Heat treatment quenching technology 
 

Reduction and Elimination of Quench Distortion by CRB Method

Zhang Ke-Jian (Beijing HuaLi Fine Chemical Company Ltd. Beijing 102200, China)


Abstract: A new practical method and its explanations about quench distortion and its prevention were reported in this paper. A curve which relates hardness and cooling rate (HCRC) was derived from the end quench (Jominy test ) curve. The HCRC was divided into four parts£ºI rapid cooling rate zone, ¢òsuitable cooling rate zone, ¢ó insufficient cooling rate zone, ¢ô too slow cooling rate zone, which characterize the different quench effect. A cooling rate band (CRB) of the distortion participating region (DPR) in the workpiece was defined by hardness measurement. And the location of the CRB relative to the different parts of HCRC was decided according to the quench distortion and the measured hardness of the workpiece. Quench distortion can be reduced only if the location of CRB of the workpiece is controlled totally within the range of the ¢ò cooling rate zone. The role and effect of the heat treating technical parameters, of chemical compositions of the steels and of other conventional methods and measures to reduce quench distortion on the adjustment of CRB and of the position of the HCRC were discussed. How to select and utilize these factors to make the CRB totally entering the ¢ò cooling rate zone so as to solve the quench distortion problems were illustrated.

Keywords: quench distortion; heat treating parameters; quench media

 

Quench distortion has been one of the most difficult problems facing the heat treaters. Theoretically, one of the main cause of quench distortion is thermal stress, the other one is transformations stress. However, up to now, there has been no practical systematic method which is effective for analyzing and solving distortion problems.

In the present paper a new method, being based on the end quench test curve, by means of the cooling rate band (CRB) which was derived from the analysis of the variation in hardness values in the quench distorted workpieces is proposed. It is called CRB method.

The Cooling Rate Band (CRB) and the Method for Reducing Distortion

Quench distortion is defined as dimensional change (i.e. changes in both shape and size)at a certain part of the workpiece that exceeds the tolerance limit. The present paper defines the distorted portions and the related regions as the quench distortion participation region (DPR) of the workpiece.

The hardness-cooling rate curves (HCRC), cooling rate band (CRB) and overlapping zones are the basic concept of the present method.

(1). Partition of the HCRC and it¡¯s relation to quench distortion. Fig.1 shows a typical end quenching (Jominy test) curve. The abscissa is changed into the cooling rate in quenching, the curve is divided into four zones (see Fig.1) according to the cooling rate and its relation to the hardness after quenching. such a curve is named HCRC.

Division
Designation
Quenching effect in the zones
 Zone I
Excessive cooling rate zone
High hardness, cracking, distortion
 Zone II
Adequate cooling rate zone
High and uniform hardness, no cracking, small distortion
 Zone III
Insufficient cooling rate zone
Insufficient and non-uniform hardness, severe distortion
 Zone IV
Excessive slow cooling rate zone
Not hardened at all, small distortion
Figure 1  End-quench curve and its four zones

In the first zone shown in Fig.1, the workpiece can be fully hardened. However, it may be distorted and quench cracking may occur due to too rapid cooling.

In zone II, the cooling rate is just adequate for sufficient hardening. The uniform hardness obtained indicates that the products of phase transformations at the DPR are primarily the same. Consequently, the stress level induced during the quenching process is low, and so the final distortion is little, usually below the allowable tolerance. Zone II is also called the slight distortion zone.

Because the HCRC is very steep (see Fig.1) in zone III, a slight difference in the cooling rate may induce a great change in hardness when quenched in this zone. That is to say, a relatively large difference in the microstructures and there corresponding specific volumes will be resulted. Thus, the factors that may induce quench distortion are quite complicated; some of them may occur during the quenching process and the others may be caused after the quenching. This is the reason why the distortion is always severe when quenched in the 3rd zone. In summary, severe distortion and non-uniform hardness are expected when quenched in this zone, and usually the workpiece is insufficiently hardened.

When quenched in the 4th zone, the cooling rate at the DPR is quite low and the temperature difference among these regions is little, these portions are not hardened at all; therefore, the phase transformation occurs nearly at the same time and the products are basically the same and so distortion is minimized.

However, the utilization of the end-quench curve and its partition are qualitative, and it should be properly adapted according to the actual conditions of distortion. In production, a median line of the end quenching curves of steels belonging to the name type from a handbook is sufficient for the analysis.

