Residual Stress Measurement by the Introduction of Slots Or Cracks I

Residual stress measurement by the introduction of slots or cracks

I. Finnic, W.Cheng

Department of Mechanical Engineering, University of California,

Abstract

A relatively new approach for the experimental measurement of residual stress is outlined and extended. In essence, it makes use of strain measurement as a slot of progressively increasing depth is introduced into a body. For near- surface residual stress measurement the Nisitani "body force method" is used to determine residual stresses from strain measurement In cases in which through-the-thickness stress measurement is desired it is shown that many solutions may be obtained from procedures based on linear elastic fracture mechanics. Although the method requires more extensive prior numerical computation than traditional methods, it involves a simple experimental procedure and in many cases leads to much greater precision in prediction of residual stresses than traditional methods.

1 Introduction

Residual stress is a topic of major importance in any discussion of the mechanical behavior of materials. In many important applications residual stresses have led to premature failure. By contrast, compressive residual stresses are often introduced deliberately to extend the life of parts. As a result there is an extensive literature on the measurement of residual stress. More recently, with enhanced computational ability, the numerical simulation of residual stress has received increasing attention. However, experience teaches that any simulation requires experimental validation before being applied in practice. In this paper we review and extend an approach to residual stress measurement which we have developed over the past decade and originally referred to as the "crack compliance method." In many, but certainly not all applications this procedure is experimentally more convenient and more accurate for rapidly varying stresses than traditional methods such as "hole drilling" or

38 Localized Damage methods which require layer removal. As we will point out, the crack compliance method requires an extensive amount of numerical computation before experimental measurements can be converted to residual stresses. This may well be the reason why this approach, which appears obvious once explained, has only been implemented in the present era of generally available inexpen- sive high speed computation.

2 The crack compliance method

This approach to residual stress measurement had its roots in linear elastic fracture mechanics. In essence, instead of using known stresses and the geometry of a cracked part to compute a stress intensity factor or displacements due to a crack, an inverse approach is taken. Residual stresses are determined from measurements of strain, displacement, or, less conveniently, stress intensity factor as a cut of progressively increasing dimensions is introduced into a part.

The first attempt to implement this approach appears to be due to Vaidyanathan and Finnie [1] in 1971. They showed that measurements of stress intensity factor as a function of crack length, using a photoelastic coat- ing, could be used to deduce the residual stresses due to a butt-weld between two flat plates. However, the experimental technique required special equip- ment, was time consuming and was unsuited to general application. A more generally useful procedure, which was subsequently extended to a variety of configurations was introduced by Cheng and Finnie [2] in 1985. This involved measurements of strain as a function of crack depth to deduce the axial residual stress distribution in a circumferentially welded cylinder. A similar procedure was later proposed by Fett [3] in 1987. Other approaches which involved introducing a cut to measure residual stresses were proposed by Ritchie and Leggatt [4] in 1987 and Kang et al. [5] in 1989. However, these two papers presented procedures which, in essence, followed a "layer removal" approach in estimating the stresses in layers as a slot was extended into the material. This procedure can be shown to lead to the accumulation of error which is minimized in the approach proposed by the authors and Fett which involves a "least squares fit" of all the data for different slot depths.

To explain the basis of our present approach we consider the plane body shown in Fig. 1 which contains normal residual stresses on the plane y' = 0. If only near surface residual stresses are of interest, then the part may be treated as a semi-infinite body and the dimensions x',y' and a' are normalized by the final depth of cut a^, i.e., x = x'/af, y = y'/a^, a = a'/af. Alternatively, for through- the-thickness measurement in a beam-like member the thickness t would be used to normalize the dimensions a',x',y' and s'. In this case, a strain gage is located on the back face of the strip. Typically the strain readings e (a,s' = 0) allow the stress

located close to the mouth of the cut as shown on the left side of the figure. There are important differences which must be considered in carrying out through-the-thickness or near-surf ace stress measurement but to outline the method of analysis our discussion can be quite general.

£(a',S) o (x) = I A,P(x) ' '

S s' e(a',s'=0)

Figure 1: A thin cut is made in a semi-infinite body or strip. The strain is measured as a function of the depth of cut. The fictitious forces F will be discussed later in the paper.

