Rare Metals2018年第11期

Material flow behavior modeling with consideration of size effects

Zhen-Wu Ma Zi-Yang Cao Jin-Bin Lu Hua Li Yuan-Jing Zhang Wei Liu Zhen Yin

College of Mechanical Engineering, Suzhou University of Science and Technology

Suzhou Key Laboratory of Precision and Efficient MachiningTechnology

作者简介：*Zi-Yang Cao,e-mail: dukeczy@nuaa.edu.cn;

收稿日期：10 June 2018

基金：financially supported by the Foundation of Suzhou University of Science and Technology (No. XKQ2017005);

Material flow behavior modeling with consideration of size effects

Zhen-Wu Ma Zi-Yang Cao Jin-Bin Lu Hua Li Yuan-Jing Zhang Wei Liu Zhen Yin

College of Mechanical Engineering, Suzhou University of Science and Technology

Suzhou Key Laboratory of Precision and Efficient MachiningTechnology

Abstract：

Size effects make traditional forming theories infeasible in analyzing the micro-forming process, so it is necessary to develop an accurate material model to describe the material flow behavior with consideration of size effects. By studying the size effects of the flow behavior of H80 foils experimentally, it is found that the foil flow stress and strain hardening ability reduce significantly with the decrease of foil thickness. The reduction of the proportion of internal grains which own complete grain boundaries is the main cause of size effects of foil flow behavior. Moreover, grain refinement can reduce the size effects on material flow behavior. On these bases, a phenomenological material model has been developed to mathematically describe the material flow behavior with consideration of the effects of geometry size, grain size and strain hardening behavior. The reasonability and accuracy of this new model are verified by comparing the calculation values with experimental results in metal foil tensile and micro-bulk upsetting experiments. These experimental results and the proposed model lay a solid foundation for understanding and further exploring the material flow behavior in the micro-forming process.

Keyword：

Size effects; Internal grains; Material behavior; Modeling;

Received： 10 June 2018

1 Introduction

With the drastic development of miniaturization in the fields of biomedicine,communications and unmanned aerial vehicles,high-quality and high-volume production become new challenges in micro-manufacturing [ 1] .Various micro-manufacturing technologies are widely applied for manufacturing micro parts including LIGA (lithographie,galvanoformung,abformung) [ 2] ,micromachining [ 3] ,micro-injection molding [ 4] and powder injection molding [ 5] ,etc.Currently,high-precision production can be achieved by these technologies,but high cost and low efficiency severely limit their further development.Microforming technology has attracted extensive attention due to its advantages of excellent mechanical properties,high efficiency and net shape [ 6] .However,when the geometry size of the specimen reduces to the same order of magnitude as the size of the microstructure,the material behavior including the flow behavior,fracture behavior and tribological behavior shows a strong dependence on size,that is,the size effect phenomenon [ 7, 8, 9] .Therefore,mature forming theories cannot be directly applied to microforming processes because of the size effects [ 10, 11, 12] .Currently,the micro-forming process analysis and control is still a trial-and-error procedure based on personal experience,lacking necessary theoretical models as a support [ 13, 14] .

In order to analyze the material flow behavior in microforming more concisely and accurately,many material models have been developed.Geiger et al. [ 15] performed an upsetting experiment on CuZn15 micro-cylinder and found that the flow stress decreased with the decrease of the micro-cylindrical geometry size.Considering that the mechanical behavior of surface grains is weaker than that of internal grains,a classical model called“surface grain model”is put forward by adding items to describe the influence of specimen geometry size on material behavior in the traditional model.This model has been widely adopted by researchers [ 16, 17, 18, 19, 20] .Chan et al. [ 18] proposed methodologies to estimate the properties of grains which own different crystallographic orientations,based on which a model that can describe the inhomogeneous deformation was developed.Wang et al. [ 21] established a“multi-region model”according to the inference that as the geometry size of the specimen decreases,each surface grain in the deformation zone can affect the flow behavior of the entire specimen.In addition,the“composite model”theory holds that materials can be subpided into two sections:“intragranular region”and“grain boundary region” [ 22] .Combining the“surface grain model”with the“composite model”theory,Liu et al. [ 23] developed a model with consideration of grain boundary strengthening effects.Furthermore,Ma et al. [ 24] proposed a size-dependent model based on the implicit assumption that size effects will gradually disappear when there are enough grains in micro-parts.Although these models consider both effects of geometry size and grain size on the material behavior,it cannot be flexibly used for the complicated construction process.

