A constitutive model for modeling of the deformation behavior in microforming with a consideration of grain boundary strengthening

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Abstract

Micro-manufacturing technology is now getting more and more important in industries due to product miniaturization. Microforming, as one of the micro-manufacturing processes, has attracted much attention in the past decade. In microforming, the traditional metal-forming theories are often crippled or rendered totally inefficient in analysis of micro-deformation behavior due to the size effect arising from the part geometry scaling down from macro- to micro-scale. To reveal its deformation behavior, the size effect needs to be considered and appropriate constitutive models to be developed. In this research, by employing the surface layer and composite models, each grain in the microforming workpiece is divided into two portions, viz., grain interior and grain boundary. The flow stress of grain is determined by the ‘law of mixtures’ based on the flow stresses of grain interior and boundary, and in such a way that a new constitutive model is developed with a consideration of grain boundary strengthening. This new constitutive model considers both the grain and geometry size effects simultaneously. Using this newly developed model, the micro-bulk upsetting and the micro-tensile test of sheet metal are numerically simulated by Finite Element Method (FEM). The grain and geometry size effect on the flow stress of workpiece is thus investigated. The model is further applied to the micro-extrusion of CuZn30 alloy with three different grain sizes. The validity of the model is thus verified through physical experiments and FEM simulations of three different micro-plastic deformation processes.

Highlights

► A new constitute model for microforming is developed. ► Micro-bulk upsetting and micro-tensile processes are conducted. ► Micro-extrusion processes of CuZn30 alloy are analyzed. ► Grain and geometry size effects are investigated.

Introduction

Microforming, a newly developed micro-manufacturing technology for fabrication of microparts, is now getting more and more important due to product minimization [1], [2]. For macro-scaled metal forming process, the analyses have been extensively conducted and widely used in metal forming industries to design and evaluate metal forming systems [3]. However, when the part size is reduced to micro-scale such as in sub millimeter range, the deformation behavior changes due to size effect [4]. In the past decade, a lot of researches have been conducted to have an in-depth understanding of this process [5]. In general, a microforming system comprises four main elements, viz., material, process, tool and machine tool, and equipment [6]. Among them, the material and its properties is critical as it affects the deformation behavior of microforming process due to the size effect arising from the part geometry scaling down from macro- to micro-scale. To explore the relationship of deformation behavior with these size effects, the experiments on microforming of copper and aluminum alloys have been conducted [6], [7], [8], [9], [10], [11], [12]. The results show that the flow stress decreases with the increasing miniaturization when the grain size of the test specimen is held constant. Different from the conventional macroscale parts, microparts have only a very limited number of grains. The deformation behavior of each grain plays a significant role in the entire deformation behavior of the deformation body. The investigation on the decreasing flow stress with the scaling down of metal-formed components and the development of appropriate constitutive models to represent those physical phenomena in microforming process is thus critical in design and development of micro-deformed parts, microforming process, and microtooling.

To address this issue, many investigations were conducted and the corresponding constitutive models were developed in the past decade [9], [13], [14], [15], [16], [17], [18], [19], [20]. Among these researches, Miyazaki et al. [21] revealed that the flow stress decreases with the reduction of specimen size and explained this phenomenon by using an affected zone model. Based on this model and Hall–Petch relation, Leu [8] proposed a simple flow stress model via introducing a function with a ratio (t/d) between the sheet thickness t and grain size d to represent the size effect. Geiger et al. [22] and Engel et al. [12] proposed the surface layer model, which is the most commonly used model for representing size effect. Based on this model, the deforming workpiece is divided into two portions, viz., inner and surface layer portions. The share of surface grains increases with the increasing grain size or the decreasing part geometrical dimensions, as shown in Fig. 1 [12].

