Anisotropic Yield Criterion Research Articles

A study on anisotropic hardening of 7075 aluminum alloy based on non-associated flow rules

The accurate description of anisotropic plastic deformation is key to accurately predicting the stamping forming of metal sheets. The anisotropic yield criterion, when based on the assumption of isotropic hardening, often leads to significant inaccuracies. To address this issue, this paper observes the anisotropic hardening phenomenon through non-associated flow rules and evaluates the anisotropy of 7075-O aluminum alloy. Through a series of tensile tests, we determined the mechanical properties of 7075-O aluminum alloy in three distinct orientations. To describe the metal hardening behavior, we employed the Swift-Voce hardening criterion. From the hardening curves in three different directions, it was found that AA7075-O exhibits plastic anisotropy. Based on the VUMAT subroutine, finite element simulation of AA7075-O tensile tests was conducted through Abaqus. Compared with the Hill48 model, it was found that the simulated values of the S–Y2009 anisotropic hardening model have a higher degree of agreement with the experimental curves. The S–Y2009 anisotropic hardening model was adopted to predict the earing behavior of AA7075-O during circular cup deep drawing. The root mean square error between the predicted values of the S–Y2009 model and the experimental values was only 0.1795, which is far smaller than that of the Hill48 yield model. Therefore, the SY2009 model has important guiding significance for the stamping forming of metal sheets.

Effects of anisotropy and infill pattern on compression properties of 3D printed CFRP: mechanical analysis and elasto-plastic finite element modelling

PurposeFinite element (FE) analysis can be used for both design and verification of components. In the case of 3D-printed materials, a proper characterization of properties, accounting for anisotropy and raster angles, can help develop efficient material models. This study aims to use compression tests to characterize short carbon-reinforced PA12 made by fused filament fabrication (FFF) and to model its behaviour by the FE method.Design/methodology/approachIn this work, the authors focus on compression tests, using post-processed specimens to overcome external defects introduced by the FFF process. The material’s elastoplastic mechanical behaviour is modelled by an elastic stiffness matrix, Hill’s anisotropic yield criterion and Voce’s isotropic hardening law, considering the stacking sequence of raster angles. A FE analysis is conducted to reproduce the material’s compressive behaviour through the LS-DYNA software.FindingsThe proposed model can capture stress values at different deformation levels and peculiar aspects of deformed shapes until the onset of damage mechanisms. Deformation and damage mechanisms are strictly correlated to orientation and raster angle.Originality/valueThe paper aims to contribute to the understanding of 3D-printed material’s behaviour through compression tests on bulk 3D-printed material. The methodology proposed, enriched with an anisotropic damage criterion, could be effectively used for design and verification purposes in the field of 3D-printed components through FE analysis.

Mechanical characterization and constitutive modeling of additively-manufactured polymeric materials and lattice structures

Additively manufactured polymeric lattice structures are being extensively studied, primarily because their mechanical properties can be tailored by controlling the unit cell geometry, giving them higher designability than stochastic materials. However, the inherent layer-wise additive manufacturing process affects the base material properties related to the printing direction, which in turn affects the macroscopic responses of the entire lattice materials. A robust understanding and modeling of lattice structures' elastic and plastic yield behavior in a homogenized approach are essential to enhance their design and analysis efficiency in engineering applications. In pursuit of this goal, a unified printing angle-dependent constitutive model of base materials is proposed in line with the tensile experimental data. The elastic material properties (elastic modulus, shear modulus, and Poisson's ratio), obtained through numerical simulations of one unit-cell with periodic boundary conditions, exhibit anisotropic properties, with the degree of anisotropy determined by the angle of the constituent members and base materials. Furthermore, both experimental and numerical results of lattices demonstrate anisotropic mechanical response under horizontal and vertical compression. Virtual multiaxial experiments are conducted through multi-cell numerical simulations, enabling the determination of initial yielding points of two different lattice structures (Kelvin and Simple cubic and body-centered cubic hybrid structures) under various loading conditions using a dissipation energy-based criterion. Overall, the multiaxial yield surface of the investigated lattices under various stress states, except for the isotropic principal stress plane, can be properly depicted by the Extended-Hill anisotropic yield criterion.

