A note on the expression for the rotation tensor in terms of Euler-angles

It is pointed out in this paper that the derivation of the expression for the rotation tensor in terms of Euler angles by N. E, Kochin^[1] is erroneous. To point out Kochin’s deficit on this problem is advantageous for perfecting the concept of the rotation tensor and its method of expression.     

Some discussions on finite deformation of continuous media

Three problems are discussed in this paper:1.The physical significance of the characteristic tensorθ~ii of finitedeformation is discussed as a complement to the literature.2.The four characterietic tensors of finite deformation,introduced inthe.literature,ard analyzed and discussed more thoroughly.3.The representation of the general finite deformation through simple.loading process is not always possible,the condition for its realization beingthat the given finite deformation satisfies the equations of compatibility.It is pointed out in this paper that for the finite deformation set forth byL.I.Sedov in an illustrative example in[9],the equations of compatibilitycannot be satisfied even for k=1,henceforth for whatever value of k,thefinite deformation set forth by Sedov can not be represented through simpleloading process.     

A new approximate analytical solution of the ideal potential flow around a circular cylinder between two parallel flat plates

In ref. [1], Lin obtained an approximate analytical solution of the ideal potential flow around a circular cylinder between two parallel flat flates.In this paper, the author shows that one may obtain the result coinciding with that obtained in ref. [1] by making use of the Shvez's method^[2]. Morever, we can obtain a more accurate result than that obtained in ref. [1], if we make use of the improved Shvez's method^[2]. Some calculating examples are presented.     

On the torsion of a cylinder with several cracks

In this paper, based on paper [1], the analytic expression of the torsion function for a cylinder containing arbitrary oriented cracks is obtained. The problem is reduced to solve a system of singular integral equations for the unknown dislocation density functions. Using the numerical method of the singular integral equations^[2,7] the torsional rigidities and stress intensity factors are evaluated for several multicracked cylinders. Next, the creak-cutting method^[5] is firstly extended to lve the torsion problem for a rectangular prism. The numerical results show that the method presented here is successful. Projects Supported by the Science Fund of the Chinese Academy of Sciences.     

Bending integral expressions of a cylinder with cracks

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Bending of thick plates with a concentrated load

In this paper, according to the simplified theory of [1], the bending of rectangular plates with two opposite edges simply supported and other two opposite edges being arbitrary under the action of a concentrated load is treated by means of properties of two-variable-function and the method of series^[2]. The effect of transverse shearing forces on the bending of plates is considered. When the thickness h of plates is small, and the term, whose orders are more than order of h² are neglected, then the results agree with the solutions corresponding to the problem of thin plates^[3]. At the end, the solutions of the bending problem of plates with arbitrary linear distributed load are also obtained.     

Fixed pansystems theorems and pansystems logic conservation of chaos

The study of nonlinear problems was developed in works ^[1,2] by means of the pansystems methodology (PM) which does not need the condition related to differential manifolds and linear space. In work [2], within the framework of PM, we proved that study of panchaos, panattractor and strange panattractor can be transformed conditionally into some forms of fixed subsets. As a continuation of work [2], we now research the pansystems logic conservation of panchaos, panattractor, strange panattractor and some other fixed subsets.     

Coseismic stress variation and numerical analysis of 2011 Japan–Honshu 9.0 earthquake

Using hybrid hypersingular integral equation-Lattice Boltzmann method [1], [2] and [3], the 2011 Japan–Honshu 9.0 earthquake coseismic stress tensor has been explored. A hypothesis based on slip elastic bound back and transient elastic–dynamic ultra low frequency shock plates assumes has been proposed and chaos phenomena in the results has been explained by using this hypothesis.     

