Boundary Value Problems Of Elasticity Theory Research Articles

Asymptotic Solution of the First 3D Dynamic Elasticity Theory Problem on Forced Vibrations of a Three-Layer Plate with an Asymmetric Structure

The first 3D dynamic problem on forced vibrations of an orthotropic three-layer plate with an asymmetric structure is solved asymptotically. On faces of the package, conditions of the first boundary-value problem of elasticity theory, i.e., the values of corresponding components of the stress tensor, are set. It is assumed that they vary harmonically in time. An asymptotic solution of the internal (external) problem is found. Conditions for the origination of resonance are established. The cases where the solution of the internal problem becomes mathematically exact are indicated, and an illustrative example is given. The question about the conjugation of solutions of the inner and boundary-layer problems is discussed.

Contact Problem on the Interaction of Two Lap Plates, Absolutely Rigid in Tension and Flexible in Bending, with a Thin Circular Sector

Using the Fourier method, a solution is constructed for the boundary-value problem of elasticity theory for a circular sector whose radial sides are reinforced by two lap plates absolutely rigid in tension and flexible in bending. On the arc part of its contour, external conditions are given. The stress singularity in the vicinity of top of the circular sector and the behavior of coefficients of the singularity are investigated. It is established that stresses in this vicinity have a singularity of the type r −1+e (e > 0; e → 0 at α → π or α → 2π), whose coefficients, in the general case of loading of the arc part of the sector, differ from zero, which is inadmissible from the viewpoint of the mechanics of brittle fracture. However, an appropriate selection of external conditions on the arc part of sector allows one to equate these coefficients to zero.

Asymptotic Solutions of Boundary Value Problems of Electroelasticity for Transversely Isotropic Toroidal Shells Made of Piezoceramic Materials

By asymptotic integration of the equations and relations of a three-dimensional problem of electroelasticity theory for transversely isotropic ceramics, boundary-value problems for a thin toroidal shell are solved in the cases where the conditions of the first, second, or mixed dynamic boundary-value problems of elasticity theory are given on its surface. Recurrent formulas for calculating components of the displacement vector and stress tensor and the electric field potential in thickness-polarized ceramics are deduced. The resonance frequencies are also found.

An odd-symmetric boundary-value problem of elasticity theory for a semi-strip: Exact solution

An odd-symmetric boundary-value problem of elasticity theory for a semi-strip: Exact solution

Exact solution of a mixed boundary-value problem of elasticity theory for an infinite cone truncated along an axis

Exact solution of a mixed boundary-value problem of elasticity theory for an infinite cone truncated along an axis

Axisymmetrical boundary-value problems of elasticity theory for finite-length cylinders and cones

Axisymmetrical boundary-value problems of elasticity theory for finite-length cylinders and cones

An exact stiffness theory for unidirectional xFRP composites

UD xFRP composites, i.e., isotropic plastics reinforced with long transversely isotropic fibres packed unidirectionally according to the hexagonal scheme are considered. The constituent materials are geometrically and physically linear. The previous formulations of the exact stiffness theory of such composites are revised, and the theory is developed further based on selected boundary-value problems of elasticity theory. The numerical examples presented are focussed on testing the theory with account of previous variants of this theory and experimental values of the effective elastic constants. The authors have pointed out that the exact stiffness theory of UD xFRP composites, with the modifications proposed in our study, will be useful in the engineering practice and in solving the current problems of the mechanics of composite materials.

A plane periodic boundary-value problem of elasticity theory for a half-plane with curvilinear edge

We deduce a singular integral equation of a plane periodic problem of the theory of elasticity for a half plane with loaded curvilinear edge. The numerical solution of the integral equation is obtained by the method of quadratures for various configurations of the boundary of the half plane. The stress concentration factors are computed for a half plane with sinusoidal edge and for a periodic system of rounded V-shaped notches. On this basis, as a result of the limit transition, we get the stress intensity factors at the tips of sharp V-shaped notches and analyze their dependence on the opening angle of the notch.

On homogenization of a variational inequality for an elastic body with periodically distributed fissures

We study the problem of small deformations of an elastic body with periodically distributed fissures, where one-sided constraints are imposed on the sides of the fissures; this problem is equivalent to a variational inequality. We prove that if the linear size of the period of the distribution of the fissures tends to zero, then the solutions of this problem converge in the -norm to the solution of the homogenized problem, which is a non-linear boundary-value problem of elasticity theory for a domain without fissures.

Solution of the stochastic boundary-value problem of elasticity theory for composites with disordered structures in the correlative approximation of the method of quasi-periodic components

The method of quasi-periodic components, a new method of statistical mechanics of composites, is presented. In correlative approximation, this method makes it possible to reduce the problem of solving the stochastic boundary-value problem of elasticity theory for composites with disordered structures to a simpler problem for an individual cell with one inclusion in a homogeneous elastic medium. The generalized volumetric forces on the cell boundary take into account the random distribution of inclusions in the composite fragment studied. The problem for one inclusion cell can be solved, for example, by the boundary element method. The numerical solution in the correlative approximation of the method of quasi-periodic components for inhomogeneous interphase stress fields in the matrix of an epoxy composite containing randomly distributed unidirectional fibers is given. A comparison with the known analytical solutions obtained by other authors confirms the high accuracy of the correlative approximation.

