Stress Update Algorithm Research Articles

Study on unified strength theory and elastic–plastic stress update algorithm

The unified strength theory with the two-piecewise linear equations is more convenient and concise to calculate the strength of materials. It can fully explore the potential in the strength of materials and improve the economic benefits of engineering design. This study combines the semi-implicit return mapping algorithm and the Aitken accelerated iteration scheme and develops a plastic constitutive algorithm for isotropic softening materials based on the unified strength theory. The combining method can simplify the stress update and make the calculation of consistent tangent modulus easier. Furthermore, it can avoid solving the partial derivatives of the plastic flow rule and overcome the stress-deviating problem. The self-developed constitutive algorithm is used to simulate the elastic–plastic excavation process of a deep-lying circular tunnel. The numerical simulation results match well with the theoretical solution, verifying the correctness of the self-developed constitutive algorithm. Based on the self-developed constitutive algorithm, the stability of an underground mining stope is comprehensively analyzed, and its structural parameters are optimized. The research reveals the mechanism of stope instability, provides a reliable scientific basis for the mining design and decision-making, ensures the safe and efficient production of the stope, and achieves the expected goal.

Correction to: In the original article there are several transcription errors in appendix A: the stress update algorithm

Correction to: In the original article there are several transcription errors in appendix A: the stress update algorithm

Modeling confined concrete behavior in finite element with a non-orthogonal elastoplastic model

The accuracy of a constitutive model for confined concrete largely relies on its capability to capture concrete's dilatancy behavior. In this paper, a non-orthogonal flow rule (NFR) is used to reasonably characterize the concrete's volume change law in relation to the stress state without the necessity for a plastic potential function. Then, the non-orthogonal plastic model is implemented in the finite element (FE) software ABAQUS using the implicit stress update algorithm, which employs the line search method and the numerical consistent tangent stiffness matrix to ensure the robustness and computational efficiency of FE analysis. Finally, FE simulations are performed to evaluate the constitutive model's capabilities in actively and passively confined concrete. In the latter case, steel tubes restrict the concrete's lateral deformation. An analysis and discussion are conducted regarding the impact of dilatancy behavior on concrete-filled steel tube (CFST) columns. The consistency between experimental data and simulation results demonstrates that the FE modeling with a non-orthogonal constitutive model provides an effective tool to describe the behavior of confined concrete.

A damage mechanics model under dynamic thermal loads and its application to pressure vessels under fire invasion

The accurate prediction of pressure vessel rupture under fire invasion conditions is crucial for identifying boiling liquid expanding vapour explosions (BLEVEs). This study proposes a material damage model based on a calculus strategy to predict the evolution of material damage under dynamic thermal loads. In this model, the dynamic thermal load was differentiated, and the Gurson–Tvergaard-Needleman (GTN) model was combined with the Johnson-Cook (J-C) hardening function to calculate the damage within each differential time step. The stress update algorithm was used to calculate the cumulative damage. The damage characteristics of the materials at different temperatures were obtained, and the key parameters were identified as functions of temperature to describe the constitutive relationship during the dynamic damage process. The model was used to simulate and predict the failure morphology, failure time, and failure pressure of pressure vessels under fire invasion conditions. The results showed that it had good prediction accuracy and could be used to analyse BLEVE accidents involving pressure vessels.

Reconstruction of Cam-clay model based on multiplicative decomposition of the deformation gradient

This study formulated the Cam-clay model, an elastoplastic constitutive model for fully remolded and normally consolidated clay, using the multiplicative decomposition of the deformation gradient and a hyperelastic body to correspond to the finite deformation theory. Finite deformation elastoplastic theory in this framework has been developed for metallic materials that do not yield plastic volume change. If this theory is directly applied to a model that yields plastic volume change, the Cauchy stress cannot be determined only from the current elastic deformation. Therefore, the existing multiplicative decomposition Cam-clay models cannot accurately reproduce the experimental facts underlying the core of critical-state soil mechanics. Through thermodynamic considerations that consider the volume changes occurring in the intermediate configuration, this study determined a common stress suitable for describing the elastic and plastic components, in addition to forms of hyperelastic body and plastic flow rule. Consequently, the proposed model can draw the state boundary surface in the Cauchy stress-specific volume space, which previous models could not achieve. In addition, an implicit stress-update algorithm for the proposed model was constructed, and a tangent modulus consistent with the algorithm was derived. The proposed model demonstrated the same numerical calculation convergence as that of existing models.

