The exponential asymptotic solution of differential equation

In this paper, the exponential asymptotic solution (E.A.S.) of differential equation is discussed. Firstly, E.A.S. of the second-order differential equation is studied and the orthogonal conditions of the uniformly valid E.A.S. are found out. Next, E.A.S. in matched asymptotic method is discussed. Finally, some examples are given.     

Interaction of continuous-spectrum waves with each other and with discrete-spectrum waves

The nonlinear problem of the evolution of an initial perturbation in Couette flow is solved in the quadratic approximation and it is shown that the energy of the initial perturbation is transmitted to the main flow so that its profile is somewhat modified. The evolution of the initial perturbation in a fluid with a very simple model flow profile which, in addition to continuous-spectrum waves, also admits the existence of a single neutral mode of the discrete spectrum is then investigated. It is shown that as a result of the linear resonant interaction of the discrete-spectrum and continuous-spectrum waves disturbances that grow linearly with time may be formed. A flow that does not contain exponentially growing modes will be unstable with respect to certain initial disturbances; this instability is called algebraic [6, 7]. A physical interpretation of this effect is given. From this interpretation it is clear that algebraic instability is possible in a fluid with flow profiles of a more general type, in which there are neutral or weakly damped discrete-spectrum modes having a critical layer.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 116–123, July–August, 1989.The author is grateful to G. I. Barenblatt, S. Ya. Gertsenshtein, M. A. Mironov, S. A. Rybak, O. S. Ryzhov, and E. D. Terent'ev for their interest and useful comments.     

各向同性弹性损伤的双标量描述

损伤状态的描述是损伤力学中仍未完善解决的基本问题．我们旨在对此问题就最简单的一种情形——各向同性弹性损伤，进行较为全面的研究．首先，指出了古典各向同性损伤理论中，基于应变等效假设，用单个标量损伤变量描述损伤状态的局限性．然后，建立了一个用两个标量损伤变量描述的各向同性弹性损伤模型．此模型解除了古典理论的局限，能完全描述各向同性弹性损伤，并且得到本文数值实验的验证．最后，将本文模型与现有细观力学结果连接，给出了宏细观损伤变量之间的关系，使得细观量可以通过宏观量来反映，建立了一个用细观损伤材料常数描述细观缺陷特征的损伤本构模型     

Exact solutions for stationary responses of several classes of nonlinear systems to parametric and/or external white noise excitations

The exact solutions for stationary responses of one class of the second order and three classes of higher order nonlinear systems to parametric and/or external white noise excitations are constructed by using Fokker-Planck-Kolmogorov equation approach. The conditions for the existence and uniqueness and the behavior of the solutions are discussed. All the systems under consideration are characterized by the dependence of nonconservative forces on the first integrals of the corresponding conservative systems and are called generalized-energy-dependent (G.E.D.) systems. It is shown taht for each of the four classes of G.E.D. nonlinear stochastic systems there is a family of non-G.E.D. systems which are equivalent to the G.E.D. system in the sense of having identical stationary solution. The way to find the equivalent stochastic systems for a given G.E.D. system is indicated and as an example, the equivalent stochastic systems for the second order G.E.D. nonlinear stochastic system are given. It is pointed out and illustrated with example that the exact stationary solutions for many non-G.E.D. nonlinear stochastic systems may be found by searching the equivalent G.E.D. systems.Project Supported by The National Natural Science Foundation of China. Accepted by XVIIth International Congress of Theoretical and Applied Mechanics.     

ANALYSIS OF FINANCIAL DERIVATIVES BY MECHANICAL METHOD (Ⅰ)—BASIC EQUATION OF PRICE OF INDEX FUTURES

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ANALYSIS OF FINANCIAL DERIVATIVES BY MECHANICAL METHOD (Ⅰ)-BASIC EQUATION OF PRICE OF INDEX FUTURES

Similar to the method of continuum mechanics, the variation of the price of index futures is viewed to be continuous and regular. According to the characteristic of index futures, a basic equation of price of index futures was established. It is a differential equation, its solution shows that the relation between time and price forms a logarithmic circle. If the time is thought of as the probability of its corresponding price, then such a relation is perfectly coincided with the main assumption of the famous formula of option pricing, based on statistical theory, established by Black and Scholes, winner of 1997 Nobel’ prize on economy. In that formula, the probability of price of basic assets (they stand for index futures here) is assummed to be a logarithmic normal distribution. This agreement shows that the same result may be obtained by two analytic methods with different bases. However, the result, given by assumption by Black-Scholes, is derived from the solution of the differential equation.     

