Journal of Advanced Research (2013) 4, 83–92

Cairo University

Journal of Advanced Research

ORIGINAL ARTICLE

Analytical study for deformability of laminated sheet metal

Mohammed H. Serror *

Department of Structural Engineering, Faculty of Engineering, Cairo University, Egypt

Received 11 October 2011; revised 24 December 2011; accepted 23 January 2012Available online 3 March 2012

*

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El

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KEYWORDS

Sheet metal laminate;

Necking instability;

Deformability;

Power-law model

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Abstract While a freestanding high-strength sheet metal subject to tension will rupture at a small

strain, it is anticipated that lamination with a ductile sheet metal will retard this instability to an

extent that depends on the relative thickness, the relative stiffness, and the hardening exponent

of the ductile sheet. This paper presents an analytical study for the deformability of such laminate

within the context of necking instability. Laminates of high-strength sheet metal and ductile low-

strength sheet metal are studied assuming: (1) sheets are fully bonded; and (2) metals obey the

power law material model. The effect of hardening exponent, volume fraction and relative stiffness

of the ductile component has been studied. In addition, stability of both uniform and nonuniform

deformations has been investigated under plane strain condition. The results have shown the retar-

dation of the high-strength layer instability by lamination with the ductile layer. This has been

achieved through controlling the aforementioned key parameters of the ductile component, while

the laminate exhibits marked enhancement in strength–ductility combination that is essential for

metal forming applications.ª 2012 Cairo University. Production and hosting by Elsevier B.V. All rights reserved.

Introduction

There has been a strong demand for high-strength steel havingan exceptionally good strength–ductility combination. Forconventional steel in bulk form, however, there exists a clear

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boundary drawn in the space of strength and ductility combi-nation, beyond which no conventional steel can go. Consider-

ing the rupture modes of high-strength steel, two kinds ofelongation limit exist, one is associated with fracture due tolack of toughness and the other is induced by the mechanismso called ‘‘plastic instability’’. The plastic instability itself has

two deformation modes, one is called diffuse necking andthe other is localized necking. In order to retard both rupturemechanisms and to overcome the boundary for conventional

steel, we have been studying the introduction of a laminatedstructure composed of brittle high-strength (BHS) and ductilelow-strength (DLS) steels, and it has been clarified that the

brittle rupture of BHS steel can be suppressed by laminatingit with a DLS steel with an appropriate selection of layer thick-ness, interface fracture toughness, and mechanical propertiesof the more ductile constituent layer. The experimental studies

that have been conducted by Inoue et al. [1] and Nambu et al.

84 M.H. Serror

[2] support the context of this work. They have presented

examples of currently developed laminated sheet metal thatcould achieve high-strength and ductility combination. Suchcombination could not be achieved before [3,4]. The concernof this paper is to clarify the deformability of the laminate in

association with plastic instability.As a freestanding metal sheet loaded in tension elongates,

the tensile force increases due to the hardening of the metal,

but decreases due to the reduction in the cross-sectional area.The tensile force peaks at some strain, at which a neck sets in.The uniform deformation in the metal becomes unstable when

the geometric softening prevails over material hardening [5,6].Grote and Antonsson [7] studied the deformability of a

freestanding metal sheet described by power hardening with

strain hardening exponent, N. They have reported that if thestress state is not uniaxial, as usual in most sheet forming pro-cesses, the diffuse necking criterion does not set the limitstrain. The localized necking criterion, however, sets the limit

strain in practical sheet forming while it predicts well the neg-ative minor strain region of the forming limit diagram. Thelocalized necking limit strain is equal to [N/(1 + e2/e1)]; while[e2/e1] is the ratio between minor and major strains in metalforming. It is worth noting that in the forming limit diagramthe plane strain state represents the critical state where the

minor strain vanishes and the localized necking strain (eNecking)becomes equal to N, (eNecking = N). The localized necking ofthe laminate has been analyzed [8] within a forming limit anal-ysis. It was found that the plane strain assumption is still valid

in observing the deformability of the laminate, where the min-imum necking strain in the forming limit diagram is associatedwith the plane strain condition.

Hence, in this study the plastic instability associated with thelocalized necking under plane strain condition has been adoptedto evaluate the laminate deformability. In addition, the lami-

nate layers are assumed to be fully bonded; consequently, theinterface delamination is beyond the scope of this paper.

