Two techniques, speckle photography and holographic interferometry, were used to test three-dimensional finite-element calculations in an internally pressurized cylinder with an external part-circular crack. Opening displacements along the crack line were measured by speckle photography. Radial displacements were obtained from holographic fringe patterns. Good agreement between experimental and numerical data is obtained. Stress-intensity factor variations along the crack front are calculated from numerical results

Barbero, Ever

Idelsohn, Sergio

Kaufmann, GH

Lopergolo, A.

English

Scipedia

Evaluation of Finite-Element Calculations in a Part-Circular Crack by Coherent Optics Techniques by G.H. Kaufmann, A.M. Lopergolo, S.R. Idelsohn and E.J. Barbero ABSTRACT--Two techniques, speckle photography and holographic interferometry, were used to test three~limensional finite-element calculations in an internally pressurized cylinder with an external part-circular crack. Opening displacements along the crack line were measured by speckle photography. Radial displacements were obtained from holographic fringe patterns. Good agreement between experimental and numerical data is obtained. Stress-intensity factor variations along the crack front are calculated from numerical results. Introduction Engineering estimates for stress-intensity factors in cracked cylinders under internal pressure have been trans- formed in a very important problem in fracture mechanics. Failure of nuclear piping systems ubjected to a cyclic load condition has been traced to fatigue growth of surface cracks. The stress-intensity factor is required to predict he rate of fatigue crack growth for a crack shape which changes continuously because of its growth. For this reason extensive work has been recently done to calculate stress-intensity factors for various shapes of semi-elliptical nd semicircular surface cracks in pressurized cylinders. Three-dimensional umerical analysis using the boundary-integral equation method is reported by Heliot et al.' Atluri and Kathiresan, 2 Miyazaki et aL 3 and Raju and Newman 4 use three-dimensional finite-element methods. Variations of the stress-intensity factor along the crack front have been obtained for a range of crack shapes and sizes for external and internal cracks. In spite of the considerable importance of the experimental evaluation of numerical methods, only a few works are available in the literature. In the past few years, coherent optics techniques such as holographic interferometry and speckle photography have been used to determine the displacement fields ad- jacent to the crack tip. 5-' Several authors use these measurements o calculate stress-intensity factors for different ypes of cracks. The use of both techniques in fracture mechanics offers several advantages. As other optical methods, they are noncontact techniques and the model surface needs no preparation. They do not require G.H. Kaufmann is Professor, and A.M. Lopergolo is Research Fellow, Instituto de Fisica de Rosario (CONICET-UNR), Avda, Pellegrini 250, 2000 Rosario, Argentina. S.R. ldelsohn is Professor, and E.J. Barbero is Research Fellow, Instituto de Desarrollo Tecnol6gico para la Industria Qu[mica (CONICET-UNL), P.O. Box 91, 3000 Santa Fe, Argentina. Original manuscript submitted: June 26, 1985. Final manuscript received: September 2, 1986. materials with special optical characteristics, so they can be directly applied to any opaque materials uch as metals or composites. The purpose of this paper is to show the usefulness of both techniques in fracture mechanics to evaluate calcula- tions obtained by means of numerical methods. As an example, the case of an external surface part-circular crack in a thick cylinder is analyzed. Finite-element calculations are performed; good agreement is found between experimental nd numerical results. Variations of the stress-intensity factor along the crack front calculated by the finite-element idealization are compared with results by Raju and Newman: Experimental Test Specimen The circular cylinder used in this study was made of PMMA. This material was chosen because it is a brittle one; so no plastic flow was expected around the crack tip. Hence, experimental results can be compared with those calculated using linear-elastic fracture mechanics. The cylinder had a wall thickness t of 0.5 cm. Its internal radius Ri was 2.98 cm. The thickness to radius ratio t /R i  was 0.168. Its length 2b was 20 cm, with a 1-cm clamping on both edges, as shown in Fig. 1. After the cylinder was cut into size, a central axial part- circular slot was inserted on the external surface of the specimen with a circular saw. The crack length 2c was 1.36 cm. The crack depth a was 0.23 cm, which gave an a/c  ratio of 0.338 and an a/ t  ratio of 0.46. The slot was 0.02-cm wide and terminated in a