Evaluation of non-linear strain paths using Generalized Forming Limit Concept and a modification of the Time Dependent Evaluation Method

The prediction of formability is one of the most important tasks in sheet metal forming process simulation. The common criterion for ductile fracture in industrial applications is the Forming Limit Diagram (FLD). This is only applicable for linear strain paths. However, in most industrial simulation cases non-linear strain paths occur. To resolve this problem, a phenomenological approach is introduced, the so-called Generalized Forming Limit Concept (GFLC). The GFLC enables prediction of localized necking on arbitrary non-linear strain paths. Another possibility is the use of the Time Dependent Evaluation Method (TDEM) within the simulation as a failure criteria. During the Numisheet Benchmark 1 (2014) a two-stage forming process was performed with three typical sheet materials (AA5182, DP600 and TRIP 780) and three different blank shapes. The task was to determinate the point in time and space of local instability. Therefore the strain path for the point of maximum local thinning is evaluated. To predict the start of local necking the Generalized Forming Limit Concept (GFLC), the Time Dependent Evaluation Method (TDEM) and the modified TDEM were applied. The results of the simulation are compared with the results of the Benchmark experiment.     

Stress based prediction of formability and failure in incremental sheet forming

A strain-based forming limit criterion is widely used in sheet-metal forming industry to predict necking. However, this criterion is usually valid when the strain path is linear throughout the deformation process [1]. Strain path in incremental sheet forming is often found to be severely nonlinear throughout the deformation history. Therefore, the practice of using a strain-based forming limit criterion often leads to erroneous assessments of formability and failure prediction. On the other hands, stress-based forming limit is insensitive against any changes in the strain path and hence it is first used to model the necking limit in incremental sheet forming. The stress-based forming limit is also combined with the fracture limit based on maximum shear stress criterion to show necking and fracture together. A derivation for a general mapping method from strain-based FLC to stress-based FLC using a non-quadratic yield function has been made. Simulation model is evaluated for a single point incremental forming using AA 6022-T43, and checked the accuracy against experiments. By using the path-independent necking and fracture limits, it is able to explain the deformation mechanism successfully in incremental sheet forming. The proposed model has given a good scientific basis for the development of ISF under nonlinear strain path and its usability over conventional sheet forming process as well.     

Modeling of localization and fracture phenomena in strain and stress space for sheet metal forming

In this study, a model based on a strain localization level to overcome the shortcomings of the well-established Forming Limit Diagram (FLD) in predicting the physical phenomenon of necking is introduced. An optical measurement system was used to capture the strain history of the Nakazima experiment until rupture occurred. In order to measure the fracture strain more accurately, a further method is introduced, which is based on the microscopic measurements of ruptured regions. This model is validated using a 3-point bending test. The results show the ability of the method to predict failure under bending conditions as well. Additionally, failure is investigated based on the pressure sensitivity and the Lode dependency. The results show that the triaxiality at the failure point is independent of the loading path.     

Shape accuracy analysis of multi-point forming process for sheet metal under normal full constrained conditions

In order to study the shape accuracy of multi-point forming (MPF) process for sheet metal under normal full constrained conditions, the in-depth analysis of shape accuracy of workpieces in multi-point forming with individually controlled force-displacement (MPF-ICFD) process is conducted in this paper by combining experiment, theoretical analysis and numerical simulation. The influences of normal force, material thickness and material properties on the shape accuracy of the feature surface are studied, and the shape accuracy characteristics of the sheet under different parameters are obtained. Afterwards, the stress and strain characteristics of sheets are obtained by numerical simulation. Finally, the effect of normal force on shape accuracy was revealed by establishing a mechanical model of the sheet metal under normal full constrained conditions. Moreover, the amount of springback reduction in MPF-ICFD is defined quantitatively. Compared with the normal unconstrained conditions, the shape accuracy of sheet metal is improved significantly under normal full constrained conditions. The stress and plastic deformation are more uniform and the amount of springback is smaller. For Q 295 steel plate with thickness of 2.0 mm, the difference between the maximum value and the minimum value of the reaction force of punch decreases from 4515.9 N to 1475 N when the forming force is 2500 N. Besides, the bending moment of the sheet on the unit width decreases from 357.9 N???mm to 328.1 N???mm. The average shape error E _rr and the amount of springback Δk decreases by 60.05% and 16.03%, respectively.     

