A mathematical boundary integral equation analysis of standard viscoelastic solid polymers

Structural analysis of viscoelastic solid polymers is one of the most important subjects in engineering structures. Several attempts have been so far made for the integral equation approach to viscoelastic problems. From the basic assumptions of viscoelastic constitutive equations and weighted residual techniques, a simple but effective boundary element formulation (BEF) is implemented for the standard linear solid (SLS) viscoelastic models. The SLS model provides an approximate representation of the observed behavior of a real polymer in its viscoelastic range. This formulation needs only Kelvin’s fundamental solution of isotropic elastostatics with material constants prescribed as explicit functions of time. This approach avoids the use of relaxation functions and mathematical transformations, and it is able to solve the quasistatic viscoelastic problems with any load time-dependence and boundary conditions. As an application, a numerical example is provided to validate the proposed formulation. The problem of the pressurization of thick-walled cylindrical viscoelastic tanks made of PMMA polymer is completely analyzed by this approach.     

Prediction of the relaxation properties of materials based on heat-resisting polymers

Conclusions The PB0, P0D, and PI materials were used to show the possibilities of simulating the long-term relaxation processes for a new group of polymer materials using the time analogies in the linear and nonlinear regions of the viscoelastic properties. The results show that the generalized dependences of compliance and relaxation modulus widen the time range of lead of the relaxation processes by approximately six to seven decimal orders. Satisfactory results were obtained in comparing the generalized dependences with the results of the long-term reference experiment carried out in the same basic conditions.Translated from Mekhanika Kompozitnykh Materialov, No. 5, pp. 778–783, September–October, 1984.     

Relaxation of normal stresses in flowing polymer systems

The authors discuss the relaxation of the first difference of the normal stresses σ(t) after cessation of steady shear flow in viscoelastic polymer systems in the linear region of their mechanical behavior. They show that this phenomenon is associated with a group of relaxation characteristics of the material. they discuss a method of calculating the relaxation spectrum from σ(t); the validity of this suggested method is verified experimentally.     

A model of weak viscoelastic nematodynamics

The paper develops a continuum theory of weak viscoelastic nematodynamics of Maxwell type. It can describe the molecular elasticity effects in mono-domain flows of liquid crystalline polymers as well as the viscoelastic effects in suspensions of uniaxially symmetric particles in polymer fluids. Along with viscoelastic and nematic kinematics, the theory employs a general form of weakly elastic thermodynamic potential and the Leslie–Ericksen–Parodi type constitutive equations for viscous nematic liquids, while ignoring inertia effects and the Frank (orientation) elasticity in liquid crystal polymers. In general case, even the simplest Maxwell model has many basic parameters. Nevertheless, recently discovered algebraic properties of nematic operations reveal a general structure of the theory and present it in a simple form. It is shown that the evolution equation for director is also viscoelastic. An example of magnetization exemplifies the action of non-symmetric stresses. When the magnetic field is absent, the theory is reduced to the symmetric, fluid mechanical case with relaxation properties for both the stress and director. Our recent analyses of elastic and viscous soft deformation modes are also extended to the viscoelastic case. The occurrence of possible soft modes minimizes both the free energy and dissipation, and also significantly decreases the number of material parameters. In symmetric linear case, the theory is explicitly presented in terms of anisotropic linear memory functionals. Several analytical results demonstrate a rich behavior predicted by the developed model for steady and unsteady flows in simple shearing and simple elongation.     

A model of weak viscoelastic nematodynamics

The paper develops a continuum theory of weak viscoelastic nematodynamics of Maxwell type. It can describe the molecular elasticity effects in mono-domain flows of liquid crystalline polymers as well as the viscoelastic effects in suspensions of uniaxially symmetric particles in polymer fluids. Along with viscoelastic and nematic kinematics, the theory employs a general form of weakly elastic thermodynamic potential and the Leslie–Ericksen–Parodi type constitutive equations for viscous nematic liquids, while ignoring inertia effects and the Frank (orientation) elasticity in liquid crystal polymers. In general case, even the simplest Maxwell model has many basic parameters. Nevertheless, recently discovered algebraic properties of nematic operations reveal a general structure of the theory and present it in a simple form. It is shown that the evolution equation for director is also viscoelastic. An example of magnetization exemplifies the action of non-symmetric stresses. When the magnetic field is absent, the theory is reduced to the symmetric, fluid mechanical case with relaxation properties for both the stress and director. Our recent analyses of elastic and viscous soft deformation modes are also extended to the viscoelastic case. The occurrence of possible soft modes minimizes both the free energy and dissipation, and also significantly decreases the number of material parameters. In symmetric linear case, the theory is explicitly presented in terms of anisotropic linear memory functionals. Several analytical results demonstrate a rich behavior predicted by the developed model for steady and unsteady flows in simple shearing and simple elongation.     