(2). The cooling rate band (CRB) of the DPR. The reason for the final distortion after quenching is that the cooling conditions of the DPR are different. The workpiece is an entity, so the different cooling rates at the DPR must fall into a certain range of the HCRC. The range of the cooling rates realized at the DPR is defined as the CRB of the DPR of the workpiece. The CRB is narrow if the difference between the cooling rates of the DPR is restricted. On the contrary, the CRB will be wide.

The CRB can be determined in the following way:

For a given quench distorted workpiece, first, find out the DPR of the workpiece. Next, measure the hardness at the outer and inner surface in DPR and based on the minimum and maximum hardness values the cooling rate can be located in the HCRC of the steel. Finally, connecting the two points on the HCRC by a straight time, the CRB is determined. A hollow cylinder made of 50Cr steel, heated at 770¡æ and quenched in water for 1.5 seconds and followed by oil cooling was taken as an example. After quenching, no cracking was found, but it was visibly distorted: the orifices at the two ends were expanded like a double bell (Fig.2). Since the shape of this workpiece was relatively simple and the distortions have occurred in everywhere of the workpiece; therefore, the whole workpiece can be regarded as the DPR. The hardness at both ends of the cylinder was ca 55HRC, while the hardness at the surface of the interior of the hole was only 35HRC. The end-quench curves of 50Cr steel can be found from a handbook, HCRC can be determined along the median of the curves. Then the cooling rate ¡°b¡± for maximum hardness value of 55HRC and the cooling rate ¡°a¡± for minimum hardness value of 35HRC can be fixed on the HCRC. The CRB is determined by the straight line connecting ¡°a¡± and ¡°b¡±. The CRB for other workpieces can also be determined in a similar way. If distortion occurred in interior holes or hollow portions, then these portions and their adjacent regions should be taken as DPR. The workpiece has to be sectioned through these regions in order to measure the hardness.

Figure 2   Distortion of the workpiece and Determination of HCRC

(3). CRB across more than one zone in HCRC. In production, the width of the CRB varies with steel type, shape, size, and heat treatment conditions. Consequently, the widths of their CRBs are different: some are very wide and some may be very narrow. When the CRB is relatively narrow, it may completely fall into one HCRC, However, if the CRB is relatively wide, it may cross two or more cooling rate zones. For example, the CRB of the workpiece shown in Fig.2 crosses zone II and III. Taking another example, a large circular saw blade of 65Mn with a diameter of 1600mm and thickness of 8mm, after being heated in a specially designed furnace and hanging vertically, was directly dipped into a quenching tank with stirring circulator. The quenching medium was a polymeric aqueous solution at 25¡æ. Severe warp distortion was found after quenching. By inspection, there were several quenching cracks near the trench of the teeth. The maximum hardness on the edge (B) was 62HRC while the minimum hardness near the center (A) was only 30HRC. For such a thin and large workpiece with severe distortion, it is appropriate to take the whole workpiece as the DPR. Similarly, the CRB can be determined as shown in Fig.3. Because quench cracking had occurred near the bottom of the trench, it indicates that the cooling rate at that place had already entered the first cooling rate distribution zone. However, the hardness in the center was only 30HRC, not sufficiently hardened, indicating that the cooling rate at that portion had already extended into Zone III. Thus the cooling rate of the whole circular saw blade cover three zones (Zone I, II and III). As shown in this example, a thin and big workpiece, experienced a cooling rate covering three zones, it was liable to be severely distorted.

(4). The objectives for reducing quench distortion. To obtain high and uniform hardness, sufficient hardening depth, no quenching cracking and no unacceptable distortion, the CRB of the DPR should fall completely into the second cooling rate zone. When this goal is achieved, the distortion can be controlled with ease. The measures which are necessary to realize this objective are summarized below.

Figure 3   The CRB for the big circular saw blade of 65Mn extended these zones( I¡¢II¡¢III)

Methods to Reduce Quench Distortion

(1). Shift the CRB totally. It is possible to move the CRB of the workpiece totally into the second zone by increasing or decreasing the cooling rate as a whole. This method works for the workpieces with a relatively narrow CRB, and the quench distortion is caused by the CRB extending partly or entirely into zone I or zone III.

