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.     

Long time response of aging glassy polymers

Aging amorphous polymeric materials undergo free volume relaxation, which causes slowing down of the relaxation dynamics as a function of time. The resulting time dependency poses difficulties in predicting their long time physical behavior. In this work, we apply an effective time domain approach to the experimental data on aging amorphous polymers and demonstrate that it enables prediction of long time behavior over the extraordinary timescales. We demonstrate that, unlike the conventional methods, the proposed effective time domain approach can account for physical aging that occurs over the duration of the experiments. Furthermore, this procedure successfully describes time–temperature and time–stress superpositions. It can also allow incorporation of varying dependences of relaxation time on aging time as well as complicated but known deformation history in the same experiments. This work strongly suggests that the effective time domain approach can act as an important tool to analyze the long time physical behavior of aging amorphous polymeric materials.     

La30Ce30Al15Co25金属玻璃应力松弛行为

作为潜在的工程材料, 金属玻璃在材料科学和凝聚态物理等领域引起广泛的研究兴趣. 金属玻璃结构与性能的关系表明, 金属玻璃的动态非均匀性与其黏弹性和塑性紧密相关. 然而, 宏观应力松弛行为与动态弛豫之间的物理图像并不清晰. 与传统金属材料不同, 金属玻璃的变形机理非常复杂. 应力松弛是一种表征玻璃体系黏弹性和塑性变形机制的有效手段, 从而探索结构和动态非均匀性. 本研究以La₃₀Ce₃₀Al₁₅Co₂₅金属玻璃为模型体系, 在较宽的温度窗口研究了其应力松弛行为. 研究结果表明, 与传统金属玻璃不同, La₃₀Ce₃₀Al₁₅Co₂₅金属玻璃具有明显的β弛豫行为. 基于Kohlarausch-Willams-Watts (KWW)方程的分析表明, 金属玻璃应力松弛为动态不均匀过程; 热动力学分析发现La₃₀Ce₃₀Al₁₅Co₂₅金属玻璃应力松弛存在显著的双阶段行为, 即从高应力条件下应力驱动为主导的松弛行为, 向低应力下热激活为主导的松弛行为发生转变. 通过激活能谱模型分析表明, 应力松弛单元的激活并非均匀, 而是存在能量上的起伏, 金属玻璃对于外力响应是一个渐进过程, 具有动力学不均匀性. 本研究进一步构建了金属玻璃的结构和动态非均匀性之间的关联, 为研究金属玻璃的α弛豫和β弛豫提供了强有力的支撑.      

Comparison between the relaxation spectra obtained from homogenization models and finite elements simulation for the same composite

The objective of this work is to determine the relaxation spectrum of spherical particles reinforced viscoelastic and isotropic composites from 3D Finite Elements (FE) simulations of the microstructure. The matrix and the reinforcements are assumed to be incompressible and Maxwellian. The spectra obtained from the FE simulations are compared with those obtained from analytical homogenization models. This paper presents the procedure used for generating the FE models as well as the procedure used for obtaining relaxation spectra meeting the thermodynamics requirements imposed on linear viscoelastic materials. It seems that the relaxation spectrum for the microstructure studied in this paper is composed of a negligible continuous part and a discrete part of higher intensity. In any case, the resulting material does not have a Maxwellian behavior.     

Dynamic stiffness of chemically and physically ageing rubber vibration isolators in the audible frequency range

The constitutive equations of chemically and physically ageing rubber in the audible frequency range are modelled as a function of ageing temperature, ageing time, actual temperature, time and frequency. The constitutive equations are derived by assuming nearly incompressible material with elastic spherical response and viscoelastic deviatoric response, using Mittag-Leffler relaxation function of fractional derivative type, the main advantage being the minimum material parameters needed to successfully fit experimental data over a broad frequency range. The material is furthermore assumed essentially entropic and thermo-mechanically simple while using a modified William–Landel–Ferry shift function to take into account temperature dependence and physical ageing, with fractional free volume evolution modelled by a nonlinear, fractional differential equation with relaxation time identical to that of the stress response and related to the fractional free volume by Doolittle equation. Physical ageing is a reversible ageing process, including trapping and freeing of polymer chain ends, polymer chain reorganizations and free volume changes. In contrast, chemical ageing is an irreversible process, mainly attributed to oxygen reaction with polymer network either damaging the network by scission or reformation of new polymer links. The chemical ageing is modelled by inner variables that are determined by inner fractional evolution equations. Finally, the model parameters are fitted to measurements results of natural rubber over a broad audible frequency range, and various parameter studies are performed including comparison with results obtained by ordinary, non-fractional ageing evolution differential equations.     

