Let us analyze semi-infinite Region S in Plane Z.
The Function, mapping the lower half-plane ImZ<0 onto the investigated Region S, is accepted in the form of a polynomial, as it was done in the work [1]:
(1)
where 
are real numbers.
Let us assign the boundary values of Displacement Component 
and 
at Segment L´ of Boundary L of Region S, occupied by an elastic body, and at the rest Segment L´´ - boundary values of Stress Component N(t) and T(t). It is necessary to determine the stressed state of Region S.
Along with the given problem we will analyze the problem of stressed state determination, presuming that the main external force vectors 
are assigned, which are applied to every Segment Lk separately.
We will refer to the above-formulated problems as problems A and B, according to [2]. Thus to obtain a common solution of the given problems let us use the methods developed by N.I. Muskhelishvili [2], as well as in the paper of I.N. Kartsivadze [3]. Let us extend here the approach of the author of paper [3] to the case of a half-plane, subjected to conformal mapping.
As it is known [2], the formulas, expressing the displacement and stress components through Functions 
and 
of Complex Variable 
take the form of
(2)
(3)
Summing up the left and right parts of these equations, and, then proceeding to conjugate values, we get
(4)
The formula, expressing Displacement Components u and ν in Cartesian coordinates, is given by
(5)
where: 
If Functions 
and 
are assigned, then Functions and are entirely determined. And in case Functions and are assigned, Functions and are determined with accuracy to arbitrary constants. Therefore Eq.(5) may be written in the form of
(6)
Assuming now, that 
is a rational function, let us apply Function determination to Region 
, supposing that
(7)
where: 
lies in the upper half-plane.
This operation must be carried out so that the values of Function , determined in the upper half-plane, can be analytically extended to the lower half-plane through the unoccupied segments of boundaries (if there are any).
Taking into account (7), having changed for 
and proceeded to conjugate values, we obtain
(8)
Eq. (8) expresses the values of for the lower half-plane through values of , for the lower half-plane, as well as for the upper half-plane.
The determination of Function can be extended to the upper half-plane, provided that the relation holds in the upper half-plane
Integrating both parts of expression (7) over , and, dropping the arbitrary constant value, we have
(9)
where: lies in the upper half-plane.
Similarly,
(10)
where: lies in the lower half-plane.
Thus, Stress and Displacement Components can be expressed through one and the same Function , determined in the lower half-plane, as well as for the upper half-plane.
To fulfill further solution it is necessary to employ Expression (2) and Expression (4), taken in the form of:
(11)
It is necessary to explain that Expression (11) is obtained after Function 
is subtracted from the right part of (4) and after the same function replaced with its expression obtained from Eq. (8) after having proceeded to conjugate values is added there. Let us notice that the expression determined by Eq. (8) is meant by Function .
For Displacement Components u and ν, replacing in Eq. (6) with expression (10), we obtain the following relation
From this point on we will also need the expression for 
, where,
It can be obtained if we differentiate both parts of Eq. (5) over . Then we have
Having added Function 
to the right part of above relation and subtracted the same Function, replaced with its expression, obtained from Eq. (8) after proceeding to conjugate values, we have
(12)
Let us note that both of Functions and are holomorphic in the lower half-plane.
Now it is necessary to ascertain the behaviour of Function , extended by Eq. (7) to the upper half-plane. To do that it is enough to analyze the behaviour of Product 
, referring to Eq. (7) which determines Function when .
Now let us proceed to the analysis of problems A and B.
Let us denote the points of Real Axis 
which correspond to the points 
of the Cutout L as 
, a segment of a real line which corresponds to L´ as 
, and the rest segment of the line as 
.
As we know [1] the solution of the first main problem, it is more convenient to take account of the influence of the forces assigned at Segment L´´ separately. Consequently, it can be supposed that Segment L´ of Boundary Lis free from external stresses.
Then, based on Eqs. (11) and (12) the boundary conditions in both problems are given by
(13)
(14)
Eq. (13) proves that Segment of the real axis is not saltus function for Function , i.e. that Function is a holomorphic one on a cut-down plane except for finite number of points where it may have poles. The same refers to Function 
.
