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.     

玻璃微珠增强超高分子量聚乙烯的耐磨耐热性能研究

采用热压成型工艺制备了偶联玻璃微珠填充的超高分子量聚乙烯(UHMWPE)复合材料,通过万能材料试验机、洛氏硬度计考察了玻璃微珠含量对复合材料力学性能、硬度的影响,对复合材料进行了X射线衍射分析(XRD)、差示扫描量热分析(DSC)和静态热机械分析(TMA),通过热变形、维卡温度测定仪考察了玻璃微珠含量对复合材料维卡软化温度的影响,通过高速环块磨损试验机和自制砂浆磨损水浴试验装置考察了玻璃微珠含量对复合材料耐摩擦磨损性能的影响,通过扫描电子显微镜观察复合材料磨损表面形貌并分析了其磨损机理.结果表明:偶联玻璃微珠在UHMWPE体系中起填充增强作用,能有效提高复合材料的硬度、维卡软化温度、熔点和玻璃化转变温度,从而降低材料的摩擦系数与磨耗.维卡软化温度最多能提高12.5%,而磨损质量最多能降低62%,玻璃微珠质量分数在5%~20%之间效果明显,材料的力学性能随玻璃微珠的含量增加而有所下降.     

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].     

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.     

On the structure of the Hugoniot relation for a shock-induced martensitic phase transition

The Hugoniot curve relates the pressure and volume behind a shock wave, with the temperature having been eliminated. This paper studies the Hugoniot curve behind a propagating sharp interface between two material phases for a solid in which an impact-induced phase transition has taken place. For a solid capable of existing in only one phase, compressive impact produces a shock wave moving into material, say, at rest in an unstressed state at the ambient temperature. If the specimen can exist in either of two material phases, sufficiently severe impact may produce a disturbance with a two-wave structure: a shock wave in the low-pressure phase of the material, followed by a phase boundary separating the low- and high-pressure phases. We use a theory of phase transitions in thermoelastic materials to construct the Hugoniot curve behind the phase boundary in this two-wave circumstance. The kinetic relation controlling the evolution of the phase transition is an essential ingredient in this process.      

Modelling effects of degree of crystallinity on mechanical behavior of semicrystalline polymers

Viscoplasticity theory based on overstress (VBO) which is one of the unified state variable theories is extended to account for crystallinity ratio () on mechanical behavior of semicrystalline polymers. The modifications on VBO are done considering the semicrystalline polymeric materials somewhat as a composite material since it consists of amorphous and crystalline phases. Amorphous and crystalline phase resistances are arranged in two different analog models: amorphous stiffness and flow are in parallel and series with crystalline phase. Apart from many existing work in the literature, not only uniaxial loading are modeled but also creep and relaxation behaviors are simulated for a hypothetical material. It is shown that when amorphous and crystalline phase resistances acting in parallel are considered in the model, creep, relaxation and uniaxial loading and unloading behaviors can be simulated well using the modified VBO. In addition, uniaxial compression loading and unloading behavior of highly crosslinked ultra-high molecular weight polyethylene (UHMWPE) and creep behavior of polytetrafluoroethylene (PTFE) with different crystallinity ratios are simulated using the proposed VBO model where amorphous and crystalline phases are parallel. Simulation results are compared to the experimental data by Kurtz et al. (2002) and Sun et al. (2005) [Kurtz, S.M., Villarragaa, M.L., Herra, M.P., Bergström, J.S., Rimnacc, C.M., Edidin, A.A., 2002. Thermomechanical behavior of virgin and highly crosslinked ultra-high molecular weight polyethylene used in total joint replacements. Biomaterials 23, 3681–3697; Sun, H., Cooke, R. S., Bates, W. D., Wynne, K.J., 2005. Supercritical CO₂ processing and annealing of polytetrafluoroethylene (PTFE) and modified PTFE for enhancement of crystallinity and creep resistance. Polymer 46, 8872–8882] respectively and good match with experimental data is obtained.     

Heat capacities and volumetric changes in the glass transition range: a constitutive approach based on the standard linear solid

A novel approach to represent the glass transition is proposed. It is based on a physically motivated extension of the linear viscoelastic Poynting–Thomson model. In addition to a temperature-dependent damping element and two linear springs, two thermal strain elements are introduced. In order to take the process dependence of the specific heat into account and to model its characteristic behaviour below and above the glass transition, the Helmholtz free energy contains an additional contribution which depends on the temperature history and on the current temperature. The model describes the process-dependent volumetric and caloric behaviour of glass-forming materials, and defines a functional relationship between pressure, volumetric strain, and temperature. If a model for the isochoric part of the material behaviour is already available, for example a model of finite viscoelasticity, the caloric and volumetric behaviour can be represented with the current approach. The proposed model allows computing the isobaric and isochoric heat capacities in closed form. The difference \(c_\mathrm{p} -c_\mathrm{v} \) is process-dependent and tends towards the classical expression in the glassy and equilibrium ranges. Simulations and theoretical studies demonstrate the physical significance of the model.     