If the hardness at the DPR, partly or wholly, is pretty low, this indicates that the cooling rate at that portion(s) had entered zone III. Accordingly, in order to avoid quench distortion, one or more of the following measures can be taken: increase the cooling rate of the workpiece wholly; increase appropriately the heating temperature; or change steel with higher hardenability. All these adaptations would help to move the CRB into the second zone and thus avoid quench distortion.

If the workpiece suffers quench distortion and the hardness is high and at the same time cracking has also occurred, it indicates that part or all of the DPR has entered zone I. Methods of solving such problems are: properly decrease the quenching temperature, decrease the cooling rate of the whole workpiece or change the steel for a lower carbon and lower alloyed one. The purpose for this is to move the CRB entirely into the second zone.

For example, one factory woked on 11 ¡Á 75mm 65Si2Mn steel plates to produce plate spring for automobiles by using PAG quenchant at 25¡æ. Severe distortion as large as 4mm in side bending and large change in the height of the arc were found after quenching. In the beginning the heat treaters thought it was induced by large quenching stress, Therefore, measures were taken to reduce the quenching stress by properly decreasing the heating temperature, increasing the temperature of the liquid, stopping the swinging of the quenching machine to decrease the relative flow speed, and increasing the concentration of the PAG quenchant up to 15%. All the efforts were made to reduce the cooling speed of the plate so as to avoid distortion via decreasing quenching stress. However, the distortion became even more severe, at the same time the hardness of spring sheets was low and could not meet the technical requirement.

Afterwards, it was found that the hardness in the range of 28¡«52HRC was insufficient and non-uniform, these facts indicate that their CRB fell right in zone III as shown in Fig.4. The right way to solve this problem is to increase the cooling rate so that the CRB of the sheets move entirely to the left and fall into zone II. This can be practically done by supplying tap water into the quenching tank so as to reduce the concentration of PAG down to 10% and keep the quenching machine swinging continuously to increase further the cooling rate during quenching. Eventually, spring sheets of the same batch were successfully quenched with extremely little distortion. At the same time the hardness increased to 59¡«61HRC, and there was no cracking.

Figure 4   To avoid quench distortion of the spring sheets by shifting the cooling rate band totally

(2). Squeeze the CRB into zone II. When the CRB of the workpiece spans over more than two zones, the solution to the distortion is to make the two ends of the CRB shrink toward their middle point. If the CRB of some portions extends into zone I, one must reduce the cooling rate so that the left end of the band shrink toward the right until it enters zone II. Meanwhile, if the cooling rates of some other portions are too low so as to make the CRB extending into zone III, one need to increase the cooling rates of these portions so that the CRB of the workpiece shrinks toward the left and falls into zone II.

For example, because the CRB of the big circular saw blade shown in Fig.3 spans over the 1st, 2nd and the 3rd zones, one should make its CRB shrink toward the center from both ends in order to make it fall into the 2nd zone entirely.

Main Measures of Adjusting the CRB

It can be seen from the previous discussion that the approach to solving the problems of quench distortion is, basically, to apply certain measures to shift or squeeze the CRB of the workpiece into the 2nd zone.

The measures available are: (1) change the heating temperature; (2) alter the local cooling conditions; (3) change the temperature, concentration and flow of the quenching medium; (4) change the medium to meet the fluctuation in composition and quality of the steels; (5) change the steel type or the designing of the workpiece. In production one can choose one or several methods to solve quench distortion problems. More details will be given as follows.

(1). Heating temperature adjustment. When the CRB of the workpiece extends into the 1st zone, one can make the left end of the CRB shrink to a certain extent to the right by decreasing the quenching temperature. The technical measures also include reducing the local temperature before quenching. When the CRB of the workpiece extends into the 3rd zone, the right end of the band can be brought to the left to a certain extent by properly increasing the heating temperature of the workpiece wholly or locally. If the soaking time was not sufficient, then prolonging the soaking time will have the same effect.

(2). Alternation of quenching medium and cooling conditions. To reduce the extent of the quench distortion through the change of quenching medium is widely adapted in heat treatment. However, the methods described in the present paper are different from the traditional ones at two main points. Firstly, the technical measures in the present paper, which include the change of the application conditions or the change of quenching medium, are guided by the principles established previously. Secondly, the goal of the traditional methods for adjustments of quenching medium and its application conditions are often limited to reducing the cooling rate so as to decrease the internal stresses. However, the present methods are based on how the CRB spans the cooling rate zones. Whether decreasing or increasing the cooling rate depends upon the actual cases. In fact, in most cases, the cause of the distortion was that the cooling rates were insufficient in DPR so that their CRB entered the 3rd zone. Therefore, measures for increasing the cooling rate must be used. The consequence of this adjustment in that the cooling rate differences experienced among the DPR reduced while the external cooling rate increased because they all reside in the 2nd zone. Thus, the internal stresses are eventually reduced, and so is the tendency for quench distortion.

