Smooth Elastic-plastic Transition Research Articles

Elucidating the deformation characteristics of Ti–Ni–Fe based pseudo-binary B2 intermetallics

The room temperature uniaxial compression behavior of homogenized Ti50Ni50-xFex (5≤x ≤ 45) pseudo-binary B2 intermetallics has been investigated. The compressive stress-strain curves of these intermetallics exhibited three different characteristics (a) intermetallics with low Fe content (i.e., Ti50Ni45Fe5 and Ti50Ni40Fe10): exhibited stress induced martensite (SIM) transformation (B2 → B19 ') (marked by a plateau with non-linear elastic regime) followed by significant plastic deformation up to ∼ 40 % strain marked with significant strain hardening, (b) intermetallics with moderate Fe content (Ti50Ni35Fe15) showed no SIM transformation, but a typical smooth elastic-plastic transition with maximum strain to failure of ⁓ 11–12 % was observed, and (c) intermetallics with high Fe content (≥ 25 at. %) displayed a very small strain to failure (≤ 5 %) without visible plastic deformation. The TEM and XRD analysis of the deformed (15 % and 40 % strain) Ti50Ni45Fe5 and Ti50Ni40Fe10 intermetallics indicated the presence of internally twinned martensite variants (MVs) along with severely dislocated banded B2 matrix. The observed contrast in the deformation characteristics has been attributed to the variation in the values of elastic constants (C' and C44) and Zener anisotropy (A) with Fe content. The SIM transformation in Ti50Ni45Fe5 and Ti50Ni40Fe10 occurred at lower values of C' and C44 (due to elastic softening) and relatively low Zener anisotropy A. The significant plastic strain in these two intermetallics was accommodated by the presence and deformation of super intrinsic stacking faults (SISFs) formed by dissociation of 12⟨111⟩ screw super-partials. In contrast, the deformation in the intermetallics with high Fe content (≥ 25 at. %) was governed by the activation of ⟨100⟩ screw dislocations.

Subloading-elastoplastic constitutive equation of glass

Elastoplastic constitutive equation of glass is proposed in this article, which is formulated based on the subloading surface model which possesses the various distinguished advantages for the description of the plastic deformation, i.e., the smooth elastic-plastic transition, the continuous variation of the tangent stiffness modulus tensor, the automatic controlling function to attract the stress to the yield surface during the plastic deformation process, etc. It would be the firstly proposed three-dimensional elastoplastic constitutive equation of glass, which is furnished with the basic properties, e.g. the ellipsoidal yield surface with the dependence of the third deviatoric stress invariant, undergoing the flattering to the deviatoric direction with the plastic compression and the plastic volumetric hardening with the associated flow rule. The validity of the description of the deformation behavior of glass will be verified by comparisons with some test data for silica glass specimens.

SANISAND-H: A sand bounding surface model for high pressures

This paper presents a new bounding surface constitutive model for sand, named SANISAND-H, capable of predicting the shear and volumetric behavior of the material under extra-high range of pressures, in addition to normal pressures. The model is an extension of the SANISAND08 model (Taiebat and Dafalias, 2008) with a closed cone-type yield surface that obeys rotational and isotropic hardening. The critical state line and the limiting isotropic compression curve of the SANISAND08 model are modified to accommodate a wide range of pressures. A new constitutive ingredient is the definition of a reference compression curve that serves as bounding surface for establishing the isotropic hardening rule of the yield surface. The kinematic hardening rule and stress-dilatancy relationship of the SANISAND08 model are significantly modified, to better capture the response of sands under high pressure. The value of the back stress ratio at the initiation of a new loading process is re-introduced, ensuring a smooth elastic-plastic transition necessary for accurate simulation of the response under loading reversals. The SANISAND-H model simulations are in very good agreement with test data for Toyoura and Cambria sands under various drained and undrained monotonic stress loading paths in a wide range of densities and pressures.

