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

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

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

A tube concept in rubber viscoelasticity

A constitutive model is derived for the time-dependent response of particle-reinforced elastomers at finite strains. An amorphous rubbery polymer is treated as a network of long chains linked by permanent junctions (chemical crosslinks, entanglements and filler particles). A strand between two neighboring junctions is thought of as a sequence of mers whose motion is restricted to some tube by surrounding macromolecules. Unlike the conventional approach that presumes the cross-section of the tube to be constant, we postulate that its radius strongly depends on the longitudinal coordinate. This implies that a strand may be modeled as a sequence of segments whose thermal motion is totally frozen by the environment (bottle-neck points of the tube) bridged by threads of mers which go through all possible configurations during the characteristic time of a test. Thermal fluctuations affect the tube's radius, which results in freezing and activation of regions with high molecular mobility (RHMs). The viscoelastic response of an elastomer is associated with thermally activated changes in the number of RHMs in strands. Stress-strain relations for a rubbery polymer at finite strains and kinetic equations for the concentrations of RHMs are developed by using the laws of thermodynamics. At small strains these relations are reduced to the conventional integral constitutive equation in linear viscoelasticity with a novel scaling law for relaxation times. The governing equation is determined by 5 adjustable parameters which are found by fitting experimental data in tensile dynamic tests on a carbon black filled natural rubber vulcanizate.Comment: 28 pages, 14 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

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

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 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

Elasticity of thin rods with spontaneous curvature and torsion - beyond geometrical lines

We study three-dimensional deformations of thin inextensible elastic rods with non-vanishing spontaneous curvature and torsion. In addition to the usual description in terms of curvature and torsion which considers only the configuration of the centerline of the rod, we allow deformations that involve the rotation of the rod's cross-section around its centerline. We derive new expressions for the mechanical energy and for the force and moment balance conditions for the equilibrium of a rod under the action of arbitrary external loads. Several illustrative examples are studied and the connection between our results and recent experiments on stretching of supercoiled DNA molecules is discussed.Comment: 28 pages, 2 figure