Radial Return Mapping Algorithm Research Articles

Implementation of a modified Graham-Walles viscosity function within a Chaboche viscoplastic constitutive model

This work provides a numerical framework for the accurate prediction of operational life of metallic components exhibiting a non-classical creep behavior under constant loadings and very high temperature. A modified Graham-Walles type analytical viscoplastic function is implemented into a Chaboche unified viscoplastic constitutive model. The numerical model is integrated into the finite element software Lagamine following a fully-implicit two-step radial return mapping algorithm. The non-linear system of equations is solved using a robust Newton-Raphson method. The computational efficiency of the model is enhanced by implementing a sub-step routine, thereby decreasing the average number of iterations of the finite element software. The validation of the model is performed using experimental data available in the literature on the non-classical creep behavior of Incoloy 800H, a Ni-superalloy exhibiting a two-step creep strain rate minima attributed to multiple complex dislocation-precipitate interactions.

Mechanical Characterization of Fatigue and Cyclic Plasticity of 304L Stainless Steel at Elevated Temperature

BackgroundThe mechanical characterization of the cyclic elastoplastic response of structural materials at elevated temperatures is crucial for understanding and predicting the fatigue life of components in nuclear reactors.ObjectiveIn this study, a comprehensive mechanical characterization of 304L stainless steel has been performed including metallography, tensile tests, fatigue tests, fatigue crack growth tests and cyclic stress-strain tests.MethodsIsothermal tests were conducted at both room temperature and 300 °C for both the rolling direction and the transverse direction of the hot rolled steel. Mechanical properties were extracted from the uniaxial experiments by fitting relevant material models to the data. The cyclic plasticity behavior has been modelled with a radial return-mapping algorithm that utilizes the Voce nonlinear isotropic hardening model in combination with the Armstrong-Frederick nonlinear kinematic hardening model. The plasticity models are available in commercial FE software and accurately capture the stabilized hysteresis loops, including a substantial Bauschinger effect.ResultsThe material exhibits near isotropic properties, but its mechanical performance is generally reduced at high temperatures. Specifically, in the rolling direction, the Young’s modulus is reduced by 16 % at 300 °C, the yield strength at 0.2 % plastic strain is lower by 23 %, and the ultimate tensile strength is lower by 30 % compared to room temperature. Fatigue life is also decreased, leading to an accelerated fatigue crack growth rate compared to room temperature. A von Mises radial return mapping algorithm proves to be effective in accurately modelling the cyclic plasticity of the material. The algorithm has also been used to establish a clear correlation between energy dissipation per cycle and cycles to failure, leading to the proposal of an energy-based fatigue life prediction model.ConclusionsThe material exhibits reduced mechanical performance at elevated temperatures, with decreased monotonic strength, compared to room temperature. Fatigue life is also compromised, resulting in accelerated fatigue crack growth. The material’s hardening behavior differs at room temperature and elevated temperature, with lower peak stress values observed at higher temperatures. The radial return mapping algorithm can be used to determine the dissipated energy per cycle which together with fatigue testing has been used to propose a low cycle fatigue life prediction model at both temperatures.

Finite element implementation of a certain class of elasto-viscoplastic constitutive models

In this paper a selected type of elasto-viscoplastic constitutive equations is considered. The viscoplastic model which was proposed by Marquis is generalized by allowing multiple terms describing the isotropic and the kinematic hardening. Furthermore, the presented model formulation enables one to use an arbitrary equivalent plastic strain function to describe the isotropic hardening behavior. What is more, the backstress evolution equation which was proposed by Marquis was modified so that any equivalent plastic strain function can be used in the recovery term now. The general form of the generalized constitutive model obtained this way was subsequently implemented into the finite element method (FEM). For that purpose, the radial-return mapping algorithm was utilized. The consistent tangent operator was derived for the considered class of models and is presented in the paper. A developed user material subroutine (UMAT) which allows one to use the viscoplastic models under consideration in the FEM program CalculiX is attached in the appendix section. A number of numerical simulations were conducted in order to verify the performance of the developed UMAT code.