The above review indicates that current material models in micro-forming mainly focus on the description of the effects of geometry size on material behavior.In fact,as the geometry size of materials decreases continuously,its geometry size,grain size and strain hardening property will have combined effects on the material flow behavior.Therefore,it is necessary to research all these size influencing factors and include them when modeling.Besides,the modeling process should be simple and accurate.In this study,H80 foils that have been widely used in microelectronic devices are selected to study the size effects on the material flow behavior in micro-forming,and a phenomenological model that can accurately and concisely describe the material behavior in micro-forming is developed with the overall consideration of the effects of geometry size,grain size and strain hardening behavior.

2 Experimental

The H80 foils with a purity of 99.85%and different thicknesses from 30 to 200μm were used to conduct uniaxial tensile tests along the rolling direction to obtain the flow behavior of materials.Based on the ASTM-E8 standard,the initial measuring length and width for all specimens were kept constant at 50.0 and 12.5 mm,respectively.The strain rate was in the order of1×10^-3 s^-1 to ensure that all foils were deformed under a quasi-static condition.The true stress strain curves of the foils were obtained by averaging the data of six tests for each group.It is known that when the number of grains in the thickness direction of foils is less than a critical value,the change of grain numbers has a marked effect on the foil flow behavior,and the material deformation behavior can be regarded as micro-forming.This critical value is near 10for copper foils [ 19, 24, 25, 26] .In order to remove the rolling hardening and obtain the desired grain number in thickness direction of foils,H80 foils were recrystallized annealed at different combinations of temperatures and time in a vacuum furnace with a vacuum of 1.5×10^-3 Pa,after which the recrystallized foils were inlaid,polished and etched.The microstructures of recrystallized H80 foils were observed with an optical microscope (OM,Novel MR5000) and the average grain size of foils was measured by transversal method.The annealing conditions and material parameters are listed in Table 1.

3 Results and discussion

3.1 Size effects on flow stress of foils

The flow stress curves of H80 foil specimens with similar grain sizes (～35 (μm) but different thickness dimensions are shown in Fig.1,which reveals that the flow stress decreases significantly with the decrease of thickness dimension,and as the strain value increases,the decreasing amplitude presents a growing trend.

The grains consist of intragranular regions and grain boundary regions [ 22] .The atomic arrangement in grain boundary regions is disordered due to enriched impurities,so the strain hardening ability of grain boundary regions is higher than that of intragranular regions.Then the flow stress of grain boundary regions will be greater than that of intragranular regions when dislocation pile-up and tangle begin.Assuming that the grain size (diameter) is d,the relationship of grain boundary region thickness (T) and grain size (d) is as follows [ 22] :

An idealized schematic illustration of the grain is illustrated in Fig.2.Grains are assumed to be regular hexagon with d and the thickness of grain-boundary regions (T).Then the area fraction (A) of intragranular regions can be calculated as:

