Photoviscoelastic behavior of amorphous polymers during transition from the glassy to rubbery state

Tensile stress‐relaxation experiments with simultaneous measurements of Young's relaxation modulus, E, and the strain‐optical coefficient, C_?, were performed on two amorphous polymers—polystyrene (PS) and polycarbonate (PC)—over a wide range of temperatures and times. Master curves of these material functions were obtained via the time‐temperature superposition principle. The value of C_? of PS is positive in the glassy state at low temperature and time; then it relaxes and becomes negative and passes through a minimum in the transition zone from the glassy to rubbery state at an intermediate temperature and time and then monotonically increases with time, approaching zero at a large time. The stress‐optical coefficient of PS is calculated from the value of C_?. It is positive at low temperature and time, decreases, passes through zero, becomes negative with increasing temperature and time in the transition zone from the glassy to rubbery state, and finally reaches a constant large negative value in the rubbery state. In contrast, the value of C_? of PC is always positive being a constant in the glassy state and continuously relaxes to zero at high temperature and time. The value of C_σ of PC is also positive being a constant in the glassy state and increases to a constant value in the rubbery state. The obtained information on the photoelastic behavior of PS and PC is useful for calculating the residual birefringence and stresses in plastic products. © 2001 John Wiley & Sons, Inc. J Polym Sci Part B: Polym Phys 39: 2252–2262, 2001     

Relaxation modulus in the fitting of polycarbonate and poly(vinyl chloride) viscoelastic polymers by a fractional Maxwell model

 The stress relaxation of two different polymers under a constant strain has been studied and approached by using a fractional Maxwell model in which the stress appears as a noninteger-order derivative of the strain. To obtain an accurate approximation of the experimental data for the model, two noninteger values for the derivative order are required. These values are related to two relaxation types. For short times, the derivative order is smaller and near zero, which indicates behavior close to the ideal elastic solid. For long times the derivative order is higher, showing more plastic behavior. In this work some classic models are revised and the fractional Maxwell model is used to fit the experimental data. Finally, the complex fractional modulus, the two derivative orders, and the relaxation times for samples of polycarbonate and poly(vinyl chloride) are obtained. Received: 16 June 2001 Accepted: 19 October 2001     

Infrared dichroism of glassy polycarbonate at small strains

The dichroism of the 889, 1364, and 3063 cm^?1 infrared absorption bands of glassy, amorphous polycarbonate has been measured as a function of the strain in the range 0 to 2% at 23°C. The data obtained for these three bands superpose rather well over this strain range. Negligible dichroism is observed up to about 0.6% strain; above this level, the dichroic ratio increases in an approximately linear manner. Independent mechanical data, obtained under comparable conditions of time-scale and temperature, are cited which show that a transition from approximately linear to marked non-linear viscoelastic behavior occurs with glassy polycarbonate in the range 0.7 to 1.0% strain. The coincidence on the strain axis of a relatively abrupt increase in optical absorption anisotropy with a distinct change in Young's modulus is discussed in terms of a recent molecular theory of deformation of glassy polymers. It is suggested that the data are consistent with the view that the transition from linear to nonlinear viscoelastic behavior in glassy polycarbonate is marked by the onset of significant rotation around backbone bonds.     

Athermal simulation of plastic deformation in amorphous solids at constant pressure

An algorithm is introduced for the molecular simulation of constant-pressure plastic deformation in amorphous solids at zero temperature. This allows to directly study the volume changes associated with plastic deformation (dilatancy) in glassy solids. In particular, the dilatancy of polymer glasses is an important aspect of their mechanical behavior. The new method is closely related to Berendsen's barostat, which is widely used for molecular dynamics simulations at constant pressure. The new algorithm is applied to plane strain compression of a binary Lennard-Jones glass. Conditions of constant volume lead to an increase of pressure with strain, and to a concommitant increase in shear stress. At constant (zero) pressure, by contrast, the shear stress remains constant up to the largest strains investigated (ε = 1), while the system density decreases linearly with strain. The linearity of this decrease suggests that each elementary shear relaxation event brings about an increase in volume which is proportional to the amount of shear. In contrast to the stress–strain behavior, the strain-induced structural relaxation, as measured by the self-part of the intermediate structure factor, was found to be the same in both cases. This suggests that the energy barriers that must be overcome for their nucleation continually grow in the case of constant-volume deformation, but remain the same if the deformation is carried out at constant pressure. © 2004 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 42: 2057–2065, 2004     