A frontal method based solution of the quasi-three-dimensional finite element model for interconnected aquifer systems. Steady and unsteady equations,with the E.N.S. method for fluid mass balance evaluation

The quasi-three-dimensional equations controlling the groundwater flow in heterogeneous and interconnected aquifer systems are discretized by finite elements, considering also the aquifer branching. A new method for fluid mass balance evaluation based on the equivalent nodal source (E.N.S.) concept allows one to express the balance in conservative terms, and interpret finite element equations as nodal balance equations. The solution of the system is based on the frontal method. Use of substructures limits the frontal increase in correspondence to the aquifer branching. In the steady state, the frontal method is integrated with an iterative solution technique to eliminate the frontal increase caused by the presence of aquitards. It converges very rapidly, using a forcing technique with an automatic parameter definition. In the unsteady case the same scope is achieved using a predictor-corrector procedure which employs the Crank-Nicolson method in the corrector phase. This very stable procedure permits use of fairly long time-steps and concerns the case of source terms depending on piezometry (problem of interaction between water table and river). This method has been tested with several fairly complex cases.     

Shakedown analyses for rolling and sliding contact problems

There is a range of problems where repeated rolling and sliding contact occurs over a half space of an elastic–perfectly plastic material. For such problems shakedown and limit analysis provide significant advantages over other forms of analysis when a global understanding of deformation behaviour is required. In this paper, a recently developed numerical upper bound method, the Linear Matching Method (LMM), for shakedown analyses is applied to the solution of a problem previously considered by Ponter et al. [Ponter, A.R.S., Hearle, A.D., Johnson, K.L., 1985. J. Mech. Phys. Solids 33 (4), 339–362] for a moving Hertzian contact, with sliding friction. This semi-analytic solution is an upper bound based on certain specific kinematic assumptions. We show that the Ponter, Hearle and Johnson solution is a reasonable approximate solution for a circular contact area but is less accurate for an elliptic contact area. For an elliptic contact area LLM solutions converge to the line contact solution. The effect of the non-coincidence of the direction of travel and slide is also investigated.     

Microscopic and macroscopic instabilities in finitely strained fiber-reinforced elastomers

The present work is a detailed study of the connections between microstructural instabilities and their macroscopic manifestations — as captured through the effective properties — in finitely strained fiber-reinforced elastomers, subjected to finite, plane-strain deformations normal to the fiber direction. The work, which is a complement to a previous and analogous investigation by the same authors on porous elastomers, (Michel et al., 2007), uses the linear comparison, second-order homogenization (S.O.H.) technique, initially developed for random media, to study the onset of failure in periodic fiber-reinforced elastomers and to compare the results to more accurate finite element method (F.E.M.) calculations. The influence of different fiber distributions (random and periodic), initial fiber volume fraction, matrix constitutive law and fiber cross-section on the microscopic buckling (for periodic microgeometries) and macroscopic loss of ellipticity (for all microgeometries) is investigated in detail. In addition, constraints to the principal solution due to fiber/matrix interface decohesion, matrix cavitation and fiber contact are also addressed. It is found that both microscopic and macroscopic instabilities can occur for periodic microstructures, due to a symmetry breaking in the periodic arrangement of the fibers. On the other hand, no instabilities are found for the case of random microstructures with circular section fibers, while only macroscopic instabilities are found for the case of elliptical section fibers, due to a symmetry breaking in their orientation.     

The performance of free rolling rigid and flexible wheels on sand

Experimental results are presented for a towed 6–16 smooth tyre and the same size rigid steel wheel in three types of sands covering a wide range of particle size distribution, two dry and one submerged sands. Their performance was compared at high and low tyre inflation pressures, two vertical loads and a wide range of soil compaction for each sand. The sand performance prediction number, Ns, proposed by the U.S. Army Engineer Waterways Experiment Station (W.E.S.) was then applied to compare with the measured results for the tyre. It was found that in all the three sands the coefficient of rolling resistance was substantially underestimated by the W.E.S. method. However Ns = 10–20 was found to be very important overall criterion for towed tyres on sand. The correlation between the skid and the fractional sinkage of the rigid wheel and the tyre was also examined.     

Energy-optimal time-dependent regimes of viscous incompressible fluid flow

The hydrodynamic equations of a viscous incompressible fluid are modified for axisymmetric flows in a pipe of time-varying radius. A new exact time-dependent solution of these equations which generalizes the well-known classic steady-state Hagen–Poiseuille solution for flow in a pipe of constant radius (independent of time) is obtained. It is shown that the law of time variation in the pipe radius can be determined from the condition of the minimum work done to pump a given fluid volume through such a pipe during the radius variation cycle period. A generalization of the optimal branching pipeline in which, instead of the Poiseuille law, its modification based on the use of the exact solution corresponding to the time-dependent M-shaped regime is employed is suggested. It is shown that the hydraulic resistance can be reduced over a certain range of the parameters of the time-dependent flow regime as compared with the steady-state pipe flow regime. The conclusion obtained can be used for the development of the hydrodynamic basis for simulating the optimal hydrodynamic blood flow regime.     

