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Effect of the size of the sheet with sheared protrusions on the deformed shape after springback
Materials and Design 95 (2016) 348–357
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Effect of the size of the sheet with sheared protrusions on the deformed shape after springback Chang-Whan Lee a,b, Dong-Yol Yang a,⁎ a b
School of Mechanical Engineering & Aerospace System, Korea Advanced Institute of Science and Technology (KAIST), 291 Daehak-ro, Yuseong-gu, Daejeon 305-701, Republic of Korea Fuel Cell Research Center, Korea Institute of Science and Technology (KIST), Hwarangno 14-gil 5, Seongbuk-gu, Seoul 136-791, Republic of Korea
a r t i c l e
i n f o
Article history: Received 12 October 2015 Received in revised form 5 January 2016 Accepted 7 January 2016 Available online 12 January 2016 Keywords: Biaxial bending Sheet with sheared protrusions Metallic bipolar plate Molten carbonate fuel cell Hexahedral mesh coarsening
a b s t r a c t In this work, the effect of the size of a sheet with sheared protrusions on the deformed shape was investigated. After the forming process, a sheet with sheared protrusions was bent to result in specific curvatures with respect to the size of the plate due to springback. Springback deformation was assumed to be biaxial bending of an orthotropic plate. Resultant material properties of the plate were determined by using finite element analysis with hexahedral mesh coarsening. An analytic model was used to calculate deformed shapes with respect to the size of the plate. The results calculated with the proposed method showed excellent agreement with the measured results. Increasing the length in the transverse direction caused the curvature in the longitudinal direction to converge to the curvature of uniaxial bending. Increasing the length in the longitudinal direction suppressed bending in the transverse direction. © 2016 Elsevier Ltd. All rights reserved.
1. Introduction Molten carbonate fuel cells (MCFCs) are high-temperature fuel cells that operate at temperatures of around 650 °C and were developed for the cogeneration of electricity and heat with its high efficiency [1–3]. A sheet with sheared protrusions is utilized for the cathode current collector and anode current collector in MCFCs [4]. The sheet with sheared protrusions forms the gas flow channel and supports other components mechanically. Fig. 1(a) shows components of MCFCs. Fig. 1(b) shows the metallic bipolar plates used in MCFCs. The sheet with sheared protrusions is bent with a specific curvature in the direction in which trapezoidal protrusions are formed due to springback [5] as shown in Fig. 2(a). After the sheet is cut into a variety of sizes for fuel cell applications, the deformation modes change with respect to the size of the sheet with sheared protrusions. The deformed shape after springback depends on the size of the sheet with sheared protrusions after cutting. As shown in Fig. 2(b), when the sheet is cut with dimensions of 200 mm in the longitudinal direction (i.e. the lengthwise pattern-aligned direction) × 30 mm in the transverse direction (i.e. transverse to the lengthwise pattern-aligned direction), the sheet is bent in the longitudinal direction as shown in Fig. 2(b). On the other hand, when the sheet is cut with dimensions of 30 mm in the longitudinal direction × 200 mm in the transverse direction, the sheet is bent in the transverse direction as shown in Fig. 2(c). ⁎ Corresponding author. E-mail addresses: [email protected] (C.-W. Lee), [email protected] (D.-Y. Yang).
http://dx.doi.org/10.1016/j.matdes.2016.01.021 0264-1275/© 2016 Elsevier Ltd. All rights reserved.
In this work, an analytic model was employed to predict the deformed shape after springback because full simulation utilizing the simulation results of the sheared protrusions is not feasible [6]. Springback of the sheet with sheared protrusions was considered to be biaxial bending deformation of an orthotropic plate because the main deformation mode is bending deformation both in the longitudinal direction and in the transverse direction. When a bending moment is applied to the edges of the plate in the longitudinal direction (i.e., uniaxial bending), the plate is deformed into a cylindrical shape in the longitudinal direction. At the same time, the deformation in the transverse direction shows an anticlastic curvature with the value of νκ, where ν is Poisson's ratio and κ is the curvature in the longitudinal direction [7]. When a bending moment is applied to the all edges of the plate (i.e., biaxial bending), the plate is deformed into a partially spherical shape, i.e. a spherical segment [8–10]. Increasing bending deformation in one direction causes the plate to deform into a cylindrical shape. Levy [11] presented a solution for biaxial bending deformation of a rectangular plate with von Karman's large deflection theory with a trigonometric series. Ashwell [12] analyzed the type of instability in large deflections of elastic plates. He investigated the characteristic instability suppressing anticlastic curvature when a flat plate is loaded by bending moments applied to all four edges. Bellow et al. [13] experimentally investigated the anticlastic behavior of flat plates and showed that the transverse distortion of a rectangular plate subjected to large longitudinal curvatures depends on the dimensionless parameter W2κ/t, where W, κ, and t are the width, curvature, and thickness, respectively, of the plate. Pomeroy [14] investigated the effect of anticlastic bending on the curvature of beams. Pao [15] presented a
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characteristics of biaxial bending deformation were discussed using moment-curvature relationship. 2. Analytic model for prediction of the deformed shape after springback 2.1. Assumptions for the proposed method
Fig. 1. Schematic figure of MCFCs: (a) components and (b) metallic bipolar plates.
closed-form solution for the problem of laminated plates subjected to simple bending of heterogeneous anisotropic plate based on the large deflection theory. Hyer and Bhavani [16] discussed the suppression of the anticlastic curvature in composite plates by using the large deflection theory analytically and experimentally with a graphite–epoxy plate and aluminum plate. Wang [17] investigated the anticlastic curvature in draw-bend springback. Gigliotti et al. [18] predicted the deformed shape and multi-stable behavior of rectangular asymmetric plates subjected to thermal and environmental loads by using the large deformation theory and Rayleigh–Ritz method. The previous studies showed that the effect of the longitudinal membrane forces on deformation becomes significant [13] and the plate is deformed into a cylindrical shape as bending deformation increases in one direction (longitudinal direction). At the same time in the transverse direction, bending deformation is suppressed, and distortion occurs. This paper presents the effect of the size of a sheet with sheared protrusions on the deformed shape after springback. Springback deformation of the sheet with sheared protrusions was assumed to be biaxial bending of an orthotropic plate [19]. First, an analytic model for biaxial bending of an orthotropic plate was introduced. Next, the material properties of the sheet with sheared protrusions were obtained using finite element analysis [20–22] where the simulation model was constructed by using hexahedral mesh coarsening [6]. The bending moments in the longitudinal and transverse directions were obtained from the deformed shape of two specimens. Then, the deformed shape of the sheet with sheared protrusions of various dimensions was calculated by using the analytic model. To verify the proposed method, the calculated deformed shapes were compared with the measured results. Finally,
A sheet with sheared protrusions undergoes large deformation during the forming process. This process also produces residual stresses that bend the sheet either in the longitudinal or in the transverse direction, as shown in Fig. 2. The main deformation modes for springback of the sheet with sheared protrusions are bending in two directions. Springback deformation of the sheet with sheared protrusions was assumed to be biaxial bending of a rectangular plate subjected to bending moments along all edges. In addition, the sheet with sheared protrusions was assumed to be an orthotropic plate [19] that has the same mechanical properties with the sheet with sheared protrusions. In short, springback deformation of the sheet with sheared protrusions was considered to be biaxial bending of an orthotropic plate. An analytic model for biaxial bending of an orthotropic plate [12] was employed to predict the deformed shape after springback. 2.2. Biaxial bending of an orthotropic plate Fig. 3 illustrates the biaxial bending of a plate. A bending moment in the x-direction (Mx) and a bending moment in the y-direction (My) are applied to the edges. If the deflection of the plate (δ) is small compared with the thickness of the plate, the bending moment in each direction and the deflection of the orthotropic plate are expressed as follows [23]: ! Z t=2 2 2 ∂ δ ∂ δ σ xx zdz ¼ − Dx 2 þ Dxy 2 ; My ¼ σ yy zdz ∂x ∂y −t=2 −t=2 ! 2 2 ∂ δ ∂ δ ¼ − Dy 2 þ Dxy 2 ∂y ∂x Z
Mx ¼
t=2
δðx; yÞ ¼ −
1 Mx Dy −My D1 2 1 My Dx −Mx Dxy 2 x − y þ C1x þ C2y þ C3 2 Dx Dy −D1 2 2 Dx Dy −D1 2
ð1Þ
ð2Þ
where Dx, Dy, and Dxy are the bending moduli of the orthotropic plate and t is the thickness of the plate. C1, C2, and C3 are the constants of integration. In this case, the plate is bent into a spherical shape. If the deformation is small compared with the thickness of the plates, the effect of the longitudinal membrane forces is small. When the bending moment in the transverse direction is zero, the transverse curvature is νκ due to the Poisson effect. However, as the bending moment in each direction increases, the effect of the longitudinal membrane forces becomes significant and must be considered [13]. Eq. (1) does not hold for the deformation of a plate subjected to biaxial
Fig. 2. Springback of the sheet with sheared protrusions and cut specimens in two major directions: (a) deformed shape of the sheet with sheared protrusions after initial cutting, (b) springback of the cut specimen aligned with the longitudinal direction, and (c) springback of the cut specimen aligned with the transverse direction.
