Armstrong Frederick Type Research Articles

A cyclic plasticity model for secondary hardening due to strain-induced martensitic transformation

Abstract A model for secondary cyclic hardening due to strain-induced martensitic transformation in metastable austenitic stainless steels is proposed. The model is built on an Armstrong–Frederick type kinematic hardening framework. Secondary hardening is described by relating the size of the limiting surface of each backstress to an evolution rule for the martensite content. A memory surface is used to account for the influence of the plastic strain amplitude on martensitic transformation. The model properly estimated the stress response of three austenitic stainless steels (types 304/304L, 321, and 348) subjected to strain-controlled axial, torsional, and proportional loading. The characteristics of the model, the determination of material constants, and suggestions for future research are discussed.

Incremental variational homogenization of elastoplastic composites with isotropic and Armstrong-Frederick type nonlinear kinematic hardening

In order to investigate the behavior of elastoplastic composites exhibiting both isotropic and nonlinear kinematic hardening, we extend the Double Incremental Variational (DIV) formulation of Lucchetta et al. (2019), based on both the incremental variational principles introduced by Lahellec and Suquet (2007) and the formulation proposed by Agoras et al. (2016). However, the Armstrong-Frederick model (Armstrong and Frederick), which is very often used to describe nonlinear kinematic hardening and refers to the framework of non-associated plasticity (Chaboche, 1977), cannot be handled within the framework of generalized standard materials as required by the incremental variational principles on which the DIV formulation relies. That is why we work with an approximation of this model, namely the modified Chaboche model (Chaboche, 1983). As the dissipation potential associated with this model depends on internal variables, we have to extend the incremental variational principles of Lahellec and Suquet to such a situation. Then, we apply twice the variational procedure of Ponte Castañeda (1991), first to linearize the local behavior and then to deal with the intraphase heterogeneity of the thermoelastic Linear Comparison Composite (LCC) induced by the linearisation step. The resulting thermoelastic LCC with per-phase homogeneous properties is homogenized by classical linear schemes. We develop and implement this new incremental variational procedure for two-phase matrix-inclusions composites with an isotropic elastoplastic matrix exhibiting combined isotropic and nonlinear kinematic hardening. For various cyclic loadings, the predictions of the proposed DIV formulation compare favorably with Finite Element simulations based on the Armstrong-Frederick model.

On the ratcheting of the VT6 alloy in a range of loading scenarios

A set of experimental results concerning the accumulation of inelastic strain in the VT6 titanium alloy is presented. The considered tests include cyclic stress-controlled loading with monotonically increasing stress amplitude and constant mean stress. The focus of the study is on the accurate phenomenological modelling of the complex material behaviour. Four different material models are employed to capture the phenomena of ratcheting and nonlinear kinematic hardening. Two models are of the Armstrong-Frederick type and the remaining two models are of the Ohno-Wang type. Since all the considered models contain a big number of material parameters, a sophisticated nested procedure is used for the parameter identification; the obtained material parameters are validated by additional experiments. It is shown that all the models yield nearly the same accuracy; the limitations of the phenomenological approach are highlighted and discussed.

Two-time-scales and time-averaging approaches for the analysis of cyclic creep based on Armstrong–Frederick type constitutive model

The aim of this paper is the analysis of inelastic behavior under periodic cyclic loading regimes for materials showing recovery effects. Starting with the Armstrong–Frederick type model and applying the two-time-scale asymptotic technique, the constitutive equation for the mean inelastic strain rate and the evolution equation for the mean backstress variable are derived. The advantage of the presented technique is the closed analytical form of the solutions such that the influence of the loading profiles on the resulting slow cycle-by-cycle strain accumulation can be analyzed explicitly. To validate the derived equations, the original constitutive model is integrated numerically by applying the time-step procedure for various cyclic loading profiles. Numerical examples are presented to illustrate cyclic creep behavior of X20CrMoV12-1 heat resistant steel.

A Fine Grain, High Mn Steel with Excellent Cryogenic Temperature Properties and Corresponding Constitutive Behaviour.

