Return-mapping Scheme Research Articles

Rate-independent gradient-enhanced crystal plasticity theory — Robust algorithmic formulations based on incremental energy minimization

Numerically robust algorithmic formulations suitable for rate-independent crystal plasticity are presented. They cover classic local models as well as gradient-enhanced theories in which the gradients of the plastic slips are incorporated by means of the micromorphic approach. The elaborated algorithmic formulations rely on the underlying variational structure of (associative) crystal plasticity. To be more precise and in line with so-called variational constitutive updates or incremental energy minimization principles, an incrementally defined energy derived from the underlying time-continuous constitutive model represents the starting point of the novel numerically robust algorithmic formulations. This incrementally defined potential allows to compute all variables jointly as minimizers of this energy. While such discrete variational constitutive updates are not new in general, they are considered here in order to employ powerful techniques from non-linear constrained optimization theory in order to compute robustly the aforementioned minimizers. The analyzed prototype models are based on (1) nonlinear complementarity problem (NCP) functions as well as on (2) the augmented Lagrangian formulation. Numerical experiments show the numerical robustness of the resulting algorithmic formulations. Furthermore, it is shown that the novel algorithmic ideas can also be integrated into classic, non-variational, return-mapping schemes.

An Explicit Modeling for the Pseudoelastic Response of Porous SMA Thick-Walled Cylinders under Internal and External Pressure

Shape-memory alloy (SMA) structures, either dense or porous, are widely used because of their functional characteristics arising from shape-memory effect and pseudoelasticity. Hence, the precise modeling of these materials is essential due to design issues. In this study, the solution algorithm based on an explicit phenomenological constitutive model is investigated to analyze pressurized cylinders. The proposed method is extracted following return mapping scheme to find the evolution of phase transformation in these structures. The formulation is validated by comparing uniaxial tension–compression curves during loading, unloading and thermal actuation against published literature. By the aid of this model, assuming elastic and phase transformation strains, a comprehensive study is presented to inspect stress, strain and displacement fields along the thickness of internally and externally pressurized thick-walled cylinders. To this end, the governing nonlinear equation in elastic–inelastic zones is discretized by the efficient method of generalized differential quadrature (GDQ) and the resultant relations are solved numerically. The influences of internal and external pressure on stress distribution along the cylinder wall during loading–unloading for different temperatures including phase transformation and saturation states are discussed. Also, the closed-form solution for critical pressure at the onset of the phase transformation is studied.

Reliability of inelastic wind excited structures by dynamic shakedown and adaptive fast nonlinear analysis (AFNA)

The ever-growing interest in performance-based wind engineering has created a need for assessment frameworks that can efficiently deal with inelasticity. The computationally efficient strain-driven dynamic shakedown approach has provided a solution that is not only capable of identifying failure mechanisms that are potentially critical during extreme winds, e.g., low cycle fatigue and ratcheting, but also allows direct estimation of inelastic deformations. This approach, however, can only solve problems at dynamic shakedown, i.e., with limited nonlinearity, and is not capable of providing response time histories. To address these limitations, this paper presents an efficient framework for reliability assessment of inelastic structures at dynamic shakedown and beyond. To this end, a novel step-by-step integration algorithm is developed for rapid response time history analysis within the setting of dynamic shakedown. The method is based on advancing fast nonlinear analysis through introducing schemes for enabling at each time step the adaptive selection of the step size, number of modes to be included, and number of potentially nonlinear elements. Inelasticity is modeled as distributed at the level of the stress resultants through a return mapping scheme based on the Haar–Kàrmàn principle, therefore enabling the integrated estimation of the state of dynamic shakedown. The scheme is seen to preserve the efficiency of recently developed strain-driven dynamic shakedown algorithms while providing a full range of response time histories at and beyond the state of dynamic shakedown. To enable reliability analysis, the scheme is embedded in a general uncertainty propagation framework.

Return mapping scheme for the Hoek-Brown model with tension cut-off

The present paper revisits a stress update procedure for the Hoek-Brown plasticity model enhanced by a Rankine type of tension cut-off failure criterion. Limiting tensile stresses not only supports the behavior of weak rock masses with a very low tensile strength but in combination with the original Hoek-Brown model improves robustness of the stress update procedure, i.e. the stress return mapping algorithm. Herein, the primary focus is on the stress return from a space of inadmissible trial stresses for which neither the Hoek-Brown yield surface nor its derivative can be evaluated. All potential scenarios are thoroughly discussed both in the framework of single- and multi-surface plasticity. The presented procedures were implemented and verified with the help of the Geo5 FEM software.