An Eulerian thermodynamical formulation of size-dependent plasticity

The objective of this work is to develop a finite-deformation, thermodynamically-consistent theory to model the size-dependent irreversible behavior of metals at the micron-scale. In contrast with other approaches, the proposed Eulerian formulation introduces an evolution equation directly for an elastic distortional deformation measure without the need for defining total or plastic deformation measures. Motivated both by experimental observations and by the literature on generalized continua, the proposed theory assumes tensorial fields separately describing macro-plasticity and micro-plasticity. An additional evolution equation defines a Nye–Kröner-like tensor that depends on the curl of the non-symmetric micro-plasticity distortional rate. By drawing inspiration from higher-order strain gradient plasticity, the micro-plasticity distortional rate is determined by a micro-plasticity balance equation with associated boundary conditions. A quadratic function of the Nye–Kröner-like tensor contributes to the Helmholtz free energy, which leads to an energetic stress entering the micro-plasticity balance that controls the material size-dependent response. The theory is designed to allow the onset of micro-plasticity to occur at a stress level below that triggering macro-plasticity and, regarding size-effects, the response characterized by macro-plasticity alone is the strongest one. In addition, the formulation uses a smooth elastic–plastic transition model for rate-independent response. The theory is specialized to the case of small strains and rotations and applied to the constrained simple shear problem, which allows the derivation of an analytical solution able to demonstrate the predictive capabilities ensuing from the rich interplay between micro- and macro-plasticity.

Subloading-friction model with saturation of tangential contact stress

The incorporation of the saturation of the tangential contact stress with the increase of the normal contact stress is required for the analysis of the friction phenomenon of solids and structures subjected to a high normal contact stress, which cannot be described by the Coulomb friction condition, in which the tangential contact stress increases linearly with the increase of the normal contact stress. In this article, the subloading-friction model, which is capable of describing the smooth elastic—plastic transition, the static—kinetic transition, and the recovery of the static friction during the cease of sliding, is extended to describe this property. Further, some numerical examples are shown, and the validity of the present model will be verified by the simulation of the test data on the linear sliding of metals.

Model for description of nonlinear unloading-reloading stress-strain response with special reference to plastic-strain dependent chord modulus

To describe the nonlinear stress-strain response in the unloading-reloading process, a model was proposed for the variation of the elastic modulus during the process. This model is directly associated with the widely used equation for the plastic-strain dependent chord modulus, and no additional material parameters are needed for modeling. In this model, nonlinearly changing elastic modulus is explicitly determined from the current stress state, instead of using an accumulated-strain measure. Therefore, it is robust against the numerical noise (e.g., stress oscillation) in computation. Based on the same elastic modulus model, two types of calculation models were constructed, namely, a micro-plasticity model and nonlinear-elasticity model. From the numerical simulations of the non-proportional unloading-reloading processes, it was found that the stress-strain responses calculated by the two models were very similar to each other. The model was validated by comparing the calculated stress-strain results with the corresponding experimental observations on a DP980 advanced high-strength steel sheet (AHSS). The influence of the nonlinear unloading behavior on the springback prediction was discussed by performing two cases of springback simulations (a uniform bending-unbending and a hat-shaped draw-bending) on two types of AHSSs (dual-phase DP980 and precipitation strengthen 780 R), by using several types of elastic modulus models. The best choice for springback simulation is to use the nonlinear elastic model to describe the realistic stress-strain responses in the yield surface, along with an appropriate kinematic hardening model of plasticity to express smooth elastic-plastic transition behavior. In comparison with the springback calculation by the nonlinear model, a certain amount of errors were found when using the plastic-strain dependent linear models, however, they were not very large. Thus the second best option in springback simulation would be to use the chord modulus model because of its simple modeling and high computation efficiency.

A simple procedure to simulate a smooth elastic-plastic transition in Cam-Clay models

A simple procedure is proposed to simulate a smooth transition from elastic to elastoplastic behaviour in Cam-Clay models. The procedure consists of the definition of an external constitutive surface where full yield is assumed, and an internal one that allows the definition of the position in which plastic strains start to appear before the external yield surface is reached. The comparison of the model results with different laboratory tests shows the validity of the procedure. The method considers one additional parameter with regard to a “standard” critical state model, and it can easily be implemented in existing integration modules.

