Modeling the viscoelastoplastic response of amorphous glassy polymers

Constitutive equations are derived for the viscoelastoplastic response of amorphous glassy polymers at isothermal loading with small strains. A polymer is treated as an ensemble of cooperatively relaxing regions (CRR) which rearrange at random times as they are thermally agitated. Rearrangement of CRRs reflects the viscoelastic response of the bulk medium. At low stresses, CRRs are connected with each other, which implies that the macro-strain in a specimen coincides with micro-strains in individual relaxing regions. When the average stress exceeds some threshold level, links between CRRs break and relaxing domains begin to slide one with respect to another. Sliding of micro-domains is associated with the viscoplastic behavior of polymers. Kinetic equations are proposed for viscoplastic strains and for the evolution of the threshold stress. These equations are validated by comparison with experimental data in tensile relaxation tests and in tests with constant strain rates. Fair agreement is demonstrated between results of numerical simulation and observations for a polyurethane resin and poly(methyl methacrylate).Comment: 19 pages, 12 figure

Non-entropic theory of rubber elasticity: flexible chains grafted on a rigid surface

The elastic response is studied of a single flexible chain grafted on a rigid plane and an ensemble of non-interacting tethered chains. It is demonstrated that the entropic theory of rubber elasticity leads to conclusions that disagree with experimental data. A modification of the conventional approach is proposed, where the end-to-end distribution function (treated as the governing parameter) is replaced by the average energy of a chain. It is revealed that this refinement ensures an adequate description of the mechanical behavior of flexible chains. Results of numerical simulation are compared with observations on uniaxial compression of a layer of grafted chains, and an acceptable agreement is shown between the model predictions and the experimental data. Based on the analysis of combined compression and shear, a novel micro-mechanism is proposed for the reduction of friction of polymer melts at rigid walls.Comment: 16 pages, 2 figure

Modelling structural relaxation in polymeric glasses using the aggregation-fragmentation concept

Governing equations are derived for the kinetics of physical aging in polymeric glasses. An amorphous polymer is treated as an ensemble of cooperatively rearranged regions (CRR). Any CRR is thought of as a string of elementary clusters (EC). Fragmentation of the string may occur at random time at any border between ECs. Two string can aggregate at random time to produce a new string. The processes of aggregation and fragmentation are treated as thermally activated, and the rate of fragmentation is assumed to grow with temperature more rapidly than that for coalescence. This implies that only elementary clusters are stable at the glass transition temperature, whereas below this temperature, CRRs containing several ECs remain stable as well. A nonlinear differential equation is developed for the distribution of CRRs with various numbers of ECs. Adjustable parameters of the model are found by fitting experimental data in calorimetric tests for polycarbonate, poly(methyl methacrylate), polystyrene and poly(vinyl acetate). For all materials, fair agreement is established between observations and results of numerical simulation. For PVAc, the relaxation spectrum found by matching data in a calorimetric test is successfully employed to predict experimental data in a shear relaxation test.Comment: 25 pages, 15 figure

Thermal degradation and viscoelasticity of polypropylene-clay nanocomposites

Results of torsional oscillation tests are reported that were performed at the temperature T=230C on melts of a hybrid nanocomposite consisting of isotactic polypropylene reinforced with 5 wt.% of montmorillonite clay. Prior to mechanical testing, specimens were annealed at temperatures ranging from 250 to 310C for various amounts of time (from 15 to 420 min). Thermal treatment induced degradation of the matrix and a pronounced decrease in its molecular weight. An integro-differential equation is derived for the evolution of molecular weight based on the fragmentation-aggregation concept. This relation involves two adjustable parameters that are found by fitting observations. With reference to the theory of transient networks, constitutive equations are developed for the viscoelastic response of nanocomposite melts. The stress-strain relations are characterized by three material constants (the shear modulus, the average energy for rearrangement of strands and the standard deviation of activation energies) that are determined by matching the dependencies of storage and loss moduli on frequency of oscillations. Good agreement is demonstrated between the experimental data and the results of numerical simulation. It is revealed that the average energy for separation of strands from temporary junctions is independent of molecular weight, whereas the elastic modulus and the standard deviation of activation energies linearly increase with mass-average molecular weight.Comment: 24 pages and 18 figure

The viscoelastic and anelastic responses of amorphous polymers in the vicinity of the glass transition temperature

