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

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...

A new computational procedure for the steady state elastoplastic analysis of structures under cyclic loading is presented. The procedure is based on the decomposition of the unknown steady state residual stress distribution into Fourier series. The coefficients of the series are evaluated in an iterative way by satisfying equilibrium and compatibility at some preselected time points inside the cycle. The procedure in the present work is applied to a simple 1-D three bar structure and to a 2-D plate. Various load cases are examined which may lead to elastic adaptation, alternating plasticity or incremental collapse.

International Journal of Solids and Structures, 2017

In this work, we investigate the numerical convergence of a set of plasticity models with different kinematic and directional distortional hardening rules under cyclic plastic loading. In particular, we revisit the results presented in Feigenbaum et al. [1] in order to more robustly check for convergence during the numerical integration procedure, and show that the results presented in the previous work do not converge. We investigate the role of the stepsize and numerical scheme on the convergence of these models when predicting ratcheting. By reducing step-sizes and using a forward Euler scheme during numerical integration, converged solutions are obtained. The new converged results lead to new conclusions. Results still suggest that directional distortional hardening can improve ratcheting predictions, however the addition of directional distortional hardening yields less improvements compared to kinematic hardening alone than previously thought. This new conclusion, strongly suggests the need for additional modeling developments in order accurately predict ratcheting strains under a wide variety of cyclic plastic loadings.

International Journal of Plasticity, 1992

The multiple backstress nonlinear kinematic hardening model of MOOSBRUGGER and McDoWELL [1990] is extended to thermomechanical cyclic loading conditions. The model employs a decomposition of the isotropic hardening between the yield surface and backstress saturation amplitudes, with certain components independent of the degree of isotropic hardening. General forms are presented for thermoplasticity and thermoviscoplasticity that include temperature rate terms in both the kinematic and isotropic hardening parameters. General forms are presented for temperature path history-dependent and -independent materials; it is shown that the latter case is an important feature in thermoplasticity, since the flow rule cannot exhibit the necessary degree of temperature dependence. In the thermoviscoplastic case, the ratedependence is decomposed between the flow rule and backstress saturation amplitudes, a unique feature consistent with dislocation cross slip. Thermodynamical restrictions are discussed for both cases, and isothermal and nonisothermal cyclic loading experiments are correlated with both theories.

International Journal of Solids and Structures, 2006

The class of generalized standard materials is not relevant to model the nonassociative constitutive equations. The bipotential approach, based on a possible generalization of FenchelÕs inequality, allows the recovery of the flow rule normality in a weak form of an implicit relation. This defines the class of implicit standard materials. For such behaviours, this leads to a weak extension of the classical bound theorems of the shakedown analysis. In the present paper, we recall the relevant features of this theory. Considering an elastoplastic material with nonlinear kinematic hardening rule, we apply it to the problem of a sample in plane strain conditions under constant traction and alternating torsion in order to determine analytically the interaction curve bounding the shakedown domain. The aim of the paper is to prove the exactness of the solution for this example by comparing it to step-by-step computations of the elastoplastic response of the body under repeated cyclic loads of increasing level. A reliable criterion to stop the computations is proposed. The analytical and numerical solutions are compared and found to be closed one of each other. Moreover, the method allows uncovering an additional Ô2 cycle shakedown curveÕ that could be useful for the shakedown design of structure.

Computer Methods in Applied Mechanics and Engineering, 2004

On the theoretical level, the present paper presents a detailed comparison of recent finite strain models for Armstrong-Frederick kinematic hardening. Thereby two strategies are discussed: (1) ''Chaboche-type'' concepts, considering the back stress as internal variable, (2) continuum mechanical extensions of the classical rheological model, using only strain-like internal variables. It is shown in the paper that models of the second kind can be recast in the format of anisotropic inelasticity with structure tensors. Second, the work focuses on the algorithmic treatment of the kinematic hardening concepts presented before. This problem has been tackled up to now only in the context of linearized models. In contrast to isotropic finite elastoplasticity, the integration cannot be carried out with respect to principal axes. Therefore, a new integration algorithm is developed which is suitable for the anisotropic case but still retains plastic incompressibility. In the case of small elastic deformation, the algorithm reduces to a system of only one non-linear equation and twelve linear equations. In general, the computational effort of the new scheme does not exceed the one of the backward Euler scheme which has the disadvantage that plastic incompressibility is not fulfilled automatically. Several numerical examples show that the representatives of both approaches, (1) and (2), yield similar results, if physically reasonable material parameters are chosen.

