Computing the asymptotic cyclic response of elastoplastic solids with nonlinear kinematic hardening

Journal of Siberian Federal University. Mathematics & Physics, 2021

The paper gives insights into modeling and well-posedness analysis driven by cyclic behavior of particular rate-independent constitutive equations based on the framework of hypoplasticity and on the elastoplastic concept with nonlinear kinematic hardening. Compared to the classical concept of elastoplasticity, in hypoplasticity there is no need to decompose the deformation into elastic and plastic parts. The two different types of nonlinear approaches show some similarities in the structure of the constitutive relations, which are relevant for describing irreversible material properties. These models exhibit unlimited ratchetting under cyclic loading. In numerical simulation it will be demonstrated, how a shakedown behavior under cyclic loading can be achieved with a slightly enhanced simple hypoplastic equations proposed by Bauer

Acta Mechanica, 1994

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],

International Conference of …, 2008

It has been a long standing problem to solve the cyclic loading phenomena by a reasonably accurate cyclic model. Most of the existing cyclic models are unable to reproduce the memory effect i.e., material's memory about its last load reversal point in the shear stress-strain plane, which produces a closed hysteretic loop. In this paper, a novel formulation of kinematic hardening rule is developed by extending the onedimensional Masing's rule to general three dimensional stress-space. The cyclic behaviour is simulated by introducing a new framework in which the dimensionless kinematic hardening rate is varied according to the instantaneous stress value at that point along the stress path. When the direction of the loading is reversed, the initial rate of hardening is restored and the rate of variation of hardening is scaled according to extended Masing's law. As a result, a closed hysteretic stress-strain loop is obtained due to cyclic loading. Also, a new hyperbolic growth function is used to simulate the rate of kinematic hardening.

Chinese Journal of Mechanical Engineering, 2021

Mechanical engineering structures and structural components are often subjected to cyclic thermomechanical loading which stresses their material beyond its elastic limits well inside the inelastic regime. Depending on the level of loading inelastic strains may lead either to failure, due to low cycle fatigue or ratcheting, or to safety, through elastic shakedown. Thus, it is important to estimate the asymptotic stress state of such structures. This state may be determined by cumbersome incremental time-stepping calculations. Direct methods, alternatively, have big computational advantages as they focus on the characteristics of these states and try to establish them, in a direct way, right from the beginning of the calculations. Among the very few such general-purpose direct methods, a powerful direct method which has been called RSDM has appeared in the literature. The method may directly predict any asymptotic state when the exact time history of the loading is known. The advantag...

International Journal of Plasticity, 2019

The second part of the study presents development of the Dirac delta functions framework to modelling of cyclic hardening and softening of material during cyclic loading conditions for the investigated in Part I low carbon S355J2 steel. A new criterion of plastic strain range change is formulated. This provides more certainty in the cyclic plasticity modelling framework compared to classical plastic strain memorization modelling. Two hardening parameters from the developed kinematic hardening rule are written as functions of both plastic strain range and previously accumulated plastic strain. This representation of hardening parameters is able to accurately match experimental results with different types of loading programs including random loading conditions and considering initial monotonic behavior with yield plateau deformation. Ratcheting behaviour is simulated by the developed cyclic plasticity framework by considering an approximated form of the Dirac delta function for modelling the deviation effect and introducing an additional supersurface for better prediction of ratcheting rate. The proposed cyclic plasticity model requires up to 21 material constants, depending on application. A clear and straightforward calibration procedure, where sets of material constants are determined for each plasticity phenomenon considered, is presented. Application of the model to different materials under various tension-compression and non-proportional axial-torsion cycles shows very close agreement with test results.