3 Residual stress estimation

The unknown residual stress distribution which is to be determined may be expressed in terms of the polynomial series shown in Fig. 1

O(x) = (1) j=o where Aj is the amplitude factor to be determined for the j^ order polynomial.

The choice of Pj(x) is dictated primarily by a desire to reduce the influence of truncational errors as the order of the polynomial series is increased. We have found that Legendre polynomials are superior to a power series in this respect.

40 Localized Damage

They also have the advantage for through-the-thickness measurement in a plane body of satisfying the equilibrium conditions of zero net force and moment for n > 2 while P<> which is a constant equal to unity and Pj which varies linearly from -1 at x = 0 to 1 at x = 1 should have coefficients AQ = AI = 0. To obtain the coefficients Aj of eqn (1) it is necessary to calculate the strain at the location of the strain gage due to a residual stress Pj(x) for m values of the dimensionless depth of cut a% where 1 < k < m. From the classic paper of Bueckner [6] it is known that this calculation may be carried out by applying the stress Pj(x) to the faces of the cut as shown in Fig. 1. The direction of the loading shown corresponds to a tensile residual stress. We refer to the strain produced by Pj(x) acting on the crack of length a% as the "compliance" Cj(a%) and return later to discuss the calculation of these quantities. The strain may now be expressed as

(2) j=o • •

In principle if the number of strain measurements m equals n+1, the unknown coefficients Aj may be obtained. In practice improved estimates are obtained if the number of cut depths m for which strain is measured is much greater than n+1. The problem is now over-determined (m > n + 1) and the method of least squares is used to determine the Aj from experimental values of e(a%) and the computed compliances Cj(a%). This leads to

= 0 (3) o- k=l j=0 where i = 0,1, ...,n. The subscript i is used to emphasize the fact that the partial derivative involves only one term in the second summation of eqn (3). After differentiation, eqn (3) becomes

= 0 (4) k=l k=l j=0 where i = 0,1,.. .,n.

These (n+1) equations may now be solved for the unknown Aj. One can also write these equations in matrix form, which is convenient for computation. Equation 2 may be rewritten as

[C]A = e (5) where A is an (n+1) x 1 column vector of the coefficients Aj, [C] is the m x (n+1) compliance matrix C%j, and e is the m x 1 column vector of the measured strains e%. To find the A as shown by Mason [7] the equation is expressed as [Cf[C]A = [Cfe (6)

Localized Damage 41

Solving for A yields:

A={[Cf[C]}-M[Cf e) (7)

Once the compliance matrix is obtained and strain measurements are available, manipulation can readily be carried out using software such as Mathcad or Mathlab.

As shown by Gremaud et al. [8] a modification of the procedure based on continuous polynomials Pj(x) leads to improved predictions when stresses vary rapidly as in shot peening or may be discontinuous as in some cladding pro-

cedures. In these cases we use piecewise functions which are a low order polynomial series to fit the data in regions. These regions may overlap if the material is continuous as in shot peening. Alternatively, non overlapping regions may be appropriate for a clad material in which there is no physical

reason for the stress to be continuous. The only difference in computational procedure using piecewise functions is that the strains due to the stresses in the first layer have to subtracted from measured strains before the stresses in the second layer can be computed and so on for successive layers.

4 Some practical considerations

Ideally, to minimize disturbance to the residual stress field and to facilitate computation of compliances the slit shown in Fig. 1 should be a flat crack. However, a crack is not easily controlled and a machining process has to be employed. Our first experiments about a decade ago were made with thin mil- ling cutters. Later researchers [3,4,5] used saws. It was reported [4] that errors of ±20 MN/irr^ could arise depending on the condition of the saw. As a result of an extended visit to Japan by one of the authors in 1990 we became aware of the advantages of electric discharge wire machining (EDWM) as well as conventional electric discharge machining (EDM) for making thin slots. We now use these methods extensively for conducting materials. Commercial EDWM can be used with wires as small as 0.002 inch (50 \im) diameter. We have used 0.001 inch (25 |im) wire for some tests and cutting with 10 |im wire has been reported by Kinoshita and Hayashi [9] although not in connection with residual stress measurement. In addition to providing thin slots EDWM generally is carried out with a copious flow of temperature controlled deion- ized water. This essentially eliminates any problems of temperature control for the foil resistance strain gages we use for strain measurement. For cutting in remote locations, such as the valve seat inside a valve or on a curved surface, EDWM is not practical so conventional EDM with a thin electrode in kerosene can be used. In this case the wear of the electrode has to be considered in estimating the depth of cut. One concern with EDWM or EDM is that residual stresses may be produced by the cutting process. Cheng et al. [10] have developed a method by which the strains produced by cutting may be elim- inated from the measurements before computing residual stresses. For through-the-thickness stress measurement or when the residual stress levels are

