Through-thickness Normal Stress Research Articles

A novel three-dimensional M-K model by integrating GTN model for accurately identifying limit strains of sheet metal

To accurately identify the limit strains and improve the prediction accuracy of forming limits for sheet metal, a novel three-dimensional M-K model by integrating GTN model is developed and named 3D M-K-GTN model. By introducing the through-thickness normal stress and friction stress, the original M-K model is extended for adapting to the three-dimensional stress state. Furthermore, the empirical failure criterion is replaced by integrating the GTN model, and thus the limit strain is identified when the void volume fraction within the groove accumulates to a critical value. Besides, the voids nucleation and growth expressions are also re-derived and modified by embedding the friction stress and normal stress. Finally, the comparison between predicted forming limit curves and experimental limit strains of AA6016-T4 sheet indicates that the 3D M-K-GTN model solves the significant overestimation of prediction results under large strain paths, and always provides a high prediction accuracy for each strain path given.

Investigation on the forming limit diagram of AA5754-O alloy by considering strain hardening model, strain path, and through-thickness normal stress

This study clarified the effects of the chosen hardening model, through-thickness normal stress, and non-linear strain path on the precision of forming limit diagram (FLD). The Marciniak-Kuczynski (M-K) instability theory was employed to calculate the FLD of AA5754-O based on the different hardening models. Swift, Voce, modified Voce, Kim-Tuan, and modified Kim-Tuan hardening models were adopted to predict the forming limits, and their constant parameters were presented. Predicted FLDs indicate that modified Kim-Tuan has a better agreement with experimental data than other models. After that the influences of through-thickness normal stress on the forming limit diagram were examined. The three-dimensional stress state was converted to plane-stress condition according to the assumption that hydrostatic stress is ineffective on the plastic deformation. The results show that formability improves when normal stress increases. However, the effect of the normal stress on FLD for various hardening models is different. Due to the complexity of sheet metal forming, the strain path of the material may change during the deformation process. The results indicated that strain path has a significant effect on the FLD, and the path dependence of the forming limit stress diagram (FLSD) will be examined in detail. Finally, the forming limits were computed to analyze the effect of the normal stress on the non-linear loading path deformation. It was observed that by increasing the amount of pre-strain, the effect of the normal stress on the formability will diminish. Moreover, results showed that normal stress will affect the path dependence of the FLSD.

The determination of friction coefficient in Nakazima test and its application in predicting FLCs with modified M-K model

In this work, the friction coefficient between the punch and the specimen during the Nakazima tests is determined by comparing the experimental and the numerical strain paths. Afterwards, the FLCs of the Al-Mg-Li alloy sheet are predicted with the modified M-K model, which considers the through-thickness normal stress and the friction through introducing the expressions of the normal stress and the friction stress into M-K model. Finally, the predicted FLCs of the sheet are compared with the forming limits determined by Nakazima tests.

Influence of thickness reduction on forming limits of mild steel DC01

The Forming Limit Diagram (FLD) is an essential tool to assess sheet metal formability in sheet metal deep drawing. In FLDs/FLC (Forming Limit Curve) is representation of material formability limits at which material is not able to withstand higher deformation. In this work, different methodologies for FLC determination of mild steel DC01 are investigated. The Nakajima test is a well-known experiment for FLC determination but, however, contact conditions that can appear in the form of either friction or pressure may perturb the results. The former may cause the need for repetitions for different widths by altering the necking location, whereas the latter may be responsible for overestimation of the material formability. Another method for FLC determination without those effects is to use the cruciform specimen test under biaxial loading conditions. As the first step, Necking has to be induced at the center of cruciform specimen by thickness reduction or groove. Therefore, in this work, the cruciform geometry under biaxial loading is optimized using FEM through the comparative analysis. The goal of simulation was to induce the necking just in the thinner gauge, not adjacent zone. Thereupon, different loads on biaxial machine’s axes can provide different strain paths at the thinner gauge of the optimized specimen. After reviewing of various researches in connection with thickness reduction effect on sheet metal deformation, it was found that thickness reduction might not influence the material formability during uniaxial and plane strain tensions, but it was observed that there is gradual decrease of limit strains with decreasing the thickness for equi-biaxial tension.Manufacturing process like milling, cause small defects on material surface and local instability takes place at lower level of deformation. Besides, it was found that existence of through thickness normal stress leads to higher value of FLC. Deformation during Nakajima and cruciform experiments were captured by Digital Image Correlation (DIC) system. The execution and evaluation of those experiments are explained in detail and Time Dependent Method (TDM) was applied to detect the beginning time of instability. Experimental strain paths are compared with consideration of strain rates and, as it was expected, FLC from cruciform specimen is lower than Nakajima one, especially on the right hand side of FLD. The remarkable difference can be attributed to defects in the material and existence of through-thickness normal stress, which leads to overestimation of the material formability during Nakajima experiment.

