Limit strains in the processes of stretch-forming sheet metal
01 Sep 1967-International Journal of Mechanical Sciences (Pergamon)-Vol. 9, Iss: 9, pp 609-620
Abstract: The process of the loss of stability is analysed for sheet metal subjected to biaxial tension when the ratio of the principal stresses 0.5 ⩽ σ 2 /σ 1 ⩽ 1 . The loss of stability manifests itself by a groove running in a direction perpendicular to the larger principal stress. In this groove local strains begin to concentrate gradually. In the initial stage of the process the deepening of the groove is associated with a gradually fading strain in the regions adjacent to the groove. This fading strain attains a certain limiting value e∗. This paper contains both experimental results and a theoretical analysis of the process of the generation of the groove based on anisotropic plasticity theory. The system of equations derived was solved numerically with the aid of a computer, which enabled the limiting strain of the sheet metal to be determined as a function of the following properties of the material: (i) Initial inhomogeneity of the sheet metal, (ii) exponent of the strain-hardening function, (iii) coefficient of normal anisotropy, (iv) initial plastic strain, (v) strain at which the fracture occurs. The results are discussed and the properties are described that influence the drawability of sheet metal used in the stretch-forming process.
Abstract: The forming limit diagram of high-purity niobium sheets used for the manufacturing of superconducting radiofrequency (SRF) cavities is presented. The Marciniak (in-plane) test was used with niobium blanks with a thickness of 1 mm and blank carriers of annealed oxygen-free electronic copper. A high formability was measured, with an approximate true major strain at necking for plane-strain of 0.441. The high formability of high-purity niobium is likely caused by its high strain rate sensitivity of 0.112. Plastic strain anisotropies (r-values) of 1.66, 1.00, and 2.30 were measured in the 0°, 45°, and 90° directions. However, stress–strain curves at a nominal strain rate of ~10−3 s−1 showed similar mechanical properties in the three directions. Theoretical calculations of the forming limit curves (FLCs) were conducted using an analytical two-zone model. The obtained results indicate that the anisotropy and strain rate sensitivity of niobium affect its formability. The model was used to investigate the influence of strain rate on strains at necking. The obtained results suggest that the use of high-speed sheet forming should further increase the formability of niobium.
Abstract: This study presents a method for finite element (FE) simulation of a deep drawing process of a cold-rolled carbon steel (SPCC) sheet material based on the graphical method. First, uniaxial tensile specimens were prepared and experimental tests were conducted to determine the flow stress curves. The calculation of the fracture points at special strain modes (plane strain, uniaxial tensile strain, and biaxial tensile strain) was presented using the modified maximum force criterion (MMFC). After that, the graphical method was adopted for the estimation of the forming limit curve (FLC) based on several hardening laws. FE models for a deep drawing process of the SPCC sheet were then built using the calculated FLCs. Using FE simulations, the fracture heights of cylinder cups formed by the deep drawing process were finally determined and compared with those from experiments. The results showed a good agreement between simulated and measured fracture height with a maximum of 3.6 % deviation. Additionally, simulations and corresponding experiments were performed to investigate the effects of the blank holder force, punch corner radius, and drawing ratio on the fracture height of cylinder cups.
Abstract: An isotropic ductile fracture criterion considering nucleation, growth, shear coalescence and necking coalescence of voids during plastic deformation is developed to model fracture behavior of ductile metals. The innovation of this fracture criterion is that it considers the mechanism of necking coalescence of voids on the basis of a shear-controlled ductile fracture criterion developed by the micro-mechanism ( Lou et al., 2014 ). The damage from coalescence of voids is expressed as the sum of the damages from the shear coalescence of voids and necking coalescence of voids. We assume that the damage from necking coalescence of voids is controlled by the normalized normal stress at the maximum shear plane, the larger positive normalized normal stress at the maximum shear plane will promote the necking coalescence of voids, while the negative gives no effect on the necking coalescence of voids. After the establishment of the proposed fracture criterion, a detailed parametric study is performed to demonstrate the flexibility of the proposed criterion. Then, the proposed ductile criterion is employed to construct fracture loci for three different materials (AA 2024-T351, AA 6060-T6 and A710 steel). And fracture loci constructed are compared with those experimental data points to validate the performance of the proposed fracture criterion. For the purpose of comparison, the predicted results are compared with those predicted by the micro-mechanism-motivated criterion ( Lou et al., 2014 ), the modified Mohr-Coulomb criterion ( Bai and Wierzbicki, 2010 ). The better agreement of the predictions with the experimental data demonstrates that the proposed fracture criterion is able to improve the prediction accuracy for different metals under various stress states and the mechanism of necking coalescence exhibits a noticeable role in predicting the ductile fracture. Moreover, the proposed criterion is expected to be employed in finite element simulation to calculate the ductile fracture for different metals due to the high accuracy.
