YANMAR Technical Review

Abstract

Quenching is a hardening technique for heat treating materials to improve strength and wear resistance. It is widely used for engine and other parts subject to heavy loads. However, as quenching is an extremely complex process involving the interaction of phase transformations and deformation, elucidating how it works is not easy. As the deformations and residual stress left after quenching have the potential to adversely affect part dimensions or strength, recent years have seen considerable work going into using simulation to study the quenching process.
In this instance, Yanmar has devised a new mathematical model for the elasto-plastic behavior of quenching that offers higher accuracy than the model used in the past. The model was implemented using standard finite element analysis software and provides a technique for the reliable and highly accurate mathematical analysis of the quenching process. The accuracy of the technique was evaluated by using it to analyze the induction hardening of crankshafts, confirming that it is suitable for use in assessing actual parts.

1. Introduction

Quenching is a means of heat treating materials to improve their strength and wear resistance. As shown in Fig. 1, it toughens the microstructure of a metal by first raising it to a prescribed temperature and then rapidly cooling it. Quenching is used for the parts of Yanmar products that require high durability. Unfortunately, as the rapid cooling associated with quenching causes thermal strain and phase transformations, it also has the potential to deform parts or to weaken them in unexpected parts. Understanding what is going on during quenching is vital to ensuring product quality and this makes quenching analysis a very useful tool.

Use of quenching analysis offers the following benefits.

• Calculation of optimal quenching conditions. For example, avoiding excessive heat treatment can provide considerable energy savings and lower CO₂ emissions, making the production process more environmentally friendly.
• By reviewing quenching conditions beforehand to reduce the amount of trial and error when determining conditions, the product development time can be shortened and products brought to market more quickly.
• Whereas measurement of internal stress is extremely difficult, the ability of analysis to visualize the stress inside parts can make the quality of heat-treated parts more reliable.

Referencing a previously proposed mathematical model for quenching¹⁾, the development project sought to realize these benefits through more precise calculations that would improve the speed and accuracy of analysis. This new model was then used to analyze the induction hardening of diesel engine crankshafts and its accuracy verified by comparing the results with actual stress measurements.

2. Formularization of Elasto-Plastic Analysis for Quenching

2.1. Constitutive Equation

The following describes the constitutive equation for an analytical solution that models the quenching process. The strain calculation expresses the total strain as the sum of the elastic strain , thermal strain , transformation strain , plastic strain , and transformation plastic strain . The formulas for each of these strain components are given in equations (1) to (5). Here, is the coefficient of linear expansion, is the coefficient of transformation expansion, is the proportion of the emergent phase in the microstructure, is the second-order unit tensor, is the cumulative equivalent plastic strain, is the unit normal tensor pointing out of the yield surface, is the transformation plasticity coefficient, and is the deviatoric stress tensor.

To replicate the smooth material behavior, a compound hardening model that combines isotropic hardening with the Chaboche kinematic hardening model²⁾ was adopted in the hardening rule used to characterize the plastic behavior of the metal. Equation (6) is the evolution equation for the kinematic hardening model, where is the divided back stress and and are material parameters for kinematic hardening.

While the direct solution of evolution equations like those in equations (1) to (6) is extremely difficult, analysis typically involves discretization with the equations then being solved by approximation. A variety of different discretization methods are available. In this work, a more exact discretization than has been used in the past was adopted for equation (6) in particular in an effort to make the overall analysis faster and more accurate. Omitting intermediate steps, equation (7) was derived by the discretization of equation (6). Here, the physical quantities for steps n and n+1 are indicated by the and subscripts.

2.2. Return Mapping and Calculation of Consistent Tangent Modulus

Return mapping³⁾ is widely used to calculate stress from strain in static analysis. This works by assigning an initial trial value to the strain and then calculating the stress by an iterative calculation. The Mises yield criterion widely used for metals is used to determine whether or not yield has occurred. If so, the stress and strain are updated by solving equation (8) for the given trial strain^{4), 5)}.

Once the stress and strain have been obtained, the next step is to obtain the equilibrium of internal and external forces for the entire model. Here, the speed of the calculation can be improved provided that the consistent tangent modulus obtained by differentiating the stress tensor with respect to the strain tensor is calculated accurately, in which case the calculation should converge at a reasonably fast rate. Omitting the derivation, this is done by calculating the respective derivative terms in equation (9) below^{4), 5)}.

3. Analysis Results

3.1. Verification of Calculation Speed and Accuracy

The benefits of the new mathematical model derived above were verified by comparing it against the previous model^{5), 6)}. To assess computation speed, a simple quenching analysis was performed in which a model of a cube that is physically constrained in the Z direction was quenched from 900°C to 20°C. In Fig. 2, the vertical axis represents the “residual force” after each step and the horizontal axis represents the number of iterations. Here, “residual force” means the difference from the obtained solution. That is, the smaller the residual force, the more the calculation has converged. The improved analysis speed is demonstrated by how the residual force for the new model takes fewer iterations than its predecessor to reduce in size.