On the evolution law for the frozen fraction in linear theories of shape memory polymers

Shape memory properties of thermally responsive polymeric materials are due mainly to a phase transition from the rubbery phase above the transition temperature (glass transition or melting temperature) to the glassy or semicrystalline phase below this temperature. Within constitutive models of shape memory polymers (SMPs), this phase transition is mathematically accounted for by the frozen volume fraction for which a suitable evolution law must be postulated or derived. In this paper, the evolution laws that have been proposed in the literature are examined both from the experimental and from the theoretical point of view. It is found that the predictive capabilities of the phenomenological laws may be improved by admitting involved material constants to depend on parameters such as pre-strain, rate of heating and cooling, and other quantities characterizing thermomechanical cyclic tests. It is next shown that for a wide class of linear constitutive models of SMPs, the evolution law for the frozen volume fraction may be derived in a systematic way from strain and stress profiles experimentally obtained in the standard thermomechanical test.     

Thermal contraction and volume relaxation of amorphous polymers

Two experimental techniques are described for the determination of the change of specific volume of polymers with temperature and aging time, which allow measurements between – 160 °C and + 200 °C. Four technical amorphous polymers, PS, PVC, PMMA and PC have been investigated. Volume-temperature curves under constant rate of cooling are presented and interpreted with respect to relaxation processes known from other physical investigations. The rate dependence of dilatometric glass transition temperatures is compared with the time dependence of rheometric glass transition temperatures from shear creep data. Volume relaxation data at constant aging temperature are presented. Aging is found to proceed until very low temperatures in the glassy state for e.g. PMMA.For polystyrene, a comparison is made between the predictions of a very simple theory of volume relaxation due to Kovacs with experimental data, using additional information from volume temperature curves and the time temperature shift of the shear creep transition. The theory predicts a rate of volume relaxation which is much lower than that found by experiment.     

水蒸汽驱动高聚物形状记忆效应的力学模拟

部分形状记忆高聚物在相对湿度较高的环境中会从其临时形状恢复到永久形状,这种效应被称之为水蒸汽驱动形状记忆效应。由于不需要升高温度就可实现形状恢复,水蒸汽驱动的形状记忆效应在多个领域都有着潜在的应用价值。本文拟建立一个热-力-化学多场耦合的理论模型来模拟非晶态高聚物的水蒸汽驱动形状记忆行为。该理论模型采用自由体积的概念来模拟玻璃态转变,采用Fick定律来模拟水蒸汽在高聚物基体中的扩散行为。相关有限元模拟结果表明,该模型能定性地描述文献中观察到的恢复温度、相对湿度以及溶剂分子扩散速度对形状恢复行为的影响,也能模拟复杂变形条件下水蒸汽驱动的形状记忆效应。     

水蒸汽驱动高聚物形状记忆效应的力学模拟

部分形状记忆高聚物在相对湿度较高的环境中会从其临时形状恢复到永久形状,这种效应被称之为水蒸汽驱动形状记忆效应.由于不需要升高温度就可实现形状恢复,水蒸汽驱动的形状记忆效应在多个领域都有着潜在的应用价值.本文拟建立一个热-力-化学多场耦合的理论模型来模拟非晶态高聚物的水蒸汽驱动形状记忆行为.该理论模型采用自由体积的概念来模拟玻璃态转变,采用Fick定律来模拟水蒸汽在高聚物基体中的扩散行为.相关有限元模拟结果表明,该模型能定性地描述文献中观察到的恢复温度、相对湿度以及溶剂分子扩散速度对形状恢复行为的影响,也能模拟复杂变形条件下水蒸汽驱动的形状记忆效应.     

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     

Ein Modell zur Beschreibung der Stoffübertragung zwischen fluiden Phasen auf der Grundlage molekularer Vorgänge

The known phenomena of the mass transfer are the result of the molecular motions. A system of equations is represented, what describes mass- and heat transfer on this base and under consideration of the influence of molecular interactions. The change of concentration and temperature in the phase of the balance room is the result of moleculare streams over all limits of this room. It is considered a molecular mass transfer resistance on the interfacial area. In this way the mass transfer and the equilibrium of the phases is represented as an unitary process. Possibilities and limits of this model were discussed.     

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.     

