Modeling of Residual Stress

Finite Element Analysis represents a numerical technique for finding approximate solutions to partial differential equations as well as integral equations, permitting the numerical analysis of complex structures based on their material properties. This book presents 20 different chapters in the application of Finite Elements, ranging from Biomedical Engineering to Manufacturing Industry and Industrial Developments. It has been written at a level suitable for use in a graduate course on applications of finite element modelling and analysis (mechanical, civil and biomedical engineering studies, for instance), without excluding its use by researchers or professional engineers interested in the field, seeking to gain a deeper understanding concerning Finite Element Analysis.


Introduction
In many industries, nickel-base alloys represent an important segment of structural materials. Critical components made of these alloys are relied upon to function satisfactorily in corrosive services. The demand for safe, reliable and cost-effective performance requires that these nickel-base alloys provide the anticipated corrosion resistance. Corrosion-resistant high alloy castings are often the subject of major concern because failures of cast components have led to significant downtime costs and operating problems [1]. Over the years, the nickel-chromium-molybdenum / tungsten alloys have proven to be among the most reliable and cost effective materials for aggressive seawater application and excellent resistance to localized corrosive attack (pitting, crevice corrosion).
Among these alloys, Hestelloy C-types (C, C-4, C-276, and C-22) are used to serve the above mentioned purposes. As these alloys are commonly subject to further machining after casting, it becomes very vital to have an idea about the change in properties imparted to the machined surfaces after such cutting operations as end milling. For this reason, finite element methodology is used in this study to determine the machined surface stress characteristics.
In the past decade, finite element method based on the updated-Lagrangian formulation has been developed to analyze metal cutting processes [1][2][3][4][5][6][7]. Several special finite element techniques, such as the element separation [1][2][3][4][5][6][7], modeling of worn cutting tool geometry [1,2,4,5,6], mesh rezoning [3,5], friction modeling [1][2][3][4][5][6][7], etc. have been implemented to improve the accuracy and efficiency of the finite element modeling. Detailed work-material modeling, which includes the coupling of temperature, strain-rate, and strain hardening effects, has been applied to model material deformation [3,5,6]. An early analytical model for predict ing resi dual stre s ses was prop osed by Okushima and Kakino [8], in wh ich residual stresses were related to the cutting force and temperature distribution during machining. In another analytical model [9] a connection was made between residual stresses and the hardness of the workpiece. Shih and Yang [10] conducted a combined experimental/computational study of the distribution of residual stresses in a machined workpiece. More recently, Liu and Guo [11] used the finite element method to evaluate residual stresses in a workpiece. They also observed that the magnitude of residual stress reduces when a second cut is made on the cut surface. Liu and Barash [12] measured the www.intechopen.com Finite Element Analysis -From Biomedical Applications to Industrial Developments 370 residual stress on the workpiece subsurface with consideration of tool flank wear. Their findings indicated that under the condition of a lower cutting speed, the mechanical load had a greater impact on residual stress, while the thermal effect become the major factor effecting residual stress at higher cutting speed. Lee and Shaffer [13] proposed a shear-angle model based on the slip-line field theory, which assumes a rigid-perfectly plastic material behavior and a straight shear plane. Kudo [14] modified the slip-line model by introducing a curved shear plane to account for the controlled contact between the curved chip and straight tool face. Henriksen [15] conducted a series of tests to understand residual stresses in the machined surface of steel and cast iron parts under various cutting conditions. Kono et al. [16] and Tonsoff et al. [17] revealed that residual stresses are dependent on the cutting speed. Matsumoto et al. [18] and Wu and Matsumoto [19] observed that the hardness of the workpiece material has a significant influence on the residual stress field. Konig et al. [20] showed that friction in metal cutting also contributes to the formation of residual stresses.

Finite element model
The finite element model is composed of a deformable workpiece and a rigid tool. The tool penetrates through the workpiece at a constant speed and constant feed rate. The model assumes plane-strain condition since generally depth of cut is much greater than feed rate. The finite model used in this study is based on the commercial finite element software. The software, called "Thirdwave AdvantEdge" uses six-noded quadratic triangular elements by default.

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AdvantEdge is an automated program and it is enough to input process parameters to make a two-dimensional simulation of orthogonal cutting operation. The boundary conditions are hidden to the user. Figure 1 shows the Thirdwave AdvantEdge model for milling operation and Figure 2 shows an example of visual simulation of residual stresses induced after milling.

Workpiece and tool material modeling
The workpiece material used for simulation is HASTELLOY C-22HS and the cutting tool is carbide coated with TiALN and 20 o rake angle. Every one pass (80mm), the simulation was stopped. AdvantEdge uses an analytical formulation for material modeling. In a typical machining event, in the primary and secondary shear zones very high strain rates are achieved, while the remainder of the workpiece deforms at moderate or low strain rates. In order to account for this, Thirdwave AdvantEdge incorporates a stepwise variation of the rate sensitivity exponent: where T is the current temperature, 0  is the initial yield stress at the reference temperature T 0 , 0 p  is the reference plastic strain, n is the hardening exponent and   T  is the thermal softening factor. In the present study, it is assumed that the tool is not plastifying. Hence, it is considered as an absolutely rigid body.

