Subsequent Yield Surfaces Research Articles

On thermal and strain-rate dependences of polymethacrylimide (PMI) foam materials

Polymethacrylimide (PMI) foams exhibit great potentials in engineering applications thanks to its outstanding lightweight and superior mechanical properties. However, the temperature and loading rate can greatly affect its mechanical behaviors, which are the key factors in analysis and design of the PMI foam structures for practical applications. Unfortunately, there is lack of fundamental data and constitutive models available concerning the elastic-plastic properties of the PMI foam in various stress states at different ambient temperatures and different strain rates. Therefore, this study aims to fill this knowledge gap by generating basic experimental data and providing a generic modeling solution for PMI foam materials and structures. To better reflect the actual service conditions in practice, different stress states are considered under different temperatures and strain rates. The cyclic loads are applied to characterize the actual mechanical responses of the PMI foam where the initial and subsequent yield surfaces were calibrated and analyzed. Following the experimental results, the elasticity, plasticity, viscosity and damage of the PMI foam are investigated thoroughly, especially for the initial and subsequent yielding behaviors. Further, the thermal and strain rate dependent constitutive models are established and calibrated based on the in-house experimental data for finite element (FE) modeling. A novel damage model is proposed to replicate the phenomenon of the stiffness degradation in compression. It is found that accurate predictions on the mechanical responses of PMI foam can be achieved under different temperatures and different strain rates. This study is anticipated to provide fundamental experimental data and effective constitutive models of PMI foams for real engineering applications, such as lightweight design of sandwich structures under some harsh conditions.

A new plastic flow theoretical model and verification for non-dense metals

The mechanical behavior and constitutive equations of isotropic non-dense metals, such as metal foams, porous metals, and lattice metals, have been extensively studied, but the subsequent yield surfaces depicted by different theoretical models are somewhat controversial and have not been fully validated in the whole permissible loading space. Based on two accepted assumptions for isotropic non-dense metals, we proposed a new plastic flow theoretical model. In order to verify its rationality, we established two mesoscopic models with different initial relative densities and different meso-structures. Then, the large amount of numerical simulation experimental data was established, which covers enough multiaxial loadings in the permissible principle-strain space. Our model solves some of the controversies in current models and adapts the equivalent stress, equivalent strain, and constitutive equations seamlessly to deformation from non-dense to dense state. Numerical results from two mesoscopic models show the relations between equivalent stress and plastic strain in our theoretical model have better consistency under all multiaxial loadings than those in some known models. We checked the topology of subsequent yield surfaces in the plastic principle-strain space and the results turn out the subsequent yield surfaces are not self-similar. The large amount of numerical test data not only well validates our theoretical model but also will be beneficial to the mechanical study of non-dense metals under multiaxial loadings.

Yield surface identification of CP-Ti and its evolution reflecting pre-deformation under complex loading

Uniaxial testing methods to characterize materials provide only limited data that is insufficient to fully understand all aspects of their behaviour, such as initial texture or anisotropy. Therefore, this research aims to conduct complex stress loading experiments to understand the physical mechanism accountable for plastic deformation caused by monotonic tension and tension assisted by proportional cyclic torsion in the CP-Ti (Commercially Pure Titanium). The yield surface approach was applied to assess the variation of mechanical properties in the as-received and pre-deformed material. It was found, that such monotonic tension associated with cyclic torsion caused a significant decrease of the tensile stress. The initial yield surface obtained for the as-received material exhibits anisotropic behaviour, whereas, the sizes of subsequent yield surface reflecting pre-deformation were reduced in all directions with exception of the tension direction.

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.

A non-associated flow rule with simple non-branching form representing the apparent non-normality effects after abrupt strain-path change

Although the existed ‘corner models’ are capable of reasonably predicting experimental observations of the corner effects including the formation of a sharp vertex on the subsequent yield surface and the apparent non-normality of plastic flows after abrupt strain-path change, these models have not been widely used in metal forming computation due to their complexity. The present work proposes a non-associated flow rule with simple non-branching form to represent the non-normality effects after abrupt strain-path change. Then a stable stress update algorithm for the present rule is developed, based on the fully implicit backward Euler method. The present rule is implemented in ABAQUS and its effectiveness is verified through the FEM simulations of two typical strain-path change experiments. Furthermore, some concerned influencing factors of the non-normality effects after strain-path change are discussed. The results show that the new rule has good prediction performance by comparing with the previous works by Simo, Yoshida et al. and the associated flow rule (AFR). And the combination of the non-associated flow rule and the visco-plastic model can closely reproduce the complete experimental observations of corner effects. Moreover, the plane strain tension simulations reveal that the proposed rule predicts the earlier and sharper strain localization than AFR.