A Fe-34.5 wt % Mn-0.04 wt % C ultra-high Mn steel with a fully recrystallised fine-grained structure was produced by cold rolling and subsequent annealing. The steel exhibited excellent cryogenic temperature properties with enhanced work hardening rate, high tensile strength, and high uniform elongation. In order to capture the unique mechanical behaviour, a constitutive model within finite strain plasticity framework based on Hill-type yield function was established with standard Armstrong-Frederick type isotropic hardening. In particular, the evolution of isotropic hardening was determined by the content of martensite; thus, a relationship between model parameters and martensite content is built explicitly.

A Coupled Armstrong-Frederick Type Plasticity Correction Methodology for Calculating Multiaxial Notch Stresses and Strains

Based on the pseudo-strain method, a computational modeling technique coupling with Armstrong-Frederick type nonlinear kinematic hardening rule (Jiang-Sehitoglu model) is developed to calculate the multiaxial stress-strain responses of notched components. The pseudo-strain-true notch stress curve is determined using Neuber’s rule. The material constants in Jiang-Sehitoglu model are calculated using the Ramberg-Osgood curve. The presented method is applied to simulate the notch-tip deformations of circumferentially notched 1070 steel and S460N steel shafts subjected to various loadings, including box, circle, V-shape, zigzag-shape, and butterfly-shape loading paths. The calculated strain loops are in accord with experimental data and show reasonable accuracy.

Sensitivity of polycrystal plasticity to slip system kinematic hardening laws for Al 7075-T6

Abstract The prediction of formation and early growth of microstructurally small fatigue cracks requires use of constitutive models that accurately estimate local states of stress, strain, and cyclic plastic strain. However, few research efforts have attempted to systematically consider the sensitivity of overall cyclic stress-strain hysteresis and higher order mean stress relaxation and plastic strain ratcheting responses introduced by the slip system back-stress formulation in crystal plasticity, even for face centered cubic (FCC) crystal systems. This paper explores the performance of two slip system level kinematic hardening models using a finite element crystal plasticity implementation as a User Material Subroutine (UMAT) within ABAQUS (Abaqus unified FEA, 2016) [1] , with fully implicit numerical integration. The two kinematic hardening formulations aim to reproduce the cyclic deformation of polycrystalline Al 7075-T6 in terms of both macroscopic cyclic stress-strain hysteresis loop shape, as well as ratcheting and mean stress relaxation under strain- or stress-controlled loading with mean strain or stress, respectively. The first formulation is an Armstrong-Frederick type hardening-dynamic recovery law for evolution of the back stress [2]. This approach is capable of reproducing observed deformation under completely reversed uniaxial loading conditions, but overpredicts the rate of cyclic ratcheting and associated mean stress relaxation. The second formulation corresponds to a multiple back stress Ohno-Wang type hardening law [3] with nonlinear dynamic recovery. The adoption of this back stress evolution law greatly improves the capability to model experimental results for polycrystalline specimens subjected to cycling with mean stress or strain. The relation of such nonlinear dynamic recovery effects are related to slip system interactions with dislocation substructures.

Modeling of I + II mixed mode crack initiation and growth from the notch

Abstract The fatigue crack initiation and growth from the notches under combined loading are modeled by the finite element method. The compact tension shear (CTS) specimens of Q345R steel are used to study the I + II mixed mode crack growth behavior. To consider the effect of the residual stress and loading history induced by fatigue crack growth, the dynamic crack propagation model is employed in the finite element simulation. A multiaxial fatigue damage criterion is employed and an Armstrong–Frederick type cyclic plasticity model was inputted as a UMAT to describe the material behavior. The fatigue crack initiation and growth from the notch root of the CTS specimen is determined by the stabilized stress–strain responses of the material points near the notch root and the fatigue damage of different material planes. The material point with maximum fatigue damage corresponds to the fatigue initiation position, and the crack growth orientation from the initiation position is identical to the material plane on which maximum fatigue damage arises. The fatigue crack propagates on the material plane with maximum crack growth rate. The results of prediction are in excellent agreement with the experimental observations in terms of crack initiation orientation, crack growth rate and crack growth path.

