Ratcheting of cyclically hardening and softening materials: I. Uniaxial behavior

The phenomenon of ratcheting of materials cyclically loaded in the plastic range is studied through combined experimental and analytical efforts (ratcheting here describes the cyclic accumulation of deformation). In particular the work seeks to illustrate how cyclic hardening and softening influence ratcheting. To this end, systematic sets of experiments were performed on stainless steel 304 and carbon steel 1018 which, respectively, exhibit cyclic hardening and softening. Due to the wide variety of behavior observed, and to better illustrate the modelling challenges, the results are divided into uniaxial and multiaxial behavior, and are presented in Parts I and II, respectively. In Part I, the results from a series of uniaxial stress-controlled experiments are presented, which illustrate the parametric dependence of ratcheting in the two materials examined. Results from a set of auxiliary strain-controlled experiments required for quantifying the cyclic hardening and softening characteristics of the materials are also presented. In a preceding publication, the authors demonstrated that ratcheting in cyclically stable materials could be simulated with consistent accuracy by allowing the bounds of the two-surface model of Dafalias-Popov to translate in the direction of ratcheting at the rate of ratcheting. This modified model, coupled with previously developed schemes for simulating cyclic hardening in strain-controlled cycling, are used to simulate the experimental results developed. Strengths, weaknesses and plausible alternatives are critically presented. The results are quite promising.     

Plastic shearing of materials exhibiting strain hardening or strain softening

We consider the plastic shearing of a strain-rate dependent material exhibiting strain hardening or strain softening, subjected to steady shearing. We establish the existence of classical solutions and study the stability of uniform shearing. For materials exhibiting strain hardening or a moderate degree of strain softening we show that, as t , every solution approaches, at specific rates of convergence, uniform shearing; thus shear bands do not form.     

Nonlinear softening and hardening behavior in Helmholtz resonators for nonlinear regimes

The governing equation of a Helmholtz resonator considering nonlinear terms of restoring and damping forces is studied. The system is treated by a time multi-scales method, and the softening and hardening behaviors of such resonators are detected as functions of amplitudes. Analytically detected responses of the system are compared with obtained results via direct numerical integration of its equation, with continuation and also with experimental results.     

Near tip quasi-static crack growth behavior in strain hardening and softening material

Analyzed in this work is a semi-infinite crack that grows slowly in a steady-state. The assumed constitutive relation for the material permits strain hardening and softening as it is damaged in time. Four distinct regions divided angularly are identified for the asymptotic expressions of the quasi-static crack-tip stress field. They refer to material degraded in front of the crack; undergone elastic unloading; reloading of degraded material; and material completely by exhausted in its load carrying capacity.     

Cyclic plastic deformation and cyclic hardening/softening behavior in 304LN stainless steel

Low cycle fatigue experiments have been conducted on 304LN stainless steel in ambient air at room temperature. Uniaxial ratcheting behavior has also been studied on this material and in both engineering and true stress controlling modes. It is shown that material’s cyclic hardening/softening behavior in low cycle fatigue and in ratcheting is dependent not only on material but also on the loading condition. Improvement of ratcheting life and mean stress dependent hardening are observed in the presence of mean stress. A method based on the strain energy density (SED) is used to represent cyclic hardening/softening behavior of the material in this work. The decrease of SED with cycles is an indication that the life in low cycle fatigue and in ratcheting is improved. The SED represents the area of the hysteresis loops.     

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

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.     

