Subloading Surface Model Research Articles

Application of Dimension Extending Technique to Unified Hardening Model

This paper provides the process of incremental constitutive integration for the unified hardening model combined with the transformation stress method. The dimension-extending technique takes the hardening function of the hardening/softening model as the same position as the stress components, so that the constitutive integration of the plasticity can be reduced to an initial value problem of differential–complementarity equations, which is solved using the Gauss–Seidel algorithm-based Projection–Correction for the mixed complementarity problem. The Gauss–Seidel based Projection–Correction algorithm does not require the calculation of the Jacobean matrix of the potential function, making it relatively easy to implement in programming. The unified hardening model is proposed based on the modified Cam–Clay model and the sub-loading surface model, and the elastic properties are pressure-dependent. Two processing methods, backward Euler integration and exact elastic property, are used for the variable elasticity properties. The constitutive integration of the increased dimensional unified hardening model is reduced to a special mixed complementarity problem and solved by the proposed algorithm, which does not need to calculate the Jacobean matrix of the potential function, and greatly simplifies the derivation process. Several numerical examples are given to verify the feasibility of the incremental constitutive integration in the unified hardening model, including the single integral point and the boundary value problems. The research results have expanded the scope of use of the Gauss–Seidel based Projection–Correction algorithm.

Foundations of Elastoplasticity: Subloading Surface Model (Fourth Edition)

Foundations of Elastoplasticity: Subloading Surface Model (Fourth Edition)

The subloading surface model in hyperelastic-based plasticity with time integration algorithms in intermediate and current configurations

This paper introduces a novel application of subloading surface concept to effectively model the elastoplastic behavior of materials in the finite strain regime. The subloading surface approach allows for a smoother and more seamless transition from the elastic to the plastic region. Two time integration schemes are presented in this study, one in the intermediate configuration and the other in the current configuration. In the first time integration scheme, co-rotational objective time rates are not required, and the formulation utilizes the Mandel stress. In the second time integration scheme, implemented in Eulerian description, the Kirchhoff stress defined in the current configuration is utilized.The results obtained for the simple shear deformation case demonstrate the absence of oscillations. Additionally, these results show excellent agreement with experimental data and existing findings from the literature. Furthermore, a computational comparison between the time integration schemes using the subloading surface model and a model discussed in the literature reveals that the proposed algorithms in this paper require significantly fewer increments in the given time integrations. This indicates a higher computational efficiency and reduced computational time for the proposed methods. Overall, the predicted responses of all the analyzed cases from the provided time integrations exhibit excellent agreement with each other, confirming the accuracy and effectiveness of the presented model.

Subloading-elastoplastic constitutive equation of glass

Elastoplastic constitutive equation of glass is proposed in this article, which is formulated based on the subloading surface model which possesses the various distinguished advantages for the description of the plastic deformation, i.e., the smooth elastic-plastic transition, the continuous variation of the tangent stiffness modulus tensor, the automatic controlling function to attract the stress to the yield surface during the plastic deformation process, etc. It would be the firstly proposed three-dimensional elastoplastic constitutive equation of glass, which is furnished with the basic properties, e.g. the ellipsoidal yield surface with the dependence of the third deviatoric stress invariant, undergoing the flattering to the deviatoric direction with the plastic compression and the plastic volumetric hardening with the associated flow rule. The validity of the description of the deformation behavior of glass will be verified by comparisons with some test data for silica glass specimens.

Implementation of subloading surface model for hyperelastoplasticity with nonlinear kinematic/isotropic hardening based on reference and intermediate configurations

In this paper, the elastoplastic behavior of isotropic materials under finite strains is modelled and simulated under consideration of the subloading surface concept. Using the subloading surface model has many advantages, such as the ability of modeling more smoothly transition from elastic to plastic state and also more realistic modeling of the Bauschinger effect. For this purpose, three time integration algorithms are utilized in the reference and intermediate configurations in the present study. The first algorithm is based on a time integration scheme, available in the literature, and the second time integration is proposed by tensor exponential-based time integration and the third one is represented by a standard backward Euler time integration scheme. In the second proposed time integration, which is defined in the intermediate configuration, there is no need to use co-rotational objective time rates. However, in the third one, although it is based on the intermediate configuration, the Zaremba–Jaumann co-rotational rate is employed. Besides, in the second two time integrations, the constitutive equations are derived based on the unknown tensors defined in the intermediate configuration and as a consequence there is no restriction of using different hyperelastic models. For case studies, diverse deformation gradient tensors with and without volume changes are considered. These studied cases are accomplished for SUS 304 stainless steel and a glassy polymer (oriented PET). The obtained results of some cases are compared with available results in the literature. The agreement among the predicted results in this paper and the results from literature is good.

