Advances in Materials Science and Engineering

Volume 2018, Article ID 5287945, 14 pages

https://doi.org/10.1155/2018/5287945

## Analysis of Springback Behaviour in Micro Flexible Rolling of Crystalline Materials

^{1}School of Mechanical, Materials, Mechatronic and Biomedical Engineering, University of Wollongong, Northfields Avenue, Wollongong 2522, Australia^{2}Shanxi Province Metallurgical Equipment Design Theory and Technology Key Laboratory (Provincial Department to Build National Key Laboratory Training Base), Taiyuan University of Science and Technology, Taiyuan 030024, China

Correspondence should be addressed to Xiaogang Wang; nc.ude.tsuyt@gxw

Received 8 May 2018; Revised 18 September 2018; Accepted 19 September 2018; Published 15 November 2018

Academic Editor: Donato Sorgente

Copyright © 2018 Feijun Qu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper presents a constitutive modelling of the polycrystalline thin metal strip under a state of combined loading in microflexible rolling. The concept of grained inhomogeneity is incorporated into the classic Chaboche hardening model that accounts for the Bauschinger effect, in order to provide more precise description and analysis of the springback mechanism in the particular forming operation. The model is first implemented in the finite element program ABAQUS to numerically predict the stress-strain relationship of 304 stainless steel specimens over a range of average grain sizes. After validation of the developed model by comparison of predicted curves and actual stress-strain data points, it is further applied to predict the thickness directional springback in microflexible rolling of 304 stainless steel strips with initial thickness of 250 *µ*m and reduction changing from 5 to 10%. The model predictions show a reasonable agreement with the experimental measurements and have proven to be more accurate than those obtained from the conventional multilinear isotropic hardening model in combination with the Voronoi tessellation technique. In addition, the variation of thickness directional springback along with the scatter effect is compared and analysed in regard to the average grain size utilising both qualitative and quantitative approaches in respect of distinct types of data at different reductions.

#### 1. Introduction

Rolling technology has been investigated extensively in the last few decades, during which theory and research have been formulated by using a combined analytical and numerical approach [1, 2]. However, thanks to the increasing popularity of miniaturisation in modern industrial development over recent years, it has been promising to look into the flexible rolling process of micrometer thick materials, whereof the overall dimensions as well as the longitudinal thickness profiles of the workpieces may be adapted to the requirements of the microapparatus and systems. Pioneering investigations into the microflexible rolling process have been conducted with respect to the rolling force, the elastic recovery in thickness direction of the workpiece, the surface roughness evolution related to different process variables, and so forth [3–6]. Due to the thinness of the material, the thickness directional springback of the rolled workpiece, mainly caused by the elastic redistribution of the internal stresses upon removal of the rolling force, can lead to failure to produce a predefined thickness profile along rolling direction [7]. Consequently, an appropriate control for the springback is worthy of more attention and deeper examination.

One basic approach to the regulation of springback attempts to reduce or compensate for it through modification of process parameters or adjustment in tool design. For instance, Anggono et al. [8] developed a hybrid algorithm to rectify the springback error by iterative comparison between the deformed and target shapes, which provided a faster correction with a smaller deviation from the desired dimension during a U-channel forming case. Wang et al. [9] presented an incremental bending methodology to minimise the springback error in air bending process by utilising the data gathered from loading-unloading cycles, based on which the workpiece thickness, and material properties could be more accurately assessed and then used to calculate the final punch position to achieve the target bend angle. An irritation with a high-power diode laser was deployed by Gisario et al. [10] to control the extent of the springback by selectively heating the bent area in V-shape bending of aluminium alloy sheets. With regard to the flat rolling process, automatic gauge control technology has been widely adopted to create real-time adjustments of the roll gap in accordance with the feedback of the workpiece exit thickness, so as to correct the difference between the desired and actual thicknesses induced by springback [11–13]. Nevertheless, the cost and unpredictability of springback compensation can be reduced providing that the springback potential is recognised prior to the actual manufacturing [14].