Multiaxial ratcheting with advanced kinematic and directional distortional hardening rules

Ratcheting is defined as the accumulation of plastic strains during cyclic plastic loading. Modeling this behavior is extremely difficult because any small error in plastic strain during a single cycle will add to become a large error after many cycles. As is typical with metals, most constitutive models use the associative flow rule which states that the plastic strain increment is in the direction normal to the yield surface. When the associative flow rule is used, it is important to have the shape of the yield surface modeled accurately because small deviations in shape may result in large deviations in the normal to the yield surface and thus the plastic strain increment in multi-axial loading. During cyclic plastic loading these deviations will accumulate and may result in large errors to predicted strains.This paper compares the bi-axial ratcheting simulations of two classes of plasticity models. The first class of models consists of the classical von Mises model with various kinematic hardening (KH) rules. The second class of models introduce directional distortional hardening (DDH) in addition to these various kinematic hardening rules. Directional distortion describes the formation of a region of high curvature on the yield surface approximately in the direction of loading and a region of flattened curvature approximately in the opposite direction. Results indicate that the addition of directional distortional hardening improves ratcheting predictions, particularly under biaxial stress controlled loading, over kinematic hardening alone.     

A theory of inelastic behavior of materials Part II. Inelastic materials

Archive for Rational Mechanics and Analysis -     

Constitutive model for stretch-induced softening of the stress-stretch behavior of elastomeric materials

Elastomeric materials experience stretch-induced softening as evidenced by a pre-stretched material exhibiting a significantly more compliant response than that of the virgin material. In this paper, we propose a fully three-dimensional constitutive model for the observed softening of the stress-strain behavior. The model adopts the Mullins and Tobin concept of an evolution in the underlying hard and soft domain microstructure whereby the effective volume fraction of the soft domain increases with stretch. The concept of amplified strain is then utilized in a mapping of the macroscopic deformation to the deformation experienced by the soft domain. The strain energy density function of the material is then determined from the strain energy of the soft domain and thus evolves as the volume fraction of soft domain evolves with deformation. Comparisons of model results for cyclic simple extension with the experimental data of Mullins and Tobin show the efficacy of the model and suggest that an evolution in the underlying soft/hard domain microstructure of the elastomer captures the fundamental features of stretch-induced softening. Model simulations of the cyclic stress-strain behavior and corresponding evolution in structure with strain for uniaxial tension, biaxial tension and plane strain tension are also presented and demonstrate three-dimensional features of the constitutive model.     

Representation of cyclic hardening and softening properties using continuous variables

Our aim is the modeling of cyclic hardening, cyclic softening, cyclic mean stress relaxation, and additional nonproportional cyclic hardening. We do so by means of hardening functionals for back stress and yield stress without employing additional memory surfaces. Rather, we suppose all quantities to evolve simultaneously during elastic-plastic loading in a continuous manner. The basic idea is to formulate evolution equations for the hardening variables, which are of the “hardening/dynamic recovery” format with respect to a transformed arc length. The corresponding transformation is influenced by continuously evolving parameters, measuring strain amplitude and nonproportionality during the recent process history. Although the resulting, model has a very simple structure, it is capable of describing the basic phenomena under quite general loading conditions.     

ECAP对纯铜循环硬化/软化特性的改变

以纯铜棒材试样为研究对象,通过试验研究1、4、8道次等通道转角挤压(ECAP)后材料的单轴拉压循环行为,探讨ECAP后材料循环特性的变化,得到以下结论:(1)具有循环硬化特性的纯铜进行ECAP挤压后,其循环特性可转变为循环软化;(2)第一道次ECAP挤压对材料循环应力应变响应的强化作用最大,后续道次挤压对强化的效用迅速降低,4道次以后挤压的强化作用似可忽略不计;(3)因循环软化,纯铜经ECAP挤压后的循环应力应变曲线大大低于其单调拉伸应力应变曲线,与未经ECAP挤压的结果相反.论文研究表明,评估ECAP对材料的强化效果需同时考察材料单调加载和循环加载的力学性能.     

Experimental identification of hardening and softening nonlinearity in circular laminated plates

This work investigates nonlinear characteristics of a circular laminated plate. A nonparametric identification method based on the Hilbert transform is applied to identify the nonlinear system. The results demonstrate that the force–displacement curve has a soft nonlinear characteristic under small displacements and a hard nonlinear characteristic under large displacements. The force–velocity curve also has a soft nonlinear characteristic. A circular isotropic plate is treated to test the method. The force-state method is adopted to confirm the identification results. The effects of the plate diameter are examined. A combination of a cubic polynomial and a hyperbolic tangent function is proposed to fit the experiment data. The fitting results are verified by time domain simulations under random excitations. The work illustrates some novel nonlinear characteristics in transverse vibration of a circular laminated plate via a nonlinear system identification process.     

