engleski

Modelling of Cohesive Stresses in a Plastic Zone around the Crack Tips in a Strain-Hardening Material

A contribution to modelling of cohesive stresses in a yield zone around the crack tips is presented in this paper. We meet the same type of a problem in modelling of a crack in ductile materials. Plastic zones around the crack tips appear by loading of a solid body with an inherent crack if the body is built-up from a ductile material. Determination of the shape and magnitude of a plastic zone, determination of stresses, strains and displacements in a yield area and in its neighbourhood, as well as a crack opening are often stated as a problem. One of the first models by which it was possible to determine these parameters was the Dugdale strip yield model. Although it simplified the real physical picture of occurrences around a crack tip, it was very often used and even more often cited. This model describes a yield zone as a narrow band lying in the direction of a crack plane. Instead of a real physical blunt crack it introduces an imaginary sharp elastic crack. The original Dugdale strip yield model was formulated for a special case of biaxial loading of a plate with a crack. The model implied the application of the Tresca yield criterion and an elastic-perfectly plastic material model of a plate. It assumed constant cohesive stresses in the whole yielding zone whose magnitude is equal to the yield strength of a material, i.e. . In their previous investigations, the authors were very intensively engaged in the application of the Dugdale strip yield model to the investigation of the fracture mechanics parameters. The results of those investigations were published on the 11th and 12th European Conference on Fracture [1], [2] and in the journal [3] as well. In those investigations the authors had kept the assumption about constant cohesive stresses within the yield zone and the assumption about elastic-perfectly plastic model of a material as well. However, they built into the model not only Tresca's yield criterion but also the Mises'. Furthermore, the authors investigated, according to both yield criteria, how different combinations of biaxial loading of a plate with a crack (the load biaxiality factor k) influence the magnitude of the plastic zone around the crack tip and on the magnitude of the crack tip opening displacement CTOD, as well. In this paper we wish to define a model which will be closer to the real elastic-plastic state of a material within a yield zone. An assumption of variable cohesive stresses within a yield zone is introduced. However, the lack of knowledge about the distribution law of cohesive stresses is the biggest problem. We assumed the linear distribution of cohesive stresses σ Y(x) in the yielded zone, as shown in Fig. 1b). Also, it is possible to introduce an arbitrary non-linear cohesive stresses distribution in the yielded zone, according to second order parabola, for example, or a general polynomial function of arbitrary order, respectively. If we wish to obtain the law of distribution of cohesive stresses, it is necessary to carry out an iteration process. We have incorporated both yield criteria, Tresca and Mises, in the model. In this way we would like to get a model for describing plastic yielding around a crack tip in a isotropic hardening material (linear strain hardening and a non-linear strain hardening law). FIGURE 1. a) Plastic zone around the crack tip, b) linear distribution of cohesive stress in the yielded zone We applied the above described micromechanical model to the analysis of the fracture mechanics parameters in a thin infinite plate with a central, straight crack. The plate is biaxially loaded, while the crack surface is free from loading. The loading in the direction of the x axis (parallel to the crack surface) is dependent on the loading in the direction of the y axis (the direction perpendicular to the crack surface). This dependence is established through the load biaxiality factor k. Parameters such as the magnitude of a plastic zone around a crack tip , the crack tip opening displacement CTOD, the displacements of the points on a crack surface etc., are anallysed. A computation of the mentioned parameters is carried out by analytical methods, i.e. by the application of the theory of the complex variable functions. References 1. Pustaić, D. and Štok, B., Crack Tip Plasticity Investigations using Dugdale Strip Yield Model Approach, In Proceedings of the 11th European Conference on Fracture (ECF-11), Engineering Materials Advisory Services (EMAS), Poitiers-Futuroscope, 1996, Vol. I, p. 151-156. 2. Pustaić, D. and Štok, B., Some Critical Remarks on the Dugdale Strip Yield Model for the Crack Tip Plasticity, In Proceedings of the 12th European Conference on Fracture (ECF-12), edited by M. W. Brown, E.R. de los Rios and K. J. Miller, Engineering Materials Advisory Services (EMAS), Sheffield, 1998, Vol. II. p. 889-894, 3. Štok, B. and Pustaić, D., On the Influence of the Loads, Acting Parallel to the Crack Surface, on the Crack Tip Opening Displacement (CTOD) and on the Magnitude of the Plastic Zone around the Crack Tip, Strojarstvo, Journal for the Theory and Application in the Mechanical Engineering, vol. 38, 2/3, 73-87, 1996, [in Croatian].

modelling of; cohesive stresses; plastic zone; crack; crack tip; strain - hardening material; elastic-perfectly plastic material; constant cohesive stresses; variable cohesive stresses; magnitude of the plastic zone; crack tip opening displacement

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