Quadratic Yield Function Research Articles

Strict convexity of yield surfaces of some weakly-textured materials

Let Sym0 be the space of traceless symmetric second-order tensors. We say that a polycrystalline elastic–plastic material is weakly-textured if its yield function f:Sym0→R is the sum of a texture-independent isotropic part fiso and an anisotropic part which is linear in the relevant texture coefficients. Let c>0 and S≔f−1(c)⊂Sym0 be the yield surface of the weakly-textured material in question. We present a sufficient condition (*), namely that ∇2f(S) be positive definite for each S∈S, for a smooth yield surface S to be strictly convex in Sym0. We apply this sufficient condition to weakly-textured materials with yield functions that satisfy the following conditions: (i) the yield functions f and fiso are smooth; (ii) ∇2fiso(S) is positive definite for each S in Siso≔fiso−1(c)⊂Sym0. We prove that the yield surface S⊂Sym0 of such weakly-textured material is strictly convex if the texture coefficients in f are sufficiently small. As illustration for practical applications, by appealing to condition (*) we study the strict convexity of the yield surface pertaining to a weakly-textured orthorhombic aggregate of cubic crystallites which has a quadratic yield function of the type proposed by Hill in 1948. Moreover, we show that all 35 samples of cold-rolled and annealed low-carbon steel sheets studied by Stickels and Mould have their quadratic yield functions and corresponding yield surfaces strictly convex.

Ductile fracture of anisotropic QP980 steel sheet by using the isotropic/anisotropic modified Mohr-Coulomb models

The modified isotropic Mohr-Coulomb (iso-MMC) fracture model, which utilizes the Mises yield function and isotropic hardening law, was calibrated through tensile tests on various specimens. The resulting calibrated model was constructed in (η, L, ε¯f) space to represent the isotropic fracture locus of QP980 steel sheet. Subsequently, this calibrated model was utilized to predict the ductile fracture behavior of QP980 steel sheet loaded along rolling direction under different stress states. To comprehensively consider the impact of anisotropic properties on fracture evolution, an anisotropic MMC (aniso-MMC) ductile fracture model was developed in terms of strain space. Under the assumption of plane stress, the Mohr-Coulomb (MC) criterion was converted from stress space to strain space through an anisotropic constitutive model in order to characterize ductile fracture in QP980 steel sheet. The quadratic yield function of Hill's 48 yield function with the isotropic hardening law was employed to characterize the plastic behavior during loading. The anisotropic fracture loci of QP980 steel sheet were obtained in various loading directions. The scanning electron microscope (SEM) was utilized to characterize the microscopic morphology of the fracture surfaces of QP980 steel sheet, investigating the fracture mechanism of anisotropic ductility under different stress states. The results of both tensile experiments and simulations have confirmed that the plastic anisotropy of materials plays a critical role in determining ductile fracture behavior. The aniso-MMC fracture model was ultimately validated through hole expansion tests and equi-biaxial tensile tests, demonstrating its capability to predict ductile fracture behavior under both proportional and non-proportional loading conditions.

A fabric tensor based small strain constitutive law for the elastoplastic behavior of snow

The mechanical behavior of snow depends on its density and microstructure. The anisotropy in the microstructure is expressed in terms of fabric tensor that leads to an anisotropic stress-strain relation. Lately, fabric-based relations have successfully estimated the elastic properties of snow. Motivated by this, we propose a fabric-based macroscopic elasto-plastic constitutive law for snow, which can be used to study avalanche initiation. Mean Intercept Length tensor used as a measure of material fabric is determined by X-ray tomography. Fabric tensor and density-dependent yield surface with a provision for isotropic hardening/softening are used in this process. Beyond the initial yield, the yield function grows till the strength of the snow is reached and then softens. Since snow exhibits tension and compression behavior asymmetry, a piece-wise quadratic yield function is used. Stress-strain curves needed for determining the equation of the hardening/softening law and other parameters of the proposed macroscopic constitutive law are obtained through micro-Finite Element (μ-FE) simulations. The macroscopic constitutive law has been implemented as a user subroutine in a FE code and can predict the snow's multiaxial behavior.

