart and the nonlinear

behavior is due primarily to stable crack growth emanating from the

existing notch; (3) the driving "force" for the crack extension is

derived from the strain energy released in the near notch tip region

rather than from diffusional dissipation of energy in the case of long

term creep tests 5

; (4) for the sake of simplicity it is assumed that the

contribution of compliance from the loading system is negligible so that

the output of the experiment is a true curve. (Realistically, a true

curve for the specimen can always be obtained by subtracting the

system's compliance from the gross curve. 3 On the other hand, the

analysis could be extended to handle the gross output directly including

the machine's compliance); (5) the notch is assumed to be sharp and

straight through, although the solution scheme is developed in a general
4

way and is thus capable of solving other crack geometries, including the

chevron notch.

2 . 1 Problem Definition

Let us consider a notched beam of rectangular cross-section (width

B x height W) under a load P applied in a four-point bend fashion as

illustrated in Fig. 1. Temperature is held constant throughout the

test. It is assumed that a kinetic law of the following form adequately

describes the subcritical crack growth behavior of the material:

u umax <VK

IC>

N

for Kth < K

x < Kk ( 1 )

Here u is the crack tip velocity (=da/dt, crack length a and time t)

is the applied stress intensity in the opening mode; u is the

critical velocity at which rapid propagation occurs and N is the power

law crack growth exponent, is the fracture toughness, is the

threshold below which no crack growth can take place. The last four

parameters are presumed to be the material's intrinsic properties.

Thus, by definition,

u = 0

and

when K t < K,

I th

u = infinity (indeterminate) if

( 2 )

(3)

Fig. 2 demonstrates a typical u-K^ relationship on a logarithmic scale.

A set of data for a typical ceramic material is also plotted for the

sake of comparison. Eq. 1-3 are the fundamental equations on which the

present analysis is based. Note that by setting different values of K

^
5

and those equations are capable of representing a wide range of

crack growth behaviors. Now, if the notched specimen is loaded in pure

bending at a constant crosshead speed such that the load-point

displacement is in direct proportion to time, then

6 = 6 t (4)

where <5 is the constant loading rate which is controllable. The problem

is to find the time history of the applied load P and other important

variables under such arrangements.

2 . 2 Derivation of Governing Equations

In the case of a pure bend test as illustrated in Fig. 1 the crack

tip stress intensity factor, , can be related to the applied load P,

crack length a and other geometric constants by the following

expression: 6

K

I _ 3 P (L-A)

yr

a -

2 B w

3/2 V (5a)

where L and 2 are the major and minor spans respectively (see Fig. 1)

and Y is a function depending only on the normalized crack length. A

typical form for the expression of Y is 6

Y (x) = (2 tan(^))^ [0.923 + 0.199 ( l-sin(/tx/2)

)

4

] /cos (ttx/ 2) (5b)

Eq. (5b) is claimed to have an accuracy within of the exact value

for all crack lengths (0 < — < 1). Various expressions for Y other than

W
6

Eq. (5b) are also available which yield different degrees of accuracy. 7

In any event, any form of Y can be handled and Eq. (5b) will be used

hereafter for sake of demonstration.

The compliance C of the beam specimen is defined at any time by

C = 5/P (6)

where 5 is the recorded load-point displacement. Now, since we assume

the energy release rate G is in the form of strain energy released in

the crack tip zone, the following equality can be established:

G = K

2

/E'

where E' is the

conditions; and

ratio)

2 P2 dC

2B da

effective Young’s

E’ = E/ ( 1 - v> 2

) for

modulus (E' = E for

the plane-strain ca

(7)

plane stress

e, v is Poissons

Eqs . (1), (4-7) form the control equations of the time-dependent

problem for the unknown variables a, C, and P. However, after a

careful study of this system of equations, it is found that only two

dependent variables are needed to fully characterize the problem.

Mathematically, this means that two nontrivial equations can be derived

from these five control equations. For example, either and a or C

and a can be chosen as a pair of dependent variables; but we found the

latter pair will give more accurate results based on the numerical

solution scheme, because a numerical integration inside each solution

loop can be eliminated in this case. Furthermore, it is desirable for
7

the sake of numerical solution to derive these two equations in a

nondimens ional form.