A dual decomposition of the closest point projection in incremental elasto‐plasticity using a mixed shell finite element

AbstractMixed assumed stress finite elements (FEs) have shown good advantages over traditional displacement‐based formulations in various contexts. However, their use in incremental elasto‐plasticity is limited by the need for return mapping schemes which preserve the assumed stress interpolation. For elastic‐perfectly plastic materials and small deformation problems, the integration of the constitutive equation furnishes a closest point projection (CPP) involving all the element stress parameters. In this work, a dual decomposition strategy is adopted to split this problem into a series of CPPs at the integration points level and in a nonlinear system of equations over the element, in order to simplify its solution. The strategy is tested with a four nodes mixed shell FE, named MISS‐4, characterized by an equilibrated stress interpolation which improves the accuracy. Two decomposition strategies are tested to express the plastic admissibility either in terms of stress resultants or point‐wise Cauchy stresses. The recovered elasto‐plastic solution preserves all the advantages of MISS‐4, namely it is accurate for coarse meshes in recovering the equilibrium path and evaluating the limit load showing a quadratic rate of convergence, as demonstrated by the numerical results.

A variational minimization formulation for hydraulically induced fracturing in elastic-plastic solids

A variational modeling framework for hydraulically induced fracturing of elastic-plastic solids is developed in the present work. The developed variational structure provides a global minimization problem. While fracture propagation is modeled by means of a phase-field approach to fracture, plastic effects are taken into account by using a Drucker–Prager-type yield-criterion function. This yield-criterion function governs the plastic evolution of the fluid-solid mixture. Fluid storage and transport are described by a Darcy–Biot-type formulation. Thereby the fluid storage is decomposed into a contribution due to the elastic deformations and one due to the plastic deformations. A local return mapping scheme is used for the update of the plastic quantities. The global minimization structure demands a $$H(\mathrm{div})$$ -conforming finite-element formulation. Furthermore this is combined with an enhanced-assumed-strain formulation in order to overcome locking phenomena arising from the plastic deformations. The robustness and capabilities of the presented framework will be shown in a sequence of numerical examples.

A division method for shallow tunnels and deep tunnels considering soil stress path dependency

Critical burial depth plays an important role in tunnel design and analysis. So, the division method for deep tunnels and shallow tunnels is perceived as a main concern. Considering the soil stress path dependency, this paper investigates the construction mechanical behavior of soil mass at different burial depths, and aims to explore a method for identifying critical burial depth. Firstly, using the Eulerian method and stress return mapping scheme, an elastoplastic algorithm applicable to double loading criteria was developed for a soil constitutive model considering complex stress paths. Then, the developed algorithm was implemented into a user-defined material subroutine (UMAT) and validated through numerical triaxial tests under different stress paths. Subsequently, the UMAT was used to model tunnel excavation in soils. The simulated results were compared with the results that from practical engineering, which shows that the UMAT could accurately predict the ground surface settlement. On the basis of this, the stress distribution and deformation performance in soil during tunneling at different burial depths were explored. Finally, the division method for differentiating shallow tunnels and deep tunnels in soil was discussed and a method for determining the critical depth was proposed.

Numerical study on the rate-dependent behavior of geogrid reinforced sand retaining walls

Time effect on the deformation and strength characteristics of geogrid reinforced sand retaining wall has become an important issue in geotechnical and transportation engineering. Three physical model tests on geogrid reinforced sand retaining walls performed under various loading conditions were simulated to study their rate-dependent behaviors, using the presented nonlinear finite element method (FEM) analysis procedure. This FEM was based on the dynamic relaxation method and return mapping scheme, in which the combined effects of the rate-dependent behaviors of both the backfill soil and the geosynthetic reinforcement have been included. The rate-dependent behaviors of sands and geogrids should be attributed to the viscous property of materials, which can be described by the unified three-component elasto-viscoplastic constitutive model. By comparing the FEM simulations and the test results, it can be found that the present FEM was able to be successfully extended to the boundary value problems of geosynthetic reinforced soil retaining walls. The deformation and strength characteristics of the geogrid reinforced sand retaining walls can be well reproduced. Loading rate effect, the trends of jump in footing pressure upon the step-changes in the loading rate, occurred not only on sands and geogrids but also on geogrid reinforced sands retaining walls. The lateral earth pressure distributions against the back of retaining wall, the local tensile force in the geogrid arranged in the retaining wall and the local stresses beneath the footing under various loading conditions can also be predicted well in the FEM simulations.

