Life-span of classical solutions for one-dimensional hydromagnetic flow

The paper concerns Cauchy, problem for one-dimensional hydromagnetic dynamics with dissipative terms. When the dissipation coefficient is equal to zero it is shown that the smooth solutions develop shocks in the finite time if the initial amounts of entropy and magnetic field are smaller than those of sound waves; when it is larger than zero, and the initial amounts of entropy, this dissipation coefficient and the magnetic field in each period are smaller than those of sound waves, then the smooth solutions blow up in the finite time. Moreover, the life-span of the smooth solution is given.     

STABILITY OF SYSTEM OF TWO-DIMENSIONAL NON-HYDROSTATIC REVOLVING FLUIDS

Applying the theory of stratification, it is proved that the system of the two-dimensional non-hydrostatic revolving fluids is unstable in the two-order continuous function class. The construction of solution space is given and the solution approach is offered. The sufficient and necessary conditions of the existence of formal solutions are expressed for some typical initial and boundary value problems and the calculating formulae to formal solutions are presented in detail.     

ANALYSIS OF SYMMETRIC LAMINATED RECTANGULAR PLATES IN PLANE STRESS

Symmetric laminated plates used usually are anisotropic plates. Based on the fundamental equation for anisotropic rectangular plates in plane stress problem, a general analytical solution is established accurately by method of stress function. Therefore the general formula of stress and displacement in plane is given. The integral constants in general formula can be determined by boundary conditions. This general solution is composed of solutions made by trigonometric function and hyperbolic function, which can satisfy the problem of arbitrary boundary conditions along four edges, and the algebraic polynomial solutions which can satisfy the problem of boundary conditions at four corners. Consequently this general solution can be used to solve the plane stress problem with arbitrary boundary conditions. For example, a symmetric laminated square plate acted with uniform normal load, tangential load and nonuniform normal load on four edges is calculated and analyzed.     

Two exact solutions of the DPL non-Fourier heat conduction equation with special conditions

This paper presents two exact explicit solutions for the three dimensional dual-phase lag （DLP） heat conduction equation, during the derivation of which the method of trial and error and the authors＇ previous experiences are utilized. To the authors＇ knowledge, most solutions of 2D or 3D DPL models available in the literature are obtained by numerical methods, and there are few exact solutions up to now. The exact solutions in this paper can be used as benchmarks to validate numerical solutions and to develop numerical schemes, grid generation methods and so forth. In addition, they are of theoretical significance since they correspond to physically possible situations. The main goal of this paper is to obtain some possible exact explicit solutions of the dual-phase lag heat conduction equation as the benchmark solutions for computational heat transfer, rather than specific solutions for some given initial and boundary conditions. Therefore, the initial and boundary conditions are indeterminate before derivation and can be deduced from the solutions afterwards. Actually, all solutions given in this paper can be easily proven by substituting them into the governing equation.     

Analytical solutions for inflation of pre-stretched elastomeric circular membranes under uniform pressure

Elastomeric membranes are frequently used in several emerging fields such as soft robotics and flexible electronics. For convenience of the structural design, it is very attractive to find simple analytical solutions to well describe their elastic deformations in response to external loadings. However, both the material/geometrical nonlinearity and the deformation inhomogeneity due to boundary constraints make it much challenging to get an exact analytical solution. In this paper, we focus on the inflation of a prestretched elastomeric circular membrane under uniform pressure, and derive an approximate analytical solution of the pressure–volume curve based upon a reasonable assumption on the shape of the inflated membrane. Such an explicit expression enables us to quantitatively design the material and geometrical parameters of the pre-stretched membrane to generate a target pressure–volume curve with prescribed peak point and initial slope. This work would be of help in the simplified mechanical design of structures involving elastomeric membranes.     

