Issue 43

F. Majid et alii, Frattura ed Integrità Strutturale, 43 (2018) 79-89; DOI: 10.3221/IGF-ESIS.43.05 82 stress [8]. In 1953, Faupel developed a formula that takes into account the ultimate stresses and the internal and external diameters of a pipe, Eq. (1) [6]. Lately in 2006, Klever developed a relationship based on thickness, mean diameter and ultimate stress [9]. More recently, Brabin has presented in his works a new formula using the same data as Faupel in addition to a parameter λ, which depends on the characteristics of the studied material [10]. Finally, in 2010, the DNV standards simplified the formulas proposed in the literature by introducing a new formula based only on the yield stress, the thickness and the mean diameter [11].                 0 2 2 ln 3 y y UTS i D P D (1) • D 0 and D i are the external and internal diameters of the cylinder ; • σ y is the yield stress ; • σ UTS is the ultimate stress. Static damage Kachanov formulated the approach of continuum damage mechanics in 1958 [12]. From a physical point of view, the author considered the initiation of damage as an internal phenomenon of progressive deterioration of the material reflected by the presence of cavities and micro-cracks under the effect of repetitive loadings generating the reduction of the area of the cross section [13]. Chaboche who proposed a law of fatigue damage with nonlinear evolution took up this approach to quantify the level of damage. The damage value D takes imposed values varying from zero, for the undamaged material, to a value equal to one corresponding to the appearance of a detectable crack or rupture. These damage theories, developed in the literature, are elaborated based either on the concepts of Palmgren, Langer and Kommers or on Kachanov's continuum damage approach. The most common and used model, adopted by the international codes like ASME and ISO, is the linear damage model which is directly proportional to the life fraction. Miner has presented the damage as equal to the life fraction. This law is expressed as below: D=β=n/N f (2) The life fraction presented by Miner is referring to the ratio between the instantaneous number of cycles under an applied stress (n) and the total number of cycles at the rupture (N f ) in the fatigue case. In this paper, as per the fatigue phenomenon, the notch level is considered as the preloading for fatigue because it is consuming a number of cycle of the material that corresponds to the level of weakening of it. Therefore, the ratio between the thickness fluctuation, which is measuring the notch impact and the thickness, which represents the full strength of the HDPE pipe, can be considered as the life fraction for the material. In this paper, we evaluated the damage through a combined theory using the static damage of the unified theory [14–16] and burst pressure equations herein developed. Indeed, we exposed neat and notched HDPE pipes to an increasing internal pressure until rupture (Burst). Each notch level is corresponding to a life fraction calculated through the thickness fluctuation. This fluctuation is influencing the burst pressure of the pipes. Therefore, each (β) is corresponding to a burst pressure. For the neat pipe, this pressure is considered as the ultimate one (P u ), while they are considered as the ultimate residual pressures (P ur1 , Pur2 , …, P a ) for the others.    1 1 ur u s a u P P D P P (3) where, • P ur is the burst pressure for damaged pipes ; • P u is the burst pressure for a neat pipe ; • P a is the pressure just before rupture