NASA/TM-1998-208453

_7

/

A Back Face Strain Compliance Expression

for the Compact Tension Specimen

William T. Riddell and Robert S. Piascik

Langley Research Center, Hampton, Virginia

National Aeronautics and

Space Administration

Langley Research Center

Hampton, Virgirtia 23681-2199

October 1998

https://ntrs.nasa.gov/search.jsp?R=19990008182 2018-05-11T17:13:51+00:00Z
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The effect of a finite-sized back-face-mounted strain gage was determined by taking the difference in ydirection displacements at two symmetric points on the back face, and dividing by the gage length

between the two points. The J-integrals were determined from the elasticity solutions [9]. These values

were used to calculate the stress intensity factors.

Results

Predicted stress intensity factor and crack mouth opening relationships are compared to corresponding

relationships from the literature. Good agreement with these well established relationships increases the

confidence in the numerical analyses, allowing a reliable relationship for back faced strain to be

developed.

Stress Intensity Factors

The FRANC2D-calculated and previously published [5,6] relationship between K and crack length for

the compact tension specimen is shown in Figure 3. Here, K is normalized by BW_a/P, where B is the

specimen thickness, W is defined in Figure 1, and P is the applied load. For the range in which the

published solution is valid (a/W > 0.2) the values agree within 0.3%. The FRANC2D-calculated stress

intensity factor is 8.7% greater than that predicted by the published relationship for a/W = 0.1. However,

a/W = 0.1 is outside the stated range of validity for the equation, and in most cases, will be less than the

initial notch length. It should be noted that there is good agreement between the FRANC2D-calculated

and published value of stress intensity factor for a/W = 0.9, even though the ratio of ligament length to

crack length is small enough that the validity of linear elastic fracture mechanics might be questioned.

140

120

I00

_._.80

_6o

40

20

0

FRANC2D

Strawley [6]

I I I I I I t I I

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

a/W

Figure 3. Dimensionless stress intensity factor versus a/W.

3
Crack Mouth Opening

Saxena and Hudak [4] have presented an expression for crack length to width ratio (a/W) as a function

of crack mouth opening. Their expression is of the form

a/W= Co +GU+CzU 2 +C3 U3 +C4 U4 +C5U5 (1)

where the coefficients C_ depend on the specific clip gage location and U is a non-dimensional crack

mouth opening, defined by

(2)

where E is Young's modulus, v is the crack mouth opening displacement, P is the pin load, and B is the

thickness of the specimen. This solution is valid for 0.2 _<a/W < 0.975. The FRANC2D-calculated

relationship is compared to the Saxena and Hudak solution in Figure 4. For a given a/W, FRANC2Dpredicted crack mouth opening agrees well with the published solution.

0.25

0.20

"7

+

e o.15

;.rd0.10

0.05

0.00

FRANC2D

Saxena and Hudak [4]

I I I I I I I I I

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

a/W

Figure 4. Crack mouth opening parameter (LO versus aJW.

Back Face Strain

The dimensionless back face strain parameter typically presented in the literature is

A* = eEBW/P (3)

where _ is back face strain, E is Young's modulus, B is thickness, P is pin load, and W is defined in

Figure 1. Results from the FRANC2D calculations and the previously published A' versus a/W

relationship are shown in Figure 5, and given in Table 1.

4
The relationship between A" and a/W is fitted to a polynomial of the form

a/W = N O+ Nl(logA *)+ N2 (logA *) 2 +N3(logA*) 3 +N4(logA*) 4 +Ns(IogA*) 5 (4)

This is the form presented by Piascik, et al. for a back face strain expression for the extended compact

tension specimen [10]. The compact tension specimen polynomial coefficients, based on the numerical

calculations presented herein, are given in Table 2.

100

_10

II

-<

1 i i

0.0 0.1 O.2

[] • FRANC2D

O Deans and Richards [2]

[] Shaw and Zhao [7]

I I I I I I I

0.3 0.4 0.5 0.6 0.7 0.8 0.9

a/W

1.0

Figure 5. Back face strain parameter (A') versus alW.

a/W

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0.55

0.60

0.65

0.70

0.75

0.80

0.85

0.90

Table l--Back face strain parameter (A') versus aJW.

