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].     

A theoretical resistance-curve based on nonproportional plastic strain

Using an exact solution for an elliptical hole in a perfectly plastic material, an expression is derived for the resistance of a ductile material undergoing subcritical crack propagation in the plane. This resistance curve is based on an analogy to the J-integral where an energy dissipation rate is determined rather than an energy release rate. The Tresca yield condition under plane stress loading conditions is employed in this derivation as well as finite deformation theory. This resistance curve is applicable to the initial stage of subcritical crack growth for a ductile material subject to crack tip blunting.     

Fatigue behavior and dislocation substructures for 6063 aluminum alloy under nonproportional loadings

In this paper, the fatigue behavior and dislocation substructures of 6063 aluminum alloy were studied under several nonproportional path loadings, which were circle, ellipse, rectangle and square paths. After fatigue test the micro-structure especially the dislocation substructures of the failure materials was carefully observed with the transmission electron microscope (TEM) method. Under the same 93 MPa equivalent stress amplitude loading, the alloy has the shortest life and the most severe cyclic additional hardening with circle path loading among all the loading paths. This attributes to the complicated dislocation substructures and severe stress concentration of the alloy during the cycling process. While under the ellipse path loading, the alloy has a comparably long life and light cyclic additional hardening. The deformation of the alloy and the morphology of the dislocation substructures determine the fatigue behavior of 6063 alloy under the same equivalent stress amplitude loading. Under the circle path loading, the fatigue life decreases while the cyclic strain increases as the loading stress amplitude increases from 47 MPa to 163 MPa. The dislocation evolution of 6063 alloy during the cycling process under circle path loading was examined with TEM. It was found that the dislocation merges with each other and changes from single lines to crossed bands. The movability of dislocation reduces and the stress concentration degree rises during the cycling process.     

A nonuniform hardening plasticity model for concrete materials

For the most part, the developments of constitutive models for for the concrete by the theory of plasticity have in the past been made to scarch for a suitable failure surface. The initial yield surface is usually assumed to have the same shape as the failure surface but with a reduced size. The subsequent loading surfaces are then obtained by the uniform expansion of the initial one. This approach is found generally inadequate in predicting the deformational behavior of concrete for a wide range of loading conditions.

The present non-uniform hardening plasticity model adopts the most sophisticated failure model of Willam-Warnke or Hsieh-Ting-Chen as the bounding surface; assume an initial yield surface with a shape that is different from the failure surface; proposes a nonuniform hardening rule for the subsequent loading surfaces with a hydrostatic pressure and Lode-angle dependent plasticity modulus; and utilizes a nonassociated flow rule for a general formulation.

The work-hardening stress-strain behaviors of concrete based on the present model are found in good agreement with experimental results involving a wide range of stress states and different types of concrete material. The important features of inelastic behavior of concrete, including brittle failure in tension; ductile behavior in compression; hydrostatic sensitivities; and volumetric dilation under compressive loadings can all be represented by this improved constitutive model.     

A new strain hardening model for rate-dependent crystal plasticity

This paper presents a crystal plasticity based finite element analysis employing the new microstructure-based strain hardening model recently presented by Saimoto and Van Houtte (2011) [7] to simulate formability and texture evolution in the commercial aluminum alloy 5754. Simulations are performed to compare the predictive capability of the new hardening model against the common work hardening models using a rate-dependent plasticity formulation. The parameters in the numerical models are calibrated using the X-ray data published by Iadicola et al. (2008) [9] for the aluminum sheet alloy 5754. The predictions of the model for balanced biaxial tension and in-plane plane-strain tests are compared against experimental observations presented in Iadicola et al. (2008) [9]. It is concluded that the new model provides the best predictions of the large strain behavior of Aluminum sheet alloy 5754 subjected to various strain paths.     

Application of endochronic theory of plasticity for modeling nonproportional repetitive-variable hard loading

To evaluate the effectiveness of the previously proposed constitutive equations of the endochronic theory of plasticity, we examine the deformation of thin-walled tubular specimens along square paths in deviatoric strain space with the combined action of axial force and torque. We obtain the resolving system of equations in application to strain-controlled nonproportional cyclic loading. Good agreement of the modeling results with the experimental data is noted.Translated from Problemy Prochnosti, No. 5, pp. 3–10, May, 1994.     