42 Localized Damage

high we have found the strain induced by cutting to be insignificant. Most machines provide settings for "fine cutting" which minimize the "recast" layer and residual stresses at the expense of reduced cutting speed which is not an issue in our experiments. Also the recent development of "anti-electrolysis" machines for EDWM is said to reduce the heat affected layer. In testing a car- burized steel with such a machine and a 0.002 inch (50 jLim) wire it was found by Prime [11] that no correction had to be made for strains induced by cutting.

The displacements or strains induced by cutting could be measured by a wide variety of techniques. We chose to use strain gages because of the general availability of gages and associated instrumentation.

5 Near surface stress measurement

Stresses near the surface are of interest since failures often initiate in this region. Strain measurements from a gage such as shown on the left side of Fig. 1 cease to be useful when the depth of cut a' reaches the distance S from the centerline of the strain gage to the edge of the cut. For this reason several gages may be installed at different distances from the cut. As shown by Cheng and Finnie [12] it is necessary to consider the finite width of cut to calculate the compliances when the depth of cut is less than about ten times its width. After consideration of different computational techniques we decided to use the body force method proposed in 1967 by Nisitani [14] which he used to obtain stress fields near notches [15]. This approach makes use of stress solutions for point forces acting in an unnotched semi-infinite body. The surface tractions on the vertical faces of the slot shown in Fig. 2 are, as in Fig. 1, taken to be the same as those in the unnotched body but with an opposite sign. The rest of the boundary is taken as traction free which is always satisfied by the point force solutions. It is therefore particularly convenient to apply Nisitani's method to a body with only residual stresses. The body force method is similar to the boundary integral method, but its formulation is simpler for prescribed loading on the slot faces. It is preferable also to the widely used finite element method for the following reasons: (1) the body force method only models stresses on the boundary rather than throughout the volume which is a much simpler approach; (2) the boundary conditions on the free surface and at infinity are satisfied exactly while these are only approximated in a finite element program; (3) the discretization of the loading conditions on the slot faces is easily modified in the body force method; and finally (4) it is simpler to apply the calculations to each depth of cut unless a finite element program which automatically generates a new mesh is used.

Localized Damage 43

Figure 2: Configuration of a rectangular slot loaded by normal stresses. The center of the strain gage is located a distance S

from the edge of the cut whose final depth is a^.

In essence, Nisitani's approach requires the introduction of distributed forces on the contour of the cut in the x and y directions so that the resulting stresses coincide with the desired surface tractions on the faces of the slot. A detailed discussion of the computation of strain on the free surface as a result of this procedure was given by Cheng and Finnic [12]. As an example, Fig. 3 shows the ratio of the strains on the surface for a slot relative to a crack for uniform surface loading. It should be noted that the distance S corresponds to that from the edge of the slot to the center of the strain gage. At least for uniform loading of the slot surface, it is seen that the finite width may be neglected when the slot depth is about ten times its depth. In practice an EDWM cut has a semi-circular rather than a flat bottom. Cheng et al. [15] analyzed this geometry, again using Nisitani's method and have shown that taking a rectangular slot with a width which gives the same area as a slot of the same depth with a semi-circular bottom leads to a very accurate estimate of compliances.

1.6 T ^ I ^- |

C/) S = 1.27 mm (/) 0) S = 1.78mm -t-» CO S = 2.29 mm

E 1.4 J5

C/) c 1.2

-Ct—f 1.0 OC

0 10

Depth to width ratio (a/2w)

Figure 3: The ratio of the strains due to a cut of aspect ratio

(a/2w) to those for a line crack of the same depth when the crack and cut faces are loaded by a uniform stress.