Influence of unobservable overstress in a rate-independent inelastic loading curve on dynamic necking of a bar

A nonlinear rate-independent overstress model with a smooth elastic-inelastic transition is used to analyze instabilities during dynamic necking of a bar. In the simplified model the elastic strain εe determines the value of stress and the hardening parameter κ determines the onset of inelasticity. These quantities {εe, κ} are obtained by integrating time evolution equations. The main and perhaps surprising result of this paper is that, based on the critical growth rate ωcr of a perturbation, two rate-independent materials with a smooth elastic-plastic transition due to overstress and nearly the same loading curve (elastic strain or stress versus total strain) can have different susceptibilities to tensile instabilities. Specifically, increase in overstress causes decreased material instability near the onset of the smooth elastic-inelastic transition and increased instability when the elastic strain approaches its saturated value. To the authors' knowledge, this new insight has not been reported in the literature.

A rate-independent crystal plasticity model with a smooth elastic–plastic transition and no slip indeterminacy

A new crystal plasticity constitutive model is proposed that combines, for the first time, the following features: (i) multi-criterion formulation, (ii) exact strain rate-independence, (iii) absence of a consistency condition and (iv) smooth elastic–plastic transition. It is characterised by the existence of a rate-independent overstress that removes the usual possible indeterminacy of simultaneously activated slip systems. The performance of the new model compared to a reference crystal plasticity model by Meric and Cailletaud (1991) in its quasi-rate-independent limit was evaluated in the case of two significantly different materials, namely single crystalline copper and a nickel-based superalloy at room temperature. The provided material point simulations of complex loading paths and large scale finite element simulations do not show any spurious effect of the model, provided that the overstress remains small enough. The improved computational efficiency of the model is discussed.

Elastoplastic model of metals with smooth elastic–plastic transition

The subloading surface model is based on the simple and natural postulate that the plastic strain rate develops as the stress approaches the yield surface. It therefore always describes the continuous variation of the tangent modulus. It requires no incorporation of an algorithm for the judgment of yielding, i.e., a judgment of whether or not the stress reaches the yield surface. Furthermore, the calculation is controlled to fulfill the consistency condition. Consequently, the stress is attracted automatically to the normal-yield surface in the plastic loading process even if it goes out from that surface. The model has been adopted widely to the description of deformation behavior of geomaterials and friction behavior. In this article, it is applied to the formulation of the constitutive equation of metals by modifying the past formulation of this model. This modification enables to avoid the indetermination of the subloading surface, to make the reloading curve recover promptly to the preceding loading curve, and to describe the cyclic stagnation of isotropic hardening. The applicability of the present model to the description of actual metal deformation behavior is verified through comparison with various cyclic loading test data.

Material laws denoting the influence of textures due to spring back simulations

Most calculations of the spring back of sheet metals fail especially if the influence of bending processes is dominant, because at the end of a bending and a subsequent re-bending process followed by a complete un-loading, an extremely precise description of the influence of the first steps of deformation on the final behaviour is necessary. Most people try to get reasonable results using traditional material laws with yield loci within which the material behaves purely elastically. The yield locus itself may include isotropic or kinematical hardening. Also any kind of Hill’s anisotropy possibly combined with kinematical hardening is often included. But all of that is not enough since the local variations of the lattice orientations of the neighbouring crystallites induce a much softer transition from the purely elastic to the (also) plastic regime. This is extremely interesting in the never purely elastic re-bending and un-loading phase that occurs during spring back processes of sheet metals.It was often tried to denote these effects by appropriate phenomenological laws, sometimes also types that did not include any yield locus at all or had only a very small purely elastic regime, whereas a very rapidly changing kinematical (tensor) parameter denoted the smooth elastic–plastic transition. These laws have, in general, no possibility to let the basic inner process enter into the final denotation. So lots and lots of experiments are necessary.A better approach is based on the behaviour of the single crystals or crystallites within the assembly. Unfortunately, the use of reference volumes is extremely expensive and time-consuming. So – as a compromise – an enhanced Taylor’s theory for elastic plastic media including also some aspects of the Sachs’ theory is proposed. Here, it is necessary to use a manifold of representatives of the crystallites with their locally variable orientations. If their number is too high, calculations become too expensive. But with a relatively small number of representative starting orientations accompanied by a sophisticated choice of them, the results become very reasonable and can be applied in present numerical codes. The method remains to be applicable also if real yield loci do no longer exist. Those which are found according to defined bounds (offsets of deformations) at the end of a total un-loading situations will be demonstrated as well as the transitional behaviour of the material immediately after the forming process which are possibly not really comparable.