Multiaxial yield behavior of 2D re-entrant auxetic cellular materials

A series of re-entrant honeycombs (RH) with different off-axis angles and perturbation parameters are constructed and utilized to numerically explore the multiaxial yield behavior for 2D auxetic cellular materials. The off-axis angle significantly affects the anisotropic properties and Poisson's ratio of RH. The effects of microstructural irregularity on the multiaxial yield surfaces are also evaluated for all of the topologies. Simulation results under principal stress space, mean stress-effective stress space and normal stress-shear stress space reveal that the unique microstructure of auxetic cellular materials makes its mechanical properties significantly different from conventional cellular materials with positive Poisson's ratio. Furthermore, a pressure-dependent anisotropic phenomenological yield criterion is proposed for 2D auxetic cellular materials, which can effectively capture the multiaxial yield behavior of RH with different topologies in principal stress space. The present conclusions provide fresh insights into the microstructural design of auxetic cellular materials and pave the way for their future engineering applications.

Deep drawing simulation of dual phase steel using hardening curves and anisotropic parameters from uniaxial and biaxial tensile tests

Among the various factors, the accuracy of the predictions from the numerical simulation of sheet metal forming processes depends on the material model used to define the mechanical behavior of the blank material. The coefficients of the hardening model to define the flow curve and the plastic strain ratios are commonly determined using the uniaxial tensile tests. The advanced anisotropic yield criteria incorporate material flow behaviour and plastic strain ratio in equi-biaxial tension. In this work, deep drawing of a flat bottom cylindrical cup has been simulated using dual-phase steel (DP600) sheets. The biaxial material properties obtained by conducting hydraulic bulge test and cruciform specimen test are used in the anisotropic yield criteria in the simulations. The hardening curves are extrapolated using different hardening laws in which the coefficients are determined from the stress-strain curves obtained from both uniaxial tensile and hydraulic bulge tests. The predicted peak drawing load and thickness variation in the drawn cups are compared with the experimental results of deep drawing.

Ductile fracture of tubular specimens under non-proportional loading condition

The ductile fracture behaviours of two tubular specimens, a AA6260-T4 macrotube (60 mm diameter and 2 mm thickness), and a SS-304L microtube (2.38 mm diameter and 0.15 mm thickness) under non-proportional loading condition are studied using a combined experimental-numerical approach. The experiments are conducted by loading the tubes under axial force and internal pressure along various non-proportional (i.e., corner) paths until failure by controlling the force/pressure ratio. The plastic behaviours of the tubes are characterized using the non-quadratic anisotropic yield criteria Yld2000-2D and Yld2004-3D. The material models are employed in the finite element (FE) simulation of the tube experiments using Abaqus/Standard (implicit). The FE models, which include thickness imperfection to capture the failure modes observed in the experiments, are then used to probe the fracture parameters inside the tube wall where fracture is likely to initiate. It is observed that the fracture strains from the non-proportional loading results are noticeably different when introduced to the proportional fracture strains, revealing the path-dependence fracture behaviour in tubular specimens.

Experimental investigation and numerical modelling of the cyclic plasticity and fatigue behavior of additively manufactured 316 L stainless steel

This study addresses the critical need for a constitutive model to analyze the cyclic plasticity of additively manufactured 316L stainless steel. The anisotropic behavior at both room temperature and 300 °C is investigated experimentally based on cyclic hysteresis loops performed in different orientations with respect to the build direction. A comprehensive constitutive model is proposed, that integrates the Armstrong-Frederick nonlinear kinematic hardening, Voce nonlinear isotropic hardening and Hill's anisotropic yield criterion within a 3D return mapping algorithm. The model was calibrated to specimens in the 0° and 90° orientations and validated with specimens in the 45° orientation. A single set of hardening parameters successfully represented the elastoplastic response for all orientations at room temperature. The algorithm effectively captured the full cyclic hysteresis loops, including historical effects observed in experimental tests. A consistent trend of reduced hardening was observed at elevated temperature, while the 45° specimen orientation consistently exhibited the highest degree of strain hardening. The applicability of the model was demonstrated by computing energy dissipation for stabilized hysteresis loops, which was combined with fatigue tests to propose an energy-based fatigue life prediction model.

