Plastic Bending Theory Research Articles

Residual stress analysis and its effect on springback for multi-step pressbrake bending

Pressbrake bending is a multi-step metal forming process and widely applied in the fields of aerospace, shipbuilding, and offshore industry. However, the bending residual stress that arise from the previous-pass bending step will affect the stress distribution of the next-pass bending step, and finally affect the springback after unloading, which results in the difficulty to guarantee the dimensional accuracy of the forming part. In this study, the residual stresses for multi-pass pressbrake forming and its influence on springback are systematically investigated. First, the analytical model for the residual stress of the bent specimen with initial curvature is deduced based on geometric relationship and elastic–plastic bending theory. Second, the calculation method of rigid body position redefine to simulate the multi-pass bending forming process is developed, and the numerical method is confirmed by comparison with experimental method. The analytical model to predict the residual stress is shown to be in close agreement with results from numerical method, demonstrating the validity of the analytical model. Finally, the impact of residual stress in single-pass and multi-pass bending forming on springback are discussed thoroughly. It is found that, as the number of forming passes increases, the equivalent plastic strain will increase to 9% of three-pass loading from 7.8% of one-pass loading, which explains the reason of increasing the number of bending passes can not only reduce springback, but also reduce the residual effect of bending forming.

Study on the mechanism of wrinkle flattening under internal pressure and the critical condition of dead wrinkle

Wrinkling is a common defect in sheet metal forming. With the wide use of thin-walled components, wrinkling has become a more serious problem in the forming process. The conventional wisdom is that the forming part is failed once the wrinkle occurs. In some cases, wrinkles can be flattened and the forming quality can be improved, and there is a few studies about the wrinkle flattening. In this paper, based on the plastic bending theory, an analytical model of wrinkle flattening is established by analyzed the force balance and bending moment of wrinkles under the internal pressure. By analyzing the force and geometric character, it is concluded that the direction of the bending moment at the wave crest is the mechanics character for wrinkle flattening, and the rising tendency of wave height-length ratio δ/L is a striking geometric characteristic of dead wrinkle. At last, the maximum length difference umax that wrinkle can be flattened as the critical condition of dead wrinkle was proposed, and it has a linear correlation with the wall thicknesses, not related with the wavelength. A special experimental device was designed to verify that the accuracy of the theoretical calculation is enough satisfactory for the judgment of dead wrinkle when the ratio δ/L > 0.1. This research can be used to guide the part correction after forming, and can develop into a special process.

Failure mechanisms and bending strength of Fuchsia magellanica var. gracilis stems

In the course of biological evolution, plant stems have evolved mechanical properties and an internal structure that makes them resistant to various types of failure. The mechanisms involved during damage development and failure in bending are complex and incompletely understood. The work presented builds on a theoretical framework outlined by Ennos and van Casteren, who applied engineering mechanics theory to explain why different woody stems fail in different ways. Our work has extended this approach, applying it to a detailed analysis of one particular species: Fuchsia magellanica var. gracilis. When subjected to three-point bending, stems of this species exhibited one of two failure mechanisms: a plastic hinge or a greenstick fracture. We developed a predictive model using a computer simulation and a mathematical analysis using the theory of plastic bending. Required material properties were obtained from tests, the literature and imaging techniques. We found that greenstick fractures are more likely to occur in more lignified stems with a higher density. We discovered a new failure mode: an internal crack caused by tensile transverse stress. This work helps in understanding how plants have evolved their bending resistance and may assist in the creation of novel engineering structures inspired by these principles.

Residual stresses and strains analysis in press-braking bending parts considering multi-step forming effect

Press-braking bending is a multi-step bending process and widely applied in the aerospace industry. Residual stresses and strains generated during the forming process play an important role in determining its forming parameters and bending path. This work aims to analyze the residual stresses and strains in press-braking bending parts using both the theoretical method and numerical method. First, the analytical model of residual stress and strain is established based on the elastic–plastic bending theory. Second, a fully finite element model of press-braking bending has been developed, and a procedure to simulate the multi-step bending process is presented by using the elastic–plastic large deformation finite element method. The simulation results are then compared with three-point bending experiments in terms of forming force and final shapes of the bent specimens, and excellent agreement is achieved. Finally, the results calculated from the analytical model are compared with the numerical results. The distributions of residual stresses and strains on the finished plate along the length and thickness direction, and particularly the multi-step forming effect on residual stresses and strains, are discussed. It is found that the residual stresses and strains decrease at the initial loading position along the thickness direction during the forming process of subsequent loading positions. With the same punch displacement, the residual stresses and strains at the initial loading position are less than those at the subsequent bending position.