The Nonlinear Viscoelastic Response of Glassy Polymers Subjected to Physical Aging

Constitutive equations are developed for the nonlinear viscoelastic behavior of amorphous glassy polymers in the sub‐yield region. A polymeric glass is treated as an ensemble of cooperatively rearranging regions bridged by links. Stress‐strain relations are derived and verified by comparison with experimental data in static mechanical tests on polycarbonate and poly(methyl methacrylate). We analyze the effects of the straining state (tension, compression and torsion), strain intensity, temperature and time of annealing on stress relaxation. Fair agreement is demonstrated between observations and results of numerical simulation.     

Mechanisms of anelastic deformation in solid polymers: Solidlike and liquidlike processes

Several aspects of anelastic deformation of glassy polymers that cannot be explained in terms of existing theories are considered. Resemblance in the stress-strain response for solids of various natures and structures, including semicrystalline and glassy polymers, organic and inorganic solids, and low-molecular-mass and high-molecular-mass compounds, is analyzed. It was pointed out that the phenomena of the yield peak, strain softening, strain concentration (localization) in narrow shear bands, and transient effects are characteristic of the plastic deformation of any solid. The same is true for differences in the kinetics and mechanism of deformation at low (T _def < 0.7T _g) and high deformation temperatures (T _def > 0.7T _g). The mechanism of plastic deformation is discussed in detail for glassy polymers; at microscopic and nanoscale levels, plastic deformation proceeds via two stages: initial nucleation of small-scale shear transformations and their further coalescence. This coalescence leads to the advance of the shear front in the sample and to the nucleation and displacement of classical shear bands. The heat of plastic deformation is released out at the coalescence of shear transformations. It was assumed that shear transformations are responsible for the development and evolution of the yield peak in glassy polymers, strain softening, and other phenomena. The proposed mechanism of deformation in glasses fully agrees with the results of thermodynamic measurements and other experimental data reported in the literature. Computer simulation makes it possible to visualize the scenario of nucleation and evolution of shear transformations at the atomic level.     

Constrained Rouse model of rubber viscoelasticity

In this work we use a new approach to investigate the equilibrium and linear dynamic-mechanical response of a polymer network. The classical Rouse model is extended to incorporate quenched constraints on its end-boundary conditions; a microscopic stress tensor for the network system is then derived in the affine deformation limit. To test the model we calculate the macroscopic stress in equilibrium, corresponding to the long-time limit of relaxation. Particular attention is paid to the treatment of compressibility and hydrostatic pressure in a sample with open boundaries. Although quite different in general, for small strains the model compares well with the classic equilibrium rubber-elasticity models. The dynamic shear modulus is obtained for a network relaxing after an instantaneous step strain by keeping track of relaxation of consecutive Rouse modes of constrained network strands. The results naturally cover the whole time range--from the dynamic glassy state down to the equilibrium incompressible rubber plateau.     

Strain-induced fibrillation of glassy polymers

Literature data on structural rearrangements taking place in amorphous glassy polymers upon their plastic deformation are analyzed. This deformation is shown to be primarily accompanied by polymer self-dispersion into fibrillar aggregates composed of oriented macromolecules with a diameter of 1—10 nm. The above structural rearrangements proceed independently of the deformation mode of polymers (cold drawing, crazing, or shear banding of polymers under the conditions of uniaxial drawing or uniaxial compression). Principal characteristics of the formed fibrils and the conditions providing their development are considered. Information on the properties of the fibrillated glassy polymers is presented, and the pathways of their possible practical application are highlighted.     