Machine learning-based constitutive model for J2- plasticity

This research aims to propose a machine learning (ML)-based constitutive model to predict elastoplastic behavior for J2-plasticity. An artificial neural network (ANN) was constructed to replace the nonlinear stress-integration scheme conducted in the conventional theoretical constitutive model under isotropic hardening and associated flow rule. The training dataset required for the ANN Model was numerically generated based on the conventional return mapping scheme in the principal stress space. The training has been effectively carried out with one element simulation along all the possible plastic loading paths for problem independent training. A conventional theoretical method is used for the unloading procedure. Therefore, ANN is selectively utilized only for nonlinear plastic loading while keeping linear elastic loading and the unloading with a physics-based model. After one element training, the ML-based constitutive model was implemented in Abaqus User MATerial (UMAT) and its performance was verified. For this purpose, one element and tensile test simulations were applied to examine the accuracy of the ANN-based model. Also, for fully nonlinear strain-paths, a circular cup drawing simulation was applied to predict the cup profiles which was compared with that to the conventional J2 plasticity. It was concluded that the simulation results predicted from the ANN-based model show good agreement with those from the conventional J2-based constitutive model. Also, according to simulation time, the ANN-based model shows an advantage in computational efficiency.

Identifying elasto-plastic damage coupling model of laser-welded aluminum alloy by virtual field method and digital image correlation

The local high temperature of laser welding affects the mechanical damage properties of aluminum alloy. The full-field strain near the weld of the laser-welded aluminum alloy 6061 under uniaxial tension was obtained using digital image correlation. An iterative stress increment method for the isotropically hardened elasto-plastic damage coupling constitutive model is proposed herein based on the combination of a return mapping scheme and the nonlinear virtual field method. The elasto-plastic damage coupling constitutive parameters and its corresponding stress–strain relationship in each subzone around the weld are established. The results show that the plastic strain accumulating in the heat-affected zone is the fastest, where the damage initiation occurs earlier, and the damage evolution speed is faster than that other zones.

Using neural networks to represent von Mises plasticity with isotropic hardening

Neural networks are universal function approximators that form the backbone of most modern machine learning based models. Starting from a conventional return-mapping scheme, the algorithmic description of von Mises plasticity with isotropic hardening is mathematically reformulated such that the relationship between the strain and stress histories may be modeled through a neural network function. In essence, the neural network provides an estimate of the instantaneous elasto-plastic tangent matrix as a function of the current stress and plastic work density. For plane stress conditions, it thus describes a non-linear mapping from ℝ4 to ℝ6. The neural network function is first developed for uniaxial stress conditions including loading histories with tension-compression reversal. Special attention is paid to the identification of the network architecture and artifacts related to overfitting. Furthermore, the performance of networks featuring the same number of total parameters, but different levels of non-linearity, is compared. It is found that a fully-connected feedforward network with five hidden layers and 15 neurons per layer can describe the plane stress plasticity problem with good accuracy. The results also show that a high density of training data (of the order of ten to hundred thousand points) is needed to obtain reasonable estimates for arbitrary loading paths with strains of up to 0.2. The final neural network model is implemented into finite element software through the user material subroutine interface. A simulation of a notched tension experiment is performed to demonstrate that the neural network model yields the same (heterogeneous) mechanical fields as a conventional J2 plasticity model. The present work demonstrates that it is feasible to describe the stress-strain response of a von Mises material through a neural network model without any explicit representation of the yield function, flow rule, hardening law or evolution constraints. It is emphasized that the demonstration of feasibility is the focus of the present work, while the assessment of potential computational advantages is deferred to future research.

A reformulated hardening soil model

The hardening soil (HS) model is an advanced soil plasticity model, which incorporates many features including stiffness stress dependency, hardening from initial loading and soil dilatancy. In this paper, the HS model is explored in depth and two improvements are proposed. The first is a new shear yield surface and hardening rule that have been reformulated to remove singularities. The second is a robust implicit return mapping scheme. Options for improving the global convergence of finite-element analyses are also explored. Single-elements tests replicate results from experimental triaxial data and previous versions of the HS model very closely, and at excellent convergence rates. In addition, a slope stability analysis is performed using the φ − c strength reduction method in two-dimensional plane strain. The results from the slope analysis showed good agreement with analytical and graphical slope stability methods. A three-dimensional (3D) slope stability analysis was also conducted with modified boundary conditions, in order to demonstrate the 3D capabilities of the model.

