J. Cent. South Univ. Technol. (2009) 16: 0349-0353

DOI: 10.1007/s11771-009-0059-6

A new method of characterizing equivalent strain for

equal channel angular processing

ZHAO Jun(赵 军)^{1, 2, 3}, WANG Zhen-hua(王振华)^{1, 2}, SUN Shu-hua(孙淑华)^{1, 2},

ZHAO De-li(赵德利)^{1, 2}, REN Li-guo(任利国)^{1, 2}, FU Wan-tang(傅万堂)^{1, 2}

(1. State Key Laboratory of Metastable Materials Science and Technology, Yanshan University,

Qinhuangdao 066004, China;

2. College of Materials Science and Engineering, Yanshan University, Qinhuangdao 066004, China;

3. North China Institute of Aerospace Engineering, Langfang 065000, China)

Abstract: In order to establish the quantitative relationship between equivalent strain and the performance index of the deformed material within the range of certain passes for equal channel angular processing (ECAP), a new approach to characterize the equivalent strain was proposed. The results show that there exists better accordance between mechanical property (such as hardness or strength) and equivalent strain after rolling and ECAP in a certain range[A1]  of deformation amount, and Gauss equation can be satisfied among the equivalent strain and the mechanical properties for ECAP. Through regression analysis on the data of hardness and strength after the deformation, a more generalized expression of equivalent strain for ECAP is proposed as: ε=k₀exp[-(k₁M-k₂)²], where M is the strength or hardness of the material, k₁ is the modified coefficient (k₁∈(0, 1)), k₀ and k₂ are two parameters dependent on the critical strain and mechanical property that reaches saturation state for the material, respectively. In this expression the equivalent strain for ECAP is characterized novelly through the mechanical parameter relating to material property rather than the classical geometry equation.

Key words: equal channel angular processing (ECAP); equivalent strain; mechanical property; characterizing method

1 Introduction

Pure shear deformation can be achieved through equal channel angular processing (ECAP) without changing the basic geometry of the material, and the material will acquire a considerable amount of strain after the repeated extrusion. Although the strain expressions given by SEGAL [1], IWAHASHI et al [2] and GOFORTH et al [3] have disclosed the relationships between equivalent strain, extrusion pass and the channel angle of the ECAP die, they just indicate the simple geometric relationships rather than the performance index of the material. Since high performance materials can be acquired through ECAP, it is of great significance to quantify the relationship between the strain and mechanical properties of the material especially in work hardening stage so as to forecast mechanical properties of the material after ECAP under different conditions. As to the process of drawing and rolling, equivalent strain can be derived from the geometric changes before and after deformation. However, it is hard to measure ECAP equivalent strain directly because the geometry of the specimen undergoes little change after ECAP, which is usually conducted in a relatively closed environment. On the other hand, for the repeated ECAP, two distinct stages including work hardening and saturation of mechanical property for material can be observed from the curve of hardness (or strength) vs equivalent strain [4]. Based on the combination of ECAP and rolling, LEE et al [4] put forward dissimilar channel angular processing (DCAP), and apply it to studying 1050 Al alloy. Through a comprehensive analysis of the microhardness and strain, it is found that there exists better coherence with the mechanical properties (such as microhardness), no matter whether the material is affected through DCAP or cold rolling (CR). That is to say, there exists an overlap in the relationship curves between microhardness and strain. This makes it possible to predict DCAP equivalent strain through rolling equivalent strain.

In this work, a new method of characterizing ECAP equivalent strain is proposed, which contains the performance indexes of the deformed material, based on an integrative analysis of the mechanical property (microhardness or strength) of the materials subjected to rolling and ECAP. Thus, the explicit and quantitative relationship between equivalent strain and the performance index of the deformed material within the range of certain passes for ECAP is exhibited.