Computer Methods in Applied Mechanics and Engineering, 2012

The asymptotic steady state behavior of an elastic-perfectly plastic structure under cyclic loading may be determined by time consuming incremental time-stepping calculations. Direct methods, alternatively, have a big computational advantage as they attempt to find the characteristics of the cyclic state right from the start of the calculations. Most of these methods address an elastic shakedown state through the shakedown theorems and on the basis of mathematical programming algorithms. In the present paper, a novel direct method that has a more physical basis and may predict any cyclic stress state of a structure under a given loading is presented. The method exploits the cyclic nature of the expected residual stress distribution at the steady cycle. Thus, after equilibrating the elastic part of the total stress with the external load, the unknown residual stress part is decomposed into Fourier series whose coefficients are evaluated iteratively by satisfying compatibility and equilibrium with zero loads at time points inside the cycle and then integrating over the cycle. A computationally simple way to account for plasticity is proposed. The procedure converges uniformly to the true cyclic residual stress for a loading below the elastic shakedown limit or to an unsafe cyclic total stress, which may be used to mark the regions with plastic straining inside the cycle. The method then continues to determine whether the applied loading would lead the structure to ratcheting or to regions that alternate plastically. The procedure is formulated within the finite element method. A von Mises yield surface is typically used. Examples of application of one and two dimensional structures are included.

International Journal of Plasticity, 1993

Kinematic hardening rules formulated in a hardening/dynamic recovery format are examined for simulating rachetting behavior. These rules, characterized by decomposition of the kinematic hardening variable into components, are based on the assumption that each component has a critical state for its dynamic recovery to be activated fully. Discussing their basic features, the authors show that they can predict much less accumulation of uniaxial and multiaxial ratchetting strains than the Armstrong and Frederick rule. Comparisons with multilayer and multisurface models are made also, resulting in a finding that the simple one in the present rules is similar to the multilayer model with total strain rate replaced by inelastic (or plastic) strain rate. Part II of this work deals with applications to experiments.

Computer Methods in Applied Mechanics and Engineering, 1977

A finite element approach for cyclic elastic-plastic dynamic analysis is presented. A hardening model suited for cyclic plasticity behavior is incorporated. It is composed of several yield surfaces, and nonlinear stress-strain curves can be included. The central difference timewise operator is employed to solve the equations of motion. Comparison is made with the Newmark operator. Numerical examples illustrate the effect of cyclic plastic deformations on the dynamic response of simple problems. Comparison is presented for the structural behavior as predicted by the present hardening model and by the isotropic hardening model.

International Journal of Mechanical Sciences, 2002

The kinematic hardening theory of plasticity based on the Prager and Frederick-Armstrong models are used to evaluate the cyclic loading behavior of a beam under the axial, bending, and thermal loads. The beam material is assumed to follow non-linear strain hardening property. The material's strain hardening curves in tension and compression are assumed to be both identical for the isotropic material and di erent for the anisotropic material. A numerical iterative method is used to calculate the stresses and plastic strains in the beam due to cyclic loadings. The results of the analysis are checked with the known experimental tests. It is concluded that the Prager kinematic hardening theory under deformation controlled conditions, excluding creep, results into reversed plasticity. The load controlled cyclic loading under the Prager kinematic hardening model with isotropy assumption results into reversed plasticity. Under anisotropy assumption of tension=compression curve, this model predicts ratcheting. On the other hand, the Frederick-Armstrong model predicts ratcheting behavior of the beam under load controlled cyclic loading with non-zero mean load. This model predicts reversed plasticity under the load controlled cyclic loading with zero mean load, and deformation controlled cyclic loading.

2009

KRISHNA, SHREE. Unified Constitutive Modeling for Proportional and Nonproportional Cyclic Plasticity Responses. (Under the supervision of Dr. Tasnim Hassan.) Several features of cyclic plasticity, e.g. cyclic hardening/softening, ratcheting, relaxation, and their dependence on strain range, nonproportionality of loading, time, and temperature determine the stress-strain responses of materials under cyclic loading. Numerous efforts have been made in the past decades to characterize and model these responses. Many of these responses can be simulated reasonably by the existing constitutive models, but the same models would fail in simulating the structural responses, local stress-strain or global deformation. One of the reasons for this deficiency is that the constitutive models are not robust enough to simulate the cyclic plasticity responses when they interact with each other. This deficiency can be understood better or resolved by developing and validating constitutive models agains...