(3). Improvement of local cooling conditions. When the cooling rate of some portions of the workpiece is too fast, it can be reduced by installing shields so that the CRB shrinks from zone I into zone II. On the other hand, when the cooling rate at certain DPR is too slow, it is also possible to make the CRB at the corresponding positions shrink leftward from the 3rd zone into the 2nd zone.

(4).Control of quench distortion by martempering. According to the method discussed above thereason why martempering can be used for the control of quench distortion can also be explained by the position and width of the CRB of the workpiece. Fig.5 represents the cooling processes at the ¡°fast cooling end¡± and ¡°slow cooling end¡± of the DPR. Since the workpiece had been cooled all the way down, the fast cooling end of the CRB extended into the 1st zone on the corresponding HCRC. Accordingly, quench deformation in excess of the tolerance is induced.

Figure 5  Ordinary quench hardening process£¨a£©, and the corresponding cooling rate band£¨b£©

The problem of quench distortion can be solved by making the fast cooling end of the CRB retreating rightward into zone II. The role of martempering is to reduce the width of the CRB so that it can retract completely into zone II. Fig.6 (a) and (b) show the principle how martempering at temperatures higher or lower than Ms point is used for the control of quench distortion.

(a) Martempering higher than Ms temperature (b) Martempering higher than Ms temperature (c) Both process makes the cooling rate band retreat into zone II
Figure 6  Analysis for the control of quench distortion by martempering

It¡¯s clear that martempering can bring the CRB rightward until it falls entirely into the 2nd zone, and so the deformation problem is solved. Here fast cooling end and slow cooling end of the CRB are concerned, but not the surface and interior of the workpiece. In fact, during quenching, differences in cooling rates at different locations of the surface of the workpiece are very large, especially for those with complicated shapes, except for standard spheres.

Mastempering is suitable for such cases: Workpieces have complicated shapes and stringent tolerable distortion is required, and their second cooling rate zones are relatively narrow, When quenched all the way down in one medium, the fast cooling end of the CRB will extend into zone I, while the slow end still remains in zone I or zone II. In cases where the slow cooling end of the CRB spanned into the excessively slow cooling zone (zone III), martempering is not suitable for such cases. This is why martempering is only suitable for small workpieces and steels with high hardenability. It can be inferred that using a quenching medium with a faster cooling rate, it is possible to control the distortion of larger workpieces or workpieces made of low-hardenability steels by mastempering.

(5). Material¡¯s properties. The Jominy test curves are different with different steel types. For high-hardenability steels, the hardness-distance curves are relatively smooth, while for low hardenabilty-steels abrupt changes can be observed on the curves. The 2nd zone of a high-hardenability steel is wide while that of a low-hardenability steel is relatively narrow. The 3rd zone of high-hardenability steel is wider than that of a steel with low hardenability. Also, the hardness-cooling curve of the former is smoother than that of the latter, as shown in Fig. 7. It can be inferred that the more hardenable the steels are, the easier it is to control the quenching distortion of the workpieces. However, with the increase in hardenability, the width of the 1st zone is widened, too; therefore, it is necessary to use quench media with slower cooling rate to avoid cracking.

Figure 7 Relationship between the shape of the HCRC and the hardenability of the steel
Figure 8 Effect of fluctuations in hardenability of steels on the partition of the cooling rate zones

Let us see the hardenability bands of steel of the same sort as shown in Fig.8. If the upper limit is chosen for the reason of compositional fluctuations in different heats, then the upper curve illustrates the relationship between hardness and the cooling rate. If the lower limit is chosen, then the lower curve has to be used. For the upper curve, the 1st cooling rate zone is slightly wider, the 2nd and the 3rd zone are widened further; the opposite is true for the lower curve. It can be inferred that if variations in hardenability of the steels are induced by its compositional fluctuation, it will be easier to control the quench distortion of workpieces made of higher-hardenability steels. On the other hand, it will be much more difficult for workpieces made of low hardenability steels. However, if the hardenability is too high especially when it is caused by too high carbon content, the 1st zone of the workpiece is relatively wide, and the steel is more prone to quench cracking.