A Novel Approach to Predict Wrinkling of Aluminum Alloy During Warm/Hot Sheet Hydroforming Based on an Improved Yoshida Buckling Test.

In order to predict the wrinkling of sheet metal under the influence of fluid pressure and temperature during warm/hot hydroforming, a numerical simulation model for sheet wrinkling prediction was established, taking into account through-thickness normal stress induced by fluid pressure. From simulations using linear and quadratic elements, respectively, it was found that the latter gave results that were much closer to experimental data. A novel experimental method based on an improved Yoshida Buckling Test (YBT) was proposed for testing the wrinkling properties of sheets under the through-thickness normal stress. A wrinkling coefficient suitable for predicting wrinkling was also presented. Based on the numerical simulations, an experimental validation of wrinkling performance was conducted. Ridge-height curves measured along the main diagonal tensile direction of the sheet were presented and showed that the wrinkling prediction criterion provided good discrimination. Furthermore, the wrinkling properties of several different materials were simulated to evaluate the accuracy of the prediction method, and the results revealed that the improved YBT gave good predictions for wrinkling in the conventional sheet metal forming process, while the prediction results for wrinkling in warm/hot sheet hydroforming were also accurate with the fluid pressure of zero.

Mechanics analysis of aluminum alloy cylindrical cup during warm sheet hydromechanical deep drawing

To obtain the stress states of the sheet metal and understand the potential forming mechanism under the influences of fluid pressure and temperature during the warm hydroforming of 5A06-O aluminum alloy sheets, mechanics analysis was conducted for the warm hydromechanical deep drawing of a cylindrical cup, taking into account through-thickness normal stress induced by fluid pressure. By considering the additional stress during bending using a variable-stiffness pure bending model as equal to that determined using a bending moment model, the additional stress at the die corner during bending was solved. The force equilibrium equations for the cup near the die corner, which were classified by whether the blank contacted the die, were also established and the results were in good agreement with those obtained via the finite element method. Furthermore, mechanics analysis was also performed for the straight-wall region, and the results revealed that the error was slightly greater when the liquid pressure was used to approximately equal to through-thickness normal stress because the ratio of thickness to radius was not sufficiently large.

Finite element analysis of a modified progressive damage model for composite laminates under low-velocity impact

In this paper, a 3D finite element model is established in ABAQUS/Explicit based on a modified progressive damage model to study the dynamic mechanical response and damage development in cross-ply composite laminates subjected to low-velocity impact. The 3D Hashin criterion and the damage evolution model with the through-thickness normal stress component σ33 are applied to predict the intra-laminar damage initiation and evolution. The cohesive elements with the bilinear traction-separation relationship are inserted between layers to predict the inter-laminar delamination induced by impact loading. A user-material subroutine VUMAT involving the modified progressive damage del of intra-laminar and inter-laminar damage is coded and implemented in the finite element package ABAQUS/Explicit. The numerical results of three different impact energies (7.35, 11.03 and 14.70 J) are analyzed by the impact force–time, force-displacement and energy-time histories curves as well as different damage modes. The respectable relationship between numerical simulation and experimental result indicates that the proposed modified method is more suitable for low-velocity impact on composite laminates under different impact energies than the previous method without σ33. Moreover, the effects of Smt and Smc on global mechanical response and local damage predictions for laminates are discussed in detail. It can be concluded that both of the coefficients should be adopted between 0.93 and 0.96 when using this damage model to simulate composite laminates under low-velocity impact.

