UDC 539. 3

 

Vakhonina L.

Potryvaieva N.

Sadovyi О.

Modern problems of dynamic fracture mechanics, improvement of means of non-destructive testing and flaw detection require further development and improvement of methods for solving problems of dynamic interaction of thin-walled inclusions with the environment. An important case of inclusions is a circular (disc-shaped) inclusion. This is primarily due to the fact that thin disc-shaped reinforcements are quite common in machine parts and building structures. Thin inclusions are not only stress concentrators, but are also used as fillers in composites. When creating composite materials, the matrix is often filled with coin-like reinforcing elements of high rigidity. Therefore, it is inclusions of this shape that have always been given a lot of attention, which requires the solution of problems on the stress-strain state of bodies with inhomogeneities such as thin inclusions.

Methodology. The solution method is based on the representation of displacements in the matrix through discontinuous solutions of the Lamé equations for harmonic vibrations. This made it possible to reduce the problem to Fredholm integral equations of the second kind with respect to functions associated with jumps of normal stress and radial displacement to included ones. After the realization of the boundary conditions on the sides of the inclusion, a system of singular integral equations is obtained to determine these jumps.

Results. In the case of real materials, taking into account the elasticity of the inclusions significantly affects the value of the stress intensity factors. The values of the stress intensity factors obtained taking into account the elasticity for some materials may exceed, and for some be much smaller than those corresponding to the absolutely rigid inclusion. Taking into account the stiffness of the inclusion also significantly changes the dependence of the voltage intensity coefficients on the wave number. It becomes more complex with many highs and lows. Moreover, the maximum values of voltage intensity factors can be several times higher than the corresponding values for absolutely rigid inclusions.

 Originality. Determination of problems about harmonious communication of non-interconnected body with disk-like inclusions in the minds of smooth contact. Previously, such a task was tied for an absolutely hard inclusionю.

Practical value. The data obtained can be used in the calculations of machine parts and structures in which it is necessary to take into account elastic inclusions.

Key words: elastic inclusions, cylindrical waves, matrix, stress intensity factor

 

References:

1. Litvin O.V. Popov V.G. (2002) Izgibnye kolebanija tonkogo uprugogo vkljuchenija v neogranichennoj srede pri vzaimodejstvii s uprugimi volnami.  Teoreticheskaja i prikladnaja mehanika. Vip.38. S. 131-140.
2. Kit H.S., Kunets Ya. I., Yemets V.F. (1999) Elastodynamics scattering from a thinwallad inclusions of low rigidity.International Journal of Engineering Science. 37. P. 331-343.
3. Kit G.S., Kunec V.V., Mihas’kiv V.V. (2004) Vzaimodejstvie stacionarnoj volny s tonkim diskoobraznym vkljucheniem maloj zhestkosti. Izvestija RAN.   Mehanika tverdogo tela.  №5. S. 83-89.
4. My`xas`kiv V.V., Kaly`nyak O.I. (2005) Nestacionarni zburennya try`vy`mirnoyi pruzhnoyi matry`ci z zhorstky`m dy`skovy`m vklyuchennyam. Fizy`ko–ximichna mexanika materialiv. T. 41, # 2.  S. 7-15.
5. Vahonina L.V., Popov V.G. (2002) Osesimmetrichnye kolebanija prostranstva s tonkim zhestkim krugovym vkljucheniem // Teorija i praktika processov izmel’chenija, razdelenija, smeshenija i uplotnenija. Odessa: OGMA.  Vyp. 9. S. 28-34.
6. Vahonina L.V., Popov V.G. (2003) Vzaimodejstvie uprugih voln s tonkim zhestkim krugovym vkljucheniem v sluchae gladkogo kontakta. Teoreticheskaja i prikladnaja mehanika.    Vyp. 38. S. 158-166.
7. Popov G.Ja. (1999) Postroenie razryvnyh reshenij differencial’nyh uravnenij teorii uprugosti dlja sloistoj sredy s mezhfaznymi defektami. Doklady RAN. T.364, № 6. S.769-773.
8. Kit G.S., Mihas’kiv V.V., Haj O.M. (2002) Analiz ustanovivshihsja kolebanij ploskogo absoljutno zhestkogo vkljuchenija v trehmernom uprugom tele metodom granichnyh jelementov. Prikladnaja matematika i mehanika. T. 66, Vyp. 5. S. 855-863.
9. Gradshtejn I.S., Ryzhik I.M. (2011) Tablicy integralov, summ rjadov i proizvedenij. M. 1108s.
10. Rakhmanov Evguenii A. (2016) Orthogonal Polynomials Walter De Gruyter Incorporated. 510 с.
11. Suetin P.K.(2005) Klasicheskie ortogonal’nie mnogochleny – 3-e izd., pererab. i dop. M.: Fizmat. 480s.
12. Zemanian A.H. (2010) Distribution Theory and Transform Analysis: An Introduction to Generalized Functions, with Applications (Dover Books on Mathematics). 400 с.
13. My`xas`kiv V.V., Kunecz` Ya.I., Mishhenko V.O. (2003) Napruzhennya u try`vy`mirnomu tili z tonky`m podatly`vy`m vklyuchennyam za frontom impul`sny`x xvy`l`.  Fizy`ko–ximichna mexanika materialiv.  T. 39, # 3.  S. 63–68.