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CONSTRUCTION OF A DISCONTINUOS SOLUTION OF THE WAWE EQUATION FOR A
SPHERICAL DEFECT
Nazarenko O.
PhD, Assistant Professor, Odesa State Academy of Civil Engineering and Architecture https://doi.org/10.5281/zenodo.6778204
Abstract
The study of the interaction of undeformed shells with the surrounding elastic medium is of practical value, due to the increase in the impact resistance of ships against underwater and air explosions, the improvement of methods and methods of underwater acoustics, and the provision of seismic resistance of hydraulic structures and their components. Thus, the development of mathematical methods for solving problems on the interaction of non-stationary (stationary) waves with different objects, including shell type, is relevant.
Among the analytical methods, the following can be distinguished: the method of integral equations (the potential method), the method of separation of variables and its various modifications (the Fourier method and its generalizations in vector and scalar forms, as well as reduction to infinite systems of algebraic equations), the method of the theory of functions of a complex variable. These methods have proven themselves well when applied to canonical domains (the equations of their boundary surfaces are reduced to standard canonical forms).
At present, various numerical methods of finite differences, finite elements, etc. are widely used to solve spatial problems. The proposed work is devoted to solving a spatial problem of elasticity theory for a spherical segment by the discontinuous solution method [1, 2].
Work goals. Generalization of the method of discontinuous solutions [1, 2] to the case of spherical defects (cracks or thin rigid spherical inclusions). A method for constructing a discontinuous solution of the wave equation for a spherical coordinate system is proposed.
Keywords: wave equation, elasticity theory, defect, inclusion, crack, discontinuous solution, jump, spherical coordinates, stress, displacement.
Main material. Under the defect (from the point of view of mechanics) we mean [1, 2] a part of the surface, at the intersection of which the stresses and displacements of the first kind suffer discontinuities. As a classical defect, we can consider some mathematical cut along the specified part of the surface (crack). A certain rigid inclusion in the form of a shell (cavity), the middle surface of which coincides with the same part of the surface, can also be attributed to such defects. Consider, as one of the special cases, when a part of a spherical surface serves as a defect.
Let's set its geometric parameters in the form: r = R,0<6 <u,-n<y<n,
where r, 9,(p are the parameters of the spherical coordinate system. It is widely known that the solution of the equations of motion of an elastic isotropic medium can be expressed in terms of wave functions [3, 4]. Therefore, before proceeding with the construction of a discontinuous solution for the equations of motion, one should construct a solution for the wave equation 1 d2
hib----= 0,0<r<m,0<9 <
c2 dt2
n,\<p\ <n,t>0, (1)
where h is the Laplace operator expressed in spherical coordinates.
Under the discontinuous solution of equation (1), which is given in the entire space for a spherical defect r = R,0<6<u,-n<y<n (2) one should understand such a solution to equation (1), which must satisfy it everywhere, excluding only the points of the defect itself (2) (R is the radius of a
spherical defect). At these points, the function and its normal (to the surface of the considered defect) derivative suffer discontinuities of the first kind and their jumps are given, for which we introduce special notation
ip(R - 0,9, <p, t) - ip(R + 0,9, <p, t) = {tp), ^'(R - 0,9,y,t) - ^'(R + 0,6,y,t) = {$'). In addition, here and everywhere below in the text we will denote the derivative with respect to the variable r by a prime, with respect to 9 by a dot, and with respect to the variable cp by a comma. To construct such a solution, we use the same scheme as in the materials [1, 2].
By successively applying to equation (1) the integral transformations of Laplace (with respect to the variable t), Fourier (with respect to the variable (p)
% = 10° V (r, 8,9, t)e-ptdt, %n = S*nxPp(r,e,cp)e-in<rdcp (3) and Legendre (with respect to the variable 9), %nk(r) = j"sin8Pln\cos8)d8 (4) (P£ (cos9) is the adjointed Legendre polynomial), we reduce equation (1) to the following one-dimensional form:
(r2 ^pnk(r))
k(k + l) %nk(r)
2
^pnk(r) = 0, (5)
where 0 < r < m.
At this stage, it is necessary to construct a discontinuous solution of this equation with predetermined jumps
{ %nk) = %nk(R -0)- %nk(R + 0), (6)
{Vpnk) = VpnkiR -0)- Vpnk(R + 0).
The values of these jumps will be determined based on the boundary conditions of the problem.
If in (5) we make a change of variables of the form XPnk(r) = ^^pnk(r), then this equation is transformed into the Bessel equation. Let us apply the Hankel transformation to the resulting equation
Xpnk = J r/k+i^(ar)zPnk(r)dr
0
to get rid of the variable r according to the generalized scheme [1] (in this formula, Jk+y (ar) is the cylindrical Bessel function).