Effect of Plastic Anisotropy on Ironing Formability of 3104 Aluminum Alloy Hard Sheets

3104 aluminum alloy hard sheets are widely used for aluminum beverage can body stock (CBS) mainly because of their high ironing formability. The CBS has large anisotropy of r-value (Lankford value) and n-value (work hardening exponent), however, the effect of such anisotropy on the ironing formability remains unclear. In this study, cup and DI forming behavior was investigated using cold-rolled 3104 aluminum alloy hard sheets with different anisotropy of r and n-values. Magnitude of the anisotropy changes during drawing and ironing. Ironing fracture tests reveal that fracture occurs at angles between 10 and 20° in the can circumferential direction. From the strain distribution measurement of the can wall, it is found that the axial strain of the can wall at the fracture angle is larger. A theoretical analysis method of ironing stress introducing r and n-value anisotropy is devised based on the Hill’s anisotropic plasticity theory (quadratic yield function). The devised analysis reveals that as the anisotropy of r-value in the can wall becomes smaller, (1) the fracture is less likely occur, (2) the margin of stress which calculated from the ratio of the can axial tensile stress to fracture stress is larger.

On the effect of anisotropy on the performance and simulation of shrinking tubes used as energy absorbers for railway vehicles

The standard BS EN 15227 requires accurate numerical modelling of railway vehicle energy absorbers that must be correlated against experimental data. Although thin-walled tubes can exhibit anisotropy, such numerical models have traditionally included isotropic material properties. Thus, this work investigates whether anisotropic material models may increase the accuracy of numerical models of shrinking tube energy absorbers.Tensile testing of extruded AW-6082 aluminium alloy shrinking tubes showed the yield strength of the tubes was 10% lower in the hoop direction than in the longitudinal direction. Further, to assess the effect of incorporating anisotropic material behaviour in the numerical model, the tubes were compressed under quasi-static conditions.Numerical models of the shrinking tubes, including isotropic (von Mises yield function) and anisotropic (Hill’s quadratic yield function) material properties, were compared to the experimental data. The isotropic numerical models overestimated the steady-state reaction force, even without the inclusion of friction, indicating that such models do not fulfil the requirements of the standard. Conversely, incorporating anisotropic material models predicted a lower reaction force and enabled the inclusion of energy dissipation by friction, by means of a coefficient of friction μ ​= ​0.03. Although these results demonstrate the need to include anisotropy in the numerical simulation, the friction value was lower than expected due to the methodology of the material characterization and the accuracy of the anisotropic model implemented.

On ductile fracture of 316L stainless steels at room and cryogenic temperature level: An engineering approach to determine material parameters

Two austenitic 316L stainless steels from different provider are tested mechanically. Uniaxial tensile experiment serves the basis to determine material parameters for elastic-plastic response at different temperature levels considering from room (23 ∘ C) to cryogenic (-163 ∘ C) temperatures. The typical plastic strain induced kinetic transformation of martensite content becomes appreciable as temperature decreases yielding an apparent improvement in material strength with its irreversible loss in ductility. The evolution of martensitic content inside the austenitic matrix would increase the macroscopic strain hardening significantly and thus delaying localization by preventing through thickness necking. The present work summarizes the experimental/computational program performed on metastable austenitic 316L stainless steel under a number of different tensile stress states at low temperatures. The results indicate that at room temperature the martensite kinetic transformation is negligible and it is surpassed by slip mechanism of austenite grain matrix and a simple isotropic hardening law is sufficient to describe plastic flow. On the other hand, at low temperatures the phase transformation process of FCC γ -austenite into BCC α ’-martensite is triggered leading to a significant material strength enhancement, but with subsequent loss of ductility. A suitable constitutive model with pressure-sensitive quadratic yield function and temperature dependent hardening law that relates the increasing of yield strength to the thermal martensitic transformation kinetic process is proposed. Likewise, the Modified Mohr-Coulomb (MMC) model is then extended to incorporate thermal effects on ductile fracture initiation at different temperature levels, stress state and loading conditions.