An Anisotropic Continuum Damage Mechanics Model for the Simulation of Yield Surface Evolution and Ratcheting

A new anisotropic continuum damage model was presented on the basis of irreversible thermodynamics. Second rank symmetric damage tensor, kinematic hardening tensor, and damage potential surface were introduced. The yield surface in this study has been modeled to distort in the stress space by the evolution of the damage tensor. The new model was applied in yield surface characterization and led to a well description of the subsequent yield surfaces. This model is then applied to predict experimental results of the cyclic nonproportional loading paths obtained by Kang et al. [2004], Hassan et al. [2002] and Taleb and Cailletaud [2011]. It is demonstrated that the new model can predict the yield surface distortion and the ratcheting strains of multiaxial nonproportional cyclic loading with acceptable accuracy and by using less material constants with respect to the previous models.

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.

Anisotropic hardening and evolution of r-values for sheet metal based on evolving non-associated Hill48 model

The uniaxial tensile tests and biaxial tensile tests were carried out to investigate the deformation behavior and yield loci of DC06 steel sheet. The continuous evolution of r-values was calculated by the proposed inverse method, which has higher accuracy and simpler form compared with the previous methods. Based on the equivalent plastic work principle, the anisotropic hardening, the evolution of r-values and subsequent yield loci were investigated. Then, an evolving non-associated Hill48 anisotropic plasticity model was formulated. Where, based on the constructed stress cost-function and constraint conditions of r-values, the anisotropic parameters coupling the distortional hardening and evolution of r-values were developed to characterize the anisotropy evolution. Moreover, the relationship between the uniaxial tensile plastic strain and the equivalent plastic strain was derived. Based on this, the continuous variation curves of anisotropic parameters with real equivalent plastic strain were obtained. The comparison between the experimental and prediction results indicates that the developed model can accurately capture the distortional hardening, evolution of r-values and yield loci simultaneously. Furthermore, the effect of distortional hardening and evolution of r-values on the subsequent yield surfaces and plastic potential is analyzed.

Investigation on Plastic Flow Behaviors of FCC Polycrystalline Aluminum under Pre-Cyclic Tension-Compression Loading: Experiments and Crystal Plasticity Modeling.

The plastic flow behaviors of FCC polycrystalline aluminum after pre-cyclic tension-compression deformation are mainly investigated in tension–torsion stress space by the physically based crystal plasticity model introducing a back-stress. A global finite element model (GFEM) constructed of sufficient grains was established to simulate the same-size thin-walled tube specimen constrained and loaded as the experiments of yield surfaces. The computational results showed that the shape of subsequent yield surfaces and the plastic flow directions directly depended on the given offset strain levels and the applied re-loading paths under different pre-cyclic deformations. The angle deviation between the plastic flow direction and the theoretical orthogonal direction further indicated that there was a large difference between them in the inverse pre-straining direction, but the difference was negligible in the pre-straining direction. From the influence of the anisotropic evolution of the subsequent yield surfaces on plastic flow, we found that the plastic normality rule followed the smooth yield locus; conversely, the significant non-associated flow was attributed to the distorted yield locus. Furthermore, it was also demonstrated that the anisotropic evolution and the plastic flow trend of the subsequent yield surfaces obtained by experiments can be better reproduced by the crystal plasticity model.

Biaxial ratcheting analysis based on real behaviors of subsequent yield surface

To develop a plastic constitutive equation for general plastic deformation analysis, it is useful to confirm that the constitutive equation is viable for analyzing mechanical ratcheting in which a small strain accumulates along with cyclic loading. Many constitutive models and analyses on mechanical ratcheting have been proposed. However, most models do not consider distortion and rotation of the subsequent yield surface observed in the process of ratcheting deformations. In this study, behaviors of both the subsequent yield surface and the ratcheting deformations have been experimentally evaluated for the initially isotropic metals of SUS316FR stainless steel, 6-4 brass and S25C mild steel, which were subjected to biaxial mechanical ratcheting. Simulation analysis of biaxial ratcheting was conducted based on real behaviors of the subsequent yield surfaces such as swing rotation, translation, and distortion observed in ratcheting, and the two-surface hardening rule. The reliability of the proposed plastic constitutive model was verified by comparing the numerical results with the corresponding experimental results.

