A plasticity model for multiaxial cyclic loading and ratchetting

Summary The nonlinear behavior of metals when subjected to monotonic and cyclic non-proportional loading is modeled using the proposed hardening rule. The model is based on the Chaboche [1], [2] and Voyiadjis and Sivakumar [3], [4] models incorporating the bounding surface concept. The evolution of the backstress is governed by the deviatoric stress rate direction, the plastic strain rate, the backstress, and the proximity of the yield surface from the bounding surface. In order to ensure uniqueness of the solution, nesting of the yield surface with the bounding surface is ensured. The prediction of the model in uniaxial cyclic loading is compared with the experimental results obtained by Chaboche [1], [2]. The behavior of the model in multiaxial stress space is tested by comparing it with the experimental results in axial and torsional loadings performed by Shiratori et al. [5] for different stress trajectories. The amount of hardening of the material is tested for different complex stress paths. The model gives a very satisfactory result under uniaxial, cyclic and biaxial non-proportional loadings. Ratchetting is also illustrated using a non-proportional loading history.     

A robust kinematic hardening rule for cyclic plasticity with ratchetting effects Part II. Application to nonproportional loading cases

Summary Performance of the proposed kinematic hardening rule is examined using several examples of cyclic plasticity phenomena observed in experiments. Results obtained and compared with experimental observations on various loading histories are presented. With the memory effects added to the model, impressive results are obtained without using an anisotropic yield model. Drifting of the yield surface occurs during the numerical computation of the plastic response due to nonproportional loading paths. The drift due to the finite increments of stress or strain is corrected using a simple and efficient method proposed in this paper. The new kinematic hardening rule proposed for the limit surface as being related directly to the yield surface kinematic hardening rule ensures nesting using the blended rule discussed in the part presenting the theoretical formulation [14].     

Simulations of stress distributions in crankshaft sections under fillet rolling and bending fatigue tests

The residual stresses due to fillet rolling and the bending stresses near the fillets of crankshaft sections under bending fatigue tests are important driving forces to determine the bending fatigue limits of crankshafts. In this paper, the residual stresses and the bending stresses near the fillet of a crankshaft section under fillet rolling and subsequent bending fatigue tests are investigated by a two-dimensional plane strain finite element analysis based on the anisotropic hardening rule of Choi and Pan [Choi KS, Pan J. A generalized anisotropic hardening rule based on the Mroz multi-yield-surface model for pressure insensitive and sensitive materials (in preparation)]. The evolution equation for the active yield surface during the unloading/reloading process is first presented based on the anisotropic hardening rule of Choi and Pan (in preparation) and the Mises yield function. The tangent modulus procedure of Peirce et al. [Peirce D, Shih CF, Needleman A. A tangent modulus method for rate dependent solids. Comput Struct 1984;18:875–87] for rate-sensitive materials is adopted to derive the constitutive relation. A user material subroutine based on the anisotropic hardening rule and the constitutive relation was written and implemented into ABAQUS. Computations were first conducted for a simple plane strain finite element model under uniaxial monotonic and cyclic loading conditions based on the anisotropic hardening rule, the isotropic and nonlinear kinematic hardening rules of ABAQUS. The results indicate that the plastic response of the material follows the intended input stress–strain data for the anisotropic hardening rule whereas the plastic response depends upon the input strain ranges of the stress–strain data for the nonlinear kinematic hardening rule. Then, a two-dimensional plane-strain finite element analysis of a crankshaft section under fillet rolling and subsequent bending was conducted based on the anisotropic hardening rule of Choi and Pan (in preparation) and the nonlinear kinematic hardening rule of ABAQUS. In general, the trends of the stress distributions based on the two hardening rules are quite similar after the release of roller and under bending. However, the compressive hoop stress based on the anisotropic hardening rule is larger than that based on the nonlinear kinematic hardening rule within the depth of 2 mm from the fillet surface under bending with consideration of the residual stresses of fillet rolling. The critical locations for fatigue crack initiation according to the stress distributions based on the anisotropic hardening rule appear to agree with the experimental observations in bending fatigue tests of crankshaft sections.     