Of interest in this paper is the role in retarding the onset of

necking of a DLS sheet metal bonded to a BHS sheet metal.Neck retardation allows the laminate to be stretched to largeroverall strains. In the range of strains relevant to the BHSsheet metal necking, we anticipate that the incremental modu-

lus of the nominal stress–strain curve of the DLS sheet metalremains constant with stretching while that of the BHS sheetmetal decreases steadily. Accordingly, compared to a single

(freestanding) BHS sheet metal [9,10] at a given level of stretch,the laminate has lower average stress and higher tangent mod-ulus, both of which promote necking retardation. This is the

essence of the phenomenon as similarly introduced [5,11,12]for the polymer substrate-bonded metal film.

There are initiatives that have been investigating the shear

and normal stresses in laminated sheet metal [13–15]. The flex-ural response has been examined [16] for the vibration-damp-ing type of laminated steel (steel/polymer/steel laminate). Acomparison has been performed for beam theory predictions

with the experimental results, and good agreement has beenobserved in case of using two layers of shells in the finite ele-ment analysis. The formability of multilayer metallic sheets

has been evaluated by tensile, V-bending, hat bending andhemming tests [17,18]. Marked enhancement of the bendingformability was observed in the bending of type-420J2 stainless

steel sheets when they are layered by type-304 stainless steelsheets and composed into a multilayer metallic sheet.

In the present study, an idealized structure is considered: a

BHS sheet metal bonded to a DLS sheet metal in two-layer ormultilayer laminate that is subject to a tensile plane strain. Keytwo questions are: whether the BHS sheet can survive largerstrains without rupture, and how much would be the associ-

ated ultimate strength of the laminate. This is to introduce en-hanced strength–ductility combination.

Oya et al. [18] conducted uniaxial tension tests on WT780C

as brittle martensitic steel (BHS sheet) and SUS304 as ductileaustenitic stainless steel (DLS sheet). The chemical composi-tions for WT780C and SUS304 are shown therein [18] with a

yield strength of 1080 [MPa] and 226 [MPa], respectively. Un-der uniaxial tension, the metal deforms according to the powerlaw r = K eN, where r is the true stress, e is the strain, K and N

are constants determined from known true stress–strain databefore necking. N is also known as the metal hardening expo-nent. It has been reported that WT780C could achieve a hard-ening exponent N = 0.05 associated with K= 1663 [MPa]; on

the other hand, SUS304 could achieve a hardening exponentN= 0.5 associated with K= 1611 [MPa]. It is worth notingthat the range of laminate parameters that have been consid-

ered in this study intends to cover a broad spectrum of poten-tial material combinations for BHS/DLS laminates.

Uniform deformation stability

The laminate in question has two different hardening expo-

nents, a low hardening exponent of the BHS sheet and a highhardening exponent of the DLS sheet. In a freestanding BHSsheet metal, the geometric softening predominates the materialhardening at a small strain, and the uniform deformation be-

comes unstable. The behavior is similar for a freestandingDLS sheet metal; however, the onset of instability takes placeat much higher limit strain. Consequently, at the onset of BHS

sheet instability the DLS sheet stiffens steeply and the tensileforce increases with deformation by material hardening. Sothe question is what will happen to a BHS/DLS laminate?

Fig. 1 describes the model, a freestanding BHS sheet metalalong with BHS/DLS laminate are analyzed. For the laminate,different values of volume fraction of the DLS component f

are studied.Under uniaxial plane strain stretching, the stress in x-direc-

tion is related to the applied strain as:

rBHSDLS

¼ KBHSDLS

eNBHSDLS ð1Þ

in the BHS and DLS layers. By volume conservation, as thelaminate elongates in the x-direction, both the BHS and

DLS layers thin by a factor of exp(�e) in the y-direction[5,19]. Consequently, the normalized nominal stress rNorm is gi-ven as follows:

rNorm ¼rn

ru avg

¼ ½ð1� fÞeNBHS þ fkeNDLS � expð�eÞ½ð1� fÞNNBHS

BHS expð�NBHSÞ þ fkNNDLS

DLS expð�NDLSÞ�

rn ¼F

ðHDLS þHBHSÞ¼ ðrBHSHBHS þ rDLSHDLSÞ expð�eÞ

ðHDLS þHBHSÞ¼ ðð1� fÞKBHSe

NBHS þ fKDLSeNDLSÞ expð�eÞ

¼ KBHS½ð1� fÞeNBHS þ fkeNDLS � expð�eÞ

Freestanding BHS sheet

HBHS

HDLS

BHS/DLS Laminate

f =1/2, (HDLS/HBHS = 1)

