Creep Kernels Research Articles

Influence of creep of the material of inclusions on the stress concentration in the body

We solve the problem of determination of the stress-strain state of an elastic plate with thin viscoelastic inclusion. The problem is reduced to a singular integrodifferential equation whose exact solution is obtained for the case of elliptic inclusion. Several creep kernels used to describe the rheological properties of the material of the inclusion are analyzed. The stresses acting in the inclusion and their concentration in the matrix in the vicinity of the inclusion are determined.

Critical time of a long multilayer viscoelastic shell

The buckling in stability of a long multilayer linearly viscoelastic shell, composed of different materials and loaded with a uniformly distributed external pressure of given intensity, is investigated. By neglecting the influence of fastening of its end faces, the initial problem is reduced to an analysis of the loss of load-carrying capacity of a ring of unit width separated from the shell. The new problem is solved by using a mixed-type variational method, allowing for the geometric nonlinearity, together with the Rayleigh-Ritz method. The creep kernels are taken exponential with equal indices of creep. As an example, a three-layer ring with a structure symmetric about its midsurface is considered, and the effect of its physicomechanical and geometrical parameters, as well as of wave formation, on the critical time of buckling in stability of the ring is determined. It is found that, by selecting appropriate materials, more efficient multilayer shell-type structural members can be created.

Application of time–stress superposition to nonlinear creep of polyamide 66 filled with nanoparticles of various sizes

The long-term tensile creep of polyamide 66 and its nanocomposites filled with 1 vol.% TiO 2 nanoparticles 21 and 300 nm in diameter is studied. It is assumed that the dominant mechanisms of creep deformation are of viscoelastic nature, while the contribution of plastic strains is not essential in the stress (< 0.6 of the ultimate stress) and time (about 100 hours) ranges considered. The creep isochrones obtained show that the materials exhibit a nonlinear viscoelastic behaviour and the degree of nonlinearity is reduced significantly by incorporation of the nanoparticles. The evolution of viscoelastic strains is less pronounced for the nanocomposite filled with smaller nanoparticles. Smooth master curves are constructed by applying the time–stress superposition (TSS). The Boltzmann–Volterra hereditary theory is used for the creep modeling. The nonlinearity of viscoelastic behaviour is taken into account by using the TSS principles and introducing a stress reduction function into an exponential creep kernel. The master curves are employed to predict the creep for time periods more than 60 times exceeding the test time. A comparison of relaxation spectra of the polymers shows that the incorporation of nanoparticles restricts the mobility of polymer chains. The smaller the nanoparticles, the greater the enhancement in the creep resistance. An empirical approach and a three-parameter law is also used for creep approximation. The efficiency of two models is evaluated by comparing their prediction validity.

Optimizing selection of a model of the viscoelasticity of synthetic fibres

A method was obtained for numerical solution of the problem concerning the resolvent of the determining integral relaxation and creep equations for the case of nonlinearity of the viscoelastic properties of synthetic fibres. Numerical rotation of the integral relaxation and creep kernels and the corresponding normalized functions in the form of the normalized arctangent of the logarithm of the reduced time allows drawing a conclusion concerning the validity of the choice of a given function as the basis for the model of the viscoelastic properties. The optimality criteria for selection of the mathematical model of the viscoelasticity of synthetic fibres are reported.

Equivalent rheological and restoring force models for predicting the harmonic response of elastomer specimens

This article presents two theoretical approaches that simulate the visco-elastic behaviour of elastomer specimens. The first approach, based on an equivalent rheological model, provides a dynamic modulus extracted from a Volterra development of the visco-elastic constitutive law using either relaxation or creep kernels. The second approach establishes a restoring force model based on a first-order differential equation that relates the restoring force to the deflection, the forcing frequency and deflection amplitude dependence being taken into account by the envelope curves of the force–deflection loop. The two models proposed are first applied to an elastomer cylinder mount made of a small quantity of carbon black filler and then to elastomer plates made of a large quantity of black filler. The cylinder and plates specimens are subjected to traction–compression and shear tests, respectively. In order to compare the two approaches, the dynamic modulus of the second approach is extracted by applying classical formulae to the force–deflection loop obtained with the restoring force model. Moreover experimental investigations permit comparing the simulated and measured dynamic modulus and validating the two theoretical approaches proposed.

