numero25

P. Lazzarin et alii, Frattura ed Integrità Strutturale, 25 (2013) 61-68; DOI: 10.3221/IGF-ESIS.25.10 67 problem, the latter that of the corresponding out-of-plane notch problem. The two mentioned differential equations must be solved simultaneously, thus justifying by a theoretical point of view the mutual interaction between loading modes shown numerically by other authors. The capability of the new frame to describe the local stress fields in plates of arbitrary thickness under various loading conditions has been verified by some practical examples. 0.1 1 10 100 0.001 0.01 0.1 1 10 In plane shear stress  xy (MPa) Distance from the crack tip (mm)  yy =   yz = 0  III = 0.5 h=10 mm h=2.5 mm h=40 mm scale factor 2 h-decreasing Figure 7 : Scale effect on in-plane shear stresses (plate thickness h=40, 10 and 2.5 mm) induced by Mode III loading with K III =1 MPa mm 0.5 .  xy was plotted at z/h=0 (free surface) where the maximum effect due to induced mode II occurs. The plate sizes are scaled by a factor 1/4 while the resulting stresses are scaled by a factor 2. R EFERENCES [1] Dougall, J., An analytical theory of the equilibrium of an isotropic elastic plate, Trans. Roy. Soc., Edinburgh, 41 (1904) 129–228. [2] Green, A.E., Three-dimensional stress systems in isotropic plates, I. Philos. Trans. R. Soc., London, Ser A 240 (1948) 561-97. [3] Sternberg, E., Sadowsky, M.D., Three-Dimensional Solution for the Stress Concentration Around a Circular Hole in a Plate of Arbitrary Thickness, J. Appl. Mech., 16 (1949) 27-38. [4] Folias, E.S., Wang, J-J., On the three-dimensional stress field around a circular hole in a plate of arbitrary thickness. Comput. Mech., 6 (1990) 379-91. [5] Papkovich, P.F., Solution Générale des équations differentielles fondamentales d'élasticité exprimée par trois fonctions harmoniques, Compt. Rend. Acad. Sci. Paris, 195 (1932) 513–15. [6] Neuber, H., Theory of notch stresses, Berlin: Springer-Verlag; (1958). [7] Hartranft, R.J., Sih G.C., An approximate three-dimensional theory of plates with application to crack problems, Int. J. Eng., Science 8 (1970) 711-29. [8] Hartranft, R.J., Sih G.C., Effect of Plate Thickness on the Bending Stress Distribution Around Through Cracks, J. Math. Phys., 47 (1968) 276-91. [9] Hartranft, R.J., Sih, G.C., The use of eigenfunction expansions in the general solution of three-dimensional crack problems, J. Math. Mech., 19 (1969) 123-38. [10] Williams, ML., Stress singularities resulting from various boundary conditions in angular corners of plate in extension, J. Appl. Mech., 19 (1952) 526–534. [11] Kassir, M.K., Sih G.C., Application of Papkovich-Neuber potentials to a crack problem, Int. J. Solids Struct., 9 (1973) 643-54. [12] Yang, W., Freund, L.B., Transverse shear effects for through-cracks in an elastic plate, Int. J. Solids Struct., 21 (1985) 977–94. [13] Kane, T.R., Mindlin, R.D., High frequency extensional vibrations of plates, J. Appl. Mech., 23 (1956) 277–83. [14] Nakamura, T, Parks, DM., Three-dimensional stress field near the crack front of a thin elastic plate, J. Appl. Mech., 55 (1988) 805–13.