Modeling of bending characteristics of symmetric tri

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ScienceDirect Materials Today: Proceedings 4 (2017) 10704–10713

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AMMMT 2016

Modeling of bending characteristics of symmetric tri-layer laminated Sheet materials Ganesh Govindasamya *, Mukesh K. Jainb b

a Aleris Rolled Products Inc. 26677, West 12 Mile Road, Southfield 48034, Michigan, USA. Department of Mechanical Engineering, McMaster University, 1280 Main St West, Hamilton, L8S4L7, Ontario, Canada.

Abstract Bending of sheet materials is widely applied to shaping many automotive, aerospace and other industrial components. Tailored laminate sheet materials are desirable in bending applications where monolithic sheets do not meet the design requirements or manufacturing demands. An accurate analytical model can be effectively utilized for rapid design of components for laminated sheet materials for a given application. Many analytical models based on advanced theory of bending have been proposed in the literature to predict plane strain bending characteristics of monolithic sheet materials. However, there are very few such models for laminated sheet materials. In this study, an analytical model for tri-layer laminated sheet material is developed based on advanced theory of bending. The model considers Mises yielding and Ludwik non-linear plastic hardening with Bauschinger effect for various laminate thickness ratios. Also, a 3D FE model based on Marciniak-Kuczynski (MK) bend test design is developed to assess pure bending characteristics of a symmetric tri-layer aluminum sheet laminate with softer and thinner clad layers on both surfaces and a harder thicker core in the middle. The through-thickness tangential stresses from the analytical model are compared with those from FE model for different clad to matrix thickness ratios. The tangential stresses decrease in magnitude with increasing aluminum clad thickness ratios in the analytical model. This behavior is in good general agreement with the results from the mid-width section for the FE model. The analytical and FE models also yield similar order trend in relative thickness change with increasing clad thickness ratios and with increasing specimen radius of curvature. The 3D FE model exhibited anticlastic curvature at the edges as a result of strain inhomogeneity across the width. Unlike the tangential stress distribution, the tangential strain is maximum at the specimen edges than at the mid-width. The inhomogeneity in stress and strain across the bend line shows that plane strain condition is not consistent across the width of the sheet. © 2017 Elsevier Ltd. All rights reserved. Selection and Peer-review under responsibility of Advanced Materials, Manufacturing, Management and Thermal Science (AMMMT 2016). Keywords: Bending, laminate sheets, Analytical modeling, FE analysis

2214-7853 © 2017 Elsevier Ltd. All rights reserved. Selection and Peer-review under responsibility of Advanced Materials, Manufacturing, Management and Thermal Science (AMMMT 2016).

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1. Introduction Extensive research efforts have been made towards developing engineering materials of superior mechanical properties and performance capabilities in recent years. Performance limitations of monolithic materials in corrosive environments and increasing demands of lightweight materials for automotive applications without compromising much on the strength and stiffness are some of the driving factors for developing multi-layered materials. The increased interest and fast multiplying usage of multi-layered materials is discussed in the recent review work of Gibson [1]. Different alloys are typically roll-bonded to have a multi-layered clad/core structure which provides a unique combination of formability and corrosion resistance, as well as mechanical properties such as strength, ductility, fatigue, impact resistance [2]. Some applications of laminate sheet materials include steel and aluminum bonded sheets for corrosion resistance and weight reduction designed for automotive body [3], roll bonded high strength aluminum alloy coated with soft corrosive resistant pure aluminum for aerospace applications [4] and brazed dissimilar sheets for heat exchangers with differential thermal conductivity and corrosion resistance [5]. Other than roll bonding, a recently developed method that has significantly facilitated large-scale production of multi-alloy, multi-layered Al laminates is the direct co-casting ingot solidification (or fusion) process [6]. With the advances in the manufacturing of laminated sheets, it is apparent that a comprehensive study of the mechanics of laminate sheets would be useful for optimizing the material and geometric composition of the laminate systems. Joining of dissimilar materials in a single component creates inhomogeneity which may result in discontinuity of stress distributions across the sheet thickness when such components are formed by plastic deformation processes, and thus place limitation on their design. Bending of laminated sheet metals is clearly an important area of study because of its wide spread application in sheet metal forming industry. Several approaches to predict bending characteristics of monolithic sheets materials exist in the literature but only a few for laminated sheets. The objective of this work is to develop a mathematical model for symmetric tri-layer laminate sheet material from an existing model for monolithic material. Comparison of model results with 3D finite element plane strain bending in terms of stresses and strains across the mid-section of the thickness area of bending is also carried out to assess its usefulness and limitations. Nomenclature σ01, σ02 yield strength of matrix and clad K1, K2 strength coefficient of matrix and clad nL1, nL2 strain hardening exponent of matrix and clad (Ludwik hardening law) r general radius of curvature of a fiber rm radius of curvature of the mid surface. ri, ry inside and outside radius of curvature of bent sheet, respectively rn current radius of curvature of neutral surface ru radius of curvature of unstretched fiber ra radius of curvature of inner laminate boundary fiber rb radius of curvature of outer laminate boundary fiber t, to deformed and original laminate thickness, respectively tc1, tc2 thickness of inner and outer laminate layers q1, q2 inner and outer laminate to matrix thickness ratios (tc1/t, tc2/t) effective strain effective stress εθ, εr tangential strain εθ = ln (r/ru); radial strain εr, = -εθ σθ, σr tangential and radial stress components, respectively η relative thickness, t/to ρ relative radius of curvature of neutral surface, rn/ru, κ relative curvature, t/rm