On the description of ductile fracture in metals by the strain localization theory

Numerical simulations based on the bifurcation and imperfection versions of the strain localization theory are used in this paper to predict the failure loci of metals and applied to an advanced high strength steel subjected to proportional loading paths. The results are evaluated against the 3D unit cell analyses of Dunand and Mohr (J Mech Phys Solids 66(1):133–153, 2014. doi: 10.1016/j.jmps.2014.01.008) available in the literature. The Gurson porous plasticity model (Gurson in J Eng Mater Technol 99(1):2–15, 1977. doi: 10.1115/1.344340) is used to induce strain softening and drive the localization process. The effects of the void growth, void nucleation and void softening in shear are investigated over a large range of stress triaxialities and Lode parameters. A correlation between the imperfection and bifurcation results is established.     

Modelling the correlation between the geometrical features and the forming limit strains of perforated Al 8011 sheets using artificial neural network

In the recent years, sheet metals are produced with perforations in various shapes and patterns to improve the appearance of sheet and to save weight of components. As in conventional metal sheets, it is important to form the perforated sheet metals also within their safe strain regions to avoid the forming failures like necking, fracture and wrinkling. The Forming Limit Diagram (FLD) is an appropriate tool to determine the forming limit strains. The limiting strains of perforated sheet metals mainly depend on the geometry of the perforations and forming variables. This leads to large increase in number of test to be conducted with various geometry and forming variables for determining the forming limit strain for perforated sheets. Aiming to reduce the number of experiments needed, in this work, an Artificial Neural Network (ANN) model has been developed for forming limit diagram of perforated Al 8011 sheets based on experimental results and correlated with the geometrical features of the perforated sheets. This model is a feed forward back propagation neural network (BPNN) with a set of geometrical variables as its inputs and the safe true strains as its output. This ANN model can be applied for prediction of FLD of perforated sheet having any geometry.     

Test on the linear combinations of mean vectors in high-dimensional data

In this study, we propose a procedure for simultaneous testing \(l (l\ge 1)\) linear relations on \(k(k\ge 2)\) high-dimensional mean vectors with heterogeneous covariance matrices, which extends the result derived by Nishiyama et al. (J Stat Plan Inference 143(11):1898–1911, 2013) and does not need the normality assumption. The newly proposed test statistic is motivated by Bai and Saranadasa (Statistica Sinica 6(2):311–329, 1996) and Chen and Qin (Ann Stat 38(2):808–835, 2010). As a special case, our result could be applied to multivariate analysis of variance, that is, testing the equality of k high-dimensional mean vectors.     

A moment generating function of a combination of linear rank tests and its asymptotic efficiency

When testing hypotheses in two-sample problems, the Lepage test has often been used to jointly test the location and scale parameters, and has been discussed by many authors over the years. The Lepage test was a combination of the Wilcoxon statistic and the Ansari–Bradley statistic. Various Lepage-type tests were proposed with discussions of an asymptotic relative efficiency (Duran et al., Biometrika 63:173–176, 1976; Goria, Stat Neerl 36:3–13, 1982), a robustness and a power comparison (Neuhäuser, Commun Stat Theory Methods 29:67–78, 2000; Büning, J Appl Stat 29:907–924, 2002) and an adaptive test (Büning and Thadewald, J Stat Comput Sim 65:287–310, 2000). We derive an expression for the moment generating function of a linear combination of two linear rank statistics. As a suggested Lepage-type test, we use a combination of the generalized Wilcoxon statistic and the generalized Mood statistic. Deriving the exact critical value of the statistic can be difficult when the sample sizes are increasing. In this situation, an approximation method to the distribution function of the test statistic can be useful with a higher order moment. We use a moment-based approximation with an adjusted gamma polynomial to evaluate the upper tail probability of a Lepage-type test for a finite sample size. We determine the asymptotic efficiencies of the Lepage and Lepage-type tests for various distributions.     

Constitutive modeling and the effects of strain-rate and temperature on the formability of Ti–6Al–4V alloy sheet

The constitutive model considering the strain-rate and temperature effects was presented by fitting the true stress–strain curves of Ti–6Al–4V alloy over a wide range of strain-rates (0.0005–0.05 s⁻¹) and temperatures (923–1023 K). The Forming Limit Curve (FLC) of Ti–6Al–4V alloy at 973 K was measured by conducting the hemispherical dome test with specimens of different widths. The forming limit prediction model of Ti–6Al–4V alloy, which takes strain-rate and temperature sensitivity into account, was predicted based on Marciniak and Kuczynski (M–K) theory along with Von Mises yield criterion. The comparison shows that the limit strain decreases with temperature lowering but strain-rate increasing. The comparison between theoretical analysis and experiment of FLC verifies the accuracy and reliability of the proposed methodology, which considers the strain-rate and temperature effects, to predict limit strains in the positive minor strain region of Forming Limit Diagram (FLD).     