As a consequence, springback prediction is becoming progressively more significant in metal forming processes and has been increasingly dependent upon numerical modelling and simulation techniques in recent years. For instance, Ankėnas and Barauskas [15] presented numerical evaluation of springback of sidewall and flange under various process conditions in U-shaped deep drawing with mild steel materials. Wagoner and Li [16] estimated the suitability of different numerical integration schemes as well as the number of through-thickness integration points in terms of springback accuracy in a beam bending model, which was reported to vary with material properties, *R*/*t* values, and sheet tensions. Sumikawa et al. [17] led a probe into the influence of material behaviour on the accuracy of springback analysis, and then developed a material model with a consideration of elastoplastic anisotropy and Bauschinger effect to further improve the accuracy of springback prediction during the hat-shaped forming using high-strength steel sheets. Furthermore, Wang et al. [18] proposed an analysis model based on the surface layer model and composite model to predict the unloading springback of copper alloy sheets in microscaled U-bending process, taking into account the interactive influence of geometry size, grain size, and punch radius. The pattern of the size effects affecting the springback was disclosed by the numerical simulations, physical experiments, and the comparison in-between. Diehl et al. [19] performed finite element simulations to reveal the variation of springback angle in relation to different material grain sizes and strain gradients in free bending of thin aluminium foils with thickness to the order of micrometers. Jiang and Chen [20] numerically investigated the grain size effect on the springback behaviour of microtubes in press bending process, and they discovered that the springback amount decreased with a decrease in thickness to average grain size (T/D) ratio for microtubes with a constant wall thickness and varying average grain size, whereas it decreased with an increase in T/D ratio for microtubes with constant average grain size. Since the springback phenomenon is closely related to the residual stress after the part has been released from the forming tools, a proper characterisation of residual stress may enhance the accuracy of springback prediction, which nonetheless highly depends on the modelling of material behaviour during the forming process [21].

For decades, numerous investigators have observed the constitutive behaviour of different materials subjected to an assortment of forming operations involving cyclic loading or complex loading conditions, under which the Bauschinger effect has widely been acknowledged, namely, the reduction of yield stress upon load reversals, which is usually accounted for by the release of dislocations piling up at the grain boundaries during initial deformation [22]. Prager [23] and Ziegler [24] first developed the linear kinematic hardening models to describe the Bauschinger effect, and the difference between their models was reflected by the translation direction of the yield surface. Later, Armstrong and Frederick [25] and Chaboche [26] modified these linear models into nonlinear ones by adding an extra term so as to capture the transient behaviour and reproduce the ratcheting in fatigue. Since then, increasing and diverse demands have encouraged improvements in either the Armstrong–Frederick model or the Chaboche model, such as to control the evolution of the decomposed kinematic hardening rules by Ohno and Wang [27, 28], define effective quantities in stress and plastic strain rate by Chung and Richmond [29], or incorporate the permanent softening along with the Bauschinger effect and transient behaviour by Chun et al. [30, 31]. Besides, two-surface models have been formulated by Krieg [32] and Dafalias and Popov [33] to define a continuous variation of hardening between these two surfaces, namely, that the evolution of the inner surface characterises the transient response of the material, whereas the evolution of the outer surface represents the long-term response of the material such as the Bauschinger and softening phenomena. McDowell [34] introduced a two-surface stress space model by incorporating the effects of changes in plastic strain range and nonproportionality of loading in the evolution of isotropic hardening in order to enhance the prediction accuracy of complex nonproportional deformation behaviours like cyclic ratcheting, mean stress relaxation, etc. D. Kim and J. Kim [35] developed a two-surface model for rate-dependent plasticity by combining both isotropic and kinematic hardening rules. This model was validated through both monotonic and cyclic loading cases using structural steels, where simulation results exhibited an excellent agreement with experimental values in terms of the maximum stress and shape of hysteresis. Hashiguchi [36] incorporated a subloading surface within the inner surface of the two-surface model so as to express the smooth elastic-plastic transition, avoiding an abrupt change of the strain rate/stress rate relationship when loaded from the stress state within the inner surface. Moreover, a multisurface model was proposed by Mroz [37] to combine properties of isotropic and kinematic workhardening via a configuration of surfaces of constant workhardening moduli, which specified an expression for the general features of the cyclic loading behaviour as well as a smooth transition from the elastic to plastic deformation.

