Evolution Of Yield Surface Research Articles

Study on plastic yield surfaces of pyramidal lattice materials with circular and rectangular cross-sections

The yield surfaces and the evolution of yield surfaces of a bending-dominated pyramidal lattice material under complex stress states considering the coupling effect of axial force and bending moment of a strut are studied theoretically. The two-dimensional primary yield surfaces, secondary yield surfaces, and complete yield surfaces of pyramidal lattice materials of two kinds of cross-section struts are obtained, and the analytical expressions of yield surfaces are established. The effects of the inclination angle and diameter-to-length ratio of the struts on the yield surfaces of pyramidal lattice materials with circular cross-sections are discussed. The results provide a basis for analyzing homogenized plastic properties of complex lattice material structures.

Towards improved description of plastic anisotropy in sheet metals under biaxial loading: A novel generalization of Hill48 yield criterion

Lightweight design and development of automobiles are strongly dependent on lightweight high-strength sheet metals. An accurate prediction of anisotropy and yield surface evolution of these materials under different sampling directions of biaxial tension (BT) is a necessary condition for the numerical analyses of stamping forming. In this work, the Hill48 model considering the principal stress direction is generalized to describe the anisotropic evolution and the yield loci on the normal and diagonal planes. The material parameters of the generalized M-Hill48 model (GM-Hill48) have analytical form and can be calibrated by the yield stresses and r-values expressed by the continuous equivalent plastic strain (EPS). A quadratic function interpolation method based on the near-plane strain (NPS) stress is established to control the yield locus under different plane stress states. In addition, a method for predicting the yield loci in arbitrary sampling directions under BT is proposed by tensor coordinate transformation. Different from the traditional model which can only determine material parameters by using equi-biaxial tension (EBT) or NPS stress along the rolling direction (RD) and transverse direction (TD), the yield stresses of BT in different sampling directions can be incorporated into the generalized model. The predictive capability of the GM-Hill48 model is verified by capturing the continuous evolution of yield stresses and strain for DP590 and AA6016-T4 materials. The convexity condition of the established modified model is given and discussed. Finally, the advantages of new model are further expounded in terms of the prediction accuracy and identified parameters compared with the advanced yield models.

A physically-based mixed hardening model for the prediction of the ductility limits of thin metal sheets using a CPFE approach

An advanced Crystal Plasticity Finite Element (CPFE) approach is developed to accurately predict the ductility limit strains of thin metal sheets. This method uses polycrystalline unit cells to represent the metal sheets at the macroscopic level. The macroscopic behavior of these unit cells is determined based on that of the constituent single crystals using the periodic homogenization multiscale scheme. At the single crystal scale, the constitutive framework follows a finite strain rate-independent formulation, with the flow rule governed by the Schmid law. The evolution of the single crystal yield surface is described through a physically-based mixed hardening model, where isotropic hardening is characterized by a dislocation density-based formulation, while kinematic hardening is described by the nonlinear Armstrong–Frederick model. The unit cell ductility limit strains are predicted by the Rice bifurcation criterion. The reliability of the mixed hardening model in accurately reproducing mechanical behavior is confirmed through simulations of uniaxial tension/compression loading. Then, the developed computational strategy is used to investigate the impact of key microstructural hardening parameters on the initiation of localized necking under linear strain paths. The numerical predictions reveal the significant influence of these parameters on the formability of thin metal sheets. Additionally, the analysis of ductility limits under non-linear strain paths demonstrates a strong dependency of the numerical predictions on strain path changes. The numerical predictions obtained by the developed CPFE multiscale strategy are compared with experimental results from the literature. In summary, the proposed approach provides a reliable tool for accurately predicting the ductility limits of thin metal sheets, offering valuable insights for engineering applications.

