Multiaxial cyclic deformation and non-proportional hardening employing discriminating load paths

Abstract

Some novel discriminating multiaxial cyclic strain paths with incremental and random sequences were used to investigate cyclic deformation behavior of materials with low and high sensitivity to non-proportional loadings. Tubular specimens made of 1050 QT steel with no non-proportional hardening and 304L stainless steel with significant non-proportional hardening were used. 1050 QT steel was found to exhibit very similar behavior under various multiaxial loading paths, whereas significant effects of loading sequence were observed for 304L stainless steel. In-phase cycles with a random sequence of axial-torsion cycles on an equivalent strain circle were found to cause cyclic hardening levels similar to 90° out-of-phase loading of 304L stainless steel. In contrast, straining with a small increment of axial-torsion on an equivalent strain circle results in higher stress than for in-phase loading of 304L stainless steel, but the level of hardening is lower than for 90° out-of-phase loading. Tanaka’s non-proportionality parameter coupled with a Armstrong–Fredrick incremental plasticity model, and Kanazawa et al.’s empirical formulation as a representative of such empirical models were used to predict the stabilized stress response of the two materials under variable amplitude axial-torsion strain paths. Consistent results between experimental observations and predictions were obtained by employing the Tanaka’s non-proportionality parameter. In contrast, the empirical model resulted in significant over-prediction of stresses for 304L stainless steel.

Introduction

Most service load histories of components and structures are multiaxial, some proportional and others non-proportional. Some materials exhibit additional strain hardening due to the non-proportionality of cyclic loading. This phenomenon was first observed by Taira et al. (1968), and then explained by Lamba and Sidebottom, 1978, Kanazawa et al., 1979 in the late 1970s.

Subsequently, the mathematical modeling of non-proportional cyclic loading has been a topic of interest in solid mechanics (McDowell, 1985, Benallal and Marquis, 1987, Doong and Socie, 1991, Fan and Peng, 1991, Jiang and Kurath, 1997) and presents a challenging issue for design engineers due to the effect of the non-proportional hardening on fatigue life (Jordan et al., 1984, Wu and Yang, 1987, Socie, 1987, Itoh et al., 1995, Kurath et al., 1999). As a result, multiaxial fatigue models reflecting the constitutive behavior of material, such as those in (Fatemi and Socie, 1988, Park and Nelson, 2000) have been proposed over the last two decades.

Although there are several definitions for proportional and non-proportional loadings, the definition often used is in connection with the interaction of multiple slip systems. Any cyclic loading resulting in the rotation of principal axes in time, and consequently interaction of multiple slip systems in different directions, is considered as non-proportional loading (Socie and Marquis, 2000). If the principal directions of cyclic loading remain fixed, then the corresponding loading history is considered as proportional.

Studies on the non-proportional hardening can be categorized into three groups consisting of experimental studies, constitutive modeling, and empirical methods. Although this work is not intended to provide a comprehensive review of such studies, several studies are reviewed to complement this investigation.

A number of studies have investigated the effect of strain path shape, load amplitude, and load sequence on additional non-proportional hardening behavior by experimental observations. For instance, Tanaka et al., 1985a, Tanaka et al., 1985b conducted a series of plastic strain-controlled proportional and non-proportional axial-torsional tests on 316 stainless steel. They observed that the stress response of a 90° out-of-phase path is only affected by prior cycles of the larger amplitude (Tanaka et al., 1985a). On the other hand, the stress response of torsional path was observed to be independent of the prior cycles regardless of the amplitude level, indicating a non-hardening strain region, as previously proposed by Ohno (1982).

Tanaka et al. also reported that the deformation memory effect of the prior larger amplitude cycles was erased in presence of sufficient accumulated plastic strain for non-proportional loading. However, they observed the memory effect not to vanish for non-proportional loading, if the prior cycles have very large amplitudes, which is similar to the observations of Chaboche et al. (1979).