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Fig. 3. Biaxial bending of plates: (a) spherical shape when deformation is small compared with the thickness and (b) cylindrical shape when major bending deformation occurs in one direction.
bending. The spherical deformed shape (Fig. 3(a)) cannot be sustained. The final deformed shape of the plate is very close to a cylindrical shape (Fig. 3(b)) [12] due to the longitudinal membrane forces. The distortion of the plate when bending moments are applied to all edges of the plate was derived by Ashwell [12]. From von Karman's large deflection theory, the governing equation of the deflection (δ) of the isotropic plate in the transverse direction was derived [12]. For an orthotropic plate with major bending in the x-direction, as shown in Fig. 3(b), the governing equation of the deflection (δ) is given by. 4
4
d δ Exx κ x d δ Exx κ x 2 þ δ ¼ 4 þ 4α 2 δ ¼ 0; 4α 4 ¼ Dy t 2 dy4 Dy t 2 dy
ð3Þ
where tx is the tension per unit length along the edges x = ±L/2 and mx is the equivalent moment per unit length along the edges x = ± L/2. And, λ ¼ ½ðA þ BÞ sinhðαW=2Þ cosðαL=2Þ−ðA−BÞ coshðαW=2Þ sinðαW=2Þ; μ ¼ ½2 A2 þ B2 ð sinh αW þ sin αWÞ þ A2 −B2 þ 2AB cosh αW sin αW þ A2 −B2 −2AB sinh αW cos αW þ 2 A2 −B2 αW: ð6Þ
When a plate is loaded by uniformly distributed biaxial bending moments, Eq. (5) can be rewritten by using biaxial moments such as mx and my. The moment–curvature relationship is given by mx ¼ Dx κ x þ
where κx is the principal curvature in the x-direction, Exx is the elastic modulus in the x-direction, t is the thickness of the plate, and Dy is the flexural rigidity in the y-direction. The general solution of Eq. (3) for a rectangular plate is a series of sinusoidal functions. The initial lengths of the plate in the x- and ydirections are denoted by L and W, respectively. By using the boundary conditions for the moment about four edges and the shearing force along the four edges, the solution to Eq. (3) can be obtained as follows: δ ¼ kðA cosh α y cos α y þ B sinh α y sin α yÞ t
ð4Þ
2Dxy αλ my 1 Dxy Exx tμ my 1 Dxy 2 κx − − : þ W 8αW Dy κ x Dy Dy κ x Dy
ð7Þ
If mx, my and other variables are known, the variable in Eq. (7) is only κx. By solving Eq. (7), κx was calculated. The equation was solved using MATLAB (v2014a, MathWorks, Natick, MA, USA). After calculating κx, δ was calculated using Eq. (4). 2.3. Calculation procedure Fig. 4 shows the calculation procedure for the deformed shape that was used in this study. First, the mechanical properties of the sheet with sheared protrusions were obtained from finite element analysis, where the simulation model was constructed by using hexahedral
where k¼
My 1 Dxy my 1 Dxy − − ¼ ; Dy κ x W Dy Dy κ x Dy
1 sinhðαW=2Þ cosðαW=2Þ− coshðαW=2Þ sinðαW=2Þ A ¼ sffiffiffiffiffiffiffiffiffiffi sinhαW þ sinαW 1 Exx 4 Dy and 1 sinhðαW=2Þ cosðαW=2Þ þ coshðαW=2Þ sinðαW=2Þ : B ¼ sffiffiffiffiffiffiffiffiffiffi sinh αW þ sin αW 1 Exx 4 Dy The total moment applied along each edge is the integration of the tension per unit length times width and moment along the edges. The equivalent moment is given by Z Mx ¼
þ12L −12L
Z t x δdy þ
þ12L −12L
mx dy ¼ Dx κ x a þ 2Dxy αλk þ Exx Dx
1 2 μtk ð5Þ 8α
Fig. 4. Flow chart for the calculation of the deformed shape of the sheet with sheared protrusions.
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mesh coarsening. Next, the system of equations with respect to mx and my was obtained by using Eq. (7), the measured curvatures of two specimens, and the calculated mechanical properties. mx and my were obtained by solving the system of equations. Then, the principal curvature of the sheet with sheared protrusions (κx) was obtained by using Eq. (7) with given L and W. The distortion of the sheet with sheared protrusions (δ) normal to the principal curvature was obtained by using Eq. (4). Finally, the predicted deformed shapes of various dimensions were compared with measured results to verify the proposed method.
351
with isotropic hardening was used. The flow stress is given in Eq. (8) [24]. σ ¼ 1192:57ð0:0314 þ εÞ0:455
ð8Þ
Fig. 5 presents the experimental and simulation results of the threestage forming process. The length of the sheared protrusion was 6 mm, and the height of the sheared protrusion was 2.4 mm. The sheet with sheared protrusions experiences large deformation during the threestage forming process. In the experiment and simulation, the minimum thickness occurred in the upper round region. The ratio of the minimum thickness (tmin) to the initial thickness (t0) was 0.77.
3. Mechanical properties of the sheet with sheared protrusions 3.1. Manufacturing process of the sheet with sheared protrusions A sheet with sheared protrusions was manufactured from the threestage forming process consisted of the slitting process, the preforming process, and the final forming process [24]. The slitting process tears the sheet and forms it into sheared protrusions of low height. The preforming process stretches the sheet with the designed shape of the preform to distribute the plastic deformation uniformly. The final forming process forms the sheet into sheared protrusions of larger height. Using the steepest descent method Yang et al. [24] optimized the preforming process. In the simulation of the forming process, a 1/4 model with symmetric boundary conditions was utilized. For the simulation of the slitting process, the ductile fracture criterion of Cockcroft-Latham [25,26] was employed in the commercial FEM software ABAQUS/Explicit v6.13 with the user material subroutine VUMAT [27]. The sheet with sheared protrusions was manufactured from cold-rolled AISI310S stainless steel sheets. The thickness of sheets (t0) was 0.4 mm. Material properties of AISI310S were obtained from the uniaxial tensile test in accordance with ASTM E8/E8M [28]. The Young's modulus was 140 GPa. The yield stress of the material was 246.9 MPa. The von Mises yield criterion
3.2. Hexahedral mesh coarsening of the sheet with sheared protrusions The sheet with sheared protrusions was assumed to be an orthotropic plate. In the calculation of the deformed shape after springback, homogenized resultant material properties of the sheet with sheared protrusions are required. These resultant material properties were determined in virtual experiments using finite element analysis [20]. In the finite element analysis of the formed structures, results of the forming process such as the deformed geometry and distributions of the plastic strain and stress influence the mechanical behavior. In order to simulate the mechanical responses for formed structures more precisely, the results of the forming process was directly used in the finite element analysis [21,22]. However, the finite element analysis directly using the results of the forming process is hardly a simple task due to the large size of the problem. In this work, a simulation model was constructed using hexahedral mesh coarsening [6] for precise and effective simulation. Hexahedral mesh coarsening gives an efficient and precise simulation model by reducing the number of elements and transferring the results of the forming process. Hexahedral mesh coarsening can be described as follows [6]. First, the external geometry of the sheared protrusion is extracted from the simulation results of the forming process (old mesh). Next, a new mesh with the same geometry as the initial design of the sheared protrusion is constructed. Then, the nodes on the new mesh are repositioned to correspond to the external geometry of the old mesh. Then, the nodal values of state variables such as the stress and strains are mapped from the old mesh onto the new mesh by interpolation. Finally, a finite element model of the sheet with sheared protrusions is constructed by translating and reflecting the new mesh of the unit structure. In the simulation of the three-stage forming process, 19,876 8-node hexahedral elements (Fig. 6(a)) were used for the 1/4 model of the sheared protrusion to describe the complex deformation of the sheet at the sheared edge. This was employed as the old mesh. For the hexahedral mesh coarsening, regular hexahedral elements with a size of 0.1 mm were utilized for the new mesh. As a result of hexahedral mesh coarsening (Fig. 6(b)), 1398 8-node hexahedral elements were created. In the coarsened model composed of 16 sheared protrusions (Fig. 6(c)), 89,472 hexahedral elements were used. If the simulation results were used directly, the number of elements would be over 1,200,000, which would have imposed an excessive computational burden. 3.3. In-plane mechanical properties of the sheet with sheared protrusions
Fig. 5. Experimental and simulation results of the forming process for the sheared protrusion: (a) sectional view and (b) perspective view.
