Molecular theories of nonlinear viscoelasticity of polymers

It is emphasized that considerable advances have been made recently in the development of the molecular theories of nonlinear viscoelasticity of concentrated solutions and melts of linear polymers. The new ideas in this exceptionally important field of the rheology of polymers are analysed. The methods by which the constraints (entanglements) imposed on the motion of macromolecules by the polymer environment are taken into account are also considered in the paper. The most detailed discussion is devoted to the model of topological constraints in the form of a tube and to the self-consistent theory of anisotropic micro-viscoelasticity which takes into account the relaxation nature of the interaction of macromolecules with their surrounding medium as well as the anisotropy of their mobility.Invited paper, presented at the XII-th All-Union Symposium on Rheology held in Riga (USSR), December 7–9, 1982.     

The mirror relations and nonlinear viscoelasticity of polymer melts

An attempt is made to incorporate into a quasilinear viscoelastic constitutive equation of the Boltzmann superposition type the two mirror relations of Gleissle, as well as his relation between the steady-state first normal-stress difference and the shear viscosity curve. It is shown that the three relations can hold separately within this constitutive model, but not simultaneously, because they require a different nonlinear strain measure, namelyS ₁₂ () = – a ( – 1) (a = 0 for 1,a = 1 for 1) for the mirroring of the viscosities,S ₁₂ () = – a (–k ²/) (a = 0 for k, a = 1 for k) for the mirroring of the first normal-stress coefficients, and for the third relation. Here denotes the shear strain and erf the error function. Experimental data on melts of a low-density polyethylene, a high-density polyethylene and a polypropylene show that the mirror relations are passable approximations, but that the third relation meets reality surprisingly close if the right value ofk is used.     

Effects of viscoelasticity on the stability and bifurcations of nonlinear energy sinks

Due to the increasing use of passive absorbers to control unwanted vibrations,many studies have been done on energy absorbers ideally, but the lack of studies of real environmental conditions on these absorbers is felt. The present work investigates the effect of viscoelasticity on the stability and bifurcations of a system attached to a nonlinear energy sink(NES). In this paper, the Burgers model is assumed for the viscoelasticity in an NES, and a linear oscillator system is considered for inve...     

A molecular network model to describe the nonlinear viscoelastic behavior of polymers

A transient molecular network model is built to describe the nonlinear viscoelasticity of polymers by considering the effect of entanglement loss and regeneration on the relaxation of molecular strands. It is an extension of previous network theories. The experimental data on three thermoplastic polymers (ABS, PVC and PA6) obtained under various loading conditions are used to test the model. Agreement between the theoretical and experimental curves shows that the suggested model can describe successfully the relaxation behavior of the thermoplastic polymers under different loading rates by using relatively few relaxation modes. Thus the micromechanism responsible for strain-rate dependence of relaxation process and the origin of nonlinear viscoelasticity may be disclosed. The project supported by the National Natural Science Foundation of China and Doctorial Fund     

Molecular origin of viscoelasticity and chain orientation of glassy polymers

Dynamic birefringence and dynamic viscoelasticity of poly(4-methyl styrene) and poly(4-t-butyl styrene) were measured to investigate the molecular origin of viscoelasticity around the glass transition zone. The data were analyzed with a modified stress-optical rule: The birefringence and the stress were separated into two component functions of different molecular origins. One component is related to the orientation of the main chain axis and the other one to the rotation of the repeating units about the main chain axis. The strain dependence of the two characteristic orientation functions in the glassy zone was estimated and the orientation mechanism of repeating units was discussed.Dedicated to Prof. John D. Ferry on the occasion of his 85th birthday.     

Application of fractional exponential hereditary kernels in the nonlinear theory of viscoelasticity

It is proved that fractional exponential hereditary kernels of nonlinear viscoelasticity can be used to evaluate creep strains and stress relaxation. A nonlinear theory of viscoelasticity with time-independent nonlinearity described as a nonlinear curve of instantaneous elastoplastic deformation is used. The calculated results are validated against experimental data on the viscoelastic deformation of laminated and unidirectional fibrous composites and their components under the conditions of constant stresses, complete unloading, incremental loading, pure torsion, and constant strains     

Strain-coupling effects in branched polymers

The applicability of a strain-coupling integral constitutive equation to the analysis of the nonlinear viscoelasticity of branched polymers is evaluated. A simpler and more efficient method is proposed for evaluating the material functions of the strain-coupling constitutive equation.     

