On Sanders' energy-release rate integral and conservation laws in finite elastostatics

Sanders showed in 1960, within the framework of two-dimensional elasticity, that in any body a certain integral I around a closed curve containing a crack is path-independent. I is equal to the rate of release of potential energy of the body with respect to crack length. Here we first derive, in a simple way, Sanders' integral I for a loaded elastic body undergoing finite deformations and containing an arbitrary void. The strain energy density need not be homogeneous nor isotropic and there may be body forces. In the absence of body forces, for flat continua, and for special forms of the strain energy density, it is shown that I reduces to the well-known vector and scalar path-independent integrals often denoted by J, L, and M.     

On the Existence and Conditional Energetic Stability of Solitary Gravity-Capillary Surface Waves on Deep Water

This paper presents an existence and stability theory for gravity-capillary solitary waves on the surface of a body of water of infinite depth. Exploiting a classical variational principle, we prove the existence of a minimiser of the wave energy E{{\mathcal E}} subject to the constraint I=?2m{{\mathcal I}=\sqrt{2}\mu}, where I{{\mathcal I}} is the wave momentum and 0 < m << 1{0 < \mu \ll 1} . Since E{{\mathcal E}} and I{{\mathcal I}} are both conserved quantities a standard argument asserts the stability of the set D _μ of minimisers: solutions starting near D _μ remain close to D _μ in a suitably defined energy space over their interval of existence. In the applied mathematics literature solitary water waves of the present kind are modelled as solutions of the nonlinear Schr?dinger equation with cubic focussing nonlinearity. We show that the waves detected by our variational method converge (after an appropriate rescaling) to solutions of this model equation as mˉ 0{\mu \downarrow 0} .     

A thermo-mechanical continuum theory with internal length for cohesionless granular materials

This article continues Part I. Here the non-equilibrium responses of the constitutive variables t (Cauchy stress tensor), q (heat flux vector), h (equilibrated stress vector), Γ (flux term associated with the internal length ℓ), Π (production term associated with ℓ) and f (equilibrated intrinsic body force) as well as the Helmholtz free energy Ψ are postulated by use of a quasi-linear theory for three of four models deduced in Part I. In so doing, together with the equilibrium responses gained in Part I, a complete set of constitutive equations for the constitutive quantities for each model is obtained. The implemented models are applied to investigate typical isothermal steady granular shearing flows with incompressible grains, namely, simple plane shear flow, inclined gravity-driven flow and vertical channel-flow. The emphasis is on the models in which ℓ is considered a material constant (Model I) and an independent dynamic field quantity (Model III). Numerical results show that Model III is more appropriate than Model I since in the former model the effect of the motion of an individual grain can better be taken into account. Such a result is in particular significant for avalanches, since it verifies the existence of a thin layer immediately above the base of an avalanche, in which the grains are colliding strongly with one another, and provides a quantitative means to measure such a thin layer.     

Application of the material force concept to formation of discontinuities in soft sedimentary rocks

The possibility of rock joint formation from discontinuities which originate in poorly lithified sediments is presented. The mechanism for the development of these discontinuities along the maximum compressive stress T _max is discussed and quantitatively investigated. Micro-defects (holes, cavities) may exist in sediment due to non-ideal packing. It is shown that in certain situations, a physically warranted tendency to reduce the total energy of the system splits into two separate holes which move away relative to material particles (grains) along the T _max axis. The trace left by the moving hole is interpreted as a macro-discontinuity evolving with time into the closed mode I macro-crack. Some justifications for the mechanism are presented.     