Let a bar on the top of a parameter be denoted as its counterpart

in the nondimensional version. We can identify the specimen height W as

a normalization factor for length and for K such that

| = 6/W; B = B/W ; a = a/W

K = K

X /KIC

To choose normalizing parameters for C and P, a criterion is found which

enables the expression for K in Eq. (5a) to be translated to the

following simple form in the nondimensional version

K = P Y(a) (8)

if we define

C* = | ((L-£)/B)/(K

IC VW) (9)

and

P* = W/C* = |(B/(L-£)) W

3/2

Kk

such that C = C/C and P = P/P .

( 10 )

Now, with the well-defined nondimensional system, we can derive the

required equations. First, the kinetic law of crack growth (Eq. (1))

reads

da

dt

= umax

ic"

if the applied falls in [K^, • But dt = dS/<5 according to Eq.

(4)

Hence
8

6 — = 6 — = u K

N

d6 d6

maX

or N — = PK

d<5

where 8 = u /6. After substitution of K = P Y = (6/C) Y, this

max

equation finally becomes

— = p (-)

N /d) , d6 C

1 ;

This is a first order nonlinear differential equation for a as a

function of 6.

Next, the expression of the strain energy release rate G, Eq. (7),

can be used to derive another control equation. We rewrite dC/da and

Eq. (7) respectively in the following way

dC _ C_ dC _ C_ ^.dC^

i

^ da

da W da W d6 d6

and

dC _ 2B r K

2 _ 2B (K Y/P~)2

da E’ V E’

1

Combination of these two equations result in

( <L)( dC} / ( da

} = 2B (K^Y/p'V

W d6 d<5 E’

2 2

dc _ 2BW KIC

Y

da

d6 E ' C(P")

2

d<5

or
9

^ 2BW K

IC ^ ^+2

E ' C

? (P ") 2

C

after substitution of da/d6 from Eq. (11). This equation can be

rewritten in the following simple form if we substitute the expressions

JL. for C and P from Eqs . (9-10)

& - 3a g (I)"

dS C

N+2

Y

where a = [ (L-£) /W] (K /E ' Vw)

2.3 Initial conditions

( 12 )

We have derived the necessary control equations for the unknown

variables C and a as a function of 6, Eqs. (11-12), which are coupled

differential equations. Clearly, the initial conditions C(o) and a(o)

are required in order to obtain a unique solution.

At the onset of the test when t = 0, 6=0 and

a = a(0) = a (13)

o where a

Q

is the normalized initial crack length, a

Q

/W, which is known

and controllable.

To evaluate the initial compliance, we note that in general the

elastic compliance of a cracked beam is dependent on the crack length,

a, and is a sum of two contributions: C

q

, the compliance of an

equivalent unnotched beam and C(a), the additional compliance due to the

presence of a crack with a length a. The expression for C

q can be
10

derived from the conventional strength of materials method . 8 The result

is a function of cross-section, major and minor spans, and effective

elastic modulus.

C = 7 (L -£)

2 (L+2£)/BW3

E' (14)

O 4

The increased compliance due to the existence of a crack of length

a can^be found by integration of Eq. (7) to give

_ 9 (L -£) 2 /a/w 2

2 £

, bw2 / [Y(x) ] dx (15)

Thus, the true nondimensional compliance at the onset of the test

(5 = 0) is the sum of Eq. (14) and (15)

C(o) = [C + C (a ) ] /

C* =

o o

a (L+2£)

6 W

( 16 )

Eas. (11) and (12) together with the initial conditions Eq. (13)

and (16) constitute a well defined initial value problem. Unique

solutions for a typical case are to be pursued in the subsequent

section.

Ill . Numerical Solutions

The nonlinear first order differential equations, Eqs . (11) and

(12) for the unknown variables a and C are solved numerically in this

section as a function of <5 by a Runge-Kutta method with the initial
11

conditions given by Eqs . (13) and (16). Note that all parameters

involved are dimensionless. Descriptions of the solution methodology

and how the computer program is constructed to solve this initial value

problem are given in the Appendix. Fig. 3 shows a typical set of

solutions for the normalized crack length, applied load, stress

intensity and compliance as a function of normalized displacement using

a typical input data compiled in Table I.