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2. Limitations of modeling and experimental tests methods for sheet bending Several analytical models have been proposed to predict pure bending characteristics of monolithic sheets. Analytical models of sheet bending based on advanced bending theory based on plane strain assumption were first developed by Hill [7] and Lubahn and Sachs [8] and by many others in recent years [9] [10] [11]. These models provide interesting through-thickness tangential and radial stress and strain distributions in plastic bending as a function of curvature that could not be obtained from the older elementary bending theory. Two early studies on modeling of plane strain bending of bi-layer and tri-layer laminate sheets were carried out by Verguts and Sowerby [10] and Majlessi and Dadras [14]. Most importantly, the model did not consider the unstretched fiber (ru) in the analysis. Majlessi and Dadras model, on the other hand, assumed rigid-strain hardening material and considered the unstretched fiber in the analysis. However, the model’s resultant stress equation (σθ - σr) was formulated in terms of strength coefficient and strain ratios, whereas the Verguts and Sowerby model directly utilized the material hardening law. Consequently, Majlessi and Dadras formulation required radii of neutral and unstretched fiber which could not be easily obtained through experimental or FE based methods. In this study, a monolithic pure plane strain bending model of Tan et al. [13] in its isotropic form was extended to symmetric tri-layer laminate. A 3D finite element model of pure bending was also developed based on a bend test design concept proposed by MarciniakKuczynski (MK) [15]. The MK bend test is potentially a useful design to simulate pure bending by applying moment action and by eliminating reaction forces and friction that is commonly observed in V-bending. For this modeling study, a tri-layer laminate from higher strength aluminum alloy core and softer aluminum clad layers was chosen. The laminate consisted of two thinner and softer AA1100 aluminum clad layers on two sheet surfaces and a thicker and harder AA2024 aluminum core in the middle. For analysis purposes, different clad to matrix thickness ratios were considered to compare the 2D analytical and 3D FE model results in terms of through-thickness stresses and strains as well as total relative thickness changes as a function of bend curvature. 3. Proposed analytical model The analytical model for plane strain sheet bending for monolithic specimen considers three zones across the specimen thickness [13]. The zones are categorized by dividing the thickness with respect to outer radius (ry), unstretched fiber (ru), neutral fiber (rn) and inner radius (ri). For tri-layer laminate sheets, additional boundary radii parameters (ra and rb) between the clad and the matrix are defined, making it a five zone model. The radial stress and tangential stresses are obtained from the governing equilibrium equation and Hill’s quadratic anisotropic yield criterion [16]. The model uses Ludwik hardening law for the tension, compression and reverse loaded zones. The Bauschinger effect in reverse loaded zone is captured by modified Ludwik law as described by Tan et al. [17]. A 3 dimensional representation of bend specimen and the three principal stresses are shown in Figure 1. A schematic diagram for the tri-layer specimen types is shown in Figure 2. The differential expression for radial stress at each plastic zone for tri-layer specimen with boundary condition at zone interfaces is shown in Table 1.

Fig. 1. Schematic of a bent specimen.

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Fig. 2. Classification of zones in analytical pure bending model for tri-layer laminate. Table 1. Differential expression for radial stress at each plastic zone for tri-layer specimen with boundary condition at zone interfaces.