Tanaka et al. (1985b) also conducted tests with several combinations of proportional and non-proportional strain paths on 316 stainless steel. Results indicate that proportional paths with a fixed principal direction resulted in the least cyclic strain hardening. Strain paths with intersecting proportional segments such as cruciform and stellate paths resulted in more strain hardening due to the interaction of dislocation structures in different directions. They suggest that the intersection of the stable dislocation structures explains the significant discontinuous cyclic hardening observed in cross hardening, such as when a proportional loading suddenly changes direction.

Taleb and Hauet (2009) explained significant cross hardening observed in 304L stainless steel under alternating axial and shear cycles by generation of high defects density caused by multiple slip systems, intersecting stacking faults and twins, formation of dislocation heterogeneous structures, and nucleation of martensite structure. The concept of cross hardening was also discussed by Krempl and Lu (1984).

Tanaka et al. (1985b), however, observed the highest level of cyclic hardening for square and circular strain paths. They explained this significant cyclic strain hardening by the requirement that the plastic strain rate vector and the corresponding plastic strain vector are not in the same direction thus activating additional slip systems within the material. They reported that the saturated stress level is independent of previous less significant hardening cycles due to the tendency of materials to rearrange the dislocation structures under the larger strain hardening cycle. Similar behaviors were also observed by Ohashi et al. (1985) for the same material.

Another group of non-proportional hardening cyclic plasticity studies focus on developing constitutive models, which relate stress to strain or plastic strain, by means of continuum mechanics. These models commonly involve either the Mroz, 1967, Mroz, 1969 multiple surface, or the Armstrong and Frederick (1966) plasticity formulations.

Mroz multiple surface plasticity model consists of several surfaces, similar to the initial yield surface, in deviatoric stress space. The translation of the yield surface in Mroz model is defined in the direction of the vector connecting the current stress point with the stress point on the next surface in a way that both stress points have the same outward normal. Armstrong and Frederick considered the movement of the yield surface in deviatoric stress space by a nonlinear kinematic hardening rule, taking into account the strain memory effect by a recovery term.

McDowell (1985) proposed a non-proportionality parameter based on the time derivative of the principal strain on a two-surface Mroz type model, which uses the kinematic hardening rules of Mroz and Prager for the yield and limit surfaces, respectively. Benallal and Marquis (1987) considered the non-proportionality parameter to be the angle between the stress and the plastic strain rate.

Fan and Peng (1991) employed two parameters based on the Benallal and Marquis’ non-proportionality factor to distinguish lateral hardening due to sudden change in loading direction and the hardening associated with the non-proportional loading. They also addressed the different load history effects for wavy slip and planar slip materials by introducing deformation memory parameters.

Doong and Socie (1991) proposed a constitutive model to address the anisotropic deformation behavior of metals after plastic deformation, based on a two-surface form of Mroz plasticity model. They considered the non-proportionality parameter to be a function of the plastic strain history.

Chaboche and Rousselier (1983) modified the Armstrong–Frederick model by decomposing the total backstress parameter into several parts, each of which independently satisfies the Armstrong–Frederick relation. Due to considerable contributions to the Armstrong–Frederick model by Chaboche (Chaboche et al., 1979, Chaboche, 1987, Chaboche, 1991), this incremental plasticity model is sometimes called Armstrong–Frederick–Chaboche model (Ohno, 2008).

Tanaka (1994) incorporated a non-proportionality parameter based on the state of internal dislocation structures into Chaboche’s viscoplastic model to explain the experimental observations from several studies (Tanaka et al., 1985a, Tanaka et al., 1985b, Benallal et al., 1989, Nishino et al., 1986), investigating load path shape and amplitude effects on non-proportional hardening.

Tanaka considered the non-proportionality parameter to be a function of the normalized plastic strain rate vector and the internal microstructure of material, which is represented by a fourth rank tensor in a 5D plastic strain vector space (Tanaka, 1984). Based on this model, he reported consistent results between experimental observations and predictions for a wide variety of load paths involving 316 stainless steel (Tanaka, 1994).