In general, if the initial crack size is shallow (a

Q

<<W) and the

loading rate is slow (6<<u ) then the load-displacement records will

show a complete curve without abrupt interruption and the entire test

period can be divided into three stages: (1) no crack growth period in

which the crack front remains stationary, resulting in linear elastic

behavior, (2) rapid crack growth period during which the peak load and

maximum stress intensity occur and finally (3) a slow crack growth period

when an elastic hinge forms at the remaining uncracked ligament. In

terms of <5, the transition points separating those three stages would be

at 6xl0 4 ~1.0 and 3.0 respectively in reference to Fig. 3. Of course, any

change in the crack growth parameters (e.g. u , K , etc.) would cause

these boundary lines to shift.

In the initial stage when the applied load is sufficiently small

such that K < K a stationary crack tip results. This means that the

specimen behaves in a linear elastic manner: both compliance and crack

length remain fixed at their initial values; the applied load increases

linearly with increasing displacement, as does the stress intensity so

long as its magnitude stays below K .
12

As the test proceeds with increasing load, a point is reached where

K has increased to its threshold level K , and the displacement is th r

6 = K

th VKIC

Y( V (17)

where C

q is the initial compliance given by Eq. (16). Then the crack

starts to grow according to the governing kinetic law of Eq. (1) and the

second stage ensues. In this region, the crack tip first moves very

slowly and then begins to take off. This crack growth behavior is

solely responsible for causing both P and K curves to become nonlinear,

reach their peaks and then to decrease. This phenomenon can be easily

explained if we write the slope of the P vs 6 curve in the following way

dP =

P

C 1 . P

dC

) d6 d6 (18)

It becomes clear then that there are two components contributing to the

slope: the first term is just the stiffness of the beam and the second

term represents a decreased slope which is attributable to the increase

of compliance due to crack growth. Note that in the first stage, C =

const (= C ) so that the second term vanishes and dP/d<5 = P/6 or P =

o (const) 5, a consistent result of linear behavior. In the second stage

of rapid crack growth, dC/d<5 increases rapidly from zero causing dP/d<5

to deviate from its linearity. The peak load, P is achieved when the

right side of Eq. (18) vanishes, or

max

dC

dS

-1

6 = 6

(19)
13

where 6 is the normalized 6 at which P = P (see Fig. 6). With

max increasing displacement the slope becomes negative, resulting in a rapid

decrease of load. Note that the peak load and peak stress intensity do

not coincide in time; rather the former always occurs earlier than the

latterThis can be supported by the following argument. It was shown

in Eq. (8) that K consists of two competing multiplicative factors,

namely the applied load and the instantaneous crack length. Immediately

after the maximum load is reached, the decrease of K attributable to

reducing P is far less than the increase of K by crack growth so that

the true K values continue to increase until the former factor outweighs

the latter, at which point the peak K is observed. It has been

conventional to compute KTO based on the measured P . This is correct

only if there is no subcritical crack growth in the entire test period

so that peak load coincides with peak K in time. We have witnessed that

this practice will give a nonconservative estimate of which is

always higher than the true value, if stable crack growth does indeed

take place.

Finally, in the third stage after the crack length has grown to 195

pet of the specimen's height, a "hinge" mechanism forms where the

uncracked ligament is so small that it serves as a rotational hinge

allowing continued displacement at exceedingly small loads. As a

result, the applied load in this region has dropped to <<2% of the peak

load. Meanwhile, the compliance of the specimen increases exponentially

(see Fig. 3)
14

IV. Discussion

We have presented a typical set of complete time-history solutions

of crack length, applied load, stress intensity and compliance for a

four-point bend beam specimen from simultaneous differential equations

for materials exhibiting power law crack growth. Realistically, there

always exist some degrees of uncertainty in determining the crack growth

parameters from a given set of crack growth (u vs K^) data. This

prompts a need to carry out a parametric study wherein the effects of

variations in the crack growth parameters on the load-displacement

records can be assessed. This study can be used to determine the

applicability of the flexural test methods for a given material. The

solution scheme developed in the preceding section makes such a study

possible. In the following subsections, the behavior of load as well as

other important variables will be analyzed individually by variation of

crack growth parameters.

