PSI - Issue 16

Volodymyr Panasyuk / Procedia Structural Integrity 16 (2019) 3–10 Volodymyr Panasyuk / Structural Integrity Procedia 00 (2019) 000 – 000

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In the middle of the 20 th century, in view of the creation of new military equipment, in particular, famous American military transport ships “Liberty”, it turned out that these classical approaches are insufficient. Ships began to destroy in ports under their fluctuations in the sea wave (although during their construction everything was calculated according to classical methods). Sharp stress concentrators – cracks in some elements of the ship structure were the reason for their destruction. Such cracks grow during fluctuations and cause the ship destruction. The classical methods of structural strength evaluation do not take into account this phenomenon (Fig. 1 a ).

Fig. 1. Material fracture models: ( а ) classical; (b) non-classical; (c) non-classical at the crack tip; (d) Griffith concept.

This phenomenon has become an impulse for the increased attention of scientists in the field of materials and materials science to the cracked solids strength and fracture. Then they recalled the researches by A. A. Griffith done in the 20’s of the last century. He was the first to solve the problem of the boundary-equilibrium state of a cracked plate (Fig. 1d) and set the value of the fracture stress   p p . He used the energy balance of a cracked deformed body and costs for creating a unit of its new surface. The formula for the critical stress has the form   0 2 γ / π    p l , where Ε is Young’s modulus, γ is the surface energy , 2 l 0 is the length of initial crack (Fig. 1d). So, in the middle of the 20 th century, a new direction in the science on materials strength was formed, that is the well-known fracture mechanics and strength of materials. The basis of this science lies in the non-classical concept (Fig. 1b) of deformed solid fracture described by Panasyuk (1991, 2002). Such an approach and results by A. A. Griffith have become an impulse to solve the problems of elastic bodies with cracks. 2. Griffith – Irwin concept Griffith (1924) was first who took into account the I-states in a stress-strained solid near the crack tip and formulated the criterion (conditions) of crack growth. After him, an important achievement in the development of materials fracture mechanics was the establishment of the structure stress field and displacements near the crack-cut in the deformed elastic body, in particular, it was found that stress tensor components (Fig. 2) in the crack tip vicinity are represented by formula (1):   I0 I II0 2 III0 3 1 σ (θ) (θ) (θ) 0(1) 2 π        ij ij ij ij K f K f K f r (1) III0 III0 = ( , )  K K p l – stress intensity factors (SIF). These factors are the functions of body configuration, crack size ( l ) and level of loading p ; 0(1) – limited value if r  0; f kij (  ) – known functions ( k = 1, 2, 3). Taking into account formula (1) G. Irwin (1957) formulated new criterion equation of limiting-equilibrium state of the deformed cracked body in the form of equation (2) and showed that this formula is equivalent to Griffith formula   I0 I ,   c K p l K (2) where i, j = x, у, z – in Cartesian coordinate system, or i, j = r ,  , z – in polar (cylindrical); I0 I0 = ( , )  K K p l , II0 II0 = ( , )  K K p l ,