Mathematical Modelling Stress Distribution in a Metallic Plate with an Asymmetrical Notch

The paper is focused on analyzing , , , arising in a steel plate with stress raisers when alternate bend fatigue. The research based on the finite-element method is carried out by software ANSYS. Stress distribution is studied in two planes.


Introduction
A sample stressed in a cantilevered manner was used for the purpose of research into high-and low-cycle fatigue of various structural steel classes [1][2][3][4]. Stress values around the notch-shaped stress raiser were calculated according to elasticity theory, methods of which are characterized by certain inaccuracy.
Numerous works have been addressed to the behavior of materials with various stress raisers. However, investigation into the behavior of materials with asymmetrical stress concentrators is a complex issue, which is to be solved according to the theory of strength of materials. A lot of software has been developed recently to solve mechanics problems of strength of materials, making the matters at issue more simplified.

Problem statement and analysis of results
This work is focused on mathematical modelling stress distribution in a metallic plate with an asymmetrical notch. For this purpose stress distribution in a steel plate with asymmetrical semi-cylindrical notch is analyzed. An assumption is made that a plate body is subject to alternate bending. The data were analyzed by software CȺȿ ANSYS on the base of finite-element method [5 -8].
A model of the sample was given in a two-dimensional view to simplify the calculation (Fig.1). The problem was solved on assumption of plate elasticity, with one its end fastened, while the other end is subject to the vertically acting force of 1000 N.
As the defects are accumulated most intensively around the stress raisers the values ıɯ, ıɭ, Ĳɯɭ were analyzed at distances ±10 mm from the stress raiser in the X-direction and in the minimal section of the sample in the Y-direction (Fig.1).
The results of calculation by the software ANSYS are given below, the finite element scheme is assigned according to notch shape and dimensions.
The values ı ɯ on the sample surface ( Fig.2ɚ) vary parabolically, moreover, the minimal value -1175 ɆPɚ is registered in the center of the stress raiser, rising to -200ɆPɚ symmetrically with respect to the distance from the centre. The values ı ɯ in the center are -390ɆPɚ at a distance of 1.5 mm (Figure 2b) from the surface. Therefore, the values increase on the whole parabola, it is not symmetrical though: for ɯ=-10mm ı ɯ =-225ɆPɚ, whereas at a distance of +10 mm ı ɯ =-100ɆPɚ. At a distance of 2.4 mm from the surface (Fig.2c) the values ı ɯ have an opposite sign. At the same time ı ɯ in the stress raiser center equals 200ɆPɚ, ı ɯ is -125ɆPɚ for ɯ=-10mm, and -10ɆPa for ɯ=+10mm, respectively. At a distance 3.1 mm ( Figure 2d) the values ı ɯ for ɯ=+10 mm and ɯ=-10 mm are quite the same. However, the increase up to ı ɯ =610ɆPɚ is registered in the stress raiser center. The stresses on the sample surface in the minimal section of the stress raiser ( Fig.2ɚ and 2e) have a quite similar modulus, but their nature becomes different: tension stresses transform into compression ones. Focusing on the varying values ı ɭ an emphasis is to be laid on proximity of the plot given in Fig.2 a, since the values ı ɭ are zero on the surface due to the dispersion of values when calculating ɋȺȿ ANSYS.
Analyzing the distribution ı ɭ at a distance of 1.5 mm from the surface (Fig.3b) a parabolic distribution is detected with the maximum in the center (ı ɭ =38ɆPɚ) and minimums at the extremities of intervals under consideration (ı ɭ =-10ɆPɚ). The distribution ı ɭ is nearly the same at a distance of 2.4 mm from the surface (Fig.3c), the maximum increases up to 60 ɆPɚ and the minimum decreases to -20ɆPɚ. It is similar for the distance of 3.1 mm: -73ɆPɚ and 28ɆPɚ, respectively (Fig.3d).
Strong changes are detected at a distance of 4 mm (Fig.3e). Two maximums are apparent around the stress raiser at distances of ɯ=-3 mm and ɯ=3 mm, equaling 80ɆPɚ and 42ɆPɚ, respectively. In the center of a stress raiser there is a minimum with the value 21ɆPɚ. A probable cause of it is the surficial position of the curve to be analyzed in this area. The distribution of tangential stresses Ĳ ɯɭ seems to be more interesting (Fig.4). The values Ĳ ɯɭ approximate to zero on the surface at the distances ɯ=-10 mm, in the center and for ɯ=10 mm (Fig.4ɚ). The minimum is detected at a distance ɯ=-3 mm (Ĳ ɯɭ =-20ɆPɚ), and the maximum 18ɆPɚ is located symmetrically with the center at a distance ɯ=4 mm. At a distance of 1.5 mm from the surface (Figure 4 b) the distribution Ĳ ɯɭ differs in numerical values only. In the stress raiser center the values are in the range of zero, being 3ɆPɚ. However, at a distance of 2.4 mm deep into the surface (Fig.4c) the plot is displaced Ĳ ɯɭ (ɏ) because the value in the concentrator center Ĳ ɯɭ =50ɆPɚ, and the Ĳ ɯɭ plot -X-axis cross point is displaced too. Positions and values of the maximums and minimums are not displaced significantly (Fig.4c). The second highly pronounced maximum 50ɆPɚ is registered for ɯ=-1 mm at a distance 3.1 mm from the surface (Fig.3d). The maxi-mum, which was seen previously in Fig.4c, is displaced to ɯ=7 mm (Ĳ ɯɭ =78ɆPɚ). At a distance of 2 mm from the center is the minimum (Ĳ ɯɭ =18ɆPɚ) registered.
At a distance 4 mm from the surface (Fig.3e) specifics of tangential stress distribution is similar to that registered when considering ı ɯ and ı ɭ . In this case negative area of the plot is almost mirrored relative to the axis X (Fig.3c), but the maximum on the plot is 150 MPa, corresponding to ɏ=-2mm, whereas the minimum is -95ɆPɚ for ɯ=+2 mm.
This paper provides consideration of stress distribution over the minimal section of the sample (Fig.4), since there the maximal stresses are registered.
The ı ɯ versus Y curve is a linear one (Fig.5ɚ) and expressed by the interpolation equation ı ɯ =592Y-1188, where Y is given in millimeters, and ı ɯ in MPa. The ı ɭ and ı ɯɭ versus Y dependences ( Fig.5b and 5c) are of the same type with maximums in the range 2.5-3 mm. For instance, the maximal value ı ɭ equals 60ɆPɚ and Ĳ ɯɭ is 49ɆPɚ, respectively.

Conclusions
To sum up, a computer model of stress distribution in a sample with asymmetrical notch makes it obvious that notch displacement is the cause of inflection points and even double peaks on stress distribution curves. Therefore, various rupture points are possible when cyclic stressing.