4 . 1 Effect of Critical Velocity on Load-Displacement Curves

In this study, the P-5 curves are solved for several values of

u , wki-^- e keeping the remaining parameters constant, as given in Table

I. The results are shown in Fig. 4 where the three curves are seen to

differ considerably in the rapid crack growth region (stage II). It is

observed that there are no differences in the linear parts because

linear elastic behavior governs and hence u plavs no role in this

region. Note that the peak load decreases as u increases. Indeed, ° r max

in the nonlinear regime, the load at any given displacement also
15

decreases as u increases. However, for a given material, u is an max max intrinsic property and hence must remain fixed. But the loading rate,

5, can be manually adjusted to give the same results, since it is the

ratio of u /S which characterizes the problem so that increasing u max max

has the same effect as reducing 5. The implication is that by raising

the loading rate, the peak load would be predicted to increase

monotonically until it reaches a level corresponding to K^. At this

point, a sharp drop in load-displacement curve will be observed and an

incomplete curve will result. In contrast, the solutions given in Fig.

4 show complete curves because the loading rate applied in these cases

is very slow, where K always remains below K^.

4 . 2 Effect of Crack Growth Exponent (N) on Maximum Load

The change in maximum load (P ) as a result of changing the crack

growth exponent (N) is studied in this subsection for u = 10 mm/ s and

other geometric and loading constants listed in Table I. For each value

of N, the value of P can be obtained from the present work. To study max

the effect of changing N, a series of solution schemes has to be applied

in a discrete manner in N. Hence we begin with N = 0 and repeat the

computer program, each time with the increment of AN = 0.5 until the

desired range of N is wholly covered. The program produced Pmaxvs N

curves for 0^N^50. Fig. 5 shows the curves of P vs N for a = 0.2,

0.5 and 0.7 respectively. For comparison, the limiting values of P^

(i.e. the asymtotic value at which N = infinity) are indicated for each

curve. We found at N = 50, the maximum N we computed, P has arrived

to 92%, 91% and 90% of their limiting values for a^ = 0.2, 0.5 and 0.7
16

respectively. The following two observations can also be made from Fig.

5: (1) for a given initial crack length, materials with higher N can

sustain higher load although the increase of load levels off after N

> 20. (2) for a given material where N is fixed, a shallow initial

cracked specimen can carry higher load; as can be seen the differences

between the three curves corresponding to a

q = 0.2, 0.5 and 0.7 are

substantial

4 . 3 Effect of Crack Growth Exponent on Peak Load Decrement

In carrying out a flexural test for a new material to ascertain its

crack growth behavior, one often is confronted with a question as to

whether the crack growth is observable from the output of the loaddisplacement records with a built-in inaccuracy (albeit a small

percentage). In this regard, it may be useful to introduce a new

parameter which can measure the degree of observability of crack growth.

This new parameter, the peak load decrement, R, is defined as the

percentage of deviation of stiffness at the peak load point on the P-5

curve from the initial stiffness, namely

VCo - 1/C

£ _ _ R (pet) = 5 x 100% = [1 - (C /C )] x 100% (20)

1/C

° o

For physical interpretation of R, a schematic sketch of a loaddisplacement curve is shown in Fig. 6. Here 5 is the nondimensional 6

at which peak load occurs. The parameter R can be expressed in terms of

load by multiplying 5 on both numerator and denominator of the right

side of Eq. (20)
17

R (pet)

P 0 - P

£ max

x 100% ( 21 )

We can interpret the physical meaning of R in the following way.

It is a measure of deviation of P-6 curves from linearity. Thus, a

higher value of R implies a heavier influence of crack growth on the P-6

curves. For large R's, one can determine crack growth parameters

accurately from these curves. On the contrary, low R-values would yield

a wide range of uncertainty, for a slight variation in the P-5 curve.

