Determination of mechanical response of non-linear viscoelastic solids based on frechet expansion

This paper deals with the determination of mechanical response of a non-linear viscoelastic solid based on a Fréchet expansion. In the first portion, a modified superposition principle, suggested earlier by various authors, is arrived at from a slightly different point of view. The effect of such a principle on the kernel functions and a geometric interpretation are discussed. In the later portion, the tests required for the experimental determination of the kernel functions are discussed. It is seen that besides the tests suggested by Onaran and Findley for the two-dimensional case, three additional tests are needed for the three-dimensional case to compute the response under constant combined stress.     

The strain energy density failure criterion applied to notched elastic solids

Application of the strain energy density failure criterion is made to plane notch problems, where the crack now becomes a special case of a more generalized approach to failure. The specific case considered is that of the plane elliptical cavity under remote tension and compression. Both failure loads and fracture trajectories are discussed. It is shown that an additional characteristic dimension provides satisfactory agreement of the theory with available data. Finally, known characteristics of fracture trajectories from a notch tip are shown to be predicted for unstable fracture conditions.     

A class of linear viscoelastic models based on Bessel functions

In this paper we investigate a general class of linear viscoelastic models whose creep and relaxation memory functions are expressed in Laplace domain by suitable ratios of modified Bessel functions of contiguous order. In time domain these functions are shown to be expressed by Dirichlet series (that is infinite Prony series). It follows that the corresponding creep compliance and relaxation modulus turn out to be characterized by infinite discrete spectra of retardation and relaxation time respectively. As a matter of fact, we get a class of viscoelastic models depending on a real parameter \(\nu > -1\). Such models exhibit rheological properties akin to those of a fractional Maxwell model (of order 1/2) for short times and of a standard Maxwell model for long times.     

Prediction of forming limit diagrams based on shear failure criterion

Maximum shear stress theory, also called the ‘Third Strength Theory’, is a classical theory used to predict the failure of common metal, but it cannot be used directly to predict sheet metal failure due to anisotropy and the loading path. Therefore, this paper proposes a maximum shear stress calculating method, which has been named “shear failure criterion” for the purpose of this paper. In order to validate the shear failure criterion, a general program was developed, and two typical materials, steel, and aluminum alloy, were used to study the new shear failure criterion in this study. The two materials were modeled by advanced constitutive models, including Hill1948 and Yld2000-2d yield functions and several types of isotropic hardening models. Experimental validation has indicated the accuracy of predicted FLD using shear failure criterion, which is able to provide a new alternative method to numerically predict FLD.     

Effective properties of a linear viscoelastic composite

The relaxation moduli of a composite are determined. The relaxation of its components is described by various few-parameter kernels: Mittag-Leffler functions of different orders and Rzhanitsyn kernel. It is assumed that the composite components are made of model materials with volume relaxation. The Laplace transform and fractional rational approximation are used to develop an algorithm for reducing the relaxation functions of the composite to one class (series of decreasing exponents or exponents of fractional order). The relaxation moduli of a unidirectionally reinforced composite are determined as an example     

A method of analysis for compressible viscoelastic solids

Most of the solutions in the development of methods of viscoelastic stress analysis have dealt with incompressible materials or materials with restrictions in dilatation. It is influenced in part by the increasing complexity due to the additional operator which represents the viscoelastic characteristics in dilatation. A simple procedure of solution is suggested in this paper which shows that a certain class of problems for compressible materials can be solved with the similar simplicity as the analysis of corresponding incompressible solids. Examples are given for problems with spherical boundaries.     

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).     

Penny-shaped crack in a linear viscoelastic medium under shear

The problem of a linear viscoelastic body, containing a penny-shaped crack subjected to the shear parallel to the edge of the crack is considered in this paper. Closed form expressions for the displacements over the surface of the crack, the shear components in the plane of the crack and the stress intensity factors are determined. The various expressions are then specialized for two particular linear viscoelastic materials and the effect of viscoelasticity, wherever possible, is pointed out.     

Finite amplitude waves superimposed on pseudoplanar motions for Mooney-Rivlin viscoelastic solids

This paper deals exclusively with finite amplitude motions in viscoelastic materials for which the stress is the sum of a part corresponding to the classical Mooney-Rivlin incompressible isotropic elastic solid and of a dissipative part corresponding to the classical viscous incompressible fluid. Of particular interest is a finite pseudoplanar elliptical motion which is an exact solution of the equations of motion. Superposed on this motion is a finite shearing motion. An explicit exact solution is presented. It is seen that the basic pseudoplanar motion is stable with respect to the finite superposed shearing motion. Particular exact solutions are obtained for the classical neo-Hookean solid and also for the classical Navier-Stokes equations. Finally, it is noted that parallel results may be obtained for a basic pseudoplanar hyperbolic motion.     

