Back Stress Component Research Articles

Springback behavior after air bending of pre-strained AA 6016-T4 sheets: Influence of dislocation density and backstress on model accuracy

Predicting springback in sheets of aluminum alloys is challenging, particularly for complex strain path deformation used to form the material. This paper presents experimental and modeling results for air bending of AA 6016-T4 sheet material. Prior to bending, pre-strains were applied to the specimens in uniaxial tension, plane-strain tension, and biaxial tension. Specimens were then cut from the pre-strained material and were subjected to air bending using three different bend angles in a v-die, after which springback was measured. It was found that greater levels of pre-strain result in larger springback magnitudes, and that the biaxial pre-strained specimens generally exhibited greater springback than the others. A CPFE – EPSC model with phenomenological backstress component in the hardening law was used to predict springback magnitude across the different pre-strain paths, pre-strain levels, and imposed air bend angles. It was observed that the more significant the strain path change, as was seen in the case of biaxial tension pre-strain followed by bending, the greater the influence of backstress on model accuracy, owing to the activation of new slip systems. When backstress was excluded from the model, a 3.2 % prediction error in springback angle was observed in the worst case, versus an error of 0.8 % when backstress was included. Backstress was correlated to more rapid dislocation density development on both the tensile surface of the sheet, and also through the thickness, after the strain path change. On the other hand, for the case of plane-strain pre-strain followed by bending, dislocation density development was more gradual and the backstress impact on model prediction was less significant, with the same slip systems active during both pre-strain and bending. Therefore, it is seen that the influence of backstress on accuracy springback modeling in AA 6016-T4 varies significantly depending on the strain path. Thus, the use of a phenomenological backstress law within the CPFE-EPSC framework is seen to be an accurate and efficient approach for modeling springback after strain path changes that can occur during industrial forming applications.

Simulations of the effect of shot peening backstress on nanoindentation

Shot peening is a mechanical surface treatment that can introduce compressive residual stress and work hardening simultaneously. This work hardening, considered as a modification of the elastic region with plastic strain, can be modeled with two types of contributions: isotropic hardening and kinematic hardening. In order to characterize the mechanical properties of the treated surface using the instrumented indentation technique, the effect of the backstress associated with kinematic hardening should be studied, especially for works related to fatigue loading. In this paper, the distribution of three backstress components is obtained by shot peening simulations on a nickel-based alloy, Inconel 718, commonly used in the aerospace industry, and a series of indentation simulations are carried out using a spherical tip with different equivalent backstress levels. For Inconel 718, the third backstress component, which has the slowest evolution rate, is found to have the most significant influence on the response. However, compared to the effect of residual stress and cumulated plastic strain, the effect of backstress can be neglected.

Traversing with quantitative fidelity through the glass transition of amorphous polymers: Modeling the thermodynamic dilatational flow of polycarbonate

We follow the assumption that the dilatational response of glassy polymers can be characterized by a back-stress type analog that includes a thermal expansion for each elastic component and with a viscosity that is dependent on the expansion of the elastic back-stress component. To this, we add the assumption of an unloaded equilibrium temperature that correlates to the past processing through the viscous flow. After setting this in a thermodynamically consistent structure, elastic, elastic back-stress, thermal expansion, back-stress thermal expansion, heat capacity, and viscous damping are evaluated using existing experiments for the response of polycarbonate over the glassy and rubbery ranges. For the demonstration, this is done entirely using a WLF shift factor that is augmented to include, in addition, back strain superposition. We then examine the resulting model under different thermal and mechanical loadings that have the material passing through the glass transition.

Trans-twin dislocations in nanotwinned metals

The strength-controlling dislocation mechanism of a material can be illuminated by partition of the flow stress into its effective stress and back stress components. Recent experiments report a nearly constant saturated effective stress of about 100 MPa for nanotwinned Cu with a range of nanotwin thicknesses less than 100 nm [Z. Cheng et.al., Proc. Natl. Acad. Sci. U.S.A. 119 (2022) e2116808119]. This surprising result implies that the effective stress is controlled by the long dislocations spanning multiple nanotwin lamellae, termed trans-twin dislocations, rather than the commonly thought threading dislocations confined within individual nanotwin lamellae. Here we use molecular dynamics to simulate the formation and motion of trans-twin dislocations across multiple nanotwin lamellae. We show that the interconnected segments of a trans-twin dislocation can glide concertedly on the corrugated {111} slip planes in consecutive nanotwin lamellae with little resistance from coherent twin boundaries. Our results indicate that the finite resistance to slip transmission across coherent twin boundaries can set a lower limit of the effective obstacle spacing to hinder dislocation glide and thus dictate the upper limit of the saturated effective stress of nanotwinned metals. This work provides new understanding of the strength-controlling mechanism in nanotwinned metals.