In the investigation of the in-plane mechanical properties, 4 (in the longitudinal direction) × 4 (in the transverse direction) sheared protrusions were considered in the virtual experiments using finite element analysis in order to consider the effect of the arrangement of sheared protrusions. The sheet with sheared protrusions was assumed to be an
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Fig. 6. Simulation model for determining the homogenized property: (a) simulation results of the three-stage forming process (19,876 elements), (b) remeshed model (1398 elements), and (c) simulation model composed of 16 sheared protrusions (89,472 elements).
orthotropic plate [19]. The constitutive equation of an orthotropic plate is 2
3 2 3 2 σL εL 1=EL 4 σ T 5 ¼ C 4 εT 5 ¼ 4 −vTL =EL 0 σ LT εLT
−vLT =ET 1=ET 0
3−1 2 3 0 εL 0 5 4 εT 5: 1=GLT εLT
ð9Þ
Homogenized resultant material properties can be obtained by using the boundary conditions summarized in Table 1. The subscripts L and T in Eq. (9) and Table 1 denote the longitudinal and transverse directions, respectively. Here, the elastic modulus in the longitudinal direction (EL) is calculated as an example of the calculation procedure. The boundary conditions in the second row of Table 1 were utilized in the simulation. The finite element simulation employed ABAQUS/standard v6.13 [27]. In the simulation results, the reaction force in the longitudinal direction (FL) with respect to displacement (Δu) was obtained. The effective stress in the longitudinal direction (σL) and strain in the longitudinal direction (εL) are denoted by FL/Wt and Δu/L, respectively. EL is the ratio of σL to εL for elastic deformation. The other material properties were obtained by using the same method. 3.4. Bending properties of the sheet with sheared protrusions The bending properties of the plate are usually represented by the flexural rigidity (D). This is given by EI/W where E is the Young's modulus, I is the second moment of inertia and W is the width of the plate. However, the flexural rigidity determined from the constitutive model of an homogeneous orthotropic plate (Et3/12) cannot accurately describe the actual bending deformation because the homogenized properties cannot describe the actual bending deformation of the plate
Table 1 Boundary conditions for determining the homogenized properties.
EL ET GLT
x=0
x=L
y=0
uL = 0 ⁎
uL = Δu ⁎
⁎
⁎
Fixed
uT = Δν
uT = 0 free
uT = Δv free
⁎ Normal displacements of the nodes in this surface are coupled.
y=w
with sheared protrusions due to the limitation of the homogenized model [29]. In this work, the flexural rigidities of the sheet with sheared protrusions in the longitudinal and transverse directions were obtained from a three-point bending simulation. The three-point bending simulation models were constructed by using hexahedral mesh coarsening. Fig. 7 shows the results. For the simulation of the longitudinal direction, planes in the transverse direction were defined as symmetric planes. For the simulation of the transverse direction, planes in the longitudinal direction were defined as symmetric planes. The simulation was conducted by using ABAQUS/standard v6.13 [27]. In the three-point bending tests, the span length (2S) was 120 mm. The radii of the punch and die (R) were 12.7 mm. The simulation and experiments were conducted in the elastic deformation region. In three-point bending, the relationships between the deflection of the plate (δP), load of the punch (P), and the flexural rigidity of the plate are given by [30]. δP ¼
Px 1 2 3S −x2 ; 0 ≤x≤S 12W D
ð10Þ
D¼
1 P S3 6 W δP
ð11Þ
where D is the flexural rigidity of the plate, x is the distance from the die, W is the width of the plate and S is the distance between the punch and die. The flexural rigidity (D) of the sheet with sheared protrusions can be obtained from Eq. (11). The results of three-point bending can be represented by a load– displacement curve. To verify the simulation model, the experimental results for three-point bending of a 200 mm × 30 mm plate were compared with the simulation results, as shown in Fig. 8. Here, the load– displacement curves are linear because the simulation and experiments were conducted in the elastic deformation region. The simulation results showed excellent agreement with the experimental results because the newly generated mesh with hexahedral mesh coarsening correctly accounted for the result of the forming process. Table 2 gives the homogenized resultant material properties of the sheet with sheared protrusions. The elastic modulus and flexural rigidity were greater in the longitudinal direction than in the transverse
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Fig. 7. Simulation result of three-point bending (δP/S = 0.1): (a) in the longitudinal direction and (b) in the transverse direction.
direction due to the geometric properties of the sheet with sheared protrusions. The calculated resultant material properties of the sheet with sheared protrusions were used to calculate the deformed shape. 3.5. Calculation of the moments applied along edges The moments per length applied in the longitudinal direction (mL) and transverse direction (mT) were obtained from the deformed shape of two specimens. The measured curvatures of plates with different sizes were used to construct the system of equations with respect to mL and mT. The two specimens employed in this work are shown in Fig. 2(b) and (c). When the sheet with sheared protrusions was cut into dimensions of 200 mm (longitudinal direction, L′) × 30 mm (transverse direction, W′), the curvature in the longitudinal direction (κL′) was 0.00268 mm−1. When the sheet was cut into dimensions of 30 mm (transverse direction, L″) × 30 mm (transverse direction, W″), the curvature in the transverse direction (κT″) was 0.00144 mm− 1. The curvatures of the two measured specimens (κL′, κT″) were used to construct the following system of equations: 0 2DLT α 0 λ mT 1 DLT EL tμ 0 mT 1 DLT 2 − − þ 0 0 κL 0 0 0 0 DT κ L DT DT κ L DT W 8α W ″ ″ ″ 2D α λ m 1 D E tμ m 1 DLT 2 LT L LT T L mT ¼ DT κ T 00 þ − κ T 00 − : þ ″ ″ 00 0 ″ DL κ T DL DL κ T DL L 8α L
mL ¼ DL κ L 0 þ
ð12Þ
The mechanical properties of the plates (DL, DT, DLT, EL, ET, ELT, νLT) listed in Table 2 were utilized. The system of equations was solved by using MATLAB (v2014a, MathWorks, Natick, MA, USA). The moments in the longitudinal direction (mL) and transverse direction (mT) were 3.18 N mm/mm and 0.74 N mm/mm, respectively. 4. Prediction of the deformed shape after springback The size of the original specimen was 1000 mm (longitudinal direction) × 600 mm (transverse direction). The sheet with sheared protrusions was cut into various sizes. External geometries of the sheet with sheared protrusions were measured using an optical 3D scanner (ATOS III Triple Scan, GOM mbH). Curvatures in the longitudinal and transverse directions were measured by using least-squares fitting of the measured three-dimensional geometry. Fig. 9 presents some example specimens. In order to verify the proposed method and investigate the effect of the size on the deformed shape, two cases were investigated. In case A, the length in the longitudinal direction (L) was fixed in order to investigate the effect of the length of the specimen in the transverse direction (W) on the deformed shape after springback. In case B, W was fixed in order to investigate the effect of L on the deformed shape after springback. 4.1. Computed results with respect to the length of the transverse direction (case A) Fig. 10(a) shows the curvature of the sheet with respect to W, when L was fixed at 200 mm. The measured curvatures were expressed as symbols. Curvatures of plates were measured when W = 40, 80, 120, 200, and 400 mm. Table 3 summarizes the values of the curvature in the longitudinal direction. The calculated results showed excellent agreement with the measured results. When W was very small, the curvature in the longitudinal direction (κL) remained unchanged, as shown in Fig. 10(a). In this case, the effect of transverse bending was very small, and the plate showed biaxial bending deformation. As W increased, the effects of the membrane stress and transverse bending increased. Finally, κL converged to Table 2 Homogenized resultant material properties of the sheet with sheared protrusions. EL ET GLT νLT (GPa) (GPa) (GPa)
Fig. 8. Simulation and experimental results of three-point bending.
13.21 5.83
1.4
DL DT DLT t (N·mm2/mm) (N·mm2/mm) (N·mm2/mm) (mm)
0.22 1163.25
406.61
116.75
2.4
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Fig. 9. Deformed shapes of the sheet with sheared protrusions after springback: (a) with respect to W (L = 200 mm) and (b) with respect to L (W = 200 mm).
(κL)f = mL/DL = 0.00273 mm−1, which was the curvature in the case of uniaxial bending. Fig. 10(b) illustrates the computed deflection at the center along the transverse direction (δT) with respect to W. As W increased, distortion occurred near the edge of the plate. 4.2. Computed results with respect to the length of the longitudinal direction (case B) The curvature distribution was obtained by using Eq. (7) under the assumption that the curvature in the transverse direction is the principal curvature. Fig. 11(a) shows the calculated results of the curvature in the transverse direction (κT) with respect to L when W was fixed to 200 mm. The experimental results are shown with symbols. The specimen was measured when L = 40, 80, 120, 200, and 400 mm. Table 4 summarizes the values of κT. Transverse bending remained when L was very small. As L increased, κT decreased drastically, as shown in Fig. 11(a). Finally, κT converged to zero, and the plate bent in the longitudinal direction as L increased. As
discussed in previous studies on the suppression of transverse bending [16], bending in the transverse direction was suppressed due to the membrane stress in the longitudinal direction. In this work, the critical ratio of κT to κL when the plate was mainly bent in the longitudinal direction was defined as 0.1. When L was larger than 92.7 mm, κT was 0.000271 mm−1, and bending in the transverse direction was suppressed. While bending in the transverse direction was suppressed, longitudinal bending remained. In this case, the plate was bent in the longitudinal direction. Fig. 11(b) shows the computed deflection at the center along the longitudinal direction (δL) of the plate with respect to L. As L increased, the curvature increased by a small amount. When L was 200 mm, κL was calculated to be 0.00271 mm−1. Fig. 12(a) shows the three-dimensional scanning results of the sheet with sheared protrusions (W = 200 mm and L = 200 mm). The calculated and measured curvatures in the longitudinal direction were 0.00271 and 0.0027 mm−1, respectively. Fig. 12(b) shows the deflection of the sheet with sheared protrusions along A–A′. The calculated and
Fig. 10. Curvature distribution and deformed shape of the sheet with sheared protrusions with respect to W when L was 200 mm: (a) variation of κL with respect to W and (b) computed deflection at x = 0.
