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.     

Modelling large deformation behaviour under loading–unloading of semicrystalline polymers: Application to a high density polyethylene

In this work, the large deformation behaviour under monotonic loading and unloading of a high density polyethylene (HDPE) is studied. To analyze the nonlinear time-dependent response of the material, mechanical tests were conducted at room temperature under constant true strain rates and stress relaxation conditions. A physically-based inelastic model written under finite strain formulation is proposed to describe the mechanical behaviour of HDPE. In the model, the inelastic mechanisms involve two parallel elements: a visco-hyperelastic network resistance acting in parallel with a viscoelastic–viscoplastic intermolecular resistance where the amorphous and crystalline phases are explicitly taken into consideration. The semicrystalline polymer is considered as a two-phase composite. The influence of the crystallinity on the loading and unloading behaviour is investigated. Numerical results are compared to experimental data. It is shown that the model is able to accurately reproduce the experimental observations corresponding to monotonic loading, unloading and stress relaxation behaviours at different strain levels. Finally, the model capabilities to capture cyclic loading–unloading behaviour up to large strains are discussed. To demonstrate the improved modelling capabilities, simulations are also performed using the original model of Boyce et al. [Boyce, M.C., Socrate, S., Llana, P.G., 2000. Constitutive model for the finite deformation stress–strain behavior of poly(ethylene terephthalate) above the glass transition. Polymer 41, 2183–2201] modified by Ahzi et al. [Ahzi, S., Makradi, A., Gregory, R.V., Edie, D.D., 2003. Modeling of deformation behavior and strain-induced crystallization in poly(ethylene terephthalate) above the glass transition temperature. Mechanics of Materials 35, 1139–1148].     

Ageing and rejuvenation in glassy amorphous polymers

Physical ageing of amorphous polymers well below their glass transition temperature leads to changes in almost all physical properties. Of particular interest is the increase in yield stress and post-yield strain softening that accompanies ageing of these materials. Moreover, at larger strain polymers seem to rejuvenate, i.e. aged and non-aged samples have identical stress-strain responses. Also, plastically deforming an aged sample seems to rejuvenate the polymer. In this work we use molecular dynamic simulations with a detailed force field suitable for macromolecular ensembles to simulate and understand the effects of ageing on the mechanical response of these materials. We show that within the timescales of these simulations it is possible to simulate both ageing and rejuvenation. The short range potentials play an important role in ageing and rejuvenation. A typical yield drop exhibited by glassy polymers is a manifestation of a sudden relaxation in the short range structure of an aged polymer. Moreover, the aged polymers are known to be brittle. We show that this is intimately related to its typical stress-strain response which allows it to carry arbitrarily large mean stresses ahead of a notch.     

超高分子量聚乙烯材料的应力松弛研究

采用实验方法研究超高分子量聚乙烯(UHMWPE)材料,在不同温度、应变率和初始应变的条件下进行单轴压缩应力松弛实验,得出松弛应力与时间成非线性关系,且温度越高、应变率越大、初始应变越小,则最终稳定的应力值越小的结论．采用时间分数阶粘弹性模型,结合Boltzmann叠加原理推导出UHMWPE材料在整个加载段及松弛段的应力响应函数,并与实验数据最小二乘拟合．结果表明,时间分数阶Scott-Blair模型能很好地描述UHMWPE材料的粘弹性行为．     

A thermoviscoelastic model for amorphous shape memory polymers: Incorporating structural and stress relaxation

A thermoviscoelastic constitutive model is developed for amorphous shape memory polymers (SMP) based on the hypothesis that structural and stress relaxation are the primary molecular mechanisms of the shape memory effect and its time-dependence. This work represents a new and fundamentally different approach to modeling amorphous SMPs. A principal feature of the constitutive model is the incorporation of the nonlinear Adam–Gibbs model of structural relaxation and a modified Eyring model of viscous flow into a continuum finite–deformation thermoviscoelastic framework. Comparisons with experiments show that the model can reproduce the strain–temperature response, the temperature and strain-rate dependent stress–strain response, and important features of the temperature dependence of the shape memory response. Because the model includes structural relaxation, the shape memory response also exhibits a dependence on the cooling and heating rates.     