Effect of microstructure on the hardening and softening behaviors of polycrystalline shape memory alloys Part II: Numerical simulation under axisymmetrical loading

Based on the microstructure-based constitutive model established in Part I, a detailed numerical investigation on the role of each microstructure parameter in the kinematical and kinetic evolution of polycrystalline SMA under axisymmetrical tension loading is performed. Some macroscopic constitutive features of stress-induced martensite transformation are discussed. The subject supported by the Research Grant Committee (RGC) of Hong Kong SAR, the National Natural Science Foundation of China and the Provincial Natural Science Foundation of Jiangxi Province of China     

Unified simulation of hardening and softening effects for metals up to failure

Applied Mathematics and Mechanics - Toward accurately simulating both hardening and softening effects for metals up to failure, a new finite strain elastoplastic J2-flow model is proposed with the...     

Kinematic hardening of granular materials

Summary Some recently developed constitutive equations (yield function, loading criteria, flow rule) for kinematic hardening of granular materials are discussed and some relevant material dependent parameters are determined on the basis of test results.

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Kinematische Verfestigung granularer Stoffe

Übersicht Einige in jüngster Zeit entwickelte konstitutive Beziehungen (Fließbedingung, Belastungsbedingung und Fließregel) für kinematisch verfestigende granulare Stoffe werden diskutiert und einige relevante materialabhängige Parameter werden auf der Grundlage experimenteller Ergebnisse bestimmt.

     

Nonlinear softening and hardening nonlocal bending stiffness of an initially curved monolayer graphene

In the present article, the governing nonlinear nonlocal elastic equations are obtained for a monolayer graphene with an initial curvature and the related softening and hardening bending stiffness is analytically calculated. The effects of large deformation, initial curvature, discreteness and direction of chiral vector on the bending stiffness of the monolayer graphene are discussed in detail. A behavior more complex than previously reported in the literature emerges. It is found that the bending stiffness of graphene strongly depends on the initial configuration, showing not obvious maxima and minima, and suggesting the possibility of a smart tuning.     

Modelling of stress softening in elastomeric materials: foundations of simple theories

This work is concerned with formulation of constitutive relations for materials exhibiting the stress softening phenomenon (known as the Mullins effect) typical observed in elastomeric and other amorphous materials during loading–reloading cycles. It is assumed that microstructural changes in such materials during the deformation process can be represented by a single scalar-valued softening variable whose evolution is accompanied by microforces satisfying their own law of balance, besides the classical laws of mechanics underlying macroscopic deformation of a material. The constitutive equations are then derived in consistency with thermodynamics of irreversible processes with the restriction to purely mechanical theory. The general form of the derived constitutive equations is subsequently simplified through introduction of additional assumptions leading to various models of the stress softening phenomenon. As an illustration of the general theory, it is shown that the so-called pseudo-elastic model proposed in the literature may be derived without an ad hoc postulate of the variational principle.     

Nonlinear hardening and softening resonances in micromechanical cantilever-nanotube systems originated from nanoscale geometric nonlinearities

Micro/nanomechanical resonators often exhibit nonlinear behaviors due to their small size and their ease to realize relatively large amplitude oscillation. In this work, we design a nonlinear micromechanical cantilever system with intentionally integrated geometric nonlinearity realized through a nanotube coupling. Multiple scales analysis was applied to study the nonlinear dynamics which was compared favorably with experimental results. The geometrically positioned nanotube introduced nonlinearity efficiently into the otherwise linear micromechanical cantilever oscillator, evident from the acquired responses showing the representative hysteresis loop of a nonlinear dynamic system. It was further shown that a small change in the geometry parameters of the system produced a complete transition of the nonlinear behavior from hardening to softening resonance.