Thermal stresses in plane elasticity: Numerical solutions based only on harmonic functions

Summary A theoretical treatment is outlined allowing solution of thermal stress problems in plane elasticity by using only numerical methods suited to solving — in 2 D — the Laplace equation. Only one type of element matrix (supposing for the sake of simplicity F.E.M.s are used) and only one mesh would thus be required, both for the determination of the thermal field and of the displacement/stress field. The numerical solutions required in the plane domain of interest entail, consequently, only one variable per node in place of two. Even if numerous unit solutions are required in order to impose arbitrary boundary conditions, this reduction of nodal variables allows to spend less computation time in solving linear systems, at least for problems of a certain extent.

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Sommario Si delinea una tecnica che permette di risolvere i problemi di coazioni termiche in elasticità piana facendo uso solo di metodi numerici atti a risolvere l'equazione di Laplace. Un solo tipo di matrice degli elementi (nel caso si usi una formulazione a E. F.) e una sola reticolazione sono pertanto richiesti tanto per il problema termico come per quello elastico. Ne consegue altresì il vantaggio che le soluzioni numeriche richieste nel dominio di interesse cornportano una sola variabile per nodo, anzichè due; anche tenuto conto che sono richieste numerose soluzioni unitarie per poter imporre condizioni al contorno comunque definite, ne deriva per problemi di una certa ampiezza una riduzione del tempo di calcolo speso nella soluzione di sistemi.

     

Uniform strength design of beams with non-zero minimum flexural rigidity under multiple loadings

The analytical method for uniform strength design (U.S.D.) of statically indeterminate beams is extended to deal with a preassigned constraint of non-zero minimum flexural rigidity and multiple load cases. And a numerical method of finding the U.S.D. is presented. Therefore we have a uniform procedure of solution of U.S.D. to beams with arbitrary cross-sectional shape and acted upon by several loadings and subjected to the constraint of minimum flexural rigidity.     

Electro-elastomers: Large deformation analysis of silicone membranes

Electro-elastomers are large strain smart materials capable of both sensing and actuation. Typical electro-elastomer setups consist of either a silicone or acrylic membrane sandwiched between two compliant grease electrodes. Silicone electro-elastomers have maximum elastic strains between 200% and 350%. Acrylic electro-elastomers are more widely employed due to larger actuation strains but are softer than silicone and have a lower force output [Goulbourne, N.C., Frecker, M., Mockensturm, E.M., Snyder, A.J., 2003. Modeling of a dielectric elastomer diaphragm for a prosthetic blood pump. In: Proceedings of SPIE, Smart Structures and Materials: EAPAD, San Diego; Goulbourne, N.C., Mockensturm, E.M., Frecker, M., 2005b. Quasi-static and dynamic inflation of a dielectric elastomer membrane. In: Proceedings of SPIE, Smart Structures and Materials: EAPAD, San Diego]. A numerical formulation for the large deformation response of electro-elastomer membranes subject to electromechanical loading is derived in this paper. The approach is based on modifying the elastic membrane theory of Green, Adkins, and Rivlin [Adkins, J.E., Rivlin, R.S., 1952. Large elastic deformations of isotropic materials IX. The deformation of thin shells. Philosophical Transactions of the Royal Society of London. Series A Mathematical and Physical Sciences 244, 505–531; Green, A.E., Adkins, J.E., 1970. Large Elastic Deformations. Oxford University Press, London]. The electro-elastic stress state is defined as the combination of the electrical Maxwell stress and the mechanical stress for hyperelastic materials [Goulbourne, N.C., Mockensturm, E.M., Frecker, M., 2005a. A nonlinear model for dielectric elastomer membranes. ASME Journal of Applied Mechanics 72, (6) 899–906]. This paper augments our previous work by presenting a mathematical solution procedure for simulating the field responsive behavior of silicone electro-elastomers configured for both in-plane and out-of-plane deformation. Thin axisymmetric membranes subject to electromechanical loads are the focus of this investigation. The numerical analysis shows that there is a delicate balance between the electrical and the mechanical portions of the stress, which must be maintained for the overall stress to remain tensile and by extension the electro-elastomer to remain stable. It is shown that at very high voltages the stress can become negative ultimately leading to transducer failure. For sensing applications, the varying capacitive behavior of electro-elastomers is used to extract information about the membrane’s deformed state.