Constitutive recasting of macromolecular-based thermoviscoplasticity as yield function-based formulation

The rate- and temperature-dependent “macromolecular” polymer plasticity relies on a thermally-activated model of the non-linear strain measure, and is thus deficient in an explicit definition of the yield surface. Such an approach, typically written in the rate-form, is only suitable for the explicit finite element solver where the consistent tangent modulus and the return mapping process are unnecessary. This paper presents a procedure to deduce the pressure-dependent yield function of the rate- and thermal-based plastic constitutive models in the continuum thermodynamics. The recasted yield model aids in deriving its consistent elasto-thermoviscoplastic tangent modulus and implementing it in the implicit finite element solver. The recasted expressions are suitable for incremental formulation using the classical radial return mapping algorithm, regardless of the complexities of the original macromolecular expression. Four viscoplastic macromolecular models are used to illustrate the recasting process: three of them are based on the classical Boyce-Parks-Argon model and one on the Bergström-Boyce model. Numerical simulations are performed on real-sized specimen geometries to establish the functionality and capabilities of the methodology. The practical advantage of the implicit-oriented forms is studied in comparison to the explicit-based implementation under quasi-static and isothermal conditions.

Numerical Simulation of Conical and Linear-Shaped Charges Using an Eulerian Elasto-Plastic Multi-Material Multi-Phase Flow Model with Detonation.

This study developed a hydrocode to numerically simulate both conical and linear-shaped charges using an Eulerian multi-material and multi-phase flow model. Elasto-plastic solids and the detonation of a high explosive charge were modeled using a Johnson–Cook material model and the programmed burn model, respectively. Further, the plasticity of the solids was calculated using a radial return mapping algorithm. The model was solved using a high-resolution computational fluid dynamics (CFD) technique on Cartesian grids. Material interfaces were tracked using the level-set method, and the boundary conditions were imposed using the ghost fluid method. The developed hydrocode was validated using high-speed impact problems. Consequently, the developed hydrocode was used to successfully simulate the evolution and penetration of metal jets in shaped charges after a detonation.

On finite element implementation of cyclic elastoplasticity: theory, coding, and exemplary problems

In this work the finite element (FE) implementation of the small strain cyclic plasticity is discussed. The family of elastoplastic constitutive models is considered which uses the mixed, kinematic-isotropic hardening rule. It is assumed that the kinematic hardening is governed by the Armstrong–Frederick law. The radial return mapping algorithm is utilized to discretize the general form of the constitutive equation. A relation for the consistent elastoplastic tangent operator is derived. To the best of the author’s knowledge, this formula has not been presented in the literature yet. The obtained set of equations can be used to implement the cyclic plasticity models into numerous commercial or non-commercial FE packages. A user subroutine UMAT (User’s MATerial) has been developed in order to implement the cyclic plasticity model by Yoshida into the open-source FE program CalculiX. The coding is included in the Appendix. It can be easily modified to implement any isotropic hardening rule for which the yield stress is a function of the effective plastic strain. The number of the utilized backstress variables can be easily increased as well. Several validation tests which have been performed in order to verify the code’s performance are discussed.

An HLLC-type approximate Riemann solver for two-dimensional elastic-perfectly plastic model

In this work, an elastic-perfectly plastic model in two-dimensional planar geometry is studied and a new HLLC-type approximate Riemann solver (HLLCN) is put forward. The main feature of the new approximate Riemann solver is that it almost includes all stress waves, such as elastic, plastic, longitudinal and shear waves simultaneously in the presence of elastic-plastic phase transition. The analyses of the Jacobian matrix of governing equations are carried out for elasticity and plasticity separately, and the complicate order in the light of magnitude of characteristic speeds is simplified when constructing the approximate Riemann solver. The radial return mapping algorithm originally proposed by Wilkins is not only applied for the plastic correction in the discretization of the constitutive law, but also used to determine the elastic limit state in the approximate Riemann solver. A cell-centered Lagrangian method equipped with this new HLLC-type approximate Riemann solver is developed. Typical and new devised test cases are provided to demonstrate the performance of proposed method.

Prediction of Surface Wrinkle Defect of Welding Wire Steel ER70S-6 in Hot Bar Rolling Process Using Finite Element Method and Experiments

High quality products are demanded due to increasingly fierce market competition. In this paper, the generation of surface wrinkle defect of welding wire steel ER70S-6 was studied by the combination of the experimental method and finite element simulation. Firstly, a thermal compression test was conducted on the Gleeble-3500 thermosimulator under different strain rates and temperatures and a strain dependent Arrhenius-type constitutive function was employed to fit the flow stress–strain curves obtained from the experiments. Then, the elastoplastic constitutive relationship was implemented using radial return mapping algorithm by means of the user subroutine VUMAT of Abaqus/Explicit. A new instability criterion was proposed to predict the possibility of the surface wrinkle defect during the multipass hot bar rolling process. In order to verify the reliability of the finite element model of the six-pass continuous rolling process, the simulated results were compared with experimental data. Finally, the effects of groove width and groove radius on the billet were investigated by the orthogonal test method, and the friction coefficient and rolling temperature. The results show that the groove width and groove radius are key factors to suppress the surface wrinkle defect. Decreasing the groove width can be beneficial for improving the surface quality and reducing the fillet radius. The optimized combination of the rolling process parameters was further applied in an industrial test and the surface quality of the billet was greatly improved.

