Unified Hardening 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.

Geotechnical analysis involving strain localization of overconsolidated soils based on unified hardening model with hardening variable updated by a composite scheme

AbstractStrain localization simulation of overconsolidated soils with high overconsolidation ratio (OCR) has been a long‐standing challenge. Some critical state soil models, including the modified Cam‐clay (MCC) model, have been widely applied, but they may not predict the shear dilatancy of overconsolidated soils well in some cases. Hence, the unified hardening (UH) model, which may be viewed as a generalized version of the MCC model, is implemented. It has been recognized, nonetheless, that without resorting to the regularization mechanism, the standard finite element method (FEM) or the second‐order cone programming optimized finite element method (FEM‐SOCP) often experiences instability or interruption of calculating the hardening‐softening responses of overconsolidated soils. To resolve the aforementioned difficulty, the UH model is developed and implemented in the framework of FEM‐SOCP based on the micropolar continuum (mpcFEM‐SOCP) to predict strain localizations of overconsolidated soils. Furthermore, to obviate non‐convexity of mpcFEM‐SOCP induced by material softening, an effective composite update scheme of hardening variable pc, which refers to the implicit variable (IV) scheme for the hardening stage and then refers to the explicit variable (EV) scheme for the softening stage, is proposed. Based on one biaxial compression problem and one rigid strip footing problem, numerical analyses disclose that by applying mpcFEM‐SOCP in conjunction with the composite update scheme of pc, the UH model of micropolar continuum can effectively predict the strain localization behavior of overconsolidated soil during its failure stage, and the stable hardening‐softening responses of overconsolidated soils can be readily attained.

A multiscale study on the microstructure and hardening models of the irradiation defects on reactor pressure vessel steels: Modelling and experiment

Irradiation defects are known to impede the dislocation motion in irradiated reactor pressure vessel (RPV) steels. However, consistent experimental studies of the respective irradiation defects and relevant theoretical studies of the interaction mechanism and hardening model are still lacking. In this study, the microstructure and irradiation-induced hardening of RPV steels are investigated using transmission electron microscopy (TEM)/energy-dispersive X-ray spectrometry (EDS) and molecular dynamics (MD) simulations. In A508-Ⅲ steel, 9R, twinned fcc, and other typical Cu-rich cluster structures are observed using TEM after the irradiation experiment. Based on the experimental results, MD simulations are performed to study the interactions between the dislocations and Cu clusters or voids. A mechanism diagram of the interaction mechanism coupling the effects of temperature and defect size is shown for both the Cu clusters and voids. A relevant hardening model is proposed based on the MD data, which is further discussed in relation to the macroscopic mechanical properties. Furthermore, the irradiation-induced hardening of the RPV steel is successfully predicted, which further demonstrates the reliability of the proposed unified hardening model.

Unified hardening (UH) model for unsaturated expansive clays

Unified hardening (UH) model for unsaturated expansive clays

Approaches to improve the convergence and efficiency of the implicit stress integration algorithm for the unified hardening model

To realize the numerical calculation of the unified hardening (UH) model for soils, the closest point projection method (CPPM) is used to do the stress integration in the finite element analysis. A package of approaches is proposed to improve the convergence and efficiency of the algorithm. An implicit differentiation method, which can greatly save the computational cost and storage space, is employed to calculate the second derivatives of the 3D yield function generalized by the transformed stress (TS) method. In addition, the denominator of the UH law is equal to 0 at the characteristic state and critical state, while the TS equation contains a square root and may be invalid for tensile stress. These singular problems are solved by introducing subtle treatments without changing the calculation results of the UH model. The consistent tangent operator corresponding to the stress integration can be expressed in a concise form. Finally, a series of simple loading tests and finite element simulations of the bearing capacity of a rigid circular foundation are conducted to evaluate the accuracy, convergence and efficiency of the proposed algorithm.

Elastic-viscoplastic model for coarse-grained soil considering particle breakage

This paper presents a novel elastic-viscoplastic constitutive model for reproducing the time-dependent behaviour of coarse-grained soil considering particle breakage, by integrating the Unified Hardening (UH) model, the elastic-viscoplastic (EVP) model and the overstress theory. The relationship between the degree of particle breakage and the loading rate is established, and the state variables associated with the critical state of coarse-grained soil are derived to jointly consider both time and particle breakage. A new three-dimensional elastic-viscoplastic constitutive model is then constructed through combining the one-dimensional viscoplastic hardening parameter with a secondary consolidation coefficient considering particle breakage. The proposed model requires 19 parameters and it can well describe the influence of time-dependency and particle breakage to the shear, dilatancy, and compression behaviours of coarse-grained soil with various confining pressures or initial void ratios. Comparisons between the model predictions and experimental data are employed to validate the capability of the proposed model to replicate the time-dependent behaviour of coarse-grained soil.

