Investigation+on+hot+forming+limits+of+high+strength+steel22MnB5

Investigation on hot forming limits of high strength steel 22MnB5

Junying Min a,*,Jianping Lin a ,Jiayue Li a ,Wenhua Bao b

a College of Mechanical Engineering,Tongji University,Shanghai 201804,China b

Shanghai Volkswagen Co.,Ltd.,Shanghai 201805,China

a r t i c l e i n f o Article history:

Received 30March 2010

Received in revised form 2May 2010Accepted 7May 2010

Available online 9June 2010Keywords:22MnB5Hot forming

Localized necking

Forming limit diagram

a b s t r a c t

As hot stamped steel sheet 22MnB5deforms at elevated temperatures,it accords to the 0-angle necking type on both sides of forming limit diagram (FLD),which is not in accordance with the classical hypothesis of Hill’s localized necking theory.A prediction model for hot forming limits of steel 22MnB5is derived based on Storen and Rice’s Vertex theory and Logan–Hosford yield criterion.According to M–K theory,a calculation model for hot forming limits is also established based on Logan–Hosford yield criterion.Tests for hot forming limits of steel 22MnB5are performed to validate the prediction models.By compar-ison,the calculated FLD based on the Vertex theory and four-order Logan–Hosford yield criterion is in good accordance with the measured FLD,but the FLDs based on M–K theory are less consistent with it.

1.Introduction

Steel USIBOR1500(22MnB5)is a kind of high strength steel (HSS)developed by Arcelor.Vehicle light-weighting and improving of crashworthiness in auto industry greatly increase the require-ment to high strength steel.Due to better performance,such as softening and improving of plasticity of steel sheets at elevated temperatures,hot stamping process is usually applied to HSS sheet 22MnB5in order to obtain structural components with compli-cated shape,and the strength of ?nal components can reach about 1500MPa [1].

Steel 22MnB5is rate-sensitive and planar isotropic but has through-thickness anisotropy [2]at high temperatures,and its necking behavior is different from common sheet metals.It accords to the 0-angle necking type [3]whatever the strain state is on the left-or right-hand side of FLD.This is not in accordance with the hypothesis of Hill’s localized necking theory.Many scholars ana-lyzed and predicted FLD of sheet metals with analytical method.Currently,there are a large mount of references about forming lim-its theory of sheet metals [4–8].For a sheet metal under tensile load,diffuse necking and localized necking may successively occur [9].The diffuse necking is a state of in-plane instability of the sheet metal;while on the other hand,the localized necking is a state of through-thickness instability which leads to severe reduction in the local thickness.The following consequence of localized necking is crack of the sheet metal.Therefore,the critical strains value cor-responding to occurrence of localized necking are regarded as

forming limits of the sheet metal [10].At present,there are mainly three theoretical models describing localized necking.The ?rst is Hill’s localized necking theory [11].It supposes that a necking band,between whose normal direction and the direction of major strain exists an angle,forms along zero-extension direction.The second is groove hypothesis raised by Marciniak and Kuczynski [12]from the aspect of material damage,which is the well known M–K theory.The theory postulates an initial groove,based on which a localized necking band develops,due to the inhomogene-ity of physical property or thickness.The M–K theory has clear physical signi?cance and simple mathematical form and is widely employed.But its prediction model is over sensitive to the initial inhomogeneity of thickness,namely,the value f 0,which is usually determined by ?tting the prediction results with testing data.The third is the Vertex theory from Storen and Rice [13].It supposes the initial state of sheet is uniform.Storen and Rice consider that a ver-tex may form on a yield surface under a loading path.At the vertex,the direction of plastic ?ow shows uncertainty and it causes local-ized necking band on the sheet metal.Across the localized band,the plastic stress rate and strain rate are discontinuous.The Vertex theory was modi?ed to be more simple mathematic form by Zhu et al.[14],and forming limits of several kinds of sheet metals were analyzed combined with von Mises yield criterion.

The available investigations on steel 22MnB5focus on materials mechanical properties,microstructure transformation,stamping process,and so on [1,2,15,16].But there are few reports as regards investigation on forming limits of steel 22MnB5.Dahan et al.[17]have developed a new experimental apparatus on which hot stamping tests can be performed.The analysis scheme to deter-mine the critical strain values is based on the Bragard method.It

*Corresponding author.