A generalized thermodynamically consistent distortional hardening model for Mg alloys

Abstract In the current work, a generalized distortional hardening model is developed by assuming each part of dissipation non-negative. More precisely, quadratic forms are assumed for dissipation and thermodynamical consistency is fulfilled naturally. In order to balance the contributions of isotropic and distortional hardening into dissipation, a weighting coefficient w is introduced in contrast to the work in Feigenbaum and Dafalias (2007). It is found that the saturation directions of distortional hardening are governed by this coefficient w , which is related with deformation modes. In particular, by setting the coefficient w = 2, the current generalized model recover to the one based on convex plastic potential, cf. Shi and Mosler (2013), and thus Armstrong–Frederick type distortional hardening law is obtained. The predictive ability of the current distortional hardening model is verified by comparison with experimental observations in three cases: stress-strain behaviours along different directions, biaxial tensile test and biaxial compression test. The evolution of normalized yield surfaces for the three cases is briefly discussed.

Inherent and induced anisotropic finite visco-plasticity with applications to the forming of DC06 sheets

In the current work we present a finite visco-plasticity model accounting for inherent and induced plastic anisotropy as well as Bauschinger effect for the interstitial free (IF) steels and its application to a forming process simulation of DC06 sheets. The inherent plastic anisotropy uses a Hill-48 type structural tensor whereas the induced anisotropy is modeled via its evolution accounting for dynamic (active) and latent (inactive) parts. The latter appears to be an eminent requirement for predicting the qualitative effect of the evolving dislocation microstructures under orthogonal loading path changes, i.e., the cross hardening. A nonlinear isotropic and Armstrong–Frederick type kinematic hardening is also involved. Finally, the rate dependence of the plastic response is incorporated using Johnson–Cook type formulation. The model is implemented as VUMAT user defined material subroutine for ABAQUS and used in a set of sensitivity analyses to present mentioned model features. The model parameters are identified based on a set of experiments involving monotonic shear, uniaxial tension, forward to reverse shear and plane strain tension followed by shear tests. Finally, the channel forming process of a DC06 sheet is simulated. A good agreement with the experimental findings is observed, in both the tool response history curves and the extent of spring-back which is conclusive on the final product geometry.

Numerical comparison of isotropic hypo- and hyperelastic-based plasticity models with application to industrial forming processes

Abstract Industrial forming processes are usually characterized by large plastic strains and rotations of material elements. This emphasizes the importance of reliable finite strain elastoplasticity models in corresponding finite element simulations. The aim of this work is to review and numerically compare two inherently different types of formulations of finite strain elastoplasticity, namely hypoelastic- and hyperelastic-based plasticity models, with special reference to their applicability in forming processes. Both models allow for nonlinear isotropic and kinematic hardening of Voce and Armstrong–Frederick type and were implemented as user material subroutines (UMAT) into ABAQUS/Standard. Several numerical tests were conducted to assess their respective capabilities. Interestingly enough, although both models led to remarkably different results in shear-dominated single element deformation tests, the structural simulations of a deep drawing, draw bending and thermoforming process delivered nearly congruent results. This suggests that both models are well-suited for modeling elastoplastic materials in common industrial forming processes.

Closed‐form matrix exponential and its application in finite‐strain plasticity

SUMMARYA new method to compute numerically efficient closed‐form representation of matrix exponential and its derivative is developed for 3 × 3 matrices with real eigenvalues. The matrix exponential is obtained by automatic differentiation of an appropriate scalar generating function in a general case, and highly accurate asymptotic expansions are derived for special cases in which the general formulation exhibits ill‐conditioning, for instance, for almost equal eigenvalues. Accuracy and numerical efficiency of the closed‐form matrix exponential as compared with the truncated series approximation are studied. The application of the closed‐form matrix exponential in the finite‐strain elastoplasticity is also presented. To this end, several time‐discrete evolution laws employing the exponential map are discussed for J2 plasticity with isotropic hardening and nonlinear kinematic hardening of Armstrong–Frederick type. The discussion is restricted to the case of elastic isotropy and implicit time integration schemes. In this part, the focus is on a general automatic differentiation‐based formulation of finite‐strain plasticity models. Numerical efficiency of the corresponding incremental schemes is studied in the context of the FEM. Copyright © 2014 John Wiley & Sons, Ltd.

