PSI - Issue 2_B

Pasquale Gallo et al. / Procedia Structural Integrity 2 (2016) 809–816 P. Gallo et al. / Structural Integrity Procedia 00 (2016) 000–000 5 The stress σ y (r p ) is considered to be constant inside the plastic zone, that means elastic-perfectly plastic behavior is assumed. The lower integration limit is ݎ ଴ , that depends on the opening angle and notch tip radius. Due to the plastic yielding at the notch tip, the force F 1 cannot be carried through by the material in the plastic zone r p . But in order to satisfy the equilibrium conditions of the notched body, the force F 1 has to be carried through by the material beyond the plastic zone r p . As a result, stress redistribution occurs, increasing the plastic zone r p by an increment ∆ r p . If the plastic zone is small in comparison to the surrounding elastic stress field, the redistribution is not significant, and it can be interpreted as a shift of the elastic field over the distance ∆ r p away from the notch tip. Therefore the force F 1 is mainly carried through the material over the distance ∆ r p , and therefore the force F 2 (represented by the area depicted in the Fig. 1-b) must be equal to F 1 . For this reasons, ܨ ଵ ൌ ܨ ଶ ൌ ߪ ఏ ሺ ݎ ௣ ሻ ή ȟ ݎ ௣ , and the plastic zone increment can be expressed as the ratio between ܨ ଵ and σ θ evaluated (through Lazzarin-Tovo equations) at a distance equal to the previously calculated ݎ ௣ : ο ݎ ௣ ൌ ܨ ଵ ߪ ఏ ൫ ݎ ௣ ൯ (6) Substituting in Eq. (6) the formula given by Eq. (5) for F 1 and the explicit form of σ θ , the expression for the evaluation of Δ r p is obtained: ο ݎ ௣ ൌ ൞൬ ݎ ௣ ݎ ଴ ൰ ଵିఒ భ ൦൫ ݎ ଴ െ ݎ ௣ ൯ ൬ ݎ ௣ ݎ ଴ ൰ ఒ భ ିଵ ቈሺ ߣ ଵ ൅ ͳሻ ൅ ߯ ଵ ሺͳ െ ߣ ଵ ሻ ቈͳ െ ൬ ݎ ௣ ݎ ଴ ൰ ఓ భ ିఒ భ ቉ ൅ ሺ͵ െ ߣ ଵ ሻ ൬ ݎ ௣ ݎ ଴ ൰ ఓ భ ିఒ భ ቉ െ ሾሺ ߣ ଵ ൅ ͳሻ ൅ ߯ ଵ ሺͳ െ ߣ ଵ ሻሿ ൤ ݎ ଴ െ ݎ ௣ ቀ ݎ ௣ ݎ ଴ ቁ ఒ భ ିଵ ൨ ߣ ଵ ൅ ሾ߯ ଵ ሺͳ െ ߣ ଵ ሻ െ ሺ͵ െ ߣ ଵ ሻሿ ൤ ݎ ଴ െ ݎ ௣ ቀ ݎ ௣ ݎ ଴ ቁ ఓ భ ିଵ ൨ ߤ ଵ ൪ ൢ Ȁ ቊሺ ߣ ଵ ൅ ͳሻ ൅ ߯ ଵ ሺͳ െ ߣ ଵ ሻ ቈͳ െ ൬ ݎ ௣ ݎ ଴ ൰ ఓ భ ିఒ భ ቉ ൅ ሺ͵ െ ߣ ଵ ሻ ൬ ݎ ௣ ݎ ଴ ൰ ఓ భ ିఒ భ ቋ (7) The last step consists in the definition of the plastic zone correction factor C p , which is according to Glinka (1985) but introducing the Lazzarin-Tovo equations: ܥ ௣ ൌ ͳ ൅ ୼௥ ೛ ௥ ೛ ൌ ͳ ൅ ቐቀ ௥ ೛ ௥ బ ቁ ଵିఒ భ ቎൫ ݎ ଴ െ ݎ ௣ ൯ ቀ ௥ ೛ ௥ బ ቁ ఒ భ ିଵ ൤ሺ ߣ ଵ ൅ ͳሻ ൅ ߯ ଵ ሺͳ െ ߣ ଵ ሻ ൤ͳ െ ቀ ௥ ೛ ௥ బ ቁ ఓ భ ିఒ భ ൨ ൅ ሺ͵ െ ߣ ଵ ሻ ቀ ௥ ೛ ௥ బ ቁ ఓ భ ିఒ భ ൨ െ ሾሺఒ భ ାଵሻାఞ భ ሺଵିఒ భ ሻሿቈ௥ బ ି௥ ೛ ቀ ೝ ೝ ೛ బ ቁ ഊ భ షభ ቉ ఒ భ ൅ ሾఞ భ ሺଵିఒ భ ሻିሺଷିఒ భ ሻሿ൤௥ బ ି௥ ೛ ቀ ೝ ೝ ೛ బ ቁ ഋ భ షభ ൨ ఓ భ ቏ቑ Ȁ ቊ ݎ ௣ ቈሺ ߣ ଵ ൅ ͳሻ ൅ ߯ ଵ ሺͳ െ ߣ ଵ ሻ ൤ͳ െ ቀ ௥ ೛ ௥ బ ቁ ఓ భ ିఒ భ ൅ ሺ͵ െ ߣ ଵ ሻ ቀ ௥ ೛ ௥ బ ቁ ఓ భ ିఒ భ ൨቉ቋ (8) At this point, the general stepwise procedure to be followed to generate a solution is identical to that proposed by Nuñez and Glinka (2004): 1. Determine the notch tip stress, ߪ ଶ௘ ଶ , and strain, ߝ ଶ௘ ଶ , using the linear-elastic analysis. 813