E-mail address:dabao_0701@https://www.360docs.net/doc/905432308.html, (J.Min).

uses the extrema of the second derivative of the major strain to determine the critical major strain value and thus one point of the FLC through a polynomial function.In this paper,based on Sto-ren and Rice’s Vertex theory,a forming limits prediction model of steel 22MnB5at elevated temperatures is deduced combined with Logan–Hosford yield criterion.A calculation model of hot forming limits is also established according to M–K theory.The prediction results of two models are compared with testing data and conclu-sions are drawn.

2.The prediction model for hot forming limits based on the Vertex theory

According to the Vertex theory of Storen and Rice [13],the localized necking of a sheet metal corresponds with the appear-ance of a vertex on yield surface;and the stress and strain in-and-out the necking band are considered continuous,but the stress rate and strain rate are discontinuous.It is supposed that the angle between the normal direction of necking band and the direction of major strain is h (see Fig.1).Then,the discontinuity in-and-out necking band can be expressed as follows,

D v i ?f i en k x k Tei ;k ?1;2T

e1T

where n k (k =1,2)is the component of the unit normal vector n *

of necking band,n 1=cos h ,n 2=sin h .The gradient of deformation rate in-and-out necking band is

D @v i @x j

?@eD v i T

@x j ?f 0i en k x k Tn j ?g i n j

e2T

where v i is velocity of mass point on sheet metal,and g i ?f 0i en k x k T,

(i ,k =1,2).

Due to _e

ij ?@v i =@x j ,Eq.(2)can be expressed as Eq.(3)under primary axes.

D _e 1?g 1n 1;D _e

2?g 2n 2e3T

According to the modi?ed Vertex theory [14],the equilibrium equa-tion in primary axes can be described as

D _r

1àr 1eg 1n 1tg 2n 2T?0D _r

2àr 2eg 1n 1tg 2n 2T?0&

e4T

When steel 22MnB5deforms at elevated temperatures under austenite state,on the left-hand side (LHS)or right-hand side (RHS)of FLD,the normal direction of necking band always parallels to the major strain direction (see Fig.2),namely,n 1=1and n 2=0.Then the localized necking criterion on both sides of FLD can be uni?ed as

D _r

1àr 1g 1?0;namely ;D _r

1_e ?D _r 1?r 1e5T

f er ij T?

r 1tr er 1àr 2Tm t11tr

r m

1tr m 2àá? r m e6T

where m is an even equivalent to or greater than 2.When m equals

to 2,Logan–Hosford yield criterion becomes Hill’s two-order yield criterion obviously.

The elastic strain can be ignored comparing with limit strain of a sheet metal.The deformation theory of plasticity is applied to predicting https://www.360docs.net/doc/905432308.html,bined with Eq.(6),there is

d e ?

e1tr T r

m à1d e 1r e1à2Ttm à1

e1tr T r

m à1d e 2r e2à1Ttm à1

e7T

In consideration of loading path being radial or proportional loading,Eq.(8)can be obtained after performing integration on both sides of Eq.(7).

e 1? e r er 1àr 2Tm à1tr m à11e1tr T r

m à1e 2? e r er 2àr 1Tm à1tr m à12 r

><>:e8T

If r =1and m =2,it becomes the J 2deformation theory based on

von Mises yield criterion

e i ?3 e

S i

Taking differential operations on both equations in Eq.(8)with respect to time,there is

e1tr T r m _e 1??_ e r àem à1T e _ r r er 1àr 2Tm à1tr m à11h i t r e em à1Tr er 1àr 2Tm à2e_r 1à_r 2Ttr m à21_r 1h i e1tr T r m _e 2??_ e r àem à1T e _ r r er 2àr 1Tm à1tr m à12h i t r e em à1Tr er 2àr 1Tm à2e_r 2à_r 1Ttr m à22

_r 2h i 8>>>>>>>>>><>>>>>>>>>>:e9T

After transposing terms of Eqs.(8)and (9)can be obtained.