Constitutive model of ferroelectric composites with a viscoelastic and dielectric relaxation matrix I

Based on micromechanics and Laplace transformation, a constitutive model of ferroelectric composites with a linear elastic and linear dielectric matrix is developed and extended to the ferroelectric composites with a viscoelastic and dielectric relaxation matrix. Thus, a constitutive model for ferroelectric composites with a viscoelastic and dielectric relaxation matrix has been set up Project supported by the National Natural Science Foundation of China (Grant No. 19891180).     

Modelling an anomalous stress relaxation in glassy polymers (the Kitagawa effect)

Constitutive relations are derived for the nonlinear viscoelastic response of glassy polymers. The model is based on the concept of adaptive links (a version of the theory of transient networks), where the rates of annihilation of active links are determined by the Eyring formula. Adjustable parameters in the constitutive equations are found by using experimental data for a polyester resin in standard relaxation tests. The stress-strain relations are applied to describe an anomalous stress relaxation in solid polymers observed in tensile test after strain reversal (the Kitagawa effect). Qualitative agreement is demonstrated between results of numerical simulation and observations.     

Effect of viscoelastic relaxation modes on stability of extrusion film casting process modeled using multi-mode Phan-Thien-Tanner constitutive equation

Extrusion film casting (EFC) is a commercially important process that is used to produce a significant quantity of polymer films, sheets and coatings for both industrial and household applications. Recently, we have demonstrated the influence of polymer chain architecture on the extent of necking under isothermal as well as non-isothermal film casting operation for commercially relevant polyolefin based materials [1], [2], [3], [4]. In the present research, we focus on another instability that frequently occurs in high-speed EFC process called as draw resonance. Draw resonance manifests itself as an instability that causes periodic fluctuations in both the width as well as thickness of the extruded molten film above a critical draw ratio (DR). In this work, we have carried out a linear stability analysis of the isothermal EFC process using a multi-mode Phan-Thien-Tanner (PTT) constitutive equation to determine the onset of draw resonance. We show that as the number of relaxation modes is increased there is a dramatic change in the stability regions. In particular, there is a marked variation in the stability regions obtained by simulating the multi-mode model and those obtained by taking averaged relaxation time of the modes. Additionally, as the number of faster-relaxing modes in a multi-mode spectrum is progressively increased, the process becomes increasingly stable as the level of elasticity in the melt decreases. Finally, the addition of a long relaxation mode in a multi-mode spectrum is akin to adding a long chain branch to a linear polymer that leads to a reduction in film necking and in many cases to enhanced process stability.     

Limiting stretch ratios of polymers. Role of molecular mobility

Orientation stretching stops before the polymer chains reach full orientation. The reason for the cessation of stretching in linear polymers is considered to be the decreased kinetic flexibility of the macromolecules. With Kapron as an example, it has been demonstrated, using NMR, that on attainment of the limiting stretch ratio under the action of the stretching (orienting) forces, segmental motion in the amorphous regions is almost completely stopped: the amorphous regions are glassy at any orientation temperature, even those close to the melting point of the crystals. The orientation process stops because the polymer under load behaves like a rigid body devoid of rubber-like elasticity.A. F. Ioffe Physicotechnical Institute, Academy of Sciences of the USSR, Leningrad. Translated from Mekhanika Polimerov, Vol. 9, No. 3, pp. 387–391, 1973.     

Equations of three-dimensional deformation of linear viscoelastic media

An equation is proposed relating the stress and strain tensors in time for the three-dimensional deformation of linear viscoelastic media whose properties are determined by four parameters. A method is indicated for determining these parameters based on simple creep or relaxation tests. Values of the parameters are given for certain polymer materials. For such materials, there is an upper limit of the stress state below which an equilibrium limit state is possible. A creep kernel is proposed for the case of no equilibrium limit state, and a resolvent constructed with account for the variation of stress and strain in time.Mekhanika Polimerov, Vol. 1, No. 4, pp. 35–42, 1965     

Convergence of the Equations for a Maxwell Fluid

The equations for the flow of a viscoelastic fluid of the Maxwell type are analyzed in a linear approximation. First, we establish that the solution depends continuously on changes in the relaxation time. Next, we investigate how the solution to the linearized Maxwell system converges to the solution to Stokes flow as the relaxation time tends to zero. Convergence in different measures is examined and specific a priori bounds are derived.     