As a final point in connection with the Nisitani body force method, we should mention that this is a central feature of the correction procedure we have developed [10] for the effect of residual stresses introduced by EDM or EDWM.

Two examples which have been discussed in more detail [15] illustrate the results which may be obtained for near surface stresses. Figure 4 shows residual stress measurements on a shot-peened specimen of Ti-6Al-4V. The

agreement with X-ray measurement is very satisfactory. Figure 5 shows results for a carbon steel clad with Stellite 6 using laser melting of powder. X-ray measurements and an approximate numerical simulation are also shown. An interesting feature of this test is that metallography shows a layer of martensite in the base material adjacent to the cladding. The volume expansion associated with the transformation to martensite would be expected to produce compressive stresses. This is revealed by our experimental procedure and the numerical simulations but was not detected by X-ray measurements.

Localized Damage 45

200

05 CL

CD O 03 -200 -

CO -400 - 0) Slit No. 1 Slit No. 2

CD X-ray results -4—' -600 - (/) "cd

CD -800 tr

-1000 200 400 600

Depth (jim)

Figure 4: Residual stress distributions measured by the present method and those obtained by the X-ray method for a shot-peened part.

The data points correspond to the depth at which strains are measured.

6 Through the thickness stress measurement

For measurements through the thickness of a part, the finite width of cut may usually be neglected. Also the body force method is much less convenient for computation than for the case of a semi-infinite solid. Fortunately, we have been able to obtain the compliance for many two dimensional or axisymmetric configurations using solutions based on linear elastic fracture mechanics. A summary of the solutions obtained is given in the following table.

46 Localized Damage

1000 - 03 0_

500

"cti 13

CD X-ray (a) oc A X-ray (b) -500 - Compliance method Numerical result

200 400 600 800 1000

Distance (pm)

Figure 5: Comparison of the results obtained by the X-ray method from two independent laboratories (data points), the present method (bold data line) and numerical simulation (dotted line). The data

points in the curves correspond to the locations where strains were recorded.

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Table 1 Configurations for which residual stress measurements have been carried out. See Cheng and Finnic [15] for a list of references.

Axisymmetric stresses Thin-walled rings hoop stresses

Thick-walled rings hoop and radial stresses Thin-walled cylinders axial stresses (non-uniform in the axial direction) Thick-walled cylinders axial, hoop and radial stresses

(uniform in the axial direction) Solid cylinders axial, hoop and radial stresses (uniform in the axial direction)

Stresses in plane stress/plane strain Plates or beams with rectangular cross-section longitudinal stresses

Disks normal stresses on a diametrical plane Afille twel d or a T-butt weld normal stresses on a plane at the toe of the weld

The approach is based on Castigliano's theorem which may be used to obtain the displacement due to loads applied to a cracked body. For two dimensional parts of unit dimension perpendicular to the x-y plane subjected to mode I loading, a general expression for the displacements v on the surface at a distance s from the crack plane can be obtained by introducing a pair of virtual line forces F as shown in Fig. 1. This leads to

, v(a,s) = -- = - K,(a) - da (8) b 0 dF where E' = E for plane stress and E/(l-|i^) for the plane strain with E and |i being the elastic modulus and Poisson's ratio, respectively, U is half of the change of strain energy due to the crack, Kj and K/ are the stress intensity fac- tors for an arbitrary stress on the crack faces and the virtual line force F respectively. The normal strain e(a,y) at a location y = s produced by introducing a crack of depth a is given by differentiating eqn (8) to obtain

, , 9v(a,s) a%U 1 , \ ^ e(a,s) = ' = — - = — J Kj(a) — — - da (9) os dFdS|p=o E 3F9s

Solutions are available for K/ and for Kj(a) for stresses corresponding to the crack face loading Pj(x). With these available, compliance may be obtained from eqn (9).