Return mapping algorithm for elasto-plastic deformation of thin walled pipes

Return mapping algorithms are popular procedures for evaluating elasto-plastic material behaviour. This paper presents a dedicated return mapping algorithm for thin walled pipes and vessels with Von Mises plasticity and isotropic hardening, for which it can be assumed that the hoop stress is constant. This applies for example to pipe line sections during offshore laying operations. In the algorithm a nonlinear scalar equation is solved iteratively. A general 3D radial return mapping algorithm with an iteration loop to fulfil the assumptions on hoop and radial stress is used to verify results obtained with the dedicated algorithm. Results of both algorithms are identical, while the new algorithm is more efficient.

A fully coupled thermo-viscoelastic-viscoplastic-damage framework to study the cyclic variability of the Taylor-Quinney coefficient for semi-crystalline polymers

In the present work, a rigorous and consistent thermo-viscoelastic-viscoplastic ductile damage model is proposed to address the variability of the Taylor-Quinney coefficient (the storage to anelastic energy rate ratio) during the anelastic deformation of thermoplastic polymers. More specifically, the constitutive model developed is mainly dedicated to the description of recoverable viscoelastic effects occurring on different time scales, irreversible strains observed above the stress threshold and energy responses arising from the anelastic deformation process (storage of energy, dissipation of energy, thermomechanical coupling, etc.). One of the aims of this paper is to evaluate the partitioning between stored and heat energy without assuming any Taylor-Quinney values. The proposed approach is incorporated into the framework of Thermodynamics of Irreversible Processes and Generalized Standard Materials formalism to offer the thermodynamic consistency of all the constitutive equations. The numerical algorithm for the proposed model is implemented in the well-known finite element code Abaqus via the user material subroutine UMAT using an implicit formulation of the constitutive equations coupled with a radial return mapping algorithm. The model is then calibrated and validated across monotonic tensile and cyclic tensile-tensile tests by comparing predicted and experimentally observed energy responses. This comparison shows a good level of accordance between experimental findings and model predictions in terms of stress-strain responses for both monotonic and cyclic thermomechanical loading conditions. The model can also numerically capture the cyclic kinetics of the storage and dissipation energies. The capabilities of the fully-coupled model have been demonstrated through simulating the thermo-mechanical response of a complex 3D structure. The numerical analysis establishes the model’s capability to accurately render the spatio-temporal patterns of the Taylor-Quinney coefficient and the self-heating induced part of the ductile damage.

Finite Element Implementation of a Temperature-Dependent Cyclic Plastic Model for SA508-3 Steel

A new temperature-dependent cyclic plastic model, combining the nonlinear cyclic softening and kinematic hardening rules is established for a nuclear material of SA508-3 steel. A modified isotropic hardening rule is proposed to capture the temperature-dependent cyclic softening, and a modified kinematic hardening rule is established to improve the prediction of the ratcheting behavior by introducing an exponential function related to the accumulated plastic strain. The stress is updated by the radial return mapping algorithm based on the backward Euler integration. A new expression of consistent tangent modulus for the equilibrium iteration is derived, and then the proposed model is implemented into the finite element software ABAQUS by using the user material subroutine (UMAT) to simulate the temperature-dependent ratcheting behaviors of SA508-3 steel. Finally, the ratcheting evolutions of notched bars at elevated temperature are obtained by uniaxial stress-controlled cyclic tests, and the nonuniform strain fields on the surface of plates with a center hole is measured by using the digital image correlation (DIC) technology. Comparisons between experimental and simulated results of a material point and structural examples show that the implemented model can provide reasonable predictions for ratcheting behaviors and nonuniform strain fields of structures at different temperatures for SA508-3 steel.