Approximate controllability results for viscoelastic flows with infinitely many relaxation modes

We prove results on approximate controllability for linear viscoelastic flows, with a localized distributed control in the momentum balance equation. The constitutive law is a multimode Maxwell or Jeffreys model with an infinite number of relaxation modes.     

Kinetic Equations for Describing the Liquid-Glass Transition in Polymers

We present a theoretical approach based on nonequilibrium thermodynamics and used to describe the kinetics of the transition from the liquid to the glassy state (glass transition). In the framework of this approach, we construct kinetic equations describing the time and temperature evolution of the structural parameter. We discuss modifications of the equations required for taking the nonexponential, nonlinear character of the relaxation in the vitrification region into account. To describe the formation of polymer glasses, we present modified expressions for the system relaxation time. We compare the obtained results with experimental data, measurements of the polystyrene glass transition for different cooling rates using the method of differential scanning calorimetry. We discuss prospects for developing a method for describing the polymer glass transition.     

The theory of orientational drawing of linear amorphous polymers

On the basis of modern ideas on the structure of linear, amorphous polymers a model of a molecular, three-dimensional network with temporary cross-links and van der Waals interaction between the chains is presented. Using this model as a starting point, a differential equation for the deformation of a linear, amorphous polymer in the viscoelastic state is derived. The differential equation for the orientation of a linear, amorphous polymer, which describes the evolution of distribution functions for statistical segments, is obtained. Previously obtained results follow from this theory as special cases.V. I. Lenin State Pedagogic Institute, Moscow. Translated from Mekhanika Polimerov, No. 6, pp. 1021–1028, November–December, 1974.     

A novel nonlinear viscoelastic solid model

The standard linear solid (SLS) model has been widely used to describe linear viscoelastic materials. Although the SLS can be extended to describe more complex phenomena through the inclusion of additional components, a compact model of nonlinear viscoelasticity remains elusive. Here, using the framework of the three component SLS model, the stress-strain relationships of the individual components have been generalised. This work describes the derivation of the new generalised model, termed nonlinear viscoelastic solid model, or NVS, that can potentially describe nonlinear viscoelastic phenomena in a compact differential form and reduces to the familiar SLS model if linear components are selected. As a proof of concept, exponential functions of strain and strain rate were selected for the three components and major viscoelastic phenomena such as stress relaxation, creep and sawtooth strain loading were simulated. Finally, to demonstrate its efficacy in describing biological tissue, the NVS model was used to simulate the cyclic loading of mammalian stomach and cardiac muscle tissues.     

Features of viscoelastic behavior of materials based on heat-resistant polymers

Conclusions In the example of polybenzoxazole, the relationships of viscoelastic behavior have been investigated for monolithic heat-resistant polymers over a wide range of temperatures (from 22° to 230°C) and initial deformations (from 0.5 to 2.5%). The experimental data on stress relaxation, covering the regions of linear and nonlinear viscoelasticity, are approximated by the basic equation of the primary cubic theory of Il'yushin, using the singular kernel of Rzhanitsyn. For the first time, the region of stable mechanical capability of monolithic heat-resistant polymers has been defined with due regard for relaxation characteristics.Paper presented at Fourth All-Union Conference on Polymer Mechanics and Composite Materials (Riga, October 1980).Translated from Mekhanika Kompozitnykh Materialov, No. 6, pp. 978–983, November–December, 1980.     

Energy dissipation calculations for a linear viscoelastic body under impact loading

An expression is obtained for the dissipated energy of a linear viscoelastic body with a continuous relaxation spectrum in the case of an arbitrary law of variation of the strains with time. Lower bounds for the total and dissipated energies are constructed by the method proposed in [3]. The calculated values of the dissipated energy are compared with experiment.     

General decay rate estimates for viscoelastic dissipative systems

The linear viscoelastic equation is considered. We prove uniform decay rates of the energy by assuming a nonlinear feedback acting on the boundary, without imposing any restrictive growth assumption on the damping term and strongly weakening the usual assumptions on the relaxation function. Our estimate depends both on the behavior of the damping term near zero and on the behavior of the relaxation function at infinity.     

A boundary element methodology for viscoelastic analysis: Part II without cells

In this study Kelvin and Boltzmann viscoelastic models are implemented in a two-dimensional boundary element atmosphere. This general methodology is based on differential constitutive relations for viscoelasticity, avoiding the use of relaxation functions. In this part of the study, important algebraic operations are introduced into the formulation allowing analysing viscoelastic problems without using internal cells. This improvement is very important to model infinite and semi-infinite regions. The formulation is verified comparing the numerical results with analytical solutions. An extension of the formulation to consider soil–structure interaction is presented in order to demonstrate the vast applicability of the technique.