Maxwell Type Research Articles

On the explicit differential operator formulae for the Oldroyd rate type constitutive equation

We have established the conditions under which the implicit Oldroyd type constitutive equation is equivalant to a differential type constitutive equation. Physical interpretations of these conditions are given. Finally some explicit forms for Maxwell type and Oldroyd type constitutive equations are given as illustrations.

Numerical experiments and theoretical analysis of the flow of an elastic liquid of the upper-convected Maxwell type in the presence of geometrical discontinuities

The upper-convected Maxwell model of an elastic liquid is solved by means of the finite element method, using a penalty function approach. It is shown that the algorithm produces oscillations in the stresses in the presence of geometrical discontinuities, for increasing Deborah number De. The cause of this problem is explored and an indication of how to solve this problem is given.

Stability analysis of a tangentially loaded column with a maxwell type viscoelastic foundation

The stability of a uniform cantilever column supported by a Maxwell type viscoelastic foundation and subjected to a constant tangential force is investigated. Stability conditions are obtained for the entire range of system parameters through an application of Routh-Hurwitz criteria. Unlike the case of conservative loading, the Maxwell foundation is shown to produce a stabilizing effect on this nonconservative problem. Furthermore, an optimum combination of foundation parameters exist to yield the maximum flutter load. An approximate analysis is also presented through an application of the Galerkin's method. It is shown that a two-term approximation may not be adequate to yield meaning-ful results in a certain range of system parameters.

Physical charged sectors in quantum electrodynamics. II. The charge operator

AbstractThe properties of Maxwell type equations and the related problem of approximation of physical charged states are studied in the framework of axiomatic gauge quantum field theory. The charge operator and the physical charged sectors of quantum electrodynamics are constructed under a set of hypotheses which can be verified in perturbation theory and are supported by some nonperturbative arguments.

Some energetic properties of smooth solutions in rate-type viscoelasticity

A free energy function compatible with the second law of thermodynamics is constructed for the semilinear rate-type viscoelasticity. Under physically acceptable conditions this energy function is a positive and convex function of stress and strain. It is shown that for a class of one dimensional initial-boundary value problems, the total energy at any time is bounded by the sum of the total energy of the initial data and the energy supplied to the body by nonvanishing body forces. A Maxwell type viscosity approach to non-linear elasticity for an isolated body is also discussed. The stability in total energy and uniqueness of the smooth solutions of some initial-boundary value problems is discussed for the one dimensional case as well as for the three dimensional case of small deformations.

Silicon Melts During Pulsed Ruby Laser Annealing

ABSTRACTThe lattice temperature of silicon was measured during pulsed ruby laser annealing (-20 ns pulse, spot diameter ≥ 5 mm) using a classical time of flight method to determine the velocity distribution of Si atoms evaporated from the hot surface. The maximum of this Maxwell type distribution was used to calculate the temperature of the Si surface. The resulting lattice temperatures vary between 1200 and 2500 K for energy densities between 1.0 and 2.0 J/cm2 , i.e., Si is molten for energy densities ≥ 1.4 J/cm2 . This result clearly supports the strictly thermal annealing model [1] and contradicts the non-thermal-equilibrium model [2] as well as Raman measurements [3].

Growth of a dent on an isothermal fluid filament in uniaxial extension a spinnability model in Lagrangian coordinate

Growth of an arbitrarily shaped dent made on a filament of power law and Maxwell type fluids being uniaxially extended was analysed in the Lagrangian coordinate assuming inertia, surface tension, gravity and air drag forces to be negligible in comparison to the rheological force.

Transfer properties of the slowly adapting stretch receptor of the crayfish abdomen.

The slowly adapting stretch receptor in the abdomen of freshwater crayfish (Astacus fluviatilis) was investigated to determine its properties under dynamic conditions. An in situ preparation was used; the necessary dissection did not involve the receptor organ or its immediate surroundings. Sinusoidal variations in the angle of flexion in the joint to which the receptor organ was connected, were generated by a feed-back controlled stretcher. Nerve spiked recorded from the axon of the receptor neurone and information about angle of flexion in the joint obtained by position transducers, were fed into a computer. Fourier transforms were performed on both input and output data to determine the amplitude of the 0. and 1. harmonic together with the phase of the 1. harmonic. The receptor organ was investigated for linearity up to 1.5 degrees input amplitude, and proved to be surprisingly linear within this range. In addition, the transfer function of the receptor organ was determined by stimulating it with small-amplitude sinusoidals with different frequencies. With a steady flexion of 35-40 degrees in the joint, the gain of the receptor organ increased 5-6 times when the modulation frequency of the input signal was increased from 0.1 to 5 cycles/s. A maximum in gain was constantly found at about 5 cycles/s, with a rapid fall towards 0 when the modulation frequency was increased further. A change in phase lead from positive (leading output) to negative with change in sign about 1 cycle/s was also found. These results resemble the results found by investigators of isolated preparations. A "hold" property is probably a part of the overall property of the receptor organ together with an element of Maxwell type. An element of the form h(s) = ksn with n approximately 0.45 is also a part of the transfer function of the receptor organ, although the physiological parallel to this element is uncertain.