Precision Forming Technology for Crimping of Large Straight Welded Pipes

Crimping plays a pivotal role in the production of large-diameter submerged-arc welded pipes. In the crimping forming process, predicting the springback while considering the real-time variation of material parameters is a major challenge faced by many researchers, which has a direct impact on the quality of the welded pipe. To address this problem, the precision forming technology of crimping was developed. The engineering theory of plastic bending was used to investigate the crimping forming process. Two methods, namely, the slope reverse method and the optimization method, were adopted to identify the material parameters. The results showed that the inverted material parameters can be evaluated in real time based on the analytical model of crimping. The maximum relative error of the identification value is less than 4%. Therefore, the springback and displacement during crimping can be predicted dynamically to control the crimping forming quality. Thus, this project provides an important opportunity to achieve precise forming through crimping.

On bending of single crystal beam with continuously distributed dislocations

The theory of plastic bending of single crystal beam taking into account continuously distributed dislocations is proposed. Applying the variational-asymptotic method we reduce the energy functional of the beam to the one-dimensional energy functional which admits analytical solutions. The threshold value at the onset of plastic yielding as well as the dislocation density are found in terms of the applied bending moment. We consider also the polygonization of the bent beam after unloading and annealing and show that such state is energetically preferable. The number of polygons is estimated by comparing the surface energy of small angle tilt boundaries and the gradient terms in the bulk energy.

Theoretical and experimental study on the effects of explosive forming parameters on plastic wrinkling of annular plates

Wrinkling is increasingly becoming one of the most common and troublesome modes of unacceptable deformation in sheet metal-forming prediction that is very important on the design of die geometry and processing parameters. In an effort to provide a reliable and efficient tool to predict the critical blank holding force for prevention of wrinkling, an analytical model for flange wrinkling in high-velocity forming processes, such as explosive forming, is presented here. With consideration of constant blank holder force and using a combination of energy method and plastic bending theory, the critical radial displacement and number of wrinkling waves are obtained. For validation, some experimental tests have been performed that their results have adequate agreements with the analytical ones. Moreover, the effects of process parameters such as blank holding force, radii ratio, and material mechanical properties on wrinkling behavior has been discussed.

Influence of Material Properties on Air Bending Based on RBF

Based on the plastic bending theory, the influence of material properties fluctuations of pipeline steel on the air bending is discussed by the computer simulation technology and design of the orthogonal was adopted to define scientific simulation experiment scheme. Among the material properties, initial yield strength is the most critical issue. The relation of the initial yield strength, the key process parameter, punch penetration, and the tube forming quality was studied and the variation of bending angle after springback and forming force with the punch insertion was analyzed. The parameters prediction model, which was established by using RBF network technology, is proved rational and precise since the error between predicting results and simulation value is very small. The method can be used to optimize the relevant process parameters and to design the reasonable process scheme.

Improvement of springback prediction of wide sheet metal air bending process

Accurate springback prediction of wide sheet metal air bending process is important to improve product quality and ensure the precision in dimension. The definition of elastic limit bend angle was proposed. Based on cantilever beam elastic deforming theory, the geometrical parameters of forming tools, sheet thickness and the material yielding strain were derived and validated by the finite element method (FEM). Employing the degree of elastic limit bend angle, the equation for springback prediction was constructed, the results calculated fit well with experimental data. Especially for the small bend angle, the predicted results by equation were applied to conduct the springback prediction and compensation in industries and give closer correlation to the experimental data than those calculated by engineering theory of plastic bending.