Evolution of subsequent yield surfaces with plastic deformation along proportional and non-proportional loading paths on annealed AA6061 alloy: Experiments and crystal plasticity finite element modeling

This work presents an experimental and numerical investigation of annealed aluminum alloy AA6061 subjected to proportional and non-proportional loading paths at various finite levels of pre-strain. A careful mechanical characterization program is conducted for subsequent yield surfaces in the σ11−3σ12 stress space using an equivalent 10με offset linearity definition of yield. Electron backscatter diffraction (EBSD) is also used to characterize the microstructure of the material in the annealed condition. Each subsequent surface shows a very well-known shape of being nearly flat in unloading and having a prominent “nose” in the loading direction that deviates substantially from a generalized isotropic behavior. All proportional loading conditions show a rapid evolution of the mixed hardening effect that was characterized by the first load reversal in the first yield surface probe. Differences associated with the path-dependent effect in shifting the non-proportional yield surfaces are also observed in experiments. The experimental program is followed by Taylor-type crystal plasticity finite element (CPFE) simulations of the observed mechanical phenomenon. The Taylor-type CPFE model is calibrated using only the proportional uniaxial tension response and the EBSD imaging to predict all subsequent proportional and non-proportional loading paths. All CPFE show good predictive capabilities in capturing the experimental yield surface measurements for both proportional and non-proportional loading conditions. The CPFE simulations predict both the resulting shape and magnitude of the “nose” in the yield surfaces, as well as the mixed hardening, distortion, and yield surface translations. A systematic study is presented to investigate the effects of latent hardening on the yield surface predictions. This work highlights the promising results of advanced material model developments for larger-scale component simulations of lightweight aluminum alloys.

Exploration of subsequent yield surfaces through unit cell simulations

Subsequent yield surfaces of ductile solids, pre-strained in shear and/or tension, exhibit various characteristic features that depend on the proof strain used to detect yield. These features include kinematic hardening when the proof strain denoting yield is low, isotropic hardening when it is high, formation of a ‘nose’ in the loading direction and flattening of the rear part of the yield surface. With the aid of a computational homogenisation scheme that allows application of arbitrary macro stress or macro deformation gradient to a unit cell, we show that essential features of the subsequent yield surfaces owe their origin to the extent of plasticity accumulated within the unit cell. For instance, a simple unit cell containing a spherical void, where the void serves to create a heterogenous distribution of plastic strain, suffices to computationally reproduce the experimentally observed features. Moreover, in contrast to experiments which generally allow plotting of the yield surface in a normal stress – shear stress plane, the computational scheme allows fuller exploration of the subsequent macro yield surface in the macro stress space. We find that what seems to be a translation and distortion of the yield surface in experiments, is actually most pronounced near some critical octahedral planes characterised by levels of triaxiality induced by the pre-straining process. The evolution of the yield surface in regions where the triaxiality is higher is considerably different. Moreover, as homogeneous yielding is approached with loading of the unit cell, the material seems to forget the nature of the pre-straining process and undergoes isotropic hardening.

Distortional Hardening Behavior and Strength Different Effect of Pure Titanium Grade 1 Sheets: Experimental Observation and Constitutive Modeling

This study presents a systematic methodology that used to identify the subsequent yield surfaces of a pure titanium grade 1 sheet at different levels of equivalent plastic work. Several experimental tests including uniaxial tensile tests (UT), hydraulic bulge test (BT), simple shear tests (SS), and uniaxial compressive tests (UC) have been conducted for samples prepared in different orientations to achieve a comprehensive experimental data of yielding behaviors observed in different forming modes. Under the condition of equivalent plastic work, the yielding behaviors are characterized and normalized to clarify the distortional hardening behavior and strength different effect for the tested material. It is seen that the yielding surface of the tested material distorts largely during plastic deformations and approaches to its final shape at an equivalent plastic work value of 40 MPa. Under the plane-stress assumption, experimental data obtained from on-axis tests (UT, BT, UC) are used to calibrate three constitutive models: Yld2k, CPB06, and CB04. Finite element analyses for a simple shear test have been performed in Abaqus/explicit software to validate the developed material models. In conclusion, the CPB06 model provides the best prediction for plastic yielding behaviors of the tested material.