Extended Lagrangian formalism and the corresponding energy relations

In this paper an extended Lagrangian formalism for the rheonomic systems with the nonstationary constraints is formulated, with the aim to examine more completely the energy relations for such systems in any generalized coordinates, which in this case always refer to some moving frame of reference. Introducing new quantities, which change according to the law τ_a=φ_a(t), it is demonstrated that these quantities determine the position of this moving reference frame with respect to an immobile one. In the transition to the generalized coordinates qⁱ they are taken as the additional generalized coordinates q^a=τ_a, whose dependence on time is given a priori. In this way the position of the considered mechanical system relative to this immobile frame of reference is determined completely.Based on this and using the corresponding d'Alembert–Lagrange's principle, an extended system of the Lagrangian equations is obtained. It is demonstrated that they give the same equations of motion qⁱ=qⁱ(t) as in the usual Lagrangian formulation, but substantially different energy relations. Namely, in this formulation two different types of the energy change law dE/dt and the corresponding conservation laws are obtained, which are more general than in the usual formulation. So, under certain conditions the energy conservation law has the form E=T+U+P=const, where the last term, so-called rheonomic potential expresses the influence of the nonstationary constraints.Afterwards, a detailed analysis of the obtained results and their connection with the usual formulation of mechanics are given. It is demonstrated that so formulated energy relations are in full accordance with the corresponding ones in the usual vector formulation, when they are expressed in terms of the rheonomic potential. Finally, the obtained results are illustrated by several simple, but characteristic examples.     

Brittle fracture: Variation of fracture toughness with constraint and crack curving under mode I conditions

The effect of constraint on brittle fracture of solids under predominantly elastic deformation and mode I loading conditions is studied. Using different cracked specimen geometry, the variation of constraint is achieved in this work. Fracture tests of polymethyl methacrylate were performed using single edge notch, compact tension and double cantilever beam specimens to cover a bread range of constraint. The test data demonstrate that the apparent fracture toughness of the material varies with the specimen geometry or the constraint level. Theory is developed using the critical stress (strain) as the fracture criterion to show that this variation can be interpreted using the critical stress intensity factorK _Cand a second parameterT orA ₃,whereT andA ₃are the amplitudes of the second and the third term in the Williams series solution, respectively. The implication of this constraint effect to the ASTM fracture toughness value, crack tip opening displacement fracture criterion and energy release rateG _Cis discussed. Using the same critical stress (strain) as the fracture criterion, the theory further predicts crack curving or instability under mode I loading conditions. Experimental data are presented and compared with the theory.     

Energy dissipation and contour integral characterizing fracture behavior of incremental plasticity

Jep -integral is derived for characterizing the frac- ture behavior of elastic-plastic materials. The J ep -integral differs from Rice’s J-integral in that the free energy density rather than the stress working density is employed to define energy-momentum tensor. The J ep -integral is proved to be path-dependent regardless of incremental plasticity and deformation plasticity. The J epintegral possesses clearly clear physical meaning: (1) the value J ep tip evaluated on the infinitely small contour surrounding the crack tip represents the crack tip energy dissipation; (2) when the global steadystate crack growth condition is approached, the value of J ep farss calculated along the boundary contour equals to the sum of crack tip dissipation and bulk dissipation of plastic zone. The theoretical results are verified by simulating mode I crack problems.     

Mixed mode fatigue crack growth and the strain energy density factor

The strain energy density factor S was first proposed by Sih for the prediction of the critical of the load and failure direction under monotonic, mixed mode loading condition. It seems a natural extension to apply the same concept to fatigue crack propagation. However, a close examination of the existing theory indicates that the Strain Energy Density Factor cannot logically account for the phenomena of the R-ratio effect and crack arrest. Thus, modification is necessary before the concept can be applied successfully for the prediction of mixed mode fatigue crack propagation.Based on the concept of hysteresis energy dissipation, an effective strain energy density factor range, ΔS_p,eff, is proposed for the correlation of fatigue crack growth data. ΔS_p,eff is consistent with the concept of crack closure. Experimental investigation indicates that it could predict the crack growth rates and trajectories.     

Transient growth in the asymptotic suction boundary layer

In the present experimental setup, the transient disturbance growth in a spatially invariant boundary layer flow, i.e., the asymptotic suction boundary layer (ASBL), has been investigated. The choice of the ASBL brings along several advantages compared with an ordinary spatially growing boundary layer. A unique feature of the ASBL is that the Reynolds number (Re) can be varied without changing the boundary layer thickness, which in turn allows for parameter variations not possible to carry out in traditional boundary layer flows. A spanwise array of discrete surface roughness elements was mounted on the surface to trigger modes with different spanwise wavenumbers (β). It is concluded that for each mode there exists a threshold roughness Reynolds number (Re _k ), below which no significant transient growth is present. The experimental data suggests that this threshold Re _k is both a function of β and Re. An interesting result is that the energy growth curves respond differently to a change in Re _k when caused by a change in roughness height k, implying that Re remains constant, compared with a change in the free-stream velocity U_￥U_\infty, which also affects the Re. The scaling of the energy growth curves both in level and the downstream direction is treated and appropriate scalings are found. The result shows a complex non-linear receptivity mechanism. Optimal perturbation theory, which has failed to predict the energy evolution in growing boundary layers, is tested for the ASBL and shows that it may satisfactorily predict the evolution of all transiently growing modes that are triggered by the roughness elements.     