高サイクル疲労過程の繰返し応力－ひずみ関係

In order to simulate mechanical fatigue phenomena represented by cyclic plasticity such as ratcheting and hysterisis loop, the plastic stretching within a yield surface has to be described, whilst the plastic strain is induced remarkably as the stress approaches the dominant yielding state. The traditional plastic constitutive equation, however, is capable of describing deformation behavior for the stress path only near the monotonic/proportional loading, since the inside of the yield surface is assumed to be an elastic state. In this study, an unconventional plasticity model is proposed for the description of the cyclic loading behavior observed during so-called high cycle fatigue subjected to the cyclic stresses lower than the yield stress. The extended elastoplastic constitutive equation is formulated by introducing both the elastic boundary and the damage concepts. The former is introduced to describe a purely elastic response for the stresses lower than the proportional limit, and the later is to describe the damage effect represented by a progressive degradation of stiffness of materials, which is caused by the accumulation of plastic strain even under the macroscopically elastic condition. The proposed model exhibits a smooth elastic-plastic transition with increase of stress to the dominant yielding state with both plasticity and damage effects. Finally, the extended elasto-plastic model is applied for metals obeying not only isotropic but also kinematic hardening law, and the mechanical responses under cyclic loading condition are examined briefly and compared with the corresponding experimental results for SN490B.

Generation of Plastic Strain due to Constant Cyclic Loading Under Macroscopically Elastic Condition

In order to describe cyclic plasticity phenomena the plastic stretching within a yield surface has to be considered. Subloading surface model categorized in the unconventional plasticity models and describing a smooth elastic-plastic transition would be applicable to non-proportional loading process including cyclic loading behavior of materials with a smooth yield surface. In this study the model is extended for materials exhibiting a purely elastic response under a particular state of stress state, named an elastic boundary surface, whilst they also exhibit a smooth elastic-plastic transition. A damage evolution is also described by incorporating a damage function for the conventional model. The applicability of the model to the cyclic plasticity phenomena observed for the cyclic stresses below the yield stress is verified through the comparison with the experimental data.

Overstress Model with Tangential-Viscoplastic Strain Rate by Incorporation of Overstress Tensor

The elastoplastic stress is first defined as the stress which evolves as the actual strain rate is induced in an imaginary quasi-static process of elastoplastic deformation, while internal variables evolve with the viscoplastic strain rate calculated by the viscoplastic constitutive equation. Further, the novel variable “overstress tensor” reaching the current stress from the elastoplastic stress is defined. Then, the overstress model is extended so as to describe also the tangential viscoplastic strain rate induced by the overstress tensor component tangential to the yield surface. Furthermore, the viscoplastic strain rate due to the change of stress inside the yield surface is incorporated by adopting the concept of the subloading surface which falls within the framework of the unconventional elastoplasticity describing the smooth elastic-plastic transition fulfilling the smoothness and continuity conditions.

Generalized plastic flow rule

An elastoplastic constitutive equation capable of describing the tangential-plastic strain rate induced by the component of stress rate tangential to the subloading surface, called the tangential-plastic stress rate, is proposed based on the subloading surface model [J. Appl. Mech. (ASME) 47 (1980) 266]. Here, the novel tangential-yield surface and the novel tangential-loading criterion are incorporated for the tangential-plastic strain rate. The equation is capable of describing the deformation behavior with the smooth elastic–plastic transition. Based on the equation, a constitutive equation for metals is formulated, its mechanical features are examined and some basic responses are compared with test data.

Friction Theory Based on the Subloading Surface Concept.

A constitutive model for the description of friction phenomenon is formulated by incorporating the subloading surface concept proposed for the formulation of the unconventional plastic constitutive equation. It describes the nonlinearity of the relationship between the normal and the tangential forces on the contact boundary. Further, it describes the smooth relation of the contact force and the sliding distance, i.e. the smooth elastic-plastic transition and the thus the judgment for the fulfillment of friction condition is not needed, which is required in the conventional friction theory. Thus, a rough numerical calculation with large loading steps is allowed in the present model.

