Modeling and validation of the large deformation inelastic response of amorphous polymers over a wide range of temperatures and strain rates

A robust physically consistent three-dimensional constitutive model is developed to describe the finite mechanical response of amorphous polymers over a wide range of temperatures and strain rates, including the rubbery region and for impact loading rates. This thermomechanical model is based on an elastic–viscoplastic rheological approach, wherein the effects of temperature, strain rate, and hydrostatic pressure are accounted for. Intramolecular, as well as intermolecular, interactions under large elastic–inelastic behavior are considered for the mechanisms of deformation and hardening. For a wide range of temperatures and strain rates, our simulated results for poly(methyl methacrylate) (PMMA) and polycarbonate (PC) are in good agreement with experimental observations.     

On phase transformations in shape memory alloy materials and large deformation generalized plasticity

A new version of rate-independent generalized plasticity, suitable for the derivation of general thermomechanical constitutive laws for materials undergoing phase transformations, is proposed within a finite deformation framework. More specifically, by assuming an additive decomposition of the finite strain tensor into elastic and inelastic (transformation induced) parts and by considering the fractions of the various material phases as internal variables, a multi-phase formulation of the theory is developed. The concepts presented are applied for the derivation of a three-dimensional thermomechanical model for shape memory alloy materials. The ability of the model in simulating several patterns of the extremely complex behavior of these materials, under both monotonic and cyclic loadings, is assessed by representative numerical examples.     

A viscoplastic constitutive model for nickel-base superalloy,part 2: modeling under anisothermal conditions

In Part 2 of this study, extensive deformation tests were carried out on the nickel-base polycrystalline superalloy IN738LC under isothermal and anisothermal conditions between 450 and 950 °C. Under the isothermal conditions, the material showed almost no rate/time-dependency below 700 °C, while it showed distinct rate/time-dependency above 800 °C. Regarding the cyclic deformation, slight cyclic hardening behavior was observed when the temperature was below 700 °C and the imposed strain rate was fast, whereas in the case of the temperature above 800 °C or under slower strain rate conditions, the cyclic hardening behavior was scarcely observed. Unique inelastic behavior was observed under in-phase and out-of-phase anisothermal conditions: with an increase in the number of cycles, the stress at higher temperatures became smaller and the stress at lower temperatures became larger in absolute value although the stress range was approximately constant during the cyclic loading. In other words, the mean stress continues to evolve cycle-by-cycle in the direction of the stress at lower temperatures. Based on the experimental results, it was assumed that evolution of the variable Y that had been incorporated into a kinematic hardening rule in Part 1 of this study is active under higher temperatures and is negligible under lower temperatures. The material constants used in the constitutive equations were determined with the isothermal data, and were expressed as functions of temperature empirically. The extended viscoplastic constitutive equations were applied to the anisothermal cyclic loading as well as the monotonic tension, stress relaxation, creep and cyclic loading under the isothermal conditions. It was demonstrated that the present viscoplastic constitutive model was successful in describing the inelastic behavior of the material adequately, including the anomalous inelastic behavior observed under the anisothermal conditions, owing to the consideration of the variable Y.     

A multi-axial, multimechanism based constitutive model for the comprehensive representation of the evolutionary response of SMAs under general thermomechanical loading conditions

We present a fully general, three dimensional, constitutive model for Shape Memory Alloys (SMAs), aimed at describing all of the salient features of SMA evolutionary response under complex thermomechanical loading conditions. In this, we utilize the mathematical formulation we have constructed, along with a single set of the model’s material parameters, to demonstrate the capturing of numerous responses that are experimentally observed in the available SMA literature. This includes uniaxial, multi-axial, proportional, non-proportional, monotonic, cyclic, as well as other complex thermomechanical loading conditions, in conjunction with a wide range of temperature variations. The success of the presented model is mainly attributed to the following two main factors. First, we use multiple inelastic mechanisms to organize the exchange between the energy stored and energy dissipated during the deformation history. Second, we adhere strictly to the well established mathematical and thermodynamical requirements of convexity, associativity, normality, etc. in formulating the evolution equations governing the model behavior, written in terms of the generalized internal stress/strain tensorial variables associated with the individual inelastic mechanisms. This has led to two important advantages: (a) it directly enabled us to obtain the limiting/critical transformation surfaces in the spaces of both stress and strain, as importantly required in capturing SMA behavior; (b) as a byproduct, this also led, naturally, to the exhibition of the apparent deviation from normality, when the transformation strain rate vectors are plotted together with the surfaces in the space of external/global stresses, that has been demonstrated in some recent multi-axial, non-proportional experiments.     