The time-dependent response of polystyrene and poly(methyl methacrylate) is studied in isothermal long-term shear creep tests at small strains and various temperatures in the vicinity of the glass transition point. A micromechanical model is derived to describe the experimental results. Constitutive equations are developed under the assumption that the behavior of amorphous polymers is governed by two micro-mechanisms: rearrangement of cooperatively relaxing regions (CRR) reflects the viscoelastic response, whereas displacement of CRRs with respect to each other is responsible for the anelastic response. It is demonstrated that some critical temperature exists slightly above the glass transition temperature, where the dependences of adjustable parameters on temperature are dramatically changed. The critical temperature is associated with transition from dynamic heterogeneity in amorphous polymers to static inhomogeneity.Comment: 27 pages, 14 figure

Enthalpy recovery in semicrystalline polymers

Constitutive equations are derived for enthalpy recovery in polymeric glasses after thermal jumps. The model is based on the theory of cooperative relaxation in a version of the trapping concept. It is demonstrated that some critical temperature and some critical degree of crystallinity exist above which the energy landscape becomes homogeneous and structural relaxation ceases.Comment: 13 pages, 5 figures, LATE

Stress-softening and recovery of elastomers

A constitutive model is developed for the mechanical response of elastomers at finite strains. A polymer is treated as a network of linear chains linked by permanent (chemical crosslinks) and temporary (entanglements and van der Waals forces) junctions. Temporary junctions are assumed to be in two states: loose (passive) when they impose only topological constrains on available configurations of chains, and tight (active) when their effect is tantamount to that for crosslinks. Stretching of a specimen implies that some loose junctions become active, which decreases the average length of a chain. A long chain is treated as an ensemble of inextensible strands connected in sequel. Two neighboring strands are bridged by a bond which may be in two conformations: flexed (trans) and extended (cis). A bond in the flexed conformation is modeled as a linear elastic solid, whereas the mechanical energy of a bond in the extended conformation (two rigid rods directed along a straight line) is disregarded. For a virgin specimen, all bonds are in the flexed conformation. Under loading some bonds are transformed from flexed to extended conformation. Stress-strain relations for a rubbery polymer and kinetic equations for the trans-cis transition are derived using the laws of thermodynamics. Governing equations are determined by 5 adjustable parameters which are found by fitting experimental data in uniaxial tensile tests on natural rubber vulcanizates with various amounts of crosslinks. Fair agreement is demonstrated between results of numerical simulation and observations with the elongation ratio up to $k=8$. We analyze the effects of cyclic loading, thermal annealing and recovery by swelling on the material constants.Comment: 42 pages, 18 figure

Buckling of rods with spontaneous twist and curvature

We analyze stability of a thin inextensible elastic rod which has non-vanishing spontaneous generalized torsions in its stress-free state. Two classical problems are studied, both involving spontaneously twisted rods: a rectilinear beam compressed by axial forces, and a circular ring subjected to uniform radial pressure on its outer perimeter. It is demonstrated that while spontaneous twist stabilizes a rectilinear rod against buckling, its presence has an opposite effect on a closed ring.Comment: 8 pages, 1 figur

The effect of temperature on the viscoelastic response of rubbery polymers at finite strains

Constitutive equations are derived for the viscoelastic response of rubbery polymers at finite strains. A polymer is thought of as a network of long chains connected to temporary junctions. At a random time, a chain detaches from a junction, which is treated at transition from its active state to the dangling state. A dangling chain randomly captures a new junction in the vicinity of its free end and returns to its active state. Breakage and reformation of long chains are modeled as thermo-mechanically activated processes. Stress-strain relations for a rubbery polymer are developed using the laws of thermodynamics. Adjustable parameters in the model are found by fitting observations in uniaxial tensile tests for a carbon black filled rubber at various temperatures. Fair agreement is demonstrated between experimental data and results of numerical simulation.Comment: 33 pages, 5 figure

The Payne effect for particle-reinforced elastomers

The study deals with the Payne effect (a substantial decrease in the storage modulus of a particle-reinforced elastomer with an increase in the amplitude of mechanical oscillations). The influence of temperature, concentration of filler and amplitude and frequency of strains is analyzed on the mechanical response of filled rubbery polymers. Constitutive equations are derived using the concept of two interpenetrating networks: one comprises semiflexible polymeric chains connected to temporary junctions, whereas the other is formed by aggregated filler clusters. Adjustable parameters are found by fitting experimental data for natural rubber, bromobutyl rubber and styrene-butadiene rubber reinforced by carbon black and polymeric particles. The critical concentration of particles is determined that characterizes transition from an ensemble of disjoint clusters to the network of filler. The volume fraction of filler corresponding to this transition is found to be close to theoretical predictions based on the percolation theory, as well as to experimental data for isolator-conductor transition.Comment: 34 pages, 7 figure