Semi-analytic approach for sandwich plate U-bending considering shear deformation of the core

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Abstract

The deformation mechanism of a sandwich plate during U-bending was theoretically investigated by rigid-plastic analysis considering shear deformation of the core and the frictional effect. The deformed shapes of a sandwich plate, which were described as a combination of straight lines and arcs, were calculated by Newton method to minimize the total internal work dissipation. Homogenized properties of core shear stress–shear strain curves calculated by using pure shear FE-analyses were used in the analysis. For the numerical verification of the proposed analysis, U-bending simulations were carried out for sandwich plates with various inner structures including a pyramidal truss core, a honeycomb core, corrugated cores, and a sheared dimple core. The analytic scheme has enabled the acceptable prediction of the load curves and deformed shapes during U-bending.

Highlights

► Deformation mechanism of a sandwich plate during U-bending was theoretically investigated. ► Core shear deformation and frictional effect are taken into account in rigid-plastic analysis. ► Proposed model enables the effective prediction of the deformed shapes and load curves.

Introduction

The metallic sandwich plate is composed of a relatively low-density core positioned between two face sheets to improve the specific bending strength/stiffness as a lightweight construction. Various inner structures for a metallic sandwich plate, such as a honeycomb core [1], a tetrahedral truss core [1], a pyramidal truss core [2], a Kagome truss core [3], a textile core [4], a corrugated core [5], and I-beam/Y-frame cores [6] have been introduced according to the design objective of each type of core. Due to their load-bearing properties and multifunctional characteristics, metallic sandwich plates have the potential for diverse applications, including lightweight structured materials, energy absorbers, thermal isolators, heat exchangers, sound dampers, catalyst supports, and current collectors [7], [8]. Despite their many benefits, the practical application of metallic sandwich plates is restricted within narrow limits because metallic sandwich plate is a typical difficult-to-form material. Efficient forming by using the commercial process is a challenging issue to expand their practical application and to reduce the manufacturing cost.

Mohr [9] analytically investigated the bending mechanism of sandwich plates during draw bending. Rigid-plastic material with a linear hardening model was assumed. This analysis provides insight into the bending mechanism of sandwich plate. The maximum tensile strain of the face sheet and the required shear strength of the core were calculated for the criteria of face fracture and core shear failure, respectively. The derived punch load in the steady-state condition was not a function of the punch stroke and the clearance.

A design map for a bendable sandwich plate was constructed to prevent core shear failure, face fracture, and face buckling [10]. From the design map, a sheared dimple core that satisfied the required conditions was introduced and designed to improve the bendability of metallic sandwich plate. The results of U-bending experiments at Rd/h3.33 showed that brazed sandwich plates can be successfully bent by 90° without any failure, such as face buckling, face fracture, core shear failure, or debonding failure.

However, a debonding failure was observed in a welded sandwich plate during U-bending, due to shear stress of the core [11]. The load equations that varied with respect to the change of the upper die stroke and clearance were derived for the U-bending process of a sandwich plate. The shear stress of the core was calculated from the upper die load. It has been shown by the analysis that the criterion for debonding failure can be controlled by the clearance. An analytically predicted condition to prevent debonding failure was experimentally verified. However, the deformed shape during U-bending cannot be predicted, as shear deformation of the core was not considered in the formulation.

In this work, the effects of shear deformation of the core are investigated and formulated by rigid-plastic analysis with the frictional work. The focus of this study is to investigate the U-bending process of metallic sandwich plates. A detailed theoretical formulation for sandwich plate U-bending is presented. The deformed shape and load curve calculated by the theoretical analysis are verified by comparison with the results of FEM simulations for sandwich plates with various inner structures, such as a pyramidal truss core, a honeycomb core, corrugated cores, and a sheared dimple core. In addition, the theoretical analysis of the sheared dimple core is compared with experimental results. The effects of some parameters on the shear deformation of the core are discussed.

Section snippets

Effect of the shear deformation of the core

In the previous work [11], the bending behavior of sandwich plate was analytically investigated on the assumption that shear deformation of the core did not occur during U-bending. In the present study, the results of the previous analysis were compared with those of a FEM simulation which was carried out using the ABAQUS-standard implicit code. A bendable sandwich plate with sheared dimple cores was modeled using reduced integration eight-node elements (C3D8R) and six-node elements (C3D6). The

Strain of the face sheets

Before the effects of shear deformation and friction work were formulated in U-bending analysis of the sandwich plates, plastic strains of the face sheet and the core were defined.

Theoretical model for θ<90°

The deformed shape of the sandwich plate can be modeled by the combination of straight lines and arcs using the characteristic points of B, C, and D as seen in Fig. 5. These points define the boundaries between the deformation zones, namely, zone I, zone II, zone III, and zone IV. Point B, which is located at the boundary between zones I and II, is determined by the bending angle (θ). The defined bending angle (θ) was expressed asθ=θ(δ)=tan1(xByB)

Changes in the deformed shape during the time

Verification examples

For numerical implementation of the proposed analysis, pure shear FE-analyses were carried out for various inner structures including a pyramidal truss core, a honeycomb core, corrugated cores, and a sheared dimple core. The simulations were carried out using the ABAQUS-standard implicit code(ver. 6.7). The pyramidal truss core, the honeycomb core, and the corrugated core of the longitudinal direction were modeled using 8-node linear brick elements (C3D8R) with reduced integration. In addition,

Error analysis of the assumed kinematics

The bending mechanism was investigated to take into account the core shear deformation during U-bending. This work focuses on a deformed shape and load curves determined by a combination of the shear mode and the bending mode. However, this model cannot show stress distribution and strain distribution in a deformed sandwich plate, because the face sheet is treated as a membrane. Actually, face sheets carry an overall shear force and bending moment. When a sandwich plate is subjected to shear

Conclusion

The bending mechanism in sandwich plate was investigated by theoretical analysis considering the effects of shear deformation of the core and frictional work. The deformed shapes of sandwich plate were modeled by the combination of straight lines and arcs. Three characteristic points were defined to determine the boundaries between the deformation zones. The load curves and the deformed shapes were calculated by minimized internal work dissipation under geometrical constraints. The calculated

Acknowledgments

This work was supported by the program of the Ministry of Knowledge Economy (MKE), Grant no. 10028226, and in part by the research project of “Development of Ultra-light Metal Sandwich Plate with Three-Dimensional Inner Structures”.

References (14)

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