Bao MENG, Jiejie SHI, Yiyun ZHANG, Cheng CHENG, Bolin MA, Min WAN

School of Mechanical Engineering and Automation, Beihang University, Beijing 100083, China

KEYWORDS Failure criteria;Forming limit;Metal foil;Microforming;Size effect

Abstract Forming limit of metal foil is an important index to evaluate its formability, and is of considerable significance to improve the quality of products. The ductile fracture (DF) behavior in microscale plastic deformation is remarkably affected by the geometry and grain size.To explore the size-dependent forming limit curve (FLC), the Holmberg and Marciniak tests of SUS304 foils with the thicknesses of less than 0.1 mm and diverse grain sizes were performed. In addition, the validity and feasibility of three types of existing failure models including Swift/Hill, Marciniak-Kuczynski(M-K)and DF criteria for predicting the micro-scaled FLCs were discussed.It is found that the Swift/Hill model possesses the worst accuracy with predicting deviation above 50%. Four classical DF criteria including Freudenthal, Ayada, Brozzo and Oh show great difference, and Oh model considering plastic anisotropy presents the best precision. The predicted deviation of M-K model is aggravated with increasing grain size and decreasing foil thickness, which is attributed to the intensified free surface roughening and transformation of fracture mechanism with miniaturization. This research thus provides a deeper understanding and valuable reference for the widespread application of FLC in microforming.

1. Introduction

As the demand of miniaturized and integrated products grows higher in consumer electronics, medical device, automobile and aerospace, the miniaturization trend has continued for decades.1In these application scenarios, the geometric dimensions of the fabricated workpieces are usually less than 1 mm in two dimensions, which are termed as microparts.2Diverse micro-manufacturing methods including laser-processing,micro-machining and microforming have been developed rapidly. Among them, the microforming is recognized as one of the most prospect techniques for the manufacturing of the ultrathin-walled micropart owing to its advantages of high material availability, high production rate, and desirable performance of the formed components.3,4The conventional forming approaches for metal foils mainly include microblanking,micro-bending,micro-drawing and micro-bulging.5,6

Many studies have demonstrated the size dependent deformation and fracture behaviors of metal foils. The wrinkle and rupture defects are prone to appear during the forming process, which significantly affects the quality of the final products. Forming limit curve is recognized as one of the most crucial indicators to characterize the rupture and formability of sheet metal,which can be achieved by experimental and theoretical methods.Since the existence of size effect,the fracture pattern is transformed and the forming limit changes accordingly. A mixed model considering both the crystal boundary and interior crystal effect was proposed by Fan,7and the impact of crystal size on the failure behavior of foils was revealed. It is found that the microscale failure behavior presents a great difference as the thickness of sheet decreases.Zhao et al.8conducted tensile tests of copper sheets from 0.25 to 2 mm thick with both coarse and fine grains, and they discovered that the fracture mode continually transforms from tensile failure to shear failure as the thickness reduces. Peng et al.9achieved the same conclusion and argued that the micro voids are gradually reduced or even disappeared when only very little grains involved in the deformation process. Meng and Fu10explored the impact of free surface roughening on the ductile fracture of copper sheets with the thicknesses of 0.2, 0.4 and 0.6 mm, and they pointed out that the severe coarsening and less necking appeared before the rupture with the decreasing of foil thickness and increasing of grain size.Cheng et al.11proposed a sectionalized failure criterion to capture the microscale FLC considering the coupled factors of surface roughening and fracture mechanism transform.Consequently,the internal failure mechanism of foils reveals vast differences with conventional sheet metals.It is thus necessary to have a deeper understanding of size effect on the formability and characterize the foil formability more reasonably and effectively. Since a large number of experiment samples and complicated devices are needed to attain the complete forming limit diagram,12,13the prediction of the formability is developed on to reduce the experimental cost. The failure criterion is the theoretical basis of the forecast of formability, and the prediction accuracy is significantly affected by the chosen model. The commonly-used macroscale fracture models include Swift diffuse and Hill localized instability criteria, MK model, and ductile fracture theory. Xu et al.12improved the Oyane theory by incorporating the grain numbers along the thickness direction and achieved a better prediction of FLC for the sheet metals with the thicknesses of 0.1, 0.2 and 0.4 mm. Chen et al.13performed a number of tests of metal foils with the thicknesses of 150, 100, 50 and 20 μm, and they proposed an improved Oh model by considering the strain path and grain size, which dramatically enhances the prediction accuracy of forming limit. By combining surface roughness with Parmar-Mellor-Chakrabarty (PMC) model, Bong et al.14realized a better FLC prediction of 0.1 mm thick stainless steel foil.

