Nonlinear Isotropic Hardening Research Articles

Large‐scale elasto‐plastic topology optimization

AbstractThis work presents large‐scale elasto‐plastic topology optimization for design of structures with maximized energy absorption and tailored mechanical response. The implementation uses parallel computations to address multi million element three‐dimensional problems. Design updates are generated using the gradient‐based method of moving asymptotes and the material is modeled using small strain, nonlinear isotropic hardening wherein the coaxiality between the plastic strain rate and stress is exploited. This formulation renders an efficient state solve and we demonstrate that the adjoint sensitivity scheme resembles that of the state update. Furthermore, the KKT condition is enforced directly into the path dependent adjoint sensitivity analysis which eliminates the need of monitoring the elasto‐plastic switches when calculating the gradients and provides a straight forward framework for elasto‐plastic topology optimization. Numerical examples show that structures discretized using several millions degrees of freedom and loaded in multiple load steps can be designed within a reasonable time frame.

Modeling of LCF Behaviour on AISI316L Steel Applying the Armstrong-Frederick Kinematic Hardening Model.

The combination of kinematic and isotropic hardening models makes it possible to model the behaviour of cyclic elastic-plastic steel material, though the estimation of the hardening parameters and catching the influence of those parameters on the material response is a challenging task. In the current work, an approach for the numerical simulation of the low-cycle fatigue of AISI316L steel is presented using a finite element method to study the fatigue behaviour of the steel at different strain amplitudes and operating temperatures. Fully reversed uniaxial LCF tests are performed at different strain amplitudes and operating temperatures. Based on the LCF test experimental results, the non-linear isotropic and kinematic hardening parameters are estimated for numerical simulation. On comparing, the numerical simulation results were in very good agreement with those of the experimental ones. This presented method for the numerical simulation of the low-cycle fatigue on AISI316 stainless steel can be used for the approximate prediction of the fatigue life of the components under different cyclic loading amplitudes.

Modelling the Response of Timber Beams Under Fire

AbstractA fundamental requirement for analysing timber structures under fire is to consider the degradation of material properties with temperature. Therefore, the objective of this study is to propose a model that accounts for the variation of the thermo-physical properties, the development of char, and its evolution with temperature. This model integrates a sequential coupling of heat transfer analysis with structural response. The degradation of the material properties is accounted for through the regulatory approach recommended in Eurocode 5. The stress analysis employs an elasto-plastic model with nonlinear isotropic hardening. Implementation of the model is achieved within the Abaqus suite of finite element software using external subroutines. The model's predictions align well with experimental data, accurately reproducing both thermal and structural responses. Specifically, the model accurately predicts temperature profiles, displacements, and the depth of the charred layer, which initiates above 300 °C. Additionally, for rectangular sections, it was observed that exposure of all faces to fire results in a non-rectangular residual section. Furthermore, employing the temperature-dependent thermal property curves suggested by EC5 yields satisfactory results when predicting the fire resistance of softwood timber structures.

Numerical method for the primary torsional capacity of arbitrary steel cross sections considering nonlinear plastic behaviour

In structural design, plastic capacities of cross sections are of great importance for evaluating the capacity of steel members. Moreover, when it comes to problems of member stability, states of partial plasticisation are of interest as well. Depending on the cross section geometry and the material law to be considered, the determination can be challenging. This particularly applies to load conditions characterised by shear stress states, as for instance in the case of primary torsion (St. Venant's torsion). Simplifications are therefore often made with regard to the cross section geometry and engineering models are applied if the plastic limit moment of primary torsion Mpl,xp is to be determined. However, numerical methods allow a precise determination of properties and stress distributions of arbitrary cross section. In the present study, corresponding finite element formulations are combined with a plasticity algorithm for torsional analysis considering effects of nonlinear isotropic hardening. The approach approximates the nonlinear material law by a multilinear curve, which is stepwise evaluated based on linear hardening formulations. It allows determining states of partial cross section plasticisation for any strain or deformation state as well as the identification of the plastic limit capacity of the cross section. Results of the implemented algorithm are presented and verified for different common steel profiles showing excellent agreement to comparative studies.