Full Three-Dimensional Cavitation Instabilities Using a Non-Quadratic Anisotropic Yield Function

Abstract Full three-dimensional cell models containing a small cavity are used to study the effect of plastic anisotropy on cavitation instabilities. Predictions for the Barlat-91 model (Barlat et al., 1991, “A Six-Component Yield Function for Anisotropic Materials,” Int. J. Plast. 7, 693–712), with a non-quadratic anisotropic yield function, are compared with previous results for the classical anisotropic Hill-48 quadratic yield function (Hill, 1948, “A Theory of the Yielding and Plastic Flow of a Anisotropic Metals,” Proc. R. Soc. Lond. A193, 281–297). The critical stress, at which the stored elastic energy will drive the cavity growth, is strongly affected by the anisotropy as compared with isotropic plasticity, but does not show much difference between the two models of anisotropy. While a cavity tends to remain nearly spherical during a cavitation instability in isotropic plasticity, the cavity shapes in an anisotropic material develop toward near-spheroidal elongated shapes, which differ for different values of the coefficients defining the anisotropy. The shapes found for the Barlat-91 model, with a non-quadratic anisotropic yield function, differ noticeably from the shapes found for the quadratic Hill-48 yield function. Computations are included for a high value of the exponent in the Barlat-91 model, where this model represents a Tresca-like yield surface with rounded corners.

Mechanical Properties and Formability of Heat-Treated 7000-Series High-Strength Aluminum Alloy: Experiments and Finite Element Modeling

The feasibility of room-temperature (cold) forming of peak-aged 7075 high-strength aluminum alloy (7075-T6) sheets after solution heat treatment followed by water quenching was investigated. The anisotropic plastic properties of the heat-treated aluminum sheet (7075-WT) were experimentally characterized by performing uniaxial tensile tests (along different loading directions) and a balanced biaxial tension test. Limit dome height (LDH) and cylindrical cup drawing experiments were conducted to evaluate the formability at room temperature. With the obtained anisotropic mechanical properties, different yield functions, including Yld2000-2d, Hill 1948, and von-Mises, were determined and implemented in the finite element simulations to numerically predict the formability at room temperature. The formability of the 7075-WT sheet at room temperature could be significantly enhanced owing to the prior heat treatment, and the finite element simulations performed using the anisotropic yield functions and Voce hardening law were in good agreement with the measured LDH. The LDH of the 7075-WT sheet in the plane strain and biaxial stress modes and the earing in the circular cup drawing tests could be better predicted non-quadratic Yld2000-2d function than those by isotropic and Hill 1948 quadratic yield functions.

Forming-limit strains of perforated steel sheets under biaxial stress state

The forming-limit strains of cold-rolled steel sheets perforated with regularly arranged round-, square-, and cross-shaped holes were experimentally and theoretically estimated. Two types of hole arrangements, i.e. square and triangular patterns, were considered for each hole shape and the experimental forming-limit strains were determined using the hemispherical dome test (also known as Nakazima test). The theoretical forming-limit curves were computed using the finite element method with Hill's quadratic yield function and a plastic instability criterion determined from the external force power. By comparing the experimental results and theoretical calculations under proportional loading, the formability prediction performance of the theoretical approach was evaluated. It was found that although the approach is not applicable for the triangular array of cross-shaped holes, it can achieve partially acceptable prediction for the other cases.

