极限图形映射中Lyapunov指数计算方法浅析

Escher M.C.构造出了抽象的极限图形,文献[1]与文献[2]分别提出了上半平面极限映射和极限圆映射,Lyapunov指数作为混沌的有力判据,在大量生成极限图形中起到了不可或缺的作用,通过分析极限图形映射的Lyapunov指数的计算方法,旨在为相关研究提供思路。

Forming limit of textured AZ31B magnesium alloy sheet at different temperatures

Repeated unidirectional bending （RUB） was carried out to improve the texture of commercial AZ31B magnesium alloy sheets. All specimens were prepared in the rolling direction. The forming limit diagrams （FLDs） of AZ31B magnesium alloy sheet were determined experimentally by conducting stretch-forming tests at room temperature, 100, 200 and 300 ℃ Compared with the as-received sheet, the lowest limited strain of AZ31B magnesium alloy sheet with tilted texture in the FLD increased by 79% at room temperature and 104% at 100 ℃. The texture also affected the extension of the forming limit curves （FLC） in the FLD. However, the FLCs of two kinds of sheets almost overlapped at temperature above 200 ℃. It can be concluded that the reduction of （0002） texture intensity is effective to the improvement of formability not only at room temperature but also at low-and-medium temperature. The effect of texture on FLDs becomes weak with increasing temperature.

Theoretical prediction of forming limit diagram of AZ31 magnesium alloy sheet at warm temperatures

A theoretical prediction on forming limit diagram（FLD） of AZ31 magnesium alloy sheet was developed at warm temperatures based on the M-K theory. Two different yield criteria of von Mises and Hill＇48 were applied in this model. Mechanical properties of AZ31 magnesium alloy used in the prediction were obtained by uniaxial tensile tests and the Fields-Backofen equation was incorporated in the analysis. In addition, experimental FLDs of AZ31 were acquired by conducting rigid die swell test at different temperatures to verify the prediction. It is demonstrated from a comparison between the predicted and the experimental FLDs at 473 K and 523 K that the predicted results are influenced by the type of yield criterion used in the calculation, especially at lower temperatures. Furthermore, a better agreement between the predicted results and experimental data for AZ31 magnesium alloy sheet at warm temperatures was obtained when Hill＇48 yield criterion was applied.

Formability of AA5052/polyethylene/AA5052 sandwich sheets

The formability of AA5052/polyethylene/AA5052 sandwich sheets was experimentally studied. Three kinds of AA5052/polyethylene/AA5052 sandwich specimens with different thicknesses of core materials were prepared by the hot pressing adhesive method. Then, the uniaxial tensile tests were conducted to investigate the mechanical properties of AA5052/polyethylene/ AA5052 sandwich sheets, and the stretching tests were carried out to investigate the influences of polymer core thickness on the limit dome height of the sandwich sheet. The forming limit curves for three kinds of sandwich sheets were obtained. The experimental results show that the forming limit of the AA5052/polyethylene/AA5052 sandwich sheet is higher than that of the monolithic AA5052 sheet, and it increases with increasing the thickness of polyethylene core.

Theoretical prediction and experimental investigation on formability of tailor-welded blanks

Based on the elastoplastic mechanical properties of the weld and heat affected zone metals obtained by a nanoindentation test, a theoretical calculation model was established for the forming limit diagram（FLD） of tailor-welded blanks（TWBs） on the basis of plastic constitutive relations and Hosford yield criteria. Hemispherical punch bulging tests were performed to verify the FLD theoretical calculation results. The results demonstrated that not only the FLD theoretical calculation of base materials but also that of TWBs had a good agreement with their experiments. Besides, poorer formability of TWBs caused its FLD significantly lower than that of base materials. The theoretical calculation model offers a reliable approach to obtain the specific FLD of TWBs.

Viscous warm pressure bulging process of AZ31B magnesium alloy with different ellipticity dies

Based on the bulging principle of different ellipticity dies, the methyl vinyl silicone rubber with excellent thermal stability and heat transfer performance was chosen as the viscous medium. The finite element analysis and experiments of viscous warm pressure bulging （VWPB） of AZ31B magnesium alloy were conducted to analyze the influence of different ellipticity dies on the formability of AZ31B magnesium alloy. At the same time, based on the grid strain rule, the forming limit diagram （FLD） of VWPB of AZ31B magnesium alloy was obtained through measuring the strain of bulging specimens. The results showed that at the temperature range of viscous medium thermal stability, the viscous medium can fit the geometry variation of sheet and generate non-uniform pressure field, and as the die ellipticity increases, the difference value of non-uniform pressure reduces. Meanwhile, according to the FLD, the relationship between part complexity and ultimate deformation was investigated.

Stress based forming limit diagram for formability characterization of 6061 aluminum

Two numerical criteria of forming limit diagram（FLD） criterion and ductile fracture criterion（DFC） are presented for FLD prediction of 6061 aluminum. The numerical results are compared with the experimental FLD and also punch＇s load-displacement curve of experimental samples. Experimental FLD of this study is calculated using hemispherical punch test of Hecker. Experimental FLD is converted to FLSD and imported to the Abaqus software to predict necking of samples. Numerical results for FLSD prediction were compared with experimental FLSD. Results show that ductile fracture criterion has higher accuracy for FLD and FLSD prediction of 6061 aluminum. Comparison of numerical and experimental results for force-displacement curve of punch shows that numerical results have a good agreement with experiment.