On the development of a constitutive model for steel subjected to fire and explosion

The behavior of steel under thermal and mechanical loading including high transient temperatures and strain rates (e.g., fire, creep or explosion) is extremely complex. To study this behavior, we propose a constitutive model accordance with the laws of thermodynamic, the principle of virtual power and experimental results. The thermodynamic framework considers the Clausius-Duhem inequality and maximum dissipation rate principle, as well as the theory of finite deformations, linear isotropic viscoelasticity, nonlinear isotropic and kinematic hardening, local and non-local viscoplasticity, local and non-local anisotropic viscodamage and parameters associated to material behavior dependent on loadings. The components of the thermodynamic conjugate forces are defined from the Helmholtz free energy function and the rate of energy dissipation, which are postulated considering the creep phenomenon, thermal-mechanical transient phenomenon, cyclic phenomenon and plastic wave propagation phenomenon. The balance of microforces is defined based on the principle of virtual power. A novel rule of non-associated thermo-viscoplastic flow of the Perzyna type and a novel rate dependent local and non-local damage evolution law are introduced from constitutive model. The verification of the model was carried out in two benchmark studies composed of fire tests. The predictions of the model have shown in according with the results of the experiments. A sensitivity study was also performed with a view to analysis the influence of the parameters in achieved results. From this study a simplified constitutive model is presented. Benchmark studies composed of explosion tests are part of future work.

Experimental investigation and numerical modelling of the cyclic plasticity and fatigue behavior of additively manufactured 316 L stainless steel

This study addresses the critical need for a constitutive model to analyze the cyclic plasticity of additively manufactured 316L stainless steel. The anisotropic behavior at both room temperature and 300 °C is investigated experimentally based on cyclic hysteresis loops performed in different orientations with respect to the build direction. A comprehensive constitutive model is proposed, that integrates the Armstrong-Frederick nonlinear kinematic hardening, Voce nonlinear isotropic hardening and Hill's anisotropic yield criterion within a 3D return mapping algorithm. The model was calibrated to specimens in the 0° and 90° orientations and validated with specimens in the 45° orientation. A single set of hardening parameters successfully represented the elastoplastic response for all orientations at room temperature. The algorithm effectively captured the full cyclic hysteresis loops, including historical effects observed in experimental tests. A consistent trend of reduced hardening was observed at elevated temperature, while the 45° specimen orientation consistently exhibited the highest degree of strain hardening. The applicability of the model was demonstrated by computing energy dissipation for stabilized hysteresis loops, which was combined with fatigue tests to propose an energy-based fatigue life prediction model.

Highly detailed structural integrity assessment of the reactor pressure vessel nozzle of Atucha-I during a pressurized thermal shock event

In pressurized water reactor (PWR) nuclear plants, reactor pressure vessel (RPV) nozzles are critical due to geometric stress concentrations that worsen with large temperature gradients. In the context of Atucha-I Long Term Operation (LTO) assessment, this work presents a high-fidelity thermo-mechanical simulation approach to verify the stress levels on the main nozzles of the vessel under a pressurized thermal shock event. Nucleoeléctrica Argentina S.A. (NA-SA) has provided the boundary conditions, including the thermal plume that results from computational fluid dynamics (CFD) simulations of the transient event. Then, using a one-way and non-conformal projection procedure, CFD data is coupled to a highly-detailed 3D finite element model (FEM), including the RPV, closure head, nuts and bolts, and moderator tank, all interconnected through non-linear contact condition that accounts for friction. Finally, a 3D FEM submodel assuming an elastoplastic material behavior with non-linear isotropic hardening is used to verify the structural integrity of the nozzle. Results indicate that the plastic strain and stresses on the nozzles are low enough to ensure the correct function of the reactor during the LTO period.