Prediction of Shear-Creep-Induced Variation in Parameters of Clayey-Soil Masses over Time

Theoretical bases are presented for quantitative prediction of the shape-altering deformations of the cross-sectional perimeters of arbitrary homogeneous massives over time due to the creep of cohesive soil in a gravitational field. Differential equations in partial derivatives are derived as applies both to the most frequently used rheologic models, and also for cases where arbitrary creep kernels are used. An example is cited for the analytical solution of a differential equation with the existence of initial conditions alone, and finite relationships are obtained for analysis of the pattern of variation in the transverse perimeter of a soil mass of simple configuration over time.

Comparison and minimum theorems in the time domain for viscoelasticity and other convolutive initial-boundary value problems with applications to random inhomogeneous materials

The new variational theorems presented in Huet (J. Mech. Phys. Solids 49 (2001) 675–706) for viscoelasticity problems with applications to heterogeneous materials are supplemented by considering functionals involving the rates of the relaxation and creep kernels and the creep kernel itself. As in Huet (J. Mech. Phys. Solids 49 (2001) 675–706), the results are based on a Continuum Thermodynamics approach, from which extended Clapeyron formulae are derived, leading to the use of auxiliary problems involving data continuations. Functional requirements are discussed with more details. Extended comparison theorems and a new set of minimum theorems are derived. The latter supplement the ones obtained by Huet (Mech. Mater. 31 (1999) 787–829) for other functionals under stronger admissibility conditions. The results are applied to the derivation of a further set of order relationships in the time domain for the overall properties of viscoelastic heterogeneous media supplementing the ones obtained in Huet (J. Mech. Phys. Solids 49 (2001) 675–706) for the boundary-condition effects in bodies smaller than the representative volume and in Huet (Mech. Mater. 31 (1999) 787–829) for the size-effects. This provides the viscoelastic counterpart of the hierarchies first derived by Huet (J. Mech. Phys. Solids 38 (1990) 813–841; Engng Fract. Mech. 58 (5–6) (1997) 459–556) for the elastic case without and with unloaded defects that are recovered as particular cases. A viscoelastic extension of the classical Hill (J. Mech. Phys. Solids 11 (1963) 357–372) modification theorem is also derived for a special class of kernel modifications. The obtained results apply to other kinds of time-dependent physical properties like dielectric and piezoelectric relaxation and thermal or mass diffusion problems, under appropriate choice of the variables respecting the needed positive definiteness requirements provided by the positiveness of the dissipated power and the entropy production rate.

On the stability of the solutions of linear stochastic integro-differential equations encountered in elasticity and viscoelasticity problems

The asymptotic stability, almost sure also in the mean square, of a viscoelastic system subjected to a load in the form of a random stationary broadband ergodic process is investigated. The behaviour of this system is described by integro-differential equations with stochastic parameters. The stability is considered with respect to the perturbation of the initial conditions. The governing relation is taken in an integral form with a creep (or relaxation) kernel of convolution type, which satisfies the condition of limited creep of the material. Using the fundamental solution of the corresponding deterministic integro-differential equation and its maximum Lyapunov exponent, the sufficient condition for stability of the zero solution of the initial equation or, which is the same thing, the equilibrium position of the viscoelastic system, is obtained.

Creep and damage accumulation in orthotropic composites under cyclic loading

Experimental results and theoretical prediction of the response of glassfiber-reinforced polyester under quasi-static, static (creep), and cyclic (fatigue) loading are presented. The nonlinear strain component at static loading and the strain amplitude rate at cyclic off-axis loading of an orthotropic composite are shown to follow the associated flow rule with a single-parameter quadratic potential function. The influence of fatigue damage on deformation is considerable due to the reduction in the elastic modulus of the composite and is apparently negligible with respect to its effect on the parameters of the creep kernel.