An analytical model for plate wrinkling under tri-axial loading and its application

The prediction and prevention of wrinkling have been challenging issues in sheet metal forming processes. In an effort to provide the design and process engineers a reliable and efficient tool in assessing the onset of flange wrinkling, an analytical model, based on the wrinkling criterion proposed by Cao and Boyce [1], is presented here. The critical buckling stress and wavelength as functions of normal pressure are calculated using a combination of energy conservation and plastic bending theory. The present results are in excellent agreement with those obtained from Cao and Boyce’s numerical approach which has demonstrated its excellent predictive capability by comparing the experimental study of a conical cup [1] and a square cup forming [2]. Additionally, the effects of the tension in the plane of sheet and material properties on the initiation of flange wrinkling are investigated.

An Analytical Prediction of Flange Wrinkling in Sheet Metal Forming

Wrinkling is one of the major defects in sheet metal forming. The ability to accurately predict the occurrence of wrinkling is critical to the design of tooling and processing parameters. An analytical approach for predicting the onset of flange wrinkling is presented. This method is based on the wrinkling criterion proposed by Cao and Boyce for predicting the buckling behavior of sheet metal under normal constraint. Using a combination of energy conservation and plastic bending theory, the analysis provides the critical buckling stress and wavelength as functions of normal pressure. The results are in excellent agreement with those obtained from Cao and Boyce's numerical approach, and also match well with the experimental results of a square cup forming. In addition, the effects of material properties on the wrinkling behavior are also discussed. The analytical method significantly reduces computational time and is suitable for direct engineering application.

Prediction of Plastic Wrinkling Using the Energy Method

The prediction and prevention of wrinkling during a sheet metal forming process have been challenging issues on the design of part shape, die geometry, and processing parameters. In an effort to provide a reliable and efficient tool to assess the onset of wrinkling, analytical models for flange wrinkling and straight side-wall wrinkling are presented here. Using a combination of energy conservation and plastic bending theory, the critical buckling wavelength and stress are calculated as functions of the boundary conditions (displacement constraint and binder pressure). Comparisons between the numerical and experimental results are given for both cases and excellent agreement is obtained.

Dynamics of plastic bending of whiskers, induced by bombardment by a dense electron beam

We report new results from experimental investigations of the strong plastic bending induced in whiskers by nanosecond pulsed bombardment by a dense electron beam. The bending dynamics has been traced for the first time by means of a VFU-1 high-speed movie camera synchronized with a powerful small GIN-600 electronic accelerator. The measured duration of the plastic bending process for whiskers 40–100 μm in diameter was 1–2 msec. This is of the same order of magnitude as the characteristic relaxation time of the quasistatic mechanical stresses that are formed under uneven bombardment because of temperature gradients and disappear when the temperature is evened out by heat conduction. The measured bending dynamics was found to be nonmonotonic and rather complicated. After a nanosecond pulse of bombardment by a strong electron beam the bending of the whisker at first increases, then decreases, increases once again, and finally is saturated at some final value during flexural damping vibrations. The experimental data are discussed on the basis of the dislocation theory of plastic bending.

Effect of target bending in normal impact of a flat-ended cylindrical projectile near the ballistic limit

The current work is concerned with the normal impact of flat-nosed cylindrical projectiles striking at a velocity in the vicinity of the ballistic limit on metallic targets. The ratio of target thickness to striker diameter ranges from ¼ to ½. The resulting penetration model extends existing theories to include the extensive deformation of the target resulting from the bending effect. The dynamic and static stress-strain relations of 2024-0 aluminum are obtained by means of the split-Hopkinson bar technique and an Instron machine, respectively. A series of tests with 0.8 mm diameter pre-drilled holes in each target is conducted to assess the relative effects of bending and shear on the plate deformation. A total of 20 shots are executed to examine the phenomenon of the plate response under impact loading. The 12.7 mm diameter hard-steel projectiles are fired with a pneumatic gun against 3.18, 4.76, and 6.35 mm aluminum targets at a velocity of up to 182 m s−1. A dynamic plastic bending theory is proposed that will be superposed on the previously developed one-dimensional phenomenological penetration model of Liss et at. (Int. J. Impact Engng 1, 321–324 (1983)), to permit a more accurate prediction of target response. This phenomenological model is numerically analyzed and compared with the experimental findings and the two-dimensional Lagrangian computational codes Dyna2d and Autodyn using a Cray X/MP-48 and PC-AT, respectively. Excellent correspondence with data is obtained for the projectile exit velocity when a higher impact speed is employed. The final deformation field computed from the model is not limited to a narrow zone and shows good correlation with the experimental data, especially for thinner targets.