International Journal of Plasticity, 1986

Using the concept of an internal time as related to plastic strains, a differential stressstrain relation for elastoplasticity is rederived, such that (i) the concept of a yield-surface is retained; (ii) the definitions ot~ elastic and plastic processes are analogous to those in classical plasticity theory; and (iii) its computational implementation, via a "tangent-stiffness" finite element method and a "generalized-midpoint-radial-return" stress-integration algorithm, is simple and efficient. Also, using the concept of an internal time, as related to both the inelastic strains as well as the Newtonian time, a constitutive model for creep-plasticity interaction, is discussed. The problem of modeling experimental data for plasticity and creep, by the present analytical relations, as accurately as desired, is discussed. Numerical examples which illustrate the validity of the present relations are presented for the cases of cyclic plasticity and creep.

International Journal of Plasticity, 1998

ÐThe hardening behavior of metals to non-proportional loading and ratchetting eects is investigated here using the proposed model presented by Voyiadjis and Basuroychowdhury (1998). The backstress evolution equations are modi®ed here in order to account for non-proportionality of loading. The same general form of the kinematic hardening formulation de®ned previously by Voyiadjis and Basuroychowdhury (as above) is used here. The material constant, i , is now expressed in a functional form, " i , to account for the variation in the loading direction. Numerical results are obtained using the proposed model for a series of plastic strain controlled cyclic tests due to multiaxial cyclic tests performed at room temperature on thin-walled tubural specimens of Type 316 stainless steel. The results are compared with the experimental values obtained by Tanaka, et al. (1985). The drift correction due to the ®nite increments of stress or strain is corrected using an ecient approach that corrects the backstress only. The model is also tested for cyclic creep which occurs due to a non-zero mean stress. The cyclic creep occurs initially and saturates reaching a stable cycle.The modi®ed proposed kinematic hardening rule and the associated bounding surface are used in conjunction with a robust plasticity model for metals. The hardening dependence on the plastic strain path is also investigated in this work.

This work studies how a nonlinear kine-matic model aimed for cyclic plasticity could be put into effect and used within a FEM code. A correct modeling of cyclic elasto-plastic behavior can be exploited in low-cycle fatigue life investigation as well as in manufacturing problems related to springback prediction. The chosen formulation has been proposed by Chaboche, and it is implemented in most of the commercial codes used for nonlinear FEM simulations. At first, a procedure for the proper identification of unknown material model parameters has been put forward. This calibration, based on the data collected from experimental low-cycle fatigue tests, has been performed by means of an inverse method. Laboratory tests differ according to the type of material under investigation. A classification can be operated distinguishing between specimens obtained from bulk materials or from sheet metals. For the former, standard tension-compression tests have been performed, while for the latter, a dedicated testing equipment for three-point bend cyclic tests has been devised. Then, further experimental tests have been run to check model trans-ferability: different strain per cycle amplitudes, asym-metric strain cycling and different stress triaxiality levels have been investigated. For each of these tests, G.B. Broggiato · F. Campana · L. Cortese () experimental vs. FEM results have been analyzed to show the level of agreement that has been reached.

International Journal of Plasticity, 1992

The experimental study of evolution rules of two state variables: back stress c~ij and yield stress R are first discussed. It is shown that back stress evolution is affected by the maximal prestress, and asymptotic value of yield stress depends on strain amplitude. The constitutive model is presented next, and its prediction for cyclic loading program is compared with experimental data.

Engineering Structures, 2013

In this work we present a phenomenological constitutive model which is capable of coupling two basic inelastic behavior mechanisms, plasticity and damage. The model is targeting cyclic loading applications. Thus, in either plasticity or damage part, both isotropic and linear kinematic hardening effects are taken into account. The main advantage of the model is the use of independent plasticity versus damage criteria for describing the inelastic mechanisms. Another advantage concerns the numerical implementation of such model provided in hybrid-stress variational framework, resulting with very enhanced accuracy and efficient computation of stress and internal variables in each element. Several illustrative examples are presented in order to confirm the accuracy and efficiency of the proposed formulation in application to cyclic loading.