Dynamic perforation of ultra-hard high-strength armor steel: Impact experiments and modeling

The dynamic perforation of 3 mm thick target plates extracted from ultra-high strength steel (Mars® 300) is investigated experimentally and computationally. Projectiles of 8 mm diameter are also extracted from Mars® 300 plates featuring blunt, conical and hemispherical tips. Using a single stage gas gun, impact experiments are performed with projectile velocities of up to 380 m/s. A minimum impact velocity of about 240 m/s (ballistic limit) was required to perforate the armor steel targets with conical and hemispherical projectiles of a mass of about 14 g. In all successful perforation experiments, the plate targets failed through plugging. At the same time, the conical and blunt projectiles are heavily deformed in a way that their final tip shape resembles that of the hemispherical projectiles. The latter turned out to be slightly more efficient than conical ones in the sense that their exit velocities are approximately 10% higher. Numerical simulations are performed of all impact experiments using the finite element software LS-DYNA. The plasticity model made use of a quadratic yield function with non-associated flow rule, a Swift-Voce strain hardening law and Johnson–Cook type of multipliers accounting for the effects of strain rate and temperature. The stress-triaxiality, Lode angle parameter and strain-rate dependent Hosford-Coulomb fracture initiation model is employed to predict the ductile failure of the Mars 300 steel. Overall, the simulation results are in good agreement with the experimental observations, including the mode of perforation (shear plugging) and the projectile exit velocities. The numerical simulations also show that a ring-like band of localized plastic deformation forms inside the targets with temperatures of up to 700 °C. Fracture initiates from the back face of the plate under biaxial tension, while the cracks propagate through the band of localized plastic deformation towards the impacted front face of the plate, thereby forming a plug as the projectile passes through the target.

Effect of Anisotropic Yield Functions on the Accuracy of Material Flow and its Experimental Verification

This paper evaluates the performance of anisotropic yield functions based on mathematical methods which consider the prediction accuracy of yield stresses and R-values. The anisotropic yield functions being evaluated include one quadratic yield function and six non-quadratic ones. These yield functions are applied to describe the anisotropy of steel sheets and aluminum alloy sheets to evaluate their predictability. The root-mean-square errors (RMSEs) are computed to quantitatively assess their performance. The computation results of RMSEs demonstrate that the Yld2004-18p yield function exhibits the highest accuracy but requires extensive tests to calculate anisotropic parameters. The Yld2000-2d and BBC2000 yield functions are the same and thus have the same prediction accuracy. The application to cylindrical cup drawing shows that, associated with the Yld2004-18p and Yld2000-2d yield functions, the finite element (FE) simulations of cup drawing process can predict cups with six or eight ears. Considering both efficiency and accuracy, the Yld2000-2d and BBC2000 yield functions, which have less anisotropic parameters to be calculated, are recommended for metals with intermediate anisotropy.

Second-Order Cone Programming Approach to Design of Linkage Mechanisms With Arbitrarily Inclined Hinges

An optimization approach is presented for generating linkage mechanisms consisting of frame members with arbitrarily inclined hinges. A second-order cone programming (SOCP) problem is solved to obtain the locations and directions of hinges of an infinitesimal mechanism. It is shown that the primal and dual SOCP problems correspond to the plastic limit analysis problems based on the lower-bound and upper-bound theorems, respectively, with quadratic yield functions. Constraints on displacement components are added to the dual problem, if a desirable deformation is not obtained. A finite mechanism is generated by carrying out geometrically nonlinear analysis and, if necessary, adding hinges and removing members. Effectiveness of the proposed method is demonstrated through examples of two- and three-dimensional mechanisms.

Modeling of plasticity and fracture behavior of X65 steels: seam weld and seamless pipes

A non-associated/associated flow rule coupled with an anisotropic/isotropic quadratic yield function is presented to describe the mechanical responses of two distinct X65 pipeline steels. The first as a product of the cold-rolling forming (UOE) process also known as seam weld pipes and the second as a result of high temperature piercing process called seamless tube manufacturing. The experimental settings consist of a wide range of sample types, whose geometric characteristics represent different state of stresses and loading modes. For low to intermediate stress triaxiality levels, flat specimens are extracted at different material orientations along with notched round bar samples for high stress triaxialities. The results indicate that despite the existing differences in plasticity between materials due to anisotropy induced processes, material failure can be characterized by an isotropic weighting function based on the modified Mohr–Coulomb (MMC) criterion. The non-associated flow rule allows for inclusion of strain directional dependence in the definition of equivalent plastic strain by means of scalar anisotropy (Lankford) coefficients and thus keeping the original capabilities of the MMC model.