f =2/3, (HDLS/HBHS = 2)

f =3/4, (HDLS/HBHS = 3)x

y

x

y

Freestanding BHS sheet

HBHS

HDLS

BHS/DLS Laminate

f =1/2, (HDLS/HBHS = 1)

f =2/3, (HDLS/HBHS = 2)

f =3/4, (HDLS/HBHS = 3)x

y

x

y

εε

BHS

DLS

εε

BHS

DLS

Fig. 1 BHS/DLS Metal laminate in three different cases of the volume fraction of the DLS component (f= 1/2, 2/3, and 3/4) and a

freestanding BHS sheet metal under uniaxial plane strain tension in x-direction.

Deformability of laminated sheet metal 85

ru avg ¼ KBHS½ð1� fÞeNBHS

Necking�BHS expð�eNecking�BHSÞþ fkeNDLS

Necking�DLS expð�eNecking�DLSÞ� ¼ KBHS½ð1� fÞNNBHS

BHS expð�NBHSÞ þ fkNNDLS

DLS expð�NDLSÞ� ð2Þ

where F is the resultant force in x direction, rn is the nominalstress, ru avg is the average of nominal ultimate strength,

f ¼ HDLS=ðHDLS þHBHSÞ is the volume fraction of the DLScomponent, and k ¼ KDLS=KBHS is the components stiffnessratio. It is worth noting that the dimensionless ratios f and k

quantify the effect of the DLS component in the laminate.Fig. 2a plots the normalized nominal stress rNorm as a func-

tion of the applied strain e for three different values of NDLS:

0.5, 0.3, and 0.1 and three different values of the volume frac-tion of the DLS component (f= 1/2, 2/3, and 3/4), where thelaminate components stiffness ratio k is set to unity and thehardening exponent of the BHS component NBHS is set to

0.06. When f= 0, the BHS sheet metal is in effect freestand-ing, where rNorm peaks at a small strain equals to NBHS andthen drops. In the analysis, three controlling parameters ap-

pear. The first parameter is the volume fraction of the DLScomponent f, where an increase in f leads to an increase inthe limit strain, through resisting the aforementioned geomet-

ric softening for a particular hardening exponent of the DLScomponent. For instance, in Fig. 2a(ii) the limit strain in-creases from 6% at f = 0 to 22.5% at f= 3/4. The second

parameter is the hardening exponent of the DLS componentNDLS, where an increase in NDLS leads to an increase in thelimit strain, through enhancing laminate hardening against aparticular geometric softening. For instance, in Fig. 2a(i and

iii) at the same f value that equals to 2/3, the limit strain in-creases from 8.6% at NDLS = 0.1 to 30% at NDLS = 0.5.The k ratio is the third controlling parameter, where an in-

crease in k leads to an increase in the limit strain, throughresisting the aforementioned geometric softening by stiffeningthe laminate for a particular hardening exponent of the DLS

component. Fig. 2b plots the normalized nominal stress rNorm

as a function of the applied strain e for the same values ofNDLS and NBHS, and three different values of laminate compo-nents stiffness ratio (k = 1/4, 1/2, and 1.0), where f is set to

0.5. For instance, in Fig. 2b(ii) the limit strain increases from12% at k = 0.5 to 16% at k = 1.0.

It is worth noting that the rule of averages is considered in

this paper as the scale of enhancement in laminate strength andductility. This is the same enhancement scale for the experi-mental studies of available laminated sheet metal [1–4]. The

limit strain calculated from Eq. (2) at the force maxima, asshown in Fig. 2a and b, is compared further with the predic-tion of the limit strain based on the rule of averages. As shown

in Fig. 3, the comparison with the rule of averages has beenconducted for: two cases of the hardening exponent of theBHS component (NBHS = 0.01 and 0.06), three cases of the

volume fraction of the DLS component (f= 1/3, 1/2, and 2/3), four cases of the laminate components stiffness ratio(k= 2.0, 1.0, 0.5, and 0.25), and different values of the hard-ening exponent of the DLS component (NDLS ranging from