A generalized, computationally versatile plasticity model framework - Part I: Theory and verification focusing on tension‒compression asymmetry

Microstructural characteristics and complex loading conditions in the deformation of metallic materials lead to complexity in mechanical responses. In this study, we propose a generalized constitutive framework that reproduces the plastic anisotropy and asymmetry under various loading conditions. Particularly, the developed model can accurately capture distinctive flow stress, plastic flow and strain hardening between tensile and compressive dominant loadings under a wide range of stress states. The model is based on the stress triaxiality dependence of state variable (or weighting factor) newly incorporated in the existing plasticity theory to keep the computational efficiency and versatility. For example, the new generalized framework can be applied to widely employed Hill48, Yld2k-2d, and Poly6 as a class of associated flow rule-based yield functions, as well as Stoughton-Yoon2009 and Min2016 yield functions for the non-associated flow rule. Also, the model is adaptable when selecting the yield function under tension or compression, which efficiently controls the degree of accuracy in anisotropic modeling under tension and compression. The generalized plasticity framework is validated comprehensively by demonstrating the predictive capability for anisotropy in yield stress and plastic flow of metallic materials with different crystal structures. Moreover, the model can efficiently capture the continuous evolution of asymmetric yield surfaces as functions of strain, temperature, and strain rate. Finally, the identification procedure of the model is discussed by demonstrating the analytical determination of model parameters utilizing the experimental or generic material data obtained from various loading conditions such as tension, compression, plane strain loading, and pure shear.

Characterization and modelling of evolving plasticity behaviour up to fracture for FCC and BCC metals

Yield surfaces evolve depending on loading directions (anisotropic hardening) and stress states (differential hardening) during plastic deformation of sheet metals. A new analytical yield function is proposed by coupling an enhanced pDrucker function and SY2009 anisotropic hardening functions to model both the differential and anisotropic hardening behavior. All the parameters are solved analytically for the proposed function so that the proposed function can model the differential hardening from uniaxial compression to equibiaxial tension and anisotropic hardening of uniaxial tension with three different loading directions. Its convexity during evolution is analysed from plasticity onset to ultimate fracture by a geometry-inspired numerical convex analysis approach. Several tests were conducted for an aluminium alloy 2A12-O and a high-strength steel QP1180 including uniaxial tension/compression, simple shear, notched tension and bulge tests. The proposed yield function is applied to describe the plastic behaviour of AA2A12-O and QP1180 steel under the non-associated flow rule. Parameter calibration is conducted at pre-necking and post-necking stages for modelling yield surface evolution more accurately under large deformation especially at shear and equibiaxial tension states. The applications to the two metals show that the proposed function is more suitable for pressure-sensitive face-centered cubic (FCC) and body-centered cubic (BCC) metals compared to the pressure-coupled Yld2000–2d and CQN models since both the anisotropic and differential hardening behaviours are precisely modelled by the proposed yield function. The proposed function could also be utilized as a stress-based fracture criterion to model four stress states fracture from uniaxial compression to equibiaxial tension.

Lode-dependent anisotropic-asymmetric yield function for isotropic and anisotropic hardening of pressure-insensitive materials. Part I: Quadratic function under non-associated flow rule

It is challenging to precisely model the complicated plastic deformation of metals, including anisotropy in strength and plastic deformation, strength differential effect, anisotropic hardening, etc. A new approach is proposed to extend anisotropic yield functions into asymmetric ones by introducing a Lode dependent function. The approach is applied to the Hill48 function with a Lode-dependent function in a form of the normalized third stress invariant. The proposed Lode-dependent anisotropic-asymmetric (LAA) is applied to characterize the anisotropic hardening behaviors under both tension and compression of QP1180, DP980, AA2008 T4 and α-Ti to verify its performance. The convexity of yield surface evolution with plastic deformation is investigated by a geometry-inspired numerical convex analysis method. The application shows that the proposed LAA function precisely characterizes the anisotropy in tension and compression and its evolution with plastic deformation. It is therefore suggested to model the anisotropic-asymmetric plastic behavior with its applications to metal forming.