These models have been set up and adopted primarily to describe the constitutive characteristics of materials at the macroscale; nevertheless, the so-called size effects, including geometry size, grain size, and grained inhomogeneity effects, cannot be negligible when constructing constitutive models will be applied on the microscale. For instance, Lu et al. [38] established a microscale constitutive model on the basis of grain size, shape, deformability, and specimen dimension to discuss the deformation behaviours of polycrystalline copper grain with different grain and feature sizes in microforming processes, like microcompression and microcross wedge rolling. According to the dislocation theory, Jiang et al. [39] proposed a constitutive model with consideration of forming temperature, Hall–Petch relationship and surface layer model to investigate the influences of temperature and grain size on the deformability of pure copper in microcompression process, where both factors were found to significantly affect the shape accuracy of the produced product, as well as the metal flow behaviour due to material heterogeneity. Liu et al. [40] developed a novel constitutive model to characterise the effects of grain size and boundary and geometry size on the flow stress of pure copper foils as well as on the microextrusion of CuZn30 alloy billets. Utilising the surface layer model, they divided the specimen into two portions, viz., surface layer and inner portions, while the grains of the inner portion were further divided into two portions, viz., grain interior and grain boundary. Likewise, Wang et al. [41] built a comprehensive constitutive model on the basis of surface layer model with consideration of the influences of grain size, material thickness, grain number through thickness, and surface property. The developed model was employed to predict the mechanical properties of thin sheet metal, and the results were compared with the physical experimental results, which confirmed that their model was particularly suitable for thin sheets with one or several grains across the thickness.

In this study, a nonlinear elastic-plastic constitutive model incorporating the two yield surface constitutive model as well as the Chaboche kinematic hardening rule is developed with the aim to give a more accurate mathematical description of the anisotropic workpiece responding to the complex loading during microflexible rolling process, thus enhancing the prediction accuracy of thickness directional springback after unloading. The newly constructed model is integrated into the ABAQUS geometric modelling, utilising the user-defined subroutine user material (UMAT) as an interface. Firstly, this model is validated by simulating the miniature tensile test and comparing results with experiments using 304 stainless steel specimens. Secondly, different constitutive/numerical models are employed to characterise the mechanical behaviour and predict the thickness directional springback during microflexible rolling of 250 *µ*m thick 304 stainless steel strips with reduction of 5 to 10%. Thirdly, laboratory-scale microflexible rolling experiments are carried out to further verify and validate this novel model which produces more accurate predictions than the multilinear isotropic hardening model and the Voronoi tessellation based model. Finally, both simulation and experimental results of thickness directional springback are quantitatively revealed with respect to the average grain size, and the scatter effect caused by grained inhomogeneity is analysed synchronically in regard to the average grain size in a qualitative manner.

#### 2. Constitutive Modelling Based on the Concept of Grained Inhomogeneity

##### 2.1. Elastic-Plastic Mixed-State Model

It has been stated that the diversity of grains leads to the heterogeneous mechanical response of the whole granular material, viz., that grains with low yield strength may have suffered plastic deformation while those with higher yield strength are still in their elastic realm at a certain applied stress [5, 6]. The yield anisotropy of this kind can be abstractly described using two yield surfaces metaphorically termed “initial yield surface” and “complete yield surface”, which specify the transition from elastic deformation to mixed elastic-plastic deformation and that from mixed elastic-plastic deformation to plastic deformation, respectively.

As von Mises yield criterion has been adopted to analyse the constitutive behaviour of the rolled material, these two yield surfaces can be generally expressed by (for the plane-stress condition, *σ*_{3} = 0) [42]where and , in which *α*_{ij} and are the position tensors of the centres of the initial yield surface *F*_{1} and complete yield surface *F*_{2} in the stress space, respectively, associated with the deformation history of and the hardening rule for the material, whilst and , in which *σ*_{s} and represent the sizes of these two yield surfaces, depending on the equivalent plastic strainwhere denotes the components of the plastic strain increment. The relationship between the size of the yield surface and the effective plastic strain may be acquired by the uniaxial tensile test. The two surfaces are assumed to be of the identical form, having nearly parallel normal vectors at points *σ*_{ij} and on *F*_{1} and *F*_{2}, respectively.