Development of a new failure prediction criterion in sheet metal forming

In industrial try-out processes in sheet metal forming usually the forming limit curve is used as failure criterion in order to describe the onset of localized necking. Forming limits, however, are strain-path dependent. Up to today many different approaches how the strain-path dependent behavior of the forming limit curve can be avoided have been published. Best known are the approaches based on forming limit stress curves published by Arrieux, and the approach of Müschenborn published in 1975. An overview over existing failure criteria is given in this contribution. The failure criterion forming limit stress curve as well as several failure criteria based on Müschenborn’s approach will be evaluated with newly recorded experimental data on forming limit curves for non-proportional loading. A new approach, that is in contrast to the two mentioned approaches not based on assumptions but on experimental observations, is presented herein. The suggested approach is presented as failure surface where strains above the surface indicate the onset of localized necking. The failure surface is given as function of the loading mode and the level of effective pre-stretching. Different suggestions how to use and simplify the new approach are given in this paper. The prediction accuracy of the addressed approaches as well as of the newly suggested approach is evaluated by transforming the data of the approaches back into the conventional forming limit curve for several non-linear strain paths. The comparison of the described approaches shows that forming limits for non-proportional loading can be well-predicted with the suggested approach.     

On the use of effective limit strains to evaluate the forming severity of sheet metal parts after nonlinear loading

The conventional forming limit curve (FLC) is significantly strain path-dependent and therefore is not valid for formability evaluation of sheet metal parts that undergo nonlinear loading paths during the forming process. The stress-based forming limit curve (SFLC) is path-independent for all but very large prestrains and is a promising tool for formability evaluation. The SFLC is an ideal failure criterion for virtual forming simulations but it cannot be easily used on the shop floor as there is no straightforward experimental method to measure stresses in stamped parts. This paper presents a theoretical basis for predicting the effective limit strain curve (ELSC) using the Marciniak and Kuczynski (MK) analysis (Int J Mech Sci 9:609–620, 1967, Int J Mech Sci 15:789–805, 1973). Since the in-plane strain components are sufficient to calculate the effective strain, the ELSC can easily be determined from strains measured in the stamping plant, and therefore it is a better alternative to the SFLC for formability evaluation. This model was validated using experimental data for AISI-1012 steel (Molaei 1999) and AA-2008-T4 aluminum alloys Graf and Hosford (Metall Trans 24A:2503–2512, 1993). Predicted results showed that, similar to SFLC, the ELSC remains practically unchanged for a significant range of prestrain values under various bilinear loading paths, but some strain-path dependence can be observed for significant magnitudes of the effective prestrain (ε _e ?≥?0.37 for AISI-1012 steel and ε _e ?≥?0.25 for AA-2008-T4 aluminum).     

Precursors of failure and weakening in a biaxial test

We study numerical simulations of large (\({N{\approx}10^4}\)) two-dimensional quasi-static granular assemblies subjected to a slowly increasing deviator stress. We report some peculiarities in the behavior of these packings that have not yet been addressed. The number of sliding contacts is not necessarily related to stability: first the number of sliding contacts rises linearly and smoothly with the applied stress. Then, at approximately half the peak stress, the increase slows down, a plateau develops, and a decrease follows. The spatial organization of sliding contacts also changes: during the first half of the simulation, sliding contacts are uniformly distributed throughout the packing, but in the second half, they become concentrated in certain regions. This suggests that the loss of homogeneity occurs well before the appearance of shear bands. During the second half events appear where the number of sliding contacts drops suddenly, and then rapidly recovers. We show that these events are in fact local instabilities in the packing. These events become more frequent as failure is approached. For these two reasons, we call these events precursors, since they are similar to the precursors recently observed in both numerical (Staron et al. Phys Rev Lett 89:204302, 2002; Nerone et al. Phys Rev E 67:011302, 2003) and experimental (Gibiat et al. J Acoust Soc Am 123:3142, 2009; Scheller et al. Phys Rev E 74:031311, 2006; Zaitsev et al. Eur Phys Lett 83:64003, 2008; Aguirre et al. Phys Rev E 73:041307, 2006) studies of avalanches.     