Phenomenological evolution equations for the backstress and spin tensors

Summary A formulation of a constitutive model involving a new kinematic hardening rule is presented in the Eulerian reference system. The corotational stress and backstress rates involving spin tensors are discussed and incorporated in the evolution equations. The backstress evolution model is compared with other models and is found to be decomposable into a model that consists of two backstresses whose evolution is independent of each other.The elasto-plastic stiffness tensor is derived for all the models considered. It is shown in the proposed backstress evolution model we obtain an implicit system of differential equation in the stress and backstress rates. The theory is applied to two yield functions: one of the von-Mises type and the other of the anisotropic type. It is shown that for both these yield criteria the stress evolution is independent of the stress rate. As an example, the torsion of a cylindrical bar with fixed ends is investigated.     

A theoretical evaluation of plasticity hardening algorithms for nonproportional loadings

Summary Incremental plasticity theories are being incorporated into many engineering numerical analyses. There are two basic categories of incremental plasticity algorithms, (i) multiple surface such as proposed by Mróz and Garud, and (ii) the Armstrong-Frederick type as modified by Chaboche et al. Engineering bounds on the general applicability of these models for cyclic loading have not been the subject of a detailed investigation. Similar first order stress-strain results are obtained for proportional loadings when either category of hardening rules is chosen.A -circle loading path in deviatoric stress space was identified as a severe test of the algorithms. A closed form solution can be obtained for this loading path, so that the phenomena noted cannot be attributed to numerical errors. While this loading is highly idealized, analogous results are also noted for 90 degree out-of-phase tension-torsion. Multiple surface and the associated two surface models with a stationary bounding surface are shown to have both theoretical and numerical problems for the severe cyclic loading under consideration. The Armstrong-Frederick class of models demonstrates a diminished sensitivity to modeling parameters, as well as a single-valued stress-strain representation for both severe nonproportional loading paths investigated.List of symbols b Magnitude of backstress - c, r Material constants in the Armstrong-Frederick hardening rule - c _(i),r _(i) Material constants in the Chaboche hardening rule (i=1,2,...,M) - d Prefix denoting infinitesimal increment or differentiation - f Yield surface function - f _(i) A surface in the multiple surface models - h Plastic modulus function - I Unit tensor or Kronecker Delta - k Yield stress in simple shear - K Cyclic strength coefficient - M An integer denoting the number of surfaces for a multiple surface model or the number of terms in the backstress expansion for an Armstrong-Frederick model - n Cyclic strain hardening exponent - n Unit exterior normal to the yield surface at the stress state - p Equivalent plastic strain - R Yield surface radius in deviatoric stress space - R _b Bounding surface radius in deviatoric stress space - R _(i) Radius of theith surface in the multiple surface type hardening rules - S Deviatoric stress tensor - Scalar representation for stress - Total backstress tensor - ⁽ⁱ⁾ ith backstress tensor (i=1,2,...,M) in the Armstrong-Frederick type models or the center of theith surface in the Mróz multiple surface type models - ^p Plastic strain tensor - ^p /2 Scalar representation for plastic strain - _{x x} ^p Axial plastic strain for tension-torsion loading - ^p Shear plastic strain for tension-torsion loading - Stress tensor - Angle between the translation direction of the yield surface and the exterior normal at the stress state on the yield surface - Mróz translation vector - Garud translation vector - Symbol denoting inner product - A Representing an invariant of a tensorA defined by      

Ratcheting of 304 Stainless Steel Alloys subjected to Stress‐Controlled and mixed Stress‐ and Strain‐Controlled Conditions evaluated by Kinematic Hardening Rules

The present study predicts ratcheting response of SS304 tubular stainless steel samples using kinematic hardening rules of Ohno–Wang (O–W), Chen‐Jiao‐Kim (C–J–K) and a newly modified hardening rule under various stress‐controlled, and combined stress‐ and strain‐controlled histories. The O–W hardening rule was developed based on the critical state of dynamic recovery of backstress. The C–J–K hardening rule further developed the O–W rule to include the effect of non‐proportionality in ratcheting assessment of materials. The modified rule involved terms , and in the dynamic recovery of the Ahmadzadeh–Varvani (A–V) model to respectively track different directions under multiaxial loading, account for non‐proportionality and prevent plastic shakedown of ratcheting data over multiaxial stress cycles. The O–W model persistently overestimated ratcheting strain over the multiaxial loading paths. The C–J–K model further lowered this overprediction and improved the predicted ratcheting curves. The predicted ratcheting curves based on the modified model closely agreed with experimental data under various loading paths.     