C.-W. Lee, D.-Y. Yang / Materials and Design 95 (2016) 348–357 Table 3 Curvature in the longitudinal direction (κL) with respect to W (L = 200 mm).
355
Table 4 Curvature in the transverse direction (κT) with respect to L (W = 200 mm).
W (mm)
40
80
120
200
400
L (mm)
Measured results (mm−1) Computed results (mm−1)
0.00267 0.00266
0.00267 0.00267
0.00268 0.00269
0.0027 0.00271
0.00271 0.00272
Measured results (mm−1) 0.0011 0.00037 0.00015 ≈0⁎ ≈0⁎ Computed results (mm−1) 0.0013 0.00043 0.00011 1.4 × 10−5 9.2 × 10−7
40
80
120
200
400
⁎ Curvature was too small to be measured.
measured deflections along A–A′ showed similar trends. Distortion occurred near the edge of the plate. Similar to the previous results, there were no significant differences between the calculated and measured results. The assumption that springback deformation of the sheet with sheared protrusions is biaxial bending deformation of an orthotropic plate is appropriate. By using the proposed calculation method, the deformed shape of the sheet with sheared protrusions of various dimensions can be obtained easily and accurately. In addition, the proposed method can be applied to the prediction of the deformed shape after springback when the main deformation mode of formed plates is biaxial bending. For application of the sheet with sheared protrusions, the plate should be flattened. When L is larger than 92.7 mm, bending in the transverse direction is suppressed, and the plate is only bent in the longitudinal direction. In this case, the leveling process can be applied in the longitudinal direction. In the previous work [5], a leveling process using three rolls was employed. As a result, the sheet with sheared protrusions was leveled nearly flat. The flattened sheet with sheared protrusions was employed for MCFCs without any problem, as shown in Fig. 1(b).
5. Discussion 5.1. Moment-curvature relationship of the plate with respect to the length in the transverse direction The effect of bending in the transverse direction on biaxial bending of an orthotropic plate was investigated. The material properties listed in Table 2 were utilized in the calculation. When bending moments are applied in the transverse direction, the moment-curvature relationship can be categorized into three stages [12]: Stage I, the moment increases drastically when the flexural rigidity of the plate increases due to biaxial bending deformation; Stage II, negative stiffness occurs as the cross-section flattens; and Stage III, the cross-section becomes almost flat and mL is nearly proportional to κL. Fig. 13 shows the relationship between mL and κL with respect to W, when L is 200 mm and mT is 0.74 N·mm/mm. The peak becomes more conspicuous as W increases. As κL increases, the three curves in Fig. 13 converge to one line. The effect of mT decreases as κL increases. Finally,
the moment in the longitudinal direction is nearly proportional to the curvature in the longitudinal direction (mL = DLκL). From the moment-curvature relationship in Fig. 13, the curvature in the longitudinal direction can be obtained. Some examples are described below. The horizontal dotted line represents mL = 3.18 N·mm/mm. The intersecting points between the horizontal line and the curve present the deformed shape. For W = 40 and 160 mm, the horizontal line and curve intersect at one point. In other words, there is only one solution (κL) to Eq. (7) with a given mL, mT, L and W. The plate only shows stage III bending and has the same curvature as the intersecting point. However, when W is 240 mm, the horizontal line and curve intersect at three points because the peak value of the curve is larger than mL. In this case, there are three solutions (κL) to Eq. (7) with a given mL, mT, L and W. The plate shows three bending modes: stages I, II, and III. In order to make κL = 0.00271 mm−1 from the initial state, the required mL should be larger than 4.53 N·mm/mm, which is the peak value of mL in Fig. 13.
5.2. Moment-curvature relationship of the plate when the ratio of mL to mT is constrained Springback deformation of the sheet with sheared protrusions is different from the previous case. In springback deformation, the ratio of the moment in the longitudinal direction to that in the transverse direction (mL/mT) is constant. In order to investigate springback deformation, the relationship between mL and κL with respect to W was investigated with a constrained value for mL/mT. Fig. 14 shows relationship between m L and κ L with respect to W. m L /m T is constrained to 4.3 which is the ratio for the sheet with sheared protrusions. The curves were drawn by using Eq. (7). Unlike Fig. 13, the curve in the longitudinal direction in Fig. 14 shows a nearly linear relationship between mL and κL. When mL/mT is constrained and κL increases gradually, the plate undergoes stage III bending only because the moment–curvature relation is nearly linear. There is only one solution (κL) to Eq. (7) with a given mL, mT, L and W. Springback of the sheet with sheared protrusions shows stage III bending because mL/mT is constrained.
Fig. 11. Curvature distribution and deformed shape of the sheet with sheared protrusions with respect to L when W was 200 mm: (a) variation of κT with respect to W and (b) computed deflection at x = 0.
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Fig. 12. Three-dimensional scanning results: (a) measured results of the sheet with sheared protrusions (W = 200 mm and L = 200 mm) and (b) measured and computed deflection profiles along A-A’.
The material properties of the sheet with sheared protrusions were obtained by using finite element analysis with hexahedral mesh coarsening. The moments in the longitudinal direction (mL) and transverse direction (mT) due to springback were determined to be 3.18 and 0.74 N·mm/mm, respectively. The deformed shapes after springback with respect to the size of the plate were obtained by using the moment–curvature relationship of an orthotropic plate. From the analytic models, the deformed shape after springback was precisely predicted with respect to the size of the plate. Increasing the length in the transverse direction caused the curvature in the longitudinal direction to converge to 0.00273 mm−1, which was the curvature in the case of uniaxial bending. Increasing the length in the longitudinal direction caused the curvature in the transverse direction to converge to zero asymptotically. Bending deformation was suppressed in the transverse direction.
References Fig. 13. Relationship between mL and κL with respect to W (L = 200 mm and mT = 0.74 N·mm/mm).
6. Conclusion In this work, springback of the sheet with sheared protrusions was assumed to be biaxial deformation of an orthotropic plate. In order to predict the deformed shape after springback, an analytic model for the biaxial bending of an orthotropic rectangular plate was employed.
Fig. 14. Relationship between mL and κL with respect to W (L = 200 mm and mL/mT = 4.3).
[1] A. Hilmi, C.-Y. Yuh, M. Farooque, Carbonate fuel cell anode: a review, ECS Trans. 61 (2014) 245–253. [2] U. Lucia, Overview on fuel cells, Renew. Sustain. Energy Rev. 30 (2014) 164–169. [3] M. Kawase, Manufacturing method for tubular molten carbonate fuel cells and basic cell performance, J. Power Sources 285 (2015) 260–265. [4] H. Kim, J. Bae, D. Choi, An analysis for a molten carbonate fuel cell of complex geometry using three-dimensional transport equations with electrochemical reactions, Int. J. Hydrog. Energy 38 (2013) 4782–4791. [5] C.W. Lee, D.Y. Yang, D.W. Kang, T.W. Lee, Study on the levelling process of the current collector for the molten carbonate fuel cell based on curvature integration method, Int. J. Hydrog. Energy 39 (2014) 6714–6728. [6] C.W. Lee, D.Y. Yang, J.S. Park, Y.S. KIM, T.W. Lee, Investigation into mechanical behavior of the current collector for the molten carbonate fuel cell through finite element analysis using hexahedral mesh coarsening, J. Fuel Cell Sci. Technol. 11 (2014) 061005-01–0061005-10. [7] S.P. Timoshenko, S. Woinowsky-Krieger, Theory of Plates and Shells, McGRAWHILL International Book Company, 1959 27–37. [8] W. Johnson, T.X. Yu, Springback after the biaxial elastic-plastic pure bending of a rectangular plate-I, Int. J. Mech. Sci. 23 (1981) 619–630. [9] M.H. Parsa, S.N.a. ahkami, H. Pishbin, M. Kazemi, Investigating spring back phenomena in double curved sheet metals forming, Mater. Des. 41 (2012) 326–337. [10] Q.-F. Zhang, Z.-Y. Cai, Y. Zhang, M.-Z. Li, Springback compensation method for doubly curved plate in multi-point forming, Mater. Des. 47 (2013) 377–385. [11] S. Levy, Bending of rectangular plates with large deflections, National Adviory Committee for Aeronautics Technical Note 846, 1942. [12] D.G. Ashwell, A characteristic type of instability in the large deflexions of elastic plates. I. Curved rectangular plates bent about one axis. II. Flat square plates bent about all edges, Proc. R. Soc. Lond. A Math. Phys. Sci. 214 (1952) 98–118. [13] D.G. Bellow, G. Ford, J.S. Kennedy, Anticlastic behavior of flat plates, Exp. Mech. 5 (1965) 227–232. [14] R.J. Pomeroy, The effect of anticlastic bending on the curvature of beams, Int. J. Solids Struct. 6 (1970) 277–285. [15] Y.C. Pao, Simple bending analysis of laminated plates by large-deflection theory, J. Compos. Mater. 4 (1970) 380–389. [16] M.W. Hyer, P.C. Bhavani, Suppression of anticlastic curvature in isotropic and composite plates, Int. J. Solids Struct. 20 (1984) 553–570. [17] J.F. Wang, R.H. Wagoner, D.K. Matlock, F. Barlat, Anticlastic curvature in draw–bend springback, Int. J. Solids Struct. 42 (2005) 1287–1307.