A plasticity model for calculating stress–strain sequences under multiaxial nonproportional cyclic loading

There is increasing demand for analytical methods that estimate the fatigue life of engineering components and structures with a high degree of accuracy. The fatigue life is determined by the stress–strain sequences at the critical locations. Therefore, these sequences have be calculated with sufficient accuracy for arbitrary nonproportional cyclic loading. Based on the experience with a variety of material models following macroscale continuum mechanics approaches, an improved set of constitutive equations is proposed. The stress–strain behaviour of a commercial structural steel has been investigated experimentally. Firstly, the results of this experimental study serve to identify the material parameters comprised in the model. Secondly, the predicted stress–strain paths are compared to their experimentally determined counterparts as well as to paths predicted by other models. The overall accuracy of the proposed model is quite satisfying, especially as far as calculated amplitudes are concerned.     

A robust kinematic hardening rule for cyclic plasticity with ratchetting effects

Summary A robust kinematic hardening rule is proposed which appropriately blends the deviatoric stress rate rule and the Tseng-Lee rule in order to satisfy both the experimental observations made by Phillips et al. [1]–[5] and the nesting of the yield surface to the limit surface. The work presented in Part I is confined to the theoretical formulation of kinematic hardening rule with limited correlation to experimental results. A more general expression for the plastic modulus is proposed. The expressions proposed by McDowell and by Dafalias can be obtained as a special case of the proposed expression. An additional parameter is introduced that reflects the dependence of the plastic modulus on the angle between the deviatoric stress rate tensor and the direction of the limit backstress relative to the yield backstress.     

Endochronic theory of plasticity: Prediction of nonproportional cyclic deformation of materials

The possibilities of the endochronic theory of plasticity to describe the nonproportional cyclic deformation of materials are evaluated. Certain considerations are stated as to the refinement of its equations of state. Institute for Problems of strength, National Academy of Sciences of Ukraine, Kiev, Ukraine. Translated from Problemy Prochnosti, No. 3, pp. 38–45, May–June, 1998.     

Characterization of strain hardening for a simple pressure-sensitive plasticity model

Summary Paralleling the development of strain hardening for the pressure-independent von Mises criterion, a simple plasticity model in strain space was presented to characterize strain hardening for pressure-sensitive compressible materials. Two hardening moduli,H _T andH _C , which emerged from the constitutive equations and can becalculated from uniaxial stress-strain curves in tension and compression, were used to characterize the strainhardening responses forgeneral and special stress systems. The results indicated the implications and restrictions of the yield function on the hardening responses. It was also shown that strain softening, under general stress systems, can be a natural consequence of pressure-sensitive yielding. Consequently, a strain-space formulation is recommended for most (if not all) pressure-sensitive plasticity models. Preliminary application to the yielding of polymers under hydrostatic pressure gave reasonable results for polyethylene at moderate pressure and small strains; the results for polycarbonate were generally poor. Finally, the advantages and limitations of the present approach were discussed.With 6 Figures     

A simple model of elastoplasticity for nonproportional cyclic loading

On the basis of the analysis of experimental data, we formulate requirements to the constitutive relations of plasticity under the conditions of complex cyclic loading. We propose a version of constitutive relations obtained by a simple generalization of the Mazing model to the three-dimensional case and introduction of a function of cyclic hardening. We also suggest a procedure for the identification of this function. According to the results of numerical analysis, this model adequately describes the main effects of cyclic plasticity for austenitic stainless steels. Perm State Technical University, Perm, Russia. Translated from Problemy Prochnosti, No. 1, pp. 15 – 24, January – February, 1998.     

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     

Eulerian constitutive model for finite deformation plasticity with anisotropic hardening

A constitutive model is presented for finite strain plasticity. The model incorporates both isotropic and kinematic hardening of the Ziegler type. The corotational rate used here is in line with the theory suggested by Paulun and Pecherski (1985) but not necessarily confined to the von Mises type yield criterion and the Prager hardening rule. The aspect of integration of the corotational rates is also discussed here. The use of the integration of the material rate of tensors with time as a substitute for the proper integration with time of corotational rates leads to mathematical inconsistencies of the theory of Lie derivatives. The problem of simple shear is investigated and compared with other works.     

