numero26

A. Tridello et alii, Frattura ed Integrità Strutturale, 26 (2013) 49-56; DOI: 10.3221/IGF-ESIS.26.06 50 hourglass shaped specimens with a small diameter (3-6 mm) and a small risk volume. In order to increase the risk volume, dog-bone shaped specimens have been adopted in [7-9]. However, the risk volume of tested specimens (maximum 1000 mm 3 ) is significantly limited due to the non uniform stress distribution along the specimen length with constant cross section. The paper proposes a new specimen shape (Gaussian specimen) for gigacycle fatigue tests: wave propagation equations are analytically solved in order to obtain a specimen shape characterized by a uniform stress distribution on an extended specimen length and, as a consequence, by a larger risk volume. Dog-bone and Gaussian specimens with different risk volumes are compared through Finite Element Analyses and the range of applicability of the two different specimens in terms of available risk volume is determined. The stress concentration effect due to cross section variation in the specimens is also taken into account in the analyses. Finally, the stress distribution of a dog-bone and a Gaussian specimen with a theoretical risk volume of 5000 mm 3 is experimentally validated through strain gage measurements. S PECIMEN DESIGN pecimens adopted for ultrasonic fatigue tests are designed on the basis of equations for wave propagation in an elastic material with the specimen modelled as a one dimension linear elastic body. Stresses are considered uniformly distributed on the cross section and transverse displacements are considered as negligible if compared to longitudinal displacements. In this respect, the displacement amplitude along the specimen,   u z , can be obtained by solving the Webster’s equation for a plane wave:           '' ' 2 /  0 ds z dz u z u z k u z S z      (1) where     ' / u z du z dz  ,     '' 2 2 / u z d u z dz  , and  2 / d E k f      , being f the resonance frequency, and  and d E the specimen material density and dynamic elastic modulus respectively. By inverting and integrating Eq. 1, the specimen cross-section variation for an imposed displacement   u z is expressed by the following Equation:         2 '' ' 0 k u z u z dz u z s z S e       (2) where 0 S is a constant of integration depending on the boundary conditions. In order to obtain a uniform stress distribution along the specimen, the displacement distribution must be linear:     3 u z A kz B    (3) where   3 u z denotes the displacement amplitude in part 3 of the Gaussian specimen (Fig. 1) and A and B are constant coefficients. Boundary conditions for ultrasonic specimens require   3 3 0 u L  , where 3 L is half of the total length of part 3 of the specimen (Fig. 1). The constant of integration 0 S is obtained considering that   2 2 0 / 4  s D   for 0 z  (Fig. 1) and Eq. 2 becomes:     2 2 3 3 2 2 2 2 / 4 k z L kL s z D e e                    (4) Figure 1 : Gaussian specimen. S