The impact of transient load distribution on the surface of the elastic layer

Аuthors

Kuznetsova E. L.^1*, Tarlakovsky D. V.^2**, Fedotenkov G. V.^3***, Medvedskiy A. L.^4****

1. Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia
2. Institute of Mechanics Lomonosov Moscow State University, 1, Michurinsky prospect, Moscow, 119192, Russia
3. ,
4. Central Aerohydrodynamic Institute named after N.E. Zhukovsky (TsAGI), Zhukovsky, Moscow region, Russia

*e-mail: vida_ku@mail.ru
**e-mail: tdvhome@mail.ru
***e-mail: greghome@mail.ru
****e-mail: medvedskyal@nrczh.ru

Abstract

The plane transient problem of the linear elasticity theory for the thin layer of the aircraft paneling represented as an homogeneous isotropic layer of uniform thickness is studied. An arbitrary distribution of the surficial load is considered.
A rectangular Cartesian frame is used; axe is directed deep into an elastic layer, and one is directed along its free surface z=0.It is supposed that the load depends not on the coordinate y therefore a plane problem can e formulated. Using the superposition principle for the fundamental solutions we have the displacement vector represented as follows:
where is a matrix of fundamental solutions (or a transient function matrix).
For the transient functions we have the initial-boundary value problems for the layer under the concentrated loads distributed according the law where is the delta function. To compute the transient functions analytically the Fourier integral transforms with respect to the coordinate x and the Laplace one with respect to the time variable are used. The images of the transient functions are represented as the exponential power series; each member is the sum of products of homogeneous rational functions of the degree (-1) in terms of the transformation parameters and square roots  and exponential functions in terms of the linear combination of these roots. Here  are parameters of Fourier and Laplace transformations respectively, and is the dimensionless parameter defined by the properties of the material. This approach allows one to use the joint method of Laplace and Fourier transforms’ inversion that’s based on the construction of images’ analytic representations. If the exponential function’s degree contains one root only the inverse transforms can be represented explicitly. If the degree of exponent has more than one root the modified algorithm for joint inversion of Fourier-Laplace images has to be used to obtain the inverse transforms.
To obtain the solutions the numero-analytical algorithm based on the method of quadrature and Simpson formulae is developed. Some examples of solutions are shown.

Keywords:

thin skin of the aircraft, transient effects, elastic waves, integral transformation, the influence function of the elastic layer, the numerical-analytical algorithm

References

1. Gorshkov A.G., Medvedsky A.L., Rabinsky L.N., Tarlakovsky D.V. Volny v sploshnykh sredakh (Waves in continuum media), Moscow, Fizmatlit, 2004, 472 p.
2. Gorshkov A.G., Tarlakovskii D.V., Fedotenkov G.V. Mekhanika tverdogo tela, 2000, no. 5, pp. 151-158.
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4. Suvorov Ye.M., Tarlakovskii D.V., Fedotenkov G.V. Prikladnaya matematika i mekhanika, 2012, vol. 76, no. 5, pp. 511-518.
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6. Gorshkov A.G., Amar Abdul Karim Salman, Tarlakovs D.V.,Fedotenkov G.V. Mekhanika tverdogo tela, 2004, vol. 39, no. 3, pp.65-72.

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