This can be easily envisioned in Fig. 7 where R is plotted as a function

of N for three different initial crack lengths. The same technique as

described in Sec. 4.2 was employed to produce this figure. Other

required parameters are held constant (see Table I). Suppose inertia of

the motor driven mechanism combined with the width of the drawing pen

results in ±1% of confidence margin. For a material with a true value

of N = 10, the possible range of N for a^ = 0.5 is from 8.5 to 11.5

according to Fig. 7. However, if true N increases to 25, the range of

observed N would expand from 20 to 32. This means that the prediction

of N for a typical case shown here is subject to a great degree of

uncertainty when N > 20. Table II gives a list of true N vs. its band

range for R = ±1%. Also, as indicated by Fig. 7, the confidence range

seems to be insensitive to the initial crack length, although deep

cracked specimens do yield higher values of R. Hence we are led to

conclude that at least for the cases we studied, only low N materials

yield meaningful results. For materials exhibiting high N (say N > 20)

the bend test is not a good method to obtain crack growth parameters.
18

V. CONCLUDING REMARKS :

We have presented here a general mathematical procedure to quantify

the effect of crack growth on the load-displacement characteristics of

elastic flexural specimens. By doing this, a critical assessment has

been made of the feasibility of using bend test methods to obtain crack

growth parameters by evaluating the relations between load-displacement

and u-K-r curves. In studying the effects of changing u and N on the

P-6 curves, we are able to determine the confidence margins of this

methodology for a class of materials exhibiting power law crack growth

behavior. It is concluded from this study that only for low N

materials are the flexural test methods capable of making satisfactory

predictions of crack growth behavior. When evaluation of high nominal N

materials is attempted, this technique breaks down since it results in

broad error bands. However, the merits of adopting bend tests remain in

the ease of setup and load application and expedient to obtain the data.

Hence, it is appropriate to use this method to prescreen new materials

for the purpose of quickly selecting high N materials which are

superior for structural application.

Acknowledgement

The authors thank R. de Wit for his help interfacing the FORTRAN

program with the UNIVAC 1108 computer and its related plotting

accessories
19

References

1. H. Hiibner and W. Jillek, "Sub-critical crack extension and crack

resistance in polycrystalline alumina", J. Mater. Sci. 12, [ ] 117-25 (1977).

2. D. Munz, R. T. Bubsey and J.L. Shannon, Jr., "Fracture Toughness

Determination of A1 2 0 3 Using Four-Point-Bend Specimens with

Straight-Through and Chevron Notches", J. Am. Ceram. Soc. 63, [5-6]

300-305 (1980).

3. F. W. Kleinlein and H. Hiibner, "The Evaluation of Crack Resistance

and Crack Velocity From Controlled Fracture Experiments of Ceramic

Bend Specimens" Fracture, Vol. 3, ICF4, Waterloo, Canada, June,

1977.

4. R. J. Fields, E. R. Fuller, Jr., T.-J. Chuang, L. Chuck and K.

Kobayashi, "Crack Growth in Sialon" in "Fracture Mechanics of

Ceramics", Eds. R. C. Bradt, D. H. P. Hasselman, A.G. Evans and

F. F. Lange.

5. T.-J. Chuang, "A Diffusive Crack-Growth Model for Creep Fracture",

J. Am. Ceram. Soc. 65 [2] 93-103 (1982).

6. H. Tada, P. C. Paris and G. R. Irwin, "The Stress Analysis of Cracks

Handbook" Del Research Corp., Hellertown, PA pp 2-14 (1973).

7. W. K. Wilson, "Stress Intensity Factors for Deep Cracks in Bending

and Compact Tension Specimens", Engr. Frac. Mech. 2, [ ] 169-171

(1970).

8. S. Timoshenko "Strength of Materials" Part I, p. 159, 3rd Ed.,

D. Van Nostrand Co. Inc. (1955).

9. A. G. Evans, pp 373-96 in Ceramics for High-Performance

Applications, Proceedings of the 2nd Army Materials Technology

Conference, Hyannis, Massachusetts, November, 1973.
APPENDIX

Numerical Solution Scheme in Solving the Initial Value Problem

As appeared in the text, a system of simultaneous differential equations

of the following form for the unknown variables C(5) and a(6), where 6 is

independent variable, has to be solved:

II

u|<o 'vyl'-a (6, c, a)

^ = G (6, c, a)

d5

subject to the initial conditions

C (0) = c

o a(0) = a o

where specific expressions for F,

respectively in the text.