Modeling of Fatigue Crack Growth under High-Low Sequence Loading

The fatigue crack growth behavior of one compact tension specimen of 16MnR steel under high-low sequence loading was investigated. The symmetric half finite element model under plane-stress state was used to calculate the elastic-plastic stress-strain responses, in which the Armstrong-Frederick type cyclic plasticity model was implemented as a user material subroutine UMAT of ABAQUS. A recently developed dynamic crack growth model was used to simulate the effects of high loading step on the successive low loading step. The detailed evolution process of the crack closure and cyclic plastic zone within the retardation region of fatigue crack growth was obtained. The extend of the crack closure, the size of cyclic plastic zone and the stress gradient have significant influence on the fatigue crack growth rate. The predicted fatigue crack growth rate is in good agreement with the experimental data.

Simulation of Fatigue Initiation and Non-Self-Similar Fatigue Crack Growth under Mixed Mode I/II Loading

The fatigue initiation and non-self-similar fatigue crack growth behavior of three notched compact tension and shear specimens of 16MnR steel under mixed mode I/II loading were investigated. The plane-stress finite element model with the implemented Armstrong-Frederick type cyclic plasticity model was used to calculate the elastic-plastic stress-strain responses. A recently developed dynamic crack growth model was used to simulate the effects of loading history on the successive crack growth. With the outputted numerical results, a multiaxial fatigue damage criterion based on the critical plane was used to determine the location of fatigue initiation. A formula of fatigue crack growth rate, which is based on the postulation that the fatigue initiation and crack growth have the same damage mechanism, was then used to calculate the transient crack growth rate and determine the non-self-similar crack growth path. The predicted fatigue initiation position, crack path and crack growth rate are in excellent agreement with the experimental data.

Cyclic plasticity modeling and finite element analyzes of a circumferentially notched round bar under combined axial and torsion loadings

A cyclic plasticity model suitable for fatigue damage modeling of metallic structures is presented and its finite element (FE) implementation is described within the small strain plasticity framework. The model uses the von Mises yield surface in stress space whose evaluation follows an Armstrong–Frederick type of nonlinear kinematic hardening rule, and the normality hypothesis in conjunction with the associative flow rule is assumed. An incremental implicit-iterative algorithm was employed for the numerical solution of resulting stress–strain equations, and the continuum tangent obtained from plasticity model was used in FE implementation. The developed FE computational model is applied in the cyclic deformation analysis of a circumferentially notched specimen in combined axial force-torsion loading tests. The computed notch root deformations were compared with measured notch root strain histories. An assessment of model predictions showed that non-proportional loading tests have been simulated with a good accuracy. The computed strain loops were in accord with experimental data and matched qualitatively with measured shear – axial strain histories irrespective of loading path of the test. In proportional balanced torsion-axial loading, the nonlinear shear strain – axial strain loops were also simulated properly. The errors in notch root strains were dependent on the loading path shape, and compared to axial strains, the shear strain errors were relatively greater. The computer solution times were also acceptable.

Sheet metal forming and springback simulation by means of a new reduced integration solid-shell finite element technology

Abstract The paper deals with the validation of a recently proposed hexahedral solid-shell finite element in the field of sheet metal forming. Working with one integration point in the shell plane and an arbitrary number of integration points in thickness direction, highly non-linear stress states over the sheet thickness can be incorporated in an efficient way. In order to avoid volumetric locking and Poisson thickness locking at the level of integration points the enhanced assumed strain (EAS) concept with only one EAS degree-of-freedom is implemented. A key point of the formulation is the construction of the hourglass stabilization by means of different Taylor expansions. This leads to the advantage that the sensitivity with respect to mesh distortion is noticeably reduced. The hourglass stabilization includes the assumed natural strain (ANS) concept and a kind of B-Bar method. So transverse shear locking and volumetric locking are eliminated. The finite element formulation incorporates a finite strain material model for plastic anisotropy as well as non-linear (Armstrong–Frederick type) kinematic and isotropic hardening. In this context the plastic anisotropy can be modeled by representing the yield surface and the plastic flow rule as functions of so-called structural tensors. The integration of the evolution equations is performed by means of an exponential map exploiting the spectral decomposition. The element formulation and material model have been implemented into the commercial code ABAQUS/Standard by means of the UEL interface for user-defined elements. Using an implicit time integration scheme numerical results for classical deep drawing simulations as well as springback predictions are presented in comparison to experimental measurements.