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:...     

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.     

An empirical constitutive model for complex glass-forming liquids using bitumen as a model material

While extensive research efforts have been devoted to understand the dynamics of chemically and structurally simple glass-forming liquids (SGFLs), the viscoelasticity of chemically and structurally complex glass-forming liquids (CGFLs) has received only little attention. This study explores the rheological properties of CGFLs in the vicinity of the glass transition. Bitumen is selected as the model material for CGFLs due to its extremely complex chemical composition and microstructure, fast physical aging and thermorheological simplicity, and abundant availability. A comprehensive rheological analysis reveals a significant broadening of the glass transition dynamics in bitumen as compared to SGFLs. In particular, the relaxation time spectrum of bitumen is characterized by a broad distribution of long relaxation modes. This observation leads to the development of a new constitutive equation, named the broadened power-law spectrum model. In this model, the wide distribution of long relaxation times is described by a power-law with positive exponent and a stretched exponential cut-off, with parameter β serving as a measure of the broadness of the distribution. This characteristic shape of the bitumen spectrum is attributed to the heterogeneous freezing of different molecular components of bitumen, i.e., to the coexistence of liquid and glassy micro-phases. Furthermore, as this type of heterogeneous glass transition behavior can be considered as a general feature of complex glass-forming systems, the broadened power-law spectrum model is expected to be valid for all types of CGFLs. Examples of the applicability of this model in various complex glass-forming systems are given.     

Second-order two-scale computational method for ageing linear viscoelastic problem in composite materials with periodic structure

The correspondence principle is an important mathematical technique to compute the non-ageing linear viscoelastic problem as it allows to take advantage of the computational methods originally developed for the elastic case. However, the correspondence principle becomes invalid when the materials exhibit ageing. To deal with this problem, a second-order two-scale (SOTS) computational method in the time domain is presented to predict the ageing linear viscoelastic performance of composite materials with a periodic structure. First, in the time domain, the SOTS formulation for calculating the effective relaxation modulus and displacement approximate solutions of the ageing viscoelastic problem is formally derived. Error estimates of the displacement approximate solutions for SOTS method are then given. Numerical results obtained by the SOTS method are shown and compared with those by the finite element method in a very fine mesh. Both the analytical and numerical results show that the SOTS computational method is feasible and efficient to predict the ageing linear viscoelastic performance of composite materials with a periodic structure.     

DIC Strain Measurements at the Micro-Scale in a Semi-Crystalline Polymer

To improve physical motivation of mechanical modeling for semi-crystalline polymers, a better understanding of micro-mechanisms and a quantification of the mechanical response at the microstructure scale are needed. Strain field evolutions during deformation would be useful information but experimental techniques are still lacking for these materials. In this paper, an in-situ Digital Image Correlation (DIC) technique in the Scanning Electron Microscope (SEM) was developed to access strain fields at the spherulitic scale of a polypropylene. The evolution of strain heterogeneities during deformation were analyzed and correlated to the microstructure.     

Thermodynamics of relaxation processes in the glass transition region

Relaxation processes in the glass transition region, especially the recovery of the volume and the physical ageing of polymers, do not follow the common (linear) theory of relaxation. On the contrary, they show a development which depends on the previous history, may be non-monotonous and requires a relaxation time that may have negative values and a pole. These phenomena can be explained if the single relaxation time is replaced by a spectrum of relaxation times and the relaxation times are supposed to be subjected to a feedback via certain structure- and temperature-parameters (as, for instance, in the KAHR-theory).However, the feedback and a pole of the relaxation time arise already for a single internal degree of freedom by themselves, if, in the non-equilibrium thermodynamics, a dynamic and a static temperature are strictly differentiated. In the case of the relaxation of the diffusive translational motion of the molecules in the glass transition region the dynamic temperature is identical with the socalled fictive temperature introduced by Tool.With regard to the relaxation of the volume three different temperature regions must be distinguished: A fluid region at high temperatures where the relaxation is controlled by the free volume and complies with the linear theory at least approximately; a glass-like region at low temperatures where the relaxation is controlled by the thermal expansivity of the free volume and where, under certain conditions, the statements set up by Davies and Jones are valid; an intermediate region (the glass transition region) where the free volume as well as its coefficient of expansivity are decisive. In that region the effective relaxation time of the volume may have a pole and the dynamic temperature may approach its equilibrium value by discontinuous jumps or in a chaotic manner.Dedicated to Professor Dr. J. Meissner (ETH Zürich) on the occasion of his 60th birthday     