Using the obtained results, we find the dimension-less Hankel transform from the equation (5), expressing them in terms of jumps (6). Further, applying to this expression the inversion formula for the Hankel transform, as well as using formula 6.541(1) from [5], we find the necessary discontinuous solution of equation (5) with jumps (6)
VPnk(r)=R2 ( )DKp(r,R)
pnkf
d
- ( Tppnk)^Dk,p
(r,R)
Dk,p(r,R) =
i •JrR
Iv (iL) Kv(^),r<R,v = k + 1/2,k = 0,1,2,
(7)
(Iv (z), Kv (z) are respectively modified Bessel and Macdonald functions). Further, to obtain a discontinuous solution of the original wave equation, one should use the inversion formulas for the Legendre transforms [6]
R2
^pn (8)
(r,8)= ^ ^pnkir)aknPln](cos6),
k=\n\
akn = '
(k - \n\)! (2k + 1)
2(k + lnl)! ' as well as for the Fourier and Laplace transforms. Thus, applying transformation (8) to formula (7), we obtain the following equation
%n(r,e) = R2
f
-0
-f
Tnp(8,T) simdr
Tnp(8,T)sinT dr
(9)
Tnje,T) = (ip')Mn:P(e,T;r,R),înje,T) =
( &
pn
n,p\ ' - S \ T pnr
)^MnjP(e,T-,r,R),
Mn>p(9,T;r,R) =
\n\
knrk
(cos9)Pk
\n\
(cosT)Dkp(r,R).
In the event that a steady process of medium oscillations is considered (occurring according to a harmonic law), then the potential from the wave equation (1) can be written in the following form: ^(r,8,y,t) = e-iM°t{p(r,e,ç). (10) This makes it possible to exclude the use of the direct and inverse Laplace transforms with respect to the variable t, which greatly simplifies the calculations. Then, if in equation (5), instead of the parameter p, we substitute the value
p = —iw0, then we obtain a new equation, which is the solution for the function ift(r, 9, (p).
In contrast to equation (7), the discontinuous solution in this case will take a slightly different form:
$nk(r) = R2 [( Vpnk) Dkill(r,R) - ( %nk)-^DKß(r,R)]
j(i)
DkAr,R) =
ni \jv(Rß)H^(rß),r>R,ß =
dR
(11)
2^rR \jv(rß)H^1)(Rß),r <R,v = k + 1/2,k = 0,1,2,.
0
c
If in (11) we invert the Legendre transforms, then we obtain an equation of the following form:
4>n(r, d) = R2 [/" (e, t) sinTdT -
I"Pn^(e,T)sinTdx], (12)
d
Wk(z)\z=-i? = Iv(z)Kv(z)\z=-i? =
(13)
jK1)(0Jv(0=M0,
Mn^(8,r,r,R)
= aknPlnl (cos9)Plnl (c0ST)Dk^(r, R).
When substituting the value p = -iw0 in (7), it is
(1)
necessary to choose the first Hankel function H^ ' (z) in the kernel Dk^(r,R). It is she who provides the condition of radiation at infinity. The second function Jv(z) in this kernel is the cylindrical Bessel function.
When using discontinuous solutions of the form (9) and (12) in specific problems of the theory of elasticity, it is necessary to use the integral representation for the following function:
v = k + l/2.
To obtain relation (13), it suffices to use formula 5.9.2(14) from [7], which allows us to expand the functions n0(6) = I0(8) — L0(8) (L0(8) — the second Struve function [5]) into a series in the orthogonal system of functions cos [(fc + d] and therefore
Wk(z) = t-1- f n0 (2zcoscos (k + ^jd
de.
Integrating by parts based on (13), we establish an important relationship:
^(0 = ^,^(0 =
I
nsin[(k+1)r] g
2k + 1
0 -iß^i^fecos^dT, (14)
where Y0(z) = Jv(z) - iH0(z), H0(z) is the first Struve function.
n
0
References:
1. Popov G. Ya., The Concentration of Elastic Stresses Near Punches, Cuts, Thin Inclusions and Supports. Nauka, Moscow, 1982.
2. Popov G. Ya., Problems of stress concentration in the neighbourhood of a spherical defect. Advances in Mechanics 15, 1-2, 71-110, 1992.
3. Guz' A. N., Kubenko V. D. and Cherevko M. A., The Diffraction of Elastic Waves. Naukova Dumka, Kiev, 1978.
4. Gorshkov A. G. and Tarlakovskii D. V., The
Unsteady Aerohydroelasticity of Spherical Solids. Nauka, Moscow, 1971.
5. Gradshtein and Ryzhik I. M., Tables of Integrals, Sums, Series and Products. Nauka, Moscow, 1971.
6. Titchmarsh E. C., Expansion in Eigenfunc-tions Related to Second-order Differential Equations, Pt 1. Izd. Inost. Lit., Moscow, 1960.
7. Prudnikov A. P., Brychkov Yu. A. and Marichev O. I., Integrals and Series. Nauka, Moscow, 1983.