Ludwik Model Zone I ry≥r≥rb

r

()

2  dσ r σ 02 + K 2 ε =+ dr 3  σ + K ε 1  01

dσ r 2 = dr 3 dσ r 2

σ − K ε 1  01

r

Zone III ru≥r≥rn

r r

Zone V ra≥r≥ri

r

nL 2

()

dσ r 2 = dr 3

Zone II rb≥r≥ru

Zone IV rn≥r≥ra

Tangential strain



()

n L1

 ru    r 

 

ε θ = ln

()

n L1

 

ε θ = ln

()

nL 2

 

ε θ = ln

dσ r 2  =− σ 02 + K 2 ε dr 3 

=−

r ε θ = ln   ru  r ε θ = ln   ru 

n L1 

σ + K ε 01 1 3 

dr

 

 ru    r 

 ru    r 

It is to be noted that the radial stresses are expected to be continuous across the zonal interfaces. The continuity of radial stress at the neutral fiber that shifts during the bending process is obtained by setting r = rn in the radial stress equations for those zones which meet at the neutral fiber. By equating the stresses at the neutral fiber boundary (eg: σrIII = σrIV), the Λ parameter necessary for Proksa’s thickness – curvature relation [18] is obtained. Further, the value of the radius of boundary fiber (rb) between the clad and the matrix is represented in terms of other known parameters. The thickness of clad layers and clad to matrix thickness ratio are expressed by symbols (tc1, tc2) and (q1, q2), respectively. The parameters ra and rb can be represented in terms of other known parameters as follows:  κ rb = 1 + .r m − q1.η .t0  2

(1)

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 κ ra = q2 .η .t0 + 1 − .r m  2

(2)

The radial and tangential stress equations for the 5 zones tri-layer laminate along with the Ʌ parameter for the trilayer laminate are obtained as described in the work of Tan et al [13]. 4. Finite element model Proposed analytical model of pure bending presented in the previous section involves simplifying assumptions and neglects an important specimen width consideration in the analysis. Therefore, a 3 dimensional FE model is also developed to analyze strains across different sections at the bend line (edge, center or mid-width, tension and compressive sides) of the specimen. A simplified schematic of the FE model, based on M-K bend test design [15], is shown in Figure 3(a). The FE model is developed using the general purpose FE code, Abaqus/Explicit [19]. An 8noded linear brick element with reduced integration (C3D8R element in Abaqus) is used to represent the sheet material. In the model, the test specimen is represented by 53 elements along its width and 21 elements though its thickness and 58 elements across the length. The clamp tooling is considered rigid to simplify the model. The specimen dimension considered for the model is 3.12 mm (thickness), 40 mm (width) and 25 mm (length), respectively. The MK pure bending model consists of rotary clamps tied to the specimen at its two ends such that, on rotation, the clamps allow the application of pure bending moments to the specimen. A typical deformed specimen geometry from FE simulation is shown in Figure 3(b) where a relatively uniform curvature indicative of pure bending can be noted.

Fig. 3. (a) M-K model for bending [15]; (b) FE-MK model simulation showing deformed specimen geometry including development of anticlastic curvature (edge effect) in small radius bending.

5. Material data for bending models Tri-layer laminate for this numerical study was based on commercially available Alclad 2024 sheet material [4] with a very thin clad layer of AA1100 aluminum of 80 µm thickness on both sides of AA2024 aluminum core. The initial thickness of the specimen was considered to be 3.12 mm. It is to be noted that the interface between the laminate layers is assumed to be thin and perfect. Tensile material properties of constituent layers were obtained from experiments on monolithic sheets of AA2024 and AA1100. The material properties were determined by uniaxial tensile tests using ASTM-E8 standard (25.4 mm gauge length and 6.35 mm gauge width) for specimen geometry. The tensile plot for the constituent layer materials are shown in Figure 4. Prior to testing, the specimens

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were full annealed and furnace cooled for 2 hours. True stress versus true strain plots were obtained using computer controlled servo-hydraulic MTS machine of 250 kN capacity. The tests were conducted to failure at 2 different speeds, 0.254 mm/min (corresponding to initial strain rate of 0.01 /min) and 25.4 mm/min (initial strain rate 1 /min). The true stress-strain data was processed by non-linear curve fitting to obtain Ludwik hardening law parameters (see Table 2). The properties at 1/min were used for the numerical solution to the bending model.

Fig. 4. True stress versus strain plot for laminate constituent materials obtained from uniaxial tensile tests. Table 2. Material properties of laminate materials from uniaxial tension tests.

