Theory Of Thin Bodies Research Articles

On the Iterlayer Contact Conditions in Multilayer Thin Body Theory and Some Issues of Splitting Initial-Boundary Value Problems

On the Iterlayer Contact Conditions in Multilayer Thin Body Theory and Some Issues of Splitting Initial-Boundary Value Problems

On the Decomposition of Equations of Micropolar Elasticity and Thin Body Theory

The motion equations of the micropolar theory of elasticity in displacements and rotations represented by the matrix differential tensor-operator for any inhomogeneous anisotropic materials. As a particular case the micropolar isotropic homogeneous materials with a center of symmetry is considered. In this case, the matrix differential tensor-operator of cofactors to the matrix differential tensor-operator of the motion equations is constructed. This constructed operator makes possible to decompose the equations. The equations are obtained separately with respect to the displacement and rotation vectors. Decomposed equations also obtained for a reduced medium. In this case, the equation with respect to the displacement vector is the same as the equation of the classical theory, and the equation with respect to the rotation vector has a similar form. In addition, in the absence of volume loads, the equations of the reduced medium do not depend on the properties of the material. This suggests that these equations can be used to identify the material constants of this medium. The cases under which the static boundary conditions are easily split are revealed. From the decomposed equations of the micropolar theories of elasticity, the corresponding decomposed equations of the static (quasistatic) problem of the theories of single-layer and multi-layer prismatic bodies of constant thickness in displacements and rotations are obtained. From the last systems of equations the equations in the moments of unknown vector functions with respect to any systems of orthogonal polynomials are derived. As a particular case, we obtain a system of equations of the eighth approximation in moments with respect to the system of Legendre polynomials, which decomposes into two systems. One of them is the system with respect to the even order moments of the unknown vector function, and the other system is the system with respect to odd order moments of the same functions. Based on the obtained operator of cofactors to the operator of any of these systems we get a high order (the order of the system depends on the order of approximation) elliptic type equation for each moment of the unknown vector function, which characteristic roots are easily found. Using Vekua’s method for solving such equations [66], we can obtain their analytical solution. Note also that the analytic method with the use of the orthogonal polynomial systems (Legendre and Chebyshev) in constructing the one-layer [2, 3, 7, 10, 15, 17, 18, 20–22, 63, 68, 69] and multilayer [4–6, 13, 60, 61] thin body theory was also applied by other authors. In this direction the authors had published the papers [24–31, 33–37, 41–45, 51–53], and others with the application of Legendre and Chebyshev polynomial systems. These expansions can be successfully used in constructing any thin body theory. Despite this, classical theories are far from perfect, and micropolar theories and theories of another rheology are very far from perfect.

To mathematical modeling of deformation of micropolar thin bodies with two small sizes

We consider some problems of modeling the deformation of micropolar thin bodies with two small sizes. Using the three-dimensional equations of motion, the constitutive relations and the boundary conditions of the micropolar elasticity theory [1, 2], we got the equations of motion, the constitutive relations and the boundary conditions of the micropolar theory of thin bodies under the parametrization of the thin body domain with two small sizes [3]. The boundary conditions and various representations of the system of equations of motion and the constitutive relations of physical content in moments with respect to the Legendre polynomials are obtained. Note that using this method of construction the thin bodies theory with two small sizes, we get an infinite system of ordinary differential equations. This system contains quantities which depends on one variable, namely, depends on the parameter of the base line. Thus, decreasing the number of independent variables from three to one we increase the number of equations to infinity, which, of course, has its obvious practical inconveniences. In this regard, we reduce an infinite system to the finite system. The initial-boundary value problems are formulated. To satisfy the boundary conditions on the front surfaces we constructed correcting terms [4]. As a special case, we considered a prismatic body. We used the tensor calculus to do this research [5–8].

Modeling of multilayer thin bodies

In this work, we considered the new parametrization of a multilayer thin domain. In particular, in contrast to classic approaches, we used several base surfaces and an analytic method with the application of orthogonal polynomial systems. We gave the vector parametric equation of each layer and the system of vector parametric equations of a multilayer thin domain and introduced the geometric characteristics for the proposed parametrization. We also derived the expressions for the transfer components of the second-rank identity tensor and the relations connecting the various families of bases and presented some differential operators, the system of equations of motion, the heat flow equation, the constitutive relations of the theory of the micropolar elasticity, and the Fourier heat conduction law under this parametrization of the thin-body domain. Finally, we gave the classification and statements of boundary value problems in the theory of thin bodies.

To the problem of decomposition of the initial boundary value problems in mechanics

We consider the questions on the decomposition of the initial boundary value problems of elasticity theory and thin bodies for some anisotropic media. In particular, the initial boundary problems of the micropolar (and classical) theory of elasticity are presented, i.e. to introduce tensor-block matrix operators (tensor-operators). In the case of an isotropic micropolar elastic medium (isotropic and transversally isotropic classical media) tensor-block matrix operators (tensor-operators) of cofactors corresponding to the tensor-block matrix operators (tensor-operators) of of the initial boundary value problems are obtained, which are splitting the initial boundary value problems. From three-dimensional decomposed initial boundary value problems the corresponding decomposed initial boundary value problems for the theory of thin bodies are obtained.