Jiang and Kurath, 1996a, Jiang and Kurath, 1996b provided a comprehensive evaluation of the Armstrong–Frederick and Mroz multiple surface plasticity models and reported the Armstrong–Frederick model to be suitable for modeling material behaviors such as cyclic softening and hardening as well as cyclic non-proportional hardening. Plasticity theories including modified Armstrong–Frederick rules as well as two surface and multiple surface models have also been comprehensively reviewed by Chaboche (2008), although the emphasizes of the review is on capabilities of kinematic hardening models in prediction of ratcheting effects.

Effect of non-proportional loading on ratcheting response and its simulation has also been a topic of other recent publications, such as those by Kang et al., 2008, Hassan et al., 2008, Abdel-Karim, 2009, Krishna et al., 2009. Nevertheless, cyclic plasticity models often require a large number of material constants, making them difficult to use in many industrial applications.

Estimation of non-proportional hardening based on the amplitude strain path using a phenomenological approach is the emphasis of another group of studies. Most of these methods have been proposed to be used for fatigue life predictions under non-proportional loading conditions where only the stabilized response is needed.

One such method to evaluate the factor of non-proportionality, F, was proposed by Kanazawa et al. (1979). This method is based on the interaction of slip bands on different planes due to the change in the maximum shear plane within the material. It is defined as the fraction of shear strain at 45° to maximum shear strain range. The factor of non-proportionality, F, in this model is related to a factor of ellipticity of the strain path in the γ/2–ϵ plot, which defines the ratio of the minor to major axes of the circumstantial ellipse.

Itoh et al., 1995, Kida et al., 1997 proposed a formulation for the factor of non-proportionality that is computed directly from the strain path. This allows evaluating the maximum stress directly from the strain. They considered the non-proportionality parameter to be related to the angle change from the principal strain direction and strain path length after changing direction on a γ/√3–ϵ plot. The strain path length is defined as a straight line connecting two points with the utmost distance on the strain path. Later Itoh et al. (2004) incorporated this factor of non-proportionality into an incremental multiple surface plasticity model, requiring only six material properties, and reported satisfactory predictions for 304 stainless steel and 6061 aluminum under various loading paths.

The multiple surface plasticity model, strata, had been proposed by Obataya and Kato, 1998, Obataya et al., 2001 to consider the activation state of slip system within polycrystalline metals under control of shear loading. The activation state of a slip system and the direction of the incremental plastic shear strain in this model are defined by a second order tensor.

The current investigation utilizes several novel discriminating axial-torsional strain paths designed to investigate non-proportional cyclic deformation behavior and modeling of materials with different degrees of non-proportional cyclic hardening sensitivity. Incremental and random sequence proportional loading, composed of similar star shaped loading blocks, were applied to tubular specimens of quenched and tempered (QT) 1050 steel with no sensitivity to non-proportional cyclic hardening and 304L stainless steel with significant non-proportional cyclic hardening.

To serve as reference for comparisons, constant amplitude in-phase and 90° out-of-phase strain path tests with a strain ratio of $\lambda =\gamma /\epsilon =\sqrt{3}$ were also conducted. The non-proportionality parameter proposed by Tanaka, and the empirical non-proportionality method proposed by Kanazawa et al. as a representative method of this class of approaches are used for prediction of the material response. Predicted results are then compared with the experimental observations. Although ratcheting is often an important consideration in evaluation of deformation behavior, it was not a topic of investigation in this study, as the cyclic loadings used were designed to enable study of cyclic load non-proportionality effects while avoiding ratcheting.

In this paper, first the experimental program including the material used and multiaxial strain paths applied are reviewed. Then, experimental results and discussions on the experimental observations are presented. This is followed by comparing the cyclic deformation behaviors observed from various strain paths experiments used in this study, with predictions using the Kanazawa et al.’s empirical method as well as the non-proportionality parameter proposed by Tanaka. Finally, conclusions based on observed experimental behaviors and prediction results are made.