2 Establishment of expression of ECAP equivalent strain

Fig.1 shows the relationship curves between the mechanical properties (strength and microhardness) and the deformation amount of the two alloys (1050 Al alloy and Fe-0.16C steel) subjected to CR or warm rolling (WR). From Fig.1, it can be seen that the microhardness (or tensile strength) of the alloys increases as the rolling equivalent strain increases. According to the regression analysis on the experiment data in Fig.1, the relationships between the microhardness (or tensile strength) and the rolling equivalent strain of the alloys can be expressed in form of a Gauss equation whether it undergoes CR or WR, from which an expression of equivalent strain of material is derived in relation to microhardness or the strength (see Table 1).

Fig.2 shows the results of microhardness and equivalent strain from CR and DCAP experiments conducted by LEE et al [4] with 1050 Al alloy. From Fig.2, it is found that the DCAP equivalent strain has an influence on the microhardness of 1050 Al alloy similar to the CR equivalent strain. That is to say, within a certain range of strain (Region A, or work hardening stage), the relationships between hardness and strain are similar, and there exists a linear relationship between HV and ln ε in this stage, whether it is the result of CR or DCAP. The strain resulted from CR reduction ratio of 86% is almost equal to that from a DCAP with 4 passes (see Fig.2(b)). Therefore, in the segment where the two

Fig.1 Relationship curves between mechanical properties (microhardness and strength) and equivalent strain of two alloys: (a) 1050 Al alloy, CR [4]; (b) Fe-0.16C steel, WR [5]

Table 1 Expression of rolling equivalent strain (ε) and mechanical properties (microhardness, HV; tensile strength, σ_b, MPa) of materials

Fig.2 Relationship curves between microhardness and equivalent strain of 1050 Al alloy processed by CR and DCAP [4]: (a) For equivalent strain; (b) For DCAP pass number and logarithm of equivalent strain

curves are nearly identical, the relationship between hardness and strain is also applicable to the DCAP, and as a result, the DCAP equivalent strain could be expressed through the rolling strain.

Based on the common characteristics of the expressions in Table 1, a more generalized expression of equivalent strain in the work hardening stage is proposed as follows

ε=k₀exp[-(k₁M-k₂)²]                          (1)

where  ε is the equivalent strain of the material in main deformation zone along the billet, M is a parameter of mechanical properties, which could be Vickers hardness (HV), tensile strength (σ_b, MPa) or yield strength (σ_0.2, MPa), k₀ is the critical strain when the strength or hardness of the deformed material reaches the saturation, k₁ is the modified coefficient (k₁∈(0, 1), k₀ and k₂ are two parameters dependent on the critical strain and mechanical property that reaches saturation state for the material, respectively. k₂/k₁ is equal to the value of the mechanical property at saturation state.

According to the experimental results by LEE et al [4] (see Fig.2), Eqn.(1) can be converted into ε= 5.7exp[-(0.054HV-3.737)²], which is applicable to DCAP within 4 passes for 1050 Al alloy. In fact, as DCAP is developed on the basis of ECAP conducted in a relatively closed environment, it is hard to measure the strain directly. However, if the mechanical properties of the material after ECAP could be used for expressing the deformation amount (equivalent strain), the applicable scope of Eqn.(1) should be wider, thus the equivalent strain (corresponding to ECAP passes) can be estimated through the strength or hardness after ECAP, and even the strength and hardness of the deformed material can be predicted through equivalent strain (corresponding to ECAP passes). As a result, the quantitative relationship between performance indexes and equivalent strain of the material after ECAP can be established so that equivalent strain is related to the material properties any more. This is the most distinct feature different from the other reports.

3 Applicable scope of ECAP equivalent strain expression

At present, ECAP of many materials such as pure metal, light alloy and steel has been studied [6-15], and the equivalent strain for ECAP is mostly calculated through Eqn.(2) given by IWAHASHI et al [2], which has been proven by the strain data of numerical simulation calculation [16] and the color plasticene test [17]. However, whether ECAP equivalent strain expression presented in this work is reasonable, and how wide its applicable scope is, this still need to be validated with the results of ECAP experiments further.