Frictional contact analysis of sliding joints with clearances between flexible beams and rigid holes in flexible multibody systems

Joints with clearances in flexible multibody systems have been investigated for many years. However, the previous work is mainly concerned with the lower pair joints with small clearances. The aim of this paper is to present a formulation for the sliding joint with clearance between a flexible beam and a rigid hole, with special focus on the frictional contact. We thoroughly discuss a contact detection method for a flexible beam with circular cross-section and a rigid hole with rectangular cross-section. We employ the layer-wise contact detection concept and propose an efficient and robust approach to address the two-dimensional ellipse–rectangle contact detection problem within the rigid hole cross-section. After the potential contact points on the beam surface and on the hole cross-section are determined, a beam–rigid hole frictional contact element is proposed. The measure of tangential interaction is defined in the current configuration, and the changes of both contact points are taken into consideration. The constitutive relationship for the tangential contact is modeled by the Coulomb friction law with a penalty regularization, and the real tangential force vector in the current step is determined via a trial step and a subsequent return-mapping scheme. In addition, the discontinuity problem across the beam element boundaries in the case of large sliding is circumvented by a simple and effective transformation. Finally, two numerical examples are presented to verify the proposed contact detection method and the beam–rigid hole frictional contact element, and to demonstrate the influence of the frictional contact on the dynamic performance of flexible multibody systems.

Analytical and Numerical Analyses of Tunnel Excavation Problem Using an Extended Drucker–Prager Model

This paper presents a semi-analytical solution as well as finite element numerical simulations for the tunnel excavation problem, based on an extended Drucker–Prager model and under the drained condition. The adopted plasticity model is capable of accounting for the strain-hardening behavior of soils/rocks through both the internal friction angle and the cohesion. The analytical formulations show that this problem can be reduced to solving a system of ordinary differential equations, with the radial, tangential and vertical stresses, as well as the specific volume being the basic unknowns. Parametric studies are conducted to illustrate mainly the tunnel support pressure–displacement curve and the distributions of stress components, and, in particular, the critical support pressure required to stabilize the tunnel on the basis of three different collapse criteria. With regards to the numerical aspect, an integration algorithm has been developed by employing the return mapping scheme for the extended Drucker–Prager model and subsequently implemented into ABAQUS through the user subroutine UMAT. The agreement between the ABAQUS numerical results and the analytical solution predictions is overall excellent, which demonstrates the accuracy and reliability of the proposed integration scheme.

Towards neural network models for describing the large deformation behavior of sheet metal

Neural networks provide a potentially viable alternative to a differential equation based constitutive models. Here, a neural network model is developed to describe the large deformation response of a Levy-von Mises sheet material with isotropic strain hardening. Using a conventional return-mapping scheme, virtual experiments are performed to generate stress-strain data for random monotonic biaxial loading paths (up to strains of 0.2). Subsequently, a basic feedforward neural network model is trained and validated using the results from virtual experiments. The results for a shallow network with only two hidden layers show remarkably good agreement with all experimental data. The identified neural network model is implemented into a user material subroutine and used in basic structural simulations such as uniaxial tensile and notched tension experiments. In addition to demonstrating the potential of neural networks for modeling the rate-independent plasticity of metals, their application to more complex problems involving strain-rate and temperature effects is discussed.

Return-mapping algorithms for associative isotropic hardening plasticity using conic optimization

We present a mathematical programming approach for elastoplastic constitutive initial value problems. Consideration of the associative plasticity and a linear isotropic hardening model allowed us to formulate the local discrete constitutive equations as conic programs. Specifically, we demonstrate that implicit return-mapping schemes for well-known yield criteria, such as the Rankine, von Mises, Tresca, Drucker-Prager, and Mohr–Coulomb criteria, can be expressed as second-order and semidefinite conic programs. Additionally, we propose a novel scheme for the numerical evaluation of the consistent elastoplastic tangent operator based on a first-order parameter derivative of the optimal solutions.

A finite strain incremental-secant homogenization model for elasto-plastic composites

This paper presents a finite strain extension of the incremental-secant mean-field homogenization (MFH) formulation for two-phase elasto-plastic composites. The formulation of the local finite strain elasto-plastic constitutive equations of each phase is based on a multiplicative decomposition of the deformation gradient as suggested by Simo in (Computer Methods in Applied Mechanics and Engineering, 99(1):61–112, 1992.). The latter has proposed algorithms which preserve the classical return mapping schemes of the infinitesimal theory by using principal Kirchhoff stresses and logarithmic eigenvalues of the left elastic Cauchy–Green strain. Relying on this property, we show that, by considering a quadratic logarithmic free energy and J2-flow theory at the local level, infinitesimal strain incremental-secant MFH is readily extended to finite strains. The proposed formulation and corresponding numerical algorithms are then presented. Finally, the predictions are illustrated with several numerical simulations which are verified against full-field finite element simulations of composite cells, demonstrating that the micro-mechanically based approach is able to predict the influence of the micro-structure and of its evolution on the macroscopic properties in a very cost-effective manner.