The above mentioned principles can be used to prevent quench distortion. In certain cases quench distortion can be avoided by using steels with high hardenability. However, in cases where distortion was induced by a decrease in hardenability due to compositional fluctuations of the steels, quenching media with faster cooling rates can be used for distortion control. A fast quenching oil or a fast quenching oil modified by adding special additives can be used to cope with the effects of compositional fluctuations. Another reason for the use of this method is that with most steels, cracking would not occur when quenched in oil (including fast quenching oil). For most structural steels, distortion and cracking can often be avoided by using a suitable aqueous solution because the cooling rate is higher. This guarantees that the CRB of the workpiece would not span into the 3rd zone.

The Sequence of Applying the Fundamental Measures

For prevention of quench distortion, it is possible for an experienced engineer to judge directly which category or certain categories of measures should be taken from the above ones. However, for an inexperienced engineer, he can follow the following sequence of measures to try one after another until problems are solved.

(1).Change the heat treatment parameters. According to the situation that the CRB spans over different zones, the following measures can be taken:

--- Increase or decrease the heating temperature of the whole or part of the workpiece;

--- Increase or decrease the temperature of the quenching liquid;

--- Increase or decrease the flow speed of the quenching liquid around the whole body or part of the workpiece during quenching.

Adopting one or more of these measures one can solve many distortion problems.

(2)£®Modify the concentration of the quenching liquid. This method can be used for those aqueous solutions whose concentrations can be easily measured and controlled. Because the concentration is not easy to be reversed to the original one, this method is only considered when the 1st category of methods does not work.

(3)£®Change the type of quenching medium

--- Replace tap water with a suitable aqueous solution;

--- Change one kind of aqueous solution for another aqueous solution;

--- Replace tap water or aqueous solution with an oil-like medium;

--- Change mineral oils for a special quenching oil;

--- Change one quenching oil for another quenching oil;

--- Replace the cold oil with hot oil;

--- Add special additives into mineral oils.

(4)£®Change the steel type. According to traditional methods of preventing distortion, changing the steel type will be considered if distortion occurs after water quenching. After changing for a higher- hardenability steel, the distortion can be avoided by adopting oil quenching. A conventional explanation is that the steel is cooled too rapidly in water so that quench distortion in excess of the tolerance was induced by excessive internal stresses. If the steel is changed for a higher-hardenability one, since it is cooled more slowly in oil, the internal stresses as well as the distortion will be reduced. This method is often effective. However, the conventional explanation is not so perfect.

According to the methodology of the present paper, the explanation can also be given as the following: the 2nd zone of a high-hardenability steel is wide so that the CRB of the workpiece can wholly fall into it when quenched in oil. The distortion in excess of tolerance after water quenching is due to two possible reasons. Firstly, the CRB of the workpiece had entirely fallen or partly extended into zone I. Secondly, it had spanned entirely or partly into zone III. When the distortion was due to first reason, the measures listed above can be adopted one by one until the distortion problem is solved. If none of them works, changing steel types has to be considered. Since the change of steel types is often costly, it is not recommended in the first place on considering the production management and the sparing of alloy resources. If the distortion is due to the second reason, in addition to the first category of measures, it is also possible to change for a steel with a slightly higher carbon content and higher hardenabilty and continue to quench it in water.

The partiality of the conventional explanation can also be illustrated by the following argument. If the reason for distortion was that the CRB had extended into the 1st zone, the use of higher-hardenability steel can only make the problem even worse. In this case, if a medium with a cooling rate slower than mineral oils has to be used, martempering or air-cooling is probably the only choice left. However, these measures are all too costly. What should be done in this case is to change for a fast quenching oil, or modify the original mineral oil with additives, or use suitable aqueous solutions.

In summary, according to how the CRB of the distorted workpiece spanned over different zones, and only in cases when all of the first three categories of measures do not help, changing the steel for a higher or lower hardenability can be considered. It can be changed. Usually, the quenching medium should be changed at the same time.