An enhanced void-crack-based Rousselier damage model for ductile fracture with the XFEM

This work presents a modelling strategy for ductile fracture materials by implementing the Rousselier damage model with the extended finite element method (XFEM). The implicit integration scheme and consistent tangent modulus based on a radial return mapping algorithm for this constitutive model are developed by the user-defined material subroutine UMAT in ABAQUS/Standard. To enhance the modelling of the crack development in the materials, the XFEM is used that allows modelling of arbitrary discontinuities, where the mesh does not have to be aligned with the boundaries of material interfaces. This modelling strategy, so-called Rousselier-UMAT-XFEM (RuX) model, is proposed to connect both concepts, which gives an advantage in predicting the material behaviour of ductile material in terms of voids and crack relation. This is the first contribution where XFEM is used in ductile fracture analysis for micromechanical damage problems. The results indicate that the RuX model is a promising technique for predicting the void volume fraction damage and crack extension in ductile material, which shows a good agreement in terms of stress–strain and force–displacement relationships.

An efficient and robust VUMAT implementation of elastoplastic constitutive laws in Abaqus/Explicit finite element code

This paper describes the development of an efficient and robust numerical algorithm for the implementation of elastoplastic constitutive laws in the commercial non-linear finite element software Abaqus/Explicit through a VUMAT FORTRAN subroutine. In the present paper, while the Abaqus/Explicit uses an explicit time integration scheme, the implicit radial return mapping algorithm is used to compute the plastic strain, the plastic strain rate and the temperature at the end of each increment instead of the widely used forward Euler approach. This more complex process allows us to obtain more precise results with only a slight increase of the total computational time. Corrector term of the radial return scheme is obtained through the implementation of a safe and robust Newton–Raphson algorithm able to converge even when the piecewise defined hardening curve is not derivable everywhere. The complete method of how to implement a user-defined elastoplastic material model using the radial return mapping integration scheme is presented in details with the application to the widely used Johnson–Cook constitutive law. Five benchmark tests including one element tests, necking of a circular bar and 2D and 3D Taylor impact tests show the efficiency and robustness of the proposed algorithm and confirm the improved efficiency in terms of precision, stability and solution CPU time. Finally, three alternative constitutive laws (the TANH, modified TANH and Bäker laws) are presented, implemented through our VUMAT routine and tested.

Unified viscoplasticity modelling and its application to fatigue-creep behaviour of gas turbine rotor

This paper presents an elasto-viscoplastic finite element modelling framework including the associated UMAT codes to investigate the high-temperature behaviour of a gas turbine rotor steel. The model used in the FE study is an improved and unified Chaboche–Lemaitre model which combines a power flow rule with non-linear anisothermal evolution of isotropic and kinematic hardening. The computational methodology is a three-dimensional framework following an implicit formulation and based on a radial return mapping algorithm. The UMAT is calibrated and validated across isothermal hold-time cyclic tests. The model is applied to a classical industrial gas turbine rotor to study the spatial heterogeneities and critical high stress areas within the rotor structure. The effect of thermal transients and geometry singularities on the development of residual stresses is underlined. Finally, the potential improvements and extensions of such FE viscoplastic analysis are discussed.

Simulation of finite-strain inelastic phenomena governed by creep and plasticity

Inelastic mechanical behavior plays an important role in many applications in science and engineering. Phenomenologically, this behavior is often modeled as plasticity or creep. Plasticity is used to represent the rate-independent component of inelastic deformation and creep is used to represent the rate-dependent component. In several applications, especially those at elevated temperatures and stresses, these processes occur simultaneously. In order to model these process, we develop a rate-objective, finite-deformation constitutive model for plasticity and creep. The plastic component of this model is based on rate-independent \(J_2\) plasticity, and the creep component is based on a thermally activated Norton model. We describe the implementation of this model within a finite element formulation, and present a radial return mapping algorithm for it. This approach reduces the additional complexity of modeling plasticity and creep, over thermoelasticity, to just solving one nonlinear scalar equation at each quadrature point. We implement this algorithm within a multiphysics finite element code and evaluate the consistent tangent through automatic differentiation. We verify and validate the implementation, apply it to modeling the evolution of stresses in the flip chip manufacturing process, and test its parallel strong-scaling performance.

Implicit implementation and consistent tangent modulus of a viscoplastic model for polymers

In this work, the phenomenological viscoplastic DSGZ model (Duan et al., 2001 [13]), developed for glassy or semi-crystalline polymers, is numerically implemented in a three-dimensional framework, following an implicit formulation. The computational methodology is based on the radial return mapping algorithm. This implicit formulation leads to the definition of the consistent tangent modulus which permits the implementation in incremental micromechanical scale transition analysis. The extended model is validated by simulating the polypropylene thermoplastic behavior at various strain rates (from 0.92s−1 to 258s−1) and temperatures (from 20°C to 60°C). The model parameters for the studied material are identified using a heuristic optimization strategy based on genetic algorithm. The capabilities of the new implementation framework are illustrated by performing finite element simulations for multiaxial loading.