Subloading-Overstress Model: Unified Constitutive Equation for Elasto-Plastic and Elasto-Viscoplastic Deformations Under Monotonic and Cyclic Loadings

Various elasto-plastic models for the rate-independent deformation, various elasto-viscoplastic models for the rate-dependent deformation and their combinations have been proposed during a long time more than one or more centuries. Firstly, the history of the development of the elastoplasticiy and the elasto-viscoplasticity is reviewed comprehensively. Unfortunately, each of these models possesses their own drawbacks and limitations. The unified constitutive equation of the elasto-plastic and the elasto-viscoplastic deformations is provided by incorporating the subloading surface model into the overstress model in this article, which is capable of describing the monotonic and the cyclic loadings at the general rate ranging from the quasi-static to the impact loading. The validity of the unified model is verified by the comparison with various test data of metals under various loading conditions. Consequently, the elastoplastic constitutive equation can be disused hereinafter, while it is expressed by the cumbersome formulation including the complicated plastic modulus but limited to the description of the purely static deformation which is not induced actually.

Numerical implementation of the multiplicative hyperelastic-based Extended Subloading Surface plasticity model

The work presents the numerical implementation of the multiplicative hyperelastic-based Extended Subloading Surface model. The theoretical development of the constitutive equations of the unconventional plasticity theory has already been formulated in the literature. However, its application to finite element simulations has been not proposed yet. In particular, this paper discusses in detail the adaptation of the constitutive equation from a theoretical to a computation point of view, highlighting the numerical strategies adopted by the authors. Moreover, a comparison of the predictive response between the additive hypo- and multiplicative hyperelastic-based versions of the Extended Subloading Surface model is offered through a series of numerical examples.

Constitutive Modeling of Normally and Over-Consolidated Clay with a High-Order Yield Function

In this paper, a simple sub-loading yield surface model for both normally consolidated and over-consolidated clay is proposed with emphasis on the effect of the yield surface shape. Compared with the modified Cam-clay model, only one additional material parameter is introduced to reflect geometry features of the yield surface. A higher-order stress–dilatancy relation is given in the current study, leading to a new yield function capable of offering an adequate description of the yield surface of soil samples in the p–q plane. By introducing the concept of the sub-loading yield surface and the unified hardening parameter, the proposed model can capture the main features of the over-consolidated clay with dilatancy and strain-softening behavior and the main features of the normally consolidated clay with contraction and strain-hardening behavior. The results show that adjusting the yield surface leads to more accurate predictions than the modified Cam-clay model. The proposed model can also reasonably describe its mechanical behavior for clay samples.

Extended overstress model and its implicit stress integration algorithm: Formulations, experiments, and simulations

AbstractThe extended overstress model is formulated based on the subloading surface model with the smooth elastic‐inelastic transition, which is called the subloading‐overstress model. Therein, the rigorous translation rules of the elastic‐core and the similarity‐center, the limitation in the expansion of the subloading surface, and so forth are incorporated. The model possesses the basic structure capable of describing the monotonic/cyclic loading behaviors at the general rate of deformations from the quasi‐static to the impact loading. The experiments were conducted using the spheroidal graphite cast iron under the various loading conditions. It was verified that the experimental results can be simulated accurately by the subloading‐overstress model. Further, the complete implicit stress integration algorithm based on the return‐mapping projection is formulated for the present subloading‐overstress model and implemented into Abaqus through UMAT. Then, the deformation analyses of the R‐notched cylinder were performed by the present algorithm. Consequently, the performability of the present algorithm is verified by the analyses of the boundary‐value problem under the cyclic loadings.