          (2)

where  ε is the strain of the material after ECAP for N passes, N is extrusion pass number,  is the channel inner angular, and φ is the channel outer angular.

By synthetically analyzing the results of deformation amount (pressing pass) and mechanical properties (hardness, tensile strength or yield strength) reported in Refs.[6-15], the relationship curves of mechanical properties vs equivalent strain of the materials after ECAP could be obtained as shown in Fig.3. After further regression fitting all of the curves in Fig.3, the corresponding relation expressions between the number of ECAP passes and the mechanical properties results are listed in Table 2, wherein the goodness of regression fitting and applicable scope of the expressions are also given.

From Table 2, it can be seen that the listed expressions of equivalent strain in this work is applicable to pure Cu and pure Ti within 4 passes of ECAP only. For the Al alloy and Mg alloy, it is effective to the ECAP pass within ranges from 3 to 6, and for the low carbon

Fig.3 Relationship curves of mechanical properties vs equivalent strain of materials after ECAP: (a) Microhardness;  (b) Strength

Table 2 Application scope of ECAP equivalent strain expression

steel, the applicable scope is between 4 and 10 passes. Furthermore, for the high carbon steel, the expression can be used within 5 passes so far as there is no experimental data reported beyond 5 passes.

4 Discussion

Eqn.(3) shows the equivalent strain expression for ECAP derived by GOFORTH et al [3]:

                      (3)

Fig.4 shows the relationship curves of single pass ECAP equivalent strain vs the channel angle of die, which are calculated respectively with Eqns.(2) and (3). From Fig.4, it can be seen that on the whole the strain calculated with Eqn.(3) is always a little less than that with Eqn.(2) under the same condition, but their strains will incline to be the same along with the increase in the

Fig.4 Relationship curves of equivalent strain vs channel angular after single pass ECAP

channel inner angle. Thus, compared with Eqn.(3), Eqn.(2) is put to more frequent use, as its reliability has been proven by some experiments [16-17].

From the above results it can still be found that the material factors significantly correlates with the two parameters (k₀ and k₂), but in a certain range of deformation amount (such as in Region A, work hardening stage in Fig.2(b)), the equivalent strain of different materials in the main deformation zone along the billet can also be calculated using Eqn.(1) through measuring the mechanical properties (hardness or strength) and adjusting the three parameters (k₀, k₁, k₂) in Eqn.(1). Although the applicable scope of Eqn.(1) is not so wider than that of Eqn.(2), at least it can still be acquired directly through the mechanical properties (such as hardness, tensile strength or yield strength) within certain range of ECAP equivalent strain. The essential difference between Eqn.(1) and Eqn.(2) is that the performance index of the processed material has been introduced into the expression of ECAP equivalent strain.

Furthermore, for the saturation stage of mechanical property at the curve of hardness (or strength) vs equivalent strain, the equivalent strain can be expressed as ε=(1+m/n)k₀, where m is the extrusion pass number for ECAP in the saturation stage, n is the critical extrusion pass number for ECAP before the performance saturation, parameter k₀ is the critical strain when the level of strength or hardness of the material reaches the saturation in Eqn.(1).

5 Conclusions

(1) A new method of characterizing the ECAP equivalent strain is proposed, and the Gauss equation which likes ε=k₀exp[-(k₁M-k₂)²] is satisfied between the equivalent strain and the mechanical properties of deformed materials in a certain range of deformation amount.

(2) The quantitative expression is more generalized for many metals or alloys. Either the mechanical performance indexes relative to the deformed materials essence or their saturation characteristics after ECAP are considered well in the expression.

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(Edited by YANG You-ping)

Foundation item: Projects(50471102, 50671089) supported by the National Natural Science Foundation of China

Received date: 2008-07-02; Accepted date: 2008-10-17

Corresponding author: FU Wan-tang, Professor; Tel: +86-335-8387472; Fax: +86-335-8074545; E-mail: wtfu@ysu.edu.cn