Effect of the yield surface evolution on the earing defect prediction

Although the prediction of earing in the cup drawing process is considerably related to the yield surface shape, the yield surface evolution is also essential for the final ear form. The bending-unbending issue is a fundamental subject occurring on the die and punch shoulders. Since the yield stress is loading path dependent in reversal loadings, the conventional hardening models used in the monotonic loading conditions bring about inaccurate outcomes for predicting the ultimate earing profile, and a kinematic hardening model should be incorporated into the constitutive equations. This study elucidates the yield surface evolution effect involving expansion and translation simultaneously on the ear formation. A sixth-order polynomial yield function was employed to precisely characterize the yield surface shape, while a combined isotropic-kinematic hardening model was implemented to represent the evolution of the yield surface. The translation of the yield surface position was defined by the Armstrong-Frederic hardening model. Punch force-stroke responses and the ear form profiles were predicted by the implemented plasticity model in Marc using the Hypela2 user subroutine and compared with the experimental results. The combined hardening assumption yielded an increase in the mean cup height when compared to the isotropic hardening assumption. Moreover, The HomPol6 coupled with the combined hardening showed a better agreement with the experimental results.

Ratcheting assessment of additively manufactured alloys through the hardening framework: Analysis and simulation

The present study intends to evaluate the ratcheting response of the additively manufactured (AM) samples made of Al 4043, SS316L, and Ti–6Al–4V alloy through hardening rules and finite element analysis. The hardening framework consisted of Ahmadzadeh‐Varvani (A-V) kinematic hardening rule and an isotropic hardening description by Lee-Zavrel (L-Z). Yield surface evolution was evaluated for (i) AM samples taken from mid and top sections of the 3D printed plate, and (ii) heat-treated samples. The Chaboche’s materials model was employed to assess ratcheting of AM samples through use of finite element method. The choice of hardening models affected yield surface translations over the loading process. The predicted and simulated ratcheting curves were found in close agreements falling below those of measured values.

A robust computational strategy for failure prediction of masonry structures using an improved multi‐surface damage‐plastic based interface model

AbstractIn this study, a new traction‐separation based constitutive model for use in finite element simulation of masonry joints under complex loading conditions is developed for cohesive elements. The proposed model is formulated using damage parameters and plastic deformation with mutual couplings, and can accurately simulate the complex nonlinear behaviors of masonry joints considering hardening or softening of strength and stiffness degradation. To enhance the numerical stability of the model, plasticity and damage are separated algorithmically and implemented in two phases. In the first phase, the plastic deformations are treated using a multi‐surface plasticity model composed of a smooth hyperbolic yield surface for tension‐shear mixed‐mode failure and an elliptical cap primarily for the compressive failure. This is implemented in effective stress space and helps restrict the evolution of yield surfaces with no softening, significantly enhancing the efficiency of stress return mapping by the closed point projection method. In addition, an adaptive sub‐stepping scheme is adopted to further improve the robustness of the numerical implementation. In the second phase, nominal stresses are computed from the effective stresses using damage parameters. The evolution of these damage parameters is defined in terms of plastic work which is defined by a polynomial form, and is recommended in this study for a better calibration capability. Improvements are made in the formulation of compressive cap including incorporation of hardening of strength and stiffness degradations as these are ignored in existing interface models. This approach helped improve simulation of masonry under cyclic loads with tension‐compression transitions. For the structural level applications, the interface model is implemented within a finite element program, which is utilized to simulate failure of a number of masonry specimens under in‐plane/out‐of‐plane monotonic/cyclic loading. The simulated results are rigorously validated with existing experimental data that shows a good potential in modeling masonry structures.

The role of accumulated plasticity on yield surface evolution in pearlitic steel

In this paper, we analyze one specific assumption used in many cyclic plasticity models: The dependence of the yield surface evolution on the accumulated plasticity, i.e., the time-integrated plastic multiplier. We measure this scalar variable in carefully designed experiments and track the yield surface evolution. Our results show that this variable is insufficient to describe the isotropic or distortional hardening of the investigated steel, invalidating the commonly used assumption. The results highlight requirements for developing and verifying new isotropic and distortional hardening laws.