Elaborated subloading surface model for accurate description of cyclic mobility in granular materials

The description of the cyclic mobility observed prior to the liquefaction in geomaterials requires the sophisticated constitutive formulation to describe the plastic deformation induced during the cyclic loading with the small stress amplitude inside the yield surface. This requirement is realized in the subloading surface model, in which the surface enclosing a purely elastic domain is not assumed, while a purely elastic domain is assumed in other elastoplasticity models. The subloading surface model has been applied widely to the monotonic/cyclic loading behaviors of metals, soils, rocks, concrete, etc., and the sufficient predictions have been attained to some extent. The subloading surface model will be elaborated so as to predict also the cyclic mobility accurately in this article. First, the rigorous translation rule of the similarity center of the normal yield and the subloading surfaces, i.e., elastic core, is formulated. Further, the mixed hardening rule in terms of volumetric and deviatoric plastic strain rates and the rotational hardening rule are formulated to describe the induced anisotropy of granular materials. In addition, the material functions for the elastic modulus, the yield function and the isotropic hardening/softening will be modified for the accurate description of the cyclic mobility. Then, the validity of the present formulation will be verified through comparisons with various test data of cyclic mobility.

Subloading-Damage Model

The subloading surface model for the elastoplastic deformation is extended to incorporate the damage phenomenon. It is capable of describing the cyclic loading behavior under the bilateral damage.

Research for Implicit Stress Integration Method of Cyclic Loading by Extended Subloading Surface Model

Research for Implicit Stress Integration Method of Cyclic Loading by Extended Subloading Surface Model

A densification mechanism to model the mechanical effect of methane hydrates in sandy sediments

SUMMARYRecent pore‐scale observations and geomechanical investigations suggest the lack of true cohesion in methane hydrate‐bearing sediments (MHBSs) and propose that their mechanical behavior is governed by kinematic constrictions at pore‐scale. This paper presents a constitutive model for MHBS, which does not rely on physical bonding between hydrate crystals and sediment grains but on the densification effect that pore invasion with hydrate has on the sediment mechanical properties. The Hydrate‐CASM extends the critical state model Clay and Sand Model (CASM) by implementing the subloading surface model and introducing the densification mechanism. The model suggests that the decrease of the sediment available void volume during hydrate formation stiffens its structure and has a similar mechanical effect as the increase of sediment density. In particular, the model attributes stress‐strain changes observed in MHBS to the variations in sediment available void volume with hydrate saturation and its consequent effect on isotropic yield stress and swelling line slope. The model performance is examined against published experimental data from drained triaxial tests performed at different confining stress and with distinct hydrate saturation and morphology. Overall, the simulations capture the influence of hydrate saturation in both the magnitude and trend of the stiffness, shear strength, and volumetric response of synthetic MHBS. The results are validated against those obtained from previous mechanical models for MHBS that examine the same experimental data. The Hydrate‐CASM performs similarly to previous models, but its formulation only requires one hydrate‐related empirical parameter to express changes in the sediment elastic stiffness with hydrate saturation.

Complete implicit stress integration algorithm with extended subloading surface model for elastoplastic deformation analysis

SummaryThe subloading surface model released from a purely elastic domain is capable of describing not only monotonic but also cyclic loading behaviors accurately. It is expected that elastoplastic deformation analyses can be performed with high accuracy and efficiency by the complete implicit stress integration algorithm with a return‐mapping projection. Various implicit stress integration algorithms for the subloading surface model have been proposed. However, they are applicable only to the monotonic loading behavior since they adopt the incorrect loading criterion presuming that the subloading surface expands in the plastic loading process. In fact, however, the plastic strain rate is induced even when the subloading surface contracts in the elastic trial step. Therefore, erroneous results in the descriptions of unloading‐reloading and cyclic loading behaviors are caused in the calculations with the existing algorithms. The complete implicit stress integration method is formulated adopting the rigorous loading criterion and implemented in Abaqus in this study. Numerical analyses for the elastoplastic deformation behaviors in the single‐element are performed by the present and the existing algorithms. In addition, the deformation analysis of the R‐notched cylinder is performed by the present algorithm. High performability of the present algorithm is confirmed by these numerical calculation results.