混凝土的弹塑性损伤双面本构模型

针对混凝土材料拉压应变空间下损伤机制的不同,结合连续损伤力学和塑性理论建立了一个全新的本构模型。该模型中损伤和塑性变形的演变由应变空间的同一个非弹性曲面来控制,但对拉压应变空间中非弹性曲面的演变分别采用了随动强化法则和各向同性演化规律。计算结果表明,该模型能较好地描述混凝土材料在单轴及多轴单调加载和低周反复荷载下的典型非线性特征。     

高温合金材料循环相关热机械疲劳寿命预测

在变温非线性运动强化规律所描述的高温合金材料热机械寿命应力－应变循环特性的基础上，讨论了应变控制的循环相关热机械疲劳寿命预测技术，所建模型采用了由应变以密度表示的损伤参数，并且引入了温度损伤系数，考虑了温度变化范围以及温度循环和应变循环相位关系对疲劳寿命的影响，在确定模型的一些参数，采用等温力学试验和疲劳试验的数据，为了把等温疲劳研究成果推广到变温疲劳分析领域，开辟了新的途径。     

Cyclic plasticity modeling with the distribution of non-linear relaxations approach

Motivated by the distribution of non-linear relaxation (DNLR) approach, a phenomenological model is proposed in order to describe the cyclic plasticity behavior of metals under proportional and non-proportional loading paths with strain-controlled conditions. Such a model is based on the generalization of the Gibbs's relationship outside the equilibrium of uniform system and the use of the fluctuation theory to analyze the material dissipation due to its internal reorganization. The non-linear cyclic stress–strain behavior of metals notably under complex loading is of particular interest in this study. Since the hardening effects are described appropriately and implicitly by the model, thus, a host of inelastic behavior of metals under uniaxial and multiaxial cyclic loading paths are successfully predicted such as, Bauschinger, strain memory effects as well as additional hardening. After calibrating the model parameters for two metallic materials, the model has demonstrated obviously its ability to describe the cyclic elastic-inelastic behavior of the nickel base alloy Waspaloy and the stainless steel 316L. The model is then implemented in a commercial finite element code simulating the cyclic stress–strain response of a thin-walled tube specimen. The numerical responses are in good agreement with experimental results.     

超弹性镍钛形状记忆合金单轴相变棘轮行为的宏观唯象本构模型

超弹性镍钛形状记忆合金因其良好的力学性能以及独特的超弹性和形状记忆效应已广泛应用于土木工程、航空航天和生物医疗等多个领域,在实际服役环境中超弹性镍钛合金元件不可避免地会承受不同应力水平的循环载荷作用,亟待建立描述相变棘轮行为(即峰值应变和谷值应变随着正相变和逆相变循环的进行不断累积)的循环本构模型.为此,基于已有的超弹性镍钛形状记忆合金在不同峰值应力下的单轴相变棘轮行为实验研究结果,在广义黏塑性框架下,对Graesser等提出的通过背应力非线性演化方程反映超弹性镍钛形状记忆合金超弹性行为的一维宏观唯像本构模型进行了拓展,考虑了正相变和逆相变过程中特征变量的差异及其随循环的演化,以非弹性应变的累积量为内变量引入了正相变开始应力、逆相变开始应力、相变应变和残余应变的演化方程,同时通过峰值应力与正相变完成应力的比值来确定演化方程中的相关系数,建立了描述超弹性镍钛合金单轴相变棘轮行为的本构模型.将模拟结果与对应的实验结果进行对比发现,建立的宏观唯像本构模型能够合理地描述超弹性镍钛形状记忆合金的单轴相变棘轮行为及其峰值应力依赖性,模型的预测结果和实验结果吻合得很好.     

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

Abspract 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.     

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.     

Effects of matrix inelasticity on the overall hysteretic behavior of TiNi-SMA fiber composites

The effects of the inelastic deformation of the matrix on the overall hysteretic behavior of a unidirectional titanium–nickel shape-memory alloy (TiNi-SMA) fiber composite and on the local pseudoelastic response of the embedded SMA fibers are studied under the isothermal loading and unloading condition. The multiaxial phase transformation of the SMA fibers is predicted using the phenomenological constitutive equations which can describe the two-step deformation due to the rhombohedral and martensitic transformations, and the inelastic behavior of the matrix material using the standard nonlinear viscoplastic model. The average behavior of the SMA composite is evaluated with the micromechanical method of cells. It is observed that the inelastic deformation of the matrix due to prior tension results in a compressive stress in the matrix after unloading of the SMA composite and this residual stress impedes the complete recovery of the pseudoelastic strain of the SMA fibers. This explains that a closed hysteresis behavior of the SMA composite is no longer observed in contrast with the case that an elastic behavior of matrix is assumed. The predicted local stress–strain behavior indicates that the cyclic response of matrix is crucial to the design of the hysteretic performance of the SMA composite under the repeated loading conditions.     