vee notch with a very small radius. Next the specimen was heat treated to relieve stresses. Afterwards, the area surrounding the cracks was sprayed with a thin coat of matt-white paint. Two square-end steel plates were made to clamp the edges of the cylinder. On each end-plate there was a circular groove, 1-cm deep, which fitted nicely with the cylinder. Before the two ends of the specimen were slid into the groove, a cyanoacrylate c ment was poured, thus clamping the ends of the cylinder to the end-plates. Four rods parallel to the cylinder and fastened to both end- plates made sure the clamping would not become unglued. Pressure was applied by means of an hydraulic system measured with a manometer. A valve was used to main- tain the pressure at a constant value during the exposure time. 154 9 June 1987 Values of the elastic constant, E = 2920 MPa and v = 0.33, were obtained from a calibration of a bar of the same material using electrical strain gages. Speckle Photography The speckle-photography technique '~ was used to measure the crack opening (in-plane) displacements uo along the slot line. They were obtained by measuring the crack aperture along two lines parallel to the slot direction, one at each side of the crack and at a distance of 0.025 cm from it. The u0 displacements were calculated as one half of the difference between the cracks apertures corres- ponding to a pair of points at each side of the slot. The sensitivity of this technique is limited by the average speckle size. It can be shown that displacements o be measured must be greater than '~ 6o = 1.22 XF(1 +m)/m (1) where X is the wavelength of the laser, F is the aperture ratio of the recording optical system and m is its magnifi- cation. Nevertheless displacements observed along the crack line were very small (about a few microns) which were outside the sensitivity of the technique given by eq (1). This could be increased, however, by superimposing a small known artificial displacement. For our application, a rigid-body displacement of about hree times the speckle diameter was produced by mounting the photographic camera on a vernier table. This displacement was cancelled when the difference between the crack apertures corres- ponding to a pair of opposite points along the slot was done. When the cylinder was pressurized between both exposures, its surface moved in the radial direction. It is well known that out-of-plane displacements produce speckle decorrelation which causes a decrease in fringe contrast and introduces errors in the measurements. '~ The Rayleigh criterion establishes the movement which can be tolerated without any appreciable degradation of the fringes. This movement must be less than A = 4XF2(1 + m)2/m 2 (2) For the experiments presented in this paper A -  500/zm. Taking into account that the radial expansion for the maximum pressure utilized was about 30/~m, fringes with good contrast were obtained without using any com- pensation procedure. Figure 2 shows a typical fringe pattern. To simplify data reduction, the cylinder surface was imaged on to a Kodak SO-253 photographic plate mounted parallel to the specimen by means of a well-corrected camera lens. A 21-cm focal length of F = 8 aperture was used at m = 2.8 magnification. The specimen was illuminated by an He-Ne laser. Double-exposure speckle- grams were analyzed using the pointwise technique. By measuring the spacing and inclination of the fringes, aperture displacements along the crack line were calculated. Holographic Interferometry Holographic interferometry was used to measure the radial (out-of-plane) displacement u ralong the crack line. The usual holographic setup to measure out-of-plane displacements was utilized. 11 The cylinder was normally illuminated with a collimated beam of He-Ne laser light. It was mounted parallel to the holographic plate. Several double-exposure holograms were recorded on Agfa- Gevaert 8E75 plates for different pressures. After pro- cessing the hologram, the image showing deformation fringes was reconstructed and photographed at normal incidence. Under these conditions, any point on the axial crack line has a radial displacement given by 1 t u~ = N) , /2  (3) where N takes on integer values at the center of each I • • m  m• F Ri = 2.98 cm j t =05 cm 2b = 20 cm a = 0.25 cm 2c=1.36cm t /R i= 0.'168 ale = 0 .338 2c a / t  = 046 3 (3  Fig. 1 - -C i rcu la r  cy l inder  with a part -c i rcu lar  c rack  Fig. 2 - -Typ ica l  speck le  f r inge pattern Experimental Mechanics 9 155 

Evaluation of finite-element calculations in a part-circular crack by coherent optics techniques

https://www.scipedia.com/wd/images/5/5c/Draft_Samper_270690527_7052_BF02319467.pdf

Evaluation of finite-element calculations in a part-circular crack by coherent optics techniques

Abstract

Similar works

Full text

Available Versions

Scipedia