Thermal-structural analysis for reusable thrust chamber using temperature-dependent CuZr alloy model adapted to cryogenic to elevated temperature

The inner liners of reusable regeneratively-cooled thrust chambers endure significant thermal and mechanical loads, resulting in doghouse deformation and even cracks during cyclic operations. To correctly reveal the failure mechanism of the thrust chamber inner liner, this study investigates the material properties for CuZr alloy and develops a temperature-dependent material model spanning from cryogenic to elevated temperatures. Firstly, we experimentally obtain the thermal and mechanical properties of the material in the temperature range from 77 K to 900 K. The experimental data reveal that the material properties vary considerably with temperature. Then, to precisely characterize the mechanical behavior of CuZr alloy under cyclic loading across a wide temperature range, a material model that combines Voce nonlinear isotropic hardening model with Chaboche nonlinear kinematic hardening model is proposed, and temperature-dependent parameters of this model are calibrated and verified. Finally, the developed material model is applied to perform the thermal-structural analysis on a reusable thrust chamber, with the aim of investigating its failure mechanism. The results demonstrate the capability of the developed model to accurately replicate the doghouse deformation of the inner liner and quantitatively capture the ratcheting effect.

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.

Cyclic behavior of a slit damper proposed for precast concrete beam-column connections

Since the 2017 Puebla-Morelos Earthquake, interest in limiting damage to structural components has exponentially increased in Mexico. Consequently, the use of energy dissipation devices has received renewed attention. Following that line, this paper studies an attractive solution, which is a slit damper proposed as a passive energy dissipation device for, but not limited to, precast beam-column connections. Slit dampers have been studied for their cost-effectiveness and ease of application. They present stable hysteretic behavior and dissipate substantial amounts of energy under cyclic loading while maintaining less ductile elements essentially elastic. The structural behavior of the proposed device was evaluated theoretically, followed by experimental verification. Quasi-static incremental and constant-amplitude cyclic tests were conducted on eight specimens, with two of them having non-uniform struts. Test results are presented and discussed, emphasizing ductility and energy dissipation capacity. The proposed damper showed stable hysteretic behavior and adequate seismic performance. Using non-uniform struts significantly increased the damper's deformation capacity and fatigue performance, which is desirable for seismic zones where long-duration earthquakes are expected, such as in Mexico City. Numerical modeling techniques were also validated with finite element analyses. It was shown that the proposed damper's hysteretic response could be accurately predicted using the combined non-linear isotropic and kinematic hardening model.

Damage to fracture in offshore engineering materials under several stress states: Blowout preventer valve application

This contribution proposes a study of the mechanical behavior from damage to fracture of normalized and annealed samples of AISI 4340 steel. Besides, a numerical study in a security device of offshore industry is conducted. Experimental tests were initially performed on smooth and notched cylindrical specimens subjected to monotonic tension as well as on rectangular specimens subjected to pure shear and a combination of shear and tension loads. Such experimental tests were selected to observe different combinations of the stress triaxiality and the normalized third invariant concerning the stress state. The two types of heat treatments were considered to achieve different levels of ductility and verify their influence on the fracture mode of the alloy. The approach based on the second invariant of the deviatoric stress tensor ( J 2), with nonlinear isotropic hardening, is assumed to describe the mechanical behavior of the material in axisymmetric and three-dimensional numerical simulations. Finite elements simulations were carried out to analyze the performance of a simplified version of a mechanism of a blow-out preventer-BOP valve. The experimental and numerical curves of force versus displacement were compared, highlighting the effect of the stress triaxiality, the third invariant and calibration condition on the behavior of the material. The evolution of the equivalent plastic strain was plotted, and the fracture onset compared with its maximum value. The results show that the AISI 4340 alloy is highly dependent on the stress triaxiality and third invariant. Moreover, it can be concluded that different calibration conditions for the isotropic hardening curve can result in different levels of forces for the correct performance of the BOP in a critical situation.