Measurement of Residual Stress in Bent Silicon Wafers by Means of Photoluminescence

The photoluminescence (PL) method was used to measure the residual stress in plastically-bent Si single crystal wafers. The bound-exciton PL line showed remarkable energy shift by the residual stress and the stresses measured from the PL shift agreed well with values of the stress deduced from the theory of plastic bending. It was found that the residual stress introduced by the plastic deformation is quite different in floating-zone (FZ) wafers from that in Czochralski (CZ) Si wafers, possibly because of the difference in the generation and multiplication processes of dislocations in the two crystals. Annealing of the bent specimen at 1000°C reduced the residual stress in the specimen, and the peak position of the bound-exciton PL line reverted to that before bending.

中央集中荷重時単純支持鉄筋コンクリート T 梁の有効幅と破壊性状

The purposes of this paper are to investigate the influences of the span ratios of flanges a/l_x and the thickness of flange for effective widths in elastic and plastic ranges, ultimate strengths and failing behaviour of reinforced concrete T-beams, and to discuss the application of elastic theory and the Code Requirements to effective widths of T-beams and the application of plastic bending theory to the ultimate strengths of T-beams. For these purpose, six simply supported reinforced concrete T-beams classifled by two kinds of a/l_x and three kinds of flange thickness were tested under a concentrated load, and effective widths and flexural rigidity in elastic range, effective widths for ultimate strength in plastic range and the shear cracks due to membrane shear stress at the joints of flange and web were analyzed and discussed. 1) Effective overhanging flange widths λ on either side of the web obtained from the experiments in elastic range decreased from the supports to the center of span and the distribution of λ along the span agreed with the curve calculated by two dimentional elastic theory. 2) Average values of λ along the span obtained from the experiments in elastic range agreed with the values of the Eq. (5) proposed by HIGASHI as average λ of members for the calculation of the flexural rigidities of T-beams. λ=(0.5-0.3a/l_x)a, [when a<l_x]……(5) Average values of-λ along the span obtained from the experiments for three kinds of flange thickness t/D=0.067, 0.133, 0.200 were approximately same and the influence of flange thickness for λ was not observed. 3) Ultimate strengths calculated by using full flange width as the effective flange width agreed with the values obtained from experiments in both positive and negative moments. It would be considered as the results of enough anchorage for longitudinal reinforcements and enough transverse reinforcements in flanges. 4) The shear cracks due to membrane shear stress were observed on the top surface at the joints of flange and web in plastic range. However, after such cracks occured the load did not decrease, because enough transverse reinforcements were placed at the joints. Tensile principal stress on the top surface at the joints when such cracks occured (which were calculated with the experimental results) were 1/12〜1/20 of the compressive strength of concrete F_c.

Dislocation theory of plastic bending

The macroscopic stress (average stress in a volume containing many dislocations) in a uniformly bent crystal is related to the dislocation density by a simple differential equation analogous to Poisson's equation. Differences between bending and tension tests give information about the dislocation mechanism of deformation. A material that has a flat stress-strain curve in tension may show a yield point in bending if the stress required to move dislocations is substantially less than the stress required to generate dislocations.

The Behaviour of Beams in Bending

The paper reports experiments carried out on beams in pure bending. The material used was a cast magnesium alloy AZ855. The beam sections were rectangular, circular, I‐section, T‐scction and diamond. One series of tests was carried out up to 1 per cent fibre strain. A second series of tests was carried out up to fracture. Tension and compression tests were also made on the material. The experimental results show conclusively that the usual theory of plastic bending is correct and that the tension‐compression stress‐strain curve of the material may be used to determine the bending moment‐curvature relationships, etc., for a beam. Measurements of neutral axis shift also confirm the predictions of plastic bending theory.

A Theory of Plastic Bending From a Statistical Aspect

Abstract The apparent increase of the yield stress under bending with respect to that under pure tension is explained by the lesser probability that a weak element is situated close to an edge.