Anisotropic ductile fracture of a stainless steel under biaxial loading: Experiments and predictions

The anisotropic ductile fracture behavior of a stainless steel is investigated using a combination of experiments and analysis. The material is SS-304L, an austenitic, low-carbon stainless steel, received in the form of tubes of 2.38 mm dia. and 0.15 mm thickness. The tubes are inflated under volume control in a custom apparatus. At the same time, the axial force on the tubes is kept proportional to the pressure induced by the inflation. This leads to quasi-proportional stress paths in the meridional-hoop engineering stress space. A total of 15 discrete paths are successfully tested. Stereo-type Digital Image Correlation is used to measure the strains. In every case, the tubes develop a series of instabilities before bursting. The failure is oriented along the meridional or the hoop direction of the tube, depending on which stress is greater. A mild plastic anisotropy is detected in these experiments. Hence these results are then used for calibrating the anisotropic yield criterion Yld2004-3D. This is introduced in a finite element model of the experiments, which includes a thickness imperfection designed to capture the two failure orientations observed in the experiments. The numerical model using the Yld2004-3D criterion reproduces the experiments well, e.g., it captures the experimental stress-strain and induced strain paths better than von Mises. It is then used to probe the conditions at the onset of fracture (hybrid method). It is found that most paths lead to essentially proportional loading during deformation. A significant anisotropy in the fracture behavior is detected, with the meridional-stress-dominated paths being able to develop much higher strains than the hoop-dominated ones. These results are then captured by the DF2016 ductile fracture criterion, modified to use the anisotropic yield criterion Yld91. The proposed criterion is flexible enough to represent the fracture anisotropy very well, without being unnecessarily complex. The fracture forming limit curve (i.e., the fracture envelope in strain space) predicted by the DF2016/Yld91 model is also found to be very close to the experiments. The results and findings of this work help establish a framework to reliably design components and processes when significant fracture anisotropy is expected.

Investigation on the Effect of Blank Holder Force and Initial Blank Size on the Thinning Problem of the Fuel Tank Part Made from Aluminum Alloy Sheet 5754-O under Deep Drawing Process

Deep drawing process is a common sheet metal forming technique in motor vehicle manufacturing. There are three primary defects that could be occur in deep-drawn parts: tearing, wrinkling, and thinning. When the thinning is difficulty detected by visual inspection. As a result, this study aims to address the thinning issue in a fuel tank part made from an aluminum alloy sheet AA5754-O 1.5 mm thick under cold working deep drawing process, while the manufacturer's desired upper limit for thinning is 20%. Two influential parameters viz. blank holder force and initial size of blank, were investigated and optimized by using Finite Element Analysis (FEA) through PAM-STAMP simulation software with the validated material model was based on Hill’s 1948 anisotropic yield criterion with Swift hardening law. The mechanical parameters in the mentioned model were derived from the results of uniaxial tensile tests. In conclusion, both the hydraulic cushion's blank holder pressure and the initial size of the blank were found to influence the thinning of the part, either individually or in combination. Despite optimizing both parameters, they were unable to consistently achieve the desired limit.

Fracture forming limit curve prediction by ductile fracture models

The fracture forming limit curve is an essential tool giving information about fracture initiation in metal forming applications. In order to obtain a fracture-forming limit curve, the Nakajima test should be carried out. However, these tests are costly and time-consuming processes. Therefore, companies in the automotive industry tended to use non-expensive numerical approaches to determine the material's formability limits. In this work, an anisotropic polynomial yield criterion was implemented to calibrate the different ductile fracture models, including only or both the stress triaxiality and the Lode parameter's effect. Analyses of different uniaxial tensile test geometries were carried out, and the two-dimensional fracture loci were constructed by different ductile fracture models. Correspondingly, the fracture-forming limit curves were predicted by the two-dimensional fracture loci. The results showed that the ductile fracture models, including only the stress triaxiality effect, provide an acceptable performance covering the range from uniaxial tension and the plane strain tension lines, while the models, including the stress triaxiality and Lode parameter, provide a plausible prediction performance throughout the whole range for forming limit diagram.

Simplified Characterization of Anisotropic Yield Criteria for an Injection-Molded Polymer Material.