Simple waves for two-dimensional compressible pseudo-steady Euler system

A simple wave is defined as a flow in a region whose image is a curve in the phase space. It is well known that ＂the theory of simple waves is fundamental in building up the solutions of flow problems out of elementary flow patterns＂ see Courant and Friedrichs＇s chassical book ＂Supersonic Flow and Shock Waves＂. This paper mainly concerned with the geometric construction of simple waves for the 2D pseudo-steady compressible Euler system. Based on the geometric interpretation, the expansion or compression simple wave flow construction around a pseudo-stream line with a bend part are constructed. It is a building block that appears in the global solution to four contact discontinuities Riemann problems.     

Diffraction of elastic waves in the plane multiply-connected region and dynamic stress concentration

This paper deals with the problem of diffraction of elastic waves in the plane multiply-connected regions by the theory of complex functions. The complete function series which approach the solution of the problem and general expressions for boundary conditions are given.’ Then the problem is reduced to the solution to infinite series of algebraic equations and the solution can be directly obtained by using electronic computer. In particular, for the case of weak interaction, an asymptotic method is presented here, by which the problem ofp waves diffracted by a circular cavities is discussed in detail. Based on the solution of the diffracted wave field the general formulas for calculating dynamic stress concentration factor for a cavity of arbitrary shape in multiply-connected region are given.     

Exact solutions for the flow of second grade fluid in annulus between torsionally oscillating cylinders

The velocity field and the associated shear stress corresponding to the torsional oscillatory flow of a second grade fluid, between two infinite coaxial circular cylinders, are determined by means of the Laplace and Hankel transforms. At time t = 0, the fluid and both the cylinders are at rest and at t = 0 + , cylinders suddenly begin to oscillate around their common axis in a simple harmonic way having angular frequencies ω 1 and ω 2 . The obtained solutions satisfy the governing differential equation and all imposed initial and boundary conditions. The solutions for the motion between the cylinders, when one of them is at rest, can be obtained from our general solutions. Furthermore, the corresponding solutions for Newtonian fluid are also obtained as limiting cases of our general solutions.     

ASYMPTOTIC NON-STABILITY AND BLOW-UP AT BOUNDARY FOR SOLUTIONS OF A FILTRATION EQUATION

For a class of nonlinear filtration equation with nonlinear second-third boundary value condition, it is shown that a priori boundary of the solution can be estimated and controlled by initial data and integral on the boundary of the region. The priori estimate of the solutions was established by iterative method. By using this estimate the solutions may blow-up on the boundary of the region and thus it may have asymptotic non-stability.     

Similarity solutions for non-Newtonian power-law fluid flow

The problem of the boundary layer flow of power law non-Newtonian fluids with a novel boundary condition is studied.The existence and uniqueness of the solutions are examined,which are found to depend on the curvature of the solutions for different values of the power law index n.It is established with the aid of the Picard-Lindel¨of theorem that the nonlinear boundary value problem has a unique solution in the global domain for all values of the power law index n but with certain conditions on the curvature of the solutions.This is done after a suitable transformation of the dependent and independent variables.For 0 n 1,the solution has a positive curvature,while for n 1,the solution has a negative or zero curvature on some part of the global domain.Some solutions are presented graphically to illustrate the results and the behaviors of the solutions.     

Exact expressions for transient forced internal gravity waves and spatially-localized wave packets in a Boussinesq fluid

We re-examine a simple model describing the propagation of transient forced internal gravity waves in a Boussinesq fluid with constant horizontal mean velocity which was previously studied by Nadon and Campbell (Wave Motion, 2007). The waves are generated by a horizontally-periodic lower boundary condition and propagate upwards. We derive an alternative exact expression for the solution which more readily gives insight into the behaviour of the solution at high altitude. Some special cases of lower boundary conditions are considered to illustrate the features of the solution. This form of the solution allows us to use a Fourier transform to derive the solution for the more general situation where a wave packet is generated by a horizontally-localized lower boundary condition, comprising a continuous spectrum of horizontal wavenumbers or Fourier modes. This is a more realistic representation of internal gravity waves in the atmosphere and can be used as a starting point for investigating waves generated by an obstacle of finite horizontal extent such as an isolated mountain or a mountain range.     

