Axisymmetric Vibrations of Layered Cylindrical Shells with Cracks

Vibrations of circular cylindrical shells made of layered composite materials are considered. The shells are weakened by circumferential cracks. The influence of circumferential cracks with constant depth on the vibration of the shell is prescribed with the aid of a matrix of local flexibility coupled with the coefficient of the stress intensity known in the linear elastic fracture mechanics. Numerical results are presented for the case of the shell with one circular crack. Keywords—Layered shell, axisymmetric vibration, crack. I.INTRODUCTION NTIL the fracture takes place the large class of composites behave as elastic bodies, following the Hooke's law. Usually the facture of bodies from such material is brittle, for example, destruction of bodies from fiber-glass. Therefore for investigations of small deformations in bodies from such materials, methods of the classical theory of elasticity of an anisotropic body can be used. Circular cylindrical shells, made of composite materials, are widely used in many fields of engineering, especially in civil, mechanical, aerospace, marine and chemical industry. Vibration of circular cylindrical shells from composite materials is of interest in a number of different fields. Since the cracklike defects are practically unavoidable during the manufacturing and operation of structural elements there exists the need for the information about the sensitivity of vibrational parameters of the shell with respect to defects. Vibration and stability of notched beams was investigated by Dimarogonas [1], Chondros and Dimarogonas [2], [3], Rizos et al. [4], Liang et al. [5], Kisa et al. [6], Lellep and Sakkov [7], Krawczuk, Ostachowich [8], [9] making use of the weightless rotational spring model. In [12] Lellep and Roots investigated axisymmetric vibrations of cylindrical shells with circumferential cracks. According to this concept a beam with a crack can be treated as a structure consisting of two segments. These segments are connected each other with a rotational spring which stiffness is coupled with the stress intensity coefficient of the structure with the crack. This idea was extended to composite structures and to buckling of composite columns by Nikpour and Dimarogonas [10], [11]. In this paper we will study free axisymmetric vibrations of layered cylindrical shells with cracks. L. Roots is with the Institute of Mathematics,University of Tartu, Estonia (corresponding author; e-mail: [email protected]). II. FORMULATION OF THE PROBLEM FOR LAYERED SHELLS Consider a layered, circular cylindrical shell with length l (see Fig. 1). The shell can be divided into n ring segments. The symbol n denotes the number of total ring segments separated from the rest cylindrical shell by the sections where the thickness variations take place. Every jth ring segment of shell has q layers. Each layer is isotropic with thickness hij, Young’s modulus Eij, Poisson’s ratio νij, and mass density ρij as show in Fig. 1. Fig. 1 Geometry of a layered shell Let’s denote ρij= ρ1jdij (1) where dij is a constant of proportionality, d1j=1and similarly Young’s modulus for each layer Eij=E1jeij (2) where eij is a constant of proportionality, e1j=1. We will denote the thickness of each layer hij by hij=(zi+1j-zij)h1j (3) where zij is a local coordinate of a layer with the thickness hjandz1j=0. The mass of j th ring segment will be equal 1 1 1 1 q j j ij i j ij i ρ h d (z z ) + = − ∑ (4) For the jth ring segment, the free axisymmetric vibration motion can be described by the equations [12]. Axisymmetric Vibrations of Layered Cylindrical Shells with Cracks Larissa Roots U World Academy of Science, Engineering and Technology International Journal of Mechanical, Aerospace, Industrial, Mechatronic and Manufacturing Engineering Vol:7, No:7, 2013 1376 International Scholarly and Scientific Research & Innovation 7(7) 2013 scholar.waset.org/1999.8/16373 In te rn at io na l S ci en ce I nd ex , M ec ha ni ca l a nd M ec ha tr on ic s E ng in ee ri ng V ol :7 , N o: 7, 2 01 3 w as et .o rg /P ub lic at io n/ 16 37 3 1 0 j N x ∂ = ∂ (5) 2 2 2 2 0 j j j j M N w ρ h x R t ∂ ∂ − − = ∂ ∂ where for calculation of thin shells often use following formula [13]: , 0 1 ∫ = j h j j dz N σ 1 , 0 ∫ = j h j j dz N σ , 0 1 ∫ = j h j j zdz M σ (6) where [ ][ ], ) ( ) 1 /( 1 2 1 ε ν χ ε ν σ j j j j z E + + − = [ ][ ]. ) ( ) 1 /( 1 2 χ ε ν ε ν σ z E j j j j + + − = If for all ring segments νij=ν ,by using (1) (3) and [13] , x u ε1 ∂ ∂ = , R w ε = . 2 2 x w χ ∂ ∂ − = Theforce Nj and bending moment Mj (6) can be written as , ) 1 ( 12 E , ... ... ...