Lower Bound for the Energy of Bloch Walls in Micromagnetics

We study a two-dimensional nonconvex and nonlocal energy in micromagnetics defined over S ²-valued vector fields. This energy depends on two small parameters, β and e{\varepsilon} , penalizing the divergence of the vector field and its vertical component, respectively. Our objective is to analyze the asymptotic regime b << e << 1{\beta \ll \varepsilon \ll 1} through the method of Γ-convergence. Finite energy configurations tend to become divergence-free and in-plane in the magnetic sample except in some small regions of typical width e{\varepsilon} (called Bloch walls) where the magnetization connects two directions on S ². We are interested in quantifying the limit energy of the transition layers in terms of the jump size between these directions. For one-dimensional transition layers, we show by Γ-convergence analysis that the exact line density of the energy is quadratic in the jump size. We expect the same behaviour for the two-dimensional model. In order to prove that, we investigate the concept of entropies. In the prototype case of a periodic strip, we establish a quadratic lower bound for the energy with a non-optimal constant. Then we introduce and study a special class of Lipschitz entropies and obtain lower bounds coinciding with the one-dimensional Γ-limit in some particular cases. Finally, we show that entropies are not appropriate in general for proving the expected sharp lower bound.     

Reconsideration on the role of the specific heat ratio in Arrhenius law applications

Arrhenius law implicates that only those molecules which possess the internal energy greater than the activation energy Ea can react. However, the internal energy will not be proportional to the gas temperature if the specific heat ratio y and the gas constant R vary during chemical reaction processes. The varying y may affect significantly the chemical reaction rate calculated with the Arrhenius law under the constant γ assumption, which has been widely accepted in detonation and combustion simulations for many years. In this paper, the roles of variable γ and R in Arrhenius law applications are reconsidered, and their effects on the chemical reaction rate are demonstrated by simulating one- dimensional C-J and two-dimensional cellular detonations. A new overall one-step detonation model with variable γ and R is proposed to improve the Arrhenius law. Numerical experiments demonstrate that this improved Arrhenius law works well in predicting detonation phenomena with the numerical results being in good agreement with experimental data.     

Heat transfer in the flow of a viscoelastic fluid over a stretching surface

An analysis is made of heat transfer in the boundary layer of a viscoelastic fluid flowing over a stretching surface. The velocity of the surface varies linearly with the distance x from a fixed point and the surface is held at a uniform temperature T _w higher than the temperature T _∞ of the ambient fluid. An exact analytical solution for the temperature distribution is found by solving the energy equation after taking into account strain energy stored in the fluid (due to its elastic property) and viscous dissipation. It is shown that the temperature profiles are nonsimilar in marked contrast with the case when these profiles are found to be similar in the absence of viscous dissipation and strain energy. It is also found that temperature at a point increases due to the combined influence of these two effects in comparison with its corresponding value in the absence of these two effects. A novel result of this analysis is that for small values of x, heat flows from the surface to the fluid while for moderate and large values of x, heat flows from the fluid to the surface even when T _w >T _∞. Temperature distribution and the surface heat flux are determined for various values of the Prandtl number P, the elastic parameter K ₁ and the viscous dissipation parameter a. Numerical solutions are also obtained through a fourth-order accurate compact finite difference scheme. Received on 14 October 1997     

Experimental study of thermoacoustic effects on a single plate Part II: Heat transfer

