Plane strain pure bending of sheets with damage evolution at large strains

An analysis of plane-strain bending at large strains for the rigid/plastic incompressible material model including arbitrary strain-hardening and damage evolution laws is performed. The fracture criterion is based on a critical value of the damage parameter. Numerical treatment is reduced to the system of two partial differential equations written in characteristic coordinates. The through-thickness distribution of the principal stresses and damage parameter as well as the variation of the bending moment with the radius of curvature of the concave surface are found for Swift’s hardening law and one specific damage evolution law. General tendencies in solution behaviour are in agreement with physical expectations.     

Constitutive modeling for anisotropic/asymmetric hardening behavior of magnesium alloy sheets: Application to sheet springback

The constitutive model for the unusual asymmetric hardening behavior of magnesium alloy sheet presented in a companion paper (Lee, M.G., Wagoner, R.H., Lee, J.K., Chung, K., Kim, H.Y., 2008. Constitutive modeling for anisotropic/asymmetric hardening behavior of magnesium alloy sheet, Int. J. Plasticity 24(4), 545–582) was applied to the springback prediction in sheet metal forming. The implicit finite element program ABAQUS was utilized to implement the developed constitutive equations via user material subroutine. For the verification purpose, the springback of AZ31B magnesium alloy sheet was measured using the unconstrained cylindrical bending test of Numisheet (Numisheet ’2002 Benchmark Problem, 2002. In: Yang, D.Y., Oh, S.I., Huh, H., Kim, Y.H. (Eds.), Proceedings of 5th International Conference and Workshop on Numerical Simulation of 3D Sheet Forming Processes, Jeju, Korea) and 2D draw bend test. With the specially designed draw bend test the direct restraining force and long drawn distance were attainable, thus the measurement of the springback could be made with improved accuracy comparable with conventional U channel draw bend test. Besides the developed constitutive models, other models based on isotropic constitutive equations and the Chaboche type kinematic hardening model were also considered. Comparisons were made between simulated results by the finite element analysis and corresponding experiments and the newly proposed model showed enhanced prediction capability, which was also supported by the simple bending analysis adopting asymmetric stress–strain response.     

The recovery after bending of polycarbonate sheets

Summary The recovery after bending has been extensively studied for metal sheets. The data presented in this work show that in the case of polymeric materials viscoelastic effects play a very important role. In particular the influence of deformation rate, the time the sample is held under load and the recovery time is analysed. A master curve is obtained by proper modification of a recent analysis developed for metal sheets.With 10 figures     

A note on pure bending and flexure in plane stress including the effect of moment stresses

Summary The paper presents generalizations of the problem of St. Venant flexure in plane stress, based on the assumption of a medium which supports moment stresses as well as force stresses.

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Übersicht Es werden Verallgemeinerungen für das St. Venantsche Querkraft-Biegeproblem auf der Grundlage eines Stoffes gegeben, der nicht nur Kraft-Spannungen sondern auch Momenten-Spannungen übertragen kann.

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Dedicated to Hans Ziegler on the occasion of his 60th birthday.

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This research was supported by the Office of Naval Research.     

Elastic springback in plates and beams formed by bending

A plane-strain solution is presented to predict the springback of plates formed by bending. The solution assumes that the strains are small, that the material is an isotropichardening von Mises material, and either that the stress-strain diagrams for tension and compression are identical or that an average diagram can be employed. The plastic portion of each stress-strain diagram is approximated by a finite number of straight lines. Both analytical solutions and experimental evidence indicate that the much simpler plane-stress solution can be used to predict springback for both beams and plates. Experimental data confirm that the type of die has a pronounced effect on the springback for beams and plates if the material exhibits a yield point.     

Anticlastic action of flat sheets in bending

A theoretical analysis is made of the stresses induced in thin elastic strips which are bent (by uniform bending moments) into arcs with substantially constant radii of curvatures. Stresses determined experimentally on aluminum and steel strips agree very well with the theory.     

Nonlinear bending response and buckling of ring-stiffened cylindrical shells under pure bending

In this paper, the nonlinear bending response of finite length cylindrical shells with stiffening rings is investigated by using a modified Brazier approach. The basic assumptions for the present study are that the deformation of a shell subjected to pure bending can be simplified into a two-stage process. One is that the shell ovalizes but its axis remains straight; the other is that the bending of the shell is regarded as a beam with nonuniform ovalization. The nonlinear bending response is derived by applying the minimum potential energy principle and the corresponding critical moment, associated with local buckling, is determined by employing the Seide–Weingarten approximation. Numerical results are shown and compared with those obtained from other methods, which demonstrates that the assumptions used in the present study are reasonable.     

Energy-minimizing deformations of elastic sheets with bending stiffness

Necessary conditions for energy-minimizing deformations are derived for a theory of sheets in which the strain energy function depends on the second derivatives of the deformation as well as its first derivatives. All of these conditions are extensions of well-known necessary conditions in classical calculus of variations. The interpretation of some of these conditions as material stability conditions is explained.     

Evaluation of anisotropic yield functions for aluminum sheets

The anisotropic behaviour of some rolled aluminum alloys is investigated using the phenomenological approach via some of the recently proposed 3-D yield functions. The directional variations of the yield stress and the so-called R-values in the plane of the sheet are predicted using the 3-D yield functions and the results are compared to previously reported experimental measurements. The hardening properties of the alloys examined are extracted from the experimental uniaxial curves along the rolling direction (RD). The forming limit curves are produced using the Marciniak and Kuczynski (Marciniak, Z., Kuczynski, K., 1967 Limit strains in the process of stret-forming sheet metals. Int. J. Mech. Sci. 9, 609–620) approach and the predictions of different yield functions are compared to the experimental curves. In general, although the refinements incorporated to the yield function proposals in chronological order produced better agreement with the experimental observations (at the expense of some complexity in the formulation), the overall performance of each criteria examined could nevertheless vary with the type of the application, as can be anticipated from the phenomenological nature of the approach.     

简支梁绝对最大弯矩计算方法探讨

介绍了结构力学中简支梁绝对最大弯矩的一种简单计算方法. 教学实践证明,用这种方法,学生对计算绝对最大弯矩的规律容易理解、掌握,也容易记忆.     

The plastic yielding of double notched bars due to pure bending

Conclusion It is believed that all practical cases of notch shapes are covered by the above analysis. The experiments are in favour of the upper bound solution, though, for practical cases, the mean value derived from the limit analysis differs very little from that of the bounds, and it can be used with confidence.Further tests, however, are needed to confirm the deformation pattern. It is also believed that this analysis will furnish useful informations on the study on the mechanism of fracture.     

各向异性矩形薄板弯曲问题的一般解

给出了各向异性矩形薄板弯曲问题微分方程的一般解。可以求解任意载荷作用下各种边界的弯曲问题。以四边固支的正方形板为例进行了数值计算。     

Fatigue crack closure of surface flaws loaded in pure bending

The objective of this paper is to report fatigue crack growth and closure behavior for surface flaws subjected to pure bending. An optical interference technique is employed with PMMA, a transparent polymer, to determine the three-dimensional crack-surface displacement field. The point-wise measure of crack closure along the crack front obtained from optical interference is compared with an average (bulk) crack-opening load determined via crack-mouth-displacement gage measurements. In addition, the effect of crack-closure loads on the subsequent growth of the surface flaw is examined. Dr. Ray is currently associated with The Boeing Company, Seattle, WA.