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This paper deals with the mathematical modelling of large strain magneto-viscoelastic deformations. Energy dissipation is assumed to occur both due to the mechanical viscoelastic effects as well as the resistance offered by the material to magnetisation. Existence of internal damping mechanisms in the body is considered by decomposing the deformation gradient and the magnetic induction into `elastic' and `viscous' parts. Constitutive laws for material behaviour and evolution equations for the non-equilibrium fields are derived that agree with the laws of thermodynamics. To illustrate the theory the problems of stress relaxation, magnetic field relaxation, time dependent magnetic induction and strain are formulated and solved for a specific form of the constitutive law. The results, that show the effect of several modelling parameters on the deformation and magnetisation process, are illustrated graphically.

This paper deals with the mathematical modelling of large strain magneto-viscoelastic deformations. Energy dissipation is assumed to occur both due to the mechanical viscoelastic effects as well as the resistance offered by the material to magnetisation. Existence of internal damping mechanisms in the body is considered by decomposing the deformation gradient and the magnetic induction into `elastic' and `viscous' parts. Constitutive laws for material behaviour and evolution equations for the non-equilibrium fields are derived that agree with the laws of thermodynamics. To illustrate the theory the problems of stress relaxation, magnetic field relaxation, time dependent magnetic induction and strain are formulated and solved for a specific form of the constitutive law. The results, that show the effect of several modelling parameters on the deformation and magnetisation process, are illustrated graphically.

The Rouse-Bueche-Zimm molecular theory of viscoelasticity has been extended to compute the retardation time spectra of radial block and graft copolymers of various configurations and compositions. The basic assumption of the model is that the effect of the presence of foreign blocks and grafts on the retardation spectrum is due to the difference in the friction coefficients. Mathematically this modification is accomplished by introducing into the equations of motion a frictional coefficient matrix. Retardation spectra can then be obtained by numerically solving for the eigenvalue spectrum on a computer. Results show that the viscoelastic retardation spectra and maximum retardation times of those copolymers vary as a function of the number of side chains, their lengths and the spacings between them. The model is restricted to bulk homogeneous copolymers, or those copolymers dissolved in appropriate solvents without microphase separation. (Author)

This paper, second in the series of three papers, develops a general, nonlinear, non-isothermal, compressible theory for finite rubber viscoelasticity and specifies it in a form convenient for solving problems important to the rubber, tire, automobile, and air-space industries, among others. Based on the quasi-linear approach of non-equilibrium thermodynamics, a general nonlinear theory of differential type has been developed for arbitrary non-isothermal deformations of viscoelastic solids. In this theory, the constitutive equations were presented as the sum of a rubber elastic (equilibrium) and a liquid type viscoelastic (non-equilibrium) terms. These equations have then been simplified using several modeling and simplicity arguments.

Includes bibliography

Includes bibliography

We review a new theory of viscoelasticity of a glass-forming viscous liquid near and below the glass transition. In our model we assume that each point in the material has a specific viscosity, which varies randomly in space according to a fluctuating activation free energy. We include a Maxwellian elastic term and assume that the corresponding shear modulus fluctuates as well with the same distribution as that of the activation barriers. The model is solved in coherent-potential approximation (CPA), for which a derivation is given. The theory predicts an Arrhenius-type temperature dependence of the viscosity in the vanishing-frequency limit, independent of the distribution of the activation barriers. The theory implies that this activation energy is generally different from that of a diffusing particle with the same barrier-height distribution. If the distribution of activation barriers is assumed to have Gaussian form, the finite-frequency version of the theory describes well the typical low-temperature alpha relaxation peak of glasses. Beta relaxation can be included by adding another Gaussian with center at much lower energies than that responsible for the alpha relaxation. At high frequencies our theory reduces to the description of an elastic medium with spatially fluctuating elastic moduli (heterogeneous elasticity theory), which explains the occurrence of the boson-peak-related vibrational anomalies of glasses.

We show that a viscoelastic dispersion relation for shear waves in normal $^{3}$He at low temperatures follows directly from the Landau theory of normal Fermi liquids. This theoretical result is in accord with the experimental observations of propagating high frequency shear waves that identify normal $^{3}$He as a viscoelastic substance.