APPLICATION OF FRACTIONAL OPERATORS TO THE ANALYSIS OF DAMPED VIBRATIONS OF VISCOELASTIC SINGLE-MASS SYSTEMS

Free damped vibrations of two hereditarily elastic oscillators, the hereditary properties of which are described by the Boltzmann-Volterra relationships with the weakly singular Rzhanitsyn kernel taken as the creep kernel (the first model), or as the relaxation kernel (the second model) are considered. These integral relationships are equivalent to the differential relationships involving infinite sums of various order time derivatives of excitation (force) or response (displacement) depending on whether the Rzhanitsyn kernel is used for the creep kernel or relaxation kernel. The problem is solved by the Laplace transform method. When passing from image to pre-image one is led to find the roots of an algebraic (characteristics) equation with fractional exponents. A method for solving such equations is proposed which allows one to investigate the roots' behaviour in a wide range of single-mass system parameters. It is shown that, if the Rzhanitsyn kernel is the creep kernel in the Volterra equations, then the characteristic equation does not possess real roots, but has two complex conjugate roots; i.e., the test single-mass system subjected to the impulse excitation does not pass into an aperiodic regime. On the contrary, the oscillator model with the Rzhanitsyn relaxation kernel may be both in vibrating motion and in the aperiodic regime, depending on the intervals over which the relaxation times for the given model vary, as well as on the order of fractional power and the ratio of the relaxed modulus (rubbery modulus) to the non-relaxed modulus (glassy modulus). However, contrary to the standard linear solid model with ordinary time derivatives for which the dimensions of the domain of aperiodicity as well as its existence are governed only by the magnitude of the ratio of the relaxed modulus to the non-relaxed modulus, for the model with the Rzhanitsyn relaxation kernel all of the above-listed factors essentially depend in addition on an order of a fractional operator parameter. The main characteristics of the vibratory and aperiodic motions of the single-mass system as functions of the relaxation time or creep time, which are equivalent to the temperature dependences, are constructed and analyzed for both models.

Applied rheological equations obtained on the basis of the linear-fractional creep kernel

It is proposed to represent Zaretskii's linear-fractional kernel as the sum of difference and nondifference functions and to use it in hereditary-creep equations. New applied creep equations are obtained for constant and variable loads and stress relaxation in soils.

The use of fractional exponential creep kernels for long-term behavior of laterally loaded piles in permafrost

A theory for long-term prediction of laterally loaded piles in a hereditary-elastic medium is presented. A mixed formulation approach with the aid of calculus of variations ensures continuity of pile displacements, bending moments and soil reactions. Finally, an asymptotic estimate of the pile behavior for very long time (t→∞) is obtained using a fractional exponential operator derived from pile tests.

Transformation of quasilinear equations of viscoelasticity satisfying the nonlinear principle of strain summation

1. A method was developed for transforming quasilinear equations of viscoelasticity constructed on the basis of the strain summation principle. 2. It was shown that a simplified variant of the equations containing two creep kernels can be used to describe experimental data. 3. The structure of the relations that were obtained shows that, as in the linear theory of viscoelasticity, equations having resolvent creep operators are the simplest to transform.

Relaxation and propagation of pulses in fluids

Mathematical bases for the diagnostics of relaxing fluids in probing by sound pulses based on recording these pulse shapes are developed. Analysis of the change in the pulse shape makes it possible to establish the type of relaxation and to determine its parameters: the relaxation time, the dispersion jump in the velocity, and others. To calculate the pulse shape, the creep kernels for the three main types of relaxation in fluids are found: Kneser relaxation, resonance relaxation, and relaxation associated with diffusive transfer. The shape of the pulse (initially square) near the front is calculated for these types of relaxation.

Stability of circular plates from ageing viscoelastic material

Stability conditions are obtained for circular plates of an inhomogeneouslyageing viscoelastic material for an arbitrary creep kernel and different methods of plate support. Stability in an infinite time interval corresponds to determination of the Lyapunov stability, and in a finite interval, Chetayev stability.

On the stability of inhomogeneously ageing viscoelastic plates

A method of investigation is proposed and conditions are set up for the stability of viscoelastic inhomogeneously ageing plates of arbitrary shape with a common creep kernel. The form of the stability conditions is found as a function of the surface forces. The stability problem is examined numerically in a finite time interval. The paper touches on the investigations in /1–3/. (See the bibliography of research on the stability of homogeneous viscoelastic systems in /1–5/, for example.)

One class of creep kernels in aging materials

One class of creep kernels in aging materials

An algorithm for calculating the frequency dependence of the complex dielectric permittivity and the complex compliance

1. The algorithm developed for a numerical transform of the kernel in the Boltzmann — Volterra integral equation from the time domain to the frequency domain is suitable for calculating the complex dielectric permittivity from the polarization current or for calculating the complex compliance from the creep kernel.

Determining creep kernels from results of short-time tests

Determining creep kernels from results of short-time tests

Viscoelastic characteristics of the interaction between structures and soils

The shear coefficients of a body relative to the soil and the shear creep and relaxation kernels are determined from the data of creep and relaxation tests.