Simulation of cyclic plastic deformation response in SA333 C–Mn steel by a kinematic hardening model

Cyclic plasticity deals with non-linear stress–strain response of materials subjected to external repetitive loading. An effort has been made to describe cyclic plastic deformation behavior of the SA333 C–Mn steel by finite element based plasticity model. The model has been developed on the framework using a yield surface together with Armstrong–Frederick type kinematic hardening model. No isotropic hardening is considered and yield stress is assumed to be a constant in the material. Kinematic hardening coefficient and kinematic hardening exponent reduces exponentially with plastic strain in the proposed model. The proposed model has been validated through experimental results.

A combined isotropic-kinematic hardening model for the simulation of warm forming and subsequent loading at room temperature

Abstract A coupled isotropic-kinematic hardening material model was developed based on phenomenological observations of performed two stage experiments on a medium carbon steel – SAE 1144, where the first deformation is performed at elevated temperatures and the second deformation at room temperature. Above all, deformations with orthogonal loading at various temperatures were investigated in order to determine the influence of the loading direction as well as of the temperature. Bergstrom’s theory of work hardening as well as the nonlinear kinematic hardening of an Armstrong–Frederick type were used as a basis for the model development. In the proposed model a relationship between material coefficients of the classical Bergstrom model and temperature was investigated. The aim of the new material model was to introduce the least possible amount of new parameters as well as to facilitate the mathematical determination of parameters during the fitting of the model with experimental data. The developed model was implemented in an in-house FE-Code in order to simulate the material behavior due to the dynamic strain aging and the hardening behavior after the dynamic strain aging process. Representative simulation results were compared with the experimental data in order to validate the efficiency and the application range of the model.

等辺山形鋼のロール矯正プロセスに対する３次元有限要素解析

Three-dimensional finite element analysis of a roller straightening of an equal leg angle was performed in the present paper in order to validate the usefulness of the employment of FEM to the determination of the intermeshes of rolls in actual roller straightening process. SS400 was employed for the material of equal leg angle. For the plastic constitutive equation of the material, Armstrong–Frederick type combined hardening law was used. Initial shape of the equal leg angle had a uniform curvature lengthwise. A dynamic explicit procedure was used to realize the feeding of the equal leg angle into the intermeshed rolls by rotating the bottom rolls and to accomplish the calculation under the condition of frequent contact-separation between equal leg angle–roll interface. A static implicit procedure was subsequently used to obtain the final shape of equal leg angle by using the last state of equal leg angle by dynamic explicit process. Elastic spring elements were introduced in order to support upper rolls to realize the elastic deflection of the roller straightener apparatus. The spring constant was determined so that straightness of equal leg angle after straightening can be less than 0.1% of its length. The distribution of bending stress, bending plastic strain during and after straightening were observed and discussed. Finally, the effect of intermesh of final roll on the straightness of equal leg angle was calculated.

Constitutive modeling of cyclic plasticity deformation of a pure polycrystalline copper

Abstract Cyclic plasticity experiments were conducted on a pure polycrystalline copper and the material was found to display significant cyclic hardening and nonproportional hardening. An effort was made to describe the cyclic plasticity behavior of the material. The model is based on the framework using a yield surface together with the Armstrong–Frederick type kinematic hardening rule. No isotropic hardening is considered and the yield stress is assumed to be a constant. The backstress is decomposed into additive parts with each part following the Armstrong–Frederick type hardening rule. A memory surface in the plastic strain space is used to account for the strain range effect. The Tanaka fourth order tensor is used to characterize nonproportional loading. A set of material parameters in the hardening rules are related to the strain memory surface size and they are used to capture the strain range effect and the dependence of cyclic hardening and nonproportional hardening on the loading magnitude. The constitutive model can describe well the transient behavior during cyclic hardening and nonproportional hardening of the polycrystalline copper. Modeling of long-term ratcheting deformation is a difficult task and further investigations are required.