Generalized stroh formalism for anisotropic elasticity for general boundary conditions

The Stroh formalism is most elegant when the boundary conditions are simple, namely, they are prescribed in terms of traction or displacement. For mixed boundary conditions such as there for a slippery boundary, the concise matrix expressions of the Stroh formalism are destroyed. We present a generalized Stroh formalism which is applicable to a class of general boundary conditions. The general boundary conditions include the simple and slippery boundary conditions as special cases. For Green's functions for the half space, the general solution is applicable to the case when the surface of the half-space is a fixed, a free, a slippery, or other more general boundary. For the Griffith crack in the infinite space, the crack can be a slit-like crack with free surfaces, a rigid line inclusion (which is sometimes called an anticrack), or a rigid line with slippery surface or with other general surface conditions. It is worth mention that the modifications required on the Stroh formalism are minor. The generalized formalism and the final solutions look very similar to those of unmodified version. Yet the results are applicable to a rather wide range of boundary conditions.     

各向异性弹性力学一般边值问题的广义Stroh公式

当边值问题是简单的,即是应力边值问题时,Stroh公式是很有效的。对于混合边值问题,倒如滑动边界条件,Stroh公式中的简洁的矩阵表达式就失效了。我们提出了一个广义的Stroh公式,它可应用于一大类一般的边界条件。简单的边界条件和滑动边界条件是这一类一般边界条件的特殊情形。值得指出的是,这个关于Stroh公式所作的修正并不大。广义的公式和最后的解答看起来很类似于未修正的原公式和原来的解。然而这个修正却可应用于相当广的边界条件。     

On the encounter of an acoustic shear pulse with a phase boundary in an elastic material: reflection and transmission behavior

The fully dynamical motion of a phase boundary is considered for a specific class of elastic materials whose stress-strain relation in simple shear is nonmonotone. It is shown that a preexisting stationary phase boundary in a prestressed layer composed of such a material can be set in motion by a finite amplitude shear pulse. An infinity of solutions is possible according to the present theory, each of which is characterized by different reflected and transmitted waves at the phase boundary. A global analysis gives exact bounds on the size of the solution family for different shear pulse amplitudes. For certain ranges of shear pulse amplitudes a completely reflecting solution will exist, while for an in general different range of shear pulse amplitudes a completely transmitting solution will exist. The properties of these different solutions are examined. In particular, it is observed that the ringing of a shear pulse between the external boundaries and the internal phase boundary gives rise to periodic phase boundary motion for both the case of a completely reflecting phase boundary and a completely transmitting phase boundary.     

Stress wave propagation in a rectangular bar

The process of propagation of nonstationary waves in a rectangular bar is studied from the viewpoint of three-dimensional elasticity. The motion arises owing to the action of normal impact forces applied at the end face of a half-infinite bar all of whose four lateral surfaces are force-free. Precisely these one-type conditions complicate the solution of this problem. The already known solutions were obtained under the assumption that conditions of mixed type are partially or completely posed on the lateral sides, and precisely this fact permits separating the boundary values of distinct waves on these surfaces. In the absence of this simplifying factor, it is rather problematic to construct a solution satisfying all free lateral conditions.     

Spatial Simple Waves on a Shear Flow

The paper studies simple waves of the shallowwater equations describing threedimensional wave motions of a rotational liquid in a freeboundary layer. Simple wave equations are derived for the general case. The existence of unsteady or steady simple waves adjacent continuously to a given steady shear flow along a characteristic surface is proved. Exact solutions of the equations describing steady simple waves were found. These solutions can be treated as extension of Prandtl–Mayer waves for sheared flows. For shearless flows, a general solution of the system of equations describing unsteady spatial simple waves was found.     