Investigation of Yield Surfaces Evolution for Polycrystalline Aluminum after Pre-Cyclic Loading by Experiment and Crystal Plasticity Simulation.

This study aims at introducing the back stress of anisotropic strain-hardening into the crystal plasticity theory and demonstrating the rationality of this crystal plasticity model to describe the evolution of the subsequent yield surface of polycrystalline aluminum at the mesoscopic scale under complex pre-cyclic loading paths. By using two different scale finite element models, namely a global finite element model (GFEM) as the same size of the thin-walled tube specimen used in the experiments and a 3D cubic polycrystalline aggregate representative volume element (RVE) model, the evolution of the subsequent yield surface for different unloading cases after 30 pre-cycles is further performed by experiments and numerical simulations within a crystal plasticity finite element (CPFE) frame. Results show that the size and shape of the subsequent yield surfaces are extremely sensitive to the chosen offset strain and the pre-cyclic loading direction, which present pronounced anisotropic hardening through a translation and a distortion of the yield surface characterized by the obvious “sharp corner” in the pre-deformation direction and “flat” in the reverse direction by the definition of small offset strain, while the subsequent yield surface exhibits isotropic hardening reflected by the von Mises circle to be distorted into an ellipse by the definition of large offset strain. In addition, the heterogeneous properties of equivalent plastic strain increment are further discussed under different offset strain conditions. Modeling results from this study show that the heterogeneity of plastic deformation decreases as a law of fraction exponential function with the increasing offset strain. The above analysis indicates that anisotropic hardening of the yield surface is correlated with heterogeneous deformation caused by crystal microstructure and crystal slip. The crystal plasticity model based on the above microscopic mechanism can accurately capture the directional hardening features of the yield surface.

Size effect on the subsequent yield and hardening behavior of metal foil

With the growing demand for miniaturized and integrated products, the metal foils have been extensively applied in the field of aerospace, electronics, medical equipment, and microsystem technology. However, the deformation process is usually under the complex stress status for the fabrication of the micropart. In addition, size effect dramatically impact the microforming process and therefore the well-developed metal forming theory established at macroscale is inapplicable at the grain level. To further investigate the size effect on the subsequent yield and hardening behavior, the biaxial tests of stainless-steel foils with diverse grain sizes and thicknesses were conducted under different stress states. The initial and subsequent yield loci are computed under various loading histories according to the plastic work per unit volume principle. It is found that the evolution of yield loci from initial to the subsequent state shows a remarkable difference for metal foils with different grain sizes. Based on Yld2000-2d yield criterion, the theoretical yield loci based on three different hardening models, including isotropic, kinetic, and isotropic-kinetic mixed hardening models, were acquired to explore and evaluate their applicability at microscale. Results indicated that the isotropic-kinetic mixed hardening model can better describe the evolution of subsequent yield locus than the individual hardening model. In addition, the back stress is shifted down with the increase of grain size for the foils with a certain thickness. Based on the Taylor-Bishop-Hill (TBH) polycrystal plasticity model, the evolution of subsequent yield surface is interpreted by the transformation of grain orientation.

Investigation of Anisotropic Subsequent Yield Behavior for 45 Steel by the Distortional Yield Surface Constitutive Model.

The subsequent anisotropic yield behavior of 45 steel was predicted by the distortional yield surface constitutive model, which can describe the anisotropic subsequent yield and the cross effect of metal associating with loading history. The yield characteristics and plastic hardening behaviors of the 45 steel were simulated under three preloading paths including pre-torsion, pre-tension, and pre-tension–torsion. Based on the comparison between the experimental yield stresses and the simulation by the classical Chaboche model, the proposed model can describe the remarkable anisotropic yield behavior related to the loading history, which can effectively describe the sharp point of yield surface in pre-loading direction and the smaller curvature near its opposite direction. It was successfully simulated by the constitutive model proposed that the subsequent distortional yield surface defined by small offset strain and the degradation process of the distortion feature defined by large offset strain.