A computationally fast and accurate procedure for the identification of the Chaboche isotropic-kinematic hardening model parameters based on strain-controlled cycles and asymptotic ratcheting rate

The Chaboche isotropic-kinematic hardening (CIKH) model provides a versatile and realistic description of the material stress–strain behavior under generic multiaxial cyclic loadings. However, identifying the backstress parameters is challenging, and can be formulated as an optimization problem using different approaches. Instead of a computationally expensive pointwise search, in this paper the global properties of the cyclic curves are fitted to the experimental data. The conditions introduced are the hysteresis areas, peak stress values and tangent moduli at the extreme points, however the framework can be easily adapted to other target quantities. One linear and two non-linear backstress components of the kinematic hardening model are introduced, although the analytical equations developed can be used to refine the model further, with more components. Two stabilized cycles are required to identify the main kinematic parameters. New analytical expressions for asymptotic ratcheting rates in uniaxial tests are developed and then used to tune the dynamics of the slightly non-linear (hence, slowest) backstress component. After obtaining the kinematic parameters, isotropic hardening laws can also be identified, by considering the evolution of the extreme points of the strain-controlled cycles before stabilization. Practical demonstrations of the procedure are provided by experimental tests carried out on a 7075-T6 aluminum alloy, 42CrMo4+QT steel, and a high-silicon ferritic ductile cast iron. An accurate reproduction of the material behavior is achieved, at a negligible computational cost.

Comparative assessment of strain-controlled fatigue performance of SS 316L at room and low temperatures

In the present study, strain-controlled low cycle fatigue (LCF) behaviour of SS 316L at room temperature (RT) and −80 ℃ were investigated. RT LCF specimens show cyclic softening, whereas −80 ℃ specimens show initial cyclic softening followed by hardening due to strain-induced martensitic transformation (SIMT). Fatigue damage evolution mechanism is proposed for specimens tested at −80 °C, where compressive stresses develop due to SIMT and retards the crack growth. This leads to an increase in fatigue life for −80 °C specimens. An increase in back stress component at the initiation of secondary hardening is noticed for −80 °C specimens.

Modeling of microscale internal stresses in additively manufactured stainless steel

Additively manufactured (AM) metallic materials often comprise as-printed dislocation cells inside grains. These dislocation cells can give rise to substantial microscale internal stresses in both initial undeformed and plastically deformed samples, thereby affecting the mechanical properties of AM metallic materials. Here we develop models of microscale internal stresses in AM stainless steel by focusing on their back stress components. Three sources of microscale back stresses are considered, including the printing and deformation-induced back stresses associated with as-printed dislocation cells as well as the deformation-induced back stresses associated with grain boundaries. We use a three-dimensional discrete dislocation dynamics model to demonstrate the manifestation of printing-induced back stresses. We adopt a dislocation pile-up model to evaluate the deformation-induced back stresses associated with as-printed dislocation cells. The extracted back stress relation from the pile-up model is incorporated into a crystal plasticity (CP) model that accounts for the other two sources of back stresses as well. The CP finite element simulation results agree with the experimentally measured tension–compression asymmetry and macroscopic back stress, the latter of which represents the effective resultant of microscale back stresses of different origins. Our results provide an in-depth understanding of the origins and evolution of microscale internal stresses in AM metallic materials.

Constitutive model for cyclic behavior of mild steel under various strain amplitudes

To enable accurate simulation of steel material under various loadings, cyclic behaviors of mild steel Q235 were extensively investigated through experimental analysis and constitutive modeling in this work. The studied steel was tested under cyclic loading protocols with constant strain amplitudes and variable strain amplitudes. The evolutions of stress amplitude, effective stress and back stress components, and elastic stiffness during the whole cyclic loading history are quantitatively scrutinized. The results demonstrate that mild steel exhibits cyclic capacity depended strongly on strain amplitude. Besides, the strain history effect depends on both the historical maximum strain amplitude and the target unloading strain amplitude, indicating the existence of a critical state for the incomplete evanescence of strain memory surface to be activated. An abrupt drop of effective stress at onset of plastic straining is observed in experimental tests and contributes to the early re-yielding of the reversed curve. Furthermore, elastic stiffness in the cyclic loading phase differs much from that in initial tension. Based on those observations, a new constitutive model with robust ability to predict hysteretic behavior of structural steel under complex loading paths is proposed. This model considers the strain range dependence and the strain history dependence effects. In addition, a new method to identify the yield points of hysteretic loops is proposed in this paper to enable simulations of back stress with the Armstrong-Frederick hardening model without considering the strain range dependence effect. The validity of this model is demonstrated through comprehensive comparison between simulated results and experimental data.