Evaluation of typical conventional and unconventional plasticity models for prediction of softening behaviour of soils

Conventional elastoplasticity is based on the idealisation that the interior of the yield surface is a purely elastic domain. This idealisation would not lead to unrealistic prediction of hardening behaviour, but would cause unrealistic prediction of softening behaviour, which is observed typically in overconsolidated soils, whereas metals and normally consolidated soils exhibit hardening behaviour. By contrast, unconventional elastoplasticity that does not use this idealisation makes it possible to describe the plastic deformation due to the change of stress inside the yield surface exhibiting a smooth elastic–plastic transition, and is thus expected to describe softening behaviour more accurately. In this paper the capabilities of conventional and unconventional elastoplasticity for describing softening behaviour are examined, comparing the Drucker–Prager model as the conventional one and the subloading surface model as the unconventional one for predicting the deformation behaviour of overconsolidated soils. It is concluded that unconventional elastoplasticity has to be adopted for predicting softening behaviour.

Evaluation of typical conventional and unconventional plasticity models for prediction of softening behaviour of soils

Conventional elastoplasticity is based on the idealisation that the interior of the yield surface is a purely elastic domain. This idealisation would not lead to unrealistic prediction of hardening behaviour, but would cause unrealistic prediction of softening behaviour, which is observed typically in overconsolidated soils, whereas metals and normally consolidated soils exhibit hardening behaviour. By contrast, unconventional elastoplasticity that does not use this idealisation makes it possible to describe the plastic deformation due to the change of stress inside the yield surface exhibiting a smooth elastic–plastic transition, and is thus expected to describe softening behaviour more accurately. In this paper the capabilities of conventional and unconventional elastoplasticity for describing softening behaviour are examined, comparing the Drucker–Prager model as the conventional one and the subloading surface model as the unconventional one for predicting the deformation behaviour of overconsolidated soils. It is concluded that unconventional elastoplasticity has to be adopted for predicting softening behaviour.

Elastoplastic constitutive equation with tangential stress rate effect

Abstract In traditional elastoplastic constitutive equation with a single smooth plastic potential surface, the plastic stretching is independent of the tangential stress rate , i.e. the component of stress rate which is tangential to the yield surface. This traditional model predicts an unrealistically stiff response when a loading path deviates significantly from the proportional loading. In order to overcome this defect various constitutive models have been proposed. However, a pertinent model applicable to the description of the deformation behavior in a general loading process has not been proposed up to the present. In this article, an elastoplastic constitutive equation with the inelastic stretching induced by the deviatoric stress rate component tangential to the subloading surface is formulated by extending the subloading surface model with a smooth elastic–plastic transition. This model is applicable to the analysis of deformation in a general loading process of materials with an arbitrary yield surface. Based on this equation, a constitutive equation of metals with isotropic-kinematic hardening is formulated and its basic characteristics are examined in detail.

Time-Dependent Elastoplastic Constitutive Equation Based on the Subloading Surface Model and Its Application to Soils

Various constitutive models for the description of time-dependent deformation behavior have been proposed. However, it is first verified in this article that a pertinent model applicable to the description of deformation for a wide range of stress below and over the elastic limit, i.e. the yield stress, has not been found up to the present. It should be noted that a stress goes out over the yield surface at a high rate of deformation, only elastic deformation being induced. The subloading surface model does not premise that a stress exists on the yield surface even in the plastic loading process and thus describes the plastic deformation induced by the rate of stress within the yield surface, exhibiting the smooth elastic-plastic transition. In this article the subloading surface model is extended so as to describe the time-dependence for a wide range of deformation rates by allowing the stress to go out from the yield surface based on the physical interpretation that a plastic deformation due to the mutual slip between microstructures is suppressed for the deformation at a high rate causing the increase of viscous resistance acting between the microstructures. Further, based on this, a time-dependent elastoplastic constitutive equation of soils is formulated by incorporating the secondary-consolidation phenomenon, and its ability to predict deformation behavior of soils is verified by comparisons with test data on fundamental time-dependent behavior with various deformation rates, creep and stress relaxation under the undrained condition.