NBHS to 1.0). This covers a broad spectrum of designatedBHS/DLS laminates. It is obvious that the calculated limitstrain becomes closer to that predicted by the rule of averages

by decreasing the hardening exponent of the DLS componenttill a bound of homogenous BHS sheet metal where the ratioeNecking(BHS)/eNecking(DLS) approaches to unity. It is worth

noting that changing the k ratio affects the limit strain to beless, equal, or even more than the prediction of the rule ofaverages. It is also clear that by decreasing the ratio eNecking

(BHS)/eNecking(DLS), the role of the DLS component in retard-ing the BHS component instability is decreasing. This role van-ishes when the ratio eNecking(BHS)/eNecking(DLS) becomes verysmall. These results are compliant with the experimental obser-

vations [3], where it has been noted that the tensile ductility ofmost of the laminated composites is lower than that predictedfrom the rule of averages when the difference between ductility

of the two components is large. This has been attributed to thesusceptibility of the less ductile component to an early rupture.

From Eq. (2) and drn/d e = 0, it follows that:

kf=ð1� fÞ ¼ ½eNBHS

Necking �NBHSeNBHS�1Necking �=½NDLSe

NDLS�1Necking � eNDLS

Necking� ð3Þ

where eNecking is the laminate limit strain at the onset of neck-ing. When f= 0, Eq. (3) recovers the well-known solution for

a freestanding BHS sheet metal, eNecking = NBHS, where theuniform deformation bifurcates into nonuniform deformationof a wavelength much larger than the sheet thickness [7]. For a

laminate, Eq. (3) divides the plane [eNecking, kf/(1 � f)] into tworegions, the left and right sides of the curve, as shown in Fig. 4.The curves in Fig. 4 correspond to the force maxima for the

three different values of NDLS: 0.5, 0.3, and 0.1. When the lam-inate is stretched, the uniform deformation is stable for strainsup to the curve. Hence, the necking strain is bounded by the

limit strain of the freestanding BHS sheet as a lower boundand that of the freestanding DLS sheet as an upper bound,NBHS and NDLS, respectively.

Norm

ε ε ε

σ

Freestanding f =3/4f =1/2 f =2/3k =1.0

ε ε ε

Normσ

f =1/2 Freestanding k =1.0k =0.25 k =0.5

(b)

(a)

(ii) NDLS=0.3 (iii) NDLS=0.1 (i) NDLS=0.5

(ii) NDLS=0.3 (iii) NDLS=0.1 (i) NDLS=0.5

Fig. 2 The nominal stress versus strain at three different cases of the hardening exponent of the DLS component: 0.5, 0.3 and 0.1 for: (a)

three different cases of the volume faction of the DLS component (f= 1/2, 2/3, and 3/4), while the laminate components stiffness ratio k is

set to unity; and (b) three different cases of the laminate components stiffness ratio (k = KDLS/KBHS = 1/4, 1/2, and 1), while the volume

fraction of the DLS component f is set to 0.5.

86 M.H. Serror

The results obtained in this section describe a response ofcritical limit strain associated with a perturbation of longwavelengths. For a freestanding sheet, the prediction of critical

strain for long wavelengths gives the lowest critical strain [6,7].Accordingly, it is a common practice to identify the long wave-length limit, eNecking = N, as the rupture strain of a freestand-

ing metal. For a metal laminate structure, there is still a criticalstrain associated with the long wavelength perturbation limit[2]. However, a lower bifurcation strain was observed at finite

wavelength [2,20] indicating multiple necking in associationwith interface delamination. It is noted that since the laminatelayers are assumed to be fully bonded, the critical strain asso-ciated with the finite wavelength is no longer a bifurcation

mode resulting in a rupture in the laminate [2,20]. Hence, thecritical strain associated with the long wavelength limit is stillidentified as the rupture strain of the laminate driving the

behavior into a single-necking deformation. The experimentalobservations of Inoue et al. [1] and Nambu et al. [2] supportthis analysis; while, the specimen with high bonding strength

experienced large uniform elongation that was followed by asingle-necking rupture. Table 1 shows good agreement be-tween the results obtained experimentally [1,2] and those ob-

tained based on Eqs. (2) and (3) of this study.

