Initial Thickness Imperfections Research Articles

A Modified M-K Method for Accurate Prediction of FLC of Aluminum Alloy

Forming limit curve (FLC) is an important failure criterion for sheet metals in sheet metal forming, while the M-K model is widely used for the prediction of the FLC. In the M-K model, such prediction is greatly influenced by the initial thickness imperfection factor and material properties, from which the original M-K model’s theoretical derivation is proposed as a solution to the above mentioned issue in this paper. Then the relationship between the M-K model and Keeler’s empirical formula is then studied, from which a new method to predict FLC is proposed that combines the M-K model with Keeler’s empirical formula according to the previous analyses. It turns out that this new method can simplify the calculation procedure. Finally, the experimental results of two kinds of aluminum alloys, AA6016 and AA5182, have verified the effectiveness of the proposed method.

Uncertainty of membrane wrinkling behaviors considering initial thickness imperfections

Initial thickness imperfections of thin membranes due to manufacturing are inevitable and commonly observed. This paper presents an experimental and numerical investigation on how thickness imperfections affect the wrinkling of stretched membrane structures. With the measured data from a limited number of samples, we model the uncertain deviation of membrane thickness from that of a perfect geometry as a non-probabilistic bounded field with spatial dependency. After representing the bounded field as a linear function of a reduced set of uncertain coefficients by using a series expansion, the effect of thickness imperfections on the membrane wrinkling is investigated. To this end, the upper and lower bounds of out-of-plane displacements for membranes with initial imperfections are evaluated by a mathematical programming. The proposed wrinkling evaluation problem is then solved efficiently by using the Kriging-based optimization algorithm, with validation done by Monte Carlo simulation. The results show that initial thickness imperfections play a critical role in the wrinkling behavior of membranes.

Buckling characteristics of externally pressurised toroidal shell

ABSTRACT This study investigates the buckling characteristics of toroidal shells with a closed circular cross-section and loaded by static external pressure. With the thickness imperfection considered as the initial geometrical imperfection, the nonlinear buckling numerical calculation of the toroidal shells is discussed in detail. The finite element model was used for the toroidal shell and validated accordingly. The numerical results of mean thickness, variable thickness, and mode imperfection, which was introduced to simulate the real thickness imperfection without the initial thickness imperfection being specified, were compared with the experimental results. In addition, four laboratory-scale models were manufactured and tested. Each of models was assembled by welding four 90° 304 stainless steel elbows. The geometry, toroidal shell thickness, buckling load, and final collapsed mode were measured for each toroidal shell; the material properties of the corresponding sheets were also measured. The results of the theoretical, numerical, and experimental investigations are compared in this paper.

A thermal forming limit prediction method considering material damage for 22MnB5 sheet

Hot stamping of high-strength steel is an innovative technology to achieve automobile lightweight and guarantee security simultaneously. However, the formability of high-strength steel is still limited at elevated temperatures by the evolution of void damage inside materials. Thus, the establishment of an efficient forming limit prediction method is urgently demanded. In the present work, the Gurson-Tvergaard-Needleman (GTN) model is extended by containing Hosford anisotropic yield criterion which can characterize void damage in normal anisotropic materials. The flow behavior of matrix material 22MnB5 at different forming temperatures is simulated by the modified Norton-Hoff hardening law. And the damage-related parameters are calibrated through FEM inverse analysis. Furthermore, the prediction method of forming limit combined with void damage is realized within the framework of the Marciniak and Kuczynski (M-K) model. Meanwhile, the high-temperature Nakazima test is conducted to obtain experimental thermal forming limit diagram (TFLD). The comparison of results by theoretical prediction and experimental test shows good consistency. Based on the proposed method, the effects of temperature, void damage, initial thickness imperfection, and anisotropy on the forming limit are analyzed. The formability improved with the increasing of deformation temperature but worse for other factors.

Modified MK model combined with ductile fracture criterion and its application in warm hydroforming

A modified MK model combined with ductile fracture criterion (DFC-MK model) is proposed to compute the forming limit diagrams (FLDs) of 5A06-O aluminum alloy sheet at different temperatures. The material constant (C) of ductile fracture criterion and initial thickness imperfection parameter (f0) at various temperatures are determined by using a new computing method based on wide sheet bending test. The FLDs at 20 and 200 °C are calculated through the DFC-MK model. The DFC-MK model, which includes the influence of through-thickness normal stress, is written into the subroutine VUMAT embedded in Abaqus/Explicit. The cylindrical cup hydroforming tests are carried out to verify the model. The results show that compared with experimental observations, the predicted FLDs based on DFC-MK model are more accurate than the conventional MK model; the errors between the simulations and experiments in warm hydroforming are 8.23% at 20 °C and 9.24% at 200 °C, which verify the effectiveness of the proposed model.