A*

FRANC2D Deans and

Richards [71

Shaw and Zhao [8]

1.803

2.710

4.436

7.422

12.581

22.197

43.003

100.076

420.500

4.26

5.56

7.17

9.31

12.0

15.6

20.6

27.8

38.3

3.002

3.405

4.492

5.999

7.868

10.240

13.461

18.076

24.834

34.687

48.787

5
Table 2--Coefficients for back faced strain P..altionship.

No -0.07978

N_ 0.'_3982

N2 -0+64978

N, 0._3227

N. -0+21704

N_ 0.03154

Effect of Finite Sized Strain Gages

The coefficients given in Table 2 are based on strain at the centerline (y = 0) of the back face of the

compact tension specimen. In practice, finite sized strain gages measure strains over a finite gage length.

This might cause the measured strain to be somewhat less thaz that occurring at the center line of the

specimen. The variation in strain along the back face of a specimen with a/W = 0.7 is shown in Figure 6.

These strains were calculated from the first derivative of the polynomial fitted to the nodal

displacements. The percent change in strains that are measured with finite-sized gage lengths, compared

to the value of strain at y = 0 are shown in Figure 7 for three different values of a/W. Although there is

relatively little effect of the gage length on measured strain when the gage length is on the order of 0.1 W,

larger strain gages can affect the measured value of strain, especially as a/W increases.

Summary

A numerically generated expression for crack length in a compact tension specimen that is based on

back face strain is presented. This expression is given in a form similar to that previously published for a

back faced strain expression for the extended compact tensio_ specimen. The numerically generated

expression for the compact tension specimen presented in this paper is bounded by the two

experimentally generated expressions that were previously presented in the literature.

5O

"-"30

°l.._

20

10

o

Z

I I | I I I I

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3

Location along back fac_. (y/W_

Figure 6. Variation of strain along back face of compact tension specimen with a/W = 0.7.

6
0

= -5

c_

r-

•- -10

0.)

_. -15

-2O

ZX a/W= 0.3

[] a/W= 0.5

O a/W= 0.7

o

O

A

[]

O

A

[]

o

-25 , , , ,

0.0 0.1 0.2 0.3 0.4 0.5

gage length/W

Figure 7. Effect of finite-sized strain gages on measured strain for three different a/W values.

Acknowledgments

This research was performed while one author (WTR) held a National Research Council - NASA

Langley Research Associateship in the Mechanics of Materials Branch.

References

[1] Saxena, A.; Hudak, S.J., Jr.; Donald, J.K.; and Schmidt, D.W.: "Computer-Controlled Decreasing Stress

Intensity Technique for Low Rate Fatigue Crack Growth Testing," Journal of Testing and Evaluation, vol. 6,

no. 3, 1978, 167-174.

[2] Deans, W.F., Richards, C.E., "A Simple and Sensitive Method of Monitoring Crack and Load in Compact

Fracture Mechanics Specimens Using Strain Gages," Journal of Testing and Evaluation, Vol. 7, No. 3, May,

1979, pp. 147-154.

[3] ASTM Standard E647 "Standard Test Method for Measurement of Fatigue Crack Growth Rates," American

Society of Testing and Materials, 1995.

[4] Saxena, A, Hudak, Jr., S.J., "Review and Extension of Compliance Information for Common Crack Growth

Specimens," International Journal of Fracture, Vol 14, No. 5, Oct. 1978.

[5] Newman, J.C., Jr., "Stress Analysis of the Compact Tension Specimen Including the Effects of Pin Loading"

Fracture Analysis (8th Conference), ASTM STP 560, ASTM, 1974, pp. 105-121.

[6] Strawly, J.E., "Wide Range Stress Intensity Factor Expressions for ASTM Method E 399 Standard Fracture

Toughness Specimens," International Journal of Fracture, Vol. 12, June, 1976, pp. 475-476.

[7] Shaw, W.J.D, Zhao, W., "Back Face Strain Calibration for Crack Length Measurements," Journal of Testing

and Evaluation, Vol. 22, No. 6., November 1994, pp. 512-516.

7
[8] Wawrzynek, P.A., Ingraffea, A.R., "FRANC2D: A Two-Dimensi_)nal Crack Propagation Simulator," NASA

CR-4572, March 1994.

[9] Raju, I.S., Shivakumar, K.N., "An Equivalent Domain Integral Method in the Two-Dimensional Analysis of

Mixed Mode Crack Problems," Engineering Fracture Mechanics, Vol. 37, 1990, pp. 707-725.