(A-la)

(A- lb)

(A-lc)

and G are given by Eq. (11) and (12)

This is a well defined initial value problem and unique solutions

can be pursued by the conventional Runge-Kutta numerical solution

scheme

We begin the solution process with the initial point 5 = 5

q where C

(5 ) and a(5

Q

) are given by Eq. (A-lc). Here is a reference point and

hence can be arbitrarily set to zero. Now let the independent variable

5 advances a step size A6 to a new point + AS; the task is to compute

the values of C and a at this new point. In terms of equations, how can

we compute f and g if we write
C (5 + AS) = c (5 ) + f (A-2a)

o o

a (6

q + AS) = a (6

q

) + g (A-2b)

Let f. and (i = be defined by the following equations

f

l = A<5 ' F C

oa o

) !

«! = A6 ’ G <V C

c> V

f

2 = AS-F (6o + A5/ r

C o + V2, a +

0 gj/2)

^2 = AS-G < 6

o + A5/2

, C

o + fj/2, a +0 % x n-)

f

3 = AS-F (6o + A6/

2

, C

0 + f2 /2, a +

o

g 2 /2)

g

3 = AS-G (6o + A6/

2

, C

0 + f

2/2, a +

0 g2 /2)

f

4 = AS-F (5

o + AS, C +

0

f

3 + 5

g

4 = AS-G (6

o + AS, C +

0

f

3 +

«3 }

then the Runge-Kutta scheme yields the following results for f and g

f = |(f

x + 2 f 2 + 2 f

3 + f

4

) (A-3a)

and

g = \ ( g

x + 2 g2 + 2 g

3 + g 4

) (A-3b)

Note that both f and g^ can be evaluated in sequence since both f^ and

g^ depend on (f^ ^,g_^ ^)- Once f and S are computed, the unknown

variables C and a at the new position of 6

q + AS can be found from Eqs

(A-2) . The same procedure can be repeated to calculate C(6

q + 2 AS) and

a(S

Q + 2 AS) a step further at 6

q + 2 AS from the previously solved

values of C (5 + AS) and a(5 + AS). This process can be continued

o o

r

until some criteria of termination of computation are satisfied and the

whole initial value problem is solved.

A FORTRAN computer program was developed following the above

described scheme. Fig. 8 shows the flow diagram associated with this
program. In the first loop concerned with the linear region, we let 5

increase monotonically with one step size A6 at a time. In this loop,

all new C's and a's remain unchanged at their original values, but both

P and K increase linearly until K = then the computation will jump

to another loop where essentially the main task is to solve for f and g.

This process continues until either K = or a = H then the

computation will terminate and the whole solution is obtained.

In general, the smaller the A6, the more accurate the solution that

can be obtained for problems solved by Runge-Kutta scheme. But there is

a drawback in which, as AS decreases, the cost of computation time

increases exponentially. Specifically, in the case we considered, the

convergence of this ordinary differential equation system has been

-4

assessed and the optimum size of AS has been identified to be 1 x 10

It was found that a smaller step size would not increase the accuracy

substantially although the computation time would increase rapidly.
Table I. Geometrical and materials data adopted as initial value problem

input to solve the

Parameter Value

Specimen's height, W (mm) 9.964

Specimen's width, B (mm) 4.14

Major span, L (mm) 40.0

Minor span, £ (mm) 10.0

Loading rate, 5 (mm/s) 2. llxio’

4

Integration step size A6 lxlO

-6

Normalized initial crack length a^ 0.2

Effective Youngs modulus, E' (GPa) 360

Fracture toughness, (MPa*m^) 4.0

Threshold K, K (MPa-m^) 0.4

Crack growth parameters

u (mm/s) max

10.0

N 6.0

Unless specified

otherwise

Characteristic compliance, C (mm/N) 0.02722

Characteristic load, P (N) 366
Table II. List of band range in N vs N for a confidence margin of ±1%

in R.

N Band Range in N

5 5.3 ~ 5.8

10 8.5 ~ 11.5

15 12.5 ~ 18

20 16 26

25 20 32

30 23.5 ~ 46
Figure Captions

Fig. 1.

crack.

Fig. 2.

materials

Fig. 3.