Abstract: The theory of unified visco-plastic damage constitutive equations has been frequently used to model the thermal flow behavior and damage evolution of various materials. But the anisotropic properties of materials were generally neglected in the previous studies. In this paper, a novel damage correction formula was proposed, which was expressed as a function of loading path and loading direction and used as a multiplier of a continuum damage model. By integrating a set of unified visco-plastic damage constitutive equations, the Yld2004-18p criterion and the damage correction formula, an improved anisotropic damage constitutive model was established in this paper. Anisotropic plasticity and anisotropic damage of AA7075 alloy under hot forming conditions can be properly modeled by this model. Material constants included in the three integrated functions were determined respectively based on different experimental data. The calibrated model is capable to accurately predict the forming limit curves of AA7075 alloy for different forming temperatures, strain rates and loading directions. The model was then numerically implemented in hot Nakajima simulations. The simulation results were evaluated comprehensively from the aspects of limit strain point, punch force- displacement curve, major strain field and fracture position. For further verification of the established model, hot forming simulation of a B-pillar was conducted and compared with a real B-pillar of AA7075 alloy fabricated via an industrial hot stamping process. It is concluded the anisotropic damage constitutive model can be numerically applied to accurately predict the plastic deformation and damage-induced fracture of AA7075 alloy under hot forming conditions.
Abstract: Global formability and local formability are critical in different metal forming processes. Edge cracking, controlled by the local formability, is a dominant factor limiting the application of advanced high strength steels (AHSS) in automotive industries. The local formability of a medium-Mn steel (MMnS), a promising candidate of the third generation of AHSS, is evaluated based on forming limit curves at fracture and compared with a dual-phase DP1000 steel using the damage mechanics approach. The superior tensile properties of the investigated MMnS, high tensile strength, pronounced strain hardening, large uniform and total elongation, lead to a very good global formability, which is an indicator of necking resistance. However, the local formability of the investigated MMnS, which is an indicator of fracture resistance and quantified by the plastic strain at fracture under different stress states, is worse than the DP1000 steel. By comparing the local and global formability of the two AHSS, it is confirmed that ductile fracture is the dominant failure mode in the MMnS and the onset of localized necking occurs prior to ductile fracture in the DP1000 steel. To achieve an accurate determination of the local formability, the effects of stress states need to be considered, which cannot be derived explicitly from uniaxial tensile tests. In addition to tensile properties, more attention should be paid to the local formability of new AHSS to assess their potential application in automotive industries.
H.W. Swift1•Institutions (1)
Abstract: This paper examines the conditions for instability of plastic strain under plane stress for a material conforming to the Mises-Hencky yield condition and strain-hardening according to a unique relationship between root-mean-square values of shear stress (q) and incremental strain (δψ). If, under fixed loading conditions, the material undergoes a strain increment which is consistent with the applied stress system, the conditions are stable or unstable according as the increment in representative yield stress is greater or less than the increment in representative induced stress. The strain at which instability arises is found in terms of the biaxial stress ratio p2/p1 under different conditions of applied loading, and the effect is demonstrated of strain-hardening according to an empirical relation of the type q = c (a + ψ)n. The analysis is also applied to certain cases of non-uniform stress distribution. In the case of the hydrostatic bulge results are obtained showing a critical thinning ranging from 26 per cent for a non-hardening material to about 45 per cent for typical strain-hardening materials, values in general agreement with experimental data. Conditions over the punch head in the pressing of a cylindrical shell are discussed but computations are not attempted.
Abstract: Summary Explicit formulae are obtained for the stresses in a metal diaphragm which is bulged plastically by lateral pressure. The predicted influence of work-hardening on the shape of the profile, and on the relation between polar strain and curvature, agrees well with experimental data. A simple expression is developed for the instability strain.
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