Constitutive equations for cyclic plasticity and cyclic viscoplasticity

Time-independent constitutive theories for cyclic plasticity

The stress-strain behavior of carbon black filled rubber is recognized to be nonlinearly elastic in its main part (see e.g. Gent [1]). In addition, inelastic effects occur under monotonic and cyclic processes. The inelastic behavior includes nonlinear rate dependence as well as equilibrium hysteresis. Moreover, the first periods of a stress-strain curve differ significantly from the shape of subsequent cycles; a characteristic feature, which is called the Mullins effect, because it has been pointed out by Mullins [2]. All inelastic phenomena are strongly influenced by the volume fraction of the filler particles (see e.g. Payne [3], So and Chen [4], Meinecke and Taftaf [5]). The aim of the present paper is to design a constitutive model, representing this kind of material behavior as a phenomenological theory of continuum mechanics. In order to motivate the basic structure of the constitutive theory, a series of uniaxial experiments between 100% in tension and 30% in compression are presented and analyzed. First of all, monotonic strain controlled experiments show the nonlinear rate dependence of the stress response. Then, a series of inserted relaxation periods at constant strain yields the monotonic equilibrium stress-strain curve, which is strongly nonlinear and unsymmetric with respect to the origin. Finally, cyclic experiments under strain control display pronounced hysteresis behavior. The hysteresis effects are mainly rate dependent, but there exists also a weak equilibrium hysteresis (compare to similar observations of Orschall and Peeken [6]). The Mullins effect corresponds to a softening phenomenon during the first few cycles. By means of an appropriate preprocess, this effect was excluded during the above experiments. Apart from the Mullins effect, neither hardening nor significant softening phenomena were observed in the context of cyclic loadings. These results motivate the structure of a constitutive model of finite strain viscoplasticity: The total stress is decomposed into an equilibrium stress and an overstress, where the overstress is a rate dependent functional of the strain history. The overstress represents the rate dependence of the material behavior and tends asymptotically to zero during relaxation processes. The nonlinearity of the rate dependence is incorporated by means of a stress dependent relaxation time. The equilibrium stress is assumed to be a rate independent functional of the strain history. For this quantity, we make use of an arclength representation, which was originally introduced by Valanis [7]. In case of vanishing equilibrium hysteresis and vanishing rate dependence our constitutive model reduces to finite strain hyperelasticity, which is the first approximation of the constitutive properties. In more general cases the “main shape” of a stress-strain curve is determined by hyperelasticity, superimposed by rate dependent and equilibrium hysteresis. The representation of the Mullins effect is incorporated by a continuum damage model. Some numerical simulations at the end of the paper demonstrate that the presented theory is able to represent the observed phenomena qualitatively and quantitatively with sufficient approximation.

A constitutive model for carbon black filled rubber: Experimental investigations and mathematical representation

The object of this paper is to provide a critical review of the current state of plasticity in the presence of finite deformation. In view of the controversy regarding a number of fundamental issues between several existing schools of plasticity, the areas of agreement are described separately from those of disagreement. Attention is mainly focussed on the purely mechanical, rate-independent, theory of elastic-plastic materials, although closely related topics such as rate-dependent behavior, thermal effects, experimental and computational aspects, microstructural effects and crystal plasticity are also discussed and potentially fruitful directions are identified. A substantial portion of this review is devoted to the area of disagreement that covers a detailed presentation of argument(s), bothpro andcon, for all of the basic constitutive ingredients of the rate-independent theory such as the primitive notion or definition of plastic strain, the structure of the constitutive equation for the stress response, the yield function, the loading criteria and the flow and the hardening rules. The majority of current research in finite plasticity theory, as with its infinitesimal counterpart, still utilizes a (classical) stress-based approach which inherently possesses some shortcomings for the characterization of elastic-plastic materials. These and other anomalous behavior of a stress-based formulation are contrasted with the more recent strain-based formulation of finite plasticity. A number of important features and theoretical advantages of the latter formulation, along with its computational potential and experimental interpretation, are discussed separately.

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A critical review of the state of finite plasticity

https://hub.hku.hk/bitstream/10722/200516/1/Content.pdf

The art of coupon tests

T he spuniaxial viscoplastic behavior of AISI Type 304 stainless steel was investigated by tensile tests at various strain-rates (10−8−10−2s−1), and by short-term creep and relaxation tests up to 5 h. Instantaneous large changes in strain-rate were also performed during monotoniC and cyclic loading. A servocontrolled testing machine and displacement measurement on the specimen gage length were used for all tests. The results show significant rate-sensitivity, creep and relaxation. Test histories involving loading and unloading with positive loads up to 15% strain show that the relaxation behavior in the plastic range depends only on the strain-rate preceding the relaxation test and is independent of the strain magnitude. Also, the relaxation behavior is uniquely related to the stress changes corresponding to instantaneous large changes in the strain-rate during tensile tests. Completely reversed strain-controlled loading gradually changes the stress change/strain-rate change behavior. Annealed specimens and specimens loaded to a cyclic steady-state differ not only in their work-hardening characteristics but also in their rate-dependent behavior. In the cyclic steady-state, different hysteresis loops are traced for different strain-rates with fully reversible transitions from one hysteresis loop to another under strain-rate changes. These results support the notion that the viscoplastic behavior can be represented by piecewise nonlinear viscoelasticity.