The calculation accuracy was also verified by assessing the Z-axis stress for the same analysis^{5), 6)}. As the relationship between load and displacement is non-linear when considering elasto-plastic behavior, the analysis is divided into an arbitrary number of calculation steps. While increasing the number of steps tends to deliver a more accurate result, it also lengthens the computation time. Conversely, reducing the number of steps shortens the computation time at the expense of accuracy. In other words, to perform accurate calculations in a short time, it is important to establish a method that can obtain an accurate result with a small number of steps.
Fig. 3 shows the results of three different calculations performed using the old and new methods respectively in which only the number of steps were varied. Whereas the calculation result obtained by the old method gave different results depending on the computation speed, the new method obtained the same result regardless of speed. This indicates that it can perform fast and highly accurate calculations.

3.2. Use in Analysis of Induction Hardening of Crankshafts

The new analysis method was implemented on a standard finite element analysis software package and used to analyze the induction hardening of crankshafts^{5), 6)}. As crankshafts are important engine parts that require high durability, they are heat treated (quenched) using induction heating.
Fig. 4 shows stress calculation results for a quenched crankshaft. Red regions indicate tensile stress and blue regions indicate compressive stress. As internal stress is particularly difficult to measure, the ability to evaluate what is happening analytically like this is very useful.

Fig. 5 shows the results of a test in which the residual stress on the surface of a quenched crankshaft were calculated for a number of points and compared with measured values⁶⁾. While accurate measurements of residual stress are typically difficult to acquire, the measurements here were obtained after first investigating which method of doing so would deliver the most accurate results. The comparison demonstrated that the calculation accuracy was sufficient for practical use, with results that were the same in terms of their overall trend and with reasonably good agreement in terms of the differences in absolute values considering the measurement error.

The technique was also used to assess the stress in quenched parts in actual production. Given the sort of loads these parts will be subjected to when used in engines, this ability of the technique to provide highly accurate predictions of how parts will perform under actual conditions can help to supply customers with highly reliable parts. By using the technique to determine the best quenching conditions, it should also be able to help with reducing greenhouse gases by minimizing electric power consumption and CO₂ emissions. While the technique is currently entering use in applications such as improving and selecting heat treatment conditions, primarily for industrial engines and large marine engines, the hope is that its use can be expanded more widely.

4. Conclusions

The work described in this article has succeeded in formularizing a highly accurate quenching analysis technique and has demonstrated its utility by verifying its performance in terms of computation accuracy and speed. Similarly, a comparison of measured and calculated values of stress when the computation model was used to analyze the induction hardening of crankshafts demonstrated that it could perform at a level sufficient for use in product evaluation. In the future, Yanmar intends to use the technique to review and improve quenching conditions. The goal is to extend its use to other products and to expand its scope of use.

References

• 1)Y. Kawaragi, M. Fukumoto, & K. Okamura, “Effect of Implicit Integration Scheme in Residual Stress Analysis of Quenching Considering Transformation Plasticity and Kinematic Hardening,” Journal of the Society of Materials Science, Japan, Vol. 64 (4), pp. 258-265 (2015).
• 2)Chaboche, J. L., “Time-independent constitutive theories for cyclic plasticity,” International Journal of Plasticity, Vol. 2, No. 2, pp. 149-188 (1986).
• 3)Simo, J. C. and Ortiz, M., “A unified approach to finite deformation elastoplastic analysis based on the use of hyperelastic constitutive equations,” Computational Methods in Applied Mechanics and Engineering, Vol.49, pp.221-245 (1985).
• 4)T. Ogawa, K. Yoshida, “Return Mapping Formularization of Elastoplastic Analysis of Quenching and Derivation of Consistent Tangent Elasticity,” Proceedings of the JSME Materials & Mechanics Conference 2023 (2023) in Japanese.
• 5)T. Ogawa, K. Yoshida, M. Oka “Development of Faster and More Accurate Model of Elasto-Plastic Behavior in Quenching and its Application to of Induction Hardening of Crankshafts,” Proceedings of JSMS 73rd Academic Conference (2024) in Japanese.
• 6)T. Ogawa, K. Yoshida, M. Oka “Development of Faster Analysis of Elasto-Plastic Behavior in Quenching and its Application to of Induction Hardening of Crankshafts,” Document Prepared for Inaugural Meeting of Plasticity Engineering Committee in 73rd Year of JSMS (2024) in Japanese.