A method of mapping residual stress in a compact tension specimen

Scripta METALLURGICA

Vol. 22, pp. 451-456, 1988 Printed in the U.S.A.

Pergamon Press plc All rights reserved

A METHOD OF MAPPING RESIDUAL STRESS IN A COMPACT TENSION SPECIMEN. C.N.Reid, Faculty of Technology, The Open University, Milton Keynes. U.K. (Received October 19, 1987) (Revised January 25, 1988) Introduction

If a notched testpiece of an appropriate material is loaded sufficiently, plastic deformation occurs non-uniformly with the largest plastic strains being found near the tip of the notch. On unloading, residual stresses are set up within the testpiece and in some circumstances it is desirable to know the magnitudes of these stresses. During a recent study of the nucleation and growth of fatigue cracks in regions of residual stress, the method described in this paper was used to estimate the stresses. It is based on the principle of extending the notch by sparkcutting while recording the attendant changes in strain normal to the notch, measured on the back face of the specimen. The procedure for calculating residual stresses from these strain measurements is presented below and it is shown that the stresses obtained are close to satisfying the conditions for equilibrium. Also it is found that the calculated stress values are in reasonable agreement with those derived from X-ray measurements. Calculation of Residual Stress Consider a compact tension (c.t.) specimen of thickness B containing tensile residual stresses near the notch tip (see Figure 1). Suppose that the notch is extended by a small increment 8x into a region where the residual stress normal to the notch is c, and the tensile force acting normal to the increment is (a B 8x). Notch extension relaxes this force because the new faces are "free" so the force normal to them is zero, and the faces move apart by a small displacement. To undo this effect, it would be necessary to apply to the new faces a pair of compressive forces of magnitude (c~ B 8x) to reverse this displacement. So notch extension is equivalent to the removal of a pair of compressive forces which in turn is equivalent to the application of a pair of tensile forces. The application of such forces would cause the specimen to deform during notch extension and this can be detected by a strain gauge located on the back face of the specimen with its axis normal to the plane of the notch. By reading this gauge before and after an increment of notch extension (e.g. by spark-cutting), a strain increment is obtained which provides a measure of the force relaxed by cutting. In order to interpret this strain increment in terms of residual stress, a relation must be sought between the change in back face strain and the force relaxed by cutting. In the following section a simple model is used to derive such a relation. The Linear Bendin2 Model. v

It is assumed that during cutting the uncut section coplanar with the notch undergoes linear elastic bending (i.e. the strain 8~ varies linearly with x in Figure 1). 451 0036-9748/88 $3.00' + .00 Copyright (c) 1988 Pergamon Press plc

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It was s h o w n above that the deformation of the specimen d u r i n g cutting is that d u e to the application of a pair of tensile forces (o B 8x) to the cut faces. This will cause b e n d i n g of the remaining section about the neutral axis. Suppose that this axis is located at a fraction f of the section w i d t h t, m e a s u r e d from the back face. Balancing external a n d internal forces in the y direction:

(G B 8x) -

a =

- B f tE ~ B (1-f)2t E ~ B (1-2f) t E 2 + 2f 2f (1-2f) t E 8e

(1) 2 fSx w h e r e 88 is the increment in backface strain accompanying a cut of 8x and E is Young's m o d u l u s . For equilibrium to prevail, the m o m e n t s of the external and internal forces must balance a n d this condition can be used to find f. Taking m o m e n t s about the point O in Figure 1, the m o m e n t M 1 d u e to the "external" force (oBSx) must equal the m o m e n t M 2 due to the internal forces associated with bending. M 1 = (c~BSx) (1-f)t Eliminating oSx with the aid of Equation 1

The m o m e n t 8M 2 d u e to the b e n d i n g deformation in a thin layer at position x (measured from 0) is: x2 8M 2 = -~- E~BSx The total m o m e n t is obtained b y s u m m i n g over all layers: M2 -

it

x2 dx

+

o

dx o

ESeBt2 -

3f

{f3 + (1 - 0 3}

Equating M 1 a n d M 2 and simplifying gives the result f =

1

Putting this into Equation 1 gives:

o

-

tE~ 2 8x

-

-

-

t E ~(t) 2

where g(t) is the value of d e / d x at the location t.