Inelastic deformation of glassy polyaryleneetherketone: Energy accumulation and deformation mechanism

The plastic deformation of glassy non-annealed polyaryleneetherketone (PAEK) was investigated via deformation calorimetry and thermally stimulated recovery of residual strain. Polymer samples were deformed at room temperature under uniaxial compression up to ε_def =–(40?50)% at a rate of 0.04 min^?1. It was found that PAEK behaves in the deformation process similarly to many other glassy polymers: It stores internal energy excess at loading and contains two types of different inelastic strain carriers, namely the delayed elastic (ε_de) and plastic (ε_pl) strain carriers. The maximum level of the accumulated energy in PAEK reaches ≈ 8.3 J/g, which is close to those for glassy polystyrene and polycarbonate. Nearly all the deformation energy stored in PAEK is carried by the delayed-elastic strain. The carriers of plastic strain carry no extra energy or a very small amount of it. The inelastic deformation of glassy PAEK proceeds in two stages. The carriers of ε_de are nucleated at the first stage of the deformation process, and the carriers of ε_pl are nucleated at the second stage. It was shown that, during glassy-polymer loading, the molecular level structures carrying ε_pl never appear by themselves, but appear only as a result of spontaneous reorganization of ε_de. In other words, the plastic deformation appears in PAEK owing to the two-step process. This situation is typical for all glassy polymers.     

The nature of the nonlinear anelasticity of glassy polymers

Nonlinearity in stress-strain behavior under conditions where the strain is completely recoverable is often observed in glassy amorphous polymers. In the present work a suggestion is made that the nonlinearity is often associated with a secondary (β) relaxation process. By means of a model calculation it is shown that the onset of stress saturation of the relaxed or equilibrium compliance associated with localized group motion is a reasonable explanation for the phenomenon. The effect of stress on relaxation times is predicted to have some contribution but to be less important than the effect on the relaxed compliance.     

Fundamental relationship between the nonequilibrium glassy state and yield stress of amorphous polymers

This paper reports the theoretical prediction and experimental verification of the connection between the yield stress of amorphous polymers and the physical aging phenomenon. The analysis reveals the existence of a fundamental relationship between the nonequilibrium glassy state and the thermally activated process controlling viscoelastic and plastic deformation. The results show that the volume relaxation and deformation kinetics share the same relaxation times, and that the activation energy for deformation below T_g is much smaller than previously mentioned in the literature. This indicates that the phenomenon of physical aging plays a very important role in the deformation and processing of polymers at low temperatures. The effect of quenching and annealing on the yield stress is described in terms of the mean energy of hole formation, the departure of volume from its equilibrium state, the distribution of hole energies, and lattice volume. The same set of molecular parameters obtained from the molecular kinetic theory of the glass transition and volume relaxation predicts the yield stress as a function of cooling rate, annealing time, temperature, and strain rate.     

Simulation of the plastic behavior of amorphous glassy bis-phenol-A-polycarbonate

A protocol for studying the plastic deformation of amorphous glassy polymers is presented. The protocol is based on a viable computational procedure which combines constant-stress molecular dynamics simulations and fixed-cell energy minimizations, followed by kinetic, configurational, and energy analyses. It is shown that the computational results can be accounted for within a "potential energy landscape" theoretical framework, in which the plastic transition is interpreted as a crossing between and a collapse onto each other of "ideal (thermodynamic) structures." The procedure is applied to bis-phenol-A-polycarbonate (BPA-PC), but is equally valid for a wide variety of polymeric species. Allowing for the limited size of the simulation cell, the high strain rate, and the fact that the simulation are conducted at low temperature, the values of the density, Young's modulus, yield strain, yield stress, activation energy, and activation volume are in fair agreement with the experimental data on BPA-PC. The analysis of the results shows that the plastic relaxation for this polymer has both a collective and cooperative character (as in classical percolation theories), involves a significant fraction of the simulation cell, and can be viewed as a "nanoscopic shear band."     

Recent theories of gas sorption in polymers

Theories and models are presented for gas sorption in polymers above and below the glass transition temperature. With the exception of predictive theories that do not represent the data well, the models are fit to data for the carbon dioxide/silicone rubber and carbon dioxide/polycarbonate systems for the purposes of comparison. During the past decade, a number of new models and theories have been proposed specifically for gas sorption in glassy polymers. Each new model attempts to incorporate aspects of the gas sorption process that are unique to polymers below the glass transition temperature. This review discusses these recent advances, the assumptions used in their development and their advantages and disadvantages.     