The bending of elastic circular ring of non-homogeneous and variable cross section under the actions of arbitrary loads

On the basis of the stepped reduction method suggested in [1], we investigate the problem of the bending of elastic circular ring of non-homogeneous and variable cross section under the actions of arbitrary loads. The general solution of this problem is obtained so that it can be used for the calculations of strength and rigidity of practical problems such as arch, tunnel etc. In order to examine results of this paper and explain the application of this new method, an example is brought out at the end of this paper. Circular ring and arch are commonly used structures in engineering. Timoshenko, S.^[2], Barber, J. R.^[3], Tsumura Rimitsu^[4] et al. have studied these problems of bending, but, so far as we know, it has been solely restricted to the general solution of homogeneous uniform cross section ring. The only known solution for the problems with variable cross section ones has been solely restricted to the solution of special case of flexural rigidity in linear function of coordinates. On account of fundamental equations of the non-homogeneous variable cross section problem being variable coefficients, it is very difficult to solve them. In this paper, we use the stepped reduction method suggested in [1] to transform the variable coefficient differential equation into equivalent constant coefficient one. After introducing virtual internal forces, we obtain general solution of an elastic circular ring with non-homogeneity and variable cross section under the actions of arbitrary loads.     

Calculation of stresses and deformations of bellows by initial parameter method of numerical integration

By reducing the boundary value problem in stress analysis of bellows into initial value problem, this paper presents a numerical solution of stress distribution in semi-circular arc type bellows based upon the toroidal shell equation of V. V. Novozelov^[8]. Throughout the computation, S. Gill’s method^[1O] of extrapolation is used. The stresses and deformations of bellows under axial load and internal pressure are c-alculated, the results of which agree completely with those derived from the general solution of Prof. Chien Wei-zang^[1-4]. The extrapolation formula presented in this paper greatly promotes the accuracy of discrete calculation.The computer program in BASIC language of Wang 2200 VS computer is included in the appendix.     

Nonlinear MHD Kelvin-Helmholtz instability in a pipe

Nonlinear MHD Kelvin-Helmholtz (K-H) instability in a pipe is treated with the derivative expansion method in the present paper. The linear stability problem was discussed in the past by Chandrasekhar (1961)^[1] and Xu et al. (1981).^[6]Nagano (1979)^[3] discussed the nonlinear MHD K-H instability with infinite depth. He used the singular perturbation method and extrapolated the obtained second order modifier of amplitude vs. frequency to seek the nonlinear effect on the instability growth rate γ. However, in our view, such an extrapolation is inappropriate. Because when the instability sets in, the growth rates of higher order terms on the right hand side of equations will exceed the corresponding secular producing terms, so the expansion will still become meaningless even if the secular producing terms are eliminated. Mathematically speaking, it's impossible to derive formula (39) when γ ₀ ² is negative in Nagano's paper.^[3]Moreover, even as early as γ ₀ ² → O⁺, the expansion becomes invalid because the 2nd order modifier γ₂ (in his formula (56)) tends to infinity. This weakness is removed in this paper, and the result is extended to the case of a pipe with finite depth. Theproject is supported by the National Natural Science Foundation of China.     

A variational principle for the dynamic problem of linear coupled thermoelasticity

Summary A functional is constructed by whose stationarity the mixed problem for linear and dynamic thermoelasticity is obtained. In contrast with previous results[2], [4], [8], neither constrained variations nor previous transformations upon the equations of the problem are used. The unrestricted variational formulation is made possible by means of a convolutive bilinear form.

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Sommario Si costruisce un funzionale dalla cui stazionarietà si ricava il problema misto della termoelasticità lineare accoppiata. A differenza di precedenti risultati[2], [4], [8] non si ricorre né a variazioni bloccate né a preliminari trasformazioni del problema. La formulazione variazionale non ristretta è resa possibile dall'uso di una forma bilineare convolutiva.

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Work done in the sphere of activity of the Group for Mathematical Research of the (Italian) C.N.R.     

Problems on collinear cracks between bonded dissimilar materials under concentrated loads

Following Ref. [6], this paper deals with the problem on collinear cracks between bonded dissimilar materials under a concentrated force and moment at an arbitrary point. Several typical solutions of complex stress functions in closed form are formulated and the stress intensity factors are given. These solutions include a series of results of previous researchers, and redress some errors in the researches of problems containing semi-infinite cracks^[3],[4].     