According to the metal physics theory, the grains in free surface show less strain hardening effect compared with the inner grains due to the different mechanisms of dislocation movement, pile-up and the fact that they are less subject to the compatibility restriction. Finding a way to representing the deformation behaviors of the two portions of grains in a deformation body lies in the development of appropriate constitutive models. Therefore, Kim et al. [15] treated the surface layer and the inner portion of the deformation body as single crystals and polycrystals, respectively. Lai et al. [13] developed a mixed material model by combining the surface layer model and the modified Hall–Petch relation. Through combining the composite model and the Hall–Petch equation, Geiger et al. [20] proposed a mesoscopic model. In addition, Molotnikov et al. [9] developed a physically based constitutive model, in which the flow stress of inner grains is described by the dislocation-density theory initiated by Estrin et al. [23] and the flow stress of surface layer grains is considered as the image force of free dislocations at the grain boundary being attracted to the surface. Furthermore, a few phenomenological models were also developed through fitting the strain–stress curves obtained from tensile and upsetting tests [14], [24], [25], [26], [27]. Based on the classic material models and the introduction of correction terms to consider the factors such as specimen and grain sizes, the material models developed in such a way can take into account the size-dependent factor for handling simple microforming processes.

In this research, a new approach to modeling the size effect on micro-deformation behavior is developed. By combining the surface layer model [12] with the composite model [28] of polycrystal, a new constitutive model is thus developed through considering the grain boundary strain strengthening. The flow stress of the material is determined by applying the ‘law of mixtures’ to grain interior and boundary. Based on this approach, the detailed constitutive models for the bulk specimen with round cross-section and the sheet metal with rectangular cross-section are developed. The new models take into account the grain and geometry size effects simultaneously. The models are used to simulate the micro-upsetting test of bulk specimen and the micro-tensile test of sheet metal with different scaling parameter configurations. The efficiency of the developed models is verified via comparison of the calculation results with the experimental ones. In addition, the developed model for bulk specimen is further applied to the micro-extrusion of CuZn30 alloy. The effect of grain size on the extrusion load is investigated and the simulation results are compared with the experimental data. The efficiency of the model for bulk specimen is further validated.

Section snippets

Constitutive model

There are two size effects existing in metallic materials [29]. One is grain size effect and the other is geometry size effect. The material constitutive model should consider both the two size effects simultaneously in modeling and investigating the deformation behavior in microforming process. In this section, by employing both the surface layer and the composite models, a new constitutive model is developed.

Parameters determination for the developed constitutive model

In the developed constitutive model, the flow stresses of grain interior and grain boundary (σGI and σGB) are two unknowns. Therefore, more than two strain–stress curves of the deformation material with different geometry size factors are needed to determine their values. The relationships between geometry size effect factor η and flow stress at some strain points can be built. Fig. 5 is the schematic illustration of the method for determination of this relationship for CuZn15.

Furthermore, Eq.

Validation

To validate the developed model, the micro-upsetting test of bulk specimen and the micro-tensile test of sheet metal are conducted. The effects of grain and geometry sizes on the deformation behavior of micro-specimen are investigated via physical experiment and Finite Element Method (FEM) simulation. The comparison in between is also made.

Application to micro-extrusion of CuZn30 alloy

The developed model is further employed to investigate the grain size effect on the micro-extrusion force of CuZn30 alloy. The FEM simulations based on the developed constitute model are compared with the experiments done by Cao et al. [37], [38]. A reasonable agreement is observed in between and the efficiency of the developed model is further validated.

Conclusions

A new constitutive model, which combines the surface layer model and composite model, is developed in this research. Based on this model, the grain in the micro-specimen is divided into two portions, viz., grain interior and boundary. The flow stress of material is then determined based on the ‘law of mixtures’ applied to the grain interior and boundary. The model is verified by experiment and FEM simulation via three case studies, viz., micro-upsetting and extrusion of bulk metal and

Acknowledgements

The authors would like to thank the funding support to this research from the Research Grants Council of Hong Kong government under the Project of B-Q08V and the Projects of G-U923 and A-PJ29 from The Hong Kong Polytechnic University.

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