2D Elastostatic Problems in Parabolic Coordinates

In the present chapter, the boundary value problems are considered in a parabolic coordinate system. In terms of parabolic coordinates, the equilibrium equation system and Hooke ’ s law are written, and analytical (exact) solutions of 2D problems of elasticity are constructed in the homogeneous isotropic body bounded by coordinate lines of the parabolic coordinate system. Analytical solutions are obtained using the method of separation of variables. The solution is constructed using its general representation by two harmonic functions. Using the MATLAB software, numerical results and constructed graphs of the some boundary value problems are obtained.


Introduction
In order to solve boundary value and boundary-contact problems in the areas with curvilinear border, it is purposeful to examine such problems in the relevant curvilinear coordinate system. Namely, the problems for the regions bounded by a circle or its parts are considered in the polar coordinate system [1][2][3][4], while the problems for the regions bounded by an ellipse or its parts or hyperbola are considered in the elliptic coordinate system [5][6][7][8][9][10][11][12][13], and the problems for the regions with parabolic boundaries are considered in the parabolic coordinate system [14][15][16]. The problems for the regions bounded by the circles with different centers and radiuses are considered in the bipolar coordinate system [17][18][19]. For that purpose, first the governing differential equations are expressed in terms of the relevant curvilinear coordinates. Then a number of important problems involving the relevant curvilinear coordinates are solved.
The chapter consists of five paragraphs. Many problems are very easily cast in terms of parabolic coordinates. To this end, first the governing differential equations discussed in present chapter are expressed in terms of parabolic coordinates; then two concrete (test) problems involving parabolic coordinates are solved.
The second section, following the Introduction, gives the equilibrium equations and Hooke's law written down in the parabolic coordinate system and the setting of boundary value problems in the parabolic coordinate system. Section 3 considers the method used to solve internal and external boundary value problems of elasticity for a homogeneous isotropic body bounded by parabolic curves. Section 4 solves the concrete problems, gains the numerical results, and constructs the relevant graphs.
Section 5 is a conclusion.

Problems statement 2.1 Equilibrium equations and Hooke's law in parabolic coordinates
It is known that elastic equilibrium of an isotropic homogeneous elastic body free of volume forces is described by the following differential equation [20]: are elastic Lamé constants; ν is the Poisson's ratio; E is the modulus of elasticity; and U ! is a displacement vector. By projecting Eq. (1) onto the tangent lines of the curves of the parabolic coordinate system (see Appendix A), we obtain the system of equilibrium equations in the parabolic coordinates.
In the parabolic coordinate system, the equilibrium equations with respect to the function D, K, u, v and Hooke's law can be written as [20][21][22]: are Lamé coefficients (see Appendix A), u, v are the components of the displacement vector U ! along the tangents of η, ξ curved lines, and c is the scale factor (see Appendix A). And in the present paper, we take c ¼ 1, κ À 2 ð Þ= κμ ð Þ Á D is the divergence of the displacement vector, K=μ is the rotor component of the displacement vector; σ ξξ , σ ηη and τ ξη ¼ τ ηξ are normal and tangential stresses; and sub-indexes ðÞ ,ξ and ðÞ ,η denotes partial derivatives with relevant coordinates, for example, K ,ξ ¼ ∂K ∂ξ .

Boundary conditions
In the parabolic system of coordinates ξ, η À∞ < ξ < ∞, 0 ≤ η < ∞ ð Þ , exact solutions of two-dimensional static boundary value problems of elasticity are constructed for homogeneous isotropic bodies occupying domains bounded by coordinate lines of the parabolic coordinate system (see Appendix A).
The elastic body occupies the following domain (see Figures 1 and 2): Boundary conditions on the linear parts ξ ¼ 0 and η ¼ 0 of the consideration area enable us to continue the solutions continuously (symmetrically or antisymmetrically) in the domain, that is, the mirror reflection of the consideration area in a relationship y ¼ 0 line (see Figures 1b and 2b).

Solution of stated boundary value problems
In this section we will be considered internal and external problems for a homogeneous isotropic body bounded by parabolic curves.

Interior boundary value problems
Let us find the solution of problems (2), (3), (4a) (see Figure 1a), and (7)- (10) in class C 2 D ð Þ (for D area shown in Figure 1b). The solution is presented by two harmonious φ 1 and φ 2 functions (see Appendix B). From formulas (B11)-(B13), after inserting α ¼ η 1 and making simple transformations, we will obtain: D ¼ κμ The stress tensor components can be written as From (12) by the separation of variables method, we obtain (see Appendix A) where 10 , a 02 , … , b 04 are constant coefficients. When n ¼ 0 and 0 < ξ < ξ 1 , then the terms ξ, η and ξη will not be contained in φ 10 and φ 20 . If the foregoing solutions are presented in expressions of φ 10 and φ 20 , then it would be impossible on ξ ¼ ξ 1 to satisfy the boundary conditions, and grad Þwill not be bounded in the point M 0, 0 ð Þ. Provision. We are introducing the following assumptions: 1. ξ 1 is a sufficiently great positive number (see Appendix C).
3. When stresses are given on η ¼ η 1 , the main vector and main moment equal zero.
When at η ¼ η 1 u and v are given, then it is expedient to take instead of them as their equivalent the following expressions: and if at η ¼ η 1 2μ σ ξη are given, then instead of them we have to take their equivalent following expressions: Considering the homogeneous boundary conditions of the concrete problem, we will insert φ 1 and φ 2 functions selected from the (14) in the right sides of (15) or (16), and we will expand the left sides in the Fourier series. In both sides expressions which are with identical combinations of trigonometric functions will equate to each other and will receive the infinite system of linear algebraic equations to unknown coefficients A 1n and A 2n of harmonic functions, with its main matrix having a block-diagonal form. The dimension of each block is 2 Â 2, and determinant is not equal to zero, but in infinite the determinant of block strives to the finite number different to zero.
It is very easy to establish the convergence of (11) and (13) functional series on the area D ¼ Àξ f g by construction of the corresponding uniform convergent numerical majorizing series. So we have the following: Proposal 1. The functional series corresponding to (11) and (13) are absolute and uniform by convergent series on the area D ¼ Àξ