The changing trend of the proportion of grain boundary regions with grain size is shown in Fig.3.It can be seen that the proportion of grain boundary regions increaseswith the decrease of grain size,so the flow stress of the specimen increases with the decrease of grain size,which is consistent with the description of the relationship between material strength and grain size in the Hall-Petch equation [ 27] .However,in this experiment,the grain size of all specimens is similar (～35μm),so the slightfluctuation of grain sizes is unlikely to cause the significant change of flow stress in Fig.1.Miyazaki et al. [ 28] observed the dislocation distribution of specimen surface grains and found that only the trifurcated points of surface grains have a small amount of dislocations,whereas almost no dislocations can be observed in other areas.Since the specimen surface grains have at least one free surface,their f_low stress is lower than that of the internal grains.Peng et al. [ 29] proposed that the surface grains can be regarded as single crystal grains.As shown in Table 1,with the decrease of the foil thiclkness (t),the proportion of surface grains (N_S/μ) gradually increases,which enhances the weakening effect of surface grains and ultimately leads to the decrease of specimen flow stress.

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Table 1 Annealing conditions and material parameters of H80 foils

^aμ=t/d being number of grains in thickness direction ^b N_S being number of surface grains

Fig.1 Flow stress of H80 foils with different thicknesses

Fig.2 Schematic illustration of grain

Fig.3 Proportion of grain boundary regions for different grain sizes

3.2 Size effects on strain-hardening ability of foils

The strain-hardening exponent of H80 foil specimens with similar grain sizes (～35μm) but different thickness dimensions is shown in Fig.4,demonstrating that the strain-hardening exponent decreases significantly with the decrease of thickness dimension.The H80 foils with a thickness of 30μm have very weak strain hardening ability,whereas the 200-μm H80 foils have fairly strong strain hardening ability.

Fig.4 Strain-hardening exponent of H80 foils for foils with different thicknesses

Because of the lack of strengthening effect of grain boundary regions in surface grains,the strain hardening behavior of surface grains is far less than that of internal grains.As shown in Table 1,when the thickness of H80foils is 200μm,the proportion of internal grains is relatively large,so the material has a strong strain hardening ability.As the thickness dimension of foils decreases,the proportion of surface grains increases,so the dislocation slipping and entangling ability of internal grains gradually weaken under the effects of surface grain rotation and deformation,ultimately reducing the degree of foil strain hardening.Furthermore,as shown in Table 1,the specimens with thickness of 50 and 30μm both have 100%surface grains,but the strain hardening ability of 50-μm

foils is much greater than that of 30-μm foils (Fig.4).The grain distribution characteristics of H80 foils with thickness of 50 and 30μm are shown in Fig.5.It can be observed that there are still internal grains that own complete grain boundaries (Grains A and B) in the 50-μm

specimen (Fig.5a),which can block the free movement and promote the entanglement of dislocations,resulting in stronger strain hardening ability for foils with a thickness of 50μm.However,as shown in Fig.5b,the 30-μm foils are unlikely to possess internal grains,so the dislocations can slide out of the specimen more easily and are difficult to form piles and tangles.Grain rotation and grain boundary sliding which occur as the deformation increases can change some non-activated dislocation slip from unfavorable to favorable direction.For foils owning internal grains,the entanglement degree of dislocations will be increased because some non-activated slip systems are additionally revived (Fig.6a).However,for foils without internal grains,the entanglement degree of dislocations cannot be increased because the dislocations will slip out of the specimen surface (Fig.6b).Besides,some original dislocation pile-up groups will loosen and dissolve due to the changed dislocation slip direction,further reducing the entanglement degree of dislocations.Therefore,with the continuous increase of the strain,H80 foils with a thickness of 30μm lose the ability to continue hardening due to the lack of internal grains with complete grain boundaries,resulting in a slight increase of flow stress.Through the above analysis,as the strain increases,the reason for the increase of flow stress gap between specimens with different thicknesses (Fig.1) is the increased gap of dislocation entanglement degree.