C.-W. Lee, D.-Y. Yang / Materials and Design 95 (2016) 348–357 [18] M. Gigliotti, M. Minervino, J.C. Grandidier, M.C. Lafarie-Frenot, Predicting loss of bifurcation behaviour of 0/90 unsymmetric composite plates subjected to environmental loads, Compos. Struct. 94 (2012) 2793–2808. [19] C.W. Lee, D.Y. Yang, D.W. Kang, I.G. Chang, T.W. LEE, Anisotropic behavior and homogenization of the shielded slot plate for the molten carbonate fuel cell, Steel Research International, Special ed. 2011, pp. 1102–1107. [20] C.C. Besse, D. Mohr, Plasticity of formable all-metal sandwich sheets: virtual experiments and constitutive modeling, Int. J. Solids Struct. 49 (2012) 2863–2880. [21] K. Bandyopadhyay, S. Basak, S.K. Panda, P. Saha, Use of stress based forming limit diagram to predict formability in two-stage forming of tailor welded blanks, Mater. Des. 67 (2015) 558–570. [22] A.E. Tekkaya, J.M. Allwood, P.F. Bariani, S. Bruschi, J. Cao, S. Gramlich, et al., Metal forming beyond shaping: predicting and setting product properties, CIRP Ann. Manuf. Technol. 64 (2015) 629–653. [23] S.P. Timoshenko, J.M. Gere, Theory of elastic stability, McGRAWHILL International Book Company, 1963 319–347.
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Materials and Design journal homepage: www.elsevier.com/locate/matdes
Effect of the size of the sheet with sheared protrusions on the deformed shape after springback Chang-Whan Lee a,b, Dong-Yol Yang a,⁎ a b
School of Mechanical Engineering & Aerospace System, Korea Advanced Institute of Science and Technology (KAIST), 291 Daehak-ro, Yuseong-gu, Daejeon 305-701, Republic of Korea Fuel Cell Research Center, Korea Institute of Science and Technology (KIST), Hwarangno 14-gil 5, Seongbuk-gu, Seoul 136-791, Republic of Korea
a r t i c l e
i n f o
Article history: Received 12 October 2015 Received in revised form 5 January 2016 Accepted 7 January 2016 Available online 12 January 2016 Keywords: Biaxial bending Sheet with sheared protrusions Metallic bipolar plate Molten carbonate fuel cell Hexahedral mesh coarsening
a b s t r a c t In this work, the effect of the size of a sheet with sheared protrusions on the deformed shape was investigated. After the forming process, a sheet with sheared protrusions was bent to result in specific curvatures with respect to the size of the plate due to springback. Springback deformation was assumed to be biaxial bending of an orthotropic plate. Resultant material properties of the plate were determined by using finite element analysis with hexahedral mesh coarsening. An analytic model was used to calculate deformed shapes with respect to the size of the plate. The results calculated with the proposed method showed excellent agreement with the measured results. Increasing the length in the transverse direction caused the curvature in the longitudinal direction to converge to the curvature of uniaxial bending. Increasing the length in the longitudinal direction suppressed bending in the transverse direction. © 2016 Elsevier Ltd. All rights reserved.
1. Introduction Molten carbonate fuel cells (MCFCs) are high-temperature fuel cells that operate at temperatures of around 650 °C and were developed for the cogeneration of electricity and heat with its high efficiency [1–3]. A sheet with sheared protrusions is utilized for the cathode current collector and anode current collector in MCFCs [4]. The sheet with sheared protrusions forms the gas flow channel and supports other components mechanically. Fig. 1(a) shows components of MCFCs. Fig. 1(b) shows the metallic bipolar plates used in MCFCs. The sheet with sheared protrusions is bent with a specific curvature in the direction in which trapezoidal protrusions are formed due to springback [5] as shown in Fig. 2(a). After the sheet is cut into a variety of sizes for fuel cell applications, the deformation modes change with respect to the size of the sheet with sheared protrusions. The deformed shape after springback depends on the size of the sheet with sheared protrusions after cutting. As shown in Fig. 2(b), when the sheet is cut with dimensions of 200 mm in the longitudinal direction (i.e. the lengthwise pattern-aligned direction) × 30 mm in the transverse direction (i.e. transverse to the lengthwise pattern-aligned direction), the sheet is bent in the longitudinal direction as shown in Fig. 2(b). On the other hand, when the sheet is cut with dimensions of 30 mm in the longitudinal direction × 200 mm in the transverse direction, the sheet is bent in the transverse direction as shown in Fig. 2(c). ⁎ Corresponding author. E-mail addresses: [email protected] (C.-W. Lee), [email protected] (D.-Y. Yang).
http://dx.doi.org/10.1016/j.matdes.2016.01.021 0264-1275/© 2016 Elsevier Ltd. All rights reserved.
In this work, an analytic model was employed to predict the deformed shape after springback because full simulation utilizing the simulation results of the sheared protrusions is not feasible [6]. Springback of the sheet with sheared protrusions was considered to be biaxial bending deformation of an orthotropic plate because the main deformation mode is bending deformation both in the longitudinal direction and in the transverse direction. When a bending moment is applied to the edges of the plate in the longitudinal direction (i.e., uniaxial bending), the plate is deformed into a cylindrical shape in the longitudinal direction. At the same time, the deformation in the transverse direction shows an anticlastic curvature with the value of νκ, where ν is Poisson's ratio and κ is the curvature in the longitudinal direction [7]. When a bending moment is applied to the all edges of the plate (i.e., biaxial bending), the plate is deformed into a partially spherical shape, i.e. a spherical segment [8–10]. Increasing bending deformation in one direction causes the plate to deform into a cylindrical shape. Levy [11] presented a solution for biaxial bending deformation of a rectangular plate with von Karman's large deflection theory with a trigonometric series. Ashwell [12] analyzed the type of instability in large deflections of elastic plates. He investigated the characteristic instability suppressing anticlastic curvature when a flat plate is loaded by bending moments applied to all four edges. Bellow et al. [13] experimentally investigated the anticlastic behavior of flat plates and showed that the transverse distortion of a rectangular plate subjected to large longitudinal curvatures depends on the dimensionless parameter W2κ/t, where W, κ, and t are the width, curvature, and thickness, respectively, of the plate. Pomeroy [14] investigated the effect of anticlastic bending on the curvature of beams. Pao [15] presented a
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characteristics of biaxial bending deformation were discussed using moment-curvature relationship. 2. Analytic model for prediction of the deformed shape after springback 2.1. Assumptions for the proposed method
Fig. 1. Schematic figure of MCFCs: (a) components and (b) metallic bipolar plates.