[10] Piascik, R.S., Newman, J.C., Jr., Underwood, J.H., "The Extended Compact Tension Specimen," Fatigue and

Fracture of Engineering Materials and Structures, Vol. 20, No. 4, pp. 559-563, 1997.
Abstract

A numerically generated expression to determine crack length in a

compact tension specimen from back face strain compliance is

presented. The numerically generated back face strain expression is

bounded by two experimentally determined expressions previously

published in the literature. Additionally, stress intensity factor and

crack mouth opening expressions are determined. These expressions

agree well with previously published results.

Introduction

Compliance-based techniques for crack length measurement are commonly used in automated fatigue

crack growth rate testing. Common techniques for compliance measurement are a crack mouth mounted

clip gage [1] or a back face mounted strain gage [2]. The compact tension specimen, shown in Figure 1,

is a standard specimen geometry for fatigue crack growth rate testing [3]. The crack mouth opening [4]

and stress intensity factor [5,6] expressions for this configuration are well established [3]. In contrast,

there is some disagreement in the literature regarding experimentally generated back faced strain

expressions for the compact tension specimen [2,7].

0.6 W )_ __

0.275 W

J Y'

0.6 W

._le

x

-)_

thickness = B

diameter = 0.25W

-W

1.25 W

back

face

strain

gage

¢

Figure 1. Configuration of compact tension specimen for fatigue and fracture mechanics-based testing.

The purpose of this paper is to present a numerically generated back-faced strain expression for crack

length in compact tension specimens and to resolve the differences between previously published,

experimentally derived expressions. The effect of finite sized strain gages, necessary in laboratory

procedures, is investigated. The well-established relations for crack mouth opening displacement and

stress intensity factor are used to evaluate the accuracy of the present analyses.
Numerical Procedure

The finite element code FRANC2D [8] was used to perform two-dimensional Unear-elastic stress

analyses on compact tension specimen configurations. Crack length to width ratios (a/W) from 0.1 to 0.9

were considered. The FRANC2D is an ideal tool for these simulations because the topology-based data

structure allows crack extension with only local remeshing. This feature allows many different values of

a/W to be investigated with a minimum of preprocessing time.

For each analysis, a rosette of singular quarter point elements was inserted surrounding the crack-tip.

Each side of the singular elements was approximately 0.0325W. The remainder of the body was meshed

with rectangular or triangular quadratic elements. A typical mesh, for a/W = 0.5, is shown in the (grossly

exagerated) deformed state in Figure 2. A more refined mesh for a/W = 0.5 was also generated and

analyzed. The refined mesh had approximately 4 times as many elements as the typical mesh. The

strains and stress intensity factors resulting from the typical mesh, shown in Figure 2, and the refined

mesh were practically identical. This good agreement validates the typical mesh refinement used for

these analyses.

bo

O

e_

g_

e_

o

0_

O

7£

Figure 2. Finite element mesh for analysis of compact _ension specimen with a/W = 0.5.

Nineteen salient nodes of the finite element model were identified. Seventeen of these nodes were

located along the back face surface of the specimen (Figure 2), while two were located at the crack mouth

(Figure 2). The nodal displacements at these nineteen nodes are used in the present analysis to determine

simulated back-face-mounted strain gage and crack-mouth-mounted clip gage responses, respectively.

The simulated clip gage response was determined by the relative displacement, v, between the two

designated nodes. A polynomial was fitted to the y-direction _lisplacements of the seventeen designated

nodes along the back face. A 5th order polynomial produced a good fit of the displacements for a/W <

0.6. Higher order polynomials were required to obtain a good fit to these displacements for a/W = 0.7

and 0.8 (7th order) and a/W = 0.9 (9th order). The first derivative with respect to y of each polynomial

expression yields the strain in the y-direction (e) along the back face of the specimen for the given a/W.

2
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A Back Face Strain Compliance Expression for the Compact Tension

Specimen WU 538-02-10-01

6. AUTHOR(S)

William T. Riddell, Robert S. Piascik

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NASA Langley Research Center

Hampton, VA 23681-2199

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W.T. Riddell, NRC, Langley Research Center

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13. ABSTRACT (Maximum 200 words)

A numerically generated expression to determine crack length in a compact tension specimen from back face

strain compliance is presented. The numerically generated back face strain expression is bounded by two

experimentally determined expressions previously published in the literature. Additionally, stress intensity

factor and crack mouth opening expressions are determined. These expressions agree well with previously

published results.

14. SUBJECT TERMS

compact tension specimen, back-face strain, compliance, fatigue crack growth

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