 Heat fluxes close to the edge of a heated solid plate aligned parallel to the axis of an acoustic standing wave were measured for drive ratios DR≡P _A/p _m of 1, 2 and 3. It was found that at the highest drive ratio (3), the resulting heat flux vector at the edge of the plate is directed into the plate, opposite to the direction of the heat flux imposed by the resistive heaters within the plate. This observation confirms the thermoacoustic effect previously detected in the visualized temperature fields and discussed in part I of this paper. Through the energy balance the magnitudes of the heat fluxes into the plate, caused by the thermoacoustic effect, were determined. The measured data are in good agreement with numerical and analytical predictions. Received on 18 August 1999     

On the initial stage of a point explosion in a radiating gas

A study is made of the initial stage of a point explosion in a radiating gray gas whose absorption coefficient is approximated by the dependenceK=x()e ^–n ,where is the density and e is the internal energy of the gas. It is shown that for n > —1/3 the initial stage of the process differs significantly from the solution of the problem in not only the classical adiabatic case [1, 2] but also in the case of a medium with nonlinear thermal conductivity [2–4]. The supply of energy to the medium at a point leads to instantaneous heating of the complete medium. The form of this heating is found analytically. The method of matched asymptotic expansions is used to investigate the behavior of the solution in the neighborhood of the center. It is found that for definite conditions at the center of the perturbed region there are formed a shock wave and a region of reverse flow of the gas.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 75–82, May–June, 1980.I should like to thank V. P. Korobeinikov for interest in the work and a helpful discussion of it.     

Measuring turbulence energy with PIV in a backward-facing step flow

Turbulence energy is estimated in a backward-facing step flow with three-component (3C, stereo) particle image velocimetry (PIV). Estimates of turbulence energy transport equation for convection, turbulence transport, turbulence production, viscous diffusion, and viscous dissipation in addition to Reynolds stresses are computed directly from PIV data. Almost all the turbulence energy terms in the backward-facing step case can be measured with 3C PIV, except the pressure-transport term, which is obtained by difference of the other turbulence energy terms. The effect of the velocity spatial sampling resolution in derivative estimations is investigated with four two-dimensional PIV measurement sets. This sampling resolution information is used to calibrate the turbulence energies estimated by 3C PIV measurements. The focus of this study is on the separated shear layer of the backward-facing step. The measurements with 3C PIV are carried out in a turbulent water flow at Reynolds number of about 15,000, based on the step height h and the inlet streamwise maximum mean velocity U₀. The expansion ratio (ER) is 1.5. Turbulence energy budget profiles in locations x/h=4, x/h=6, and x/h=10 are compared with DNS data of a turbulent flow. The shapes of profiles agree well with each other. Different ERs between the PIV case (1.5) and the DNS case (1.2) cause higher values for the turbulence energies measured by PIV than the energies by DNS when x/h=10 is approached. PIV results also show that the turbulence energy level in these experiments is generally higher than that of the DNS data.     

The effects of boundary to the ECRH and the energy absorption of ICRH

Electron-cyclotron resonant heating (ECRH) of Tokamak plasma is examined. When plasma is heated by waves, we must consider the distribution of incident wave energy toO andX modes as the wave is incident from vacuum to the surface of plasma as well as the absorption efficiency ofO mode andX mode. Numerical calculation shows that for small incident angle, the incident energy transfers principally intoO mode when the electric fieldE ⁱ of incident wave is parallel to the incident plane, therefore it is efficient to heat the plasma byO mode. WhenE ⁱ is perpendicular to the incident plane, the energy transfers principally intoX mode and heating the plasma byX mode is efficient. Ion-cyclotron resonant heating (ICRH) is also considered, the formula of the energy of ion-cyclotron wave absorbed by plasma is obtained.     

N-space collision model for rotation-translation energy exchange in DSMC collisions

A new discrete simulation Monte Carlo (DSMC) collision model for molecules possessing an integer number of classical degrees of freedom for molecular structure energy is proposed. The total molecular energy (translation plus molecular structure) is represented by a velocity in five-dimensional space. Each collision is an elastic N-sphere collision in N-space, where N= 3, 4, or 5. For N=5, there is a maximum chance of exchange of energy between the two components of velocity, which represent the rotation energy and the three components that represent the translational velocity. For N=3, there is no change in the rotation energy of each molecule, and for N=4, there is an intermediate chance that rotation and translation energy will be exchanged. The exchange probability ϕ can be set to give the desired rotational relaxation rate. To achieve any realistic viscosity μ(T), the N-space model must be coupled with a modified collision procedure known as ν-DSMC. The new model is shown to match the results of molecular dynamics calculations for the internal structure of a Mach 7 shock, with a saving of about 20% in CPU time compared to standard DSMC using the standard Borgnakke-Larsen exchange model.     