A return mapping algorithm based on the hyper dual step derivative approximation for elastoplastic models

Accurately evaluating derivatives poses a key challenge when numerically implementing complex constitutive models. This work presents an implicit stress update algorithm that utilizes the hyper dual step derivative approximation to address derivative evaluations in elastoplastic problems. Initially, the performance of various numerical differentiation methods is discussed and compared by examining their numerical errors in the representative example. Subsequently, the hyper dual step derivative approximation, without truncation and subtractive cancellation errors, is employed to compute the Jacobian matrix and consistent tangent operator, ensuring quadratic convergence in both local and global computations. The size of the Newton search step is optimized by the line search technique, thereby enhancing the convergence in solving nonlinear stress integral equations. Finally, the proposed stress update algorithm is used to implement the non-associated Mohr–Coulomb plastic model in the ABAQUS software using the UMAT subroutine. The stress update algorithm’s performance and its practical application in geotechnical engineering problems are demonstrated using five boundary value problems.

On the experimental characterization and response modelling of particulate composite: Application to flaky aluminium-epoxy composite

The epoxy-based composites are most commonly used now in the automobile and aerospace industries due to their low weight and high strength. The quasi-brittle nature of epoxy resin, which is not desirable in several applications, can be overcome by adding flaky aluminium additive resulting in a particulate composite. A small volume fraction of flaky aluminium influences the mechanics of epoxy resin in a significant manner (increase in tensile strength and strain at failure by 56 % and 200 %, respectively, comparing with neat epoxy) resulting in a ductile-type failure. The nonlinear, rate-independent (at low loading rate), rate-dependent (at higher loading rate), and inelastic (viscous effect) response, coupled with an internal damage, observed in epoxy-based flaky-aluminium composite is not fully understood due to the complex nature of resulting material. Much of the previous literature focused on modelling, either the rate-dependency or damage at low loading rate, but not capturing everything concurrently.The novelty of present work is a broad framework development for experimental characterization and calibration of particulate composite (flaky aluminium additives in the epoxy matrix) performing uniaxial tension, cyclic loading–unloading, and creep experiments at different displacement loading rates, and subsequent modelling of the constitutive response in a thermodynamically consistent manner. The macro-scale properties of the newly proposed particulate composite are estimated by mean-field homogenization and compared with the corresponding experimental values. The elasto-plastic-damage and visco-plastic-damage material constants are successfully obtained by a novel optimization approach coupling Levenberg–Marquardt algorithm with stress update algorithms. The practical applicability of the obtained material constants is finally demonstrated solving several boundary value problems employing Lemaitre’s ductile damage model.

An implicit stress update algorithm for the plastic nonlocal damage model of concrete

The numerical implementation of advanced plastic damage models has been challenging due to the diverse computational difficulties caused by material nonlinearity. In this study, firstly, a plastic damage model is developed by a nonsmoothed Mohr–Coulomb yield criterion. The nonorthogonal flow rule without plastic potential function is employed to describe the dilatancy behavior of concrete. The stiffness degradation is considered through two tensile and compressive damage variables. Subsequently, an implicit stress update algorithm incorporating several advanced numerical methods is proposed to tackle difficulties arising in the numerical calculation of the plastic damage model. In the model implementation, the Mohr–Coulomb yield function with corners is replaced by a single smooth expression to ensure the stability of the numerical calculation. A line search method is used to solve nonlinear systems of integral stress equations to obtain better convergence than Newton method. The complex step derivative approximation is used to evaluate the consistent tangent operator to provide quadratic convergence speed of the global equilibrium iterations, thus avoiding tedious analytical derivative operations. The integral nonlocal method is also used to regularize the finite element solution. At the material point scale, the ability of the proposed model to describe the complex mechanical behavior of concrete is verified by utilizing nine sets of monotonic and cyclic loading and unloading tests. Finally, the algorithm and model’s performance is well tested by the failure analysis of five numerical examples. The code of this work will be freely available at https://github.com/zhouxin615/PD_model.