Numerical Investigation on the Service Life of a Liquid Rocket Engine Thrust Chamber

Rocket engine thrust chambers withstand very high temperatures and thermal gradients during service that induce multiple damaging phenomena such as plasticity, low-cycle-fatigue (LCF) and creep. Numerical models can be used during the design of these mechanical components in order to simulate the main mechanical damaging processes, accounting for complex material behavior as due to non-linear hardening phenomena and viscoplasticity. This work represents an improvement upon previous research by the authors, with particular reference to the addition of the Wang–Brown fatigue criterion, to consider the effects of multiaxiality and non-proportionality of loads, and the Voce model to account for non-linear isotropic hardening. A precipitation hardened copper alloy has been considered as the material of the thrust chamber internal structure. The most critical areas resulted to be on the internal surface of the chamber and in particular in correspondence of the throat region, consistently with experimental tests available in the literature conducted on similar geometries. Results demonstrated that low cycle fatigue and thermal ratcheting (plastic instability) are the dominant damaging phenomena for the considered test case.

Formulation and Numerical Solution of the Stability Loss Problem of Elastic-Plastic Shells of Revolution with an Elastic Filler under Combined Axisymmetric and Torsional Loadings

A dynamic statement and a method for numerically solving the buckling problems of elastoplastic shells of revolution with filler in axisymmetric and non-axisymmetric shapes under quasi-static and dynamic loading are presented within the framework of two approaches. In the first approach, the problem of elastic-plastic deformation and buckling of shells of revolution with an elastic filler under combined axisymmetric loading with torsion is formulated in a two-dimensional (generalized axisymmetric) formulation based on the hypotheses of the shells theory of the Timoshenko type and the Winkler foundation. The constitutive relations are written in the cylindrical system of Euler coordinates. For each shell element, a local Lagrangian coordinate system is introduced. Kinematic relations are recorded in the current state metric. The distribution of the displacement velocity components over the shell thickness and strain rate tensors in the local basis is written as the sum of the momentless and moment components, which, in turn, are written as the sum of the symmetric and asymmetric parts in the local and in the general basis. The elastoplastic properties of the shell material are taken into account within the framework of the theory of flow with nonlinear isotropic hardening. To take into account non-axisymmetric forms of buckling, the desired functions (both displacements and forces, moments, contact pressure) are expanded into a Fourier series in the circumferential direction. The variational equations of shell motion are derived from the general equation of dynamics. The contact between the shell and the deformable filler is modeled based on the conditions of non-penetration along the normal and free slip along the tangent. The variational equations of shell dynamics for axisymmetric and nonaxisymmetric processes are interconnected through the physical relations of the theory of plasticity. They take into account large axisymmetric shape changes and the instantaneous stress-strain state of the shell. At the initial stage of the nonaxisymmetric buckling process, the deflections are small; therefore, the equations of nonaxisymmetric buckling are obtained as linearized with respect to nonaxisymmetric forms. To initiate nonaxisymmetric buckling modes, initial nonaxisymmetric deflections are introduced. To solve the defining system of equations, a finite-difference method and an explicit time integration scheme of the “cross” type are used. The second approach is based on continuum mechanics hypotheses and implemented in a three-dimensional setting. Both approaches make it possible to simulate the nonlinear subcritical deformation of shells of revolution with an elastic filler, to determine the ultimate (critical) loads in a wide range of loading rates, taking into account geometric shape imperfections, to study the processes of buckling in axisymmetric and non-axisymmetric shapes under dynamic and quasi-static complex loading by tension, compression, torsion, internal and external pressure. The results of numerical simulation are compared with experimental data on the torsion of steel cylindrical elastoplastic shells ( R / h = 1.45) with an elastic filler.

A framework for formulating and implementing non-associative plasticity models for shell buckling computations

In modelling the behavior of thick-walled metal shells under compressive loads, the use of J2 flow theory can lead to unrealistic buckling estimates, while alternative ‘corner’ models, despite offering good predictions, have not been widely adopted for structural computations due to their complexity. The present work develops a new and efficient plasticity model for predicting the structural response of compressed metal shells. It combines the simplicity of the Von Mises yield surface, with a non-associative flow rule, mimicking the effect of a yield surface corner. This allows for tracing the equilibrium path of the loaded shell and identifying consistently structural instability, employing a single constitutive model. A robust backward-Euler integration scheme, suitable for both three-dimensional (solid) and shell elements is developed, along with the corresponding consistent algorithmic moduli for nonlinear isotropic hardening materials, accounting rigorously for the nonlinear dependence of plastic straining on the direction of strain increments. The model is implemented in ABAQUS as a user material subroutine. Simulations of thick-walled metal cylinders under compression predict structural instability in good agreement with experimental data.