Effects of strong anchoring on the dynamic moduli of heterogeneous nematic polymers

We focus on the linear viscoelastic response of heterogeneous nematic polymers to small amplitude oscillatory shear, paying special attention to the macroscopic influence of strong plate anchoring conditions. The model consists of the Stokes hydrodynamic equations with viscous and nematic stresses, coupled to orientational dynamics and structure driven by the flow gradient, an excluded-volume potential, and a two-constant distortional elasticity potential. We show that the dynamical response simplifies when plate anchoring is either tangential or homeotropic, recovering explicitly solvable Leslie–Ericksen–Frank behavior together with weakly varying order parameters across the plate gap. With these plate conditions, we establish “model consistency” so that all experimental driving conditions (plate-controlled velocity [strain] or shear stress, imposed oscillatory pressure) yield identical dynamic moduli for the same material parameters and anchoring conditions, eliminating the culpability of device influence in scaling behavior. Two physical predictions emerge that imply significant macroscopic elastic and viscous effects controlled by plate anchoring relative to flow geometry: (1) The storage modulus is enhanced by two to three orders of magnitude for homeotropic relative to parallel anchoring, across all frequencies. (2) The loss modulus exhibits enhancement of a factor of two to three for homeotropic over tangential anchoring, restricted to low frequencies. We further deduce a scaling law for the dynamic moduli versus anisotropy of the distortional elasticity potential.

Microdamage of a nonlinear elastic material in combined stress state

The structural theory of short-term damage is used to study the coupled processes of deformation and microdamage of a physically nonlinear material in a combined stress state. The basis for the analysis is the stochastic elasticity equations for a physically nonlinear porous medium. Damage in a microvolume of the material is assumed to occur in accordance with the Huber-Mises failure criterion. The balance equation for damaged microvolumes is derived and added to the macrostress-macrostrain relations to produce a closed-form system of equations. It describes the coupled processes of nonlinear deformation and microdamage of the porous material. Algorithms are developed for calculating the dependence of microdamage on macrostresses and macrostrains and plotting stress-strain curves for a homogeneous material under either biaxial normal loading or combined normal and tangential loading. The plots are analyzed depending on the type of stress state __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 11, pp. 30–39, November 2006.     

Volume changes in glassy polymers

Summary Constitutive equations are derived for compressible glassy polymers at non-isothermal loading with finite strains. The model is based on the theory of temporary networks in its version of adaptive links concept. The stress–strain relations are applied to the analysis of uniaxial extension of a viscoelastic bar. Explicit formulas are developed for time-dependent Young's modulus and Poisson's ratio of the bar at small strains. Results of numerical simulation are compared with experimental data for polycarbonate, polyethylene, and poly(methyl methacrylate). It is demonstrated that (i) longitudinal stresses do not affect the specific free volume in the region of linear viscoelasticity at strains up to about 0.2%, and cause substantial changes in the free volume in the region of nonlinear viscoelasticity at strains about 1.0%; (ii) in the latter case, the increment of the free-volume fraction is proportional to the increase in the specific volume. Received 3 April 1998; accepted for publication 22 June 1998     

A model for the viscoelastic behavior of polymers at finite strains

Summary A constitutive model is derived for the isothermal nonlinear viscoelastic response in polymers, which do not possess the separability property. The model is based on the concept of transient networks, and treats a polymer as a system of nonlinear elastic springs (adaptive links), which break and emerge due to micro-Brownian motion of chains. The breakage and reformation rates for adaptive links are assumed to depend on some strain energy density. The viscoelastic behavior is described by an integral constitutive equation, where the relaxation functions satisfy partial differential equations with coefficients depending on the strain history. Adjustable parameters of the model are found by fitting experimental data for a number of polymers in tension at strains up to 400 per cent. To validate the constitutive relations, we consider loading with different strain rates, determine adjustable parameters at one rate of strains, and compare prediction of the model with observations at another rate of strains. Fair agreement between experimental data and results of numerical simulation is demonstrated when the rates of strains differ by more than a decade. Received 1 July 1997; accepted for publication 7 October 1997     

Mobility anisotropy and nonlinear effects in molecular viscoelasticity theory of linear polymers

Various methods of taking the mobility anisotropy into account in modeling of the behavior of polymer fluids in a simple shear flow are compared. Both the qualitative coincidence of the results of the analysis and the difficulties of quantitatively estimating the anisotropy microparameters on the basis of the available data concerning steady shear are noted. For axial tension the dependence of steady viscosity on tensile stress is considered. This dependence is determined by two independent scalar functions: the slip coefficient and the flow anisotropy coefficient.Barnaul. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 3–12, July–August, 1995.     

The relaxation of linear flexible polymers which are slightly poly-disperse

The relaxation of slightly poly-disperse linear flexible polymers has been expressed in a simplified blending rule which is presumed to be a weighted linear superposition of the relaxation spectra of mono-disperse components which constitute the blend. Discrete components are characterized by their molecular weight M _i,weight fraction w _i,and relaxation time spectrum H _i(). ). In contrast to broadly distributed blends in which the small molecules mobilize the large ones and vice versa, we introduce the term slightly polydisperse for blends with molecular weight distributions narrow enough to have very little change in the longest relaxation times of each molecular weight component. The properties of this blending rule are analyzed and dynamic data is calculated for slightly poly-disperse polystyrene. As an application, the blending rule is used to determine the characteristic mono-disperse parameters (BSW parameters) of two materials, poly (vinyl methyl ether) and polycarbonate, for which we could not determine their BSW parameters directly since they were not available in nearly monodisperse form. The proposed blending rule can only be applied to systems in which all components are above the entanglement molecular weight, i.e. M _iM _c.Dedicated to the memory of Professor Tasos C. Papanastasiou     