Comparison Study of Constitutive Models for Overconsolidated Clays

Widely distributed in natural deposits, the overconsolidated (OC) clays have attracted extensive experimental investigations on their mechanical behaviors, especially in the 1960s and 1970s. Based on these results, numerous constitutive models have also been established. These models generally fall into two categories: one based on the classical plasticity theory and the other the bounding surface (BS) plasticity theory, with the latter being more popular and successful. The BS concept and the subloading surface (SS) concept are the two major BS plasticity theories. The features of these two concepts and the representative models based on them are introduced, respectively. The unified hardening (UH) model for OC clays is also based on the BS plasticity theory but distinguishes itself from other models by the integration of the reference yield surface, unified hardening parameter, potential failure stress ratio, and transformed stress tensor. Modification is made to the Hvorslev envelop employed in the UH model to improve its capability of describing the behaviors of clays with extremely high overconsolidation ratio in this paper. The comparison among the BS model, SS model, and UH model is performed. Evidence shows that all these three models can characterize the fundamental behaviors of OC clays, such as the stress dilatancy, strain softening and attainment of the critical state. The UH model with the revised Hvorslev envelop has the fewest parameters which are identical to those of the modified Cam-Clay model.

Formulation for Evolution Rule of Elastic-core in Subloading Surface Model

The extended subloading surface model with the translation of the elastic-core is capable of describing rigorously the monotonic and the cyclic loading behaviors. The translation rule of the elastic-core is considered in this article. Here, the dissipative parts in the plastic strain rate for the internal variables are formulated, which is required in the multiplicative hyperelatic-based plastic constitutive equation.

Elastoplastic analysis by return-mapping method with rigorous loading criterion for extended subloading surface model

Elastoplastic analysis by return-mapping method with rigorous loading criterion for extended subloading surface model

Cyclic plasticity model for fatigue with softening behaviour below macroscopic yielding

Experimental observations have evidenced the insufficiency of yield surface definition via monotonic loading to guarantee adequate description of material behaviour under fatigue tests. Therefore, mechanical property characterisation through simple experimental tests such as uniaxial monotonic extension/compression is favoured, considering different responses due to cyclic loading. Herein, a phenomenological cyclic plasticity model is formulated to capture the correct generation of plastic deformation under cyclic loading conditions, even for stress states lower than the yield stress. The unconventional extended subloading surface model is modified to include an additional internal variable describing the strain-softening phenomenon typical of low- to high-cycle fatigue problems. Implicit integration strategies, in the form of the cutting-plane and closest projection point return mapping algorithms, are implemented to reduce the computation time. The solution accuracies are compared to that of a forward Euler integration scheme, and exhibit fast and accurate performance.

Multiplicative Hyperelastic-Based Plasticity for Finite Elastoplastic Deformation/Sliding: A Comprehensive Review

Hyperelastic-based plastic constitutive equation based on the multiplicative decomposition of the deformation gradient tensor is reviewed comprehensively and its exact formulation is given for the description of the finite deformation and rotation in this article. Further, it is extended to describe the general loading behavior including the monotonic, the cyclic and the non-proportional loading behaviors by incorporating the rigorous plastic flow rules and the subloading surface model. In addition, it is extended also to the rate-dependency based on the overstress model, and the exact hyperelastic-based plastic constitutive equation of friction is formulated by incorporating the subloading-friction model. They are the exact constitutive equations describing the monotonic and the cyclic loading behavior up to the finite deformation/rotation and the friction behavior under the finite sliding/rotation with the rate-dependency, which have remained to be unsolved for a long time, although they have been required in the history of elastoplasticity theory.

A numerical study of the return mapping application for the subloading surface model

PurposeMany practical problems in engineering require fast, accurate numerical results. In particular, in cyclic plasticity or fatigue simulations, the high number of loading cycles increases the computation effort and time. The purpose of this study is to show that the return mapping technique in the framework of unconventional plasticity theories is a good compromise between efficiency and accuracy in finite element analyses.Design/methodology/approachThe accuracy of the closest point projection method and the cutting plane method implementations for the subloading surface model are discussed under different loading conditions by analyzing the error as a function of the input step size and the efficiency of the algorithms.FindingsMonotonic tests show that the two different implicit integration schemes have the same accuracy and are in good agreement with the solution obtained using an explicit forward Euler scheme, even for large input steps. However, the closest point projection method seems to describe better the evolution of the similarity centre in the cyclic loading analyses.Practical implicationsThe purpose of this work is to show two alternative implicit integration schemes of the extended subloading surface method for metallic materials. The backward Euler integrations can guarantee a good description of the material behaviour and, at the same time, reduce the computational cost. This aspect is particularly important in the field of low or high cycle fatigue, because of the large number of cycles involved.Originality/valueA detailed description of both the cutting plane and closest point projection methods is offered in this work. In particular, the two integrations schemes are compared in terms of accuracy and computation time for monotonic and cyclic loading tests.