This is different from what was observed by Li and Suo [5]for the polymer substrate-bonded metal film where in the longwavelength limit the critical strain is infinite, then drops pre-

cipitously as the wavelength of the perturbation decreasesexhibiting a multiple-necking deformation. This difference isattributed to the increasing hardening exponent of the polymer

substrate with stretching; meanwhile, the hardening exponentof the DLS component in the BHS/DLS laminate is constant.In addition, the constitutive equation for the DLS component

is a power law; meanwhile, for the polymer substrate it is not.In the next section, the large-amplitude nonuniform defor-

mation in multilayer BHS/DLS laminate has been investi-gated. The finite element analysis has been performed to

investigate the post bifurcation behavior and to identify thedeformation mode at the limit strain of the necking, whethermultiple-necking (at finite wavelength of perturbation) or sin-

gle-necking (at long wavelength limit).

Nonuniform deformation stability

The linear stability analysis fails to identify the deformationmode corresponds to the limit strain, where the amplitude of

the nonuniform displacement is large compared to the

ε Nec

king

(Lam

inat

e) /

ε Nec

king

(DL

S Sh

eet)

0.0

0.2

0.4

0.6

0.8

1.0

Rule of Averagesk =2.0

k =1.0

k =0.5

k =0.25

Rule of Averages

k =2.0

k =1.0

k =0.5

k =0.25

f =1/3, NBHS=0.01 f =1/3, NBHS=0.06

ε Nec

king

(Lam

inat

e) /

ε Nec

king

(DL

S Sh

eet)

f =1/2, NBHS =0.01

0.0

0.2

0.4

0.6

0.8

1.0

Rule of Averages

k =2.0k =1.0

k =0.5

k =0.25

Rule of Averages

k =2.0

k =1.0

k =0.5

k =0.25

f =1/2, NBHS=0.06

ε Nec

king

(Lam

inat

e) /

ε Nec

king

(DL

S Sh

eet)

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Rule of Averagesk =2.0

k =1.0

k =0.5

k =0.25

Rule of Averagesk =2.0

k =1.0

k =0.5

k =0.25

f =2/3, NBHS=0.01 f =2/3, NBHS=0.06

εNecking (BHS Sheet) / εNecking (DLS Sheet)εNecking (BHS Sheet) / εNecking (DLS Sheet)

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

εNecking (BHS Sheet) / εNecking (DLS Sheet)εNecking (BHS Sheet) / εNecking (DLS Sheet)

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

εNecking (BHS Sheet) / εNecking (DLS Sheet)εNecking (BHS Sheet) / εNecking (DLS Sheet)

Fig. 3 The necking strain of the two-layer BHS/DLS laminate compared with prediction based on the rule of averages, for different

cases of the hardening exponent of the DLS component NDLS, the hardening exponent of the BHS component NBHS, the laminate

components stiffness ratio k, and the volume fraction of the DLS component f.

Freestanding Necking Strain = NBHS = 0.06

0.1 > [Necking Strain Range ] ≥ 0.06

NDLS=0.1

Freestanding Necking Strain = NBHS = 0.06

0.3 > [Necking Strain Range ] ≥ 0.06

NDLS=0.3

0

10

20

30

40

50

60

Freestanding Necking Strain = NBHS = 0.06

0.5 > [Necking Strain Range ] ≥ 0.06

NDLS=0.5

0.1 1

Freestanding Necking Strain = NBHS = 0.06

0.1 > [Necking Strain Range ] ≥ 0.06

NDLS=0.1

0.1 1

Freestanding Necking Strain = NBHS = 0.06

0.3 > [Necking Strain Range ] ≥ 0.06

NDLS=0.3

0

10

20

30

40

50

60

0.1 1

ε

Freestanding Necking Strain = NBHS = 0.06

0.5 > [Necking Strain Range ] ≥ 0.06

NDLS=0.5

NeckingεNeckingεNecking

k f /

(1-f

)

(c) (b) (a)

Fig. 4 The necking strain range of the two-layer BHS/DLS laminate for different combinations of the laminate components stiffness

ratio k and the volume fraction of the DLS component f at three different cases of the hardening exponent of the DLS component: (a)

NDLS = 0.5; (b) NDLS = 0.3; and (c) NDLS = 0.1.

Deformability of laminated sheet metal 87

Table

1Comparisonbetweenexperim

entalresultsofpublished

papersandanalyticalresultsofthisstudy.

Laminate

description

DLSlayer

BHSlayer

f

ru_avgEq.