Comparison of Fe and Ni opacity calculations for a better understanding of pulsating stellar envelopes

Opacity is an important ingredient of the evolution of stars. The calculation of opacity coefficients is complicated by the fact that the plasma contains partially ionized heavy ions that contribute to opacity dominated by H and He. Up to now, the astrophysical community has greatly benefited from the work of the contributions of Los Alamos [1], Livermore [2], [2a] and [2b] and the Opacity Project (OP) [3]. However unexplained differences of up to 50% in the radiative forces and Rosseland mean values for Fe have been noticed for conditions corresponding to stellar envelopes. Such uncertainty has a real impact on the understanding of pulsating stellar envelopes, on the excitation of modes, and on the identification of the mode frequencies. Temperature and density conditions equivalent to those found in stars can now be produced in laboratory experiments for various atomic species. Recently the photo-absorption spectra of nickel and iron plasmas have been measured during the LULI 2010 campaign, for temperatures between 15 and 40 eV and densities of ∼3 mg/cm³. A large theoretical collaboration, the “OPAC”, has been formed to prepare these experiments. We present here the set of opacity calculations performed by eight different groups for conditions relevant to the LULI 2010 experiment and to astrophysical stellar envelope conditions.     

Mode I crack tip field with strain gradient effects

A plane strain mode I crack tip field with strain gradient effects is investigated. A new strain gradient theory is used. An elastic-power law hardening strain gradient material is considered and two hardening laws, i. e. a separation law and an integration law are used respectively. As for the material with the separation law hardening, the angular distributions of stresses are consistent with the HRR field, which differs from the stress results^[19]; the angular distributions of couple stresses are the same as the couple stress results^[19]. For the material with the integration law hardening, the stress field and the couple stress field can not exist simultaneously, which is the same as the conclusion^[19], but for the stress dominated field, the angular distributions of stresses are consistent with the HRR field; for the couple stress dominated field, the angular distributions of couple stresses are consistent with those in Ref. [19]. However, the increase in stresses is not observed in strain gradient plasticity because the present theory is based on the rotation gradient of the deformation only, while the crack tip field of mode I is dominated by the tension gradient, which will be shown in another paper. Supported by the National Science Foundation of China (No. 19704100), Science Foundation of Chinese Academy of Sciences (Project KJ951-1-20), CAS K. C. Wong Post-doctoral Research Award Fund and the Post Doctoral Science Fund of China.     

Existence and comparison results for solutions of random integral and differential equations

This paper is the continuation of [1]. In this paper, we give another criteria of the existence of solutions for nonlinear random Volterra integral. A comparison theorem and the existence of random extremal solutions are also obtained by using the notion of ordering with respect to a cone. Our theorems generalize the corresponding results of Vaughan^[2,3] and Lakshmikantham^[4,5].     

THE PROBLEMS OF THE NONLINEAR UNSYMMETRICAL BENDING FOR CYLINDRICALLY ORTHOTROPIC CIRCULAR PLATE（II）

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Qualitative mathematical analysis of the Richards equation

The Richards equation is widely used as a model for the flow of water in unsaturated soils. For modelling one-dimensional flow in a homogeneous soil, this equation can be cast in the form of a specific nonlinear partial differential equation with a time derivative and one spatial derivative. This paper is a survey of recent progress in the pure mathematical analysis of this last equation. The emphasis is on the interpretation of the results of the analysis. These are explained in terms of the qualitative behaviour of the flow of water in an unsaturated soil which is described by the Richards equation.Nomenclature a coefficient in second-order diffusion term of equation - b coefficient in first-order advection term of equation - D soil-moisture diffusivity [L²T^-1] - h pressure head [L] - H quarter-plane domain for Cauchy-Dirichlet problem [L] x [T] - K hydraulic conductivity scalar [LT^–1] - K hydraulic conductivity tensor [LT^–1] - q soil-moisture flux scalar [LT^–1] - q soil-moisture flux vector [LT^–1] - r dummy variable - R rectangle [L] x [T] - s dummy variable - s* representative value of dummy variable - S half-plane domain for Cauchy problem [L] x [T] - t time [T] - u unknown solution of partial differential equation - u₀ initial-value function - v soil-moisture velocity scalar [LT^–1] - v soil-moisture velocity vector [LT^–1]