This work considers the propagation of steady waves in viscoelastic material for which the nonlinear strain measure is not necessarily convex. The shape of such a wave is governed by an ordinary nonlinear integro-differential equation having a possibly singular difference kernel. The existence and structure of a solution depends upon the relation of the wavespeed, a parameter in the problem, to two speeds based upon the state of the material ahead of the wave. Solutions are constructed by a monotone iterative scheme which is proven to converge to a unique solution within restricted classes of functions depending upon the wavespeed. A simple numerical approximation to the iterative scheme is used to produce graphs of solutions. An algebraic quasielastic approximation produces upper bounds on discontinuous ( shock and acceleration wave) solutions. For a material such as polymethyl methacrylate (pmma) having a small power in a power-law model of its compliance, this approximation is found to be useful for accurately predicting the structure of shock solutions.

The present series of three consecutive papers develops a general theory for linear and finite solid viscoelasticity. Because the most important object for nonlinear studies are rubber-like materials, the general approach is specified in a form convenient for solving problems important for many industries that involve rubber-like materials. General linear and nonlinear theories for non-isothermal deformations of viscoelastic solids are developed based on the quasi-linear approach of non-equilibrium thermodynamics. In this, the first paper of the series, we analyze non-isothermal linear viscoelasticity, which is applicable in a range of small strains not only to all synthetic polymers and bio-polymers but also to some non-polymeric materials. Although the linear case seems to be well developed, there still are some reasons to implement a thermodynamic derivation of constitutive equations for solid-like, non-isothermal, linear viscoelasticity. The most important is the thermodynamic modeling of thermo-rheological complexity , i.e. different temperature dependences of relaxation parameters in various parts of relaxation spectrum. A special structure of interaction matrices is established for different physical mechanisms contributed to the normal relaxation modes. This structure seems to be in accord with observations, and creates a simple mathematical framework for both continuum and molecular theories of the thermo-rheological complex relaxation phenomena. Finally, a unified approach is briefly discussed that, in principle, allows combining both the long time (discrete) and short time (continuous) descriptions of relaxation behaviors for polymers in the rubbery and glassy regions.

The present series of three consecutive papers develops a general theory for linear and finite solid viscoelasticity. Because the most important object for nonlinear studies are rubber-like materials, the general approach is specified in a form convenient for solving problems important for many industries that involve rubber-like materials. General linear and nonlinear theories for non-isothermal deformations of viscoelastic solids are developed based on the quasi-linear approach of non-equilibrium thermodynamics. In this, the first paper of the series, we analyze non-isothermal linear viscoelasticity, which is applicable in a range of small strains not only to all synthetic polymers and bio-polymers but also to some non-polymeric materials. Although the linear case seems to be well developed, there still are some reasons to implement a thermodynamic derivation of constitutive equations for solid-like, non-isothermal, linear viscoelasticity. The most important is the thermodynamic modeling of thermo-rheological complexity , i.e. different temperature dependences of relaxation parameters in various parts of relaxation spectrum. A special structure of interaction matrices is established for different physical mechanisms contributed to the normal relaxation modes. This structure seems to be in accord with observations, and creates a simple mathematical framework for both continuum and molecular theories of the thermo-rheological complex relaxation phenomena. Finally, a unified approach is briefly discussed that, in principle, allows combining both the long time (discrete) and short time (continuous) descriptions of relaxation behaviors for polymers in the rubbery and glassy regions.

The present series of three consecutive papers develops a general theory for linear and finite solid viscoelasticity. Because the most important object for nonlinear studies are rubber-like materials, the general approach is specified in a form convenient for solving problems important for many industries that involve rubber-like materials. General linear and nonlinear theories for non-isothermal deformations of viscoelastic solids are developed based on the quasi-linear approach of non-equilibrium thermodynamics. In this, the first paper of the series, we analyze non-isothermal linear viscoelasticity, which is applicable in a range of small strains not only to all synthetic polymers and bio-polymers but also to some non-polymeric materials. Although the linear case seems to be well developed, there still are some reasons to implement a thermodynamic derivation of constitutive equations for solid-like, non-isothermal, linear viscoelasticity. The most important is the thermodynamic modeling of thermo-rheological complexity , i.e. different temperature dependences of relaxation parameters in various parts of relaxation spectrum. A special structure of interaction matrices is established for different physical mechanisms contributed to the normal relaxation modes. This structure seems to be in accord with observations, and creates a simple mathematical framework for both continuum and molecular theories of the thermo-rheological complex relaxation phenomena. Finally, a unified approach is briefly discussed that, in principle, allows combining both the long time (discrete) and short time (continuous) descriptions of relaxation behaviors for polymers in the rubbery and glassy regions.