Effect of annealing temperature on interrelation between the microstructural evolution and plastic deformation in polymers

The mechanical loading induced flow of glassy polymers is triggered by the nucleation of shear transformation units, and strongly depends on the initial microstructural state of the material. Therefore, investigation of the possible relationship between the microstructural state variables and plastic deformation is required for a better understanding of the macroscopic response of this class of materials during large deformation. In this study, free volume content is considered as a state variable and thermal treatment is selected as a process through which the accelerated and forced evolution of the free volume can be imposed. For two well‐known glassy polymers, poly(methyl methacrylate) and polycarbonate, the free volume content alteration upon annealing is monitored via positron annihilation spectroscopy, and the changes of the micro‐ and macromechanical properties are also obtained by utilizing nanoindentation technique and employing the homogeneous amorphous flow theory. The correlation between the microstructural state variable, that is, free volume, and the micromechanical state variable, that is, shear activation volume, is then investigated. The results reveal opposite direction of alterations of free volume and shear activation volume with annealing temperature. Accordingly, the possibility of the existence of an interrelation between these two state variables is critically discussed. © 2017 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2017 , 55, 1286–1297     

Yielding of a notched poly(vinyl chloride) sheet

The development of plastic deformation around the crack tip of poly(vinyl chloride), a ductile glassy polymer, has been studied in relation to the Dugdale–Barenblatt model of ductile yielding. Three-dimensional observations reveal that the plastic deformation ahead of the crack tip consists of crazes, shear bands, and their intersections. The formation of the craze is due to a state of plane strain at the immediate vicinity of the crack tip and restricted to early steps of loading. The size and shape of the fully developed plastic zone can be described by the model. The influence of strain hardening beyond the yield point is discussed on the basis of comparison of the plastic zone lengths of poly(vinyl chloride) with those of polycarbonate which always shows shorter lengths than the model predicts.     

Analysis of residual stress profiles in plastic materials using the hole drilling method – Influence factors and practical aspects

The current work introduces a procedure to locally measure residual stresses in polymers. For this purpose, the hole drilling method was adapted to the prevailing boundary conditions. Particular attention is given to the analysis of the measured strain signal which is time dependent and strongly influenced by local temperature fluctuations. The different sources of heat prevailing during the measurement and their impact are outlined. By thorough analysis of the viscoelastic and thermal deformation occurring during and after drilling the material, a procedure is proposed to measure the correct strain relaxation from which residual stresses can be calculated. Using quenched polycarbonate samples with known residual stress profiles, the potential of the proposed approach was demonstrated and is discussed.     

Cooperative effects, transport and entropy in simple liquids

We systematically investigate the cooperative effects in shear stress relaxation using equilibrium molecular-dynamics simulations in periodic boundary conditions containing a variable degree of strain. We show that, even in simple liquids, shear stress relaxation is a cooperative effect associated with a correlation length that increases with isobaric decrease in temperature. If the system size is less than the correlation length, shear stress in the system is determined by the boundary strain. Transport, however, does not depend on the boundary conditions. We relate these two effects to the number and properties of the configurations accessible to the system.     

Evaluation of gas sorption parameters and prediction of sorption isotherms in glassy polymers

A model of continuous‐site distribution for gas sorption in glassy polymers is examined with sorption data of CO₂ and Ar in polycarbonate. A procedure is presented for determining from a measured isotherm the number of sorption sites in a polymer, an important parameter that previously had to be assumed. With this parameter value and solubility data obtained at zero pressure, the model can reasonably predict sorption isotherms of CO₂ in glassy polycarbonate for a wide temperature range. The number of sorption sites and the average site volume evaluated from CO₂ sorption isotherms are employed for the prediction of Ar sorption isotherms with zero‐pressure solubility data and the independently measured partial molar volume of Ar. A reasonable fit to the measured isotherms of Ar is achieved. With the proposed procedure, the continuous‐site model shows several advantages over the conventional dual‐mode sorption model. © 2000 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 38: 883–888, 2000