Fractional relaxation and the time-temperature superposition principle

Relaxation processes in complex systems like polymers or other viscoelastic materials can be described by equations containing fractional differential or integral operators. In order to give a physical motivation for fractional order equations, the fractional relaxation is discussed in the framework of statistical mechanics. We show that fractional relaxation represents a special type of a non-Markovian process. Assuming a separation condition and the validity of the thermo-rheological principle, stating that a change of the temperature only influences the time scale but not the rheological functional form, it is shown that a fractional operator equation for the underlying relaxation process results.     

Dynamic stiffness of chemically and physically ageing rubber vibration isolators in the audible frequency range: Part 2—waveguide solution

The dynamic stiffness of a chemically and physically ageing rubber vibration isolator in the audible frequency range is modelled as a function of ageing temperature, ageing time, actual temperature, time, frequency and isolator dimension. In particular, the dynamic stiffness for an axially symmetric, homogeneously aged rubber vibration isolator is derived by waveguides where the eigenmodes given by the dispersion relation for an infinite cylinder satisfying traction free radial surface boundary condition are matched to satisfy the displacement boundary conditions at the lateral surface ends of the finite rubber cylinder. The constitutive equations are derived in a companion paper (Part 1). The dynamic stiffness is calculated over the whole audible frequency range 20–20,000 Hz at several physical ageing times for a temperature history starting at thermodynamic equilibrium at \(+25\,^{\circ }\hbox {C}\) and exposed by a sudden temperature step down to \(-60\,^{\circ }\hbox {C}\) and at several chemical ageing times at temperature \(+25\,^{\circ }\hbox {C}\) with simultaneous molecular network scission and reformation. The dynamic stiffness results are displaying a strong frequency dependence at a short physical ageing time, showing stiffness magnitude peaks and troughs, and a strong physical ageing time dependence, showing a large stiffness magnitude increase with the increased physical ageing time, while the peaks and troughs are smoothed out. Likewise, stiffness magnitude peaks and troughs are frequency-shifted with increased chemical ageing time. The developed model is possible to apply for dynamic stiffness prediction of rubber vibration isolator over a broad audible frequency range under realistic environmental condition of chemical ageing, mainly attributed to oxygen exposure from outside and of physical ageing, primarily perceived at low-temperature steps.     

A model for the effect of physical ageing on the stress relaxation behavior of high density polyethylene

The stress relaxation behavior of high density polyethylene (HDPE) can be affected by ageing processes; e.g., with increasing storage time at a low temperature following a quench from a high temperature (close to the melting point) the relaxation curves change shape. More specifically, the stress level approached after very long loading times in a stress relaxation experiment increases with the ageing time. Here this stress level is denoted the internal stress _i. Struik has pointed out that physical ageing may also occur in semicrystalline polymers like HDPE. The physical ageing should then be associated with that part of the amorphous phase which is closest to the surfaces of the crystallites. This part of the amorphous phase of HDPE can be assumed to have a restricted mobility at room temperature and may have a partially glassy character. In this paper a model for explaining the increase in _i for HDPE with increasing ageing time is proposed and discussed. It is based on the separation of the amorphous phase into two parts as suggested by Struik. The glassy part of the amorphous phase ages in a way similar to that of an entirely amorphous polymer quenched to a temperature below its glass transition, while the more rubbery phase is assumed not to undergo any physical ageing.     