Evaluation of the extent of the gel particle grafting of impact polystyrene

High Impact Polystyrene (HIPS) consists of a glassy polystyrene matrix and a rubber-like particle phase (gel phase). The extent of grafting of the gel phase is known to be an important parameter in the fracture toughness of the material. [1]. A simple quantitative model is developed in this paper to determine the extent of gel-particle grafting from the observed shifts in the glass transition temperature of the gel phase of three commercial types of HIPS.Although the increase in interfacial [2] and gel-particle grafting accounts for an increase in the energy absorbed before fracture at low strain rates, above a certain amount of grafting the material becomes embrittled at high strain rates. The adhesion factor A of mesophase models [3, 19], considered between the main phases of the material, was found to correlate with the observed impact behaviour.     

Rheology of critical LCST polymer blends: poly(styrene-co-maleic anhydrite)/poly(methyl methacrylate)

The shear rheology of a binary polymer blend exhibiting a lower critical solution temperature (LCST) phase diagram and a small dynamic asymmetry (difference of glass transition temperatures between its constituents) has been investigated in the vicinity of phase separation; it is a mixture of a random copolymer of styrene and maleic anhydrite and poly(methyl methacrylate). In the linear viscoelastic regime, the material functions are sensitive to phase separation, and the effects of critical concentration fluctuations, which dominate the mechanical response, are quantified, yielding both the binodal and spinodal curves. The weak dynamic asymmetry is apparently responsible for the reduced magnitude of the observed effects, compared to blends exhibiting much larger contrast in glass transition; therefore, this property affects to some degree the accuracy of the rheologically determined phase diagram. The steady shear properties are weakly sensitive to phase separation, and suggest that shear-induced demixing may be possible. They also indicate the importance of the amount of strain energy introduced to the blend in controlling the effects of flow on phase behavior.This investigation demonstrates that the universal effects of concentration fluctuations can be detected in LCST binary polymer blends, provided that some dynamic asymmetry exists, and further they can be quantified in order to characterize the interplay between rheology and thermodynamics of these systems.Dedicated to the memory of Professor Tasos C. Papanastasiou     

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.     

高聚物玻璃化的不平衡性及流变行为

高聚物在玻璃化转变前后表现出截然不同的力学、物理及流变特性,并在玻璃态下继续松弛。本文介绍了有关经典理论的形成和发展,最后从热力学、统计力学、动理学(kinetics)3方面联合考虑提出了3维空间(PVT/PST)数学表象,这是目前处理这方面问题较完善的理论。      

Buckling of a circular plate made of a shape memory alloy due to a reverse thermoelastic martensite transformation