An experimental study of room-temperature rate-sensitivity, creep and relaxation of AISI type 304 stainless steel

Following previous work ( Krempl , 1979), a servocontrolled testing machine and strain measurement at the gage length were used to study the uniaxial rate(time)-dependent behavior of AISI Type 304 stainless steel at room temperature. The test results show that the creep strain accumulated in a given period of time depends strongly on the stress-rate preceding the creep test. In constant stress-rate zero-to-tension loading the creep strain accumulated in a fixed time-period at a given stress level is always higher during loading than during unloading. Continued cycling causes an exhaustion of creep ratchetting which depends on the stress-rate. Periods of creep and relaxation introduced during completely reversed plastic cycling show that the curved portions of the hysteresis loop exhibit most of the inelasticity. In the straight portions, creep and relaxation are small and there exists a region commencing after unloading where the behavior is similar to that at the origin for virgin materials. This region does not extend to zero stress. The results are at variance with creep theory and with viscoplasticity theories which assume that the yield surface expands with the stress. They support the theory of viscoplasticity based on total strain and overstress.

An experimental study of uniaxial creep, cyclic creep and relaxation of aisi type 304 stainless steel at room temperature

A previously proposed, uniaxial differential constitutive equation (E.R. C ernocky and E. K rempl , 1979a, b), nonlinear in the Cauchy stress and the engineering strain but linear in the stress and strain-rates, is specialized to an overstress model. It is shown by qualitative arguments that the solutions correspond to typical room-temperature viscoplastic behavior of AISI type 304 stainless steel. Two unknown coefficient functions are determined by extrapolation of room temperature relaxation data for this steel. The stiff first-order nonlinear differential equations are then numerically integrated for a variety of test histories. These include strain control with strain-rates from 10 −6 to 800s −1 , stress control with stressrates from 1.95 kPa s −1 to 19.5 MPa s −1 , instantaneous large changes in strain-rate and stress-rate, and partial unloading and reloading in strain and stress control and tension-tension cyclic creep. The computed results show good qualitative agreement with tests. Based on these results we consider that the model is a good representation of metal deformation behavior as long as the overstress does not change sign.

A uniaxial viscoplastic model based on total strain and overstress

Uniaxial zero-to-tension load controlled tests at stress rates differing by three orders of magnitude were performed on annealed AISI Type 304 stainless steel. The maximum stress was determined either as the stress reached in displacement control at one per cent strain (History I), or as the stress reached after a 1050 s relaxation test at one per cent strain (History II). The tests of History III were identical to those of History I, except that the specimens were subjected to cyclic hardening prior to the start of the ratchetting test. The ratchet strain accumulations in Histories I and II are significant and depend on stress rate. A stress rate decrease increases the accumulated ratchet strain. The final ratchet strain accumulation is independent of stress rate sequence. In History II, no ratchet strain was found at any stress rate. If the steel had behaved in a rate-independent fashion, no ratchet strain accumulation should have been found in any of the tests. They were performed with an MTS servohydraulic, computer controlled testing machine, a clip on-extensometer and digitized data acquisition.

The interaction of cyclic hardening and ratchetting for AISI type 304 stainless steel at room temperature—I. Experiments

An austenitic AISI Type 304 stainless steel, a ferritic A533B pressure vessel steel and a Ti-7Al-2Cb-1Ta alloy were tested using a servocontrolled MTS axial-torsion testing machine. Tests involved changes in strain rate between 10−8 and 10−3 s−1 and intermittent creep periods of less than 1200 s duration. The tests show that inelastic work is not a suitable repository for modeling strain (work)-hardening and the Bauschinger effect is found to be rate dependent. Upon an increase in stress level, creep rate can decrease. This anomaly can be reproduced by a theory of viscoplasticity based on overstress previously proposed by the first author and his co-workers.

E. Krempl

Papers

An experimental study of room-temperature rate-sensitivity, creep and relaxation of AISI type 304 stainless steel

An experimental study of uniaxial creep, cyclic creep and relaxation of aisi type 304 stainless steel at room temperature

A uniaxial viscoplastic model based on total strain and overstress

The interaction of cyclic hardening and ratchetting for AISI type 304 stainless steel at room temperature—I. Experiments

Some critical uniaxial experiments for viscoplasticity at room temperature