(5)£®Change the design of the workpiece. This solution can be considered in cases where all the above four steps do not work. The role of altering the design is to change the locatin of the left and right boundaries of the 2nd zone of the workpiece. For instance, the left side boundary of zone II can be shifted rightward by reducing the sharpness at the concave or convex spots so as to prevent distortion and cracking. Reduction in the difference in thickness at different places can narrow the CRB and shift it back entirely into the 2nd zone. It is also possible to divide a complex workpiece into several simple workpieces of uniform thickness.

The Reasons Why Quench Distortion Problems are so Complicated

It is a widely accepted in the field of heat treatment that the problem of quench distortion is very complicated so that simple measures do not always work, and success can not be achieved even by adopting multiple measures. This is true in many cases. Why does the problem remain so complicated?

By using the analysis and the measures discussed previously to move the CRB, the role of the basic measures in controlling quench distortion are summarized in Table 1, in which the directions of movement of the CRB corresponding to changes in the heat treatment parameters are listed.

Table 1 The basic measures in controlling the CRB
Basic measures
Moving direction
leftward
rightward
  Heating temperature
increase
¡Ì
 
decrease
 
¡Ì
  Temperature of quench oil
increase
¡Ì
 
decrease
 
¡Ì
  Temperature of water or aqueous solution
increase
 
¡Ì
decrease
¡Ì
 
  Flow speed of the medium relative to the workpiece
increase
¡Ì
 
decrease
 
¡Ì
  Change the concentration of PAG aqueous solution
increase
 
¡Ì
decrease
¡Ì
 
  Replace tap water by PAG aqueous solution
 
¡Ì
  Replace tap water by oil
 
¡Ì
  Addition of special additives into oil
¡Ì
 
  Change the oil to special quench oil
¡Ì
 

For instance, the increase in heating temperature usually will result in an increase in hardness, equivalent to a leftward movement of the CRB; while a decrease in heating temperature would move the CRB to the opposite direction. For another example, increasing the flow speed of the liquid will increase the cooling rate, which makes the CRB move to the left. Opposite results will occur if decreasing the flow speed. These parameters are effective measures for solving quench distortion problems. Apparently, if several measures are to be used simultaneously, only those that act in the same direction as shown in Table 1 can be selected. If measures which act in different directions are used simultaneously the result will not be predictable. They may or may not work or can make distortion even worse. The reason why people thought the distortion problems were very complicated is that they do not know the exact roles of the technical measures, and so do not know in which direction the CRB of the workpiece should move in order to find the solution.

Summary of the Technical Measures for Solving Quench Distortion Problems

In order to enhance understanding and to facilitate the applications with ease, it is necessary to summarize once more as follows.

(1). Approaches to solving quench distortion problems. According to the methodology in the present paper, there are only two approaches that can be adopted to solve quench distortion problems. One is to move or contract the CRB so as to make it fall entirely into the 2nd zone of the workpiece, as shown in Fig.9. Another approach is to move the boundary of zone II until it covers the CRB of the workpiece, as shown in Fig.10.

Figure 9  To move or contract the CRB to make it fall into zone II entirely
Figure 10 To move the boundary of the zone II to cover the CRB

(2)£®Technical measures. Technical measures can be summarized in three categories as shown in Table 2.

Table 2   Categories of technical measures for solving quench distortion problems
Category
Measures
Function
Type I
Heat treatment parameters
To move or contract the CRB to make it fall into the zone II
Type II
Composition control for steels and change of steel type
To move the boundaries of zone II to cover the CRB of the workpiece
Type III
Modification of the design of the workpiece
To alter the boundaries of zone II and move or contract the CRB simultaneously so that to make it fall into zone II

The first type of approaches includes the measures by which the CRB is moved entirely or partly. These measures mainly concern the conditions and parameters of the heat treatment process as shown in Table 1.

The second type of approaches involves measures designed to move the boundaries of the 2nd zone by purposely moving the boundaries until the CRB is completely covered by zone II. This type of approach includes the following technical measures: a. improvement of the microstructures prior to quenching; b£®control of the composition of the steel; c. change of the steel type; d£®relief of internal stresses.

The third type of approaches is to move the CRB and the boundaries of the 2nd zone simultaneously, which includes mainly the measures involving the design of the workpiece: a£®reduce the stress concentration in the workpiece and to move the boundary between zone I and zone II leftward£»b£®reduce the difference in thickness at the different positions of the workpiece to narrow the width of the CRB; c£®improve the symmetry of the wokpiece to widen it¡¯s zone II; d£®divide the workpiece into several simpler parts so as to simplify the problems.

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