(2)(M

pa)

Experim

entalresults

Analyticalresults

Error%

rNorm

Error%

e Necking

NDLS

ru(M

pa)

NBHS

ru(M

pa)

r n(M

pa)

rNorm

(rn/r

u_avg)

e Necking

rn(M

pa)

r Norm

(rn/r

u_avg)

e Necking

Inoueet

al.[1]

–TwelveBHSLayersofSS420:

40

lmthicknesseach

0.6

700

0.08

2240

0.52

1192.30

1180.00

0.99

0.154

1109.71

0.93

0.143

6.0

7.1

–Thirteen

DLSLayersofSS304:

40

lmthicknesseach

–Totallaminate

thickness

equals1.0

mm

–Roll-bonded

laminate

withstronginterface

Nambuet

al.[2]

–OneBHSLayer

ofSCM415:

330

lmthickness

0.695

720

0.053

1292

0.50

1200.49

960.00

0.80

0.160

1051.96

0.88

0.153

�9.6

4.4

–TwoDLSlayersofSS304:

165

lmthicknesseach

–Totallaminate

thickness

equals0.66mm

–Roll-bonded

laminate

withstronginterface

88 M.H. Serror

perturbation wavelength. The finite element method (FEM) is

used to simulate large-amplitude nonuniform deformation inmultilayer BHS/DLS laminate, and to identify the deforma-tion mode at the limit strain. Fig. 5a shows three configura-tions that have been considered in the FEM analysis,

namely: A; B; and C of 15; 11; and 7 layers, respectively. Inall configurations, the total thickness of the laminate is set to1 [mm], and the material constants: KBHS = KDLS = 1600

[MPa], NBHS = 0.06, and NDLS = 0.5 have been used repre-senting a WT780C/SUS304 laminate. Each configuration hasa typical layer thickness which is calculated by dividing the to-

tal thickness of the laminate by the total number of layers.Accordingly, the volume fraction of the DLS component hasthe values of fA = (8/15) = 0.53, fB = (6/11) = 0.55, and

fC = (4/7) = 0.57 for configurations A, B, and C, respectively.The imperfections are prescribed by perturbing the laminatetop surface into a sinusoidal shape of amplitude equals to1% of the typical layer thickness and a wavelength equals

ten times the typical layer thickness; see Fig. 5b and c.Fig. 5b shows a schematic representation for the finite elementmodels that are 2.5 wavelengths long. The displacement is set

to be zero at the left-bottom corner of the models, so is thehorizontal displacement along the left end of the models. Dis-placement in the horizontal direction, u, is prescribed along the

right end of the models. Four-node quadrilateral plane strainelements are used. In the model, each layer is modeled withten elements in the thickness direction and a comparable ele-ment size in the in-plane direction. Matching elements are used

along the interface between laminate layers, assuming no inter-face delamination.

It is found that the initial deformation mode of the BHS/

DLS laminate tends to be a multiple-neck mode with a wave-length that is equivalent to the induced imperfection (finitewave length). However, it ceases and does not develop exhib-

iting a single-neck mode as shown in Fig. 5d. This can be ex-plained that as the strain increases from the minimumcritical strain of the first multiple-neck mode, the mode no

longer be a solution satisfying the general equilibrium andboundary conditions. Hence, the next multiple-neck modestarts to emerge; however, it ceases since the same behaviorrepeatedly occurs exhibiting the single-neck mode at the corre-

sponding critical strain. The concern is that for fully bondedlayers, multiple necking is not the bifurcation mode resultingin a rupture in metal laminate. Although it might take place,

it ceases right after initiation exhibiting a single-neck modeassociated with the long wavelength limit. Fig. 5d illustratesthree snapshots along the deformation increments.

Therefore, the deformation instability is different between afreestanding metal sheet and multilayered sheet laminate be-cause the former exhibits only single-neck mode; meanwhile,

the later exhibits also multiple-neck mode. However, the mostcritical mode that is leading the final rupture in both of thesheets is the single-neck mode. It is worth noting that the pres-ent study is adequate for the design purpose of sheet metal

laminates; meanwhile, further investigation is needed for thesensitivity of deformation mode.

