A New Procedure for the Evaluation of Residual Stresses by the Hole Drilling Method Based on Newton-raphson Technique

The hole drilling method is one of the most used semi-destructive techniques for residual stress analysis of mechanical parts. In the presence of non-uniform residual stresses, the stress field can be determined from the measured relaxed strains using several methods, but the most used is the so called integral method. This method is characterized by some simplifications that lead to approximate results especially when the residual stress varies abruptly. In this paper a new calculation procedure based on the Newton-Raphson method for the determination of zeroes of functions is presented. THE HOLE DRILLING METHOD The hole drilling method is one of the more used semi-destructive techniques for the analysis of the residual stresses in mechanical components [1-3]. In the case of stresses not uniform in the thickness, a hole of radius equal to R is drilled in various steps, up to the maximum depth zM and the relaxed deformations are recorded and subsequently elaborated to calculate the residual stresses. The relationship between stresses and strains is the following ( ) ( ) 0 ( , ) , 0 , 0 z M p z F Z z P Z dZ z z Z z = ≤ ≤ ≤ ≤ ∫ , (1) being ( ) ( ) 2 3 1 ε ε + = z p , ( ) ( ) 2 3 1 σ σ + = z P , (2,3) where ε1 and ε3 are the deformations measured by the grids 1 and 3 of the rosette, σ1 and σ3 are the stresses acting in the same directions, z is the current depth of the hole, Z is the abscissa measured from the surface of the component, F(Z,z), defined in the field Z≤z, is the influence function that depends on the geometry of the analyzed component, the hole-rosette assembly and can be determined through numerical simulations by the finite element method or the boundary method [4]. Typically, the stresses are evaluated up to a maximum depth zM equal to half the value of the middle radius of the rosette Rm, i.e. zM=Rm/2. The determination of the function P(z) through eq.(1) constitutes an inverse problem for whose resolution different approximate methods have been proposed, like integral method [1], the power series method [3], the spline method [5], procedures based on the application of methods of solution of the inverse problem [6-7]. In each method, the P(Z) stresses are approximated by a function P′(z) whose parameters are determined in such a way that the differences between the measured deformations p(z) and the deformation calculated by eq.(1) introducing the P′(z) stresses p′(z) 643 Copyright ©JCPDS-International Centre for Diffraction Data 2009 ISSN 1097-0002