PSI - Issue 28

O. Pozhylenkov et al. / Procedia Structural Integrity 28 (2020) 458–463 Author name / Structural Integrity Procedia 00 (2019) 000–000

461

4

(4)

Boundary conditions (1) are reformulated in the terms of the displacements: ⎪⎨ ⎩ ⎪⎧��� � 1� � � � � � � � � � � � � � , � � � � � ∗ cos� � � � � � �0� � 0 �� � � � � � � � � 0 �� �0� � � � �0� � 0 The vectors and matrices of the vector boundary problem are derived:

, � � �1 � � ∗ �1 � � �

(5)

� � � 1 0 1 � 0 � � , � � � 0 � � � � � 0 � , � � � � �� � �� � 0 0 � �� � , � � � � � � � � � � � � , �� � � � � � ∗ sin� � � ∗ ��1 � � � 0 �

(6)

With the help of the introduced matrices, the differential operator of the second order is constructed: � � � � �� � � ��� � � � � �� � � � � � � � The vector boundary problem in the transform`s domain is formulated with the help of operator (7): � � � � �� � �� � � � � � �� � � � , � � 1,� (8) To solve this vector boundary problem, the fundamental solution matrix � � is constructed. To found the solution of the stated problem firstly the matrix �� (where the unit matrix) must be substituted into the equation (8). From the equality � � �� � � � � �� , one can derive the � � matrix: � � � � � � �� � � �� � � � � � � � � � � �� � (9) The fundamental solution matrix is found with the help of formula � � � � � �� ∮ �� �� � � � Gantmakher F. R. (1998). The calculation of the integral requires to know all poles of the under integral function. To do it the determinant of the matrix � � was found: � � � �1 � � �� � � � � � � � � � (10) After contour integration procedure the two linear independent solutions of the matrix equation were derived: � � � 1 � 1 � �� �� �� �� � ��, �� � ��� � �� �� �� � ��� � � � 1 � 1 � � � � � � � � �� � � � � � � � 4 � � � � � � � � � � � � � � � � � � � � � � � � � �� � � 4 � � � � � � � � � � � � � � � � � � � � � (7)