Fig. 6a plots the laminate nominal stress rn, normalized to

the average ultimate strength ru_avg, versus the associatedstrain e at necking location as resulted from the FEM analysis.The limit strain at the onset of necking has been identified with

the point of maximum nominal stress. It is clear that the DLSlayer has retarded the necking instability of the BHS layer

H/100

10 H

(c)Surface perturbation with H=layer thickness

y

x

Zero x-Displacement

Zero (b)

C(7 Layers)

B(11 Layers)

A(15 Layers)

DLS

BHSTotal Thickness = 1 [mm]

(a)fA =(8/15)=0.53 fB =(6/11)=0.55 fC =(4/7)=0.57

(d)

Prescribed Displacement

(u )

y-Displacement

BHS

DLS

Fig. 5 The multilayer BHS/DLS laminates: (a) configurations; (b) schematic representation of FEM model and boundary conditions; (c)

surface imperfection; and (d) three snapshots for the deformation and the corresponding plastic strain contours of the 11-Layer BHS/DLS

laminate under plane strain uniaxial tension, a single-neck mode is observed.

Deformability of laminated sheet metal 89

away beyond its freestanding limit strain (0.06). It is also evi-dent that the absolute layer thickness has insignificant effect

on necking retardation, assuming no interface delamination,where the limit strain and the associated ultimate strengthare almost identical for the three configurations. The small

difference, however, is attributed to the small difference involume fraction of the DLS component.

Upon the insignificant effect of the absolute layer thicknesson necking retardation, only the 11-layer BHS/DLS laminateis used further in the nonuniform deformation analysis. Table

εNecking=0.233 (for laminate A)

εNecking=0.245 (for laminate B)εNecking=0.263 (for laminate C)

avgu

n

_σσ

700

750

800

850

900

950

1000

1050

0.05 0.10 0.15 0.20 0.25 0.30

σnLaminate

[MPa]

Case1-FEM Case1-Analytical

Case2-FEM Case2-Analytical

Case3-FEM Case3-Analytical

Case4-FEM Case4-Analytical

0.06/0.10.06/0.2

0.06/0.4

Legend

NDLS =0.1NDLS =0.2

NDLS =0.4Legend

(b)

(a)

εNecking

Fig. 6 Results of finite element analysis for plane strain uniaxial tension of multilayer BHS/DLS laminates: (a) The nominal stress,

normalized to the average ultimate strength, versus stain for three different configurations: 15, 11, and 7 layers; and (b) The necking strain

versus the associated nominal ultimate strength of the 11 layers laminates in comparison with the analytical results, where the four cases

are described in Table 2.

90 M.H. Serror

2 illustrates four cases of study for different combinations ofultimate strength ru of the BHS and DLS components, andhardening exponent and volume fraction of the DLS compo-

nent. In all cases, the yield strength and the hardening expo-nent of the BHS component are 1000 [MPa] and 0.06,respectively; while, the yield strength of the DLS component

is 300 [MPa]. The laminate components stiffness ratio k,however, is governed by the selected ultimate strength andthe hardening exponent, from the relation: k ¼ðru�DLS=N

NDLSDLS Þ=ðru�BHS=N

NBHSBHS Þ. Each case includes three

sub-cases of investigation which correspond to three differentvalues for hardening exponent of the DLS component

(NDLS = 0.1, 0.2, and 0.4).A finite element plane strain uniaxial tension analysis has

been conducted for the four cases, where the same prescribedimperfection on the top surface of the laminate (Fig. 5c) is

used. Table 2 lists the FEM analysis results against those ob-tained analytically using Eqs. (2) and (3) of this study.Fig. 6b plots the same results, where the laminate necking

strain eNecking has been plotted versus the associated nominalultimate strength rLaminate

n for the aforementioned four cases.The figure shows good agreement between the FEM analysis

results and the analytical ones, where the error percentage is

listed in Table 2. It is clear that both results are close to theprediction of the rule of averages with a noticeable deviationusually observed at the third sub-case of each analysis case

(NDLS = 0.4 and NBHS = 0.06). This is attributed to theobservation of Figs. 2 and 3, where the ultimate strength andthe associated limit strain deviate from the averages by increas-

ing the hardening exponent of the DLS component (NDLS) rel-ative to that of the BHS component (NBHS), where the ratioeNecking(BHS)/eNecking(DLS) tends to be small. It is also clear

that the limit strain increases by increasing NDLS, when com-paring the three sub-cases of each case. Fig. 6b informs alsothat the analytical results obtained in this study represent a

conservative estimate for laminate limit strain and ultimatestrength which are adequate for the design purpose of sheetmetal laminates. Further investigation is needed for the sensi-tivity of deformation mode to material parameters and inter-

face delamination.