closed-form solution for the problem of laminated plates subjected to simple bending of heterogeneous anisotropic plate based on the large deflection theory. Hyer and Bhavani [16] discussed the suppression of the anticlastic curvature in composite plates by using the large deflection theory analytically and experimentally with a graphite–epoxy plate and aluminum plate. Wang [17] investigated the anticlastic curvature in draw-bend springback. Gigliotti et al. [18] predicted the deformed shape and multi-stable behavior of rectangular asymmetric plates subjected to thermal and environmental loads by using the large deformation theory and Rayleigh–Ritz method. The previous studies showed that the effect of the longitudinal membrane forces on deformation becomes significant [13] and the plate is deformed into a cylindrical shape as bending deformation increases in one direction (longitudinal direction). At the same time in the transverse direction, bending deformation is suppressed, and distortion occurs. This paper presents the effect of the size of a sheet with sheared protrusions on the deformed shape after springback. Springback deformation of the sheet with sheared protrusions was assumed to be biaxial bending of an orthotropic plate [19]. First, an analytic model for biaxial bending of an orthotropic plate was introduced. Next, the material properties of the sheet with sheared protrusions were obtained using finite element analysis [20–22] where the simulation model was constructed by using hexahedral mesh coarsening [6]. The bending moments in the longitudinal and transverse directions were obtained from the deformed shape of two specimens. Then, the deformed shape of the sheet with sheared protrusions of various dimensions was calculated by using the analytic model. To verify the proposed method, the calculated deformed shapes were compared with the measured results. Finally,
A sheet with sheared protrusions undergoes large deformation during the forming process. This process also produces residual stresses that bend the sheet either in the longitudinal or in the transverse direction, as shown in Fig. 2. The main deformation modes for springback of the sheet with sheared protrusions are bending in two directions. Springback deformation of the sheet with sheared protrusions was assumed to be biaxial bending of a rectangular plate subjected to bending moments along all edges. In addition, the sheet with sheared protrusions was assumed to be an orthotropic plate [19] that has the same mechanical properties with the sheet with sheared protrusions. In short, springback deformation of the sheet with sheared protrusions was considered to be biaxial bending of an orthotropic plate. An analytic model for biaxial bending of an orthotropic plate [12] was employed to predict the deformed shape after springback. 2.2. Biaxial bending of an orthotropic plate Fig. 3 illustrates the biaxial bending of a plate. A bending moment in the x-direction (Mx) and a bending moment in the y-direction (My) are applied to the edges. If the deflection of the plate (δ) is small compared with the thickness of the plate, the bending moment in each direction and the deflection of the orthotropic plate are expressed as follows [23]: ! Z t=2 2 2 ∂ δ ∂ δ σ xx zdz ¼ − Dx 2 þ Dxy 2 ; My ¼ σ yy zdz ∂x ∂y −t=2 −t=2 ! 2 2 ∂ δ ∂ δ ¼ − Dy 2 þ Dxy 2 ∂y ∂x Z
Mx ¼
t=2
δðx; yÞ ¼ −
1 Mx Dy −My D1 2 1 My Dx −Mx Dxy 2 x − y þ C1x þ C2y þ C3 2 Dx Dy −D1 2 2 Dx Dy −D1 2
ð1Þ
ð2Þ
where Dx, Dy, and Dxy are the bending moduli of the orthotropic plate and t is the thickness of the plate. C1, C2, and C3 are the constants of integration. In this case, the plate is bent into a spherical shape. If the deformation is small compared with the thickness of the plates, the effect of the longitudinal membrane forces is small. When the bending moment in the transverse direction is zero, the transverse curvature is νκ due to the Poisson effect. However, as the bending moment in each direction increases, the effect of the longitudinal membrane forces becomes significant and must be considered [13]. Eq. (1) does not hold for the deformation of a plate subjected to biaxial
Fig. 2. Springback of the sheet with sheared protrusions and cut specimens in two major directions: (a) deformed shape of the sheet with sheared protrusions after initial cutting, (b) springback of the cut specimen aligned with the longitudinal direction, and (c) springback of the cut specimen aligned with the transverse direction.
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Fig. 3. Biaxial bending of plates: (a) spherical shape when deformation is small compared with the thickness and (b) cylindrical shape when major bending deformation occurs in one direction.
bending. The spherical deformed shape (Fig. 3(a)) cannot be sustained. The final deformed shape of the plate is very close to a cylindrical shape (Fig. 3(b)) [12] due to the longitudinal membrane forces. The distortion of the plate when bending moments are applied to all edges of the plate was derived by Ashwell [12]. From von Karman's large deflection theory, the governing equation of the deflection (δ) of the isotropic plate in the transverse direction was derived [12]. For an orthotropic plate with major bending in the x-direction, as shown in Fig. 3(b), the governing equation of the deflection (δ) is given by. 4
4
d δ Exx κ x d δ Exx κ x 2 þ δ ¼ 4 þ 4α 2 δ ¼ 0; 4α 4 ¼ Dy t 2 dy4 Dy t 2 dy
ð3Þ
where tx is the tension per unit length along the edges x = ±L/2 and mx is the equivalent moment per unit length along the edges x = ± L/2. And, λ ¼ ½ðA þ BÞ sinhðαW=2Þ cosðαL=2Þ−ðA−BÞ coshðαW=2Þ sinðαW=2Þ; μ ¼ ½2 A2 þ B2 ð sinh αW þ sin αWÞ þ A2 −B2 þ 2AB cosh αW sin αW þ A2 −B2 −2AB sinh αW cos αW þ 2 A2 −B2 αW: ð6Þ
When a plate is loaded by uniformly distributed biaxial bending moments, Eq. (5) can be rewritten by using biaxial moments such as mx and my. The moment–curvature relationship is given by mx ¼ Dx κ x þ
where κx is the principal curvature in the x-direction, Exx is the elastic modulus in the x-direction, t is the thickness of the plate, and Dy is the flexural rigidity in the y-direction. The general solution of Eq. (3) for a rectangular plate is a series of sinusoidal functions. The initial lengths of the plate in the x- and ydirections are denoted by L and W, respectively. By using the boundary conditions for the moment about four edges and the shearing force along the four edges, the solution to Eq. (3) can be obtained as follows: δ ¼ kðA cosh α y cos α y þ B sinh α y sin α yÞ t
ð4Þ
2Dxy αλ my 1 Dxy Exx tμ my 1 Dxy 2 κx − − : þ W 8αW Dy κ x Dy Dy κ x Dy
ð7Þ
If mx, my and other variables are known, the variable in Eq. (7) is only κx. By solving Eq. (7), κx was calculated. The equation was solved using MATLAB (v2014a, MathWorks, Natick, MA, USA). After calculating κx, δ was calculated using Eq. (4). 2.3. Calculation procedure Fig. 4 shows the calculation procedure for the deformed shape that was used in this study. First, the mechanical properties of the sheet with sheared protrusions were obtained from finite element analysis, where the simulation model was constructed by using hexahedral
where k¼
My 1 Dxy my 1 Dxy − − ¼ ; Dy κ x W Dy Dy κ x Dy
1 sinhðαW=2Þ cosðαW=2Þ− coshðαW=2Þ sinðαW=2Þ A ¼ sffiffiffiffiffiffiffiffiffiffi sinhαW þ sinαW 1 Exx 4 Dy and 1 sinhðαW=2Þ cosðαW=2Þ þ coshðαW=2Þ sinðαW=2Þ : B ¼ sffiffiffiffiffiffiffiffiffiffi sinh αW þ sin αW 1 Exx 4 Dy The total moment applied along each edge is the integration of the tension per unit length times width and moment along the edges. The equivalent moment is given by Z Mx ¼
þ12L −12L
Z t x δdy þ
þ12L −12L
mx dy ¼ Dx κ x a þ 2Dxy αλk þ Exx Dx
1 2 μtk ð5Þ 8α
Fig. 4. Flow chart for the calculation of the deformed shape of the sheet with sheared protrusions.
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mesh coarsening. Next, the system of equations with respect to mx and my was obtained by using Eq. (7), the measured curvatures of two specimens, and the calculated mechanical properties. mx and my were obtained by solving the system of equations. Then, the principal curvature of the sheet with sheared protrusions (κx) was obtained by using Eq. (7) with given L and W. The distortion of the sheet with sheared protrusions (δ) normal to the principal curvature was obtained by using Eq. (4). Finally, the predicted deformed shapes of various dimensions were compared with measured results to verify the proposed method.
351
with isotropic hardening was used. The flow stress is given in Eq. (8) [24]. σ ¼ 1192:57ð0:0314 þ εÞ0:455
ð8Þ
Fig. 5 presents the experimental and simulation results of the threestage forming process. The length of the sheared protrusion was 6 mm, and the height of the sheared protrusion was 2.4 mm. The sheet with sheared protrusions experiences large deformation during the threestage forming process. In the experiment and simulation, the minimum thickness occurred in the upper round region. The ratio of the minimum thickness (tmin) to the initial thickness (t0) was 0.77.