Influence of small polymer additives on the behavior of a linear viscoelastic fluid

A system of equations is obtained, describing a dilute polymer solution in an incompressible linear viscoelastic fluid as a nonlinear viscoelastic medium. As an example, motion with a specified velocity gradient is analyzed in the linear approximation. An expression is given for the dynamic modulus of the system in terms of the dynamic modulus of the most viscoelastic fluid containing the macro-molecules and in terms of the polymer concentration.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 3–12, November–December, 1976.     

Dispersion of linear gravity waves on a viscoelastic fluid in an horizontal canal

A characteristic equation is derived that describes the spatial decay of linear surface gravity waves on Maxwell fluids. Except at small frequencies, the derived dispersion relation is different from the temporal decay dispersion relation which is normally studied within fluid mechanics. The implications for waves on viscous Newtonian fluids using the two different dispersion relations is briefly discussed. The wave number is measured experimentally as function of the frequency in a horizontal canal. Seven Newtonian fluids and four viscoelastic liquids with constant viscosity have been used in the experiments. The spatial decay theory for Newtonian fluids fits well to the experimental data. The model and experiments are used to determine limits for the Maxwell fluid time numbers for the four viscoelastic liquids. As a result of low viscosity it was not possible within this study to obtain these time numbers from oscillatory experiments. Therefore, a comparison of surface gravity wave experiments with theory is applicable as a method to evaluate memory times of low viscosity viscoelastic fluids.     

Prediction of stress in a linear viscoelastic solid strained while cooling

A simplethermoelastic method is proposed, and justified, for predicting stresses arising during cooling of a linearviscoelastic solid. It is equivalent to representing the material by an array of spring-switch thermoelastic elements. The final stress resulting from an increment of strain is calculated using isochronal modulus data applying to the temperature at which the strain was applied, modified to accommodate temperature dependence of the limiting moduli. The method is exact for a material whose relaxation times and limiting moduli scale uniformly with change in temperature, with time-temperature shift factora _T obeying the Arrhenius equation, cooled such that reciprocal absolute temperature is linear in time. For other cooling sequences it is useful as an approximation. In particular, it assists the computational prediction of stresses arising during cooling in polymer processes.     

Response of viscoelastic solids to the action of transient thermal pulses

L'vov University. Translated from Prikladnaya Mekhanika, Vol. 25, No. 10, pp. 75–84, October, 1989.     

On periodic flow of non linear viscoelastic liquids

Summary After brief reference to energetical principles in order to find suitable constitutive equations for non-linear viscoelastic fluids, the behaviour of stress components is examined, for a simple shear motion with sinusoidal time time varying displacements of a liquid with an oscillating fading memory.

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Sommario Dopo aver richiamato brevemente un possibile procedimento per la determinazione della equazione costitutiva per liquidi viscoelastici nel caso di grandi deformazioni, si esamina il comportamento delle componenti del tensore degli sforzi nel caso di un moto periodico a filetti rettilinei e paralleli, supposto che il liquido abbia una memoria debole e oscillante nel tempo.

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Research supported by C.N.R., Gruppo nazionale per la fisica matematica.     

On the large deflections of linear viscoelastic beams

The quasi-static response and the stored and dissipated energies due to large deflections of a slender inextensible beam made of a linear viscoelastic material and subjected to a time-dependent inclined concentrated load at the free end are investigated. The beam cross-section is considered prismatic, the self-weight is disregarded and the material is initially stress free. The set of four first-order non-linear partial integro-differential equations obtained from geometrical compatibility, equilibrium of forces and moments, and linear viscoelastic constitutive relation is numerically solved using a one-parameter shooting method combined with a fourth-order Runge-Kutta algorithm. An analytical expression is derived to divide the energy supplied by the external load into conserved and dissipated parts. For the case study presented, a three-parameter solid linear viscoelastic constitutive model is employed and a step load is applied. The variables are made non-dimensional, so four parameters govern the problem: the ratio between the final and initial relaxation moduli, the load magnitude, the angle of inclination and the unloading time. A finite-element model is also performed to compare and validate the analytical and numerical formulations. Results are presented for encastré curvature and tip displacement versus time, geometrical configuration, load versus tip displacement, total work done by the external force, stored and dissipated energies versus time, energy per unit length along arc length for three times and versus time for two beam sections.     

Effects of a bent structure on the linear viscoelastic response of diluted carbon nanotube suspensions

Commonly isolated carbon nanotubes in suspension have been modelled as a perfectly straight structure. Nevertheless, single-wall carbon nanotubes (SWNTs) contain naturally side-wall defects and, in consequence, natural bent configurations. Hence, a semi-flexile filament model with a natural bent configuration was proposed to represent physically the SWNT structure. This continuous model was discretized as a non-freely jointed multi-bead–rod system with a natural bent configuration. Using a Brownian dynamics algorithm the dynamical mechanical contribution to the linear viscoelastic response of naturally bent SWNTs in dilute suspension was simulated. The dynamics of such system shows the apparition of new relaxation processes at intermediate frequencies characterized mainly by the activation of a mild elasticity. Storage modulus evolution at those intermediate frequencies strongly depends on the flexibility of the system, given by the rigidity constant of the bending potential and the number of constitutive rods.