Rheological analysis of stress change experiments during high temperature creep

The main results of stress drop experiments during high temperature creep (i.e. the occurrence of zero and sometimes negative creep rates, the existence of two parts with different slopes in the curves of the incubation time t_r versus the stress decrement , the zero or positive values of the stress variations in stress relaxation tests) are analysed from a rheological point of view. The basis of the proposed interpretation is the existence of a creep criterion, represented by a limit surface in the stress space, which explains through the usual plastic flow rules the occurrence of zero or negative creep rates. The work hardening splits into a translation (kinematic hardening) and a deformation of this limit surface. The recovery of kinematic hardening is slower than the recovery of deformation. On this basis, the concept of internal stress used either in the activated glide of dislocations or in the Bailey-Orowan relationship and the nature of negative creep rates are discussed.     

Isotropic‐kinematic hardening framework to assess ratcheting response of steel samples undergoing asymmetric loading cycles

The present study intends to characterize ratcheting response of several steel alloys subject to asymmetric loading cycles through coupling the Ahmadzadeh‐Varvani kinematic hardening rule with isotropic hardening rules of Lee and Zavrel, Chaboche, and Kang. The Ahmadzadeh‐Varvani kinematic hardening rule was developed to address ratcheting progress over asymmetric stress cycles with relatively a simple framework and less number of coefficients. Inclusion of isotropic hardening rules to the framework improved ratcheting response of materials mainly over the first stage of ratcheting. Lee and Zavrel model (ISO‐I) developed an exponential function to account for accumulated plastic strain as yield surface is expanded over stage I and early stage II of ratcheting. Isotropic models by Chaboche (ISO‐II) and Kang (ISO‐III) encountered yield surface evolution in the framework by introducing an internal variable that takes into account the prior maximum plastic strain range. The choice of isotropic hardening model coupled to the kinematic hardening model is highly influenced by material softening/hardening response.     

THE USE OF KINEMATIC HARDENING MODELS IN MULTI-AXIAL CYCLIC PLASTICITY

Abstract— The use of kinematic hardening models are examined for tubes subjected to (a) cyclic plastic torsion with a sustained axial stress and (b) cyclic plastic tension-compression with a sustained hoop stress. It is shown that the kinematic model predicts a limit to the plastic strain accumulation resulting from the sustained loads. Experimental results show that the strain accumulation is not limited and that the mode of deformation within a cycle of plastic strain cannot be predicted using a kinematic hardening model. The development of the yield surface under cyclic loading is examined. The results indicate that contraction and expansion of the yield surface along the stress axes can occur and it is shown that the direction of the sustained stresses, relative to the direction of the cyclic stresses is an important factor in the development of any cumulative plastic strains.     

Cyclic plastic behavior analysis based on the micromorphic mixed hardening plasticity model

In this paper, a small strain micromorphic elasto-plastic model with isotropic/kinematic hardening is presented for modeling the size effect and Bauschinger effect in material with microstructure. A nonlinear kinematic hardening model is embedded into the micromorphic framework by employing a backstress, a micro-backstress and a micro-couple-backstress in a physical way. The material intrinsic length scale is introduced in the constitutive law, leading to the presence of higher order stress. The present model is further implemented into a 2D plane strain finite element frame with a fully implicit stress integration scheme. The generalized consistent tangent modulus is derived to achieve the parabolic convergence of the global nodal force equilibrium equation. Two numerical examples, including a thin film and a plate with underlying structures subjected to cyclic loading, are analyzed to verify the theoretical developments and numerical formulations. Plastic behaviors in micromorphic continuum, such as size effect, Bauschinger effect, ratcheting effect and plastic shakedown phenomenon, are investigated.     

Internal-variable constitutive model for rate-independent plasticity with hardening saturation surface