Integrating a micromechanical model for multiscale analyses

SummaryThe Chang‐Hicher micromechanical model based on a static hypothesis, not unlike other models developed separately at around the same epoch, has proved its efficiency in predicting soil behaviour. For solving boundary value problems, the model has now integrated stress‐strain relationships by considering both the micro and macro levels. The first step was to solve the linearized mixed control constraints by the introduction of a predictor–corrector scheme and then to implement the micro–macro relationships through an iterative procedure. Two return mapping schemes, consisting of the closest‐point projection method and the cutting plane algorithm, were subsequently integrated into the interparticle force‐displacement relations. Both algorithms have proved to be efficient in studies devoted to elementary tests and boundary value problems. Closest‐point projection method compared with cutting plane algorithm, however, has the advantage of being more intensive cost efficient and just as accurate in the computational task of integrating the local laws into the micromechanical model. The results obtained demonstrate that the proposed linearized method is capable of performing loadings under stress and strain control. Finally, by applying a finite element analysis with a biaxial test and a square footing, it can be recognized that the Chang‐Hicher micromechanical model performs efficiently in multiscale modelling.

Hydraulic hysteresis effects on the coupled flow–deformation processes in unsaturated soils: Numerical formulation and slope stability analysis

• A coupled flow–deformation numerical method with hydraulic hysteresis is proposed. • A return mapping algorithm is developed to implement the hydraulic hysteresis. • The performance and convergence of the numerical formulation is improved. • The significant effects of hydraulic hysteresis on slope stability is demonstrated. The hysteresis of water retention curve has a profound influence on the coupled hydro-mechanical behaviors in unsaturated soils, but numerical implementation with consideration of this property was rarely reported due to the difficulties in the integration of the coupled constitutive models. In this study, a numerical formulation is proposed for modeling the coupled flow–deformation processes with hydraulic hysteresis. A return mapping scheme is developed to integrate the water retention curve model with hydraulic hysteresis and the elasto-plastic model simultaneously within a time step, and the deformation-dependent nature of the water retention curve is considered rigorously by modifying the coefficient matrices in the discretized governing equations. The performance and efficiency of the proposed numerical formulation is validated by two existing laboratory tests and a computational example, demonstrating better performance and convergence of the proposed formulation. The proposed procedure is then applied for modeling the coupled flow–deformation processes in a soil slope under rain infiltration. The simulated results reveal the significant effects of hydraulic hysteresis on the coupled water–air two-phase flow and elasto-plastic deformation processes. The solid deformation and the evolution of the shear band would be remarkably overestimated, and the slope failure would be early predicted when neglecting hydraulic hysteresis.

On the calculation of plastic strain by simple method under non-associated flow rule

Abstract The non-associated flow rule (non-AFR), which has been proved with advantages in accurately predicting yield stress components and Lankford coefficients, has different definitions for yield function and plastic potential function. Currently, there are two typical approaches to determine the plastic strain for non-AFR, one is called simple method which still uses the equivalent plastic strain (EPS) to calculate the plastic strain tensor, while the other is called full method which adopts the plastic compliance factor. Many researchers prefer to adopt the simple method in calculating the plastic strain due to its higher computation efficiency and its acceptable accuracy like the full method for some materials. However, the limitation and application condition of the simple method are still ambiguous. In the present work, a restriction for calculating the parameters in plastic potential function for the simple method is clarified and the limitation of the simple method has also been investigated. It is found that if the relative error of the difference between the yield function value and that of plastic potential function is within ±5% and hardening exponent is in the range of 0.1–0.3, the maximum absolute value of the relative error of the stress predicted by the simple method can be controlled less than 1.6%. Besides, an effective approach is introduced to reduce the error of the stress predicted by these two methods, which can improve the applicability of the simple method for those materials with the relative error of the difference between the value of yield function and that of plastic potential function is less than 10%. In order to have deeper understanding about their applicability, the two methods are compared under uniaxial tension along different orientations for AA2090-T3 and AA2008-T4 with the fully implicit return-mapping scheme. Fracture simulations of AA2090-T3 have also been conducted by introducing different methods. The results show that during the loading in transverse direction, the difference of EPSs predicted by these two methods is very small for AA2008-T4, while for AA2090-T3, the EPS predicted by the simple method is almost twice of that calculated by the full method. This further indicates that the applicability of the simple method mainly depends on the characteristics of materials.