Interpretation of piezocone tests in normally- and over-consolidated clay using a spherical cavity expansion solution

This paper presents the derivation of closed-form expressions for the interpretation of piezocone tests in clay on the basis of an undrained cavity expansion solution. The cavity expansion solution employs the modified Unified Hardening model to describe the undrained behaviour of normally- and over-consolidated clays upon shearing. The derived expressions correlate the measured tip resistance and pore pressure with the undrained shear strength and the over-consolidation ratio, using a cone factor that, unlike empirical formulas, is calculated directly from the constitutive model parameters and the cone geometry. The cavity expansion solution is validated via comparison of its results against published studies, and accordingly the accuracy of the proposed expressions is benchmarked against field data. We show that, despite the simplifications introduced in the derivation, the calibrated expressions are capable of reproducing the field data with reasonable accuracy.

An elastic-viscoplastic model for time-dependent behavior of unsaturated soils

Modeling the time-dependent behavior of unsaturated soil is a topic that is poorly addressed by the current literature. This paper presents a novel elastic-viscoplastic constitutive model for reproducing the time-dependent behavior of unsaturated soil. The model is established using the theoretical framework of the Unified Hardening (UH) model and the Barcelona Basic Model (BBM) combined with Perzyna's overstress theory. A unified hardening parameter is introduced to modify the elastic-viscoplastic constitutive model to consider the effect of the loading history. A new expression for the secondary consolidation coefficient considering the effect of suction is presented, and the state variables related to the void ratio of unsaturated soil considering the time effect are derived. Combined with the one-dimensional viscoplastic hardening parameter, the viscoplastic scalar multiplier in overstress theory is presented, and a three-dimensional elastic-viscoplastic constitutive model is then constructed. With 17 parameters, the proposed model can describe the effects of time and suction on the shear, dilatancy, and compression behaviors of unsaturated soil with various initial void ratios and stresses. Comparisons between the model predictions and experimental data are employed to validate the capability of the proposed model in terms of reproducing the time-dependent behavior of unsaturated soil or soft rock.

Coupling Numerical Analysis Method for Primary and Secondary Consolidation of Soft Soil Foundation

Soft clay is widely distributed in our country, and this type of soil usually has the character of large compressibility and poor permeability, so the foundation composed mainly of soft clay is prone to primary and secondary consolidation coupling settlement deformation after the completion of construction. In this paper, secondary consolidation deformation is redefined in the improved time logarithmic coordinate system, and the calculation formula of secondary consolidation deformation is given. Then an extended unified hardening model is introduced, and by applying this model, the effect of time can be transformed to the increment of equivalent stress. The numerical governing equations are established in combination with the seepage consolidation theory. Thus the coupling analysis of primary and secondary consolidation of soft clay foundation can be realized. By means of a finite element case analysis and verification of subgrade construction and post-construction deformation, it is found that the effective calculation of primary and secondary consolidation deformation can be accomplished smoothly based on the extended unified hardening model, and meanwhile the permeability coefficient and secondary consolidation parameters are able to effectively control the primary consolidation deformation and secondary consolidation deformation of soft clay foundation. The research is valuable to develope the prediction method of soft clay foundation deformation and improve the prediction accuracy of soft clay foundation deformation in engineering, and it is of great significance to solve complex geotechnical engineering problems.

