Elsevier

Materials & Design

Volume 31, Issue 2, February 2010, Pages 1010-1014
Materials & Design

Short Communication
A constitutive model for thin sheet metal in micro-forming considering first order size effects

https://doi.org/10.1016/j.matdes.2009.07.037Get rights and content

Abstract

Size effects in the minimization make most know-how of traditional macro-forming not suitable for the micro-forming process. Material behaviours greatly vary in micro-forming process with different thickness and grain size. The influence of the first order size effects (size effects I) on flow stress has been studied quantitatively using the surface layer model and a new presented internal grain boundary model. It was discovered that the flow stress of micro-forming can be expressed as a function of t/d (thickness/average grain size) ratio. Then, a new constitutive model considering the size effects I for thin sheet-based micro-forming has been developed combining the Hollomon equation and Hall–Petch relation. The model has also been validated with the experimental results in the literature. A good coincidence is obtained when t/d > 1. However, it has great offset when t/d < 1 due to the orientations of the local individual grains have great influence on flow stress, which can be ignored by the new constitutive model because it only considers the influence of grain size. This investigation gives the support for theoretical and numerical study of micro-forming.

Introduction

Miniaturization is a general trend in micro system industry. With the rapid development of micro electro mechanical systems (MEMS) and electronic technology, a significant progress has been made in the fabrication of micro-parts via various methods. In these methods, micro-forming is an appropriate technology to efficiently produce large numbers of metal micro-parts at low cost. Recently, lots of electronics devices need many micro-parts of aluminium or copper sheets, with a thickness typically in the range of 10–100 μm. Micro-parts can also find their applications in chemical micro-reactors or medical devices [1]. However, to realize the mass production for micro-parts, there are many limitations on fabrication need to be overcome. When miniaturization occurs, the material behaviour is characterized by only a few grains located in the deformed area, and the material can no longer be considered as a homogeneous continuum as in the macro-scale. In addition, size effects and surface effects make most traditional theories of macro-forming not suitable for the micro-forming process [2]. Besides the forming temperature and strain rate, the thickness, processing parameters and material behaviour like grain size of material have great impact on the flow stress during micro-forming process. Therefore, it is necessary to develop a constitutive model considering the size effects in micro-forming field.

Generally, size effects are divided into the first order size effects (size effects I) and the second order size effects (size effects II) [3]. For micro-parts of ultra-thin sheet, the flow stress of material decreases with the decrease of the thickness (several millimetres to decades of microns), which is called size effects I. When the thickness reduces to the same order of dimensions of the component geometry (micron or sub-micron) as the intrinsic length, the strain gradient effects occur, which is called size effects II [4]. At present, lots of investigations have focused on the strain gradient plasticity theory which can explain size effects II and be applied successfully into numerical study [5]. However, there are only a few investigations on size effects I and the typical theory for size effects I is the surface layer model presented by Geiger and Engel [6], [7]. In this paper, the influence of size effects I on flow stress of micro-forming has been studied quantitatively using the surface layer model and a new presented internal grain boundary model. Then a new constitutive model considering the size effects I for thin sheet-based micro-forming has been developed combined with the Hollomon model and Hall–Petch relationship. The model has also been validated with the experimental results in the literature.

Section snippets

Quantitative analysis of size effects I

According to Kim et al. [8], the size effects can be divided as the feature/specimen size effect and the grain size effect. In general, the feature/specimen size can be referred to as the thickness of thin sheet metal in micro-forming. For size effects I, the grain size effect follows the Hall–Petch equation when the thickness is kept constant upon minimization [9], and the flow stress of the material decreases with the increase of the grain size. The feature/specimen size effects that the flow

The new constitutive model considering the size effects I

According to the quantitative analysis of size effects I using the internal grain boundary model and the surface layer model, it can be approved that there is one certain relationship between the t/d ratio and the flow stress of the material. It also provides the basis for studying the constitutive model considering the size effects by the index of t/d ratio. The Hollomon equation which has been widely applied to describe the stress–strain relation is used for investigating the relationship

Experiment

The micro-tension tests on C1200 in the literature [19] are used to validate the new constitutive model considering the size effects I for thin sheet-based micro-forming. The Swift model for the C1200 material obtained in the tests has the form shown in Eq. (11):σtrue=K(ε0+εtrue)nwhere εtrue is the true strain, σtrue is the true stress, ε0 is the initial strain, n is the strain-hardening coefficient and K is the strength coefficient. Table 1 showed the material parameters of 8 Swift equations

Conclusions

Size effects are the most challenging problems in micro-forming process. The influence of size effects I on flow stress of micro-forming has been studied quantitatively using the surface layer model and a new presented internal grain boundary model. Based on these models, a new constitutive model considering the size effects I for thin sheet-based micro-forming has been developed combining the Hollomon equation and Hall–Petch relationship. It has also been validated with the experimental

Acknowledgements

This paper is from the project sponsored by the National Natural Science Foundation of China under Grant 50805069 and 50605029. Also, the work is supported by Natural Science Foundation of Jiangsu Province under Award Number BK2006551, Postdoctoral Foundation of China (20060390961 and 20090451174), Postdoctoral Foundation of Jiangsu (0802024C) and Initial Foundation of Super Talents of Jiangsu University (1221110022).

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