3.3 Reducing influence of size effects on foil flow behavior

The proportion of internal grains can be increased by grain refinement.The influence of increasing the proportion of internal grains on the foil flow behavior was investigated.The foils with a thickness of 30μm and a grain size of25μm were chosen as the fine-grain specimen.The foils with a thickness of 30μm and a grain size of 35μm were selected as the reference Specimen 1,and the foils with50-μm thickness and 35-μm grain size were selected as the reference Specimen 2.As shown in Fig.7,the flow stress of fine-grain specimen is significantly larger than that of reference Specimens 1 and 2.Moreover,the t/d value of fine-grain specimen is slightly smaller than that of reference Specimen 2.However,when the flow stress of finegrain specimen and reference Specimen 2 is compared,it can be observed that when the strain value is small(≤0.2),the flow stress of fine-grain specimen is similar to that of reference Specimen 2.However,with the increase of strain,the flow stress of fine-grain specimen is gradually larger than that of reference Specimen 2.According to the“surface grain”theory,the smaller the t/d value is,the weaker the material flow behavior will be.So,these experimental results cannot be explained by the“surface grain”theory alone.The effect of grain size on the material flow behavior should also be considered.The smaller the grain size is,the greater the thickness of the grain boundary is,resulting in larger proportion of grain boundary regions.As shown in Fig.3,the percentage of grain boundary regions in 25-μm grains is significantly greater than that in35-μm grains.With the increase of deformation,dislocations continue to proliferate and tangle,forming the dislocation cell sub-structure [ 30] .The greater the proportion of grain boundary regions is,the more easily the dislocation cell sub-structure can be formed.The dislocation cell wall has similar effect on material flow behavior with the grain boundary regions,which can greatly enhance the material flow stress.This is the reason that when the strain value is large,the flow stress of fine-grain specimen is greater than that of reference Specimen 2.Therefore,the above experimental results reveal that the influence of size effects on material flow behavior caused by the decrease of specimen geometry size can be reduced by decreasing the size of grains.

Fig.5 Grain distribution of H80 foils with different thicknesses:a 50μm and b 30μm

Fig.6 Schematic diagram of dislocation movement of H80 foils with different thicknesses:a 50μm and b 30μm

Fig.7 Effect of grain refinement on foil flow behavior

4 Modeling with consideration of size effects

The above results demonstrate that the precise description of material behavior should include the effects of geometry size,grain size and strain hardening behavior.Based on the“surface grain”model which can describe the effect of geometry size on material behavior,the Hall-Petch equation which can accurately describe the effect of grain size on material behavior,and the Hollomon model which can describe the strain hardening behavior caused by dislocation slip and entanglement,a new model is developed as follows.

4.1 Effect of geometry size

For the foils with rectangular cross-section,according to the“surface grain”model,the proportion of surface grains(α) can be expressed as:

where w is the width of foils.The flow stress of foils (σ)can be calculated as:

whereσ_suf is the flow stress of surface grains andσ_in is the flow stress of internal grains.To express the coupling effects of specimen geometry size and grain size,one size parameterμis defined as:

Considering that the specimen width is much larger than the grain size,Eq.(3) can be written as:

Then Eq.(4) can be written as:

From Eq.(7) it can be concluded that the term of plays a key role on the flow stress of foils.Therefore,the size effect term in the form of 1 can be extracted to construct the new model.

4.2 Effect of grain size

Based on the Hall-Petch equation:

whereσ₀ is the descriptive term of intragranular deformation and K is the descriptive term of grain boundary deformation.From Eq.(8) it can be seen that the term of has an important influence on the flow stress of foils.Therefore,the size effect term in the form of can be extracted to construct the new model.

4.3 Strain hardening behavior

Hollomon equation:

where n is the strain hardening index,m is the material constant andεis true strain.

From Eq.(9) it can be concluded that theεⁿ term has a direct correlation with the flow stress of foils.Therefore,the size effect term ofεⁿ as the mathematical expression of strain hardening effect can be extracted to construct the model.