Modeling morphology evolution and mechanical behavior during thermo-mechanical processing of semi-crystalline polymers

A new model is proposed that combines statistical mechanics and thermodynamic aspects to characterize orientation development, nucleation and growth of crystallites, and chain entanglement slippage with interdependent relationships necessary to accurately correlate and in some cases predict the morphology and mechanical behavior of semi-crystalline polymers during various thermo-mechanical processes in the rubbery state, close to the glass transition temperature. Internal state variables (ISVs) that directly represent the underlying microstructure state are used to characterize polymer morphology and the resulting properties throughout deformation. The model uses fundamental thermodynamic coefficients for polyethylene terephthalate (PET) and is correlated to experimental data at various strain rates and temperatures just above the glass transition temperature. Experimental data are used that measure the stress, amorphous orientation, and crystallinity during uniaxial deformation of PET. The model is found to correlate well to these experimental data.     

A two-phase self-consistent model for the deformation and phase transformation behavior of polymers above the glass transition temperature: application to PET

A two-phase self-consistent model for large deformation stress–strain behavior and strain-induced crystallization in polymers at temperatures above the glass transition temperature is proposed. In this model, a composite framework is utilized to deal with the presence of the two phases, crystalline and amorphous, after the onset of strain-induced crystallization. The plastic behavior of each phase is approached by a widely used viscoplastic power law. The crystallization rate is expressed following a non-isothermal phenomenological expression based on the modified Avrami equation. Our predicted results are compared to the upper and lower bound estimates and to the existing experimental results in which good agreement is found with these experiments.     

Effect of the rate of deformation on the crystallization behavior of polymers

Deformation has a significant influence on the crystallization process in a number of polymers. In this paper, the response of a recently developed model for crystallizing polymers is investigated when subject to uni-, bi-axial and constant width extensions for a range of strain rates. Both the loading and unloading behavior are examined for these deformations. The particular model studied here was developed to capture the effect of strain induced crystallization in polymers and has been applied to model crystallization in polyethylene terephthalate at temperatures just above its glass transition temperature. The model has been formulated using the notion of multiple natural configurations within a full thermodynamic framework. The connection between micro-structural changes taking place in the polymer and the form of the model are elucidated. The interplay between the relaxation processes, the rate of deformation and their combined effect on crystallization is illustrated. The results show an earlier onset of crystallization for high strain rates due to stretching of the polymer network. At low strain rates however, crystallization is not observed as the polymer network is able to relax during the deformation. A sharp upturn in the stress is observed after the onset of crystallization due to the formation of a rigid crystalline phase. The unloading curves clearly show a hysteric behavior with the amount of dissipation increasing for increasing values of strain rate. These results compare favorably with experimental observations available in literature.     

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.     

Evolution of plastic anisotropy in amorphous polymers during finite straining

The large strain deformation response of amorphous polymers results primarily from orientation of the molecular chains within the polymeric material during plastic straining. Molecular network orientation is a highly anisotropic process, thus the observed mechanical response is strongly a function of the anisotropic state of these materials. Through mechanical testing and material characterization, the nature of the evolution of molecular orientation under different conditions of state of strain is developed. The role of developing anisotropy on the mechanical response of these materials is discussed in the context of assessing the capabilities of several models to predict the state of deformation-dependent response. A three-dimensional rubber elasticity spring system that is capable of capturing the state of deformation dependence of strain hardening is used to develop a tensorial internal state variable model of the evolving anisotropic polymer response. This fully three-dimensional constitutive model is shown to be successfully predictive of the true stress vs. true strain data obtained in our isothermal uniaxial compression and plane strain compression experiments on amorphous polycarbonate (PC) and polymethylmethacrylate (PMMA) at moderate strain rates. A basis is established for providing the polymer designer with the ability to predict the flow strengths and deformation patterns of highly anisotropic materials. A companion paper by Arruda, Boyce, and Quintus-Bosz [in press] shows how the model developed herein is used to predict various anisotropic aspects of the large strain mechanical response of preoriented materials. Additional work has been done to extend the model to include the effects of strain rate and temperature in Arruda, Jayachandran, and Boyce [in press].     

A GENERALIZED PIEZOELECTRIC-THERMOELASTIC THEORY WITH STRAIN RATE AND ITS APPLICATION 1)