Generalisation of Hill’s Yield Function for Planar Plastic Anisotropy

In this paper we present a new generalised quadratic yield function for plane stress analysis [7] for the description of the plastic anisotropy of metals and also for the study of the asymmetric behaviour in tension-compression typical of the Hexagonal Closed-Pack (HCP) materials. The new yield function has a quadratic form in the stress tensor and it simultaneously predicts accurately the r-values and directional flow stresses. It also describes very well the biaxial symmetric stress state which is fundamental for the accurate modelling of aluminium alloys. The new quadratic yield function represents the non-symmetric biaxial stress state by performing a linear interpolation from pure uni-axial loading to a biaxial symmetric stress state. The most advanced yield functions for plastic planar anisotropy characterise very well the biaxial symmetric stress region by using experimental flow stresses for the symmetric bi-axial stress state. However, the behaviour of the alloy between the uni-axial stress state and the symmetric bi-axial stress state is still not very well characterised. In this new yield function that behaviour can be assessed from interpolation from the uni-axial stress state into the symmetric bi-axial stress state until experimental yield locus fitting is achieved within a reasonable tolerance in that region. The main advantages of this new yield function is that it can be used for the modelling of metals with any crystallographic structure (BCC, FCC or HCP), it only has five anisotropic coefficients and also that it is a simple quadratic yield criterion that is able to accurately reproduce the plastic anisotropy of metals whilst using an associated flow rule. In the results and discussion we validate the yield locus for FCC and HCP alloys and we apply the new yield function in a cup drawing simulation for the assessment of the cup earing profile.

Implementation of anisotropic yield functions into the subroutine library “UMMDp”

To implement new yield functions into the user subroutines of advanced FE codes, professional skills and knowledge are required. The Japan Association for Nonlinear Computer Aided Engineering (JANCAE) has been acting as an organization that provides information and knowledge on finite element analysis (FEA) to the researchers and engineers in various positions involved in CAE. Since 2009, as one of the subcommittee activities for applying FEA to practical metal-forming problems, we have not only studied elastoplastic theory, but have also developed the Unified Material Model Driver for Plasticity (UMMDp), a user subroutine library. Basic yield functions, such as the von Mises and Hill’s quadratic yield functions, which almost all advanced FE code already has, were implemented in the UMMDp as a basic study at its beginning of the development. Up to now, many anisotropic yield functions, including Yoshida’s 6th-order polynomial, Barlat’s Yld2000 and Yld2004, Comsa and Banabic’s BBC2008, Cazacu’s CPB2006, and Vegter’s spline yield functions, have been implemented in the UMMDp, and it is possible to add a new yield function. In this presentation, the techniques for implementing yield functions in the UMMDp are explained. Moreover, the problem of BM1 in NUMISHEET 2016 is discussed as an example applying selected yield functions to the implementation procedure.

On the Rate-dependent Plasticity Modelling of Unidirectional Fibre-reinforced Polymeric Matrix Composites

Three different approaches to plasticity are investigated to model the experimentally-observed non-linear behaviour of unidirectional fibre-reinforced polymeric matrix materials. The first and simplest approach consists on assuming independent one-dimensional rate-dependent plasticity on in-plane (12) and through-thickness longitudinal (13) shear components of the Cauchy stress tensor. The second, employs a 3D extension of the plane stress Hill’48 anisotropic plastic surface. The third and the last is formulated as a quadratic yield function inspired by Puck’s fracture initiation criterion. It searches for a plastic localisation plane in which a certain combination of normal and shear stresses is maximum. Numerical simulations are conducted to analyse the off-axis compression behaviour of carbon fibre reinforced epoxy composite under varying loading rates. The afore-mentioned three different approaches are explored with an aim to predict the experimentally-observed non-linear response of such composites. The model parameters are determined using a deterministic inverse modelling strategy employing an iterative domain reduction optimisation technique. As far as the experiments are concerned, the quasi-static and medium rate tests were carried out in universal testing machines, while the experiments at high rate were conducted in a Split Hopkinson Pressure Bar system. The effectiveness in terms of accuracy and robustness of the three approaches are discussed.