This paper presents a nontechnical review of current research efforts in the mathematical theory of viscoelastic materials. Recent results concerning the existence, uniqueness and regularity of solutions for initial value problems as well as for steady flow problems are discussed and a number of open problems are pointed out. Originator-supplied keywords include: Viscoelastic fluids, steady flows, nonlinear evolution problems, energy estimates, development of shocks, and singular kernels.

This paper, the last in the series, continues developing the nonlinear constitutive relations for non-isothermal, compressible, solid viscoelasticity. We initially discuss a single integral approach, more suitable for the glassy state of rubber-like materials, with basic functionals involved in the thermodynamic description for this type of viscoelasticity. Then we switch our attention to analyzing stability constraints, imposed on the general formulation of the nonlinear theory of solid viscoelasticity. Finally, we discuss specific (known from the literature or new) expressions for material functions that are involved in the constitutive formulations of both the rubber-like and glassy-like, complementary parts of the theory.

This paper, the last in the series, continues developing the nonlinear constitutive relations for non-isothermal, compressible, solid viscoelasticity. We initially discuss a single integral approach, more suitable for the glassy state of rubber-like materials, with basic functionals involved in the thermodynamic description for this type of viscoelasticity. Then we switch our attention to analyzing stability constraints, imposed on the general formulation of the nonlinear theory of solid viscoelasticity. Finally, we discuss specific (known from the literature or new) expressions for material functions that are involved in the constitutive formulations of both the rubber-like and glassy-like, complementary parts of the theory.

We have advanced our previous static theory of polymer entanglement involving an extended Cahn-Hilliard functional, to include time-dependent dynamics. We go beyond the Gaussian approximation, to the one-loop level, to compute the frequency dependent storage and loss moduli of the system. The three parameters in our theory are obtained by fitting to available experimental data on polystyrene melts of various chain lengths. This provides a physical representation of the parameters in terms of the chain length of the system. We discuss the importance of the various terms in our energy functional with respect to their contribution to the viscoelastic response of the polymeric system.

We study a system of partial differential equations with integer and fractional derivatives arising in the study of forced oscillatory motion of a viscoelastic rod. We propose a new approach considering a quotient of relations appearing in the constitutive equation instead the constitutive equation itself. Both, a rod and a body are assumed to have finite mass. The motion of a body is assumed to be translatory. Existence and uniqueness for the corresponding initial-boundary value problem is proved within the spaces of functions and distributions.

We present a comprehensive rheological study of a suspension of thermosensitive particles dispersed in water. The volume fraction of these particles can be adjusted by the temperature of the system in a continuous fashion. Due to the finite polydispersity of the particles (standard deviation: 17%), crystallization is suppressed and no fluid-crystal transition intervenes. Hence, the moduli $G'$ and $G"$ in the linear viscoelastic regime as well as the flow curves (shear stress $\sigma(\dot{\gamma})$ as the function of the shear rate $\dot{\gamma}$) could be measured in the fluid region up to the vicinity of the glass transition. Moreover, flow curves could be obtained over a range of shear rates of 8 orders of magnitude while $G'$ and $G"$ could be measured spanning over 9 orders of magnitude. Special emphasis has been laid on precise measurements down to the smallest shear rates/frequencies. It is demonstrated that mode-coupling theory generalized in the integration through transients framework provides a full description of the flow curves as well as the viscoelastic behavior of concentrated suspensions with a single set of well-defined parameters.

We consider nonlinear viscoelastic materials of differential type and for some special models we derive exact solutions of initial boundary value problems. These exact solutions are used to investigate the reasons of non-existence of global solutions for such equations.