工程中大量材料的形变介于弹性与黏性之间, 既具有弹性固体特性, 又具有黏性流体特点, 即为黏弹性. 黏弹性使得材料出现很多力学松弛现象, 如应变松弛、滞后损耗等行为. 在研究受热载荷作用的多场耦合问题的瞬态响应时, 考虑此类问题中的热松弛和应变松弛现象, 对准确描述其瞬态响应尤为重要. 针对广义压电热弹问题的瞬态响应, 尽管已有学者建立了考虑热松弛的广义压电热弹模型, 但迄今, 尚未计入应变松弛. 本文中, 考虑到材料变形时的应变松弛, 通过引入应变率, 在Chandrasekharaiah广义压电热弹理论的基础之上, 经拓展, 建立了考虑应变率的广义压电热弹理论. 借助热力学定律, 给出了理论的建立过程并得到了相应的状态方程及控制方程. 在本构方程中, 引入了应变松弛时间与应变率的乘积项, 同时, 分别在本构方程和能量方程中引入了热松弛时间因子. 其后, 该理论被用于研究受移动热源作用的压电热弹一维问题的动态响应问题. 采用拉普拉斯变换及其数值反变换, 对问题进行了求解, 得到了不同应变松弛时间和热源移动速度下的瞬态响应, 即无量纲温度、位移、应力和电势的分布规律, 并重点考察了应变率对各物理量的影响效应, 将结果以图形形式进行了表示. 结果表明: 应变率对温度、位移、应力和电势的分布规律有显著影响.     

考虑应变率的广义压电热弹理论及其应用

工程中大量材料的形变介于弹性与黏性之间, 既具有弹性固体特性, 又具有黏性流体特点, 即为黏弹性. 黏弹性使得材料出现很多力学松弛现象, 如应变松弛、滞后损耗等行为. 在研究受热载荷作用的多场耦合问题的瞬态响应时, 考虑此类问题中的热松弛和应变松弛现象, 对准确描述其瞬态响应尤为重要. 针对广义压电热弹问题的瞬态响应, 尽管已有学者建立了考虑热松弛的广义压电热弹模型, 但迄今, 尚未计入应变松弛. 本文中, 考虑到材料变形时的应变松弛, 通过引入应变率, 在Chandrasekharaiah广义压电热弹理论的基础之上, 经拓展, 建立了考虑应变率的广义压电热弹理论. 借助热力学定律, 给出了理论的建立过程并得到了相应的状态方程及控制方程. 在本构方程中, 引入了应变松弛时间与应变率的乘积项, 同时, 分别在本构方程和能量方程中引入了热松弛时间因子. 其后, 该理论被用于研究受移动热源作用的压电热弹一维问题的动态响应问题. 采用拉普拉斯变换及其数值反变换, 对问题进行了求解, 得到了不同应变松弛时间和热源移动速度下的瞬态响应, 即无量纲温度、位移、应力和电势的分布规律, 并重点考察了应变率对各物理量的影响效应, 将结果以图形形式进行了表示. 结果表明: 应变率对温度、位移、应力和电势的分布规律有显著影响.      

锆基非晶合金的动态弛豫机制和高温流变行为

非晶合金的动态弛豫机制对于理解其塑性变形, 玻璃转变行为, 扩散机制以及晶化行为都至关重要. 非晶合金的力学性能与动态弛豫机制的本征关联是该领域当前重要科学问题之一. 本文借助于动态力学分析(DMA), 探索了Zramorphous alloy,dynamic mechanical analysis,high temperature deformation,structural relaxation,quasi-points defects,1)国家自然科学基金(51971178);陕西省自然科学基金(2019JM-344);中央高校基本科研业务费专项资金(3102019ghxm007);中央高校基本科研业务费专项资金(3102017JC01003)2020-01-062020-04-10非晶合金的动态弛豫机制对于理解其塑性变形, 玻璃转变行为, 扩散机制以及晶化行为都至关重要. 非晶合金的力学性能与动态弛豫机制的本征关联是该领域当前重要科学问题之一. 本文借助于动态力学分析(DMA), 探索了Zr$_{50}$Cu$_{40}$Al$_{10}$块体非晶合金从室温到过冷液相区宽温度范围内的动态力学行为. 通过单轴拉伸实验, 研究了玻璃转变温度附近的高温流变行为. 基于准点缺陷理论(quasi-point defects theory), 对两种力学行为的适用性以及宏观力学行为变化过程中微观结构的演化规律进行描述. 研究结果表明, 准点缺陷理论可以很好地描述非晶合金损耗模量$\alpha$弛豫的主曲线. 基于非晶合金的内耗行为, 玻璃转变温度以下原子运动的激活能$U_\beta$为0.63 eV. 与准点缺陷浓度对应的关联因子$\chi $在玻璃转变温度以下约为0.38,而在玻璃转变温度以上则线性增大. Zr$_{50}$Cu$_{40}$Al$_{10}$块体非晶合金在玻璃转变温度附近, 随温度和应变速率的不同而在拉伸实验中显示出均匀的或不均匀的流变行为. 非晶合金的高温流变行为不仅可以通过扩展指数函数和自由体积理论来描述, 还可以通过基于微剪切畴(shear micro-domains, SMDs)的准点缺陷理论来描述.     