Although the size effect significantly affects ductile failure behavior of thin sheet metal, the current knowledge on size effect is still far from adequate to the demand of practical application and microproduct development, especially for the metal foils with the thickness less than 0.1 mm.It is thus urgent to conduct further exploration of the size effect on forming limit of foils to promote the development and application in micro sheet forming process.Moreover,the current prediction of microscale forming limit is still utilizing macroscale plastic instability theories, and their validity at microscale is still not evaluated, especially for the meal foil thinner than 0.1 mm.In this research,the validity of diverse failure models at microscale was examined by FE simulation and physical experiment,meanwhile, the internal influence factors on the feasibility of different failure models were analyzed.

2. Experimental procedures

2.1. Specimen preparation

The SUS304 stainless steel foils with three thicknesses including 100,50 and 20 μm were utilized to explore the size effect on the microscale forming limit. To further study the grain size effect on deformation behavior and formability, the samples were heat treated under diverse temperatures and dwelling times to attain various sizes of grain, as shown in Table 1.

2.2. Uniaxial tensile test

To acquire the true stress-strain curves of metal foils, the tensile tests were performed on various material conditions. The geometries of specimens with the thicknesses of 0.1 mm and 0.05 mm were designed according to ASTM E8-08 standard,while the geometry was scaled down according to E345-93(16) for foils with the thickness of 0.02 mm, as shown in Fig. 1, where t is the thickness of specimen. The whole tensile tests were conducted at the speed of 5 mm/min on the MTS tensile testing machine.15The digital image correlation (DIC)approach was used to measure the strain in the whole tensile process. A series of images were continuously captured by a digital camera during the tensile test.By calculating the evolution of the spots in the pictures, the displacement and strain were acquired.15After processing and analyzing the data received in the uniaxial tensile tests, the basic property and deformation behaviors of metal foils can be achieved.

2.3. FLC tests

In this research, the Marciniak and Holmberg experimental approaches were employed to obtain the right and the left half of FLCs, respectively. The Holmberg tests were performed on the uniaxial testing machine, and various strain paths can be obtained by changing the geometry and dimension of samples,as shown in Fig. 2. In the Marciniak test, to increase the friction and avoid the premature failure of the ultrathin sheets,the sample was deformed indirectly through a driving sheet with a hole through the center, and the diameter of the hole is 3.2 mm. The radius of the punch and die is 5 mm and 6 mm,respectively.DIC approach was also used to capture the deformation strain, and the first-order differential of strain difference between the necked area and un-necked area was checked to determine the necking moment.16

3. Failure criteria for the prediction of FLC

3.1. Swift diffuse and Hill localized instability criteria (Hill48)

Swift17first put forward the diffuse instability theory,believing that it is the moment to neck when the imposed tensile forcereaches the peak value. The failure moment is deemed to satisfy Eq. (1), where σ1is principal stress and ε1is principal strain.

Table 1 Foils with different thicknesses and grain sizes.

Fig. 1 Geometry of metal foils for uniaxial tensile tests.

Compared to diffuse instability, the localized deformation has a stronger impact on the ultimate failure. Based on Hill’s instability criterion, the condition for the occurrence of instability at the relative growth rate of principal tensile stress is equal to the equivalent thinning of thickness, which can be expressed as follows, where σ2is the second principal stress and ε3is the third principal strain.

Hill’s hypothesis limits the application in tension-tension deformation, which means that the theory can only predict the left half of FLC.18Therefore, the Swift diffuse instability can be used to predict the right half of the FLC, while Hill localized instability is utilized to predict the left part. According to Eq. (1), the Swift diffuse instability theory can be deduced as the following equations.

Fig. 2 Experimental geometry and dimension for forming limit tests.

ε1* and ε2* is the major and minor principal strain respectively. It is observed that the forecasting outcomes according to Swift/Hill theory depend on the hardening and anisotropic coefficients, as shown in Eq. (4).

