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Residual Stress Measurement by the Introduction of Slots Or Cracks I

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Residual Stress Measurement by the Introduction of Slots Or Cracks I Transactions on Engineering Sciences vol 13, © 1996 WIT Press, www.witpress.com, ISSN 1743-3533 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 pro- cedure and in many cases leads to much greater precision in prediction of resi- dual stresses than traditional methods. 1 Introduction Residual stress is a topic of major importance in any discussion of the mechan- ical 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 simula- tion requires experimental validation before being applied in practice. In this paper we review and extend an approach to residual stress meas- urement which we have developed over the past decade and originally referred to as the "crack compliance method." In many, but certainly not all applica- tions this procedure is experimentally more convenient and more accurate for rapidly varying stresses than traditional methods such as "hole drilling" or Transactions on Engineering Sciences vol 13, © 1996 WIT Press, www.witpress.com, ISSN 1743-3533 38 Localized Damage methods which require layer removal. As we will point out, the crack compli- ance 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 displace- ments due to a crack, an inverse approach is taken. Residual stresses are deter- mined 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 resi- dual stress distribution in a circumferentially welded cylinder. A similar pro- cedure 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 <jy(x) to be obtained for values 0.025 < x < 0.975. For other shapes, such as cylinders, other strain gage locations may be more convenient. For near surface stress measurement, one or more strain gages are Transactions on Engineering Sciences vol 13, © 1996 WIT Press, www.witpress.com, ISSN 1743-3533 Localized Damage 39 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) ' ' rt 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. Transactions on Engineering Sciences vol 13, © 1996 WIT Press, www.witpress.com, ISSN 1743-3533 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 direc- tion 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 "compli- ance" 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 unk- nown 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) Transactions on Engineering Sciences vol 13, © 1996 WIT Press, www.witpress.com, ISSN 1743-3533 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 poly- nomial series to fit the data in regions.
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