Synthesis of averaging methods in stochastic boundary-value problems of the mechanics of microheterogeneous solids

The method of periodic components is further developed, which allows us, on unified positions, to predict the effective characteristics and structural strain fields in partially or fully disordered composites. The stochastic boundary-value problem of elasticity theory for microheterogeneous solids with a statistically homogeneous structure is treated. The heterogeneous solid is considered to be macroscopically homogeneous and macroanisotropic (or quasi-isotropic) with geometric form and properties of the structural components determined and given. The structural elements are assumed to have perfect interfaces, i.e., the displacements and tractions are continuous across the interface. The boundary-value problem was solved by the method of local approximation. Numerical results were obtained for composites with a stochastic structure.

Asymptotic behavior of solutions to problems of elasticity theory at infinity in flat parabolic domains

Asymptotic representations of solutions to the boundary-value problems of elasticity theory are studied in domains with parabolic exit at infinity (or in bounded domains with singularities like polynomial zero sharpness). The procedure of derivating a formal asymptotic expansion looks like the algorithm of asymptotic analysis in domains. Under the Dirichlet conditions (displacements are prescribed on the boundary of a domain), it is not hard to justify the power asymptotic series. It follows from the theorem on the unique solvability of the problem in spaces of the type L2 containing degrees of distance r=|x| as weight multipliers. For the Neumann conditions (stresses are prescribed on the boundary of a domain) an asymptotic expansion is justified by introducing the Eiry function Φ transforming the Lamé system to the biharmonic equation. Due to the appearance of the Dirichlet condition on Φ, the study of the asymptotic behavior of a solution to the last problem is simplified. The existence theorems and conditions for solvability of the “elastic” Neumann problem are presented. These results are based on the weighted Korn inequality. Bibliography: 29 titles.

Numerical solution of a boundary-value problem of elasticity theory for a body with an inclusion

We consider the problem of steady-state oscillations of a plane body with an inclusion. A solution algorithm is constructed. Computational experiments are described, which establish the applicability of the proposed procedure for analyzing the effect of defects inside the body on wave fields excited by ultrasonic sources.

Rate of convergence bound in the W 2 1 norm of the projection-difference scheme for the first boundary-value problem of elasticity theory in the anisotropic case

Exact difference scheme operators are applied to construct a projection-difference scheme for the system of equilibrium equations of a nonhomogeneous anisotropic elastic solid rigidly clamped in the rectangular region Ω. The scheme has O(h2) convergence in the W 12 (Ω) norm if the solution\(\overline u \)(x) is contained in the Sobolev space W 32 (Ω).

The first and second fundamental boundary-value problems of elasticity theory for a transversally isotropic paraboloid of revolution

We give an exact solution of the interior and exterior problems of elasticity theory for a transversally isotropic paraboloid of revolution in the case when the stresses prescribed on its surface or the displacements along one variable can be represented by a Hankel integral and those along the other variable can be expanded in a trigonometric series. It is assumed that the roots of the characteristic equation are of multiplicity greater than one.

Rate of convergence bound of difference schemes for the first boundary-value problem of elasticity theory in arbitrary domains

The fictitious domain method and exact difference scheme operators are used to construct a difference scheme for solving the first boundary-value problem of elasticity theory in arbitrary domains. The rate of convergence bound is established for this theorem, where (x) is the polylinear completion of the grid function\(\bar y(x), \bar u(x)\) is the solution of the original problem.

The equilibrium boundary element method in boundary-value problems of elasticity theory

An equilibrium boundary element method is proposed for solving boundary-value problems in the theory of elasticity, thermo-elasticity, the dynamical theory of elasticity, bar torsion calculations, and the bending of a plate. The idea is to use simultaneously the method of constructing bundles of functions which exactly satisfy the equilibrium equations, the boundary variational equations of mechanics, and the methods of discrete finite-element approximation. The variational method of constructing the resolving boundary equations ensures that the linear system is symmetric and easily coupled to the finite-element method. Since volume integrals are eliminated the dimensions of the problem are reduced by one, but, unlike the boundary element method, there is no need to know the fundamental solutions. The solution of some bar torsion and plate bending problems confirms the high numerical efficiency of the method.

Difference scheme for the first boundary-value problem of elasticity theory for bodies of revolution

Difference scheme for the first boundary-value problem of elasticity theory for bodies of revolution

Accuracy of solution of some boundary-value problems of elasticity theory by finite-difference methods

A system of nonlinear equations associated with finite-difference solution of problems of mathematical physics is considered. A so-called bounding linear system is constructed for the given nonlinear system. The solution accuracy of the bounding system is estimated by the magnitude of the discrepancy vector, which provides an estimate of the solution accuracy of the nonlinear system for the worst-case values of the nonlinear coefficients.

Stress-strain states in a multisheet surface with cuts

The first, second and mixed fundamental boundary-value problems of elasticity theory are considered on an n-sheet Riemann surface with straight-line cuts joining the branch points. The cuts are such that their edges are situated in different planes. Complex potentials are constructed, asymptotic representations of the stresses and derivatives of the displacement components are obtained near the vertices of the cuts and invariant Γ- integrals /1/ are obtained, by the method of reduction to a matrix Riemann boundary-value problem. The first and second fundamental problems for an n= 2 Riemann surface were solved /2/ by the Riemann boundary-value problem method for a Riemann surface. For n=1 the results are identical with previously known results for a plane /3/.