Experimental and theoretical investigation of forming limit diagram for Ti-6Al-4 V alloy at 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 is useful tool to access the forming severity of a drawing or stamping processes. In the present work, FLD has been determined experimentally for Ti-6Al-4 V alloy at 400 °C by conducting a hemispherical dome test with specimens of different widths. Additionally, theoretical FLDs have been determined using Marciniak Kuczynski (M-K) model. Various yield criteria namely: Von Mises, Hill 1948, Hill 1993 and Cazacu Barlat in combination with different hardening models viz., Hollomon power law (HPL), Johnson-Cook (JC), modified Johnson-Cook (m-JC), modified Arrhenius (m-Arr.), modified Zerilli–Armstrong (m-ZA) have been used in M-K analysis for theoretical FLD prediction. The material properties required for determination of yield criteria and hardening models constants have been calculated using uniaxial tensile tests. The predicted theoretical FLDs results are compared with experimental FLD. It can be observed that influence of yield criterion in M-K analysis for theoretical FLD prediction is predominant than the hardening model. Based on the results; it is observed that the theoretical FLD using Cazacu Barlat and Hill 1993 yield criteria with m-Arr. hardening model has a very good agreement with experimental FLD.     

Strain-injection and crack-path field techniques for 3D crack-propagation modelling in quasi-brittle materials

This paper presents a finite element approach for modelling three-dimensional crack propagation in quasi-brittle materials, based on the strain injection and the crack-path field techniques. These numerical techniques were already tested and validated by static and dynamic simulations in 2D classical benchmarks [Dias et al., in: Monograph CIMNE No-134. International Center for Numerical Methods in Engineering, Barcelona, (2012); Oliver et al. in Comput Methods Appl Mech Eng 274:289–348, (2014); Lloberas-Valls et al. in Comput Methods Appl Mech Eng 308:499–534, (2016)] and, also, for modelling tensile crack propagation in real concrete structures, like concrete gravity dams [Dias et al. in Eng Fract Mech 154:288–310, (2016)]. The main advantages of the methodology are the low computational cost and the independence of the results on the size and orientation of the finite element mesh. These advantages were highlighted in previous works by the authors and motivate the present extension to 3D cases. The proposed methodology is implemented in the finite element framework using continuum constitutive models equipped with strain softening and consists, essentially, in injecting the elements candidate to capture the cracks with some goal oriented strain modes for improving the performance of the injected elements for simulating propagating displacement discontinuities. The goal-oriented strain modes are introduced by resorting to mixed formulations and to the Continuum Strong Discontinuity Approach (CSDA), while the crack position inside the finite elements is retrieved by resorting to the crack-path field technique. Representative numerical simulations in 3D benchmarks show that the advantages of the methodology already pointed out in 2D are kept in 3D scenarios.     

On the identification of an effective cross section for a cruciform specimen

Biaxial tensile testing of sheet metals is becoming increasingly popular for sheet metal forming. Determining equivalent stresses in biaxial tensile specimens is more complicated than in conventional uniaxial tensile specimens. In the present study, we compare four different approaches to calculate effective stresses during biaxial tensile loading of a cruciform specimen: (a) partial unloading method, where stresses are determined based on force–strain curves; (b) identification with uniaxial tensile testing; (c) an analysis of equivalent biaxial tests; and (d) numerical simulations. Considering experimental results for an AA1050 aluminium alloy and for a low‐carbon steel DC06, we show that, for the cruciform sample studied here, two methods do not yield physically reasonable results: The uniaxial approach does not properly take into account the effect of transverse loading, and the equivalent biaxial approach exhibits uncertainties in strain measurement data. The most comprehensible approach is the numerical method, because it also yields detailed information about the local stress and strain states. The numerical results are in excellent agreement with the partial unloading method in terms of the initial flow stress and of effective stress–strain curves for strains up to 0.02, with both methods predicting a similar effective cross section of 18.0 mm² for the considered specimen.     

A computationally efficient algorithm to simulate the butt curl deformation in casting processes

The aim of this work is to present a computationally efficient algorithm to simulate the butt curl deformation. In our previous articles [1, 2] the nonlinearities due to the viscoplastic law and the contact condition with the bottom block were solved by means of duality methods involving two multipliers. In [1] these multipliers were computed with a fixed point algorithm and in [2] with a generalized Newton’s method. In this work we improve the viscoplastic algorithm by means of a generalized duality method with variable parameters. We will present numerical results showing the applicability of the resultant algorithm to casting processes.     