Toward Elucidating the Influence of Hydrostatic Pressure Dependent Swelling Behavior in the CERCER Composite

A ceramic-ceramic (CERCER) fuel with minor actinide-enriched ceramic fuel particles dispersed in a MgO ceramic matrix is chosen as a promising composite target for accelerator-driven systems (ADS). Fission swelling is a complex irradiation-induced phenomenon that involves recrystallization, resolution, and hydrostatic pressure under extreme conditions of high temperature and significant fission flux. In this study, a multiscale computational framework was developed to integrate simulations of continuum-scale thermo-mechanical behavior in the CERCER composite with a grain-scale hydrostatic pressure-dependent fission gas swelling model. Hydrostatic pressure-dependent fission welling is taken into account in the stress update algorithms for UO2 particles. Accordingly, we programmed the user subroutines to define the thermo-mechanical constitutive relations in the finite element simulations. The obtained results indicate that (1) the proposed method accurately predicts the swelling deformation at various burnup levels while taking into account hydrostatic pressure and (2) prior to recrystallization, the particle swelling is primarily influenced by temperature variation, whereas after recrystallization, the presence of hydrostatic pressure favorably suppresses the swelling deformation. This work effectively captures the swelling behavior influenced by hydrostatic pressure within the dispersed-type CERCER composite fuel in ADSs.

A robust stress update algorithm for elastoplastic models without analytical derivation of the consistent tangent operator and loading/unloading estimation

AbstractA robust and concise implicit stress integration algorithm of elastoplastic models is presented. It does not require the loading/unloading estimation and analytical derivation operation for the stress update. First, the elastoplastic stress update problem is recast into an unconstrained minimization problem by utilizing the smooth function to bypass the loading/unloading estimation. Then, the object problem is solved by the line search method instead of the Newton method for better convergence. The consistent tangent operator is evaluated by the complex step derivative approximation without the subtraction cancellation error, which provides the quadratic convergence rate of global iteration. The rationality of the numerical consistent tangent operator is validated by the one obtained by the analytical derivation. A recently developed non‐orthogonal elastoplastic (NEP) clay model is implemented using the new algorithm. The algorithm is confirmed through comparing the numerical solution and the analytical one for a cavity expansion problem. The algorithm performance is assessed based on a series of geotechnical boundary value problems. It is found that the new algorithm is more robust than the one employed by ABAQUS. The source code of the model implementation can be downloaded from https://github.com/zhouxin615.

Skeleton-creep based bubble growth model and multi-scale mechanical constitutive model for U-10Mo fuels under irradiation

The macroscale irradiation swelling deformations of nuclear fuels induced by the solid and gas fission products are one of the most concerned problems in nuclear reactor core design. Based on continuum mechanics, an innovative macroscale volume growth strain model is proposed for U-10Mo fuels, with the coupling contributions of irradiation creep, fission solid swelling and thermal expansion of fuel skeleton. Correspondingly, new three-dimensional mechanical constitutive models and stress update algorithms are also developed for the multi-scale irradiation-deformation and stress analysis. Good agreements of the predictions with diverse experimental data validate the effectiveness of newly developed models. It is obtained that the growth of fission gas bubbles is dominated by the creep deformations of the surrounding fuel skeleton, depending on the external hydrostatic pressure and the internal bubble pressure related to the intra-granular fission gas atom diffusion; the influences of temperature, external hydrostatic pressure and fuel-skeleton creep rate coefficient are found to become prominent at higher burnup levels, dominantly induced by the coupling of fuel-skeleton creep and local porosity. This study provides a new insight into the mechanism of bubble growth in the solid skeleton, revealing the coupling of irradiation swelling with fuel-skeleton creep.

Choice of the stress integration scheme for accurate large-deformation finite element analysis

The use of computational structural models that include geometrical non-linearity in many application cases may require high reliability in prediction of displacements. Nevertheless, large differences up to 60% on maximum total displacement have been found among results of static large-deformation analyses performed by means of the major commercial software packages in a simple benchmark study with linear material properties. In order to investigate the causes of such disagreement, the present work compares different finite element formulations including well-established stress update schemes. The various formulations are tested, and results are compared in three test cases. Rodriguez stress update algorithms have shown the best performance among methods reported in literature. Finally, the cause of the large differences found in the predictions of commercial codes is identified. It is linked to the energetic inconsistency of some stress update methods in the simulation of extension/compression loading conditions. Such inaccuracy is reproduced analytically by formulating and integrating the corresponding inconsistent constitutive equations. The identified problem is very important for designers, as it affects almost all the static simulations, which are the most common type of large-deformation analyses and usually involve extension/compression loading.