In accordance with the applied stress, there are probably three stress states in the material, to wit, pure elastic, mixed elastic-plastic, and complete plastic. The constitutive laws for each stress state are described as follows:(a)For the pure elastic state: The sizes and centres of *F*_{1} and *F*_{2} remain unchanged for the applied stress, ** σ** is within the initial yield surface

*F*

_{1}, and only elastic strain occurs during this state. In the light of generalised Hooke’s law, the relationship of stress to elastic strain is expressed aswhere

*D*_{e}is the constant elasticity tensor. Note that Equation (3) remains applicable for the elastic states of the material during mixed elastic-plastic loading and plastic loading.(b)For the mixed elastic-plastic state: During the mixed elastic-plastic loading, the size and centre of

*F*

_{2}maintain their originality whereas

*F*

_{1}keeps its size unaltered, but its centre evolves (i.e., the Bauschinger effect) until these two yield surfaces make contact with

*σ*_{ij}brought to coincide with , which indicates the commencement of plastic loading. Based upon Ziegler’s linear kinematic hardening rule, the evolution of the centre of

*F*

_{1}(the back-stress

*α*_{ij}) can be written as [24]where

*dμ*is dependent on the material. Equation (4) states that

*F*

_{1}moves in a translation in the direction of the vector connecting the two stress points

*σ*_{ij}and . Due to the coexistence of both elastically and plastically deformed grains, a weighted heterogeneity coefficient [5] is adopted to reflect the proportion, namely, that represents the ratio of strain occurring in the elastically deformed grains, whereas is formulated to represent the ratio of strain occurring in the grains that have entered the plastic region. From this point of view, the stress-strain relationship in the mixed elastic-plastic state follows thatwhere belongs to the interval [0, 2], and are the elastic and plastic parts of the strain occurring in the plastically deformed grain, respectively.(c)For the complete plastic state: After the material undergoes plastic deformation, it is postulated that the size of

*F*

_{1}expands but its centre remains invariable, while both the size and centre of

*F*

_{2}evolve following the Chaboche kinematic hardening rule, taking into account the complex loading conditions, such as compression and torsion, that may exist in the actual microflexible rolling process [43–45].

According to the basic form of the Chaboche model, the strain is partitioned into elastic and plastic parts, which, in the small strain hypothesis, can be formulated by [26]and Equation (3) still illustrates the relationship between the stress and elastic strain during the plastic loading.

For yield surface *F*_{2}, the kinematic hardening evolution of the back-stress is composed of three nonlinear terms as follows [46, 47]:where are material parameters, *ε*_{p} and are the plastic strain and effective plastic strain, respectively. The evolution of isotropic hardening takes the following form [46]:where *Q* defines the maximum change in the size of *F*_{2}, and *b* indicates the rate of change in the size of *F*_{2} as plastic deformation develops. Then the change in the size of *F*_{1} can be evaluated by [48]where , ensuring that no intersections appear between the two surfaces.

##### 2.2. Numerical Implementation of Constitutive Equations

The constitutive equations have been implemented via user-material (UMAT) subroutines in ABAQUS/Standard on the basis of incremental deformation approach [42, 49–52].

Firstly, it is assumed that the stress, back-stresses, elastic strain, and plastic strain are obtained as *σ*^{(n)}, *α*^{(n)}, , , and at the *n*th step, respectively, as well as a strain increment Δ** ε** is given for the (

*n*+ 1)th step.

Neglecting the plastic part in the strain increment Δ** ε**, the initial value of the stress at the (

*n*+ 1)th step can be estimated bywhich is afterwards employed to determine the current stress state of the material:

The stress state is pure elastic. Updating the stress and back-stress inside the initial yield surface gives

The stress state is mixed elastic-plastic. The renewed stress is reckoned by

The back-stress update is executed in the following manner:where Δ*μ* is evaluated through the steps given below:(i)Adopting the von Mises yield criterion for *F*_{1} produceswhere are the stress tensor components andwhere(ii)Substituting Equations (16) and (17) in Equation (15) yieldswhere

The updated back-stress *α*^{(n+1)} is thus obtained by substituting Equation (20) in Equation (14).

The stress is in the complete plastic state. The stress renewal is achieved by

Thereby the updated back-stress and size of *F*_{2} are, respectively, expressed as

The update in the size of *F*_{1} is subsequently made available from

Figure 1 presents the schematic diagram of the constitutive model with a consideration of grained inhomogeneity.