engleski

Mathematical Approach of Crack Tip Plasticity

The thin infinite cracked plate, made of ductile metallic material, is observed. The plate is loaded in-plane with uniformly distributed continuous loading at infinity, in the direction perpendicular to the crack plane, while the crack surface is free of loading [1, 4-5]. The loading is monotonously increased. If the response of the plate on the external loading is pure elastic the stress singularity at the crack tip will be appeared. But, in the case of elastic-plastic response of a plate material, there will be no stress singularity at the crack tip [1, 3-5]. Our intention is to contribute in mathematical approaching the rather complex occurrences within the cohesive zones. Some parameters of EPFM, such as, the stress intensity coefficient and the magnitude of cohesive zone, are investigated [1]. An isotropic and non-linear strain hardening of a plate material is assumed. Non-linear strain hardening at uniaxial state of stress is described by the Ramberg-Osgood equation [1, 3-4]. The investigations were carried out for the several different values of the strain hardening exponent n, i.e. for: n = 3, 5, 7, 10, 25, 50 and 1000 [1, 3-4]. One well-known cohesive model was applied in the crack tip plasticity investigating [1, 3-5]. Also, it was assumed that the cohesive stresses within the plastic zone are changed according to non-linear law [1, 3-4]. The stress intensity coefficient from the cohesive stresses is calculated using the Green functions [1, 3-4]. In the frame of applied cohesive model, the problem is formulated and solved fully exact, analytically. Some complex integrals were solved by means of software package Wolfram Mathematica 7.0 [2]. The analytical expression which gives the dependence among the plastic zone magnitude and the plate loading was derived from the condition that the resulting K(b) has to be equal zero [1, 3-5]. How it is impossible to express the dependence of explicitly, the new independent variable P was introduced and the inverse form of the equation was found in which the plate loading, was presented as a function of plastic zone magnitude and the parameter n, i.e. as [1]. The solutions are presented through the special, Gamma and the Hypergeometric functions [1]. The values of Hypergeometric function are changed in a very narrow interval, for and it can be taken, with the quite enough accuracy, that it is [1]. In that case all expressions are strongly simplified, but they are remained enough accurate. References [1] Pustaić, D. and Lovrenić-Jugović, M., Proc. of the 9th Int. Congress of Croatian Society of Mechanics, ISSN 2623-6133, Marović, P., Krstulović-Opara, L. and Galić, M., eds., Split, Croatia, 2018. [2] Wolfram Mathematica 7.0, Champaign II, 2017, http://www.wolfram.com/products/mathematica/. [3] Chen, X. G., Wu, X. R. and Yan, M. G., Engineering Fracture Mechanics, vol. 41 (6), 1992, 843-871. [4] Pustaić, D. and Lovrenić, M., Proc. of the 5th Int. Congress of Croatian Society of Mechanics, Matejiček, F. et al., eds., CD-ROM, Trogir, Croatia, 2006. [5] Pustaić, D. and Štok, B., Proc. of the 12th European Conference on Fracture, Brown, M. W., de los Rios, E. R. and Miller, K. J., Sheffield, 1998, 889-894.

EPFM, cohesive stress, isotropic and non-linear strain hardening material, Ramberg-Osgood´s equation, strain hardening exponent, plastic zone magnitude around crack tip, stress intensity coefficient, analytical methods, method of Green functions, Wolfram Mathematica 7.0.

Cjeloviti rad (6 stranica) prošao je međunarodnu recenziju (peer-review) i bit će objavljen u časopisu Structural Integrity Procedia u 2020. godini, u formatu Open access. Časopis izdaje Elsevier.