The integration of through-thickness normal stress and friction stress in the M-K model to improve the accuracy of predicted FLCs

The through-thickness normal stress and the friction stress in the forming process have significant impact on the forming limit curves (FLCs) of sheet metal. In this work, the through-thickness normal stress expression is first established based on the analysis of the stress state and the stress gradient of specimens in the Nakazima test and then introduced into the original M-K (Marciniak-Kuczynski) model. By analyzing the friction condition on the contact surface of the specimen and the punch, the friction stress expression is then set up and considered in the M-K model. On this basis, the original M-K model is modified, and a comparison of the forming limits is performed among our modified M-K model, other M-K models and the experiment results. The conclusion can be drawn from the comparison that the prediction accuracy of the FLCs is significantly improved by our modified M-K model. Moreover, the influence of the material parameters and the model parameters on the FLCs are further investigated.

Influence of thickness and initial groove angle in M–K model on limit strain of 7B04 by considering through-thickness stress

Marciniack–Kuczinski (M–K) model is widely used to predict material’s forming limit curve (FLC). The prediction of FLC traditionally neglected through-thickness normal stress. However, it cannot be neglected in some forming processes. Much work has been done to study the effect of through-thickness normal stress on FLC with constant through-thickness normal stress or constant ratio of through-thickness normal stress and maximum principal stress. In addition, based on Nakazima test process, the ratio of through-thickness normal stress and maximum principal stress has been derived, which was a function of instantaneous thickness and loading path. Here, initial groove angle in M–K model was not considered. In this paper, uniaxial tension tests and Nakazima tests were performed on 7B04 aluminum alloy. Based on Hill 48 yield criterion and M–K model, the prediction model of FLC was established. The increase of thickness can enhance FLC. Meanwhile, it is necessary to consider through-thickness normal stress and initial groove angle in prediction model. On the left side of FLC, the effect of initial groove angle on FLC is weakened by increasing sheet thickness. On the right side of FLC, the effect of initial groove angle on FLC is strengthened by increasing sheet thickness. On the right side of FLC, the relation between limit strain points with different thicknesses is linear under one certain loading path. Thickness has decisive effect on through-thickness normal stress level and the changing trendy of through-thickness normal stress during calculation is different under different stress condition.

Forming Limit Evaluation with Perturbation Approach Considering Through-Thickness Normal Stress

To predict material’s formability in the hydroforming processes, the plane stress assumption would be invalid. The instability perturbation approach proposed by Hu et al. [1] is extended with the through-thickness normal stress by combining Hill’48 and Hosford’s yield criteria. The influences of through-thickness normal stress on the predicted forming limit strains in the forms of traditional Forming Limit Diagram (FLD) and equivalent plastic strain (EPS) based FLD (epFLD) are investigated. The results show that forming limit curves (FLCs) in both forms of FLD enhance with increasing through-thickness normal stress under proportional and non-proportional loadings. This new model can be utilized to study the effects of fluid pressure on the formability of orthotropic thin sheets.

Forming limit evaluation by considering through-thickness normal stress: Theory and modeling

For hydroforming process and incremental forming, the plane stress assumption would be invalid to predict material formability. To consider the influence of normal stress on the forming limit strains, the instability perturbation approach proposed by Hu et al. is extended with normal stress. The M–K model with normal stress is chosen to compare with perturbation approach by implementing Hill’48 and Yld2000-2d yield criteria and swift and modified voce hardening laws. To guarantee the convergence of solving process in M–K model with normal stress, the modified increment method is implemented into it. Through comparing the experimental forming limit strains of AA5754-O under plane stress state and the corresponding theoretical values predicted by perturbation approach and M–K model, the suitable yield criterion and hardening law for these two methods are determined. The influences of through-thickness normal stress on the predicted forming limit strains under different stress states are investigated. The results show that forming limit curves (FLCs) in both forms of traditional Forming Limit Diagram (FLD) and equivalent plastic strain (EPS) based FLD (epFLD) enhance with increasing through-thickness normal stress under linear and nonlinear strain paths. The sensitivity of FLD to normal stress is related to the hardening law and the range of strain path in FLD is determined by the yield criterion.