Experimental investigation and modeling of the tension behavior of polycarbonate with temperature effects from low to high strain rates

The effects of strain rate and temperature on the tension stress–strain responses of polycarbonate are experimentally investigated over a wide range of strain rates (0.001–1700 s⁻¹) and temperatures (0–120 °C). A modified split Hopkinson tension bar is used for high-rate uniaxial tension tests. Experimental results indicate that the stress–strain responses of polycarbonate at high strain rates exhibit the nonlinear characteristics including the obvious yielding and strain softening. The tension behavior is strongly dependent on the strain rate and temperature. The values of yield stress and strain at yield present a dramatic increase at higher strain rates and decrease with the increase in temperature. Moreover, there exists a significant rate-sensitivity transition in the polycarbonate tension yield behavior. Based on the experimental investigation, a physically based three-dimensional elastoplastic constitutive model for the finite deformation of glassy polymers is used to characterize the rate-temperature dependent yield and post-yield behavior of polycarbonate when subjected to tension loading. The model results are shown close to the experimental data within the investigated strain-rate and temperature ranges.     

A finite deformation thermomechanical constitutive model for triple shape polymeric composites based on dual thermal transitions

Shape memory polymers (SMPs) have gained strong research interests recently due to their mechanical action that exploits their capability to fix temporary shapes and recover their permanent shape in response to an environmental stimulus such as heat, electricity, irradiation, moisture or magnetic field, among others. Along with interests in conventional “dual-shape” SMPs that can recover from one temporary shape to the permanent shape, multi-shape SMPs that can fix more than one temporary shapes and recover sequentially from one temporary shape to another and eventually to the permanent shape, have started to attract increasing attention. Two approaches have been used to achieve multi-shape shape memory effects (m-SMEs). The first approach uses polymers with a wide thermal transition temperature whilst the second method employs multiple thermal transition temperatures, most notably, uses two distinct thermal transition temperatures to obtain triple-shape memory effects (t-SMEs). Recently, one of the authors’ group reported a triple-shape polymeric composite (TSPC), which is composed of an amorphous SMP matrix (epoxy), providing the system the rubber-glass transition to fix one temporary shape, and an interpenetrating crystallizable fiber network (PCL) providing the system the melt-crystal transition to fix the other temporary shape. A one-dimensional (1D) material model developed by the authors revealed the underlying shape memory mechanism of shape memory behaviors due to dual thermal transitions. In this paper, a three-dimension (3D) finite deformation thermomechanical constitutive model is presented to enable the simulations of t-SME under more complicated deformation conditions. Simple experiments, such as uniaxial tensions, thermal expansions and stress relaxation tests were carried out to identify parameters used in the model. Using an implemented user material subroutine (UMAT), the constitutive model successfully reproduced different types of shape memory behaviors exhibited in experiments designed for shape memory behaviors. Stress distribution analyses were performed to analyze the stress distribution during those different shape memory behaviors. The model was also able to simulate complicated applications, such as a twisted sheet and a folded stick, to demonstrate t-SME.     

Coarse-grained simulation of molecular mechanisms of recovery in thermally activated shape-memory polymers

Thermally actuated shape-memory polymers (SMPs) are capable of being programmed into a temporary shape and then recovering their permanent reference shape upon exposure to heat, which facilitates a phase transition that allows dramatic increase in molecular mobility. Experimental, analytical, and computational studies have established empirical relations of the thermomechanical behavior of SMPs that have been instrumental in device design. However, the underlying mechanisms of the recovery behavior and dependence on polymer microstructure remain to be fully understood for copolymer systems. This presents an opportunity for bottom-up studies through molecular modeling; however, the limited time-scales of atomistic simulations prohibit the study of key performance metrics pertaining to recovery. In order to elucidate the effects of phase fraction, recovery temperature, and deformation temperature on shape recovery, here we investigate the shape-memory behavior in a copolymer model with coarse-grained potentials using a two-phase molecular model that reproduces physical crosslinking. Our simulation protocol allows observation of upwards of 90% strain recovery in some cases, at time-scales that are on the order of the timescale of the relevant relaxation mechanism (stress relaxation in the unentangled soft-phase). Partial disintegration of the glassy phase during mechanical deformation is found to contribute to irrecoverable strain. Temperature dependence of the recovery indicates nearly full elastic recovery above the trigger temperature, which is near the glass-transition temperature of the rubbery switching matrix. We find that the trigger temperature is also directly correlated with the deformation temperature, indicating that deformation temperature influences the recovery temperatures required to obtain a given amount of shape recovery, until the plateau regions overlap above the transition region. Increasing the fraction of glassy phase results in higher strain recovery at low to intermediate temperatures, a widening of the transition region, and an eventual crossover at high temperatures. Our results corroborate experimental findings on shape-memory behavior and provide new insight into factors governing deformation recovery that can be leveraged in biomaterials design. The established computational methodology can be extended in straightforward ways to investigate the effects of monomer chemistry, low-molecular-weight solvents, physical and chemical crosslinking, different phase-separation morphologies, and more complicated mechanical deformation toward predictive modeling capabilities for stimuli-responsive polymers.     