(2)

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This expression gives the value of the original residual stress in the region of the first incremental cut, but not for subsequent cuts. At the sites of these, the original residual stress has been modified by the deformation caused by each of the previous cuts. In general, the residual stress a at the site of an incremental cut will be the original stress at that site, a R, plus a change Aa

due to all the previous cuts

(~ = cFR + Aa

(3)

The value of A~ is calculated as follows. Consider the effect of an incremental cut 8x at the location x u p o n the stress at the location t (see Figure 2). The strain increment at x is (2 g(x) 8x) - twice that on the back face since f = 1/3, and the change of stress at x is (2 E g(x) 8x). Hence the stress change at t is: X

(2Eg(x) 8)<)

2x

- Eg(x)

1

-1 8x

3

The total change in stress A¢~ at t due to all the cuts between x--t and x=T is: T A(~ =

T Eg(x)

-1 dx = 3 E t

x

t t By combining Equations 2-4, the value of the original residual stress at the location t is obtained:

(~R(t) = E

-3t

dx + e(T)- eft)

(5)

t This expression enables the residual stress at location t to be calculated from a knowledge of Young's m o d u l u s E, specimen dimension T, and the changes in back face strain during cutting (g(x), e(T) and e(t)). In the following section, this is used to calculate residual stresses in mild steel. Results.

Standard c.t. specimens (25 m m thick) of annealed mild steel were used. The chemical analysis and mechanical properties of the material are shown in Table 1. TABLE 1. Analysis and Properties. % by weight C Si 0.16 0.05

Mn 0.81

strength/MPa yield tensile 320 460

elongation/% 20

Hardness/Hv 150

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An electrical resistance strain gauge was attached to the centre of the back face of the specimen with its axis lying normal to the notch, in the manner proposed by Deans and Richards (1). A specimen was loaded in compression to -50kN and unloaded, thereby causing yielding near the notch tip and a permanent set on the back face. A depth of 120 ~m was removed by electropolishing from the surface of the specimen, prior to examination by X-rays. A uniform diffraction ring was then obtained from the (211) planes using chromium K(z radiation, thereby indicating that the grain structure was suitable for Xray examination. The specimen was then examined for residual stress using the sin 2 V, fixed Vo technique (2) on a Rigaku Strainflex X-ray Stress Analyser. The values obtained of the residual stress normal to the notch are shown in Figure 3 as a function of the distance from the notch tip. The notch was then extended in increments of 1/4 to 1 / 2 m m by spark erosion, using a wire cutter 1 / 4 m m in diameter. The back face strain was recorded after each increment, using an identical "dummy" gauge and specimen immersed in the cutting tank. The variation in strain as a function of the depth of cut is shown in Figure 4. It can be represented to a high degree of accuracy (correlation coefficient greater than 0.999) by two sixth order polynomials with a spline fit. These polynomials were used to evaluate Equation 5 at 1 / 2 m m intervals of t and the results obtained appear in Figure 3. The stress values obtained are close to satisfying the condition of equilibrium; positive and negative forces and moments balance to within 5%. Discussion. The values of residual stress obtained from x-ray measurements clearly do not balance the forces across the plane of the cut (see Figure 3). However, it is not necessary that they should balance because the stresses refer to a thin surface layer, only about 20 ~tm thick. In contrast, the stress values derived from the results of the cutting experiment are averages over the whole specimen thickness, and therefore should balance. One would expect the residual stresses in the surface layers to be somewhat different from those in the interior because the constraint on yielding (during both loading and unloading) is reduced at the surface. In spite of this, there are similarities between the two sets of stresses in Figure 3 in certain respects: the maximum and minimum values agree well and so do the positions of the stress peak near the back face. However, there is a significant difference between the positions of the stress minima. It is concluded that this analysis of the cutting experiment gives near-equilibrium stresses that have similar magnitudes to the X-ray stresses at the surface. It is proposed that the analysis can be used with some confidence, despite the simplicity of the model.

References. 1. 2.

W.F. Deans and C. E. Richards, J. Testing and Evaluation, 7, 147, 1979. B.D. Cullity, Elements of X-ray Diffraction, Addison-Wesley, Reading, Mass, 1956.

Acknow]edgments. The X-ray measurements were carried out at B.S.C. Swinden Laboratories by Mr. J. C. Middleton. My colleague Mr. J. Moffatt undertook the cutting experiments.

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455

- StrainHeasurements I~ X-Ray Heasurement~

m el

~

0

m

lb

zb

2's

3o

Distance fromNotch x I mm

FIG. 3. Values of residual stress normal to the notch plane plotted against the distance from the notch tip; the full line was obtained from the cutting experiment and the points were obtained by X-ray diffraction.

600, '~0 I

4. 4.

0"

,~

÷

.#.

-600. - 600

"|

0

10

I

x/ram

20

30

FIG. 4. The back face strain plotted against the position of the incremental cut (expressed in terms of the distance from the back face).

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/'~

_.____t~x__~ I ~ ft

oJ------'l I1-fl i;E 0

FIG. 1. A sketch of the residual stress (top) and the bending strain (bottom) involved in a small incremental cut, 6x.

Point of interest Strain E(x )-,.- .---- x - - ~

0

<~

------ T------~

0

FIG. 2. A sketch of the point of interest in relation to the notch and the incremental cut 8x.