No abstract

The mixed problem for the fixed semi-strip is investigated in this article for the three cases of the applied mechanical load. The solution of the boundary problem is reduced to the solution of the singular integral equation (SIE) with regard to the unknown displacements derivative. Three cases of SIE are investigated: when the mechanical load is applied on the center of the semi-strips edge, when the mechanical load is distributed near the left lateral side and when the mechanical load is distributed on the whole semi-strip's edge. In the first case SIE is solved by the using of the orthogonal polynomials method. In the second and third cases the characteristical equations to SIE are constructed, and the SIE are solved with the help of the generalized method. The stress state of the semi-strip is investigated for the three cases. Keywords semi-strip • singular integral equation • fixed singularity • orthogonal polynomials method • generalized method

No abstract

No abstract

No abstract

The three-dimensional dynamic theory of elasticity is applied to investigate the mechanical properties of virus capsid. The idealized model of the virus is based on the 3D boundary-value problem of mathematical physics formulated in spherical coordinate system for the steady-state oscillation process. The virus is modeled as a hollow elastic sphere filled by acoustic medium and is located in different acoustic medium. The stated boundary-value problem is solved with the help of the integral transform method and method of the discontinuous solutions. As a result, the exact solution of the problem is obtained. The numerical calculations of the virus elastic characteristics are carried out.The development of virus mathematical model is necessary to representing the effect of parameters variation on the behavior of a virus as a dynamic system. Such mathematical models based on the reasonable biological assumptions were obtained earlier within three main interdisciplinary approaches: 1) the hydrodynamic approach [23,21,30]; 2) the approach with the use of the numerical methods for solving the nonlinear problems of hydrodynamics and elasticity [16,18,28,29,36]; 3) the approach based on the linear elasticity models [15,34,36]. The mentioned models allowed to obtain many important characteristics of the virus, but they could not fully solve the problem of investigation of the virus as a 3D elastic object. In this paper the authors first propose to use the complete system of motion equations of linear elasticity for representation of virus wave field. It allows to take into consideration the virus 3D structure and to obtain its new qualitative characteristics.The morphology of icosahedral viruses ranges from highly spherical to highly faceted, and for some viruses a shape transition occurs during the viral life cycle. This phenomena is predicted from continuum elasticity, via the buckling transition theory by Nelson [22], in which the shape is dependent on the Foppl -von Karman number g , which is a ratio of the two-dimensional Young's modulus, Y , and the bending modulus ae : g= 2 / YR ae (R is the virus radius). However, until now, no direct calculations have been performed on atomic-level capsid structures to test the predictions of the theory.The elasticity and mechanical stability of empty and filled viral capsids under external force loading are studied in a combined analytical and numerical approach. Quantitative measurements of the mechanical response of nanosized protein shells (viral capsids) to large-scale physical deformations were reported in [29]. These measurements were compared with theoretical descriptions from continuum modeling and molecular dynamics. In [22] it was shown that the icosahedral packings of protein capsomeres proposed by Caspar and Klug for spherical viruses become unstable to faceting for sufficiently large virus size. A model, based on the nonlinear physics of thin elastic shells, produced excellent one-parameter fits in real space to the full three-dimensional shape of large spherical v...

The elastic semi-strip under the dynamic load concentrated at the centre of the semi-strip’s short edge is considered. The lateral sides of the semi-strip are fixed. The case of steady-state oscillations is considered. The initial problem is reduced to the one-dimensional problem with the help of the semi-infinite sin-, cos-Fourier’s transform. The one-dimensional problem is formulated in the vector form. Its solution is constructed as a superposition of the general solution for the homogeneous equation and the partial solution for the inhomogeneous equation. The general solution for the homogeneous vector equation is found with the help of the matrix differential calculations. The partial solution is expressed through Green’s matrixfunction, which is constructed as the bilinear expansion. The inverse Fourier’s transform is applied to the derived expressions for the displacements. The solving of the initial problem is reduced to the solving of the singular integral equation. Its solution is searched as the series of the orthogonal Chebyshev polynomials of the second kind. The orthogonalization method is used for the solving of the singular integral equation. The stress-deformable state of the semi-strip is investigated regarding both the frequency of the applied load, and the load segment’s length.

No abstract