Comparative study on forming limit prediction of AA7075-T6 sheet with M−K model and Lou−Huh criterion

In order to effectively predict the fracture of AA7075-T6 sheet, the forming limit curves of AA7075-T6 high-strength sheet were drawn according to Morciniak Kuczyski (M K) model and Lou Huh criterion, respectively. The errors between the predicted values of the two theoretical prediction models and experimental values were calculated by error analysis. The forming limit curves were verified by the punch stretch test to evaluate the prediction accuracy of M K model and Lou Huh criterion. The error analysis results show that the mean error of Lou Huh criterion with the optimal parameters for all tensile specimens is 25.04%, while the mean error of M K model for all tensile specimens is 74.24%. The prediction accuracy of Lou Huh criterion in predicting the fracture of AA7075-T6 sheet is higher. The punch stretch test results show that the forming limit curve drawn by Lou Huh criterion can effectively predict the fracture of AA7075-T6 sheet, but the prediction accuracy of M K model is relatively poor.

Influence of material models on theoretical forming limit diagram prediction for Ti-6Al-4V alloy under warm condition

Forming limit diagram (FLD) is an important performance index to describe the maximum limit of principal strains that can be sustained by sheet metals till to the onset of localized necking. It offers a convenient and useful tool to predict the forming limit in the sheet metal forming processes. In the present study, FLD has been determined experimentally for Ti?6Al?4V alloy at 400 °C by conducting a Nakazima test with specimens of different widths. Additionally, for theoretical FLD prediction, various anisotropic yield criteria (Barlat 1989, Barlat 1996, Hill 1993) and different hardening models viz., Hollomon power law (HPL), Johnson?Cook (JC), modified Zerilli–Armstrong (m-ZA), modified Arrhenius (m-Arr) models have been developed. Theoretical FLDs have been determined using Marciniak and Kuczynski (M?K) theory incorporating the developed yield criteria and constitutive models. It has been observed that the effect of yield model is more pronounced than the effect of constitutive model for theoretical FLDs prediction. However, the value of thickness imperfection factor (f0) is solely dependent on hardening model. Hill (1993) yield criterion is best suited for FLD prediction in the right hand side region. Moreover, Barlat (1989) yield criterion is best suited for FLD prediction in left hand side region. Therefore, the proposed hybrid FLD in combination with Barlat (1989) and Hill (1993) yield models with m-Arr hardening model is in the best agreement with experimental FLD.

Prediction of fracture limits of Ni-Cr based alloy under warm forming condition using ductile damage models and numerical method

The stretch forming and the deep-drawing processes were carried out at 300 and 673 K to determine the safe forming and fracture limits of IN625 alloy.The experimentally obtained strain-based fracture forming limit diagram(FFLD)was transformed into a stress-based(σ-FFLD)and effective plastic strain(EPS)vs triaxiality(η)plot to remove the excess dependency of fracture limits over the strains.For the prediction of fracture limits,seven different damage models were calibrated.The Oh model displayed the best ability to predict the fracture locus with the least absolute error.Though the experimentally obtained fracture limits have only been used for the numerical analysis,none of the considered damage models predicted the fracture strains over the entire considered range of stress triaxiality(0.33<η<0.66).The deep drawing process window helped to determine wrinkling,safe and fracture zones while drawing the cylindrical cups under different temperature and lubricating conditions.Further,the highest drawing ratio of 2 was achieved at 673 K under the lubricating condition.All the numerically predicted results of both stretch forming and deep drawing processes using the Hill 1948 anisotropic yielding function were found to be good within the acceptable range of error.

Research on technological parameters of pressure forming with hot granule medium on AA7075 sheet

Hot granule medium pressure forming （HGMF） process is a new process in which granule medium replaces the medium in existing flexible-die hot forming process, such as liquids, gases or viscous medium. Hot forming of light alloy sheet parts can be realized based on the properties of granule medium, such as withstanding high temperature and pressure, filling well, sealing and loading easily. In this work, the forming of AA7075 cylindrical parts by HGMF process is taken as an example to establish the constitutive relation and forming limit diagram （FLD） of AA7075 sheet which is related to temperature by hot uniaxial tensile test of sheet metal. Based on the assumption that granule medium is applied to extended Drucker-Prager linear material model, the finite element model of HGMF process is established and the effect of technological parameters, such as forming temperature, blank-holder gap and drawing ratio, on the sheet metal formability, is studied. The limit drawing ratio curve of AA7075 cylindrical parts at forming temperature of 175-300 ℃ is obtained by HGMF process test, and the limit drawing ratio reaches the maximum value of 1.71 at 250 ℃. The results of numerical simulation are consistent with the results of process test, and the forming force, distribution of wall thichness and form of instability are predicted correctly, which provides reference for the application of HGMF process.

Formability of the AMS 5596 Sheet in Comparison with EDDQ Steel Sheet

Some materials form better than others, moreover, a material that has the best formability for one stamping may behave very poorly in a stamping of another Configuration. The forming limit of a metal sheet is generally given in terms of the limiting principal strains under different loading conditions and represented by the so-called FLD （forming limit diagram）. In view of the difficulty to experimentally determine the forming limits, many researchers have sought to predict FLD. The formability of sheet metal has frequently been expressed by the value of strain hardening exponent and plastic anisotropy ratio. The stress-strain and hardening behaviour of a material is very important in determining its resistance to plastic instability. For these reasons, extensive test programs are often carried out in an attempt to correlate material formability with value of some mechanical properties. In this study, mechanical properties and the FLD of the AMS 5596 sheet metal was determined by using uniaxial tensile test and Marciniak＇s flat bottomed punch test respectively.