INTEGRITET I VEK KONSTRUKCIJA Vol. 19, br. 2 (2019), str. 102–108 STRUCTURAL INTEGRITY AND LIFE Vol. 19, No 2 (2019), pp. 102–108 102 Victor Iliev Rizov 1 , Holm Altenbach 2 ON THE ANALYSIS OF LENGTHWISE FRACTURE OF FUNCTIONALLY GRADED ROUND BARS ANALIZA PODUŽNOG LOMA OKRUGLIH ŠTAPOVA OD FUNKCIONALNOG KOMPOZITA Originalni naučni rad / Original scientific paper UDK /UDC: 66.017/.018:539.42 Rad primljen / Paper received: 18.03.2019 Adresa autora / Author's address: 1) University of Architecture, Civil Engineering and Geodesy, Department of Technical Mechanics, Sofia, Bulgaria, email: v_rizov_fhe@uacg.bg 2) Lehrstuhl für Technische Mechanik und Geschäfts- führender Leiter Institut für Mechanik, Fakultät für Maschinenbau, Otto-von-Guericke-Universität Magdeburg, Deutschland, email: holm.altenbach@ovgu.de Keywords • functionally graded material • lengthwise fracture • round bar • linear elastic behaviour Abstract The strain energy release rate for lengthwise circular cylindrical cracks in round bars which are functionally graded in radial direction is derived by applying linear elastic fracture mechanics. The solution obtained is valid for a crack located arbitrarily in the radial direction of the bar cross-section. Besides, the modulus of elasticity and the shear modulus can be distributed arbitrarily in the radial direction. The functionally graded bars are loaded by axial forces, bending and torsion moments. The derived solution is applied to analyse the strain energy release rate for a lengthwise crack in a clamped functionally graded bar configuration. The external load of the clamped bar consists of an axial force and a torsion moment applied at the free end of the internal crack arm and a bending moment applied at the free end of the bar. In order to verify the solution, the strain energy release rate in the clamped bar is determined also by considering the balance of energy and by applying the compliance method. The influence of various factors such as crack location in the radial direction, the material gradient and loading conditions on strain energy release rate in the clamped bar configuration are investigated and discussed. Ključne reči • funkcionalni kompozitni materijal • podužni lom • okrugli štap • linearno elastično ponašanje Izvod Primenom linearno elastične mehanike loma, izvedena je brzina oslobađanja deformacione energije kod podužnih kružnih cilindričnih prslina u okruglim štapovima od funk- cionalnog kompozitnog materijala sa slojevima u radijal- nom pravcu. Dobijeno rešenje važi za prslinu proizvoljno lociranu u radijalnom pravcu preseka štapa. Osim toga, modul elastičnosti i modul klizanja su takođe proizvoljni u radijalnom pravcu. Štapovi od funkcionalnog kompozita su opterećeni aksijalnim silama i momentima savijanja i torzije. Izvedeno rešenje je primenjeno u analizi brzine oslobađa- nja deformacione energije podužne prsline kod uklještenog štapa od funkcionalnog kompozita. Spoljašnja opterećenja uklještenog štapa su aksijalna sila i moment uvijanja, koji deluju na slobodnom kraju kraka unutrašnje prsline, i moment savijanja, koji deluje na slobodnom kraju štapa. Radi provere rešenja, brzina oslobađanja energije defor- macije uklještenog štapa se određuje razmatranjem ravno- teže energije i primenom metode popustljivosti. Razmotreni su i diskutovani uticaji pojedinih faktora, kao što su lokaci- ja prsline u radijalnom pravcu, gradijent materijala i uslovi opterećenja na brzinu oslobađanja deformacione energije kod uklještenog štapa. INTRODUCTION Functionally graded materials are inhomogeneous com- posites of two or more constituent materials. The most important feature of functionally graded materials is the smooth variation of their properties along one or more directions in the solid, /1-7/. Variation of material proper- ties can be tailored in order to improve the performance of functionally graded structural members and components to externally applied loads. In recent years, functionally graded materials have been frequently used as advanced structural materials in various engineering applications in aerospace, nuclear reactors, airplane industry and bioengineering. Accessing of structural integrity, reliability and safety of functionally graded structural members and components is closely related with their fracture behaviour. Therefore, crack problems of functionally graded materials are an important subject of research that continues to attract the attention of academic community around the globe /8-12/. Basic problems of fracture mechanics of functionally graded materials have been discussed in /8/. Methods for solving different crack problems in functionally graded materials have been developed. The fracture analyses per- formed and the results obtained can be useful for material scientists and design engineers which work in the field of functionally graded materials. Various studies of fracture in functionally graded com- posite materials have been reviewed in /9/. Analyses of cracks oriented parallel or perpendicular to the direction of material gradient have been considered. Solutions for non-