Second-order Time Derivatives Research Articles

Linear growth of a liquid droplet divided from its vapour by a “soap bubble”-like fluid interface

The theory proposed in [1,3] is particularized to describe the spherically symmetric growth, in the neighbourhood of an equilibrium state far from the critical temperature, of a liquid incompressible droplet surrounded by its vapour when the interface between the phases behaves as a fluid membrane which resembles a soap bubble. A free moving boundary problem for an integro-differential equation of parabolic type is deduced in which the second-order time derivative of the radius R(t) of the droplet appears, together with a volume source term which depends on the history of R.

Constrained generalized mechanics. The second-order case

The Dirac formalism for constrained systems is developed for systems described by a Lagrangian depending on up to second-order time derivatives of the generalized co-ordinates (accelerations). It turns out that for a Lagrangian of this kind differing by a total time derivative from a Lagrangian depending on only up to first-order time-derivatives of the generalized co-ordinates (velocities), both classical mechanics at the Lagrangian level are the same; at the Hamiltonian level the two classical mechanics differe conceptually even when the solutions to both sets of Hamiltonian equations of motion are the same.

On Analytical Evaluation of Gear Dynamic Factors Based on Rigid Body Dynamics

This paper proposes an initial step to rationalize the dynamic factor calculation and bring it under the control of the laws of mechanics. The theory is straight forward. The concept of mathematical scaling is utilized to simplify the system dynamics’ formulation. The rigid body dynamics accounts for the gear dynamic tooth loads resulting from the prescribed transmission error of each gear step—including the artificial ones. The latter converts a lumped-mass-elastic system into a rigid-bodies-transmission-error system subjected to the solution of the rigid body system dynamics according to Newton’s law. The entire concept of the solution has been implemented into a FORTRAN program approximately 600 statements in length. The results obtained through computer simulation of various test cases demonstrate the potential and effectiveness of the proposed concept. Contrary to the current practice of grossly ignoring the inertial and system effects, this paper has taken all these important factors into account. The transmission-error-induced acceleration is approximated by the second-order time derivative of one of the cubic spline curve-fitting methods. The approach is cost effective and numerically satisfactory. The model can be further improved to reduce the extent of basic assumptions, or to increase the number of conditional constraints without losing economical attractiveness.

On the Hamilton-Jacobi theory with derivatives of higher order

The Hamilton-Jacobi theory is extended to include the case of Lagrangians involving higher-order time derivatives of the degrees of freedom. In particular, Jacobi’s theorem is generalized and proved in this case. By way of application, the Hamilton-Jacobi equation for a particle constrained to turn about a translating centre is set and solved, starting from an associated Lagrangian containing the second-order time derivatives of the particle position co-ordinates.

Evaluation of a uniaxial, nonlinear, second-order differential overstress model for rate-dependence, creep and relaxation

The properties of a second-order differential equation linear in the first and second time derivatives of stress and strain but nonlinear in the overstress are investigated. The overstress is the difference between the stress and an equilibrium stress which corresponds to the limiting response in infinitely slow loading. The extension to second-order time derivatives in this nonlinear constitutive equations follows similar methods in linear viscoelasticity. The nonlinear overstress dependence of the coefficient functions is, however, maintained.The behavior of the second-order equation is compared with that of the first-order equation. Phaseplane analysis and Poincare-linearization are used to show that the first and second-order equations exhibit the same long time limits for constant stress(strain)-rate loading. Numerical integration involving strain(stress)-rates differing by four orders of magnitude and strain(stress)-rate jumps demonstrates that the limits are rapidly attained and that the transient behavior of the second-order equation can be different from that of the first-order equation. Strong and mild oscillatory behavior as well as monotonic behavior is possible in strain control. Stress control always results in monotonic behavior. For creep and relaxation the inelastic strain is a functional of the overstress history and the large time limit corresponds to that of the first-order equation.Unless very strong emphasis is placed on modeling unusual transient behavior, this second-order model does not appear to be more advantageous than the first-order model.