3.2. Ductile fracture criteria

Based on the damage theory, the DF criteria introduce variable constant, i.e., C, relevant stress and strain parameters to describe the mechanical effects of micro defects. DF criteria have realized extensive use in traditional metal forming process and have achieved satisfactory prediction results.19The wellknown criteria including Freudenthal, Ayada, Brozzo and Oh models are presented in the following:

3.3. M-K model

The Newton-Ranphson iteration method was chosen to solve the limit strain. First, once the stress ratio α of region A and equivalent strain increment of region A(dεA=0.0001) are given, and dεBcan be computed by Eq.(8). When the major strain increment of region B is far larger than that of region A (dεB/dεA≥10 in this research), the corresponding strains including ε1Aand ε2Aobtained at the moment are considered as the forming limit stains.

The initial degree of non-uniformity f0is recognized as a crucial parameter in M-K model,directly determining the prediction accuracy.However,there is still a lack of a proper satisfactory method to obtain the value of f0. Furushima et al.21and Meng and Fu10both found that the severe surface roughening occurs and develops during the microscale deformation process, which coincides well with the physical principle of the M-K model. Therefore, f0can be associated with the surface roughness, which is expressed as follows:

where Rzis the surface roughness.

Fig. 3 Illustration of M-K theory.9

4. Results and discussion

4.1. Deformation behavior of metal foil

Fig. 4 True stress-strain curves of the samples with different thicknesses along rolling direction.

The true strain-stress curves of metal foils along the rolling direction were shown in Fig. 4. The flow stress is decreased with increasing grain size, which can be well explained by the Hall-Petch relation and surface layer model.22The classic model believes that the surface grains are constrained less than the inner ones, and then cause the deformation more easily.Besides, when the thickness of metal foil remains unchanged,the surface grains occupy a larger percentage with the increasing of the grain size,directly leading to the striking decrease of flow stress. However, this decrease tendency of flow stress is not absolute during the whole deformation process.As shown in Fig.4,where d represents the grain size,the two curves with larger grain size intersect each other and the intersection point moves forward with the decrease of foil thickness, which suggests that the growth rate of flow stress is higher in the later deformation for metal foils with larger grain size. This can be explained by the fact that the ability of intergranular coordination is declined as the thickness decreases and the grain size increases. Moreover, SUS304 is a type of austenitic stainless steel, performing phase transition from austenite to martensite in the plastic deformation process and the latter possesses higher intensity and hardening rate.Hollomon equation was used to fit these curves, and the relevant parameters were listed in Table 2, where K and n represents the strength coefficient and hardening exponent, r0,r45and r90are the indexes of anisotropy along different angles with rolling direction.It is noted that the hardening coefficient is grown dramatically with increasing grain size.

In addition,the anisotropic coefficients were summarized in Table 2. It is found that the anisotropy is shifted down with increasing grain size. Since the anisotropic behavior of metal foils significantly affects the forming limit.The effect of anisotropy was considered in the prediction of FLC using diverse failure criteria.

4.2. Microscale forming limit of metal foil

The experimental microscale FLCs were shown in Fig. 5. It is suggested that the FLCs are descended with the increasing grain size, which agrees well with previous research.23The intergranular microcracks are easily extended, which explains the decline tendency of FLCs with increasing grain size.9Besides, an increasing dispersion of the experimental data was observed, especially for the foils with the thickness of 20 μm,which is considered as the results of the inhomogeneous property of individual grain when there are only a few grains involved in the deformation. When the number of grains participating in the deformation declines, the fracture behavior is gradually dominated by the properties of individual grains and strong anisotropic characteristics at microscale.1Existing studies have shown that grain size inhomogeneity and the preferred orientation of individual crystal lead to the local shear failure,which leads to the premature fracture and the growingly severe dispersion of experimental data.23

4.3. Microscale FLC prediction using diverse criteria

4.3.1. Prediction based on Swift/Hill model

The predicted results based on Swift/Hill criterion were presented in Fig. 6. It displayed apparent discrepancy between the experimental and theoretical outcomes,which suggests that the Swift/Hill instability theory is no longer credible under micro-scaled deformation. The prediction overrates the FLC under all the material conditions. It is believed that the forming limit is declined when the grain size grows, while the prediction reveals an opposite trend. The reason for the failed prediction based on Swift/Hill criterion lies in a series of ideal hypotheses.The limit strain calculated by Swift and Hill theory is only connected with the hardening coefficient and anisotropic behavior.As the crystal size grows,there tend to be one or two grains in the direction of thickness. Under this condition,the size effect can no longer be neglected. On the other hand,

the micro voids also affect the material formability during the plastic deformation,which was not considered in the Swift/Hill model. Thus, the size effect and microstructure-related factors need to be considered to acquire a more accurate prediction of microscale forming limit.