Cyclic Behavior of Material after Single Loading of Cavitation Peening

Cavitation peening has great potential to gain popularity over shot peening in industrial applications for improving the fatigue performance of materials. Improvement in fatigue performance occurs due to the introduction of compressive residual stress. However, the roughening of the surface caused by cavitation peening remains a matter of concern. It has not been investigated adequately in the literature. The cyclic behavior of a material is well known to be adversely affected by surface roughness. An experimental investigation is quite tricky for exploring the underlying role played by the surface roughness and the residual stress. Thus, the current study aims at the numerical investigation of the cyclic behavior of material after cavitation peening load. The 304L austenitic stainless steel has been considered for the present investigation. Finite element simulations (3D) were performed in ABAQUS considering combined non-linear isotropic and kinematic hardening behavior along with ductile damage model. The cavitation peening load is represented by a Gaussian profile defined by peak pressure and radial extent. The effect of cavitation peening (single load) on the material’s elasto-plastic response and subsequent fatigue behavior is studied. The increment in the peak pressure (or the decrement in the radial size) of the peening load is found to shift the maximum stress/strain and damage initiation site from the sub-surface to the surface region. Moreover, the cyclic loading is found to unify the stress distribution along the depth near the surface.

Modeling and simulation of ductile damage in forged steel used to encapsulate high-level radioactive waste

Andra, the French national radioactive waste management agency, is in charge of studying the disposal of high-level and long-lived intermediate-level waste (HLW and ILW-LL) in a deep geological repository. According to the reference concept, it is planned to encapsulate high-level waste in non-alloy P285NH steel overpacks before inserting them into horizontal steel cased micro-tunnels. This work is a part of the study about the long-term behavior of a welded steel overpack subjected to external hydrostatic pressure and several localized loading paths. Indeed, the main objective of this work is to develop the most suitable model for non-alloy steel P285NH to be used in the prediction of the long-term overpack behavior. Dealing with a ductile steel, elastoplastic constitutive equations accounting for mixed nonlinear isotropic and kinematic hardening strongly coupled with ductile isotropic damage are adopted. They are formulated based on the classical thermodynamics of irreversible processes framework with state variables at the macroscopic scale, (Germain, 1973) (Lemaitre 1985, Saanouni 2012). In this paper, a new coupling formulation between the scalar isotropic ductile damage and the deviatoric and spherical part of the Cauchy stress and elastic strain tensors is proposed. In order to calibrate the developed model on P285NH steel, multiple tensile tests are performed using classical cylindrical specimens, notched specimens and double notched specimens. In the last part, some experimental fields are measured using digital image correlation. Application is made to a simplified overpack represented by thick walled cylinder subject to compressive loading path. A FEM (Finite Element method) crushing operation of an overpack’s cylindrical part has simulated and analysed.

Experimental verification of computational and sensitivity analysis on substrate deformation and plastic strain induced by hollow thin-walled WAAM structure