A quadratic yield function with multi-involved-yield surfaces describing anisotropic behaviors of sheet metals under tension/compression

A quadratic yield function which can describe the anisotropic behaviors of sheet metals with tension/compression symmetry and asymmetry is proposed. Five mechanical properties are adopted to determine the coefficients of each part of the yield function. For particular cases, the proposed yield function can be simplified to Mises or Hill's quadratic yield function. The anisotropic mechanical properties are expressed by defining an angle between the current normalized principal stress space and the reference direction with the assumption of orthotropic anisotropy. The accuracy of the proposed yield function in describing the anisotropy under tension and compression is demonstrated.

Comparative evaluation of non-associated quadratic and associated quartic plasticity models for orthotropic sheet metals

A non-associated quadratic model and an associated quartic model are comparatively evaluated in this study for their description of anisotropic yielding and plastic flow of 21 representative sheet metals. The quartic yield stress function used in the associated model is a particularly reduced version of Gotoh’s fourth-order polynomial that has only seven independent material constants. The same experimental inputs of four yield stresses and three plastic strain ratios of a sheet metal are used via simple algebraic relations to calibrate the seven material constants in each model. The modeling results of these selected sheet metals show that both models give nearly identical performance for sheet metal plasticity under uniaxial and equal biaxial tension and similar predictions under other plane stress states. The quadratic yield and flow stress functions are shown however to have a larger domain of admissible yield stress and plastic strain ratios for the case of planar isotropy and it is much easier to verify their strict convexity conditions in general. Once it has been calibrated for each sheet metal considered in this study, Gotoh’s yield function is verified to be convex via a numerical minimization approach. A procedure is suggested to obtain an approximate but convex Gotoh’s yield function in case an as-calibrated Gotoh’s yield function is found to be non-convex. A reduced but convex 3D fourth-order polynomial yield function based on a convex Gotoh’s 2D yield function is also given that matches the total number of independent material constants of a 3D non-quadratic model as well. The 3D quartic rate-independent associated plasticity model may thus be readily implemented in a finite element analysis code for simulating 3D sheet metal forming processes.

A generalisation of the Hill's quadratic yield function for planar plastic anisotropy to consider loading direction

In this work, a new generalised quadratic yield function for plane stress analysis that is able to describe the plastic anisotropy of metals and also the asymmetric behaviour in tension-compression typical of the Hexagonal Closed-Pack (HCP) materials, is developed. The new yield function has a quadratic form in the stress tensor and it simultaneously predicts the r-values and directional flow stresses, which is shown to agree very well with experimental results. It also accurately describes the biaxial symmetric stress state which is fundamental for the accurate modelling of aluminium alloys. The new quadratic yield function represents the non-symmetric biaxial stress state by performing a linear interpolation from pure uniaxial loading to a biaxial symmetric stress state. The main advantages of this new yield function is that it can be used for the modelling of metals with any crystallographic structure (BCC, FCC or HCP), it only has five anisotropic coefficients and also that it is a simple quadratic yield criterion that is able to accurately reproduce the plastic anisotropy of metals whilst using an associated flow rule.

Compressive failure mechanism and strength of unidirectional thermoplastic composites based on modified kink band model

A modified kink band model for compressive failure in fiber direction of fiber reinforced composite polymers was investigated and established in this study. Practically modifying the previous kink band model by logically applying a quadratic yield function combining transverse tensile and shear stress to the kinking instability, the modified model was able to determine the compressive strength along with the kink band failure angle theoretically. Especially in case of unidirectional carbon fiber reinforced thermoplastic composites, the compressive test procedure the authors have developed and its results revealed that the kink band failed in out-of-plane direction with each different inclined angle depending on temperature. From the correlation of the observed kink band failure angle and the evaluated compressive strength affected by temperature condition, the modified kink band failure formula was validated.