Table 2 Material parameters under different conditions.

Fig. 5 Experimental forming limit curves of metal foils at various thicknesses.

Fig. 6 Prediction results based on Swift and Hill instability theory under different foil thicknesses.

4.3.2. Prediction based on DF criteria

The material is conceived to fracture when the material constant of C in Eq. (5) reaches the critical value according to the DF criteria. In this research, the values of C were calculated by the data from uniaxial tensile tests under each material condition, as presented in Fig. 7. It is observed that the values of C are close to each other for Oh and Brozzo criteria and emerge a significant upward trend for all the four models with the growth of λ (the ratio of foil thickness to grain size).Therefore, the geometrical and grain size have a profound effect on the material constant, and which in turn affects the fracture behavior at microscale.

By adjusting different values of ρ in Eq. (7), the various strain paths can be attained. The final predicting curves based on different DF criteria were depicted in Fig.8.It is found that the discrepancy of experimental data and predicted results using diverse models is evident. Besides, the prediction based on Freudenthal criterion exceeds the experimental data,which means the overestimate of formability for metal foil. In addition, the experimental data and predicting ones also manifest weak consistency in the intermediate portions, indicating that the prediction accuracy under the planar strain state is much lower than that under other strain paths. On the contrary,the limit strain is underestimated by Ayada criterion, and the degree of accuracy on the left-hand is higher than the right side.Furthermore,the Brozzo model presents a worse regularity on the right half side. Whereas, the Oh criterion presents the best prediction precision among all the DF criteria.In general,DF criteria still cannot achieve a satisfactory result under the planar strain path.

From the above brief analysis, the predicted left-hand FLCs based on DF criteria are consistent well with experimental data. However, the predictions under the plane strain and biaxial tension are unsatisfactory.DF criteria judge the failure behavior based on the integral form of plastic work, which belongs to a way of localized instability and fracture. However,the right half of the FLCs tend to prefer the diffuse instability,and therefore,the early fracture of the material leads to the inaccuracy of DF criteria.

4.3.3. Prediction based on M-K model

Stachowicz24believed that the surface roughening is uncorrelated with stress state,and the value of f0depends on the original surface roughness of the uniaxial tensile specimen.According to Eq. (8), the predicting results were obtained in Fig. 9 based on M-K model. It is evident that the predicted curves are far from the experimental data by utilizing the f0.In addition, the prediction deviation from the experimental results is aggravated with increasing grain size,which suggests the noticeable size effect on the failure mechanism and forming limit of metal foils.

Fig. 7 Variation of C with λ based on different DF criteria.

Fig. 8 Prediction results based on different DF criteria for metal foils with different thickness.

Given the vital position of the size effect in the theoretical prediction of forming limit, it is necessary to characterize the relationship between surface roughening and grain size. The surface roughness was measured before and after deformation,as indicated in Fig.10.It is found that the surface roughness is increased with increasing grain size and decreasing thickness,and reaches 20%of the foil thickness for the 20 μm thick foils.Thus, it is no longer suitable to use the f0in M-K model affected by surface roughness to analysis the formability of foils, which is affected significantly by size effect. Serenelli et al.25also argued that f0should be an adjustable parameter,which is related to surface quality,grain size and micro defects in the material.

In general, there are mainly two reasons leading to the invalidation of M-K model, as shown in Fig. 11. On the one hand,as the thickness of sheet continues to decrease,the ratio of surface roughness to thickness Rz/t is significantly declined,which means the influence of surface imperfection on failure behavior is expanded for micro-scaled metal foils.On the other hand, with the decrease of λ, the amount of inner voids is reduced dramatically, which suggests that the fracture mode is transformed especially when there is only one or less grain along the thickness. Therefore, the surface roughness tends to occupy a more important role than the inhomogeneous distribution affected by void growth.26

Fig. 9 Predicted results using M-K model for metal foils with diverse thicknesses.

4.3.4. Comparison of different models

To quantify the prediction accuracy of different models, the average deviation φ is defined in the Eq. (10).