Soluble Boltzmann equations for internal state and Maxwell models

We consider a class of scalar nonlinear Boltzmann equations describing the evolution of a microcanonical ensemble in which sub-systems exchange internal energy ‘randomly’ in binary interactions. In the continuous variable version these models can equally be interpreted as Boltzmann equations for Maxwell type molecules in arbitrary dimensionality. We construct general solutions in the form of a Fourier series; the expansion coefficients (Sonine or Meixner moments) satisfy the same recursive system of coupled equations as the ordinary moments, and can be solved sequentially.

Deformation mechanism of polymer spherulite by linear isothermal viscoelastic theory

A mathematical representation for the deformation mechanism of polymer spherulite is proposed through a linear isothermal viscoelastic theory to investigate the relaxation mechanism of strains and stresses within the spherulite. This description may be represented by the extension of the method proposed by Chen and Wang concerning an elastic problem. Displacements within the spherulite are obtained as solutions of equations of motion omitting an inertia term by assuming two potential functions of complex quantities. Viscoelastic stiffnesses of lamellae in the equations are obtained by applying the correspondence principle to a composite model, in which the mechanical behavior of amorphous phase and grain boundary region is assumed to be dependent upon a three-element viscoelastic model of Maxwell type. As to the results, both the relaxations of strains and stresses within the spherulite are found to be dependent upon the relaxation of viscoelastic constants in bulk.

偏光透過によるアルカリハライド単結晶の応力緩和測定

Stress relaxation of NaCl single crystal was measured by the method using transmission of polarized light. It became possible by this method to analyze the time variation of stress in the direction perpendicular to the compression axis. It was found that the mechanism of relaxation of the principal stress [σ2] in short time range was different from that in long time range. The initial process of stress relaxation was strongly influenced by the strain rate of compression. This influence was found to be strong but continue for a short time when the strain rate was high, and weak but for a long time when it was low. It was observed that σ2 decayed following the Maxwell type viscoelastic model at the early part of the initial short time range of the stress relaxation, and thereafter σ2 increased gradually by the pinning of moving dislocations and finally attained at a constant value.

The completeness of Maxwell's stress function representation

Any four times continuously differentiable, self-equilibrated stress field is shown to admit a stress function representation of the Maxwell's type. This representation is even valid for only twice differentiable stress fields provided that an extended interpretation of the operator curl curl is applied. A counterexample shows that the result is sharp in the sense that it cannot be extended to cover stress fields which are only once continuously differentiable.

Nonlinear acoustodiagnostics of viscoelastic media

One-dimensional finite longitudinal pulses propagating according to the Voigt. Maxwell, standard, and Volterra-Rabotnov hereditary type linear and nonlinear mathematical models are considered. For the linear case the exact and asymptotic solutions are derived with the aid of Laplace integral transforms. Further, making use of an iterative method, asymptotic solutions are obtained which take into consideration weak nonlinear effects. The distortion of the finite pulse's form due to viscosity and nonlinearity of the medium is analysed. Conclusions are made about how experimental data on this distortion might be used for solving the inverse problem of finding the proper mathematical model of the given nonlinear viscoelastic medium and the physical constants which describe it.

Stability Analysis of WONDY (A Hydrocode Based on the Artificial Viscosity Method of von Neumann and Richtmyer) for a Special Case of Maxwell's Law

The artification viscosity method of von Neumann and Richtmyer was originally designed and analyzed for stability in the case when the material was an ideal gas. Recently a hydrocode (WONDY) based on the von Neumann-Richymyer scheme was used in calculating wave progagation problems in materials obeying a form of Maxwell's material law; signs of an unstable difference scheme appeared. A stability analysis shows that the timestep restrictions required for stability in certain cases can be more stringent for material laws of the Maxwell type than they are for material laws of the ideal gas type.