A 3D Elasto-plastic Constitutive Model for Clay Based on the Unified Hardening Model with Insights into the Coupling Effect of Small-Strain Stiffness and Over-Consolidation Ratio

A 3D Elasto-plastic Constitutive Model for Clay Based on the Unified Hardening Model with Insights into the Coupling Effect of Small-Strain Stiffness and Over-Consolidation Ratio

Analysis of cylindrical cavity expansion in anisotropic overconsolidated clays using the Extended UH model

Anisotropy and overconsolidation are two important features of depositional clay and are not comprehensively addressed in current cavity expansion solutions. Aiming at better modeling the cavity expansion responses in the natural soils, this paper develops a novel and rigorous elastoplastic semi-analytical solution by implementing the advanced anisotropic unified hardening (UH) model in conjunction with incorporating the anisotropic SMP criterion. The novel incorporation of the anisotropic SMP criterion and the proper consideration of overconsolidated properties are the major contributions of the present solution. By utilizing the anisotropic stress transformation method, the initial stress-induced anisotropy, anisotropic yielding and anisotropic failure of soil are also incorporated into the proposed solution. Following the derived constitutive equations in incremental form and large strain theory, both undrained and the drained cases are reduced to boundary-value problems consisting of several first-order ordinary differential equations formed in the Lagrangian scheme. Detailed parametric analyses are presented and compared with other existing solutions to investigate the effects of anisotropy and initial overconsolidation on the cavity responses. The present solution could properly capture the cross-anisotropy and 3D strength of overconsolidated soil and hence provides a more advanced and generic approach for modeling the cavity expansion responses in wide ranges of soils.

Application of closest point projection method to unified hardening model

This paper presents a framework of the closest point projection method (CPPM) based on the unified hardening (UH) model. The UH model based on the Cam-clay model and the concept of the subloading yield surface was rewrote as a concise form of classical elasto-plastic model. Then the CPPM algorithm combining the exact elastic property and the spatially mobilized plane (SMP) strength criterion by the transformed stress method could be conveniently derived for the numerical implementation of the UH model. According to this algorithm, the consistent tangent operator corresponding to the final stress state of the integration increment was deduced formally. And the mean stress probed by the elastic predictor was avoided to be negative or zero for the usage of exact elastic property. Eventually, the accuracy, robustness and efficiency of the proposed integration scheme were demonstrated by the numerical examples including the boundary value problems of the single integral point and multi-elements of geotechnical engineerings, which was respectively compiled as the single integral point program and the user defined subroutine (UMAT) of the commercial finite element program ABAQUS.

Structured UH model considering induced anisotropy

Based on the structured unified hardening model (structured UH model), introducing rotational hardening, a structured UH model considering induced anisotropy is presented. Applying the transformed method, the presented model is extended in three-dimensional stress space. Comparing the test data with the model simulations, the structured UH model considering induced anisotropy is qualified to smoothly and continuously describe the behaviors of natural soils in cyclic loading.

An elastic-viscous-plastic structured UH model

On the basis of the structured unified hardening model (structured UH model) considering soil structured collapse, the time effect is take into account, and an elastic-viscous-plastic structured UH model is extended. In the presented model, the moving normal compression line (MNCL) is extended to a moving instant normal compression line (MINCL) for simulating both soil structure effect and time effect. Then the instance plastic strains irrespective of time effect are calculated by referring the MINCL, and the plastic strains due to time effect are calculated by referring the traditional instant normal compression line. Comparisons between test data and model simulation indicate that the presented elastic-viscous-plastic structured UH model is qualified to simulate the combined actions of soil structure and time effects both in compression and in shearing.

Elastoplastic Model of Sand Based on Double-State Parameters under Complex Loading Conditions

Abstract Based on the existing unified hardening model, the double-state parameters unified hardening (DPUH) model is proposed, which can take into account the initial density and confining pressur...

Expansion responses of a cylindrical cavity in overconsolidated unsaturated soils: A semi-analytical elastoplastic solution

This paper presents a semi-analytical elastoplastic solution for cylindrical cavity expansion in overconsolidated (OC) unsaturated soils under constant water content condition, in which both the OC features and the coupled hydro-mechanical behaviour of soils are properly addressed. The unified hardening model for OC unsaturated soil is further coupled with a void ratio-dependent soil-water characteristic curve to model the coupled hydro-mechanical soil responses during expansion. With the aid of an auxiliary variable, the coupled hydro-mechanical constitutive matrix for the investigated problem is finally formulated as a set of first-order ordinary differential equations in terms of Lagrangian description, with the three net stress components, the suction, and the void ratio as the basic unknowns. The governing differential equations are solved as an initial value problem with the code developed based on the Runge-Kutta algorithm. Parametric studies show that both the overconsolidation ratio and the suction have remarkable impacts on the expansion responses, including the distribution of stress components, the change of degree of saturation as well as the expansion pressures. The proposed solution could provide a reasonable tool for interpretation of the pressuremeter tests and prediction of pile installation effects in overconsolidated unsaturated soils.