Exterior boundary value problems
We have to find the solution of problems (2), (3), (5a) (see Figure 2a), (7), (8), (10), and (10 0 ), which belongs to the class C 2 Ω ð Þ (see region Ω on Figure 2b). The solution is constructed using its general representation by harmonic functions φ 1 , φ 2 (see Appendix B). From formulas (B11)-(B13), following inserting α ¼ η 1 and simple transformations, we obtain the following expressions: The stress tensor components can be written as: If u and v are given for η ¼ η 1 , then we take φ 3 ¼ 0, and when (18), by the separation of variables method, we obtain where B 2n e Ànη cos nξ ð Þ: Provision. As in the previous subsection we make the following assumptions: • ξ 1 is a sufficiently large positive number (see Appendix C).
• When stresses are given on η ¼ η 1 , the main vector and main moment will equal zero.
When u and v are given at η ¼ η 1 , then instead of them, it is expedient to take the following expressions as their equivalent: and if at η ¼ η 1 h 2 0 2μ σ ηη and h 2 0 2μ σ ξη are given, then instead of them we have to take the following expressions as their equivalent: Just like that in the previous subsection, considering the homogeneous boundary conditions of the concrete problem, we will insert φ 1 and φ 2 functions selected from (20) in Eq. (21) or (22), and we will expand the left sides in the Fourier series. Both sides of the expressions, which show the identical combinations of trigonometric functions, will equate to each other and will receive the infinite system of linear algebraic equations to unknown coefficients A 1n and A 2n of harmonic functions, with its main matrix having a block-diagonal form. The dimension of each block is 2 Â 2, and the determinant does not equate to zero, but in the infinity, the determinant of block tends to the finite number different from zero.
As in the previous subsection, we received the following: Proposition 2. The functional series corresponding to (17) and (19) are absolute and a uniformly convergent series on region Ω ¼ Àξ

Test problems
In this section we will be obtained numerical results of internal and external problems for a homogeneous isotropic body bounded by parabolic curves when normal stress distribution is applied to the parabolic border.

Internal problem
We will set and solve the concrete internal boundary value problem in stresses. Let us find the solution of equilibrium equation system (2) of the homogeneous isotropic body in the area Figure 1a), which satisfies boundary conditions (7a), (8a), (9a), and (10).
As seen, the main matrix of system (26) has a block-diagonal form, dimension of each block is 2 Â 2. Thus, two equations with two A 1n and A 2n unknown values will be solved. After solving this system, we find A 1n and A 2n coefficients, and in putting them into formulas (24) and (25), we get displacements and stresses at any points of the body.

Figure 5.
Stresses and displacements at points whereP 1n andP 2n are the coefficients of expansion into the Fourier series of functions f 1 ξ ð Þ ¼ À Pη 1 ξ 2 þη 2 1 and f 2 ξ ð Þ ¼ Pξ , respectively (f 1 ξ ð Þ, according to sinuses, and f 2 ξ ð Þ, according to cosines). As it can be seen, the main matrix of system (30) has a block-diagonal form, and the dimension of each block is 2 Â 2. Thus, two equations with two B 1n and B 2n unknown values will be solved. After solving this system, we find the values of B 1n and B 2n coefficients and put them into formulas (28) and (29) to get displacements and stresses at any points of the body.
Numerical results are obtained for some characteristic points of the body, in particular, Figure 2a), for the following data: The above-presented graphs (see Figures 5 and 6) show how displacements and stresses change at some characteristic points of body, namely, at points Þ , when 0:01 ≤ η 1 ≤ 3 (see Figure 7).
max v t j j< max v n j j: • When ξ 1 ! ∞, then displacements and stresses tend to zero, that is, the boundary conditions (10) are satisfied.
• When η 1 ! ∞, then displacements and stresses tend to zero, that is, the boundary conditions (10 0 ) are satisfied.
• When η 1 ! 0 (in this case there is a crack), then (a) at points M j ð Þ 1 0, η j ð Þ 1 tangential stresses and normal displacements tend to ∞, but other components equal to zero. It can be seen from the boundary conditions (8a) (b) at points that all components of the displacements and stresses tend to ∞.
Here superscript t and n denote the tangential and normal displacement or the stress, respectively.

Conclusion
The main results of this chapter can be formulated as follows: • The equilibrium equations and Hooke's law are written in terms of parabolic coordinates.
• The solution of the equilibrium equations is obtained by the method of separation of variables. The solution is constructed using its general representation by harmonic functions.
• In parabolic coordinates, analytical solutions of 2D static boundary value problems for the elasticity are constructed for homogeneous isotropic finite and infinite bodies occupying domains bounded by coordinate lines of parabolic coordinate system. • Two concrete internal and external boundary value problems in stresses are set and solved.
The bodies bounded by the parabola are common in practice, for example, in building, mechanical engineering, biology, medicine, etc., the study of the deformed state of such bodies is topical, and consequently, in my opinion, setting the problems considered in the chapter and the method of their solution is interesting in a practical view.

Author details
Natela Zirakashvili I. Vekua Institute of Applied Mathematics of Iv. Javakhishvili Tbilisi State University, Tbilisi, Georgia *Address all correspondence to: natzira@yahoo.com © 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.