显式模拟类橡胶材料应力软化引起的不可恢复变形及其各向异性特征

类橡胶材料在经过初次加载后会产生应力软化现象, 也就是Mullins效应. 实验证明应力软化现象会导致材料产生不可恢复变形, 同时引入各向异性特征. 本文基于对数应变构造一个多轴可压缩应变能函数, 先引入耗散来表征应力软化现象, 再引入依赖耗散大小的不可恢复变形量以及各向异性特征量, 使得新模型既可以表征Mullins效应, 又能模拟应力软化作用下产生的不可恢复变形和各向异性特征. 本文在各向同性形函数的基础上, 通过球坐标系的思想, 进一步发展并提出了一个任意方向适用的各向异性形函数. 新模型在材料尚未发生软化(耗散为0)的情况下, 表现出各向同性; 一旦发生应力软化(耗散大于0), 则变为各向异性. 随着加载?卸载循环的累积, 耗散逐渐变大, 不可恢复变形也随之变大直到达到一个稳定的值, 各向异性特性也逐渐变得明显. 新方法得到的结果可以精确匹配经典的实验数据, 并预测不同方向的应力软化现象以及由此产生的不可恢复变形和各向异性特征.      

A viscoplastic constitutive model incorporating dynamic strain aging effect during cyclic deformation conditions

In this paper, a viscoplastic constitutive model previously proposed by the authors was extended to apply to the cyclic deformation analysis of the modified 9Cr-1Mo steel. A series of cyclic deformation tests were conducted on modified 9Cr-1Mo steel at various temperatures, including those under anisothermal conditions. Furthermore, cyclic evolution of state variables used in the authors' constitutive model was experimentally measured. Based on the test results, cyclic softening behavior depending on the temperature and its history was introduced into the constitutive model. The extended model was applied to simulations of inelastic deformation behavior under monotonic tension, stress relaxation, creep, isothermal cyclic deformations including stress relaxation and anisothermal cyclic deformations. It was found that the present constitutive model has a capability of describing the inelastic deformation behavior of modified 9Cr-1Mo steel adequately at various loading conditions.     

一个考虑循环应变幅值历史效应的粘塑性本构模型

本文提出了一个考虑材料应变幅值历史效应的粘塑性本构模型。在该模型中，引入了三个具有不同演化速率的背应力演化方程；建立了非弹性应变幅值历史记忆模型，对各向同性变形阻力，引入了具有先前加载历史记忆的演化方程。将本文模型用于１Ｃｒ１８Ｎｉ９Ｔｉ不锈钢循环变形行为描述中，其预言结果与实验结果吻合得很好，表明该模型能很好地描述材料的循环应变幅值历史下的循环变形行为。     

高温合金材料循环相关和时间相关热机械疲劳循环性能模拟

从平衡热力学不可逆系统出发，用非线性粘弹塑性运动强化莱模拟高温合金材料的应变控制热机械疲劳循环特性。讨论了温度变化和应变循环的相位关系，循环相关和时间相关热机械疲劳损伤机制，蠕变和疲劳间的相互作用。在建立本构关系和状态方程时，均考虑了温度变化所产生的影响。     

An effective temperature theory for the nonequilibrium behavior of amorphous polymers

Amorphous polymers lack an organized microstructure, yet they exhibit structural evolution, where physical properties change with time, temperature, and inelastic deformation. To describe the influence of structural evolution on the mechanical behavior of amorphous polymers, we developed a thermomechanical theory that introduces the effective temperature as a thermodynamic state variable representing the nonequilibrium configurational structure. The theory couples the evolution of the effective temperature and internal state variables to describe the temperature-dependent and rate-dependent inelastic response through the glass transition. We applied the theory to model the effect of temperature, strain rate, aging time, and plastic pre-deformation on the uniaxial compression response and enthalpy change with temperature of an acrylate network. The results showed excellent agreement with experiments and demonstrate the ability of the effective temperature theory to explain the complex thermomechanical behavior of amorphous polymers.