Equations of motion and boundary conditions of physical meaning of micropolar theory of thin bodies with two small cuts

Nowadays, microcontinuous mechanics (mechanics of media with microstructure) is being developed very intensively, which is testified by recently published papers [1–14] and by many others, as well as by the symposiumdedicated to the hundredth anniversary of the brothers Cosserat monograph [15], held inParis in 2009. A survey of foreign papers is given in [16], and a special place is occupied by earlier publications of Soviet scientists on micropolar theory of elasticity [17–24]. A brief survey of Cosserat theory of elasticity and an analysis and prospects of such theories in mechanics of rigid deformable bodies is given in [21].

Mathematical modelling of multilayer thin body deformation

We construct a theory of multilayer thin bodies within the framework of the threedimensional moment theory by using an efficient parametrization of a multilayer thin domain; in contrast to classic approaches, several base surfaces and an analytic method with Legendre and Chebyshev polynomial systems are used. Geometric characteristics typical for the proposed parametrizations are introduced into consideration. A fundamental theorem for a multilayer thin domain is formulated. Various representations of the equations of motion, the heat influx, and the constitutive relations of physical and heat content are presented for the new body domain parametrization. The definition of the kth order moment of a certain quantity with respect to an orthonormal system of second-kind Chebyshev polynomials is given. The expressions of moments of first- and second-order partial derivatives of a certain tensor field are obtained, and this is also done for some important expressions required for constructing different variants of the thin body theory. Various variants of the equations of motion in moments with respect to Legendre and Chebyshev polynomial systems are also obtained. The interlayer conditions are written down under various connections of adjacent layers of a multilayer body.

Closure of the Deep Wake Hollow of High-Speed Ships

Interest has developed in recent years in the length of the "wake hollow" downstream of high-speed vessels with blunt transom sterns. Past work has shown that this wake cavity appears to increase the effective length of a vessel, and thereby reduce its effective Froude number, with the result of a reduction in wave resistance. The research presented here is development and demonstration of a simple mathematical theory for the downstream wake closure characteristics of fully ventilated wakes of fast ships with transom sterns. The theory uses the linear thin body approximations with gravity in a formulation consistent with the thin body theory of wave resistance, although wave resistance is not dealt with here. The length of the wake hollow, or "trench," predicted is compared with recent experiments. Calculations of variation of the hollow length with transom shape and Froude number are presented.

Application of Chebyshev polynomials to the theory of thin bodies

The application of shifted Chebyshev polynomials of the second kind to the construction of the thin-body theory is considered. Some basic and additional recurrence relations for Chebyshev polynomials are given. Arbitrary-order moments are obtained for the first and second derivatives of a tensor field and for some expressions. Several equations of motion expressed in terms of the moments of displacement and rotation vectors are derived for the moment theory; a number of constitutive relations of the zeroth approximation are obtained.

On the theory of thin bodies

It is shown that, in accordance with the law of plane sections representing the basis of the theory of thin bodies, two analogies are true: a nonstationary analogy and a stationary one. Within the framework of the stationary analogy, a new-type expansion reducing a three-dimensional problem to a stationary two-dimensional one is introduced. The asymptotic conditions of validity of the stationary and nonstationary analogies were determined and the boundary-layer conceptions for both cases were compared. Solutions of the internal and external problems on a two-dimensional flow in a narrow zone were obtained in the closed form. An asymptotically correct mathematical model of a flow in a film is proposed. The stationary and nonstationary analogies for a swirling flow around a body and a flow around a rotating body were determined.

THE SOLUTION ASYMPTOTICS OF CLASSICAL AND NONCLASSICAL, STATIC AND DYNAMIC BOUNDARY-VALUE PROBLEMS FOR THIN BODIES

An asymptotic method for solution of classical and nonclassical boundary-value problems of the theory of elasticity for thin bodies (beams, rods, plates, and shells) is expounded. Studies on the asymptotic theory of thin bodies are reviewed. Asymptotic results are compared with those obtained by other applied theories. The asymptotic approach has been found out to be related to Saint Venant's principle. The correctness of this principle is mathematically proved for one class of problems. A fundamentally new asymptotics in the components of the stress tensor and the displacement vector is revealed in considering new classes of problems. On their basis, the applicability domains are outlined for various models of understructures. Solutions are obtained to certain classes of dynamic problems for thin bodies, particularly, those simulating seismic effects. The resonance conditions are established and ways of preventing them are pointed out.

Solution of Boundary Value Problems by Integral Equations of the First Kind

This paper discusses an integral equation procedure for the solution of boundary value problems. The method derives from work of Fichera and differs from the more usual one by the use of integral equations of the first kind. The method here extends to equations of higher order than second. Its connection with singular perturbation theory and thin-body theory are indicated by examples. Some numerical experiments are included to indicate how the method operated in specific situations.