Modeling size effects in microwire torsion: A comparison between a Lagrange multiplier-based and a [formula omitted] gradient crystal plasticity model

The size-dependent response of metallic microwires under monotonic and cyclic torsion is modeled taking a reduced-order strain gradient crystal plasticity approach involving a single scalar-valued micromorphic variable. It is compared with the response predicted by the CurlFp model proposed in Kaiser and Menzel (2019a), which is based on the complete dislocation density tensor. It is shown that, in cyclic non-uniform plastic deformation processes, the gradient of the scalar-valued internal variable in the reduced-order model predicts isotropic hardening in contrast to kinematic-type hardening produced by the CurlFp model due to a dislocation-induced back-stress component. The arising size effect in the monotonic torsion tests is described by the normalized torque T/R3 as a function of the ratio of the microwire radius R and the characteristic length scale ℓ. In the size-dependent domain, characterized by an inflection point on the corresponding curve, the scaling law T/R3∼(R/ℓ)n can be identified, and explicit relations are found for the power n. The relative evolution of Statistically Stored Dislocation (SSD) and Geometrically Necessary Dislocation (GND) densities during torsion is described in detail.

Effective behavior of composites with combined kinematic and isotropic hardening based on additive tangent Mori–Tanaka scheme

The goal of the present work is to propose a multi-scale approach for composite materials which accounts for kinematic hardening in the phases. For that purpose, the additive/sequential interaction rule and tangent linearization of viscoplastic response proposed for elastic–viscoplastic material can be extended in a straightforward manner. A two phase composite where each phase is elastic–viscoplastic is considered. The viscoplastic flow is governed by a J2 flow theory with an overstress. To find the overall behavior of the composite, a Mori–Tanaka model is applied. Numerical validation of the proposition is carried out by considering a representative volume element with 30 inclusions. Various configurations have been tested: hard or soft inclusion cases with or without isotropic hardening. It is shown that the quality of the model predictions is not affected by the introduction of the kinematic hardening component in the local constitutive behavior. Namely, in most cases considered in the paper the overall stress–strain response as well as the average stress–strain response per phase is accurately estimated. It has been also verified that the obtained backstress components are consistent with the ones predicted by Finite element calculations with ABAQUS Software.

Concurrent ratcheting and stress relaxation at the notch root of steel samples undergoing asymmetric tensile loading cycles

AbstractThe current study intends to develop a framework model to assess ratcheting and stress relaxation at the notch root of 1045 steel samples over asymmetric loading cycles. The framework involves the Ahmadzadeh‐Varvani (A‐V) kinematic hardening rule to control ratcheting progress and Neuber rule to accommodate for local stress and strain components at the vicinity of notch root. Plastic strain at notch root was first coupled with its counterpart in the A‐V model to establish a relation between local stress and backstress components. Calculated local stress and strain values at turning points enabled the A‐V model to assess ratcheting strain over each loading cycle. The stepwise drop in stresses at peak‐valley tips of hysteresis loops at the notch root was associated to coupled framework of the A‐V model and Neuber rule through constancy in local strain while ratcheting progressed over each cycle. This relaxed out the local stresses at tips of hysteresis loops to position on Neuber hyperbolic curve. Predicted ratcheting values at notch root of various diameters closely agreed with those of measured in steel samples over stress cycles.

Finite element implementation of non-unified visco-plasticity model considering static recovery

In order to examine relaxation behaviors of materials, static recovery term is always conducted in the backstress components for kinematic hardening in time-dependent plasticity model. Implicit integration algorithm for non-unified visco-plasticity model is derived. Plastic strain is divided into two parts: the visco-plastic strain, which is calculated based on the viscosity function, and the steady strain which only depends on the equivalent deviatoric backstress. Using the Newton iterative strategy, two sets of equations for deviatoric stress and relative stress tensors are solved. Additionally, the return mapping method is applied in numerical method, and consistent tangent stiffness modulus is progressed for convergence in finite element analysis. The model is implemented into commercial software ABAQUS as user subroutine UMAT. Dwell experiments of nickel-based superalloy under strain-controlled cyclic loading with holding periods for relaxation at high temperature are selected to verify the method. The results indicate that the user subroutine attains convergence with good speed even at large increment, and the additional static recovery terms can improve the simulation for relaxation behaviors.