Thermomechanics of shape memory polymers: Uniaxial experiments and constitutive modeling

Shape memory polymers (SMPs) can retain a temporary shape after pre-deformation at an elevated temperature and subsequent cooling to a lower temperature. When reheated, the original shape can be recovered. Relatively little work in the literature has addressed the constitutive modeling of the unique thermomechanical coupling in SMPs. Constitutive models are critical for predicting the deformation and recovery of SMPs under a range of different constraints. In this study, the thermomechanics of shape storage and recovery of an epoxy resin is systematically investigated for small strains (within ±10%) in uniaxial tension and uniaxial compression. After initial pre-deformation at a high temperature, the strain is held constant for shape storage while the stress evolution is monitored. Three cases of heated recovery are selected: unconstrained free strain recovery, stress recovery under full constraint at the pre-deformation strain level (no low temperature unloading), and stress recovery under full constraint at a strain level fixed at a low temperature (low temperature unloading). The free strain recovery results indicate that the polymer can fully recover the original shape when reheated above its glass transition temperature (T_g). Due to the high stiffness in the glassy state (T < T_g), the evolution of the stress under strain constraint is strongly influenced by thermal expansion of the polymer. The relationship between the final recoverable stress and strain is governed by the stress–strain response of the polymer above T_g. Based on the experimental results and the molecular mechanism of shape memory, a three-dimensional small-strain internal state variable constitutive model is developed. The model quantifies the storage and release of the entropic deformation during thermomechanical processes. The fraction of the material freezing a temporary entropy state is a function of temperature, which can be determined by fitting the free strain recovery response. A free energy function for the model is formulated and thermodynamic consistency is ensured. The model can predict the stress evolution of the uniaxial experimental results. The model captures differences in the tensile and compressive recovery responses caused by thermal expansion. The model is used to explore strain and stress recovery responses under various flexible external constraints that would be encountered in applications of SMPs.     

Influence of temperature and strain rate on the mechanical behavior of three amorphous polymers: Characterization and modeling of the compressive yield stress

Uniaxial compression stress–strain tests were carried out on three commercial amorphous polymers: polycarbonate (PC), polymethylmethacrylate (PMMA), and polyamideimide (PAI). The experiments were conducted under a wide range of temperatures (−40 °C to 180 °C) and strain rates (0.0001 s⁻¹ up to 5000 s⁻¹). A modified split-Hopkinson pressure bar was used for high strain rate tests. Temperature and strain rate greatly influence the mechanical response of the three polymers. In particular, the yield stress is found to increase with decreasing temperature and with increasing strain rate. The experimental data for the compressive yield stress were modeled for a wide range of strain rates and temperatures according to a new formulation of the cooperative model based on a strain rate/temperature superposition principle. The modeling results of the cooperative model provide evidence on the secondary transition by linking the yield behavior to the energy associated to the β mechanical loss peak. The effect of hydrostatic pressure is also addressed from a modeling perspective.     

Phenomenological nonlinear modelling of glassy polymers

Compared with the numerous works into the constitutive equations for the mechanical behaviour of metals, very little attention has been devoted to those of polymers. However, a model is required to describe both the complex shape of the stress–strain curves and strain rate sensitivity of glassy polymers. In this Note, a unified viscoelastic-viscoplastic model is presented in which the nonlinear pre-yield behaviour, the strain softening and the strain hardening are described by internal variables, in analogy with the models developed for metals. In order to check the predictive capability of the model, the numerical results are compared with the experimental data (monotone, creep and relaxation tests) of a typical amorphous glassy polymer. To cite this article: F. Zaïri et al., C. R. Mecanique 333 (2005).