Theorems on Amplification of Fluctuations and their Biological Implications

Three theorems on amplification of fluctuations at nonequilibrium are proposed: (1) Amplification of a fluctuation is possible within a closed system. (2) The second-order time derivative of Boltzmann's H-quantity may contain terms of negative value. (3) An H-theorem for open systems.It is pointed out in the paper that the fact that microscopic transition rates are not all identical constitutes the basis of the theorems proposed. And this nonidenticalness is also a fundamental reason why phenomena defying the thermodynamic law in an apparent way (such as life) exist in the course of evolution. It is also pointed out in the paper that at states far from thermal equilibrium it is “selection” instead of “averaging” that works in the statistics of the quantum states.Finally it is speculated that certain key biomolecules such as ATP are situated at long lifetime excited states, thus at states far from thermal equilibrium. The biomolecules are large fluctuations produced or amplified during a relaxation process starting from an extreme initial state (even further from equilibrium), and stabilized by solar radiation in the present day. Again, selection takes the place of an averaging which is artificial and which Ievels out the whole set of transition rates.

Plastic Behavior of Predeformed Ice Crystals

AbstractConventional studies of plastic deformation have been complicated by the simultaneous variation of dislocation velocity and dislocation density during the tests. In the present study this difficulty was avoided by carrying out deformation tests at low stress levels on samples which had been predeformed to relatively higher stresses prior to the measurements. Creep, mechanical after effect, constant strain-rate, and stress relaxation tests were carried out as a function of measurement stress or strain-rate, temperature, and predeformation level. The results were analysed in terms of a linear stress-dependent dislocation velocity in order to determine whether that simple behavior is able to account for the macroscopic deformation behavior of ice crystals. This report is a brief summary of the results; a complete discussion is to be published elsewhere. The principal results of the study are as follows: 1.The observed behavior was particularly simple for predeformed samples. The creep strain showed a nearly linear increase with time, without the large positive curvature characteristic of conventional tests. No stress maxima were observed in the constant strain-rate tests as have been found in previous studies of non-predeformed samples. The (complete) stress relaxation curve was virtually identical in shape to the inverted constant strain-rate curve.2.The steady-state creep and constant strain-rate behavior could be described to good approximation in terms of the motion of a constant density of dislocations moving with the same linear stress-dependent velocities as have been observed directly by others. The strain-rates were linear in stress and the estimated dislocation densities (4 to 16 × 10 cm-2) varied with the magnitude of the predeformation level in a manner consistent with previous observa-tions. The strain-rates or stress levels are exponential in 1/T with an activation energy of 0.6 eV, which is approximately equal to the activation energy reported for the motion of dislocations and for the mechanical relaxation time in internal friction studies.3.A small transient creep behavior, and a small but measurable mechanical after-effect with the same (≈ 3 min) time constant were present. These effects, as well as the non-exponential behavior of the constant strain-rate and stress-relaxation stress—strain results, could be accounted for by including a second, anelastic, component in a deformation model corresponding to a restoring force in addition to the linear viscous drag force on dislocations. This leads to a differential equation which is linear in stress and strain, but involves both first-and second-order time derivatives.4.The solutions of this differential equation describe the observed mechanical response well, and provide a general internal consistency check for the model.5.A quantitative fit of the experimental test results to the dislocation model gives values of 3-4 for the ratios of total dislocation density to the recoverable component and values of 7-8 dyn/cm2 for the restoring force constant for the recoverable dislocations. The restoring force constants and recoverable dislocation densities were, within experimental error, found to be independent of temperature, measurement stress and strain-rate, and predeformation level. Values found for the parameters for creep and mechanical after-effect tests were equal within experimental error to the values found from constant strain-rate and stress-relaxation tests carried out on the same sample. If the recoverable dislocation component is identified as bowed-out dislocation segments whose ends are fixed, the restoring force could be accounted for by the elastic line tension of dislocation segments of lengths of about 8 × 10-3 cm.No feature of the experimental results was inconsistent with this dislocation model, and the results of the study are all in agreement with the theory proposed by Weertman in which the dislocation drag force is very large because of the stress-induced ordering of water molecules in the stress field of the moving dislocation. This paper is to be submitted for publication in full in another journal.