Injection-molded polyethylene plates exhibit highly anisotropic mechanical behavior due to, e.g., the uneven orientation of the polymer chains during the molding process and the differential cooling, especially in the thickness direction. Elastoplastic finite element modeling of these plates in particular is used with isotropic yield criteria like von Mises, trading off accuracy in favor of simpler constitutive characterization and faster solution. This article studies three different anisotropic yield criteria, namely, Hill 1948, Barlat Yld91, and Barlat Yld2004-18P, for the finite element modeling of low-density polyethylene (LDPE) at large uniaxial tensile deformation and compares the accuracy and computation time with von Mises. A simplified calibration technique is investigated to identify the constitutive parameters of the studied Barlat group yield criteria. The calibration process is simplified in the sense that only uniaxial tensile tests with digital image correlation measurements are used for the calibration of all the yield criteria studied in this article, although a standard calibration procedure for the Barlat group yield criteria requires additional material testing using more demanding test setups. It is concluded that both Barlat Yld91 and Barlat Yld2004-18P yield criteria can be calibrated with only a few tensile tests and still capture anisotropy in deformation-stress-strain at different levels of accuracy.

Evaluating Pile-Supported Embankment Considering the Soil Anisotropy Effect

It is an economical way to use the pile-supported embankment for the construction of the embankment over soft soil. The combined use of piles and reinforcement effectively reduces the differential settlement of the embankment surface. However, the previous analysis of embankment stress and settlement did not take into account the anisotropy in the embankment filler. In this paper, the UMAT subroutine is developed by using the material subroutine interface in ABAQUS 2016 finite element software. The anisotropy of soil cohesion and friction angle has been incorporated into the Mohr–Coulomb yield criterion so that it can consider the anisotropy of soil. The accuracy of the anisotropic yield criterion in this paper is verified by an ABAQUS source program and related engineering examples. It is found that the anisotropy value of soil cohesion is inversely proportional to the stress ratio on the pile–soil interface while being directly proportional to the tensile stress applied to the geogrid. The results show that the anisotropy of the friction angle decreases with the soil arching effect but increases by 23.1% with the tensile stress on the geogrid. The position of the settlement plane remains relatively constant at 2.3 m as the friction angle anisotropy coefficient increases. These research results provide valuable theoretical guidance for on-site construction design.

Study on the failure behavior of Nomex honeycomb sandwich panels under three-point bending load: Experiments and meso-scale simulation

The failure behavior and load-carrying capacity of Nomex honeycomb sandwich panels under three-point bending are experimentally and numerically studied. The failure modes observed in experiments are core shear failure and face-sheet/core debonding. A detailed model considering Nomex paper and resin coating is built with Abaqus/explicit simulation. The Hill 1948 quadratic anisotropic yield criteria are adopted to describe the plastic behavior of Nomex paper and the principal stress criteria are used for predicting the failure of resin coating. The effectiveness of the proposed meso-scale model is validated by comparing the experimental and the numerical results. The effects of geometric parameters on the bending behaviors are discussed in detail. It is shown that the ultimate force and bending stiffness of Nomex honeycomb sandwich panels is significantly influenced by the core orientation, resin thickness and cell size. The cutting position has no effect on the ultimate force and bending stiffness.

Modeling of Eyld2000-2d Anisotropic Yield Criterion Considering Strength Differential Effect and Analysis of Optimal Calibration Strategy.