Investigation on the Evolution of Subsequent Yield Surface of Pure Aluminum Under Changing Loading Paths Considering Microstructure Effects

The anisotropy behavior is one of the most important key issues in the simulation of sheet metal forming, which can highly affect the prediction accuracy of metal forming behavior, especially under complex loading paths. In this study, the initial and subsequent yield surfaces of the pure aluminum material are numerically investigated through the representative volume element (RVE) methodology. The crystal plasticity theory is chosen to represent the microscopic elastoplastic behavior of pure aluminum, and its polycrystalline geometrical model is built by the Voronoi algorithm. Together with user material subroutine, the initial and subsequent yield surfaces evolution of pure aluminum under different loading paths is simulated in commercial software ABAQUS/Implicit. Through the investigation, the great influence of the microtexture on the obtained inhomogeneous stress distribution of RVE model can be presented. Meanwhile, the effect of the loading directions and the predeformation is also investigated and discussed. The shape of the subsequent yield surface is also strongly dependent on both the yield definition and direction of preloading, and the high anisotropy phenomena appear corresponding to different yield definitions. At the same time, the pole figures at different forming stages are given, and the evolution of polycrystalline texture is analyzed and predicted. It can be concluded that some strong relationship between the texture evolution, the inhomogeneous stress field, and the subsequent yield surface evolution is proved and discussed. This study provides a good methodology to study the yield surface evolution of aluminum together with the crystal plasticity theory, with considering the evolution of the microtexture evolution during the plastic deformation of the metal.

Anisotropic elastic-plastic behavior of architected pyramidal lattice materials

The initial and subsequent yield surfaces for architected pyramidal lattice materials are investigated analytically. Considering lattice struts as elastic-perfectly plastic thin beams subjected to both axial force and bending moment, a set of nonlinear elastic-plastic constitutive relations for a strut is proposed. Moreover, we phenomenologically present anisotropic pressure-dependent yield functions for pyramidal lattices. Comparison of planar yield surfaces of pyramidal lattices predicted by analytical approach to the ones obtained from phenomenological models shows a good agreement for the type of external loads and range of strains investigated in this study. Investigating the normality between the plastic strain vectors and yield surfaces, the obtained results demonstrate the associative nature of the flow rule. We have also emphasized on utilizing hollow-tapered struts as the constituent of pyramidal lattices. To this end, we analytically found that hollow-tapered struts can remarkably improve the effective stiffness and yield strength of bending-dominant pyramidal lattices.

Prediction of the yield surface evolution and some apparent non-normality effects after abrupt strain-path change using classical plasticity

The two-stage abrupt strain path change method is used to investigate the subsequent yield surface evolution, as predicted by associated J2 plasticity models. Experimental results from the literature have revealed several specificities of the yield surface evolution following such a strain-path change, including formation of a vertex and apparent non-normality. The current study showed that some features of this behavior can be simulated using models based on associated plasticity. Kinematic hardening had the most important effect, together with rate sensitivity and elasticity. The results suggest that some apparent violation of the normality condition can result from the combined effects of kinematic hardening, rate sensitivity and elasticity, even when the normality rule is built into the model.

The Anisotropic Distortional Yield Surface Constitutive Model Based on the Chaboche Cyclic Plastic Model.

Considering the cross effect in the evolution of subsequent yield surfaces for metals, an anisotropic distortional yield surface constitutive model is developed. By introducing an anisotropic distortional hardening function into the isotropic hardening part of the classical Chaboche rate-dependent constitutive model, the plastic-deformation-induced distortional and anisotropic hardening behaviors of subsequent yield surfaces are characterized. The experimental data of distortional yield surfaces for T2 pure copper under three different loading paths, including pre-tension, pre-torsion, and pre-tension-torsion proportional loading of 45-degree, are simulated by implementing the models into a numerical user defined material (UMAT) procedure based on the ABAQUS finite element package. To validate the anisotropic plastic model, the simulated yield surfaces are compared with experimental observations and predicted results for a crystal plasticity model and good agreement are noted. The simulations demonstrate that the proposed model can accurately capture the characteristics of the distortional yield surface and the anisotropic hardening process of the yield surface. Moreover, the distortional shapes of experimental subsequent yield surfaces in loading direction and opposite direction can be better revealed by the anisotropic plastic constitutive model than the crystal plastic constitutive model.