We solve the axisymmetric buckling problem for a circular plate made of a shape memory alloy undergoing reverse martensite transformation under the action of a compressing load, which occurs after the direct martensite transformation under the action of a generally different (extending or compressing) load. The problem was solved without any simplifying assumptions concerning the transverse dimension of the supplementary phase transition region related to buckling. The mathematical problem was reduced to a nonlinear eigenvalue problem. An algorithm for solving this problem was proposed. It was shown that the critical buckling load under the reverse transition, which is obtained by taking into account the evolution of the phase strains, can be many times lower than the same quantity obtained under the assumption that the material behavior is elastic even for the least (martensite) values of the elastic moduli. The critical buckling force decreases with increasing modulus of the load applied at the preliminary stage of direct transition and weakly depends on whether this load was extending or compressing. In shape memory alloys (SMA), mutually related processes of strain and direct (from the austenitic into the martensite phase) or reverse thermoelastic phase transitions may occur. The direct transition occurs under cooling and (or) an increase in stresses and is accompanied by a significant decrease (nearly by a factor of three in titan nickelide) of the Young modulus. If the direct transition occurs under the action of stresses with nonzero deviator, then it is accompanied by accumulation of macroscopic phase strains, whose intensity may reach 8%. Under the reverse transition, which occurs under heating and (or) unloading, the moduli increase and the accumulated strain is removed. For plates compressed in their plane, in the case of uniform temperature distribution over the thickness, one can separate trivial processes under which the strained plate remains plane and the phase ratio has a uniform distribution over the thickness. For sufficiently high compressing loads, the trivial process of uniform compression may become unstable in the sense that, for small perturbations of the plate deflection, temperature, the phase ratio, or the load, the difference between the corresponding perturbed process and the unperturbed process may be significant. The results of several experiments concerning the buckling of SMA elements are given in [1, 2], and the statement and solution of the corresponding boundary value problems can be found in [3–11]. The experimental studies [2] and several analytic solutions obtained for the Shanley column [3, 4], rods [5–7], rectangular plates under direct [8] and reverse [9] transitions showed that the processes of thermoelastic phase transitions can significantly (by several times) decrease the critical buckling loads compared with their elastic values calculated for the less rigid martensite state of the material. Moreover, buckling does not occur in the one-phase martensite state in which the elastic moduli are minimal but in the two-phase state in which the values of the volume fractions of the austenitic and martensite phase are approximately equal to each other. This fact is most astonishing for buckling, studied in the present paper, under the reverse transition in which the Young modulus increases approximately half as much from the beginning of the phase transition to the moment of buckling. In [3–9] and in the present paper, the static buckling criterion is used. Following this criterion, the critical load is defined to be the load such that a nontrivial solution of the corresponding quasistatic problem is possible under the action of this load. If, in the problems of stability of rods and SMA plates, small perturbations of the external load are added to small perturbations of the deflection (the critical force is independent of the amplitude of the latter), then the critical forces vary depending on the value of perturbations of the external load [5, 8, 9]. Thus, in the case of small perturbations of the load, the problem of stability of SMA elements becomes indeterminate. The solution of the stability problem for SMA elements also depends on whether the small perturbations of the phase ratio and the phase strain tensor are taken into account. According to this, the problem of stability of SMA elements can be solved in the framework of several statements (concepts, hypotheses) which differ in the set of quantities whose perturbations are admissible (taken into account) in the process of solving the problem. The variety of these statements applied to the problem of buckling of SMA elements under direct martensite transformation is briefly described in [4, 5]. But, in the problem of buckling under the reverse transformation, some of these statements must be changed. The main question which we should answer when solving the problem of stability of SMA elements is whether small perturbations of the phase ratio (the volume fraction of the martensite phase q) are taken into account, because an appropriate choice significantly varies the results of solving the stability problem. If, under the transition to the adjacent form of equilibrium, the phase ratio of all points of the body is assumed to remain the same, then we deal with the “fixed phase atio” concept. The opposite approach can be classified as the “supplementary phase transition” concept (which occurs under the transition to the adjacent form of equilibrium). It should be noted that, since SMA have temperature hysteresis, the phase ratio in SMA can endure only one-sided small variations. But if we deal with buckling under the inverse transformation, then the variation in the volume fraction of the martensite phase cannot be positive. The phase ratio is not an independent variable, like loads or temperature, but, due to the constitutive relations, its variations occur together with the temperature variations and, in the framework of connected models for a majority of SMA, together with variations in the actual stresses. Therefore, the presence or absence of variations in q is determined by the presence or absence of variations in the temperature, deflection, and load, as well as by the system of constitutive relations used in this particular problem. In the framework of unconnected models which do not take the influence of actual stresses on the phase ratio into account, the “fixed phase ratio” concept corresponds to the case of absence of temperature variations. The variations in the phase ratio may also be absent in connected models in the case of specially chosen values of variations in the temperature and (or) in the external load, as well as in the case of SMA of CuMn type, for which the influence of the actual stresses on the phase compound is absent or negligible. In the framework of the “fixed phase ratio” hypothesis, the stability problem for SMA elements has a solution coinciding in form with the solution of the corresponding elastic problem, with the elastic moduli replaced by the corresponding functions of the phase ratio. In the framework of the supplementary phase transition” concept, the result of solving the stability problem essentially depends on whether the small perturbations of the external loads are taken into account in the process of solving the problem. The point is that, when solving the problem in the connected setting, the supplementary phase transition region occupies, in general, not the entire cross-section of the plate but only part of it, and the location of the boundary of this region depends on the existence and the value of these small perturbations. More precisely, the existence of arbitrarily small perturbations of the actual load can result in finite changes of the configuration of the supplementary phase transition region and hence in finite change of the critical values of the load. Here we must distinguish the “fixed load” hypothesis where no perturbations of the external loads are admitted and the “variable load” hypothesis in the opposite case. The conditions that there no variations in the external loads imply additional equations for determining the boundary of the supplementary phase transition region. If the “supplementary phase transition” concept and the “fixed load” concept are used together, then the solution of the stability problem of SMA is uniquely determined in the same sense as the solution of the elastic stability problem under the static approach. In the framework of the “variable load” concept, the result of solving the stability problem for SMA ceases to be unique. But one can find the upper and lower bounds for the critical forces which correspond to the cases of total absence of the supplementary phase transition: the upper bound corresponds to the critical load coinciding with that determined in the framework of the “fixed phase ratio” concept, and the lower bound corresponds to the case where the entire cross-section of the plate experiences the supplementary phase transition. The first version does not need any additional name, and the second version can be called as the "all-round supplementary phase transition" hypothesis. In the present paper, the above concepts are illustrated by examples of solving problems about axisymmetric buckling of a circular freely supported or rigidly fixed plate experiencing reverse martensite transformation under the action of an external force uniformly distributed over the contour. We find analytic solutions in the framework of all the above-listed statements except for the case of free support in the “fixed load” concept, for which we obtain a numerical solution.