Sheet metals usually experience various loading paths such as uniaxial tension, uniaxial compression, biaxial tension, and simple shear during the forming process. However, the existing constitutive models cannot always accurately describe blanks' anisotropic yield and plastic flow behavior of blanks under all typical stress states. Given this, this paper improves the Eyld2000-2d yield criterion by introducing hydrostatic pressure to the A-Eyld2000-2d yield criterion that can describe the strength differential effect of materials. Meanwhile, to control the curvature of the yield surface more effectively, the near-plane strain yield stresses were added in the parameter identification process to calibrate the exponent m, so that the exponent is no longer considered as a constant value. Taking the widely used AA6016-T4, AA5754-O, DP980, and QP980 blanks in the automotive stamping industry as an example, the effectiveness of the new model and different parameter identification methods was verified by predicting experimental data under various simple and complex loading paths. Subsequently, the new model employing the optimal parameter identification strategy was compared with four widely used asymmetric yield criteria under associated and non-associated flow rules, including CPB06, LHY2013, S-Y2004, and Hu & Yoon2021, to further verify the accuracy of the proposed constitutive model. The results indicate that parameter identification strategy with variable exponent can significantly improve the flexibility of the yield criterion in describing the plastic anisotropy of blanks. Compared to the other yield criteria examined in this work, the new model provides the best prediction accuracy for the yield stresses and plastic flows of all blanks, especially in the near-plane strain and simple shear stress states. Modeling under the concept of anisotropic hardening can more accurately capture the evolving plastic behavior of blanks than isotropic hardening.

Formability prediction of high‐strength steel sheet using experimental, analytical and theoretical analysis based on strain and stress forming limit curves and its application to automotive forming parts

AbstractIn this work, the high‐strength steel (HSS) sheet dual‐phase 440 (DP440) were conducted to establish the forming limit curve (FLC) and analytical forming limit stress curve (FLSC) obtained from experimental forming limit curve. First, the Nakajima stretch forming examination was carried out to obtain forming limit curve of investigated sheet. Afterwards, the theoretical Marciniak–Kuczinsky (M–K) model was developed and calculated to evaluate localized necking limits both in strain and stress spaces combination with anisotropic yield criteria. Then, the analytical forming limit stress curves were plastically calculated by using experimental forming limit curve data combination with Swift hardening model and anisotropic yield criteria namely, Hill’48 and Yld2000‐2d for representing anisotropic plastic deformation behavior on examined steel sheet. Finally, automotive stamping parts were performed in order to verify an applicability of all developed curves. It was observed that the analytical forming limit stress curves could more precisely predict the formability of automotive parts better than the forming limit curve based on strain. Particularly, the one based on Yld2000‐2d predict better than the one based on Hill’48. Simultaneously, the experimental forming limit curve and analytical forming limit stress curve were also evaluated comparing with the theoretical calculated forming limit curve and forming limit stress curve using the Marciniak–Kuczinsky model. It should be noted again that the experimental forming limit curve and analytical forming limit stress curve are the best one. Therefore, the Yld2000‐2d yield function better represented the anisotropic behavior of the high‐strength steel sheet dual‐phase 440 than Hill′ 48 yield function, and can suitable be used for the analysis prediction and design of bumper automotive parts under forming processes.

Use of uncoupled ductile damage models for fracture forming limit prediction during two-stage forming of aluminum sheet material

In this paper, failure limits of aluminum AA5052 sheet material were experimentally investigated, and subsequently, these were predicted using various uncoupled ductile fracture models. In this context, three different sets of experiments, namely two-stage stretch forming, two-stage deep drawing, and single point incremental forming experiments, were performed to estimate the failure limits under several different strain paths. Both the isotropic von Mises and anisotropic Barlat Yld2000 yield criteria were used to calibrate eight different fracture models, and found that incorporating an anisotropic model enhanced the accuracy of prediction for most of the fracture models. Moreover, it was identified that the incorporation of anisotropic yield theory in the Bao-Wierzbicki (BW) fracture model predicted the experimental fracture surface strains precisely for as-received sheet materials. This BW model-based fracture forming limit diagram (BW-FFLD) was transposed into a path-independent polar effective plastic strain (PEPS) locus to estimate the onset of fracture during two-stage forming experiments. Furthermore, the BW model was incorporated as a fracture initiation model in the finite element (FE) software, LS-Dyna, using the generalized incremental stress state dependent damage model (GISSMO) platform to predict the fracture location and dome height through the element erosion technique. It was observed that the dome height and cup height were predicted within 7 % and 5 % error margins in the FE simulation of two-stage stretch forming and deep drawing experiments, respectively. Furthermore, the damage evolution contour indicated that the accumulation of the damage value was rapid, which was responsible for the massive drop in the dome and cup heights during two-stage stretch forming and deep drawing experiments.