Multi-scale modelling of evolving plastic anisotropy during Al-alloy sheet forming

Accurate modelling of local evolving texture and yield surface evolution in complex strain path are of crucial importance to the accurate sheet metal forming simulations. This work proposed a novel multi-scale computational scheme, in which the full-field crystal plasticity (CP) modelling is mapped and real-time updated into each integration point following the local strain path in the finite element (FE) model. By the real-time close-loop feedback and update between the FE model and CP model, the evolving texture and its induced anisotropic plasticity evolution are considered utilizing on-the-fly virtual tests in the full deformation field. To carefully verify the developed computational framework, a comprehensive validation including plastic anisotropy characterization, cyclic hardening tests and deep drawing tests is conducted in an AA6016-T4 aluminium sheet with strong cube texture using ESAFORM Benchmark 2021. By comparison with experimental results and the conventional approaches with constant yield surface for the forming case, the newly developed method is thoroughly validated, showing that the significant texture evolution is observed in large plastic deformation and various strain paths lead to different texture evolution. The local evolving texture induced plastic anisotropy evolution has a considerable effect on the prediction accuracy of the deep drawing simulations, such as earing profile, wall thickness and loading history. Additionally, this method does not increase too much computation cost, enabling cost-effectively simulation of complex forming processes.

Finite element analysis on yield surface evolution of cellular materials

The yield surface is the boundary of elastic region in the theory of plasticity, therefore the area and the shape of the yield surface can represent the capability of cellular materials in the stress space. The yield point and the yield surface of one kind of 2D cellular materials, hexagonal honeycombs, is investigated in this study via the finite element analysis with a representative block approach. All components of requisites to computationally investigate yield surface evolution of 2D cellular materials are prepared. A proper determination of yield point for the honeycombs is proposed for the detection of initial and subsequent yield surfaces in the axial-shear stress space and the yield point determination is validated by the comparison between experimental result and finite element analysis. Taking into account a wide range of relative density for honeycombs, the mesh size as well as the pattern and the size of the representative block in finite element is investigated and selected based on the analysis of the mesh convergence analysis, the boundary effect, and the size effect. After probing paths and preloading paths are designed, the initial and subsequent yield surfaces of the honeycombs with different relative density are detected and the influence of relative density on the yield surface evolution is investigated. The investigation shows a shear preloading path followed by an axial preloading path strengthens the capability of honeycombs and the honeycomb with lower relative density may exhibit different behavior of yield surface evolution than those with higher relative density. Further, phenomena of cellular materials including the Bauschinger effect and the hardening behavior (isotropic, kinematic, rotation, distortional) are observed from the yield surface evolution of honeycombs.

Characterization of the asymmetric evolving yield and flow of 6016-T4 aluminum alloy and DP490 steel

6016-T4 aluminum alloy and DP490 steel were systematically tested under 24 proportional loading paths, including uniaxial tensile tests with a 15° increment, uniaxial compressive and simple shear tests with a 45° increment, and biaxial tensile tests using cruciform specimens. Cruciform specimens in the rolling/transverse and 45°/135° sampling directions were tested with seven and four different stress ratios, respectively. The normal and diagonal planes plastic work contours and the yield stresses under uniaxial tension and compression were measured to investigate the anisotropic yield. Meanwhile, the normal and diagonal planes directions of plastic strain rate and the rα-values under uniaxial tension and compression were characterized to confirm the plastic flow. Several existing asymmetric yield criteria under the associated and non-associated flow rules were comprehensively evaluated to describe the asymmetric plastic anisotropy of 6016-T4 aluminum alloy and DP490 steel. The results suggest that in the investigated yield criteria, the non-associated models can predict the tension and compression asymmetry of materials more accurately than the associated models, and the function of stress triaxiality can more effectively describe the asymmetric yield behavior than the first stress invariant. In addition, the pure shear stress states are helpful in assessing the validity and applicability of advanced asymmetric yield stress functions, and the inspection of diagonal plane plastic work contours containing more pure shear stress states should prioritized over that of normal plane plastic work contours. The evaluation of plastic potential functions should not only consider the prediction accuracy of the normal plane directions of plastic strain rate, but also further check the diagonal plane directions of plastic strain rate. Expressing mechanical properties as a function of equivalent plastic strain to calibrate parameters of the yield criterion allows the continuous capture of anisotropic evolution of the asymmetric yield surface and the changes in the asymmetric plastic potential surface.