UH model for granular soils considering low confining pressure

The mechanical behaviors including strength and isotropic compression of granular soils under low confining pressure are systematically analyzed. The compressive hardening parameter ps is introduced in semi-logarithmic coordinates to express the curved normal compression line of granular soils in the e-ln p plane, to describe the little compressibility of granular soils under low confining pressure. The nonlinear variation of critical friction angle with confining pressure is considered by introducing the interlocking strength parameter τ0 into strength criteria. Based on the unified hardening model for clays and sands (CSUH model), the unified hardening (UH model) for granular soils considering low confining pressure, called UH-L model, is established. The UH-L model behaves significantly different with the CSUH model under low confining pressure, while tends to be same as the CSUH model with increasing confining pressure. The relationship between parameter ps and τ0cotφ, as well as the effect of ps and τ0cotφ on model performance are analyzed. Good agreement with the experimental data shows that the UH-L model is capable of uniformly describing the stress–strain relations and strength of granular soils with interparticle locking over a wide confining pressure load range, especially at low confining pressure.

An elastoplastic solution for spherical cavity undrained expansion in overconsolidated soils

In this paper, a novel elastoplastic solution is developed for the undrained expansion of a spherical cavity in overconsolidated soils, with the aim of providing a mechanical model for the interpretation of cone penetration tests in overconsolidated soils. The well-established unified hardening (UH) model is adopted in the present solution to represent the unique mechanical behaviors of the overconsolidated soil during cavity expansion. The problem considered is formulated as a system of first order partial differential equations in terms of a Lagrangian description; these equations are solved as an initial value problem with the code developed based on the Runge-Kutta algorithm. The results, including the expansion-pressure curves, the distributions of the stress components, and the evolution of the potential strength during cavity expansion are presented to show the expansion responses in overconsolidated soils. The results reveal that the unique expansion responses in overconsolidated soil during cavity expansion, such as the development of a negative excess pore water pressure, the strain-hardening and softening behavior, the peak strength and the decay of the potential strength during cavity expansion could be well represented by the present solution.

Loading–unloading judgement for advanced plastic UH model

The unified hardening (UH) model proposed by Yao et al. (Geotechnique 2009) is the constitutive model which can consider the influence of the complex stress path and stress history on the deformation and strength of clays reasonably. Firstly, the loading–unloading criterion of material model is defined as the change law of the intersection of current yield surface and the p axis, which makes the loading–unloading in the process of hardening and softening can be unified considered in UH model. Then, the Newton–Raphson method is adopted to attain the nonlinear problems solution in the finite element method of UH model, and the semi-implicit return mapping method is adopted to update stress. The application of the UH model in the finite element is realized. And then, the analyses of triaxial test are performed using the unit prediction and finite element method. The results of the unit prediction method are compared with the experimental results to illustrate the rationality of the UH model. Comparing the results with the unit prediction method and the finite element method, the correctness of the finite element program of the UH model is illustrated. Further, the three-dimensional finite element analysis of embankment on soft soil is performed by the program. The comparison between the results calculated by the UH model and the modified Cam-clay (MCC) model and the experimental data shows that the UH model is rational in analyzing the actual embankment engineering on soft soil.

AN ELASTOPLASTIC CONSTITUTIVE MODEL FOR GAS HYDRATE-BEARING SEDIMENTS

The density is one of the most important factors for the mechanical behavior of soil. The content and occurrence modes of hydrates obviously affect the density of gas hydrate-bearing sediments (GHBS) because hydrates exist in the pore of sediment as solid phase. Therefore, it is necessary to consider the effect of the hydrate content and occurrences to describe the mechanical properties of hydrate sediments well. In this paper,based on the unified hardening model for clays and sands (CSUH model), the relation between the volume fraction of hydrate and the compressive hardness parameter is firstly established to reflect the influence of hydrate on the compressibility of sediments. Secondly, in order to consider the influence of hydrate content and occurrence modes on sediment density, we propose a formula to describe the effective initial void ratio, and it is then introduced into the state parameter to describe the influence of hydrate on dilatancy of sediment. Finally, combining the drop-shaped yield surface of the CSUH model, an elastoplastic constitutive model for GHBS is developed. Compared with the laboratory test results, it is verified that the model can reasonably describe the mechanical behaviors of GHBS containing hydrates with different occurrence modes and contents. For the same occurrence mode but different contents, the set of parameters is the same.