Experimental and numerical studies on damage and failure behavior of anisotropic ductile metals

AbstractThe paper deals with experiments and numerical simulations of the biaxially loaded H‐specimen to study the damage and failure in anisotropic ductile metals. The deformation and failure behavior of anisotropic ductile metals depend both on load ratio and loading direction with respect to the rolling direction. Experiments focusing on shear‐compression stress states have been performed and digital image correlation (DIC) is used to monitor the strain fields. Numerical simulations based on the Hill48 anisotropic yield criterion are used to predict the stress states of the investigated anisotropic aluminum alloy EN AW‐2017A. The fractured surfaces are visualized by scanning electron microscopy (SEM). The experimental‐numerical technique clearly shows the influence of loading direction and the stress state on the evolution of damage processes.

Research on the 2A11 Aluminum Alloy Sheet Cyclic Tension–Compression Test and Its Application in a Mixed Hardening Model

The increasing application of aluminum alloy, in combination with the growth in the complexity of components, provides new challenges for the numerical modeling of sheet materials. The material elastic–plasticity constitutive model is the most important factor affecting the accuracy of finite element simulation. The mixed hardening constitutive model can more accurately represent the real hardening characteristics of the material plastic deformation process, and the accuracy of the material property-related parameters in the constitutive model directly affects the accuracy of finite element simulation. Based on the Hill48 anisotropic yield criterion, combined with the Voce isotropic hardening model and the Armstrong–Frederic nonlinear kinematic hardening model, a mixed hardening constitutive model that considers material anisotropy and the Bauschinger effect was established. Analysis of the tension–compression experiment on the sheet using finite element method. Using the finite element model, the optimum geometry of the tension–compression experiment sample was determined. The cyclic deformation stress–strain curve of the 2A11 aluminum alloy sheet was obtained by a cyclic tensile–compression test, and the material characteristic parameters in the mixed hardening model were accurately determined. The reliability and accuracy of the established constitutive model of anisotropic mixed hardening materials were verified by the finite element simulation and by testing the cyclic tensile–compression problem, the springback problem, and the sheet in bending, unloading, and reverse bending problems. The tensile–compression experiment is an effective method to directly and accurately obtain the characteristic parameters of constitutive model materials.

Polar effective plastic strain - forming limit diagram of DP 440 steel sheet with application to two-step forming test

In this work, experimental forming limit curve (FLC) and polar effective plastic strain -forming limit diagram (PEPS-FLD) are developed in order to predict the formability of dual phase (DP) steel grade 440. The Forming Limit Diagrams (FLD) of the examined steel are first obtained using the Nakajima stretch-forming test with a decreased hemispherical punch diameter of 50 mm in combination with Auto-Grid Vialux strain measurement system. To achieve a PEPS-FLD, an experimental FLC data must be completely projected onto the polar space using the results of FLC, Hill's 48 anisotropic yield criterion, and PEPS algorithm through plasticity calculation. Finally, a two-step forming test is conducted both experiments and simulation to demonstrate the practical applicability of the PEPS-FLD. It really would appear that the PEPS-FLD could predict the formability in non-linear strain path and can be utilized to enhance formability by creating non-linear strain path of DP440 more accurately than FLD.

Some Issues with Statistical Crystal Plasticity Models: Description of the Effects Triggered in FCC Crystals by Loading with Strain-Path Changes.

The justification of the applicability of constitutive models to exploring technological processes requires a detailed analysis of their performance when they are used to describe loadings including the complex loading mode that is characteristic of these processes. This paper considers the effect of equivalent stress overshooting after the strain-path changes known to occur in metals and alloys. The macrophenomenological and multilevel models, which are based on crystal plasticity, account for this effect by applying anisotropic yield criteria at the macro- and mesolevels, respectively. We introduce a two-level constitutive statistical inelastic deformation model (identified for aluminum) that incorporates the popular simple phenomenological anisotropic hardening law for describing the behavior of FCC polycrystals. The results of the numerical simulation are in satisfactory agreement with existing experimental data. Statistical analysis of the motion of a mesostress in the stress space on the crystallite yield surface is performed. The obtained data are compared with the results found using the isotropic hardening law. The results clarify the simulation details of statistical crystal plasticity models under loading with strain-path changes in materials and demonstrate their suitability for describing the processes under consideration.