A fracture locus for a 50 volume-percent Al/SiC metal matrix composite at high temperature

The effect of the stress state on the fracture locus function of the 50 vol.% Al/SiC metal matrix composite at high temperature is studied. The value of fracture locus function is quantitatively characterized by the amount of shear strain accumulated prior to the moment of failure. Nondimensional invariant parameters are used as characteristics of the stress state, namely, the stress triaxiality k and the Lode-Nadai coefficient μ _σ showing the form of the stress state. Besides conventional testing for tension, compression and torsion of smooth cylindrical specimens, the complex of mechanical tests includes a new type of testing, namely, that for bell-shaped specimens. These kinds of testing enable one to study fracture strain under monotonic deformation in the ranges μ _σ ?=?0?…?+?1 and k?=???1.08...0 without using high-pressure technologies. The stress–strain state during specimen testing is here evaluated from the finite element simulation of testing in ANSYS. The tests were performed at a temperature of 300 °C and shear strain rate intensity Η?=?0.1;?0.3;?0.5 1/s. The test results have offered a fracture locus, which can be used in models of damage mechanics to predict fracture of the material in die forging processes.     

Censored linear model in high dimensions

Censored data are quite common in statistics and have been studied in depth in the last years [for some references, see Powell (J Econom 25(3):303–325, 1984), Murphy et al. (Math Methods Stat 8(3):407–425, 1999), Chay and Powell (J Econ Perspect 15(4):29–42, 2001)]. In this paper, we consider censored high-dimensional data. High-dimensional models are in some way more complex than their low-dimensional versions, therefore some different techniques are required. For the linear case, appropriate estimators based on penalised regression have been developed in the last years [see for example Bickel et al. (Ann Stat 37(4):1705–1732, 2009), Koltchinskii (Bernoulli 15:799–828, 2009)]. In particular, in sparse contexts, the \(l_1\)-penalised regression (also known as LASSO) [see Tibshirani (J R Stat Soc Ser B 58:267–288, 1996), Bühlmann and van de Geer (Statistics for high-dimensional data. Springer, Heidelberg, 2011) and reference therein] performs very well. Only few theoretical work was done to analyse censored linear models in a high-dimensional context. We therefore consider a high-dimensional censored linear model, where the response variable is left censored. We propose a new estimator, which aims to work with high-dimensional linear censored data. Theoretical non-asymptotic oracle inequalities are derived.     

Design optimisation of biaxial tensile test specimen using finite element analysis

One of the most restricting aspects of the biaxial tensile test for sheet metal is the design of the cruciform specimen. Although specimens of the cruciform type have been investigated quite extensively previously, no standard geometry for the cruciform specimen exists. Using a specifically designed pantograph apparatus for operation in a standard tensile testing machine, various cruciform specimens machined from low-carbon cold rolled steel sheet were analysed experimentally. Finite element modelling of the specimens was conducted in parallel to the experimental test programme to establish optimum specimen geometry. Through a process of optimisation, a standard cruciform specimen was designed which can be used to accurately predict the mechanical behaviour of the mild steel when formed in multiple directions simultaneously. This paper describes the optimisation process and the results obtained from both the experimental testing and numerical modelling.     

Theory of <Emphasis Type="Italic">d</Emphasis>-Wave High Temperature Superconductivity in the Cuprates Involving Non-linear Lattice Modes

The transition mechanism in high temperature cuprate superconductors is an outstanding puzzle. A previous suggestion on the role of non-linear local lattice instability modes on the microscopic pairing mechanism in high temperature cuprate superconductors (Lee, J. Supercond. Nov. Magn. 23(3), 333; 2009) is re-examined to provide a viable mechanism for superconductivity in these cuprates via an unusual lattice vibration in which an electron is predominantly interacting with a non-linear Q ₂ mode of the oxygen clusters in the CuO₂ planes. It is shown that the interaction has explicit d-wave symmetry and leads to an indirect coupling of d-wave symmetry between electrons. As a follow-up of Lee (J. Supercond. Nov. Magn. 23(3), 333; 2009), in this paper, we report detailed derivation of the superconducting gap equation and numerical solutions for the transition temperature as inherently integrated into the so-called extended Hubbard model (EHM). A unique feature in the EHM is that the transition temperature has an inherent k-dependence. In addition, superconducting gap solutions are restrained to specific regions in the first Brillouin zone (1BZ). It is very feasible to expect that the EHM naturally inherits a huge parameter space in which experimentally measured results, such as the well-known superconducting dome and the phase diagram from electronic Raman scattering (Sacuto et al., Rep. Prog. Phys. 76(2), 022502; 2013) can be accommodated. The EHM model hence offers a viable venue to search for or confirm any signature in k-point-sensitive experimental measurements.