Modeling stress anisotropy, strength differential, and anisotropic hardening by coupling quadratic and stress-invariant-based yield functions under non-associated flow rule

An accurate description of yield surface is essential for the high-fidelity numerical analyses of sheet metal forming processes. In this work, a simple coupling of anisotropic quadratic and isotropic stress-invariant-based yield functions is proposed under the non-associated flow rule. The quadratic part is the asymmetric Hill48 function, which describes the anisotropic hardening with explicitly calibrated parameters from stress-strain curves. A new stress-invariant-based yield function, as a second component, is developed to capture the shapes of yield surface for diverse metallic materials with enhanced flexibility. Finally, a yield function is established by multiplying the anisotropic quadratic and the isotropic stress-invariant-based parts, which satisfies the convexity requirement in terms of stress state. The proposed asymmetric anisotropic yield function is validated with automotive sheet metals including dual-phase steels, aluminum alloys and magnesium alloys. Results show that the new yield criterion can accurately predict the anisotropy in stress, strength differential effect and yield surface evolution (or anisotropic hardening) of the investigated sheet metals.

Experimental Study of Yield Surface in Polypropylene Considering Rate Effect: Stress Path Dependence.

In order to investigate the yield surface evolution of polypropylene (PP) under dynamic impact and the relationship between yield surface parameters and the strain rate, five shear-compression specimens (SCSs) with different inclination angles are designed and produced to explore the yield behavior of PP under dynamic loading. Dynamic combined stress loading paths with different compression-shear ratios are achieved by the split Hopkinson pressure bar (SHPB). The evolution laws of the compressive stress and shear stress in the measurement region during the PP SCS compressive deformation process are analyzed. In terms of mechanical response, PP under combined compression-shear loading is of visco-elasticity plasticity and its deformation undergoes a three-stage transition, namely “unyield→yield→failure”. The yield characteristics of PP are found to be affected not only by the hydrostatic pressure but also by the stress path. According to the Hu–Pae yield criterion, the dynamic yield surface and model parameters of PP are obtained, and the relationship between the yield surface and the strain rate is ascertained. These findings contribute to deepening the research on the mechanical response characteristics of PP-based materials.

Characterization of Yield Surface Evolution of AZ31 from Shear to Equibiaxial Tension

This research characterized the strain hardening behavior of AZ31 under different stress states from shear to balanced biaxial tension with a newly proposed yield function. Experiments are conducted for AZ31 magnesium alloy by in-plane shear specimens, dogbone specimens, notched specimens and bulging specimens to characterize the flow behavior under different stress states. The flow behaviors are characterized by a newly proposed yield function in a form of the three stress invariants. The proposed yield function is implemented into ABAQUS/Explicit to predict the plastic response of the alloy under different stress states. It is shown that the proposed yield function can precisely predict the distinct flow behaviors and reaction forces from shear to equibiaxial tension from the initial yielding to fracture.

Yield Surface Evolution and Elastoplastic Model with Cubic Distortional Yield Surface

Yield Surface Evolution and Elastoplastic Model with Cubic Distortional Yield Surface