加权残数配点法解正交各向异性板的积分方程

本文推导了一般各向异性板弯曲的积分方程,运用加权残数配点法求解了正交各向异性板弯曲的积发方程,本文将部分配点取在边界上,另一部分配点取在域外,只用关于找度的基本积分方程,而不用关于转角的补充积分方程,简化了方程求解和计算程序,由于正交各向异性板没有争析形式的、实用的基本解,本文提出了两种新的近似基本解;加权双三角级数;广义各向同性板解析形式的基本解和加权双三角级数的叠加,算例表明,本文提出的解法和近似基本解适用于各类边界条件的正交各向异性板,具有简单、可靠、精度高等优点。     

Theory of nonstationary interaction between pressure waves in a fluid and unclosed surfaces

Problems of nonstationary scattering of incident waves by unclosed surfaces have been solved in the general formulation under the usual assumptions of the linear mechanics of ideal compressible fluids. Such problems are encountered in a number of important hydrodynanic applications. In this case the nonstationary wave field must be known at any distance from the scatterer, and in particular in its immediate vicinity. The known methods of stationary diffraction cannot be used in nonstationary problems, when it is not possible to obtain exact expressions for the Fourier transforms of the unknown solutions and hence guarantee the unique recovery of the inverse transforms. However, the direct solution of the nonstationary problem is possible only in very simple situations: scattering of pressure waves by a plate, diffraction at the edge of a half-plane or at a slit in a flat screen, etc. These circumstances make it necessary to develop special approaches to the solution of the problem of the nonstationary scattering of pressure waves in a fluid by arbitrary unclosed surfaces. This paper outlines a method which leads to the construction of the Laplace transforms of the unknown solutions and is based on a unique means of satisfying the boundary conditions with the subsequent obtaining of exact expressions for the coefficients (densities) of the expansions employed. The class of problems solvable by this method is confined to those for which it is possible to obtain corresponding solutions by expansion in series or integrals over the complete orthogonal system of eigenfunctions on the assumption that the surface of the obstacle is closed. The Laplace transforms of the solutions can be inverted by any approximate method. The solutions constructed in accordance with the formalism developed are in satisfactory agreement with the experimental data and coincide with the classical results.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 84–91, November–December, 1985.     

Torsion in nonlinearly elastic incompressible circular cylinders

We consider a variable torsion deformation for incompressible right-circular cylinders and investigate the possibility of equilibrium states other than simple torsion in isotropic hyperelasticity. Our problem formulation also provides generalized versions of Rivlin's universal relations in torsion. In this more general setting, it is shown that simple torsion persists as the only solution for large classes of strain energies. When one allows for more general boundary conditions, variable torsion solutions are possible for special forms of the stored energy function. We derive such a form and develop general results.     

A simple and accurate mixed FE-DQ formulation for free vibration of rectangular and skew Mindlin plates with general boundary conditions

A simple and accurate mixed finite element-differential quadrature formulation is proposed to study the free vibration of rectangular and skew Mindlin plates with general boundary conditions. In this technique, the original plate problem is reduced to two simple bar (or beam) problems. One bar problem is discretized by the finite element method (FEM) while the other by the differential quadrature method (DQM). The mixed method, in general, combines the geometry flexibility of the FEM and high accuracy and efficiency of the DQM and its implementation is more easier and simpler than the case where the FEM or DQM is fully applied to the problem. Moreover, the proposed formulation is free of the shear locking phenomenon that may be encountered in the conventional shear deformable finite elements. A simple scheme is also presented to exactly implement the mixed natural boundary conditions of the plate problem. The versatility, accuracy and efficiency of the proposed method for free vibration analysis of rectangular and skew Mindlin plates are tested against other solution procedures. It is revealed that the proposed method can produce highly accurate solutions for the natural frequencies of rectangular and skew Mindlin plates with general boundary conditions.