48 Localized Damage

As an illustration of the power of this method of measurement we measured the residual stresses in a beam which was first fully stress relieved and then had stresses introduced by bending. In this case the residual stress distribution may be predicted very accurately since the stress-strain curves in tension and compression may be deduced from the bending test which induces residual stresses. Figure 6 shows the predicted residual stress distribution and that predicted by LEFM computed compliances from a single strain gage on the back face of the beam, as shown in Fig. 1. Very small strains are measured for normalized depths of cut less than about 0.025 while for large depths of cut a > 0.975 the small remaining ligament introduces problems. Nevertheless in the central 95% of the beam the residual stresses, which are quite small, are predicted with remarkable precision. In passing we note that confirmation of the predicted stresses by the X-ray approach was not possible because of the coarse grain size of the stainless steel.

An interesting and useful feature of the approach to strain prediction based on LEFM is that the stress intensity factor due to a residual stress field may be obtained without determining the residual stresses. By taking a derivative of eqn (9) with respect to the crack size we obtain

Thus, the stress intensity factor due to residual stresses can be estimated directly from the change of strain measured when a thin cut is introduced. For more complex geometries such as a welded connection we have measured residual stresses, e.g., at the toe of a fillet weld, using the crack compliance method. Since analytical solutions using LEFM are no longer feasible, we use finite element computation of compliances. For finite dimensions, this approach now becomes more tractable than the body force method. To illustrate that finite element computation can lead to precise prediction, we have also used this approach for a bent beam as shown in Fig. 6.

7 Conclusions

The approach we have discussed is capable of measuring stresses on a plane in problems in which the stresses vary rapidly in both the direction of cutting and distance from the plane. That is, in the x' and y' directions of Fig. 1. Such a situation arises at the toe of a fillet weld and other welded junctions. Tradi- tional measurement techniques are not well suited to such a configuration. As yet, we are not able to handle problems in which stresses vary rapidly in a direction perpendicular to the plane of Fig. 1. However, thickness variations in this direction can be handled. An example in which this arises is the valve body shown in Fig. 7. A strain gage is located in the circumferential direction on the surface of the hardfacing and a cut made as shown with EDM. In this application other techniques would be impossible to implement without cutting the valve apart.

Localized Damage 49

100 I ' I Compliance functions

FEM computations

05 Q_

CO CO Q) to

co 0) DC

-100

Normalized distance Figure 6: Residual stresses produced by bending a stress free beam

of type 304 stainless steel. Circles show values deduced from known applied moment and the stress-strain curves in tension and compression. Solid and dashed lines show values measured experimentally using a Legendre polynomial series with n = 7. Compliances were calculated

using linear elastic fracture mechanics solutions and finite element computation.

In situations in which strain gage rosettes may be mounted on a surface, this measurement technique has advantages and disadvantages relative to the slitting technique. The rosette method may be applied in thefiel dan d provides the biaxial stress field in the near surface region. The slitting technique provides the normal stresses on a plane. Its sensitivity is considerably greater than that for hole drilling with rosettes. Also, its ability to resolve rapidly varying stress gradients is greatly superior to hole drilling techniques. In hole drilling the hole has to be located precisely at the center of the strain gage rosette. In the slitting technique the distance of the gage from the edge of the slot and the width of the slot may be measured precisely after cutting. We conclude that the approach we have presented is a valuable addition to traditional experimental methods. The procedure appears obvious once presented but has only been developed in the past decade. As a result it is not included in the new "Hand- book on Techniques of Measurement of Residual Stresses," which has been many years in preparation and is due to be published in 1996 by the Society for Experimental Mechanics in the USA.

Hardsurfacing

1.3 mm E _E (M CD

30^

63 mm -

Figure 7: The use of EDM for measurement of the residual hoop stress in the hardsurfaced seat of a valve body.

Acknowledgement

Most of our research on residual stress measurement has been supported by the

Electric Power Research Institute of Palo Alto, California. We would also like to express appreciation to the Japan Society for the Promotion of Science for supporting an extended visit to Japan by I. Finnic in 1990. His host scientist, Professor Masaru Sakata, then with the Tokyo Institute of Technology and now at Takushoku University, knowing his research interests introduced him to Pro- fessor Nisitani and to the EDWM process. We are grateful for his thoughtful help which was of great value in advancing our research.

References

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Localized Damage 51

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