Energy release or absorption due to simultaneous expansion of many interacting nanoholes in elastic materials

The energy release or absorption due to simultaneous expansion of many interacting nanoholes in elastic materials under plane strain deformation is studied as influenced by the surface effect along rims of nanoholes. The M-integral classically used in macro mechanics with defects is extended to treat the problem with many interacting nanoholes. After some manipulations, the energy change due to the simultaneous expansion of many nanoholes represented by the M-integral is evaluated. Four different arrays of many nanoholes under a monotonically increasing tensile loading are considered. Attention is focused on the influence induced from the surface tension, the surface Lamé constants, and the interaction among many nanoholes on the M-integral. It is concluded that the surface tension yields significant influence on the M-integral, whereas the surface Lamé constants offer much smaller influence, which could be neglected with some relative errors less than 2%. It is found that, unlike those in macro mechanics with defects, the simultaneous expansion could either release energy (the positive value of the M-integral) or absorb energy (the negative value of the M-integral), depending on the loading levels. There is a neutral loading point, at which the M-integral transforms from a negative value to a positive value in all arrays of nanovoids under consideration. It is also found that the interaction among multiple nanoholes influences the value of the neutral loading point significantly because the mutual influence induced from both the interacting effect and the surface effect yields a quite different feature from those induced from the interacting effect only. That is, the surface effect always inhibits the influence of the interacting effect on the M-integral.     

A study on the forced vibrations of a class of nonlinear systems,with application to the Duffing equation Part II: Numerical treatment

Summary Numerical results on the wave form, the amplitude and the phase of the steady state solutions of the Duffing equation are obtained here. The steady state vibrations are studied:i) by a procedure of direct numerical integration based on the Runge-Kutta method;ii) by calculating the approximate solution of the equation which is obtained, as a particular case, from the theory developed in Part I. The results obtained by the two methods are compared and tested by the results of other Authors. From the comparisons it appears that the theory of Part I gives satisfactory results also in the cases of strong nonlinearity of the system.

---

Sommario Nel presente lavoro sono ricavati ed illustrati alcuni risultati numerici relativi alle forme d'onda, le ampiezze e le fasi delle soluzioni a regime dell'equazione di Duffing. I moti vibratori a regime vengono ricercati:a) mediante un procedimento di integrazione numerica diretta, basato sul metodo di Runge-Kutta;b) adottando la soluzione approssimata dell'equazione stessa, che si ottiene come caso particolare della teoria sviluppata nella Parte I. I risultati ottenuti vengono confrontati fra di loro e con i risultati di altri Autori. Dai confronti appare che la teoria sviluppata nella Parte I dà risultati soddisfacenti anche quando la nonlinearità del sistema è molto elevata.

---

---

This paper is an elaboration of a paper [10] presented at the 2nd National Congress of AIMETA (Naples, 6 October, 1974).     

Correlation between viscoelastic properties of letterpress printing inks and film splitting parameters

Summary Dynamic viscoelastic properties of letterpress inks are related to the energy requirements of film splitting. This is carried out using a modified I. G.T. printability tester and aWeissenberg Rheogoniometer. Also, the constantsb andf of theFetsko-Walker equation are reinterpreted in terms of a viscoelastic response of ink during printing.

---

Zusammenfassung Die dynamisch-viskoelastischen Eigenschaften von Schnellpressen-Druckfarben (letter press inks) stehen in Beziehung zum Energiebedarf beim Spalten des Films. Dieser Zusammenhang wird mit Hilfe eines modifizierten I.G.T.-Druckfähigkeits-Testers und einesWeissenberg-Rheogoniometers untersucht. Anschließend werden die Konstantenb undf derFetsko-Walker-Gleichung unter Berücksichtigung des viskoelastischen Verhaltens der Farben während des Druckvorgangs neu interpretiert.

---

---

With 11 figures and 1 table