269 A journal of the AMERICAN Journal of Engineering, Mechanics and Architecture www. grnjournal.us AMERICAN Journal of Engineering, Mechanics and Architecture Volume 01, Issue 10, 2023 ISSN (E): 2993-2637 Fundamentals of Determining the Stressed State of Semi-Elastic Space under the Action of a Force Applied to One Point When Solving Contact Problems Almardonov Oybek Makhmatkulovich qmii-oybek.almardonov@mail.ru Abstract: This article presents the definition of the stressed state of a semi-elastic space under the action of a force applied to one point, and also presents methods for determining the components of the stress tensor and the components of the strain tensor. Keywords: polar coordinate system, deformation, stress, displacement components, boundary conditions, yield strength. Determining the stressed state of a semi-elastic space under the action of a force applied to one point is an integral part of contact problems. Many scientists have conducted research on these issues. For example, in this problem Flamand expressed the stress function in the form of the  sin),( Arr  polar coordinate system and determined the components of the stress tensor and the components of the strain tensor. Fig.-1 In this case, A - is constant and the voltages are determined as follows (Fig-1). .0) 1 ( ,0 1 ,cos2 11 2 2 2 2 2 2 2                             rr rr r A rrr r r (1) 270 A journal of the AMERICAN Journal of Engineering, Mechanics and Architecture www. grnjournal.us To determine the constant A, we use the sum of the projections of radial stresses acting on a circle of arbitrary radius r onto the vertical axis 0z:      2 2 2 0 2cos2cos     AdArdP r . (2) From this r P r    cos2  . (3) According to the result obtained, the radial stress s_r along a circle with diameter d passing through the coordinate system point 0 remains unchanged (Fig. 1). For stress tensor components in the Cartesian coordinate system , )( 2 cossin , )( 2 cos , )( 2 sin 222 2 222 3 2 222 2 2 zx xzP zx zP zx zxP rzx rz rx             (4) we will have a relationship. Now let's calculate the components of the deformation tensor. . 1 , cos2)1(1 , cos21 )( 1 2 r u r uu r r P E u rr u r P EEr u r r r r r                                  (5) Let's integrate the resulting relations .sincossin )1)(21( cos2 )1)(21( sin2 )1( lnsin2 1 ,cossinsin )1)(21( lncos2 1 321 2 21 2 rcccP E P E P E rP E u ccP E rP E ur                                   (6) Oz since points on the minor axis 0)0( U move only along this axis, it follows that 031  cc . 2    in for movements corresponding to angles crP E uu P E uu rr      ln2 1 ) 2 () 2 ( )1)(21( ) 2 () 2 ( 2    (7) 271 A journal of the AMERICAN Journal of Engineering, Mechanics and Architecture www. grnjournal.us the result is reasonable. Results on the state of deformation of a semi-elastic medium under the action of an experimental force can be obtained as a result of rotating the medium through an angle relative to the normal force, i.e. calculation of the pole angle from the vertical force. , cos2 r Q r     0   r . (8) And the coordinate is in the Oxz plane . )( 2 , )( 2 , )( 2 222 2 222 2 222 3 zx zxQ zx xzQ zx xQ xz z x             (9) Stressed state of a half-plane under the action of uniformly distributed vertical forces. Below we will consider the stressed state of a medium consisting of a half-plane under the action of uniformly distributed vertical forces. Respectively: ;)( pxp  0)( xq , axa  (10) For the stress tensor components, substituting the constant p into the above formulas     ).2cos2(cos 2 ,)2sin2(sin)(2 2 ,)2sin2(sin)(2 2 21 2121 2121             P P P xz z x (11) ax z tg  2,1 . If you enter the designation 21   , then the main voltages are determined as follows: )sin)(/(2,1   p , (12) and acts on the plane at an angle 2/)( 21   . The maximum breakdown voltage will be  sin)/(1 p (13) Deformations for points located inside the stretch zone p Ex u x )1)(21(      ,        a a z sx ds Ex u   )1(2 2 , (14) px E ux )1)(21(    272 A journal of the AMERICAN Journal of Engineering, Mechanics and Architecture www. grnjournal.us determined by appearance. The deformation for points located inside the axa  stretch zone is determined by its appearance. According to these integrals, the function under the integral has a singularity at xs  and is called a singular integral. In the process of its integration, we divide it into two parts from to and from as  to x . Here a is a very small value. Thus, based on the results obtained by Flamand, it is possible to determine the components of the stress tensor corresponding to the deformed state of the half-plane under the action of uniformly distributed vertical forces. References: 1. Джонсон К. “Механика контактного взаимодействия”. Москва. МИР. 1989. 2. Hardy C., Baronet C.N., Tordion G.V. Elastoplastic identation of a half-space by a rigid sphere.-J.Numerical Methods in Engng., 1971, 3, p.451. 3. Аргатов И.И., Назаров С.А. Метод сращиваемых разложений для задач с малыми зонами контакта // Механика контактных взаимодействия. М. :Физматлит, 2001. С. 73- 82 4. Алмардонов О.М. Проблема кривого штампа - сборник научных и практических тезисов о роли талантливой молодежи в развитии математики, механики и информатики. Ташкент-2014. Страница 6 5. С.У. Мустапакулов О.А. Мирзаев, О.С. Нурова, О.М. Алмардонов “Динамическая изучения зон питании и дискретизации пневмомеханических прядильных машин.” Kompozitsion Materiallar. 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