3. Mechanical properties of the sheet with sheared protrusions 3.1. Manufacturing process of the sheet with sheared protrusions A sheet with sheared protrusions was manufactured from the threestage forming process consisted of the slitting process, the preforming process, and the final forming process [24]. The slitting process tears the sheet and forms it into sheared protrusions of low height. The preforming process stretches the sheet with the designed shape of the preform to distribute the plastic deformation uniformly. The final forming process forms the sheet into sheared protrusions of larger height. Using the steepest descent method Yang et al. [24] optimized the preforming process. In the simulation of the forming process, a 1/4 model with symmetric boundary conditions was utilized. For the simulation of the slitting process, the ductile fracture criterion of Cockcroft-Latham [25,26] was employed in the commercial FEM software ABAQUS/Explicit v6.13 with the user material subroutine VUMAT [27]. The sheet with sheared protrusions was manufactured from cold-rolled AISI310S stainless steel sheets. The thickness of sheets (t0) was 0.4 mm. Material properties of AISI310S were obtained from the uniaxial tensile test in accordance with ASTM E8/E8M [28]. The Young's modulus was 140 GPa. The yield stress of the material was 246.9 MPa. The von Mises yield criterion
3.2. Hexahedral mesh coarsening of the sheet with sheared protrusions The sheet with sheared protrusions was assumed to be an orthotropic plate. In the calculation of the deformed shape after springback, homogenized resultant material properties of the sheet with sheared protrusions are required. These resultant material properties were determined in virtual experiments using finite element analysis [20]. In the finite element analysis of the formed structures, results of the forming process such as the deformed geometry and distributions of the plastic strain and stress influence the mechanical behavior. In order to simulate the mechanical responses for formed structures more precisely, the results of the forming process was directly used in the finite element analysis [21,22]. However, the finite element analysis directly using the results of the forming process is hardly a simple task due to the large size of the problem. In this work, a simulation model was constructed using hexahedral mesh coarsening [6] for precise and effective simulation. Hexahedral mesh coarsening gives an efficient and precise simulation model by reducing the number of elements and transferring the results of the forming process. Hexahedral mesh coarsening can be described as follows [6]. First, the external geometry of the sheared protrusion is extracted from the simulation results of the forming process (old mesh). Next, a new mesh with the same geometry as the initial design of the sheared protrusion is constructed. Then, the nodes on the new mesh are repositioned to correspond to the external geometry of the old mesh. Then, the nodal values of state variables such as the stress and strains are mapped from the old mesh onto the new mesh by interpolation. Finally, a finite element model of the sheet with sheared protrusions is constructed by translating and reflecting the new mesh of the unit structure. In the simulation of the three-stage forming process, 19,876 8-node hexahedral elements (Fig. 6(a)) were used for the 1/4 model of the sheared protrusion to describe the complex deformation of the sheet at the sheared edge. This was employed as the old mesh. For the hexahedral mesh coarsening, regular hexahedral elements with a size of 0.1 mm were utilized for the new mesh. As a result of hexahedral mesh coarsening (Fig. 6(b)), 1398 8-node hexahedral elements were created. In the coarsened model composed of 16 sheared protrusions (Fig. 6(c)), 89,472 hexahedral elements were used. If the simulation results were used directly, the number of elements would be over 1,200,000, which would have imposed an excessive computational burden. 3.3. In-plane mechanical properties of the sheet with sheared protrusions
Fig. 5. Experimental and simulation results of the forming process for the sheared protrusion: (a) sectional view and (b) perspective view.
In the investigation of the in-plane mechanical properties, 4 (in the longitudinal direction) × 4 (in the transverse direction) sheared protrusions were considered in the virtual experiments using finite element analysis in order to consider the effect of the arrangement of sheared protrusions. The sheet with sheared protrusions was assumed to be an
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Fig. 6. Simulation model for determining the homogenized property: (a) simulation results of the three-stage forming process (19,876 elements), (b) remeshed model (1398 elements), and (c) simulation model composed of 16 sheared protrusions (89,472 elements).
orthotropic plate [19]. The constitutive equation of an orthotropic plate is 2
3 2 3 2 σL εL 1=EL 4 σ T 5 ¼ C 4 εT 5 ¼ 4 −vTL =EL 0 σ LT εLT
−vLT =ET 1=ET 0
3−1 2 3 0 εL 0 5 4 εT 5: 1=GLT εLT
ð9Þ
Homogenized resultant material properties can be obtained by using the boundary conditions summarized in Table 1. The subscripts L and T in Eq. (9) and Table 1 denote the longitudinal and transverse directions, respectively. Here, the elastic modulus in the longitudinal direction (EL) is calculated as an example of the calculation procedure. The boundary conditions in the second row of Table 1 were utilized in the simulation. The finite element simulation employed ABAQUS/standard v6.13 [27]. In the simulation results, the reaction force in the longitudinal direction (FL) with respect to displacement (Δu) was obtained. The effective stress in the longitudinal direction (σL) and strain in the longitudinal direction (εL) are denoted by FL/Wt and Δu/L, respectively. EL is the ratio of σL to εL for elastic deformation. The other material properties were obtained by using the same method. 3.4. Bending properties of the sheet with sheared protrusions The bending properties of the plate are usually represented by the flexural rigidity (D). This is given by EI/W where E is the Young's modulus, I is the second moment of inertia and W is the width of the plate. However, the flexural rigidity determined from the constitutive model of an homogeneous orthotropic plate (Et3/12) cannot accurately describe the actual bending deformation because the homogenized properties cannot describe the actual bending deformation of the plate
Table 1 Boundary conditions for determining the homogenized properties.
EL ET GLT
x=0
x=L
y=0
uL = 0 ⁎
uL = Δu ⁎
⁎
⁎
Fixed
uT = Δν
uT = 0 free
uT = Δv free
⁎ Normal displacements of the nodes in this surface are coupled.
y=w
with sheared protrusions due to the limitation of the homogenized model [29]. In this work, the flexural rigidities of the sheet with sheared protrusions in the longitudinal and transverse directions were obtained from a three-point bending simulation. The three-point bending simulation models were constructed by using hexahedral mesh coarsening. Fig. 7 shows the results. For the simulation of the longitudinal direction, planes in the transverse direction were defined as symmetric planes. For the simulation of the transverse direction, planes in the longitudinal direction were defined as symmetric planes. The simulation was conducted by using ABAQUS/standard v6.13 [27]. In the three-point bending tests, the span length (2S) was 120 mm. The radii of the punch and die (R) were 12.7 mm. The simulation and experiments were conducted in the elastic deformation region. In three-point bending, the relationships between the deflection of the plate (δP), load of the punch (P), and the flexural rigidity of the plate are given by [30]. δP ¼
Px 1 2 3S −x2 ; 0 ≤x≤S 12W D
ð10Þ
D¼
1 P S3 6 W δP
ð11Þ
where D is the flexural rigidity of the plate, x is the distance from the die, W is the width of the plate and S is the distance between the punch and die. The flexural rigidity (D) of the sheet with sheared protrusions can be obtained from Eq. (11). The results of three-point bending can be represented by a load– displacement curve. To verify the simulation model, the experimental results for three-point bending of a 200 mm × 30 mm plate were compared with the simulation results, as shown in Fig. 8. Here, the load– displacement curves are linear because the simulation and experiments were conducted in the elastic deformation region. The simulation results showed excellent agreement with the experimental results because the newly generated mesh with hexahedral mesh coarsening correctly accounted for the result of the forming process. Table 2 gives the homogenized resultant material properties of the sheet with sheared protrusions. The elastic modulus and flexural rigidity were greater in the longitudinal direction than in the transverse
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353
Fig. 7. Simulation result of three-point bending (δP/S = 0.1): (a) in the longitudinal direction and (b) in the transverse direction.
direction due to the geometric properties of the sheet with sheared protrusions. The calculated resultant material properties of the sheet with sheared protrusions were used to calculate the deformed shape. 3.5. Calculation of the moments applied along edges The moments per length applied in the longitudinal direction (mL) and transverse direction (mT) were obtained from the deformed shape of two specimens. The measured curvatures of plates with different sizes were used to construct the system of equations with respect to mL and mT. The two specimens employed in this work are shown in Fig. 2(b) and (c). When the sheet with sheared protrusions was cut into dimensions of 200 mm (longitudinal direction, L′) × 30 mm (transverse direction, W′), the curvature in the longitudinal direction (κL′) was 0.00268 mm−1. When the sheet was cut into dimensions of 30 mm (transverse direction, L″) × 30 mm (transverse direction, W″), the curvature in the transverse direction (κT″) was 0.00144 mm− 1. The curvatures of the two measured specimens (κL′, κT″) were used to construct the following system of equations: 0 2DLT α 0 λ mT 1 DLT EL tμ 0 mT 1 DLT 2 − − þ 0 0 κL 0 0 0 0 DT κ L DT DT κ L DT W 8α W ″ ″ ″ 2D α λ m 1 D E tμ m 1 DLT 2 LT L LT T L mT ¼ DT κ T 00 þ − κ T 00 − : þ ″ ″ 00 0 ″ DL κ T DL DL κ T DL L 8α L
mL ¼ DL κ L 0 þ
ð12Þ
The mechanical properties of the plates (DL, DT, DLT, EL, ET, ELT, νLT) listed in Table 2 were utilized. The system of equations was solved by using MATLAB (v2014a, MathWorks, Natick, MA, USA). The moments in the longitudinal direction (mL) and transverse direction (mT) were 3.18 N mm/mm and 0.74 N mm/mm, respectively. 4. Prediction of the deformed shape after springback The size of the original specimen was 1000 mm (longitudinal direction) × 600 mm (transverse direction). The sheet with sheared protrusions was cut into various sizes. External geometries of the sheet with sheared protrusions were measured using an optical 3D scanner (ATOS III Triple Scan, GOM mbH). Curvatures in the longitudinal and transverse directions were measured by using least-squares fitting of the measured three-dimensional geometry. Fig. 9 presents some example specimens. In order to verify the proposed method and investigate the effect of the size on the deformed shape, two cases were investigated. In case A, the length in the longitudinal direction (L) was fixed in order to investigate the effect of the length of the specimen in the transverse direction (W) on the deformed shape after springback. In case B, W was fixed in order to investigate the effect of L on the deformed shape after springback. 4.1. Computed results with respect to the length of the transverse direction (case A) Fig. 10(a) shows the curvature of the sheet with respect to W, when L was fixed at 200 mm. The measured curvatures were expressed as symbols. Curvatures of plates were measured when W = 40, 80, 120, 200, and 400 mm. Table 3 summarizes the values of the curvature in the longitudinal direction. The calculated results showed excellent agreement with the measured results. When W was very small, the curvature in the longitudinal direction (κL) remained unchanged, as shown in Fig. 10(a). In this case, the effect of transverse bending was very small, and the plate showed biaxial bending deformation. As W increased, the effects of the membrane stress and transverse bending increased. Finally, κL converged to Table 2 Homogenized resultant material properties of the sheet with sheared protrusions. EL ET GLT νLT (GPa) (GPa) (GPa)
Fig. 8. Simulation and experimental results of three-point bending.
