Introduction
Structural materials and components that are used within real life applications often incorporate changes within the geometry of the material through the material being exposed to loading. These loadings have an effect on the stress concentration points within the material. Many modern engineering products are incorporated with complex designs, which means that the stress concentration is unavoidable and therefore it is crucially important that some knowledge is gained of the stress concentrations so that the mechanical design can be a success. The analysis that was carried out was to evalutate the Stress Concentration Factor (SCF) for mode-1 loading for a crack emanating from a crack within a plate. Finite Element Analysis software, such as ANSYS, has the capability to carry out this analysis.
Methodology
The geometry of the plate that was analysed is shown below in figure 1. To determine the behaviour of the crack, the dimensions shown in table 1 below were utilised with the geometry shown.
Figure 1: Geomety of model
Table 1: relative dimensions of plate
Before the analysis was carried out, it was decided that it would be best to gain some theoretical results. The normalised stress intensity graphs which corresponded to the analysis being carried out were utilised. The results gained could then be compared to the ANSYS result gained. The theoretical value was calculated by reading the point from the graph in figure 2 below. The point of interest, where the KI value was obtained from is shown by a red box with lines, representing the value obtained.
Figure 2 - Normalised Stress intensity factors table
The interpolated point was found to lie at a value of K’1 =0.42 Pam1/2.
The finite element that was carried out was a simple two-dimensional plate and therefore it was appropriate to use PLANE 82 as the element type. The PLANE82 is eight node quadrilateral elements and were sufficient for the analysis being conducted. Furthermore, for the purpose of this analysis, linear elastic behaviour was specified.
While choosing PLANE 82, the plain strain model was chosen within this analysis. The model was therefore assumed to behave like a thick specimen. This assumption was made because it was expected that a constant stress contour pattern would be produced within the vicinity of the crack tip.
A large number of elements were utilised because the model was not meshing when a lower number of elements was inputted. A more concentrated mesh was made at the crack and this was done because this area was of most interest for the analysis being carried out. The mid plane element nodes were skewed to ¼ point of the radius length. This method was adopted as this allowed a higher degree of accuracy in the crack tip region because more nodes were situated together.
Next, the loads and the constraints could be applied. The right side of the plate, the middle node was constrained within the y-direction. Thereafter a direct tensile stress of 25MPa was applied, acting in the y-directions on the top and bottom surface of the plate. This load represented a tensile load pulling the plate ends at 25MPa.
The model was now solved; a small displacement static method was automatically used by the ANSYS software. Furthermore, after the solving had been completed, a path was created and local coordinate system was utilised level with the crack. This was established by choosing three nodes near the tip at the crack. The stress intensity factor that corresponded to the nodal path was thereafter calculated.
Results
A displacement plot for the model under the load applied is shown in figure 3 below.
Figure 3: Displacement plot of model
The maximum displacement of the model while being subjected to a pressure of 25MPa was found to be 0.214e-4 m. The displacement behaviour of the model was very realistic as the increase within the crack tip showed this. Furthermore, an increase within the length and a decrease within the width can be clearly seen, which is expected within linear strain behaviour.
A plot of the Von Mises stress distribution of the crack is shown below in figure 3.
Figure 3: Von Mises stress of crack
It is clearly evident from figure 3, that the peak stress occurs at the tip of the crack, which is found to be 0.0986 GPa. This value is far below the yield strength of steel.
Definition of Crack Path
To gain a stress intensity value from ANSYS, it was required to define specific nodes along the crack edges. The main reason why this was done was to determine results by setting up a local coordinate system which would determine the crack propagation at the crack tip. All results were then applied to this coordinate system, from which the necessary values for evaluation of the model could be determined.
The stress intensity (KI) which was calculated by ANSYS was found to be 0.22194x107. the stress intensity factor was calculated using equation 1below.
……eqn(1).
By using equation 1 above, the stress intensity factor (KI’) was found to be 0.42 Pam1/2.
Comparison of results
The results that were gained from other students were plotted on a graph so that a better comparison could be made and so that a relationship could be found. The graph plotted is show below in figure 4.
Figure 4: Comparison of results
It can be seen from figure 4 that as the crack angle increases, the stress intensity factor decreases. Furthermore, as the l/h value increased, the stress intensity factor also increased.
Conclusion
To gain further insight into fracture of plates, a finite element analysis of a cracked plate was conducted so the stress intensity present at the crack could also be established. The results gained were thereafter compared to theoretical values by using hand calculations. The stress intensity (KI) was found to be 0.42 MPam1/2 using the finite element analysis and was theoretically calculated to be 0.41MPa1/2. The difference between the answers was found to be caused through some assumptions made within the model and also through the quality of mesh utilised within the modelling process.
By gaining results for different specifications for the plate, a comparison was also done so that a relationship could be found. The results clearly showed that with an increase in crack angle, there would be an decrease within the stress intensity factor gained. Furthermore, it was found that by increasing the ratio of l/h, there was an increase of the stress intensity factor gained.
A comparison of the results obtained in several analyses conducted by students was also undertaken. Results showed that increasing crack angle θ, decreased the stress intensity factor KI, whilst using a larger value of l/h increased the stress intensity factor.