Small strain elasto-plastic multiphase-field model

A small strain plasticity model, based on the principles of continuum mechanics, is incorporated into a phase-field model for heterogeneous microstructures in polycrystalline and multiphase material systems (Nestler et al., Phys Rev 71:1---6, 2005). Thereby, the displacement field is computed by solving the local momentum balance dynamically (Spatschek et al., Phys Rev 75:1---14, 2007) using the finite difference method on a staggered grid. The elastic contribution is expressed as the linear approximation according to the Cauchy stress tensor. In order to calculate the plastic strain, the Prandtl---Reuss model is implemented consisting of an associated flow rule in combination with the von Mises yield criterion and a linear isotropic hardening approximation. Simulations are performed illustrating the evolution of the stress and plastic strain using a radial return mapping algorithm for single phase system and two phase microstructures. As an example for interface evolution coupling with elasto-plastic effects, we present crack propagation simulations in ductile material.

Revisited Simo algorithm for the plane stress state

In this paper a new numerical approach of the elasto-plastic problem with mixed hardening in-plane stress state is proposed within the classical constitutive framework of small deformation. In the formalized problem, the non-zero normal component of the strain (namely, the component in the direction perpendicular to the plane of stress) is considered to be compatible with the plane stress. We eliminate the rate of strains which are developed normal to the plane and solve an appropriate plane problem in the strain setting. The integration algorithm for solving the elasto-plastic problem with mixed hardening realizes the coupling of Radial Return Algorithm, namely an update algorithm, with the finite element method applied to the discretized equilibrium balance equation. The solution of the pseudo-elastic nonlinear problem is then solved by Newton’s method. The numerical algorithm (which is an incremental one) consists of the Radial Return Mapping Algorithm, originally proposed by Simo and Hughes (1998) adapted to the stress plane problem and coupled with a pseudo-elastic problem. We rebuild the algorithmic formula giving rise to the plastic factor in terms of the trial stress state and the algorithmic elasto-plastic tangent moduli which is requested to evaluate the Jacobian that is necessary to determine the displacement. The proposed algorithm has been named the revisited Simo algorithm and is applied to exemplify the mode of integration of the bi-dimensional problem.

Mechanistic Rutting Prediction Using a Two-Stage Viscoelastic-Viscoplastic Damage Constitutive Model of Asphalt Mixtures

A two-stage viscoelastic-viscoplastic damage constitutive model and the corresponding parameters calibration method were briefly given. Then, the numerical algorithms of the proposed constitutive model were implemented based on a radial return mapping algorithm and realized using the user material subroutine in the well-known finite-element code. A rutting prediction method of asphalt pavement was proposed based on the two-stage viscoelastic-viscoplastic damage constitutive model along with a loading equivalence method. Laboratory wheel tracking tests under different temperatures, different loading levels, once variable amplitude loads, and step variable amplitude loads were conducted to evaluate the efficiency of rutting prediction. Finally, the test results were compared with finite-element simulated results. All the results indicate that the constitutive model and the corresponding rutting prediction method proposed in this paper can effectively represent the rutting developing rules under standard loading and complicated variable amplitude loading with reliable prediction accuracy. Meanwhile, the mechanism of rutting development can be characterized through analyzing the evolution rules of internal state variables based on the proposed constitutive model.

On the Solution of the 3D Crack Surface Contact Problem Using the Boundary Element Method

The efficient solution of the 3D crack surface contact problem utilizing the boundary element method (BEM) is presented. The dual discontinuity method (DDM), a special formulation of the BEM, is applied. This method deals directly with the relative displacements and the discontinuities of the tractions at the crack. For the normal behavior a unilateral contact is assumed and for the description of the tangential behavior Coulomb’s frictional law is utilized. The hard contact formulation is regularized by the application of the penalty method. An incremental iterative procedure based on a radial return mapping algorithm is applied for the solution of this non-linear problem. Based on the stress field the fracture mechanical parameters are determined by an extrapolation method for all increments of a characteristic load cycle. By the analysis of this load cycle the cyclic fracture mechanics values are obtained. Due to the non-linear nature of crack growth the simulation is implemented in the framework of a predictor-corrector scheme. For the investigation of the influence of the crack surface roughness on the behavior of cracks two numerical examples are presented.