Understanding strain controlled low cycle fatigue response of P91 steel through experiment and cyclic plasticity modeling

In this paper, constitutive models for cyclic plasticity, namely, Chaboche and the Ohno-Wang models, are used to simulate the low cycle fatigue response of tempered ferritic-martensitic P91steel. A series of uniaxial strain-controlled low cycle fatigue tests are conducted at room temperature to evaluate various fatigue parameters. P91 steel shows cyclic softening behavior at all strain amplitudes. The analysis of the shape of hysteresis loops reveals that P91 steel deviates from Masing behavior. Plastic strain energy is used as a parameter to characterize the fatigue damage. The Morrow energy model estimates experimental average plastic strain energy density accurately. Remnant tensile properties are evaluated to quantify the damage during low cycle fatigue loading. The evaluation of back stress components and material Jacobian are implemented through a user defined subroutine UMAT into the commercial finite element package ABAQUS. Ohno-Wang material model predicts the hysteresis loop shape, area, and cyclic softening nature better than the Chaboche model. The predicted fatigue lives by simulation correlates well with the experimental fatigue lives within ± 200 cycles. These results suggest that energy based parameters can provide a reliable alternative to strain based approach in evaluating the fatigue damage of P91 steel.

Particle swarm optimization procedure in determining parameters in Chaboche kinematic hardening model to assess ratcheting under uniaxial and biaxial loading cycles

AbstractAs parameters in Chaboche model are difficult to be determined from experimental data, a single objective particle swarm optimization procedure was employed to obtain them. Hysteresis loop and uniaxial and biaxial ratcheting simulations were conducted to validate the determined models. Chaboche models determined by particle swarm optimization give more accurate simulation of ratcheting compared with the model determined by trial and error method. Chaboche models containing different backstress components were studied. Models determined considering uniaxial ratcheting can only predict uniaxial ratcheting precisely, while giving very bad simulation of biaxial ratcheting. The linear hardening rule in the N3L1 model clearly decreases the rate of the accumulation of ratcheting strain, and the N3L1 model gives the best simulations. For biaxial ratcheting, the fourth backstress component can decrease the rate of the accumulation apparently, while it has a little influence on prediction of uniaxial ratcheting.

Cyclic softening behaviors of modified 9–12%Cr steel under different loading modes: Role of loading levels

Cyclic softening behavior of modified 9–12%Cr steel was investigated at 873 K under different loading modes and stress levels. Results indicated that two domains were identified depending on the plastic strain level, in which cyclic deformations and control modes were different. Significant different cyclic softening was only observed in the macroplasticity domain at high loading levels between stress and strain controlled fatigue, companied with a relatively homogenous coarsening of subgrain. By contrast, in the microplasticity domain at low loading levels, slight cyclic softening was attributed to the fact that back stress remained a relative stable value (or decreased very slightly). An additional back stress component was produced due to the heterogeneous subgrain coarsening at the inter-packet scale. As a result, the cyclic softening was retarded due to the competition between the hardening of inter-packet back stress and the softening of intragranular back stress.

Fatigue life estimation of notched specimens of modified 9Cr-1Mo steel under uniaxial cyclic loading

Accuracy in the estimation of low cycle fatigue life of modified 9Cr-1Mo steel notched specimen by different analytical methods such as linear rule, Neuber’s rule, strain energy density method and numerical method such as finite element analysis have been studied in this investigation. The fatigue tests on notched specimens having notch radius of 1.25 mm, 2.5 mm and 5.0 mm were carried out at 823 K with net stress amplitudes of 250 MPa, 300 MPa and 350 MPa. The fatigue tests on smooth specimens were carried out with strain amplitudes ranging from ±0.3% to ±0.8% with a strain rate of 3 × 10−3 s−1 at 823 K to evaluate the fatigue life of notched specimen through strain-life approach. In order to predict the cyclic stress response of the material, Chaboche non-linear hardening model was employed considering two back stress components. Predicted hysteresis loops for smooth specimen were well in agreement with experimental results. Estimated fatigue lives of notched specimens by analytical methods and finite element analysis were within a factor ±16 and ±2.5 of the experimental lives respectively.