A method for determining the parameters of the hereditary kernels in the nonlinear theory of viscoelasticity

The analytical and experimental procedures of the method for determining the parameters of the hereditary kernels in the modified Rabotnov’s nonlinear viscoelastic model are outlined. The method is based on the similarity of the isochronous creep curves to the instantaneous-deformation curve. The parameters of the kernels are determined by fitting the discrete values of the kernels that are found by differentiating the average similarity function. The discrete values of the kernels in the domain of singularity are calculated using weight functions     

On the analytic expression of the free energy in linear viscoelasticity

Two definitions of free energy for a linear viscoelastic material, due to Graffi and to Coleman and Owen, are considered, and the compatibility of these definitions with some expressions of the free energy proposed in the literature is examined. For the expressions of Staverman and Schwarzl and of Breuer and Onat, the two definitions are proved to be equivalent, and the set of all relaxation functions for which the two expressions are indeed free energies is determined. Two more expressions, proposed by Volterra and Graffi and by Morro and Vianello, are taken into consideration. For them, only the classes of relaxation functions for which they are free energies according to the first definition, is completely characterized. All results are established under regularity assumptions weaker than those usually made in the literature.     

Effect of ionic interaction on linear and nonlinear viscoelastic properties of ethylene based ionomer melts

The effect of ionic interaction on linear and nonlinear viscoelastic properties was investigated using poly(ethylene-co-methacrylic acid) (E/MAA) and its ionomers which were partially neutralized by zinc or sodium. Dynamic shear viscosity and step-shear stress relaxation studies were performed. Stress relaxation moduli G(t, y) of the E/MAA and its sodium or zinc ionomers were factorized into linear relaxation moduli G°(t) and damping functions h(y). The relaxation modulus at the smallest strain in each ionomer agreed with the linear relaxation modulus calculated from storage modulus G and loss modulus G. In the linear region, the ionic interaction shifted the relaxation time longer with keeping the same relaxation time distribution as E/MAA. In the nonlinear region, the ionic interaction had no influence on h(y) when the ion content was low. At higher ion content, however, the ion bonding enhanced the strain softening of h(y).     

The effect of shear deformation on the post-buckling behavior of imperfect nonlinear viscoelastic columns

Summary The post-buckling behavior of imperfect columns made of nonlinear viscoelastic materials is investigated, taking into account the effect of shear deformation. The material is modeled according to the Leaderman representation of nonlinear viscoelasticity. Solutions are developed, within the elastica and the shear deformation theories, in order to calculate the growth in time of the total deflection. The numerical results establish the importance of the shear and the nonlinear viscoelasticity effects, and of the h/ℓ ratio in the column post-buckling behavior. Accepted for publication 11 November 1996     

On the admissibility criteria for linear viscoelasticity kernels

In a series of recent papers, the discrepancy of the admissibility requirements currently in use for linear viscoelasticity was noted, between the ones associated with the kernels of linear integral models and the ones resulting from the spring-dashpot representation. It is shown here that additional criteria can be obtained from the fading memory principle which completely remove the observed discrepancies between these two different representations.     

On the damping function of shear relaxation modulus for entangled polymers

Published data of the damping function of the shear relaxation modulus, h(), are reviewed. This is the ratio of the relaxation modulus measured at a finite magnitude of shear, , to that at the limit of = 0. Majority of the data are in accord with the universal function of the Doi-Edwards tube model theory, in which the damping or the decrease of h() is attributed to the contraction along the tube of extended polymer chains. The weaker damping seems to be attributed to 1) comb-branching such as in LDPE; 2) lack of entanglement in too short chains; 3) bimodal molecular weight distribution. However, a star-branching does not cause a deviation from the tube model theory and a broadness of molecular weight distribution is not a major origin of a weaker damping. A star-branched polystyrene with 15 arms exhibits no strain dependence: h() = 1. For highly entangled systems with more than 50 entanglement points per molecule, the strain dependence is stronger than that of the Doi-Edwards theory. This could be due to a slip or an instability of deformation in the material.     

A phenomenological constitutive law for the nonlinear viscoelastic behaviour of epoxy resins in the glassy state

We propose a constitutive model to describe the nonlinear viscoelasticity of epoxy resins in the glassy state. Experimental tests have shown that nonlinear viscoelasticity rules the cyclic behaviour of epoxy resins before the material strength is reached. The proposed model allows the simulation of this cyclic behaviour and, in particular, of the flex which characterises the stress–strain curve upon unloading. The consistent integration algorithm, based on the Central Difference Scheme, of the proposed constitutive law is given. As an example, the model is successfully used to numerically homogenize the cyclic behaviour of hollow sphere-filled epoxy resins (i.e., syntactic foams) by means of unit cell models.