6. Results and discussion 6.1 Stress and strain distributions Modeling results from analytical and FE-MK models for tri-layer laminate are presented in this section. Through-thickness tangential stress distributions for different thickness ratios of AA1100 clad to AA2024 matrix are shown in Figure 5(a). The data for stress values are taken from the mid-width section of the specimen that is closer to plane strain state when compared to the edges (refer to Figure 1). The stress values are compared for the inner radius (ri) of 15 mm and AA2024 monolithic case is also included. Both tangential and radial stresses decrease with increasing clad thickness ratio (i.e., increasing q values). However, the analytical model curve in the matrix layer remain largely unaffected. The tangential stress drop is considerable in the softer clad AA1100 layers in both tension and compression. The FE-MK model shows a smoother transition in tangential stress across the center of the plot and a larger tangential stress drop in the soft clad layers. The elastic component is not included in the analytical model and it causes sharp transition to plastic state. The tangential distribution across the width i.e. along bend-line of the tension side obtained from the FE model is shown in Figure 5(b). The stress distribution is compared for inner radius of 15mm. The stress profile is indicates that the maximum tangential stress is at the mid-width location where

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the material is very close to plane strain state. The stress magnitude is higher for the harder AA2024 monolithic case. The tri-layer laminates show negligible difference in stress distribution for the two thickness ratios both having similar material in its tensile surface, i.e. AA1100. This shows that that thickness ratio of the laminate material has subtle effect on the surface tangential stresses.

Fig. 5. (a) Through-thickness tangential stress distributions in tri-layer laminate for 2 different thickness ratios at a radius of curvature of 15 mm from analytical and FE models. Also included, for comparison purposes, are the curves for monolithic AA2024 sheet; (b) Tangential stress distributions across width (bend-line) in tri-layer laminate for 2 different thickness ratios at a radius of curvature of 15 mm from analytical and FE models. Also included, for comparison purposes, are the curves for monolithic AA2024 sheet.

Figure 6(a) shows tangential strain distributions across the thickness for two different laminate thickness ratios and for monolithic core material (AA2024) from the analytical and FE models. The analytical model gives virtually the same curve for all 3 cases whereas 3 distinct curves are obtained from the FE model. As for the case of monolithic material, the FE model for tri-layer predicts a lower slope through the sheet thickness in the FE model. The reduction in slope is higher at the outer and inner surfaces for both clad and monolithic specimens, which is not observed in the analytical model. This is possibly due to stress relief at the free surfaces. Figure 6(b) shows the tangential strain distribution across the specimen width. Unlike the tangential stress distribution, the tangential strain is maximum at the specimen edges than at the mid-width. The order of tangential strain compared to the monolithic case shows that the strain is maximum for the laminate with 25 % thickness ratio and decreases for 15%. 6.2 Relative thickness changes with curvature The variation of relative thickness for different outer and inner thickness ratios by keeping one layer thickness constant is shown in Figures 7(a) and (b), respectively. When the inner layer thickness ratio is varied with a constant outer layer thickness ratio (q1=0.25), the relative thickness pattern shows increase in relative thickness for increasing inner layer thickness ratio (q2) (Figure 7(a)). On the other case, when the inner layer thickness ratio is kept constant (q2=0.25), the relative thickness decreases with curvature until q1 = 0.35. For the case of q1=0.45, the relative thickness again is seen to increase (Figure 7(b)). It is to be noted that the initial total sheet thickness remains constant for all these cases.

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Fig. 6. (a) Through-thickness tangential strain distributions for 2 different laminate thickness ratios at a radius of curvature of 15 mm from analytical and FE models; (b) Tangential strain distributions across width (bend-line) in tri-layer laminate for 2 different thickness ratios at a radius of curvature of 15 mm from analytical and FE models.

Fig. 7. Comparison of relative thickness for different thickness ratios of outer an inner layers of the AA2024-AA1100 tri-layer laminate. (a) Outer layer is at constant thickness (q1=0.25); (b) inner layer is at constant thickness (q2=0.25).