Plastic Behavior of Predeformed Ice Crystals

Abstract Conventional studies of plastic deformation have been complicated by the simultaneous variation of dislocation velocity and dislocation density during the tests. In the present study this difficulty was avoided by carrying out deformation tests at low stress levels on samples which had been predeformed to relatively higher stresses prior to the measurements. Creep, mechanical after effect, constant strain-rate, and stress relaxation tests were carried out as a function of measurement stress or strain-rate, temperature, and predeformation level. The results were analysed in terms of a linear stress-dependent dislocation velocity in order to determine whether that simple behavior is able to account for the macroscopic deformation behavior of ice crystals. This report is a brief summary of the results; a complete discussion is to be published elsewhere. The principal results of the study are as follows: 1. The observed behavior was particularly simple for predeformed samples. The creep strain showed a nearly linear increase with time, without the large positive curvature characteristic of conventional tests. No stress maxima were observed in the constant strain-rate tests as have been found in previous studies of non-predeformed samples. The (complete) stress relaxation curve was virtually identical in shape to the inverted constant strain-rate curve. 2. The steady-state creep and constant strain-rate behavior could be described to good approximation in terms of the motion of a constant density of dislocations moving with the same linear stress-dependent velocities as have been observed directly by others. The strain-rates were linear in stress and the estimated dislocation densities (4 to 16 × 10 cm-2) varied with the magnitude of the predeformation level in a manner consistent with previous observa-tions. The strain-rates or stress levels are exponential in 1/T with an activation energy of 0.6 eV, which is approximately equal to the activation energy reported for the motion of dislocations and for the mechanical relaxation time in internal friction studies. 3. A small transient creep behavior, and a small but measurable mechanical after-effect with the same (≈ 3 min) time constant were present. These effects, as well as the non-exponential behavior of the constant strain-rate and stress-relaxation stress—strain results, could be accounted for by including a second, anelastic, component in a deformation model corresponding to a restoring force in addition to the linear viscous drag force on dislocations. This leads to a differential equation which is linear in stress and strain, but involves both first-and second-order time derivatives. 4. The solutions of this differential equation describe the observed mechanical response well, and provide a general internal consistency check for the model. 5. A quantitative fit of the experimental test results to the dislocation model gives values of 3-4 for the ratios of total dislocation density to the recoverable component and values of 7-8 dyn/cm2 for the restoring force constant for the recoverable dislocations. The restoring force constants and recoverable dislocation densities were, within experimental error, found to be independent of temperature, measurement stress and strain-rate, and predeformation level. Values found for the parameters for creep and mechanical after-effect tests were equal within experimental error to the values found from constant strain-rate and stress-relaxation tests carried out on the same sample. If the recoverable dislocation component is identified as bowed-out dislocation segments whose ends are fixed, the restoring force could be accounted for by the elastic line tension of dislocation segments of lengths of about 8 × 10-3 cm. No feature of the experimental results was inconsistent with this dislocation model, and the results of the study are all in agreement with the theory proposed by Weertman in which the dislocation drag force is very large because of the stress-induced ordering of water molecules in the stress field of the moving dislocation. This paper is to be submitted for publication in full in another journal.

Antireciprocity and Memory in the Statistical Approach to Irreversible Thermodynamics

The macroscopic state of a large, closed system is specified by the ensemble averages αj of a set of even dynamical variables Aj, together with the averages vj of the quantities iLAj, obtained by operating on the Aj with the self-adjoint Liouville operator L. Phenomenological equations for the time rates of change, α̇j and v̇j, are derived by a technique due to Zwanzig, in which one operates on Liouville's equation with a projection operator which projects out the part which is ``relevant'' to the phenomenological description employed. These phenomenological equations are shown to exhibit the Onsager—Casimir reciprocity relations, including antisymmetry relations whose derivation is found to require a slight modification of Zwanzig's mathematical assumptions. Since vj=α̇j, these equations also show that irreversible thermodynamics can be extended to the case where second-order time derivatives appear representing memory effects, as well as nonlinear terms in the αj, provided the equations are still required to be linear in the vj. Furthermore, Onsager's equations are obtained while allowing the phenomenological matrices to be functions of the variables αj, and not merely of constants of the motion. This serves to generalize and extend Zwanzig's earlier treatment.

Casimir Coefficients and Minimum Entropy Production

The rate equations obeyed by scalar relaxation parameters in a liquid are extended to include inertial terms in the form of second-order time derivatives, and the relations containing these terms are then written formally as first-order equations by treating first-order time derivatives as additional parameters. The resulting equations are interpreted thermodynamically as phenomenological relations containing anti-symmetric Casimir coefficients, and this interpretation leads, with application of the Onsager-Casimir reciprocity theorem, to an additional set of phenomenological equations which reduces to those used in an earlier relaxation theory when inertial effects are neglected. In this way, earlier formulas for the bulk viscosity and high-frequency bulk modulus are recovered unchanged. It is also shown why Prigogine's minimum entropy production theorem should no longer hold when one considers inertial effects.