Summary An elastic-plastic material model with internal variables and thermodynamic potential, not admitting hardening states out of a saturation surface, is presented. The existence of such a saturation surface in the internal variables space — a consequence of the boundedness of the energy that can be stored in the material's internal micro-structure — encompasses, in case of general kinematic/isotropic hardening, a one-parameter family of envelope surfaces in the stress space, which in turn is enveloped by a limit surface. In contrast to a multi-surface model, noad hoc rules are required to avoid the intersection between the yield and bounding/envelope surface. The flow laws of the proposed model are studied in case of associative plasticity with the aid of the maximum intrinsic dissipation theorem. It is shown that the material behaves like a standard one as long as its hardening state either is not saturated, or undergoes a desaturation from a saturated hardening state, whereas, for saturated hardening states not followed by desaturation, it conforms to a combined yielding law in which the static internal variable rates obey a nonlinear hardening rule similar to that of analogous models of the literature. Additionally, the material is shown to behave as a perfectly plastic material for a class of (critical) saturated hardening states for which the stress state is on the limit surface. For nonassociative material models, it is shown that, under a special choice of the plastic and saturation potentials and through a suitable parameter identification, the well-known Chaboche model is reproduced. A few numerical examples are presented to illustrate the associative material response under monotonic and cyclic loadings.Dedicated to Prof. Dr. Dr. h. c. Franz Ziegler on the occasion of his 60th birthday     

On postulates of plasticity in classical elastoplasticity

Summary. It is commonly known that Druckers postulate of plasticity in stress space is applicable to hardening materials only, but not to softening materials, and, on the other hand, that Ilyushins postulate of plasticity in strain space is applicable to both softening and hardening materials. Accordingly, it is usually thought that the latter would be less restrictive and hence more general than the former. In this work, we introduce the notions of standard elastoplastic stress and strain cycles, each starting at a point inside the yield surface and incorporating only one infinitesimal plastic strain increment. We show that these standard cycles always exist for a yield function depending continuously on stress and plastic strain history. By means of these standard cycles, we propose respective weakened forms of Druckers and Ilyushins postulates. These two weakened forms are less restrictive in two respects. One is that, unlike Druckers and Ilyushins postulates, they are not concerned with stress and strain cycles starting at points on the yield surface. The second is that they are formulated in terms of the rates of change of the net stress work and the stress work when the plastic strain increment incorporated tends to vanish. We demonstrate that either of the proposed weakened forms is adequate to result in the normality rule for the plastic strain rate and the convexity of the yield surface, as does either Druckers or Ilyushins postulate. These suggest that, in a less restrictive sense and hence in a broader scope, the two proposed weakened forms not only apply to both hardening and softening materials, but also ensure the same consequences.     

Springback investigation of anisotropic aluminum alloy sheet with a mixed hardening rule and Barlat yield criteria in sheet metal forming

A mixed hardening model has been implemented based on Lemaitre and Chaboche non-linear kinematic hardening theory to consider cyclic behavior and the Bauschinger effect. The Chaboche isotropic hardening theory is incorporated into the non-linear kinematic hardening model to introduce a surface of nonhardening in the plastic strain space. The bending and reverse bending case study has verified the effectiveness of the mixed hardening model by comparison with the proposed experiment results. Barlat’89 yielding criterion is adopted for it does not has any limitation while Hill’s non-quadratic yield criterion is for the case that the principal axes of anisotropy coincides with principal stress direction. The Backward–Euler return mapping algorithm was applied to calculate the stress and strain increment. The mixed hardening model is implemented using ABAQUS user subroutine (UMAT). The comparisons with linear kinematic hardening model and isotropic hardening model in NUMISHEET’93 benchmark show that the mixed hardening model coupled with Barlat’89 yield criteria can well reflect stress and strain distributions and give a more favorable springback angle prediction.     

Modeling of ratcheting behavior under multiaxial cyclic loading

Summary.  A two-surface plasticity theory is used to predict ratcheting strain under multiaxial loading. A kinematic hardening rule that combines the Mroz and Ziegler hardening rules is employed along with the plastic modulus given as an exponential function of the distance between the yield surface and the bounding surface. Model results are compared with the experimental data obtained on medium carbon steel under proportional and nonproportional axial-torsional loading. The model predicts reasonably well the experimental ratcheting behavior at relatively low cycles. Predictions overshoot the actual ratcheting strains at high cycles, yet the results look favorable compared with other data found in the literature. Received July 29, 2002; revised January 15, 2003 Published online: May 20, 2003 The authors gratefully acknowledge financial support for this work, in part from Brain Korea 21 Program at Pohang University of Science and Technology, and in part from National Natural Science Foundation of China and TRAPOYT.     

