PSI - Issue 68

Dragan Pustaić et al. / Procedia Structural Integrity 68 (2025) 16 – 23 Dragan Pustaić, Martina Lovrenić-Jugović / Structural Integrity Procedia 00 (2025) 000–000

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Keywords: Cohesive model; cohesive stress; isotropic and non-linear strain hardening of plate material; Ramberg-Osgood´s equation; strain hardening exponent; stress intensity factor; Green´s functions method; plastic zone magnitude around the crack tip; exact and approximate analytical solution; analytical methods.

1. Introduction The forming and spreading of the plastic zone around the crack tip in the thin, infinite plate is investigated in this paper. Two pairs of the concentrated forces, F , [N/m], act on a plate and they open a crack, according to Fig. 1a. The forces, F , act in a plane of a plate and gradually monotonously increase, and with their increasing, the plastic zone magnitude, r p , [m], also increases. It is assumed that the forces, F , are acting on a distance, c , [m], in relation to the axis, y , which is at the same time, the axis of symmetry. The distance, c , can vary in the limits 0 ≤ c ≤ a , so it can be investigated how the position of the forces on a crack surface influences on the plastic zone magnitude around its tip and on the crack tip opening displacement (CTOD). Nomenclature a half physical crack length, m longitudinal strain, - b half fictitious crack length, m strain corresponds to the yield stress according to Hooke´s law, - c distance of the forces, F , in relation to the axis of symmetry, y , m E Young´s modulus of elasticity, GPa h thickness of a plate, m material parameter, - r p plastic zone magnitude around crack tip n strain hardening exponent, - F the two pairs of the concentrated forces acting on the crack surface, N/m p (x) cohesive stress, MPa F /( σ 0 ∙ a ) non-dimensional loading of the crack, - x independent variable, m K ext stress intensity factor, (SIF), from the external loading, MPa m (x, b) Green´s function, K coh stress intensity factor from the cohesive stresses, MPa Gamma function, - yield stress, MPa Hypergeometric function, - σ y normal stress, MPa independent variable in which the plastic zone is incorporated, - It is assumed that the plate is made of ductile metal material which means that by increasing the external loading, the small plastic zones of the length, r p , will form around its tips, Fig. 1a and 1b. According to Dugdale´s idea, those small plasticized ranges can be added to the real, physical crack , which together make the fictitious elastic crack of the length 2 b = 2( a + r p ), according to Fig. 1b. As it is well-known, in case of linear elastic state of a plate material, the stress, σ y , [MPa], will be singular at the tip of a sharp physical crack. On the other hand, the stress, σ y , will not be singular, but it will obtain the finite value, at the tip of a fictitious elastic crack, i.e. for x = b = a + r p . That condition is equivalent to the condition that the resulting stress intensity factor at that point is equal to zero, i.e. K = K ext + K coh = 0, what was pointed out in many papers, for example, Chen et al. (1992), Guo (1993), Neimitz (2000), Pustaić et al. (2019., MSMF 9 and 2022., MSMF 10) and so on. The non-linear dependence between true stress and true strain in the hardening range can be well described by Ramberg-Osgood´s analytical expression which is possible to find in earlier mentioned articles. The variable cohesive stresses , according to Fig. 1.c, are acting on a part of a fictitious elastic crack, in the length r p , which corresponds to the plastic zone magnitude. They are non-linearly distributed. However, the law of distribution of those stresses isn´t known, at the beginning. So, the cohesive stress distribution can be determined, for ! ! ! ! ! ! " # ! ! ( ) ! ! ! ! ( ) ! " # # # ! " ! " # ( ! ! " ! " # " ! = # + $