A hybrid strategy blending primal‐dual interior point and return mapping methods for a class of hypoelastic‐plastic models with memory surface

AbstractBy blending the primal‐dual interior point method (PDIPM) and the return mapping algorithm, we propose a hybrid strategy of implicit stress update for a class of hypoelastic‐plastic models with the hardening rule whose evolution is restricted by the memory surface. The inequality constraint relevant to the memory surface is replaced by the equality constraint by the introduction of a slack variable, and the duality gap is gradually reduced by using path‐following method. First, we formulate an optimization problem corresponding to the principle of maximum plastic dissipation for the standard von‐Mises plasticity with isotropic and kinematic hardening rules and its variant for the model with the memory surface. Next, after function forms employed for the elastic‐plastic model are specified, the PDIPM to realize an implicit stress update is briefly reviewed and then applied to the model involving the memory surface to replace the relevant inequality constraint by the equality constraint by the introduction of a slack variable. Then, we present a hybrid scheme that combines the standard return mapping algorithm with the PDIPM. The numerical accuracy of the proposed stress update algorithm for the conventional elastic‐plastic model is verified in comparison with the standard return mapping algorithm using iso‐error map. Also, targeting a notched round steel bar under cyclic loading with three different amplitudes, we demonstrate the performance for the stress update using the elastic‐plastic model with the memory surface. Finally, the capability of the proposed algorithm is proven through a typical real‐life example such that a steel bridge is subjected to earthquake for which the residual load carrying capacity must be estimated.

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.

Efficient and robust stress integration algorithm for anisotropic distortional hardening law under cross-loading with latent hardening

A fast and robust stress-update algorithm based on the general cutting-plane method (GCPM) was developed for a distortional hardening model, known as the HAH-DPS model. It captures the anisotropic hardening behaviors such as the Bauschinger effect, transient hardening, differential permanent softening, and cross-loading effects. The lower computational efficiency of the direct application of GCPM was rectified by considering the all-evolutionary plastic state variables during iterations. The newly proposed algorithm was formulated on the dependence of the equivalent plastic strain and the other state variables defined in the distortional hardening model. And it was implemented in a commercial finite element software using a user-defined material subroutine (UMAT). Finite element simulations under strain-path change were carried out to demonstrate the performance of the new numerical algorithm in terms of the convergence behavior locally as well as globally.

On the computational treatment of fully coupled crystal plasticity slip and martensitic transformation constitutive models at finite strains

AbstractIn this article, four numerical techniques are formulated to improve the reliability and numerical efficiency of stress update algorithms of crystal‐plasticity‐like phenomena at finite strains. The resulting algorithmic setting is especially relevant for structural analyses of polycrystals and the multiscale evaluation of anisotropic microstructures in metallic materials. These techniques are exemplified with a constitutive model, which couples crystallographic slip and martensitic transformation deformation mechanisms with a viscous regularization. The model is characterized by a set of highly coupled equations expressed in a single system and solved by a monolithic solution procedure. The first technique employs a sub‐stepping procedure to generate a better initial guess for the Newton–Raphson scheme used in the iterative solution of the equilibrium problem at the Gauss quadrature point. Then, a logarithmic discretization of the exponential parameter of the viscoplastic law is proposed to approximate the target value of the exponential parameter incrementally in the return mapping algorithm. A strategy to remove the strain‐rate dependence and reduce the necessary viscoplastic parameters is also presented to help with the stiff equations that arise in the rate‐independent limit. Finally, an efficient strategy is suggested to complete the transformation process when the full martensitic transformation is approached. These techniques can be easily used in any combination and dramatically improve the model's efficiency by enabling more significant incremental steps to be used within the monolithic solution procedure. A thorough assessment of their impact is shown in a series of ablation studies.

