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This paper presents a model of a 1D–1D dynamic multi-structure, supporting propagation of a transition wave. It is used to explain the recent phenomenon of the collapse of the San Saba bridge. An analytical model is supplied with illustrative numerical simulations.
An eigenvalue problem is considered for a multi-structure consisting of a three-dimensional finite solid connected to an arbitrary smooth elastic thin shell of revolution. Two-sided estimates are obtained for the first six eigenfrequencies of the multi-structure. Explicit asymptotic formulae are given.
We consider an eigenvalue problem of three-dimensional elasticity for a multi-structure
consisting of a finite three-dimensional solid linked with a thin-walled elastic cylinder. An
asymptotic method is used to derive the junction conditions and to obtain the skeleton model
for the multi-structure. Explicit asymptotic formulae have been obtained for the first six
This work presents an asymptotic algorithm for the derivation of equations of thin elastic
shells. The algorithm is based on the analysis of a boundary value problem for the Navier
system in a thin region. The analysis covers both the membrane theory and the moment
theory of elastic shells, including the eigenvalue problems.
We present a simple and rigorous mathematical model and efficient numerical algorithm for
the three-dimensional thermal stress analysis of composite structures used in high-temperature
catalytic combustors. Numerical experiments are carried out for three types of cell geometries.
A homogenization algorithm is implemented, and asymptotic formulae are derived for the
effective elastic moduli of the periodic structures.
An algorithm, based on a discrete nonlinear model, is
presented for evaluation of the critical
shear stress required to move a dislocation through a lattice.
The stability of solutions of the
corresponding evolution problem is analysed. Numerical results provide
upper and lower
bounds for the critical shear stress.
The Pólya–Szegö dipole tensors
are employed for the analysis of plane elasticity problems in
non-homogeneous media. Classes of equivalence for defects
(cavities and rigid inclusions) are
specified for the Laplacian operator and elasticity equations:
composite materials with defects
of the same class have the same effective elastic moduli.
Explicit asymptotic formulae for
effective compliance matrices of dilute composites are
A class of three-dimensional crack problems is considered, of which a prototype example is provided by a half-space containing a long internal crack, located in a plane perpendicular to the boundary. By means of an asymptotic procedure, the original three-dimensional problem is split up into a sequence of two-dimensional formulations. Results of its numerical implementation are in good agreement with results of more computer-intensive finite-element calculations.
Integral characteristics, such as elastic polarization matrices of elastic inclusions and cavities, are described. The matrix of elastic polarization of a finite cavity is constructed in the case of the two-dimensional Lamé operator under the assumption that the geometry of the domain occupied by the cavity is defined by a conformal mapping from the unit disk. Examples and applications of these integral characteristics in the theory of cracks are considered.
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