Evolution Prediction of Hysteresis Behavior of Sand under Cyclic Loading

Soil cyclic degradation is a serious issue that should be considered in engineering design and maintenance. The hysteretic response causes strength degradation and excessive settlement of soil structure in engineered and natural geosystems. Hysteresis is essentially the coupling deformation of elastic and plastic components during reloading and unloading processes. Conventional hysteretic models for sand in the elastoplastic framework rely highly on yield surface or potential surface evolution and fall short on complexity and inaccuracy. This study proposes a decoupling method to describe the hysteretic response of sand. In contrast to the conventional elastoplastic approach, this decoupling method can directly decouple the elastic and plastic components by determining the boundary between the elastic strain extension domain and the plastic strain extension domain for each stress cycle. In this way, the elastic and plastic stress–strain branches during cyclic loading can be separately obtained, and the corresponding elastic and plastic parameters are employed to characterize mechanical behavior. With the respective evolution of elastic and plastic strains, the hysteretic behavior of sand is reproduced by combining all the branches. Finally, this decoupling method is validated by three conventional cyclic loading tests. The predictions are consistent with the experiments, indicating that the decoupling method is generally effective in describing the hysteretic behavior under cyclic loading. This decoupling method provides new insight to obtain elastic and plastic deformation behaviors separately, without recourse to complicated plastic surface and hardening law.

Experimental and numerical investigation of differential hardening of cold rolled steel sheet under non-proportional loading using biaxial tensile test

Strain path change is common phenomenon during industrial sheet metal forming processes. These strain paths can be proportional and non-proportional in nature. To enable strain path change in a single experiment is key challenge faced by many researchers. To predict deformation behavior accurately during such strain paths, it is required to represent initial yielding, the evolution of subsequent yield surface and hardening behavior precisely. This work deals with the mechanical characterization of Interstitial Free High Strength Steel (IFHS) during multi-axial proportional and non-proportional loading for the determination of yield surface evolution and strain hardening behavior. A set of tensile tests at various orientations with respect to the rolling direction and in-plane biaxial tests using cruciform specimens for various linear and bi-linear stress ratios were performed on as-received materials. It was found that at the onset of stress path change, the stress strain curve shows a change in the hardening rate for rolling and transverse directions of the biaxial specimen due to the change in active slip systems. To predict this behaviour various yield criterion like von Mises, Hill's 48 and Yld2000-2d were used. The experimental results show that the mechanical behavior of the subjected specimen deviates from isotropic behavior predicted by the von Mises and anisotropic behavior predicted by the Hill's 48 yield criterion. The initial yield loci measured by biaxial testing machine is in good agreement with what is predicted by Yld2000-2d yield criterion. However, it was observed that such phenomenological models were unable to capture the observed deformation behaviour. It is also shown that classical Crystal Plasticity Finite Element Method (CPFEM) can capture change in strain hardening rate during non-proportional deformation for few cases reasonably well while in few cases it could not predict.

Effects of Temperature on the Evolution of Yield Surface and Stress Asymmetry in A356-T7 Cast Aluminium Alloy.

As the electrification of vehicle powertrains takes prominence to meet stringent emission norms, parts of internal combustion engines like cylinder heads are subjected to an increased number of thermal load cycles. The cost-effective design of such structures subjected to cyclic thermo-mechanical loads relies on the development of accurate material models capable of describing the continuum deformation behaviour of the material. This study investigates the effect of temperature on the evolution of flow stress under cyclic loading in A356-T7 + 0.5% Cu cast aluminium alloy commonly used in modern internal combustion engine cylinder heads. The material exhibits peak stress and flow stress asymmetry with the stress response and flow stress of the material under compressive loading higher than under tension. This peak and flow stress asymmetry decrease with an increase in temperature. To compare this stress asymmetry against conventional steel, cyclic strain-controlled fatigue tests are run on fully pearlitic R260 railway steel material. To study the effect of mean strain on the cyclic mean stress evolution and fatigue behaviour of the alloy, tests with tensile and compressive mean strains of +0.2% and −0.2% are compared against fully reversed (Rε = −1) strain-controlled tests. The material exhibits greater stress asymmetry between the peak tensile and peak compressive stresses for the strain-controlled tests with a compressive mean strain than the tests with an identical magnitude tensile mean strain. The material exhibits mean stress relaxation at all temperatures. Reduced durability of the material is observed for the tests with tensile mean strains at lower test temperatures of up to 150 °C. The tensile mean strains at elevated temperatures do not exhibit such a detrimental effect on the endurance limit of the material.