Prediction of forming limit diagram of AA7075-O aluminum alloy sheet based on modified M-K model

To accurately predict the forming limit diagram of AA7075-O aluminum alloy sheet,a modified M-K model was put forward based on the ductile fracture criterion and the traditional M-K model. By using the method which combined the uniaxial tension simulation with experiment,the history of stress-strain of the dangerous element was substituted into the C-L ductile fracture criterion which was implemented into Abaqus / Explicit to get the material constant. Based on the forming limit data obtained by the uniaxial tensile,the initial thickness imperfection in the modified M-K model was obtained by the method which was programed in MATLAB. At room temperature,the experimental forming limit data of AA7075-O aluminum alloy sheet were tested through the uniaxial tensile,wide plate bending and hydraulic bulging experiments. Meanwhile,the theoretical forming limit curves obtained by the modified M-K model and the traditional M-K model were computed respectively. By comparison,the modified M-K model is proved to be feasible and more accurate.

Influence of pre-strain on the formability of 2024 aluminium alloy

Present work investigates the influence of pre-strain on the limiting strains in sheet metal forming, using the strain gradient evolution model that predict the onset of local necking from initial thickness imperfections, roughness waviness and thickness gauge. The theoretical analysis is compared with the experimental results for fully annealed and three pre-strain levels of 4%, 8% and 12% in 2024 aluminium alloy sheet with thickness of 2.5 mm presented by Leacock et al. at ESAFORM 2008. The effect of strain and strain rate hardening behaviour, pre-strain and anisotropy parameters in material formability are analyzed. The experimental forming limit curves indicate an immediate drop in the limiting strains in the pre-strained specimen when compared with the full annealed condition material as expected from the theoretical prediction by a new model. Other relevant aspects of the new approach to modeling FLC are also discussed.

Formability curves of AZ31 alloy sheet at high temperatures

Nowadays, the AZ31 magnesium alloy sheet has received great attention due to its mechanical properties, notably its low density and high specific strength, and its potential for thin-walled components and sheet parts in the automotive industry. The present work examines the influence of forming process temperature on the formability curves of AZ31 magnesium alloy sheet, using the strain gradient evolution model that predicts the onset of local necking, i.e., the forming limit curve (FLC) from material plastic properties and initial thickness imperfections. The effect of plasticity parameters such as work-hardening coefficient, strain rate sensitivity coefficient and normal anisotropy for various temperatures are investigated in AZ31 alloy sheet formability curve. The theoretical analyses for forming limit curve predictions at high temperatures are compared with the experimental results available in the literature for various temperatures ranging from 100°C up to 300°C. The correlation is reasonably good. The forming limit curve depends greatly upon the strain rate sensitivity, M-value, and the work hardening coefficient, n-value. Other relevant aspects of the mechanics of sheet metal forming modeling are also discussed.

Bulge formation and necking in a polymer tube under dynamic expansion

Bulging and necking in long thin polymer tubes subjected to increasing internal pressure are analysed numerically. The polymer is characterized by a finite strain elastic–viscoplastic constitutive relation and the calculations are carried out using a dynamic finite element program. Two types of imposed loading are prescribed: (i) a pressure that increases linearly with time and (ii) a change in enclosed volume that increases linearly with time. For both loading conditions, an axisymmetric bulge develops on the tube followed by necking in the bulge. The necks propagate in both the circumferential and the axial directions. Multiple necks form at locations given by the thin points associated with the wave number of the prescribed initial thickness imperfection. When a change in enclosed volume is prescribed, the pressure reaches a maximum, decreases and then stays approximately constant. One neck eventually becomes dominant in that it propagates further along the tube axis than any of the other necks.

Influence of thickness size in sheet metal forming

A new approach on the influence of sheet thickness and material plastic properties on the limit strains in thin sheet metal forming is developed, using the strain gradient model that predict the local necking onset from initial thickness imperfections. As a result of the analysis, the definition of a roughness concept parameter is presented: the initial roughness profile inclination to thickness ratio parameter. Also, the critical normalized strain gradient at the onset of local necking can be calculated from the initial assumed roughness profile. A brief review of stretch forming of sheet metal, the diagrams and the limiting strain curves for local necking, FLC, and the limit strain theories are presented. The main characteristics of the sheet metal forming processes are also identified and are based in the press shop practice. The limit strains for different thickness sizes are obtained using the present model and a numerical code developed by the author. Present theoretical model produced reasonable predictions about the influence of thickness size on the FLC.

Constitutive modeling and finite element methods for TRIP steels

A constitutive model that describes the mechanical behavior of steels exhibiting “TRansformation Induced Plasticity” (TRIP) during martensitic transformation is presented. Multiphase TRIP steels are considered as composite materials with a ferritic matrix containing bainite and retained austenite, which gradually transforms into martensite. The effective properties and overall behavior of TRIP steels are determined by using homogenization techniques for nonlinear composites. A methodology for the numerical integration of the resulting elastoplastic constitutive equations in the context of the finite element method is developed and the constitutive model is implemented in a general-purpose finite element program. The model is calibrated by using experimental data of uniaxial tension tests in TRIP steels. The problem of necking of a bar in uniaxial tension is studied in detail. The constitutive model is used also for the calculation of “forming limit diagrams” for sheets made of TRIP steels; it is found that the TRIP phenomenon increases the strain at which local necking results from a gradual localization of the strains at an initial thickness imperfection in the sheet.