13.21 5.83
1.4
DL DT DLT t (N·mm2/mm) (N·mm2/mm) (N·mm2/mm) (mm)
0.22 1163.25
406.61
116.75
2.4
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Fig. 9. Deformed shapes of the sheet with sheared protrusions after springback: (a) with respect to W (L = 200 mm) and (b) with respect to L (W = 200 mm).
(κL)f = mL/DL = 0.00273 mm−1, which was the curvature in the case of uniaxial bending. Fig. 10(b) illustrates the computed deflection at the center along the transverse direction (δT) with respect to W. As W increased, distortion occurred near the edge of the plate. 4.2. Computed results with respect to the length of the longitudinal direction (case B) The curvature distribution was obtained by using Eq. (7) under the assumption that the curvature in the transverse direction is the principal curvature. Fig. 11(a) shows the calculated results of the curvature in the transverse direction (κT) with respect to L when W was fixed to 200 mm. The experimental results are shown with symbols. The specimen was measured when L = 40, 80, 120, 200, and 400 mm. Table 4 summarizes the values of κT. Transverse bending remained when L was very small. As L increased, κT decreased drastically, as shown in Fig. 11(a). Finally, κT converged to zero, and the plate bent in the longitudinal direction as L increased. As
discussed in previous studies on the suppression of transverse bending [16], bending in the transverse direction was suppressed due to the membrane stress in the longitudinal direction. In this work, the critical ratio of κT to κL when the plate was mainly bent in the longitudinal direction was defined as 0.1. When L was larger than 92.7 mm, κT was 0.000271 mm−1, and bending in the transverse direction was suppressed. While bending in the transverse direction was suppressed, longitudinal bending remained. In this case, the plate was bent in the longitudinal direction. Fig. 11(b) shows the computed deflection at the center along the longitudinal direction (δL) of the plate with respect to L. As L increased, the curvature increased by a small amount. When L was 200 mm, κL was calculated to be 0.00271 mm−1. Fig. 12(a) shows the three-dimensional scanning results of the sheet with sheared protrusions (W = 200 mm and L = 200 mm). The calculated and measured curvatures in the longitudinal direction were 0.00271 and 0.0027 mm−1, respectively. Fig. 12(b) shows the deflection of the sheet with sheared protrusions along A–A′. The calculated and
Fig. 10. Curvature distribution and deformed shape of the sheet with sheared protrusions with respect to W when L was 200 mm: (a) variation of κL with respect to W and (b) computed deflection at x = 0.
C.-W. Lee, D.-Y. Yang / Materials and Design 95 (2016) 348–357 Table 3 Curvature in the longitudinal direction (κL) with respect to W (L = 200 mm).
355
Table 4 Curvature in the transverse direction (κT) with respect to L (W = 200 mm).
W (mm)
40
80
120
200
400
L (mm)
Measured results (mm−1) Computed results (mm−1)
0.00267 0.00266
0.00267 0.00267
0.00268 0.00269
0.0027 0.00271
0.00271 0.00272
Measured results (mm−1) 0.0011 0.00037 0.00015 ≈0⁎ ≈0⁎ Computed results (mm−1) 0.0013 0.00043 0.00011 1.4 × 10−5 9.2 × 10−7
40
80
120
200
400
⁎ Curvature was too small to be measured.
measured deflections along A–A′ showed similar trends. Distortion occurred near the edge of the plate. Similar to the previous results, there were no significant differences between the calculated and measured results. The assumption that springback deformation of the sheet with sheared protrusions is biaxial bending deformation of an orthotropic plate is appropriate. By using the proposed calculation method, the deformed shape of the sheet with sheared protrusions of various dimensions can be obtained easily and accurately. In addition, the proposed method can be applied to the prediction of the deformed shape after springback when the main deformation mode of formed plates is biaxial bending. For application of the sheet with sheared protrusions, the plate should be flattened. When L is larger than 92.7 mm, bending in the transverse direction is suppressed, and the plate is only bent in the longitudinal direction. In this case, the leveling process can be applied in the longitudinal direction. In the previous work [5], a leveling process using three rolls was employed. As a result, the sheet with sheared protrusions was leveled nearly flat. The flattened sheet with sheared protrusions was employed for MCFCs without any problem, as shown in Fig. 1(b).
5. Discussion 5.1. Moment-curvature relationship of the plate with respect to the length in the transverse direction The effect of bending in the transverse direction on biaxial bending of an orthotropic plate was investigated. The material properties listed in Table 2 were utilized in the calculation. When bending moments are applied in the transverse direction, the moment-curvature relationship can be categorized into three stages [12]: Stage I, the moment increases drastically when the flexural rigidity of the plate increases due to biaxial bending deformation; Stage II, negative stiffness occurs as the cross-section flattens; and Stage III, the cross-section becomes almost flat and mL is nearly proportional to κL. Fig. 13 shows the relationship between mL and κL with respect to W, when L is 200 mm and mT is 0.74 N·mm/mm. The peak becomes more conspicuous as W increases. As κL increases, the three curves in Fig. 13 converge to one line. The effect of mT decreases as κL increases. Finally,
the moment in the longitudinal direction is nearly proportional to the curvature in the longitudinal direction (mL = DLκL). From the moment-curvature relationship in Fig. 13, the curvature in the longitudinal direction can be obtained. Some examples are described below. The horizontal dotted line represents mL = 3.18 N·mm/mm. The intersecting points between the horizontal line and the curve present the deformed shape. For W = 40 and 160 mm, the horizontal line and curve intersect at one point. In other words, there is only one solution (κL) to Eq. (7) with a given mL, mT, L and W. The plate only shows stage III bending and has the same curvature as the intersecting point. However, when W is 240 mm, the horizontal line and curve intersect at three points because the peak value of the curve is larger than mL. In this case, there are three solutions (κL) to Eq. (7) with a given mL, mT, L and W. The plate shows three bending modes: stages I, II, and III. In order to make κL = 0.00271 mm−1 from the initial state, the required mL should be larger than 4.53 N·mm/mm, which is the peak value of mL in Fig. 13.
5.2. Moment-curvature relationship of the plate when the ratio of mL to mT is constrained Springback deformation of the sheet with sheared protrusions is different from the previous case. In springback deformation, the ratio of the moment in the longitudinal direction to that in the transverse direction (mL/mT) is constant. In order to investigate springback deformation, the relationship between mL and κL with respect to W was investigated with a constrained value for mL/mT. Fig. 14 shows relationship between m L and κ L with respect to W. m L /m T is constrained to 4.3 which is the ratio for the sheet with sheared protrusions. The curves were drawn by using Eq. (7). Unlike Fig. 13, the curve in the longitudinal direction in Fig. 14 shows a nearly linear relationship between mL and κL. When mL/mT is constrained and κL increases gradually, the plate undergoes stage III bending only because the moment–curvature relation is nearly linear. There is only one solution (κL) to Eq. (7) with a given mL, mT, L and W. Springback of the sheet with sheared protrusions shows stage III bending because mL/mT is constrained.
Fig. 11. Curvature distribution and deformed shape of the sheet with sheared protrusions with respect to L when W was 200 mm: (a) variation of κT with respect to W and (b) computed deflection at x = 0.
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Fig. 12. Three-dimensional scanning results: (a) measured results of the sheet with sheared protrusions (W = 200 mm and L = 200 mm) and (b) measured and computed deflection profiles along A-A’.
The material properties of the sheet with sheared protrusions were obtained by using finite element analysis with hexahedral mesh coarsening. The moments in the longitudinal direction (mL) and transverse direction (mT) due to springback were determined to be 3.18 and 0.74 N·mm/mm, respectively. The deformed shapes after springback with respect to the size of the plate were obtained by using the moment–curvature relationship of an orthotropic plate. From the analytic models, the deformed shape after springback was precisely predicted with respect to the size of the plate. Increasing the length in the transverse direction caused the curvature in the longitudinal direction to converge to 0.00273 mm−1, which was the curvature in the case of uniaxial bending. Increasing the length in the longitudinal direction caused the curvature in the transverse direction to converge to zero asymptotically. Bending deformation was suppressed in the transverse direction.
References Fig. 13. Relationship between mL and κL with respect to W (L = 200 mm and mT = 0.74 N·mm/mm).
6. Conclusion In this work, springback of the sheet with sheared protrusions was assumed to be biaxial deformation of an orthotropic plate. In order to predict the deformed shape after springback, an analytic model for the biaxial bending of an orthotropic rectangular plate was employed.
Fig. 14. Relationship between mL and κL with respect to W (L = 200 mm and mL/mT = 4.3).
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