Fig. 4.

Fig. 5.

Fig. 6.

Fig. 7.

Fig. 8.

Geometry of a typical bend specimen with a sharp midspan

N Schematic plots of u = AK^. for some typical ceramic

Typical time history solutions of crack length, applied load,

stress intensity, and specimen's compliance as a function of

load-point displacement.

P vs 6 solutions showing the effect of varying umax

Plots of normalized peak load P vs N for different initial

, , max crack lengths.

Schematic sketch of P-6 curve for physical interpretation of

the parameter R.

Plots of R vs N for a^ = 0.7, 0.5, and 0.2 respectively.

Flow diagram of solving the initial value problem numerically

by Runge-Kutta method.
LOAD

ROLLER

C\J CL

OJN

CL

Q LlI

LlI (j ^ < O Ll

CD

(TH

(TO

CLQ.

ZDCO I
Kj , MPa-m' /2

0.3 0.5 3 5 10
Normalized

Displacement,

8

(=8/w)xlO

A A A

P, K or C/IOO

o o o o

poop ro o oo A

Normalized Crack Length, a
'Normalized

Displacement

S

x

|0

Normalized Load P
p / p * ^max 7 r

1.25
~0 >
R, Percent _ ro oj

o o o o

" ’f -N

I
INPUT W.B.L,

J2 .E.Kjc.K'th*

y max' A S , a 0 ,S

' f CALCULATE

a, jS

t SET INITIAL

CONDITIONS

S , = P, = K , = 0

a,= a 0

CALCULATE

INITIAL

COMPLIANCE

C| FROM

Eq.(16)

i r

| S j + | = S

j + AS

3-

t

i

COMPUTE P & K

IN THE ELASTIC

RANGE WITHOUT

CRACK GROWTH

Pi = S l/C( .Ki =

Pj Y(a,)

i

YES

1 / INPUT /

/ 5! /

1 r

S i+ ,=J5 j + AS

1 f COMPUTE

Aa, Ac FROM

Eqs.(A-3)

i aj + 1 = aj + Aa

YES
NBSIR 82-2504

INTERNAL DISTRIBUTION

1E. R. Fuller, Jr

2. S. J. Schneider

,

3. R. P. Reed

4. N. J. Tighe

5. J. B. Wachtman, . 6. S. M. Uiederhorn

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NBS-114A 'rev. 2 -act U.S. DEPT. OF COMM. 1. PUBLICATION OR 2. Performing Organ. Report No. 3. Publication Date BIBLIOGRAPHIC DATA

REPORT NO.

SHEET (See instructions) NBSIR 82-2504 May 1982

4. TITLE AND SUBTITLE

Effect5of crack growth on the load-displacement characteristics of precracked

specimens under bending

5. AUTHOR(S)

T.-J. Chuang, E. R. Fuller, Jr., R. J. Fields, and L. Chuck

6. PERFORMING ORGANIZATION (If joint or other than N BS, see in struction s)

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

7. Con tract/ Grant No. DE-A1 05-800R20679

8. Type of Report & Period Covered

9.

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

Department" of Energy

Advanced Research & Technology Development

Fossil Energy Materials Program

Washington, DC 20545

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 here)

A critical evaluation of the feasibility of obtaining crack growth parameters

from bend tests is presented. First derived are the governing differential equations

which characterize the time-history of bend test parameters for a given elastic

material exhibiting power law crack growth behavior. A numerical solution scheme is

then developed which is capable of solving the initial value problem, thus quantitatively assessing the influence of crack growth on the load-displacement output.

The results of this analysis indicate that for high N materials (where N is the

exponent in the power law crack growth equation) the flexural test method gives a broad error band in N prediction and hence is not a reliable technique. However, it can be used by a designer to quickly screen the new materials with high N values

which are potential candidates for structural application.

12.

KEY WORDS (Six to twelve entries ; alphabetical order; capitalize only proper names; and separate key words by semicolon s) Ceramic fracture test; crack growth of ceramics, four-point bend test; fracture test;

initial value problem; load-displacement characteristics ; power-law crack growth.

13. AVAILABILITY

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20402.

IX ; Order From National Technical Information Service (NTIS), Springfield, VA. 22161

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