Results and discussion
Finite Elements simulations were carried out according Table 2. In all simulations, it is made sure that steady-state has been reached and some more data are collected after that time. Therefore, all the results presented in this work were gathered under steady-state condition. From the simulations, variables like stresses, strains, strain rates and temperatures distribution can be obtained. However, all these are very difficult to measure experimentally.  The yield strength and ultimate tensile strength for the coated carbide cutting tool used in this simulation are 600 MPa and 800 MPa respectively. Then Von Mises stress at at region 9 is 4488 MPa, which is higher than the yield strength and ultimate tensile strength of the coated cutting tool. This strees can cause permanent damage to the cutting tool since this stress is beyond the ultimate tensile strength and yield strength. Cutting speed, feedrate, and axial depth for this simulation is very high and this cause the high stress at the cutting edge, since high cutting speed, feedrate, and axial depth can cause high force in milling [22,23]. The radial depth for every simulation is 3.5 mm. This factor also contributes to higher stress. At region 1, at the cutting tool and chip contact, the Von mises stress is 501 MPa, where the yield strength and ultimate strength of the workpiece are 359 MPa and 759 MPa. The workpiece start to deform since the stress is above its yield strength. Figure 4 shows Von Mises stress for simulation no. 3 (Cutting speed 100 m/min, feedrate 0.2 mm/rev, axial depth 1.5 mm). The stress at cutting tool edge (region 5) is 1345 MPa. The Von mises stress is lower compared to that in simulation run no. 9. Even though the stress still higher than yield strength and ultimate tensile strength, but the damage should be not severe compared to that of simulation no.9. At region 3, the stress for the contact point cutting tool and chip is 577 MPa. This value is almost the same as in simulation no.9. show the stress is lower at under the cut surface and increases gradually near the cutting edge. High force is needed at the tool edge for workpiece penetration, and this is indirectly increase the stress at the tool edge. This distribution of the stress is same for both cases. The velocity vectors for simulation no.9 as shown in Figure 5 around the tool tip, clearly show the plastic flow of the material araound the cutting edge. The same trend of flow also was observed by M.S.Gadala et. al. [24]. Figure 6 shows the 3D picture for Von misses stress distribution for simulation no.9.
d increases gradually near the cutting edge. High force is needed at the tool edge for workpiece penetration, and this is indirectly increase the stress at the tool edge. This distribution of the stress is same for both cases. The velocity vectors for simulation no.9 as shown in Figure 5 around the tool tip, clearly show the plastic flow of the material araound the cutting edge. The same trend of flow also was observed by M.S.Gadala et. al. [24]. Figure 6 shows the 3D picture for Von misses stress distribution for simulation no.9. Table 2 show the average value Von mises stress at cutting tool edge for every simulation that already run. This value will be investigate through statistical method to find the relationship between variables (cutting speed, feedrate and axial depth) with response (Von mises Stress).

Conclusion
In the milling operation, cutting speed, feedrate and axial depth play the major role in producing high stresses. The Von Mises stress distribution also show the stress is lower at under the cut surface and increase gradually when come at cutting edge. The highest compressive xx  appear at the cutting edge. Most of the tensile V  appear at the cutting tool edge. The stress distribution also show the stress is lower at under the cut surface and increases gradually near the cutting edge. High force is needed at the tool edge for workpiece penetration, and this is indirectly increase the stress at the tool edge. This distribution of the stress is same for both cases. From the first order model, one can easily notice that the response y (Von Mises stress) is affected significantly by the feed rate followed by axial depth of cut and then by cutting speed. Generally, the increase in feed rate, axial depth and cutting speed will cause Von Mises stress to become larger. The increase in feed rate, axial depth and decrease in cutting speed will cause residual stress to become larger .Response surface method is very useful since with few simulations, a lot of information can be derived such as the relationship between the variables (cutting speed, feedrate and axial depth) with response (Von Mises stress and Residual stress). The combination of numerical analysis and statistical method are very useful to analysis the distribution of stresses in milling.

Acknowledgement
The financial support by Malaysian Government through MOSTI and University Tenaga Nasional is grateful acknowledged. Finite Element Analysis represents a numerical technique for finding approximate solutions to partial differential equations as well as integral equations, permitting the numerical analysis of complex structures based on their material properties. This book presents 20 different chapters in the application of Finite Elements, ranging from Biomedical Engineering to Manufacturing Industry and Industrial Developments. It has been written at a level suitable for use in a graduate course on applications of finite element modelling and analysis (mechanical, civil and biomedical engineering studies, for instance), without excluding its use by researchers or professional engineers interested in the field, seeking to gain a deeper understanding concerning Finite Element Analysis.

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