To determine it, let us refer to Eq. (14) representing a boundary condition of the problem well-known in the theory of functions of a complex variable, which is Riemann problem or Gilbert problem. It is completely investigated in the works of N.I. Muskhelishvili [4], F.D. Gakhov [5] and other authors. N.I. Muskhelishvili calls it "boundary value problem of linear conjugation", or "conjugate problem" for short.
Assuming that Function may have poles of order not higher than 
in Points 
and employing the results of the conjugate problem solution [2, p.397], we obtain for the sought function
(15)
where:
(16)
with 
implying such a branch that 
, and 
is a rational function of the form
(17)
where: 
is a polynomial of degree not higher than m+r.
Coefficients included in Expression (17) are determined based on the additional conditions of Problems A and B.
Let us consider an example of a stamping tool with a footing parallel to axis Oξ , provided that the stamping tool moves only vertically, so
(18)
Besides, suppose that external forces influencing the stamping tool have a resultant force directed vertically downwards, so
(19)
where: P is a positive constant set in advance.
In this case Eqs. (7) and (8) take the form of
(20)
(21),
as for rational Function (1) there are poles of order 
in Point 
and at infinity, and for Function 
there are poles of order 
in the same point . Thus, for Function 
there are poles of order not higher than in Point . Assuming that together with disappear at infinity, according to (15) and taking into account (18), we obtain
(22)
where: 
are to be determined.
According to (16)
For large values of 
it takes the form of
(23)
where: 
Note that Function corresponding to Function that is determined both in the lower and in the upper half planes, should be holomorphic in the lower half plane. But this is not the case, because according to (21) it has a pole in Point .
Constants are determined based on the holomorphy of Function in Point . Denote them as 
.
Now it is necessary to figure out constant value C0. Let us remark here that for large values of it follows from (22) and (23) that
On the other side, Function for large values of [2, p.339] involves
therefore
(24)
Thus, Function that is set by Eq. (22) is completely determined.
Applying (19) and taking into account (24) result in
(25)
Formulas for estimating pressure P(t) and tangential stress T(t), influencing the body under the stamping tool, are analogous to those of classical case and take the form of
(26)
From this, taking into account Eq. (22), it follows that
(27)
Now, separating the real and the complex parts it is possible to obtain formulas for pressure P(t) and tangential stress T(t).
In conclusion let us emphasize the following circumstance: if we assume for Function (1) that B=1, and put to zero all the other coefficients, we obtain a function that allows self-reflecting of the half-plane.
In this case Function (25) becomes
(28)
From this it is easy to get formulas well-known in mathematical theory of elasticity, which allow solving a classic problem of a stamping tool with a linear horizontal footing
(29)
that was obtained independently by V.M. Abramov [6] who used the method of integral transformations and by N.I. Muskhelishvili [2] who applied methods of the theory of functions of a complex variable.
References:
- Bogomolov А.N. Estimating the bearing capacity of constructions´ foundations and the stability of soil bodies in elastoplastic formulation/ A.N.Bogomolov// - Perm: PSTU, 1996.
- Muskhelishvili N.I. Some main problems of mathematical theory of elasticity/ N.I. Muskhelishvili//. -Мoscow: Nauka, 1966.
- Kartsivadze I.N. Efficient main problem solving for theory of elasticity for some regions. / I.N. Kartsivadze // Newsletter of АN Grouz. SSR, v.VII, №8, 1946, pp. 507-513.
- Muskhelishvili N.I. Singular integral equations / N.I. Muskhelishvili//. - Moscow: Nauka, 1968.
- Gakhov F.D. Boundary value problems/ F.D. Gakhov // -Moscow: Fizmatgiz, 1963.
- Abramov V.М. Problem of elastic half-plane contact with absolutely rigid foundation taking into account frictions /V.M.Abramov// Report of АN USSR, v. XVII, № 4, 1937, pp. 173 -178.
The work was submitted to III international scientific conference «Basic Research», Dominican Republic, April, 10-20, 2008, came to the editorial office 12.02.2008.
Библиографическая ссылка
Bogomolov A.N., Bogomolova O.A., Ushakov A.N., Shiyan S.I. SOLUTION OF THE MAIN MIXED PROBLEM OF THE ELASTICITY THEORY FOR A HALF-PLANE WITH CURVED BOUNDARY // European Journal of Natural History. – 2008. – № 2. – С. 79-84;URL: https://world-science.ru/ru/article/view?id=20630 (дата обращения: 25.11.2024).