The evolution of laminates in finite crystal plasticity: a variational approach

The analysis and simulation of microstructures in solids has gained crucial importance, virtue of the influence of all microstructural characteristics on a material’s macroscopic, mechanical behavior. In particular, the arrangement of dislocations and other lattice defects to particular structures and patterns on the microscale as well as the resultant inhomogeneous distribution of localized strain results in a highly altered stress–strain response. Energetic models predicting the mechanical properties are commonly based on thermodynamic variational principles. Modeling the material response in finite strain crystal plasticity very often results in a non-convex variational problem so that the minimizing deformation fields are no longer continuous but exhibit small-scale fluctuations related to probability distributions of deformation gradients to be calculated via energy relaxation. This results in fine structures that can be interpreted as the observed microstructures. In this paper, we first review the underlying variational principles for inelastic materials. We then propose an analytical partial relaxation of a Neo-Hookean energy formulation, based on the assumption of a first-order laminate microstructure, thus approximating the relaxed energy by an upper bound of the rank-one-convex hull. The semi-relaxed energy can be employed to investigate elasto-plastic models with a single as well as multiple active slip systems. Based on the minimization of a Lagrange functional (consisting of the sum of energy rate and dissipation potential), we outline an incremental strategy to model the time-continuous evolution of the laminate microstructure, then present a numerical scheme by means of which the microstructure development can be computed, and show numerical results for particular examples in single- and double-slip plasticity. We discuss the influence of hardening and of slip system orientations in the present model. In contrast to many approaches before, we do not minimize a condensed energy functional. Instead, we incrementally solve the evolution equations at each time step and account for the actual microstructural changes during each time step. Results indicate a reduction in energy when compared to those theories based on a condensed energy functional.     

Volume changes during stress relaxation in polyethylene

This paper presents the results of an experimental study of the stress relaxation behaviour of PE where the focus was on determination of the volume changes taking place during the relaxation process. The dimensions of the samples were followed using a specially designed non-disturbing extensometer. The extensometer data were confirmed in experiments where the volume was measured with a specially designed liquid stress dilatometer. The bulk of the results was obtained with LDPE and LLDPE. High density polyethylene was shown to behave similarly.The decrease in volume, corresponding to an increasing Poisson's ratio, during relaxation was approximately linear with log time. Volume vs. stress diagrams were linear; the values of the apparent bulk modulus calculated from them were only slightly higher than those obtained from the stress-strain curves. Grüneisen parameter was measured and compared with reported values. Possible similarities between the volume change during stress relaxation and that occuring during the process of physical ageing are discussed.     

Thermodynamic and rate variational formulation of models for inhomogeneous gradient materials with microstructure and application to phase field modeling

In this work,thermodynamic models for the energetics and kinetics of inhomogeneous gradient materials with microstructure are formulated in the context of continuum thermodynamics and material theory.For simplicity,attention is restricted to isothermal conditions.The materials of interest here are characterized by(1) first- and secondorder gradients of the deformation field and(2) a kinematic microstructure field and its gradient(e.g.,in the sense of director,micromorphic or Cosserat microstructure).Material inhomogeneity takes the form of multiple phases and chemical constituents,modeled here with the help of corresponding phase fields.Invariance requirements together with the dissipation principle result in the reduced model field and constitutive relations.Special cases of these include the wellknown Cahn-Hilliard and Ginzburg-Landau relations.In the last part of the work,initial boundary value problems for this class of materials are formulated with the help of rate variational methods.     

On the representation of chemical ageing of rubber in continuum mechanics

In order to represent the chemical ageing behaviour of rubber under finite deformations a three-dimensional theory is proposed. The fundamentals of this approach are different decompositions of the deformation gradient in combination with an additive split of the Helmholtz free energy into three parts. Its first part belongs to the volumetric material behaviour. The second part is a temperature-dependent hyperelasticity model which depends on an additional internal variable to consider the long-term degradation of the primary rubber network. The third contribution is a functional of the deformation history and a further internal variable; it describes the creation of a new network which remains free of stress when the deformation is constant in time. The constitutive relations for the stress tensor and the internal variables are deduced using the Clausius–Duhem inequality. In order to sketch the main properties of the model, expressions in closed form are derived with respect to continuous and intermittent relaxation tests as well as for the compression set test. Under the assumption of near incompressible material behaviour, the theory can also represent ageing-induced changes in volume and their effect on the stress relaxation. The simulations are in accordance with experimental data from literature.