Evolving structural tensor approach to model the damage induced anisotropy in viscoelastic solids

A three-dimensional anisotropic damage model, based on continuum damage mechanics, is presented capturing the damage induced anisotropy in the initially isotropic, viscoelastic (VE) solid. The second order tensor damage variable, conceived as an evolving structural tensor, is coupled with linear VE model at the finite as well as infinitesimal strains. A dissipation potential, based on a novel damage interaction tensor (fourth order tensor), is proposed from which the damage tensor evolution happens in an associative manner. The newly proposed model is numerically implemented as an implicit stress update algorithm, and the infinitesimal part of the formulation is validated against the available experimental results in the literature. A parametric study of the newly developed finite strain formulation is presented employing three different deformation histories demonstrating further the utility of the proposed anisotropic damage model. The successful experimental validation of the proposed model demonstrates its applicability and utility in the existing finite element codes.

Finite element implementation of hydrostatic pressure-sensitive plasticity and its application to distortional hardening model and sheet metal forming simulations

• The strength-differential (SD) effect of AHSS with ultimate tensile strength exceeding 1 GPa is modeled by a hydrostatic pressure-dependent distortional plasticity theory. • The finite element implementation of the pressure-dependent plasticity theory is proposed in the manuscript. The implemented theory is validated with the comparison study with the associated stand-alone code and experimental measurements. • Both the SD- and strain patch change effects of TIRP1180 are reproduced in finite element simulation by embedding the distortional plasticity model, HAH 20 , in the pressure-dependent yield condition. The effectiveness and applicability of the FE-implemented pressure-dependent distortional plasticity model in sheet metal forming simulations is validated through the springback prediction studies. In this work, the hydrostatic pressure-dependency of the distortional plasticity model HAH 20 was implemented using a finite element (FE) code to account for the strength-differential (SD) effect that has been observed in advanced high-strength steel sheets. To this end, a fully-implicit stress update algorithm formulation was introduced for the pressure-dependent plasticity theory. The implementation was validated by comparing the FE prediction of the material behavior during a number of tests with those of a stand-alone code of the constitutive description as well as with experimental data. In order to investigate the SD effect on the springback simulation results, the U-draw bending test was analyzed within this FE framework. Furthermore, in order to assess the effectiveness and stability of the formulation for a large-scale example, forming simulations of an automotive structural part were conducted. In addition to the SD effect, strain path changes and geometrical aspects were also investigated in this example. It was shown that the distortional plasticity-based pressure-dependent yield criterion well describes the asymmetric SD behavior in sheet metal forming simulations.

A coupled plasticity-damage model of K418 nickel-based superalloy with pressure and temperature dependence

The issue of containment stems from the aero-engine field, and there is little research in the turbocharger field. In order to reduce the number of tests, development cycle and cost, it is necessary to predict the burst speed of the turbine accurately. In response to this problem, a coupled plasticity-damage model is developed for turbine materials according to the stress state and operating temperature of the turbine. The effect of stress triaxiality, temperature are considered in the plastic constitutive and the failure criterion. The stress update algorithm of plasticity-damage model is given. According to the operating temperature and stress state of the turbine, the experimental scheme of K418 nikel-base superalloy is designed to calibrate the plasticity-damage model. Finally, the coupled plasticity-damage model is verified by the specimens with the notch radius R6 at 550 °C and 600 °C. The results show that the maximum error of proposed model is about 7.3%, which is better than the 14.2% prediction error of the constitutive model without any corrections. Moreover, the failure displacement can be predicted by the proposed constitutive model (the maximum error is about 8.2%), which is beyond the ability of traditional model. The location of failure initiation, fracture pattern in numerical studies have also showed close correspondence to the experimental results.

Hypoelastic-plastic-damage analysis of a tunnel in swelling rock mass

In this study, we proposed a multi-stage analysis process to simulate the mechanical behaviour of a tunnel constructed in swelling rock mass, which including self-weight analysis, excavation analysis and swelling analysis. Especially, in the swelling analysis, a hypoelastic-plastic-damage model was constructed which can express the stiffness deterioration due to the damage that occurs with the process of rock swelling, and the plastic softening and hardening behaviour based on the Modified Cam-Clay (MCC) model. Furthermore, an implicit stress-update algorithm was formulated for the swelling model to obtain an accurate and robust numerical computation. A series of numerical examples of a tunnel in swelling rocks is shown in this study to discuss the mechanical behaviour of the tunnel due to the swelling of the bedrock, aimed at evaluating the stability of a tunnel constructed in swelling rocks after some years.