_r 1?1 e em à1Th r m à1e1tr Tr m à22tr er 1àr 2Tm à2? _e 1tr er 1àr 2Tm à2_e 2r r m à21tr m à22eTer 1àr 2Tm à2tr m à21r m à22:à1àem à1T e r _ r _e

h i r 1_ e i _r 2?1 e em à1Th r m à1e1tr Tr m à21tr er 2àr 1Tm à2? _e 2tr er 2àr 1Tm à2_e 1r er m à22tr m à21Ter 2àr 1Tm à2tr m à22r m à21:à1àem à1T e r _ r _e

h i r 2_ e i 8>>>>>>>>><>>>>>>>>>:e10T

Considering the discontinuity of stress rate and strain rate be-tween the inside and outside necking bands,Eq.(10)can be rewrit-ten as

of localized necking band under localized necking band of the J.Min et al./Computational

D _r 1?1 e h r m à1e1tr Tr m à22tr er 1àr 2Tm à2? g 1n 1tr er 1àr 2Tm à2g 2n 2r e1t2Te1à2Tt12:à1àem à1T e r D _ r _ e

h i r 1D _ e i D _r 2?1 e h r m à1e1tr Tr m à21tr er 2àr 1Tm à2? g 2n 2tr er 2àr 1Tm à2g 1n 1r e2t1Te2à1Tm à2t21:à1àem à1T e r D _ r D _ e

h i r 2D _ e i 8>>>>>>>>><>>>>>>>>>:e11T

According to Cristescu [21],the quasi-linear relation of the

stress rate and strain rate can be expressed as

_ e ?/e r ; e T_ r tu e r ; e T

e12TIn consideration of the discontinuity of stress rate and strain rate between the inside and outside bands on Eq.(12),there is

D _ r ?D _ e /e r ; e T;namely ;/e r ; e T?

D _ e D _ e13T

According to the strain energy relation,there is

r d e ?r 1d e 1tr 2d e 2

e14T

Considering the discontinuity of strain rate between the inside and outside bands,Eq.(14)can be rewritten as

D _ e ?r 1 r g 1n 1tr 2 r

g 2n 2e15T

By substituting Eq.(15)into Eq.(11),it can be obtained

D _r 1?1 e h r m à1e1tr Tr m à22tr er 1àr 2Tm à2? g 1n 1tr er 1àr 2Tm à2g 2n 2r 1t2eTe1à2Tm à2t12:àr 11àem à1T e / r h i r 1 r

g 1n 1tr 2 r g 2n 2àái D _r 2?1 e h r m à1e1tr Tr m à21tr er 2àr 1Tm à2? g 2n 2tr er 2àr 1Tm à2g 1n 1r 2t1eTe2à1Tm à2t21:àr 21àem à1T e / r h i r 1 r g 1n 1tr 2 r

g 2n 2àái 8>>>>>>>>><>>>>>>>>>:e16T

Let n 1=1and n 2=0,and substitute the uni?ed localized neck-ing e R P eQ b P e

?Q m à1e1tr Tem à1TP r e1àa Tm à2ta m à2r e1ta m à2Tea à1Tm à2ta m à2à1PQ 1m à1

àR

e22T

As regards the most conventional material,the work hardening behavior of power law format and von Mises yield criterion are employed.Then,there are

/?1 e n à1? e

;R ?n ;

P ?2

???

p ??????????????????????1tb tb 2q ;

Q ?

???3p ??????????????????????

1tb tb 2

q 2tb

e23T

Then the prediction model (Eq.(22))for forming limits becomes

e ?

3b 2tn e2tb T22e2tb Te1tb tb T

e24T

which is derived on the RHS of FLD in the previous work [14]based on power law constitutive relation and von Mises yield criterion.According to Min et al.[19],the stress–strain relation of steel 22MnB5at elevated temperatures can be described as

r ?K e N _ e

M e25T

where K ,N ,M are functions of temperature,K =exp (b /T ),N =a +bT ,

M =c +dT .

Jie et al.[20]modi?ed Eq.(29)as the relation of quasi-linear stress rate and strain rate in terms of Cristescu [21],

/e r ; e T?

e eM tN T r

e N =M e26T

where c is integral constant which can be obtained from uniaxial

tensile tests.Then the expression of R is modi?ed as

R ?

M tN

x e27T

where x is àc r = e N =M t1.

Fig.3.Illustration of M–K groove model.

328J.Min et al./Computational Materials Science 49(2010)326–332

3.The calculation model theory

M–K theory was proved by many scholars.It dicting forming limits of is the famous hypothesis of poses that there is an initial As the sheet metal with a the deformation will groove because the than that out the groove difference between the sheet metal reaches its in the groove (area B)turns the strains in area A can sheet metal.