6.3 Comparison of relative thicknesses of bi-layer and tri-layer laminate models The bi-layer model developed in a similar method and reported in Govindasamy and Jain [12] was used to obtain relative thickness of AA1100-AA2024 bi-layer laminate sheet. A comparison of relative thickness change with curvature for bi-layer and tri-layer laminate with AA1100 as laminate and AA2024 as matrix is shown in Figure 8Figure 8. Different geometrical arrangement of soft clad and hard matrix in bi-layer shows different response in thickness upon bending. When the soft clad (AA1100) lies on the tensile side of the hard matrix (AA2024) (referred as clad under tension as C-T), the specimen thins with increasing curvature as experimentally confirmed in the work of Govindasamy and Jain [12]. Similar results were reported in Verguts and Sowerby [10] and Majlessi and Dadras [14], where thinning increased with increasing clad thickness ratio. The opposite trend is noticed for clad in

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compression (C-C) type bi-layer laminate that shows increased thickening with clad thickness ratio. However, the thickness change magnitudes for same clad to matrix (0.25) is lesser in the tri-layer laminate compared to bi-layer laminate. The thinning and thickening effect produced by the soft clad on either side of the specimen could counter each other to produce minor variations in thickness.

Fig. 8. Comparison of relative thickness versus curvature plots for C-T and C-C type specimens for bi-layer AA1100-AA2024 and tri-layer laminates of different thickness ratios.

7. Conclusion An analytical model based on advanced theory of bending is developed for a tri-layer laminate sheet material based on an advanced bending model for monolithic sheets. A 3D FE model is also developed to simulate pure plastic bending on tri-layer aluminum alloy laminate sheet based on M-K bend test design. The model results for an aluminum alloy three layer laminate system for different thickness ratio are compared to obtain the variation in relative thickness upon large strain bending. Analytical model provides insight about the stress distribution through the laminate thickness. The analytical model showed similar trend to mid-width data from 3D FE model in predicting the tangential and radial stress distribution for different clad to matrix thickness ratios. Its primary advantage over the FE model is in terms of its adaptability to other material systems and rapid prediction of critical bending characteristics such as tangential strain at the outer surface. The model is limited to laminate materials that exhibit similarity in strain hardening behavior. The model is inadequate in describing the stress and strain characteristics along the width direction at the bend line, owing to its two dimensional nature. The 3D FE model of a bend test design captured all of the general characteristics of the analytical model. In addition, it was able to capture anticlastic curvature effect, stress and strain variation along the specimen width at the bend line. These features of the model are useful as failures can initiate at the mid-section of the sheet. The thickness variation across the bend line shows the inhomogeneity in strain for wide specimen and inconsistent with plane strain bending.

Acknowledgements The author gratefully acknowledges the financial support from Discovery Grant program of National Science and Engineering Research Council of Canada (NSERC).

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References [1] R. Gibson, Composite Structures, vol. 92, no. 12, p. 2793–2810, 2010. [2] D. Lesuer, C. Syn, O. Sherby, J. Wadsworth, International Materials Reviews, 41(1996)169-197. [3] H. Kawase, M. Makimoto, K. Takagi, Y. Ishida, K. Tanaka, Development Transactions ISIJ, 23(1983) 628-632. [4] J. Liu, M. Li, S. Sheu, M. Karabin, R. Schultz, Materials Science and Engineering: A, 479(2008) 45-57. [5] H. Engstrom, L. Gullman, Welding Journal, 67(1988) 222-226. [6] A. Gupta, S. Lee, B. Robert, W. Wagstaff, M. Gallerneault, J. Fenton, JOM-Journal of Mineral Metals and Materials Society, 59(2007)62-65. [7] R. Hill, The mathematical theory of plasticity, London: Oxford University Press, 1950. [8] J. Lubhan, G. Sachs, Transactions of ASME, 72 (1950)201-208. [9] R. Crafoord, Annals of CIRP, 15(1967) 253-261. [10] H. Verguts, R. Sowerby, International Journal of Mechanical Sciences, 17(1975) 31-50. [11] P. Dadras, S. Majlessi, Journal of Manufacturing Science and Engineering, 104(1982) 224-230. [12] G. Govindasamy, M. Jain, Strain, 52 (2016) 191–204. [13] Z. Tan, B. Persson, C. Magnusson, International Journal of Mechanical Sciences, 37(1995) 405-421. [14] S. Majlessi, P. Dadras, International Journal of Mechanical Sciences, 25(1983) 1-14. [15] Z. Marciniak, K. Kuczynski, International Journal of Mechanical Sciences, 21(1979) 609-621. [16] R. Hill, Proceeding of Royal Society London, 193(1948) 281-297. [17] Z. Tan, C. Magnusson, B. Persson, Materials Science and Engineering: A, 183(1994) 31-38. [18] F. Proksa, Der Stahlbau, 28(1959) 29-36,. [19] Dassault Systemes, Abaqus, Velizy Villacoublay.