Exact solution of flow theory problems with isotropic-kinematic hardening. Part 2. Setting of the deformation trajectory in the spaces of total and plastic strains

For an arbitrary isotropic and linear kinematic hardening and loading paths given in the form of arbitrary multisection polygonal lines in the five-dimensional deviatoric space of total strains, we have studied analytically an initially isotropic elastoplastic material with von Mises yielding and the associated flow rule. The solutions obtained are valid for arbitrary relationships that govern the variation of the spherical component of the stress tensor. For arbitrary isotropic and kinematic hardening, we have also obtained an analytical solution to an elastoplastic problem for an arbitrary deformation trajectory given in the deviatoria space of plastic strains Institute of Problems of Strength, National Academy of Sciences of Ukraine, Kiev, Ukraine. Translated from Problemy Prochnosti, No. 1, pp. 62–71, January–February, 2000     

Ratcheting assessment of steel alloys under step-loading conditions

The present study examines the capability of a recently modified hardening rule to characterize ratcheting response of materials subjected to multi-step uniaxial stress cycles. The modified hardening rule was developed based on Armstrong–Frederick (A–F) hardening rule through implementing new ratcheting rate dependent coefficients γ₂ and δ. These coefficients were estimated by means of calibrated curves for any given stress levels defined from the uniaxial single-step ratcheting response at various cyclic stress levels. At a constant mean stress, ratcheting strain progressively increased as stress amplitude over steps of loading history increased. Similar response was also evident for step-loading with constant stress amplitude while the values of mean stress increased. For high–low histories, the trend of predicted ratcheting strain from higher to lower magnitudes found agreeable with that of experimental data. The discrepancy of the predicted and experimentally ratcheting strain values in the high–low step loading however was due to constancy in the shape and size of translating yield surface in the modified kinematic hardening rule. The modified hardening rule was employed to assess ratcheting response of SS316L, SA333, SS316L(N) and 1070 steel alloys under various step-loading conditions. Predicted ratcheting data at various stress level were found in good agreements as compared with the experimental ratcheting strains.     

Asymptotic solution for mode III crack growth in J 2-elastoplasticity with mixed isotropic-kinematic strain hardening

Mode III fracture propagation is analyzed in a J ₂-flow theory elastoplastic material characterized by a mixed isotropic/kinematic law of hardening. The asymptotic stress, back stress and velocity fields are determined under small-strain, steady-state, fracture propagation conditions. The increase in the hardening anisotropy is shown to be connected with a decrease in the thickness of the elastic sector in the crack wake and with an increase of the strength of the singularity. A second order analytical solution for the crack fields is finally proposed for the limiting case of pure kinematic hardening. It is shown that the singular terms of this solution correspond to fully plastic fields (without any elastic unloading sector), which formally are identical to the leading order terms of a crack steadily propagating in an elastic medium with shear modulus equal to the plastic tangent modulus in shear.     

Kinematic hardening rules for modeling uniaxial and multiaxial ratcheting

Ratcheting, which is the strain accumulation observed under the unsymmetrical stress controlled loading and non-proportional loadings, is modeled using the simplified viscoplasticity theory based on overstress (VBO). The influences of kinematic hardening laws on the uniaxial and multiaxial non-proportional ratcheting behavior of CS 1026 carbon steel have been investigated. The following kinematic hardening rules have been considered: the classical kinematic hardening rule, the kinematic hardening rules introduced by Armstrong–Frederick, Burlet–Cailletaud and the modified Burlet–Cailletaud. The investigated loading conditions include uniaxial stress controlled test with non-zero mean stress, and axial strain controlled cyclic test of thin-walled tubular specimen in the presence of constant pressure. Numerical results are compared with the experimental data obtained by Hassan and Kyriakides [Hassan T, Kyriakides S. Ratcheting in cyclic plasticity, part I: uniaxial behavior. Int J Plast 1992;8:91–116] and Hassan et al. [Hassan T, Corona E, Kyriakides S. Ratcheting in cyclic plasticity, part I: multiaxial behavior. Int J Plast 1992;8:117–146]. It is observed that all investigated kinematic hardening rules do not improve ratcheting behavior under multiaxial loading, but over-prediction still exists.     

A generalized Eulerian two-surface cyclic plasticity model for finite strains

Summary A cyclic theory of plasticity is formulated for finite deformation in the Eulerian reference system. A new kinematic hardening rule is proposed based on the experimental observations made by Phillips et al. [11]–[15]. The Tseng-Lee model [9] is also obtained as a special case of the proposed model.Qualitative examples are presented to demonstrate the behavior of materials under different loading paths using the proposed model. These include both proportional and nonproportional loading paths.