technical note Residual stresses in the materials by J. HANCOCK, University of Glasgow

Received August 1975

photoelastic analysis of composite

There has been much interest in the stress distribution around elastic inserts embedded in an elastic

as this problem includes those of fibre reinforced plastics and reinforced concrete. In such situations the fibres and the matrix generally have different coefficients of thermal expansion or shrinkage due to curing. This produces a self- equilibrating stress distribution which exists before load is applied to the composite. The three dimensional frozen stress technique is suitable for investigating such problems3. However it should be noted that the shrinkage stress has two components : a stress component and corresponding photoelastic response induced by shrinkage above the softening temperature; and a stress component induced by shrinkage below the softening tempera- ture. When the stresses are low these components are vectorially additive. However when the model is sliced for optical examination, the stress induced by thermal contraction below the softening tempera- ture, and which is perpendicular to the plane of the slice, is reduced to zero. Similarly the unfrozen shear stress on the plane of the slice becomes zero. In general this will result in a redistribution of the in-plane stresses and a corresponding change in the photoelastic response which will no longer corres- pond to the stress distribution before sectioning.

This problem may be overcome by removing the fibre mechanically, completely relieving the unfrozen stresses. Fig 1 shows a section of a cylinder of epoxy resin (Ciba-Geigy CT200) which was cast around an aluminium rod. In Fig 2 the insert has been carefully removed by machining. Comparison of the two figures shows the large change in fringe order due to relief of the unfrozen stresses. Mechani- cal removal of a bonded insert without damage to the matrix near the interface is difficult, however, particularly if the insert has a complicated shape. Alternatively, the insert may be removed chemically but this often leads to uncertainties due to the time edge effect. A convenient technique to overcome this problem is to examine the slice in a glass-sided oven at a temperature just below the softening temperature of the resin. In this way the unfrozen stresses are relieved as shown in Fig 3, which shows the slice containing the insert, examined at 90" C. A comparison of Figs 2 and 3 shows that this is equivalent to removing the insert as it produces a similar photoelastic pattern. In order to check that there has been no optical creep at this temperature, it is necessary to cool the model to room tempera- ture and confirm that the original photoelastic pattern returns, as shown in Fig 4, indicating that the

Fig 1. Section of an epoxy cylinder cast round an aluminium Fig 2. Same section of model after rod had been removed- rod-fringe pattern at 20" C. fringe pattern at 20" C.

174 'Strain: October 1975

Fig 3. Similar section of model as cast with rod in place- Fig 4. Section of model shown in Fig 3, after annealing at fringe pattern at 90" C. 90°C for 48 h-fringe pattern at 20" C.

frozen stress component has not changed. This 2. technique conveniently avoids spurious effects of partially relieved unfrozen stresses on the analysis of slices, although care must be taken to check that the glass sides of the oven do not become birefringent.

1. D. E. W. Stone, 'A Contribution to the Application of Photoelasitcitv to the Micromechanics of

3.

A. J. Durelli, V. J. Parks, H. C. Feng and F. Chiang, 'Strains and Stresses in Matrices with Inserts', Proc. Fifth Symposium on Naval Structural Mechanics, Philadelphia, 1967, Pergamon Press, Oxford, England (editors F. Wendt, H Liebowitz and N. Peronne). J. W. Hancock, 'Photoelastic Analysis of a Fibre Reinforced Composite', University of Glasgow, Internal Report (1975).

Composite Mateiials', J. Strain Anal., 4 (2), (1969), p. 88.

Acknowledgments are due for the work of Mr G. Falconer and the support of the S.R.C.

book reviews Free Vibration Analysis of Beams and Shafts

by D. J. Gorman, John Wiley & Sons, Ltd., London, 1975,386 pages, price f 10.75.

The objective of this book, the author says, is to simplify the general problem of free vibration analysis of beams and shafts for both the vibration specialist and the design engineer. This he achieves by using a logical approach in presenting both the subject matter and the supplied data tables for vibrational analysis. Each main section is preceded by a review of fundamental theory and the following chapters are arranged in increasing order of complexity. Within each chapter there is a comprehensive array of case studies; again these are arranged in an increasing order of complexity. In this way, beams and shafts having elastic supports, masses and torsion springs, are examined in such a way as to ensure that not only the classical but also the non-classical boundary conditions are covered. The specialist should find this extremely useful, as it should ease programming needs and reduce computer time. Also contained in each chapter are illustrative

examples and data tables. These enable frequencies and modal shapes to be calculated for a wide selection of beam and shaft problems without the need of the specialist's knowledge. Should any additional background reading be required, a list of classical texts is provided in the reference section. The last two chapters in the book examine beams having discontinuities in cross-sectional properties and miscellaneous problems in free lateral vibrations respectively. In the chapter on discontinuities in cross-sectional properties. the author uses the same logical approach that he has used throughout the book, with case studies, illustrative examples and data tables provided as before. Finally the chapter on miscellaneous problems examines struts, tapered beams and beams on elastic foundations. These topics are less comprehensively covered than pre- viously, but the chapter still provides much useful information. Overall, the book does go a long way in simplifying the general problem of free vibration analysis. For this reason, it will prove an important addition to the libraries of both universities and industrial companies The only general criticism of the book is that the illustrative examples use imperial units. E.G.

'Strain', October 1975 175