A comparative study of the extended forming limit diagrams considering strain path, through-thickness normal and shear stress

The substitute presentations of the conventional forming limit diagram (FLD) are stress-based forming limit diagram (FLSD), extended stress forming limit diagram (XSFLD), and polar effective plastic strain forming limit diagram (PEPS-FLD). These diagrams have already been proposed as alternative criteria to the conventional FLD for predicting forming limits in processes that sheet does not experience proportional or in-plane loading conditions. The present study provides a complete comparison of different forms of forming limit diagrams. For this purpose, FLD and extended FLDs are predicted based on the modified Marciniak and Kuczynski (M-K) model with Yld2011 anisotropic yield function while various factors that cause a change in the FLD of a specific alloy including initial imperfection coefficient (f0), non-linear strain path, through-thickness normal and shear stresses are considered. The results indicate that all forms of extended diagrams are independent of strain path and the diagrams in stress space (i.e., FLSD and XSFLD) are less sensitive to the strain path than PEPS-FLD. In this regard, the main weakness of XSFLD, especially on the right-hand side, is that the safety margin cannot be visualized easily. By increasing the initial imperfection coefficient, the level of all forms of diagram increases. However, the effect of through thickness stresses on different diagrams is not the same. Increase in normal and shear stresses results in a downward shift in FLSD and an upward shift in other types of forming limit diagrams.

Evaluation of the forming limit curve of medium steel plate based on non-constant through-thickness normal stress

The forming limit curve (FLC) provides a convenient and useful way of presenting the necking or facture in sheet forming process. This work focused on determining the FLC of the medium steel plate with a thickness of 5 mm. The Nakazima test and uniaxial tension test were conducted to determine the limit strain pairs and the mechanical properties of the QSTE600TM medium plate, respectively. The M-K theory was extended under non-constant through-thickness normal stress condition to predict the FLC of the medium plate with higher accuracy. Meanwhile, the numerical Nakazima test was implemented to validate the non-constant through-thickness normal stress and determine the forming limit strains. The criterion for evaluating the limit strain pairs in simulation process was presented. The comparison showed that the experimental, theoretical and numerical FLCs well coincided, and demonstrated that the non-constant through-thickness normal stress was a critical factor in the FLC prediction of medium plate.

Further assessment of the plastic instability of thin-walled tubes in double-sided hydro-bulging process with verification experiments

An interesting process of double-sided tube hydro-bulging was proposed to provide a beneficial three-dimensional stress state in the deformation zone of the tube, so as to delay the occurrence of bursting on the tube. Previously proposed forming limit theory models with consideration of through-thickness normal stress found that the external pressure could lead to a deferred occurrence of plastic instability, but it has not been confirmed experimentally. In this article, the tensile plastic instability of tubes under double-sided pressures is further assessed based on the classical plastic instability theory. However, it is seen that the occurrence of plastic instability has not found delay when high external pressure was exerted on the outside surface of tube simultaneously, which is opposite to the previous forming limit models. In addition, an experiment investigation about double-sided tube hydro-bulging is conducted to verify the theoretical results, and the experimental results show that the external pressure has hardly any influence on forming limit strain of the 2A12 aluminum alloy tube before the occurrence of necking. Moreover, the contradiction between our results and the previously proposed forming limit theory models is that the previous models ignored the thickness item in the equilibrium equation.

A damage mechanics model for low-velocity impact damage analysis of composite laminates

ABSTRACTA method was developed to predict numerically the damage of composite laminates with multiple plies under low-velocity impact loading. The Puck criterion for 3D stress states was adopted to model the intralaminar damage including matrix cracking and fibre breakage, and to obtain the orientation of the fracture plane due to matrix failure. According to interlaminar delamination mechanism, a new delamination criterion was proposed. The influence of transverse and through-thickness normal stress, interlaminar shear stress and damage conditions of adjacent plies on delamination was considered. In order to predict the impact-induced damage of composite laminates with more plies quickly and efficiently, an approach, which can predict the specific damage of several plies in a single solid element, was proposed by interpolation on the strains of element integration points. Moreover, the proposed model can predict specific failure modes. A good agreement between the predicted delamination shapes and sizes and the experimental results shows correctness of the developed numerical method for predicting low-velocity impact damage on composite laminates.