Concluding remarks

The deformability of laminated sheet metal has been studiedanalytically in this paper. The plastic instability associated

with the localized necking under plane strain condition has

Table2

Studiedcasesof11-layer

BHS/D

LSlaminate

under

planestrain

uniaxialtensionusingfiniteelem

entmethod–comparisonbetweenanalyticalresultsandfiniteelem

entanalysis

results.

Laminate

DLSlayer

BHSlayer

f

r u_avgEq.(2)

(Mpa)

FEM

results

Analyticalresults

Error%

rNorm

Error%

e Necking

NDLS

r u(M

pa)

ry(M

pa)

NBHS

ru(M

pa)

r Y(M

pa)

r n(M

pa)

rNorm

(rn/r

u_avg)

e Necking

rn

(Mpa)

r Norm

(rn/r

u_avg)

e Necking

Case-1

0.1

600

300

0.06

1400

1000

1/2

930.69

955.81

1.03

0.08

928.65

1.00

0.07

2.8

7.9

0.2

904.85

933.17

1.03

0.11

889.41

0.98

0.10

4.7

9.1

0.4

860.33

877.78

1.02

0.18

811.86

0.94

0.13

7.5

25.2

Case-2

0.1

800

1200

1/2

926.99

974.27

1.05

0.09

924.72

1.00

0.08

5.1

9.9

0.2

892.55

951.45

1.07

0.13

875.32

0.98

0.11

8.0

11.4

0.4

833.19

896.66

1.08

0.22

778.55

0.93

0.17

13.2

25.5

Case-3

0.1

800

1400

2/3

922.07

957.09

1.04

0.09

919.74

1.00

0.08

3.9

9.3

0.2

876.15

938.50

1.07

0.16

858.85

0.98

0.13

8.5

14.1

0.4

796.99

895.35

1.12

0.29

741.89

0.93

0.22

17.1

23.8

Case-4

0.1

600

1200

1/3

934.38

1027.85

1.10

0.08

932.77

1.00

0.07

9.3

7.0

0.2

917.16

1006.62

1.10

0.09

905.13

0.99

0.09

10.1

6.1

0.4

887.48

958.23

1.08

0.12

850.29

0.96

0.11

11.3

8.9

Deformability of laminated sheet metal 91

been adopted to evaluate the laminate deformability. Stability

of both uniform and nonuniform deformations has been inves-tigated for two-layer and multilayer high-strength/ductile me-tal sheets (BHS/DLS) laminates. It is assumed that laminatelayers are fully bonded, and metals obey the power law mate-

rial model. It is worth noting that the present study is adequatefor the design purpose of sheet metal laminates; meanwhile,further investigation is needed for the sensitivity of deforma-

tion mode to material parameters and interface delamination.The results are summarized as follows: (1) the key control-

ling parameters for the BHS/DLS laminate formability are: the

hardening exponent of the DLS component NDLS, the volumefraction of the DLS component f, and the laminate compo-nents stiffness ratio k, where the deformability is increased

by increasing these parameters; (2) the layer absolute thickness(component thickness) has insignificant effect on laminate sta-bility, assuming no interface delamination; (3) for differentlaminate f and k, the range of the laminate limit strain is

bounded by the freestanding limit strains of both the BHScomponent and the DLS component as lower and upperbounds, respectively; (4) the laminate limit strain becomes clo-

ser to that predicted by the rule of averages by decreasing thehardening exponent of the DLS component till homogenousBHS sheet metal where the ratio eNecking(BHS)/eNecking(DLS)

approaches to unity; (5) the laminate limit strain becomes clo-ser, equal, or even more than that predicted by the rule of aver-ages by increasing the k ratio; (6) the laminate ultimatestrength is close to the average of components ultimate

strengths as predicted by the rule of averages; (7) the deforma-tion instability is different between a freestanding sheet andmultilayered sheet because the former exhibits only single-neck

mode; meanwhile, the later exhibits also multiple-neck mode.However, the most critical mode that is leading the final rup-ture in both of the sheets is the single-neck mode.

It is found that the DLS sheet metal retards the deforma-tion instability of the BHS sheet metal to an extent that de-pends on the abovementioned three controlling parameters.

Hence, enhanced strength–ductility combination can beachieved by laminated structure compared with the freestand-ing one. Such combination is essential for metal formingapplications.

Acknowledgments

The author is grateful to Prof. Junya Inoue (Department ofMaterial Engineering, University of Tokyo) for his valuable

discussion and peer review to the paper.

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