PSI - Issue 23

Dragan Pustaić et al. / Procedia Structural Integrity 23 (2019) 27 – 32 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

32

6

Analytical expression for the magnitude of plastic zone around crack tip in that case will take the form as in (12), Pustaić and Lovrenić - Jugović (2018) .





2

    1 2 1 n n                1 n n

   π 2



2

.

p r a

 



0   

(12)





2

  



    1 2 1 n n                 1 n n

2

1 π 2

 

0   

On the basis of analytical expression (12), the magnitude of plastic zone p r was calculated in dependence of monotonous increasing external loading   and for different values of strain hardening exponent n. The diagram is in non-dimensional form presented on the Fig. 4.

Conclusion

The stress intensity coefficient from the cohesive stresses coh ( ) K b was determined by means of the Green´s functions method, (4). After transformation of expression (4) into the one (5), the solution was found, by means of program package Wolfram Mathematica 7.0, (2017). The solution is analytical, fully exact. The analytical expression (7) which gives the dependence among the plastic zone magnitude p r and the plate loading   was derived from the condition (1). How it is impossible to express the dependence of p p ( , ), r r n    explicitly, the new independent variable P was introduced and the inverse form of the equation (7) was found in which the plate loading 0 t     was presented as a function of plastic zone magnitude p r and the parameter n, the equation (8) and the Fig. 2, i.e. as ( , ), t t P n  Pustaić and Lovrenić - Jugović (2018) . The values of Hypergeometric function 2 1 F are changed in a very narrow interval for 0.00 0.30, P   Fig. 3 and it can be taken, with the quite enough accuracy, that it is 2 1 1, F  Pustaić and Lovrenić - Jugović (2018) . In that case the expressions (7) and (8) are strongly simplified, but they are remained enough accurate. Generally, it can be concluded that the plastic zone magnitude p r will be as greater as the strain hardening of a material is smaller, for the same level of external loading. It means, greater p r for greater n. The magnitude of the plastic zone p r will be greatest at elastic-perfectly plastic material. Chen, X. G., Wu, X. R., Yan, M. G., 1992. Dugdale Model for Strain Hardening Materials. Engineering Fracture Mechanics 41 (6), 843 - 871. Guo, W., 1993. Elastoplastic Three Dimensional Crack Border Field - I. Singular structure of the field, Engring. Fract. Mechanics 46 (1), 93 - 104. Guo, W., 1995. Elasto - Plastic Three - Dimensional Crack Border Field - III. Fracture Parameters. Engineering Fracture Mechanics 51 (1), 51 - 71. Hoffman, M., Seeger, T., 1985. Dugdale Solutions for Strain Hardening Materials. The Crack Tip Opening Displacement in Elastic - Plastic Fracture Mechanics. In: Proceedings of the Workshop on the CTOD Methodology. Geesthacht, 57 - 77. Neimitz, A., 2000. Dugdale Model Modification due to the Geometry Induced Plastic Constraints. Engineering Fracture Mechanics 67, 251 - 261. Neimitz, A., 2004. Modification of Dugdale Model to Include the Work Hardening and in - and out - of - Plane Constraints. Engineering Fracture Mechanics 71 (11), 1585 - 1600. Neimitz, A., Graba, M., 2008. Analytical - Numerical Hybrid Method to Determine the Stress Field in Front of the Crack in 3D Elastic - Plastic Structural Element. In: Proceedings of the 17 th European Conference of Fracture (ECF 17), Brno, Czech Republic, 514 - 521. Pustaić, D., Lovrenić - Jugović, M., 2018. Mathematical Modeling of Cohesive Zone in the Ductile Metallic Materials. In: Proceedings of the 9 th International Congress of Croatian Society of Mechanics. Marović, P. et al. (Eds.). Croatian Society of Mechanics. USB, Split, Croatia. Pustaić, D., Lovrenić, M., 2006. Analytical and Numerical Investigation of Crack Opening in Strain - Hardening Material. In: Proceedings of the 5 th International Congress of Croatian Society of Mechanics. Matejiček, F. et al. (Eds.). Croatian Society of Mechanics. CD - ROM. Trogir, Croatia. Wolfram Mathematica, Version 7.0, 2017. Wolfram Research Inc., Champaign II, http://www.wolfram.com/products/mathematica/ References