4.4 Construction of a new phenomenological model

The general modeling idea is to first extract the possible mathematical expressions with size parameterμ,and then put these mathematical expressions together to construct a new phenomenological model.After extracting the mathematical expressions of the effects of geometry size,grain size and strain hardening on material flow behavior,a phenomenological model is developed as follows:

where is the size adjustment func-tion,and a,b,c and n are adjustment factors with no physical meaning.F(μ) will adjust with the change of thickness size and grain size.Therefore,although Eq.(10)only has a single variableμ,it actually can describe the influence rule of both thickness size and grain size at the same time.

4.5 Model verification

4.5.1 Case study of metal foil forming

The accuracy of the new model in describing the foil flow behavior is verified by calculating the flow stress of H80foils shown in Fig.1.When the thicknesses of foils are 30,50,100 and 200μm,the values of size parameterμare 0.9,1.4,3.0 and 5.5,respectively.The adjustment factors in Eq.(10) were calculated using the Least Square Method combined with Global Optimization Algorithm.The calculated values of adjustment factors are:a=282.03,b=-117.93,c=175.23 and n=0.39,so Eq.(10) can be rewritten as:

The comparison between experimental results and calculated values by Eq.(11) indicates that Eq.(11) can accurately describe the flow behavior of H80 foils (Fig.8).Therefore,when performing process analysis and finite element calculations,the flow behavior of H80 foils with different values of size parameter can be described using Eq.(11).

Fig.8 Comparison of calculated values with experimental results of flow stress in foil tensile experiment

4.5.2 Case study of micro-bulk forming

In order to verify the flow behavior description of microbulk specimens by this new model,the results of a classical micro-bulk upsetting experiment performed by Engel et al. [ 15] were calculated using Eq.(10).In the micro-bulk upsetting experiment of CuZn15,the grain size is 79μm,and the diameter of the micro-bulk (D) is scaled according to the proportionality factor (λ).The values ofλare 100,0.42,0.21,and 0.10,respectively,and D is 4.8 mm whenλis 1.00.μ=D/d indicates the number of grains within the range of specimen diameter.The calculated values of size parameterμare 60.76,25.32,12.66 and 6.08,respectively.The adjustment factors in Eq.(10) were calculated using the Least Square Method combined with Global Optimization Algorithm.So,the values of adjustment factors a,b,c and n are 635.39,151.73,-12.97 and 0.49,respectively.Therefore,Eq.(10) can be rewritten as:

Figure 9 is the comparison of specimen flow stress between the experimental and calculated results by Eq.(12),revealing that the calculation results of Eq.(12) can accurately describe the material flow behavior in microbulk upsetting forming.When performing process analysis and finite element calculations,the flow behavior of CuZn15 micro-bulk specimens with different values of size parameterμcan be described using Eq.(12).

The calculation results of this experiment further confirm the rationality and sensitivity of the new phenomenological material model in capturing size effects of material flow behavior in micro-forming.Furthermore,the above case studies show that this model can accurately describe the size effects of material flow behavior with a wide range of application including various materials,shapes and deformation state.

Fig.9 Comparison of calculated values with experimental results of flow stress in micro-bulk upsetting experiment

5 Conclusion

In this study,by carrying out the tensile experiments of H80 foils with different thickness,it is found that the foil flow stress and strain hardening ability decrease significantly with the decrease of thickness dimension.The reduction of internal grains which own complete grain boundaries is the main reason of size effects of the foil flow behavior.The size effects on material flow behavior can be reduced by grain refinement.

After extracting the mathematical expressions of the effects of geometry size,grain size and strain hardening on material flow behavior,a phenomenological material model that can accurately and concisely describe the size effects of material behavior in micro-forming was developed based on the“surface grain”model,the Hall-Petch equation and the Hollomon model.The comparison between calculation results by this phenomenological model and experimental results performed in this study and in the literature perfectly proves the reasonability and accuracy of this model.

Acknowledgements This study was financially supported by the Foundation of Suzhou University of Science and Technology (No.XKQ2017005).

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