Stability analysis of WONDY (a hydrocode based on the artificial viscosity method of von Neumann and Richtmyer) for a special case of Maxwell’s law

The artificial viscosity method of von Neumann and Richtmyer was originally designed and analyzed for stability in the case when the material was an ideal gas. Recently a hydrocode (WONDY) based on the von Neumann-Richtmyer scheme was used in calculating wave propagation problems in materials obeying a form of Maxwell’s material law; signs of an unstable difference scheme appeared. A stability analysis shows that the timestep restrictions required for stability in certain cases can be more stringent for material laws of the Maxwell type than they are for material laws of the ideal gas type.

Linearized thermoviscoelasticity with high temperature variations and related periodic problems

In the first part of this paper we develop a linearization of the equations of the thermoviscoelastic field in the case of great temperature variations. The possibility of uncoupling the heat equation from the motion equation is discussed in Section 3. After recalling some results on duality and virtual work principle we then study the motion equation with temperature as data, i.e. a given function of time and space variables. More precisely we study existence, uniqueness, regularity and asymptotic stability of a T-periodic (stress) solution of the motion equation in the dynamical case (Section 7) and in the quasi-static case (Section 8), when the temperature field is T-periodic in time and with a constitutive equation of Maxwell type where the stiffness and viscosity matrix are temperature dependent and thus are T-periodic functions of time. In the proof of the theorems we use frequently an inequality of monotony which means that the material is dissipative on a period. This inequality hold if the stiffness is a slowly varying function of time (the temperature has a little effect on the stiffness), on the other hand, fortunately, there is no condition on the viscosity.

Maxwell and Wien processes as special cases of the generalized Feller diffusion process

It is shown that Maxwell and Wien type processes are special cases of the generalized Feller diffusion process. In particular, both are obtained for specific parameter values from the delta function initial condition solution of the generalized Feller equation. For specific values of the independent time variable, one obtains the well-known distribution laws of Maxwell and Wien of statistical physics.

Wave propagation in a viscoelastic rod with temperature-dependent properties

In this paper the unidirectional wave propagation in a rod made of viscoelastic material of the Maxwell type is considered. The rod is subjected to nonuniform temperature gradients and its mechanical properties are assumed to be temperature-dependent. A perturbation technique incorporated with the usual Laplace transform method is employed to obtain the analytical solution in series form. The strong temperature-history effect on the attenuation of the wavefront is demonstrated in simple examples for the case where a stress discontinuity due to constant velocity impact propagates along the rod. Subject Classification: 40.22; 35.50.

Cross-hatching: A consequence of differential deformation of a viscous solid

The hypothesis, proposed by Probstein and Gold, that cross‐hatching results from the differential deformation of an inelastic material at its surface, has been used in this analysis. Wall pressure and wall shear stress perturbations leading to the deformation were calculated with the linearized small perturbation theory. The supersonic boundary layer was composed of an inviscid outer layer and a viscous sublayer. Both laminar and turbulent mean velocity and Mach number profiles were used in the calculation. In contrast to the work of Probstein and Gold the viscous sublayer was included and wall pressure perturbations were also considered. The essential result is the prediction of the pattern cant angle, which was not possible with the inviscid analysis. The theory qualitatively confirms the experimentally observed features of cross‐hatching, similarly seen by Probstein and Gold. For instance, for cross‐hatching to occur, the boundary layer must be supersonic and turbulent, whilst the inelastic body material must be of the Maxwell type. Theory and experiment also show that the pattern is charcterized by a cant angle equal to the local Mach angle, by a wavelength inversely proportional to the static pressure and directly proportional to the viscosity of the material.

Dynamics and stability of a gas cavity in generalized viscoelastic liquids

A general theory is developed to determine the dynamic characteristics and stability of a gas cavity situated in viscoelastic liquids whose rheological state is described by the generalized model of the Kelvin or Maxwell type. The characteristic function of the bubble-liquid system is obtained by the first-order perturbation method, including the effects of surface tension, vapor pressure, and thermodynamic behavior of the gas confined in the cavity. The zeros of the characteristic function determine the types of bubble growth, bound on unbound. The Hurwitz (or Routh) stability is employed to determine the conditions for the onset of incipient cavitation. The applications of the theory are demonstrated in some liquids having simpler shear-stress-shear-strain relationship.