Fig.10 Surface roughness variation after deformation for metal foil with diverse thicknesses.

where m is the amount of strain paths,as shown in Fig.12.Δεi(i=1,2, ...,m)is the deviation between the experimental major strain and the predicted one, and εiexprepresents the experimental major strain at strain point of i.

According to Eq.(10),the relative errors of Swift/Hill instability,M-K model and Oh criterion that has highest prediction accuracy among the four DF criteria are summarized in Fig.13.It is observed that the relative deviation of the predicting results dramatically increases for both Swift/Hill and M-K models with the rise of grain size. In addition, the prediction error of Swift/Hill model exceeds fifty percent. It is because the model considers the assumption that the flow hardening produces strain concentration but neglects the important effect of micro defects on the fracture behavior.M-K model achieves great prediction results when the grain size is fine, which also confirms the applicability of M-K model at macroscopic field.However,as the grain size increases,the hardening behavior is changed correspondingly and the varieties of parameters in the constitutive model reduce the accuracy of M-K model.On the other hand,the parameter of f0has a great impact on the predictive results of M-K model and the surface roughening behavior of metal foils with different grain sizes is varied during the plastic deformation. Therefore, it is no longer suitable to adopt the same f0for foils with different grain sizes. These two reasons lead to the low precision of M-K model at micro scale and limit its application in microforming.

Fig. 11 Size effects on void nucleation and surface roughness.

Fig. 12 Schematic diagram of deviation calculation.

The feasibility of DF criteria is affected significantly not only by the proper damage evolution regulations but also a few influential factors, such as hydrostatic stress, stress triaxiality, Lode parameter, equivalent plastic strain etc. Different stress states give rise to the difference of strain increment,namely the variation of damage evolution.27Meanwhile, as the increase of grain size, the anisotropy of sheet becomes more obvious according to the previous study.16Thus, the evolution of micro voids in ductile fracture is considered to some extent by introducing the hydrostatic stress of σm,which in fact includes the effect of stress triaxiality on the growth of voids. The disunity of stress and strain in FLC tests leads to the deviation of Freudenthal model, which is only related to the equivalent stress and deformation strain.The denominator σ1-σmin Brozzo model is not a fundamental indicator to determine the stress state uniquely and also doesn’t have a clear relationship with damage behavior.28Ayada model fails to realize the accurate prediction on the right half of FLCs. The conceivable reason is that the first tensile stress occupies a more important position in the damage accumulation than hydrostatic stress under the biaxial tensile stress. In contrast, Oh model include the crucial factors including both the first principal stress and equivalent stress. The damage accumulation is thus taken into consideration and the failure positions can be described reasonably.Therefore, Swift/Hill and M-K models are no longer applicable at micro scale while the DF criteria achieve satisfactory results as a whole, especially for Oh model.

5. Conclusions

In this research, the size effect on the formability of metal foil was analyzed and discussed. In addition, the validity and applicability of Swift/Hill model, M-K theory and DF criterion for the prediction of microscale FLC were explored.The essential factors affect the validation of diverse failure model at microscale were discussed. Based on the present investigations, the main conclusions are drawn as follows:

(1) The FLC of metal foil is significantly affected by size effect and presents an apparent decreasing trend with the increase of grain size for a given foil thickness.

(2) The prediction deviation of Swift/Hill criterion for the metal foils with different grain sizes exceeds 50%, and appears an increasing tendency with the growth of grain.The variation tendency of the predicted FLCs with the rising grain size is contrary to the experimental data,revealing that the Swift/Hill model is not applicable for the prediction of microscale FLC of metal foils.

(3) The DF criteria exhibit relatively high accuracy on the left-hand of FLC and the forecasting accuracy of diverse DF criteria emerges substantial differences. The Oh model presents the best predicting accuracy under different foil thicknesses and grain sizes. In addition, the prediction based on the M-K theory overestimates the formability of metal foils, and the deviation is aggravated with increasing grain size.

(4) The invalidation M-K model for the prediction of the microscale FLCs of metal foils is attributed to the interactive effect of free surface roughening and fracture mechanism transformation when there are few grains participating in the deformation.

Acknowledgments

The authors would like to acknowledge the funding support to this research from the National Natural Science Foundation of China (Nos. 51605018 and 51635005) and Beijing Municipal Natural Science Foundation of China (No. 3172022).