Forming limit curves analysis of aluminum alloy considering the through-thickness normal stress, anisotropic yield functions and strain rate

Based on three anisotropic yield functions including Karafillis–Boyce (K–B), Yld96 and Yld2011, directional normalized uniaxial yield stresses, directional r-value and forming limit curve (FLC) for AA3104-H19 aluminum alloy under plane stress condition are numerically investigated in this article. Moreover, considering the through-thickness normal stress effect the forming limit diagram (FLD), stress-based forming limit diagram (FLSD) and extended forming limit stress diagram (XFLSD) is also studied theoretically based on Yld2011 yield criterion and modified Marciniak–Kuczynski (M–K) model. The nonlinear equations set are solved through employing Newton–Raphson numerical method to calculate limiting strains. The anisotropic plastic behavior and FLC of AA3104-H19 predicted by Yld2011 yield criterion are in good agreement with experimental data and are more accurate than those of K–B and Yld96 yield functions. In addition, according to FLD, the formability of sheet metal increases by applying the through-thickness normal stress. The effects of strain rate at quasi-static condition and temperature are theoretically investigated on the FLD of AA3104 aluminum alloy. The positive temperature sensitivity and negative strain rate sensitivity are observed of FLD of AA3104.

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.

Mechanics analysis of axisymmetric thin-walled part in warm sheet hydroforming

To obtain the influence of fluid pressure and temperature on warm hydroforming of 5A06-O aluminum alloy sheet, the unified mechanics equilibrium equations, which take through-thickness normal stress and friction into account, were established in spherical coordinate system. The distribution of through-thickness normal stress in the thickness direction was determined. The relation between through-thickness normal stress and fluid pressure was also analyzed in different regions of cylindrical cup. Based on the method of subtracting one increasing function from another, the constitutive equation of 5A06-O applied to warm hydroforming was established and in a good agreement with uniaxial tensile data. Based on whether the thickness variation was taken into account, two mechanic models were established to do the comparative study. The results for the studied case show that the calculated stress values are pretty close according to the two models and consistent with results of finite element analysis; the thickness distribution in flange computed by the second model conforms to the experimental data. Finally, the influences of fluid pressure on the flange thickness and radial stress were analyzed.

Effect of through-thickness normal stress on forming limits under Yld2003 yield criterion and M-K model

Forming limit curve considering through-thickness normal stress is theoretically obtained based on Yld2003 yield criterion and M-K model with the assumption that hydrostatic stress has no effect on plastic deformation. The suitability of Yld2003 yield criterion to describe the yield surface and forming limit under plane-stress condition for aluminum alloy AA6111-T43 is verified by the experimental data obtained from the literature. Then the effect of through-thickness normal stress on forming limit is analyzed. Different forms of forming limit presentation including traditional strain-based forming limit curve (FLC), forming limit effective strain curve (eFLC), stress-based forming limit curve (FLSC) and extended stress-based forming limit curve (XSFLC) are investigated in detail. It is found that forming limit curves in strain-space enhance with increased through-thickness normal stress under both linear and nonlinear strain paths with different pre-strains. However, the variation of forming limit curves in stress-space does not show single trend but depends on the variables of the coordinate plane. A new form of forming limit presentation named correcting FLSC is proposed. The correcting major stress enhances with through-thickness normal stress increase, and the floating range of the correcting FLSC due to through-thickness normal stress is much less than that of the traditional FLSC. The correcting FLSC can be taken as a steadier criterion for sheet metal forming limit prediction under three-dimensional stress state.

Forming limit model evaluation for anisotropic sheet metals under through-thickness normal stress

The M–K method has been used with Barlat׳s 1989 anisotropic yield surface to predicate forming limit to study the effect of the normal stress and material anisotropy. The 3D stress state is converted to plane-stress state according to the hypothesis that hydrostatic stress has no effect on plastic deformation. The predicted forming limits correlate well with experimental data for AA5XXX under plane-stress condition and AA6011 aluminum alloy under three-dimension stress condition. Analyses show that normal stress increases the forming limit described both in strain space and in stress space. Moreover, the sensitivity of forming limit curve to normal stress is associated with the material׳s hardening effect.