PurposeThis paper aims to numerical and experimental analysis on substrate deformation and plastic strain induced by wire arc additive manufacturing.Design/methodology/approachThe component has the form of a hollow, rectangular thin wall consisting of 25 deposition layers of SS316L on an SS304 substrate plate. Thermo-mechanical finite element analysis was applied with Goldak’s double-ellipsoidal heat-source model and a non-linear isotropic hardening rule based on von Mises’ yield criterion. The layer deposition was modelled using simplified geometry to minimize overall pre-processing work and computational time.FindingsA new material modelling of SS316L was obtained from the chemical composition of the evolved component characterized by scanning electron microscope/energy dispersive X-ray and further generated by an advanced material-modelling software JMatPro. In defining heat-transfer coefficients, transient thermometric analysis was first performed in the bead and on the substrate, which was followed by an adjustment of the heat-transfer coefficients to reflect the actual temperature distribution. Based on the adjusted model and boundary conditions, sensitivity analysis was conducted prior to the ultimate simulation of substrate deformation and equivalent plastic strain. Furthermore, this simulation was verified by conducting a series of automated wire + arc additive manufacturing tests using robotic gas Metal arc welding with distortion measured by coordinate-measurement machine and equivalent plastic strain measured by optical three-dimensional-metrology measurements (Gesellschaft für Optische Messtechnik).Originality/valueIt can be concluded that a proper numerical computation using the adjusted model and property-evolved material exhibits a similar trend with acceptable agreement compared to the experiment by yielding an error percentage up to 30% for deformation and up to 21% for equivalent plastic strain at each individual measurement point.

Modelling of temperature and strain rate dependent behaviour of pearlitic steel in block braked railway wheels

Block braked railway wheels are subjected to thermal and rolling contact loading. The thermal loading results in high temperatures and thermal stresses which cause slow time dependent processes such as creep, relaxation and static recovery of the wheel material. At the same time, the rolling contact loading implies a very fast mechanical load application. This paper is focused on material modeling of pearlitic steel for a wide range of loading rates at elevated temperatures. The starting point is a viscoplasticity model including nonlinear isotropic and kinematic hardening. The Delobelle overstress function is employed to capture strain rate dependent response of the material. The model also includes static recovery of the hardening to capture slower viscous (diffusion dominated) behaviour of the material. Experiments for the pearlitic wheel steel ER7 in terms of cyclic strain-controlled uniaxial tests with hold-time, uniaxial ratchetting tests including rapid cycles and biaxial cyclic tests with tension/compression and torsion are used to calibrate the material model. These experiments were performed under isothermal conditions at different temperatures. In the ratchetting tests, higher loading rates are obtained and these have been used to calibrate the high strain rate response of the viscoplasticity model. The paper is concluded with a numerical example of a block braked wheel where the importance of accounting for the viscoplasticity in modelling is highlighted.

Material model identification from set of experiments and validation by DIC

A methodology for calibration and validation of material models using more simultaneous experiments is presented. The procedure contains the selection of suitable parameters for identification using a sensitivity analysis, parameters identification, analysis of usability of the material model, and validation of the material model. Three elastoplastic models are evaluated in this study: the bilinear isotropic hardening model with Von Mises yield condition, the bilinear isotropic hardening model with the Hill anisotropic yield condition, and the nonlinear isotropic hardening model with Hill anisotropic yield condition. Five proportional as well as nonproportional monotonic loading experiments conducted on aluminium alloy 2124-T851 are used for the presentation of the inverse approach application, whereas four of them are the source of identification. The validation of calibrated material models is done based on the strain field measurement realised by the Digital Image Correlation method. The nonlinear isotropic hardening rule together with Hill yield condition brings the best description of stress–strain behaviour of the material under investigation.

A general phase-field model for fatigue failure in brittle and ductile solids

In this work, the phase-field approach to fracture is extended to model fatigue failure in high- and low-cycle regime. The fracture energy degradation due to the repeated externally applied loads is introduced as a function of a local energy accumulation variable, which takes the structural loading history into account. To this end, a novel definition of the energy accumulation variable is proposed, allowing the fracture analysis at monotonic loading without the interference of the fatigue extension, thus making the framework generalised. Moreover, this definition includes the mean load influence of implicitly. The elastoplastic material model with the combined nonlinear isotropic and nonlinear kinematic hardening is introduced to account for cyclic plasticity. The ability of the proposed phenomenological approach to naturally recover main features of fatigue, including Paris law and Wöhler curve under different load ratios is presented through numerical examples and compared with experimental data from the author’s previous work. Physical interpretation of additional fatigue material parameter is explored through the parametric study.