According to M–K theory,model as follows:

(1)The incompressibility d e 1td e 2td e 3?0

(2)The force equilibrium and B

r A nn t A ?r B nn t B ;r A nt t A where r A nn r B nn àáand r A nt components at the is real-time thickness (3)The condition of strain A and B

d e 2A ?d e 2B ?d e 2

(4)The J 2deformation d e i ?dc

@ r r i

;i ?1;2(5)The yield criterion (see Eqs.(6)and (25))The iterative equation for sheet can be obtained based 1Q A P A b A M e e A td e A TN

exp ee 3A where the initial thickness mogeneity of thickness,is through-thickness strain FLD of steel 22MnB5can (33).During iteration D e 1B requirement of is shown in Fig.4.

4.Forming limits of steel sheet 22MnB5

Forming limit tests of steel 22MnB5are performed to validate the prediction models for FLD of steel 22MnB5.

4.1.Specimens and testing ?ow

The major testing apparatus is shown in Fig.5which includes a hemi-sphere punch with diameter of 100mm.Dog-bone specimens

Fig.4.The iterative ?ow chart of limit strains based on M–K theory.

Fig.5.Testing apparatus.

Fig.6)cut from boron-steel sheets were adopted to measure strains at the LHS of FLD.The width of tested dog-bone

are20–180mm and the length of all specimens is180 According to Merklein et al.[22],steel sheets22MnB5exhibit signi?cant dependence on the rolling direction at elevated temper-The length direction of all specimens is along the direction,while the width is along the transverse direction.Ground with diameter of2.5mm were printed onto the surface specimens by applying photochemical etching method in case grids to be illegible after heat treatment.

During the tests,the specimens were heated to950°C electric furnace?rstly,and preserved heat for5min for the specimens were rapidly transferred into the testing apparatus. This transferring process took about4s.After that,tests were immediately performed with stamping velocity of30mm/s and ?nished in no more than1s.A piece of asbestos paper with thick-ness of0.8mm was laid on binder in advance to reduce the heat transferring from specimens to punch.Temperature variation of the central point of specimens was recorded by an infrared ther-mometer simultaneously.

4.2.Testing results and analysis

The temperature of specimens decreased during forming inspite of the use of asbestos paper.The initial temperature of the central point of specimens was813°C and the?nished temperature was 780°C.As the central point of specimens contacted the punch ?rstly,its temperature fell to the lowest in the round forming zone. Therefore,the tests are regarded as isothermal forming at800°C since temperature variation of specimens was not so pronounced. The samples are shown in Fig.7.The testing results are used for validating the prediction models for FLD.Three kinds of grids on

Fig.6.Specimens of boron-steel sheets for hot forming limits tests.

Fig.7.Finished samples of boron-steel sheets.

Science49(2010)326–332

2.5?10à4calculated at larger strains on the ?ow curve.The anisotropic coef?cient r at 800°C is 0.8according to Merklein al.[2].The measured FLD and calculated FLDs based on the Ver-theory are shown in Fig.11.In terms of comparing calculated results with testing data,the order of yield criterion m was varied,m =2,4,6.

It can be seen form Fig.9that the prediction results from two-order Logan–Hosford yield criterion are greater than the testing data on the RHS of FLD,and the calculation results from six-order Logan–Hosford yield criterion are smaller than the testing data the plane stain remains almost unchanged.When the order equals to 8,the FLC shows slight ?uctuation.It may be associated with over high power during the iterative calculation.Conclusions

(1)A prediction model for hot forming limits of steel 22MnB5derived based on the Vertex theory and Logan–Hosford yield criterion.According to M–K theory,a calculation model for hot forming limits is also established based on Logan–Hos-ford yield criterion.

Fig.8.Illustration of limit strains measuring.

https://www.360docs.net/doc/905432308.html,parison between the calculated FLD based on the Vetex theory Logan–Hosford yield criterion and the measured FLD.

https://www.360docs.net/doc/905432308.html,parison between the calculated FLD based on M–K theory and Mises yield criterion and the measured FLD.

https://www.360docs.net/doc/905432308.html,parison between the calculated FLD based on M–K theory and Logan–Hosford yield criterion and the measured FLD (f 0=0.998).

Acknowledgement

The authors would like to thank the?nancial support from the subproject of new energy vehicle special project of Shanghai science&technology pillar program,research on structural design and manufacturing technology based on light-weighting of crash safety components of new energy vehicle under Grant No. 08DZ1150305.

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