Aluminum Alloy 7075 Ratcheting and Plastic Shakedown Evaluation with the Multiplicative Armstrong–Frederick Model

This work investigates experimentally and computationally the uniaxial ratcheting strain and plastic shakedown of aluminum alloy 7075-T6. The experimental results illustrate the existence of both plastic shakedown and cyclic hardening, proving that both kinematic and isotropic hardening should be included in modeling. Although the multicomponent Armstrong–Frederick model with multiplier has demonstrated high accuracy in aluminum ratcheting simulation, as well implementation ease, the present results demonstrate poor performance when plastic shakedown is considered. This is attributed to the limited flexibility in varying the model parameters to balance the loading/unloading branches in the hysteresis loops and decelerate ratcheting pace. To improve the ability of the multicomponent Armstrong–Frederick model with multiplier in simulating plastic shakedown, a modification was made within the framework of the model that includes multiple backstress components, with each obeying its own kinematic hardening. A linear kinematic hardening backstress was added in the formulation, enabling the control of ratcheting pace and the occurrence of plastic shakedown. Simulations with the modified multicomponent Armstrong–Frederick model with multiplier demonstrate a significantly improved capability for ratcheting and plastic shakedown. Moreover, the modified multicomponent Armstrong–Frederick model with multiplier improved the life prediction for an actual aerospace structure. This provides a strong indication of the importance of achieving plastic shakedown accuracy when simulating cyclic elastoplastic behavior.

A mixed hardening model combined with the transformation-induced plasticity effect

A mixed isotropic–kinematic hardening model considering the transformation-induced plasticity (TRIP) effect was proposed for TRIP steels in this study. Besides the nonlinear Chaboche backstress and linear Prager–Ziegler backstress, a third backstress that depicts the contribution of TRIP effect to the kinematic hardening was presented. The TRIP effect was modeled using an equivalent plastic strain and a stress triaxiality coefficient as the variant. Forward–reverse shear experiments were performed for a cold rolled TRIP780 steel. A new clamper was designed to reduce the buckling during reverse shear. Cyclic loading–unloading–reloading uniaxial tension experiments were conducted and the degradation of elastic modulus with the plastic strain was discussed. Micro-indentation tests were performed to check the true elastic modulus. U-bending springback simulation was performed with the new hardening model combined with the Yoshida’s elastic modulus. The results showed that the investigated TRIP780 steel has prominent kinematic hardening behavior, Bauschinger effect, transient hardening and permanent softening during forward–reverse shear deformation. The percentage of the third backstress component due to the TRIP effect in the total backstress reaches 31% at 0.25 plastic strain. Springback simulation with the proposed hardening model combined with the degradation of elastic modulus is more precise than the isotropic model.

Identification of nonlinear kinematic hardening constitutive model parameters using the virtual fields method for advanced high strength steels

In this work, the nonlinear kinematic hardening combined with Voce isotropic hardening was selected to characterize the material behavior of advanced high strength steel sheet samples subjected to a few reverse loading cycles. Multi-components of backstress were considered for the combined nonlinear kinematical hardening model, namely, one, two, and three backstress components. To calibrate the model, an inverse problem solution tool, so-called virtual fields method, which takes full advantage of full-field deformation measurement, was applied to identify the material constitutive parameters. First, finite element simulations of forward-reverse simple shear were performed to validate the proposed identification method. The influence of strain noise on the identification accuracy was also evaluated. Then, the proposed method was applied to three kinds of sheet metals (DP600, TRIP780 and TWIP980) tested under two cycles of forward-reverse simple shear for parameter identification. The identification results obtained with different number of backstress components were critically discussed.

On the modelling of complex kinematic hardening and nonquadratic anisotropic yield criteria at finite strains: application to sheet metal forming

In the present paper, a finite strain model for complex combined isotropic-kinematic hardening is presented. It accounts for finite elastic and finite plastic strains and is suitable for any anisotropic yield criterion. In order to model complex cyclic hardening phenomena, the kinematic hardening is described by several back stress components. To that end, a new procedure is proposed in which several multiplicative decompositions of the plastic part of the deformation gradient are considered. The formulation incorporates a completely general format of the yield function, which means that any yield function can by employed by following a procedure that ensures the principle of material frame indifference. The constitutive equations are derived in a thermodynamically consistent way and numerically integrated by means of a backward-Euler algorithm based on the exponential map. The performance of the constitutive model is assessed via numerical simulations of industry-relevant sheet metal forming processes (U-channel forming and draw/re-draw of a panel benchmarks), the results of which are compared to experimental data. The comparison between numerical and experimental results shows that the use of multiple back stress components is very advantageous in the description of springback. This holds in particular if one carries out a comparison with the results of using only one component. Moreover, the numerically obtained results are in excellent agreement with the experimental data.