An extended crystal plasticity model for latent hardening in polycrystals

In this contribution, a computational approach to modeling size-dependent self- and latent hardening in polycrystals is presented. Latent hardening is the hardening of inactive slip systems due to active slip systems. We focus attention on the investigation of glide system interaction, latent hardening and excess dislocation development. In particular, latent hardening results in a transition to patchy slip as a first indication and expression of the development of dislocation microstructures. To this end, following Nye (Acta Metall 1:153–162, 1953), Kondo (in Proceedings of the second Japan national congress for applied mechanics. Science Council of Japan, Tokyo, pp. 41–47, 1953), and many others, local deformation incompatibility in the material is adopted as a measure of the density of geometrically necessary dislocations. Their development results in additional energy being stored in the material, leading to additional kinematic-like hardening effects. A large-deformation model for latent hardening is introduced. This approach is based on direct exploitation of the dissipation principle to derive all field relations and (sufficient) forms of the constitutive relations as based on the free energy density and dissipation potential. The numerical implementation is done via a dual-mixed finite element method. A numerical example for polycrystals is presented.     

A hardening orthotropic plasticity model for non-frictional composites: Rate formulation and integration algorithm

The rate formulation and an unconditionally stable integration algorithm for a single-surface hardening anisotropic elastoplasticity model are developed to obtain an effective medium constitutive model for a class of periodic elastoplastic composites. Because the plasticity model features an intrinsically anisotropic failure criterion and hardening law, it requires more material parameters (15) than does its isotropic analogue (2). To address this issue and to assess how well the new model serves as an effective medium constitutive model for a class of composites, results from numerical elastoplastic homogenization computations are utilized for free-parameter estimation. It is shown that the new model provides a good fit to the homogenization data, and the model's excellent performance in a two-dimensional finite element setting is demonstrated by performing computations involving in-plane loading of an elastoplastic masonry wall. The wall is modelled first as a heterogeneous medium with all of its microstructure, and subsequently as a homogeneous effective medium with the new plasticity model.     

An approximate method of determination of maximum strain hardening levels in metals under nonproportional low-cycle loading

Using the findings of analysis of deformation curves for metallic materials under static and cyclic loading, an approximate method is put forward for the determination of maximum strain hardening levels in the case of non-proportional low-cycle loading with strain monitoring. Based on the correlation between strain hardening data obtained from the static and proportional and nonproportional cyclic deformation curves, an approximate analytical relationship is built up which allows for predicting maximum strain hardening levels under nonproportional low-cycle loading. __________ Translated from Problemy Prochnosti, No. 2, pp. 29–38, March–April, 2006.     

Computational plasticity of mixed hardening pressure-dependency constitutive equations

In the present study, two semi-implicit schemes, based on the exponential maps method, are derived for integrating the pressure-sensitive constitutive equations. In spite of the fact that the consistent tangent operator is necessary to preserve the quadratic rate for the asymptotic convergence of the Newton-Raphson solution in the finite element analyses, there exists no derivation of this operator for the exponential-based integrations of the pressure-sensitive plasticity in the literature. To fulfill this need, the algorithmic tangent operators are extracted for the new semi-implicit as well as the former exponential-based integrations. Moreover, for the accurate integration presented by Rezaiee-Pajand et al. (Eur J Mech A Solids 30:345–361, 2011), the consistent tangent operator is obtained. Eventually, all the investigations are assessed by a broad range of numerical tests.     

Analysis of nonproportional cyclic deformation of materials within the scope of endochronous plasticity theory

A model is presented for strengthening of a body in order to describe processes of material nonproportional cyclic deformation on the basis of the approach of endochronous plasticity theory. New rules are introduced for isotropic and kinematic strengthening processes taking account of a nonproportionality factor for the plastic deformation cycle trajectory. Basic experiments are described for defining the equations of state. The distribution of stress fields in thin-walled tube specimens in relation to the number of the loading cycle is determined as applied to different plastic deformation trajectories. The adequacy of calculated and test data is confirmed. The values sought are compared with similar results obtained by the two-surface theory of flow.Translated from Problemy Prochnosti, No. 1, pp. 24–34, January, 1993.     

A comparative evaluation of several algorithms for phase aberrationcorrection

A common framework is presented to classify several phase correction techniques. A subset of these techniques are evaluated through simulations which utilize 2-D phase aberration profiles measured in the breast. The techniques are compared based on their ability to reduce phase errors, stability, and sensitivity to noise and missing elements in the transducer array. Significant differences are observed in these measures of performance when the size and location of the aperture area used to generate a phase reference signal are varied. Techniques that utilize a small correction reference region are more susceptible to noise and missing elements than techniques which use larger reference regions. The algorithms encounter problems in 2-D phase correction when making the transition from one row to the next, due to the low interelement correlation at the transition points. It is shown that the magnitude of the interelement correlation is the key parameter governing phase correction performance