Nonlinear Viscoelastic Theory Research Articles

Nonlinear theory of viscoelasticity with symmetrical influence functions

The characteristic equations of a physically nonlinear viscoelastic medium are investigated. A fairly simple theory of nonlinear viscoelasticity with symmetrical influence functions is constructed. This has potential applications in relation to the determination of the strength of structural elements made of materials with rheonomic properties.

Combined tension‐torsion creep experiments on polycarbonate in the nonlinear range

AbstractConstant stress creep and recovery experiments were performed in the nonlinear range under combined tension and torsion on a tubular specimen of polycarbonate. Nonlinear viscoelasticity theory was employed to describe the test results. It was found that the first three orders of stress terms and a power function of time adequately described the experimental results. The modified superposition principle satisfactorily predicted the recovery following creep. Recovery was almost complete in several days following two hours of creep.

The response of viscoelastic materials to small deformations

The main features of the theory of linear viscoelasticity are reviewed. The time-dependent behavior of any viscoelastic material can be described in terms of a complex viscosity η ( s) which must possess a certain specific analytic character, considered as a function of the complex variable s. It is shown that the complex viscosity representing the dynamic response of viscoelastic fluids to oscillatory motion superimposed upon steady shear can not be evaluated in terms of a spectrum of relaxation times and can not be compared on such basis to the complex viscosity measured in the vicinity of equilibrium. Any non-linear theory of viscoelasticity should be compatible with the linear theory. This criterion is discussed; it is shown that a non-linear constitutive equation of the Rivlin—Ericksen type involving only a finite number of derivatives of the strain rate should not be regarded as a proper constitutive equation.

Contribution to the nonlinear theory of viscoelasticity

The author presents a generalization of certain results obtained in [3] relating to the region of rigid behavior of plastics for which the nonlinearity of the viscoelastic properties is important even in the area of small strains. Certain questions connected with the derivation of the basic equations of viscoelasticity are considered for small strains. Formulas are obtained for the resolvents of kernels of arbitrary order. The general equations of the "principal quadratic theory of viscoelasticity" are derived.

On the kinematics, nonequilibrium thermodynamics, and rheological relationships in the nonlinear theory of viscoelasticity: PMM vol. 32, no. 1, 1968, pp. 70–94

On the kinematics, nonequilibrium thermodynamics, and rheological relationships in the nonlinear theory of viscoelasticity: PMM vol. 32, no. 1, 1968, pp. 70–94

Elastomer processing and application of rheological fundamentals

AbstractElastomer processing operations are discussed and classified as unit operations. The theory of nonlinear viscoelasticity is applied to processing unvulcanized amorphous rubber and the significance of the maximum relaxation time τm is emphasized.

An engineering theory of nonlinear viscoelasticity with applications

Relatively simple stress-strain equations are developed for nonlinear, initially isotropic, viscoelastic materials with constant temperature, and are shown to agree quite well with the actual behavior of unfilled and filled polymers. First, equations for a general state of strain are derived by extending Biot's linear thermodynamic theory to a restricted class of nonlinear behavior. Only two time-dependent relaxation functions appear, and these are just the familiar linear viscoelastic relaxation moduli. The general relations are then specialized to stress-strain equations applicable to uniaxial loading, radial and axial deformation problems of long, circular cylinders, and simple shear with lateral strain. The uniaxial equation yields, as a special case, a nonlinear relaxation modulus that has the familiar form for polymers wherein strain and time-dependence appear as separate factors. Results of experiments on highly-filled polymers are compared with theory for cases in which very small shear vibrations are superposed on static lateral compression and on static shear. Strong nonlinearities are observed with static strains of only a few percent.

The small-parameter method, and the theory of nonlinear viscoelasticity

The small-parameter method, and the theory of nonlinear viscoelasticity

Some fundamental problems of polymer mechanics

A survey of the present state of the science of polymer mechanics is presented with particular reference to the theory of linear and nonlinear viscoelasticity of homogeneous and heterogeneous polymers. Various approaches adopted in studies of this subject are discussed; these include methods based on the theory of creep of metals, on a statistical model composed of viscoelastic elements, and on the representation of functionals which, in a quite general sense, are sufficiently close to being linear.

An elementary theory of nonlinear viscoelasticity

AbstractA first‐order, nonlinear, one‐dimensional theory of viscoelasticity is constructed by use of an elementary perturbation technique. It is based on applying some simple hypotheses of nonlinearity to single nonlinear Maxwell and Voigt models. The response of the Maxwell model to a sinusoidally applied strain is governed by a nonlinear complex modulus whose nonlinear part is given as a function of applied strain and strain rate as well as external frequency and internal or model parameters. This is a nonlinear relaxation effect. A nonlinear creep effect is obtained by describing the nonlinear response of the Voigt model to a sinusoidally applied stress in terms of a nonlinear complex compliance.

General stress-strain relations in non-linear theory of viscoelasticity for large deformations

General phenomenoligical stress-strain relations in non-linear theory of visco-elasticity for large deformations have been presented. In the first place, according to V. V. Novozhilov 1 we express the generalized equilibrium equation for large deformations in the Lagrange representation, and we apply the generalized Hamilton's principle to the equation of energy conservation, which denotes that the sum of the elastic energy and the dissipative energy is equal to the work done by the body force and the surface on the substance; so that we obtain the required general stress-strain relations in comparison with the above two equations. On the condition that the elastic potential is a function only of the strain, and the dissipation function is a function of the rate of strain and of strain; such a substance is reduced to the Voigt material necessarily, and the stresses which act on the substance are given by the sum of elastic- and viscous stresses, and the stress-strain relations are reduced to the so-called Lagrangian form. If elongations, shears and angles of rotation are small and also the strains and rates of strain are sufficiently small, the stress-strain relations are expressed by a linear Voigt model constituting a Hookian spring in parallel with a Newtonian dashpot. Non-linearity in the theory is classified into two groups i. e. the geometrical non-linearity and the physical non-linearity. The former is introduced into the theory through the definition of the generalized strain and of the generalized stress and through the equilibrium equation for large deformation, and the latter through the general stress-strain relations. The main result of this paper is that the general stress-strain relations in viscoelasticity are deduced necessarily from the physically appropriate assumptions.

非線形粘弾性の現象論

非線形粘弾性の現象論

Phenomenological Theory of Non-Linear Viscoelasticity

Two three-dimensional models are introduced in order to describe the non-linear behavior and the three-dimensional cross effects occurring in the stress-deformation relations of viscoelastic materials. These models are the analogues of the so-called Maxwell and Voigt models in the classical one­ dimensional case. Non-linearities in the stress-deformation relations are classified into three categories: Elastic, viscous and geometrical ones, which are connected with the energy stored and dissipative mechanisms and the three-dimensional character of the materials, respectively. It is pointed out that we can actually separate these three non-linearities by observing the structural viscosity and the Weissenberg effect. The viscoelasticity of the so-called weakly coupled network model is briefly discussed as an example of the three-dimensional Maxwell model.