Effects of elastic unloading on multiple necking in tensile bars

The deformation of round tensile bars subject to dynamic loading is studied numerically using two different plasticity theories: J 2-flow theory and J 2-deformation theory. Time development of neck localization is visualized and a comparison of the necking patterns obtained using the two different constitutive models is presented. The possibility of using J 2-deformation theory as an alternative to J 2-flow theory to model the dynamic behaviour of tensile bars subject to high rate deformation is investigated and the effects of elastic unloading are discussed. Influence of specimen geometry and initial thickness imperfections is also considered. The effects of elastic unloading on multiple necking are of particular interest, since elastic unloading may serve as a restricting factor regarding the growth of multiple neck localizations. In the present analysis, multiple necking is found to appear using both J 2-flow theory and J 2-deformation theory. The use of stubby bars leads to similar necking patterns when comparing the results obtained with J 2-flow theory and J 2-deformation theory. For slender bars subject to certain loading conditions a larger number of necks is obtained when J 2-deformation theory is applied rather than J 2-flow theory. In addition, results of simulations near the ideal-plastic limit are discussed as well as results where the yield stress is varied.

Effects of inertia on dynamic neck formation in tensile bars

Abstract Neck localization during high rate extension of round bars is analyzed numerically. An axisymmetric problem formulation is given and the material is described as rate independent and elastic–plastic. The time history of neck development is investigated and effects of geometry and initial thickness imperfections are visualized. Studies of both weakly and strongly developed necks are performed, revealing the occurrence of multiple necks in some loading cases. The influence on neck formation from background inertia (lateral inertia) corresponding to the inertia originating from homogeneous deformation of the bar is examined somewhat qualitatively by introducing an artificial volume load. Calculations show that in the present analysis, background inertia does not have any noticeable influence on the necking pattern unless the effect is artificially magnified by three orders of magnitude so that it becomes comparable to the yield stress.

Nonlinear buckling of cylinders under external pressure with nonaxisymmetric thickness imperfections using the COMI axisymmetric shell element

This paper concerns the influence of thickness imperfections on the nonlinear buckling of cylinders under uniform external pressure. The material is assumed to behave elastically. A parametric study is performed using the COMI finite element, which allows to perform nonlinear analyses of shells of revolution with any kind of nonaxisymmetric initial imperfection. Let us assume that the perfect shell buckles with a Fourier mode m and that the initial thickness imperfection is on Fourier mode 1, m or 2 m. As shown in a previous study on linear buckling of cylinders with thickness imperfections, 1 and 2 m are the most harmful thickness imperfections. We show that the nonlinear pre-buckling does not affect the buckling pressures very much. We then study the coupling with geometrical imperfections. We show that the geometry and thickness imperfections have a multiplicative effect on the decrease in buckling load in the nonlinear range. Finally, we show that if one uses only the measured outer radius of a shell to input initial imperfections, very different nonlinear buckling pressures can be obtained under the assumptions that the imperfection is purely geometric, purely thickness, or a combination of both.

Flow localization in the plane strain tensile test

The deformations in a plane strain tensile test are analyzed numerically, both for a solid characterized by a phenomenological corner theory of plasticity and for a nonlinear elastic solid. As opposed to the simplest flow theory of plasticity with a smooth yield surface, both these material models exhibit shear band instabilities at a realistic level of strain. Initial imperfections are specified in the form of thickness inhomogeneities. A long-wavelength imperfection grows into the well-known necking mode and subsequently, at a sufficiently high local strain level, bands of intense shear deformations develop in the necking region. The location of these shear bands is strongly influenced by the location of small strain concentrations near the surface, induced by various short-wave patterns of initial thickness imperfections. In accord with the non-uniform straining in the neck it is found that the intensity of the localized deformations varies along the bands, and some of the shear bands end inside the material.

Necking of pressurized spherical membranes

T he necking of spherical membranes subject to a prescribed increase in enclosed volume is investigated. Attention is restricted to axisymmetric deformations. The materials considered are incompressible, isotropic, time-independent and incrementally linear. A complete set of axisymmetric bifurcation modes is considered and a simple relation is found to govern the critical stress for bifurcation into a given mode. The limiting critical stress and the corresponding mode for short wavelengths are investigated and related to the results obtained from an independent local-necking analysis. Two perturbation methods are employed to study the growth of initial imperfections: one is valid for arbitrary modes, but restricted to small deviations from sphericity, and the other is valid only for the local-necking mode, but is not restricted to small deviations. The effect of path-dependent material behavior on the onset of local necking is explored. Path-dependent material behavior is found to encourage the preferential growth of short wavelength imperfections. Path-independent materials are shown to exhibit significant sensitivity to initial imperfections in the localized-necking mode, although this sensitivity is far less than for a path-dependent material. When account is taken of initial material-property inhomogeneities as well as initial thickness imperfections, it seems that no definite conclusion can be drawn concerning the appropriateness or inappropriateness of an explanation of the onset of localized necking based on a smooth yield-surface plasticity theory and assuming the presence of such initial inhomogeneities.