( ) ( ) ( ) ( ) 0 , 2 , 2 1 1 2 1 2 = - - + ¶ ¶ - - - i i i i i i i r z r r z r r r z z t p t p p s (1) Figure 2(b) shows the inner cylinder assembly of the ith layer, where 1-is denotes the mean normal stress averaged over i-1 inner layers: å - = - - - - = 1 1 2 1 2 1 2 1 i j i j j j j i r r r E e s (2) Hence, the relationship of the stresses acting on the inner cylinder assembly is given by ( ) ( ) 0 , 2 1 1 2 1 1 = + ¶ ¶ - - - - i i i i r z r r z z p t p s (3) The shear stress is expressed by ( ) ( ) ( ) ( ) r w r z G z u r z r w r z r z G ¶ ¶ ÷@ ø ö ç è æ ¶ ¶ + ¶ ¶ = , , , , t (4) where u and w are displacements along r and z, respectively. From the Eqn.s 1-4, the following equation is obtained. ( ) ( ) [ ] ( ) ( ) ( ) 0 , ln , ln 2 , , 2 2 2 1 1 1 2 1 2 2 2 1 2 1 2 1 2 1 1 = ¶ ¶ ÷÷ ø ö çç è æ - + ¶ ¶ ú û ù ê ë é ÷÷ ø ö çç è æ - - + - - - = - - - - - - å z r z r r E r r z r z r r r r r G r z r z E j j i i i j j j j i i i i i i i i i i i i i e e e e (5) Then, the Fourier transform of ( ) r z i i , e is symbolized by ( ) r k i i ˆ , e , and ( ) r k i i ˆ , e is related to ( ) r k ˆ , 1 1 e using a transfer function ( ) H k i as ( ) ( ) ( ) r k H k r k i i i ˆ , ˆ , 1 1 e e = . The Fourier transform of Eqn. 5 yields ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) [ ] ( ) r k r r r k E r r G r r k EH k r r GH k r k i i i i i i i i j i i j j j j i i i i ˆ , 2 ln 2 2 ln 2 2 ˆ , 1 1 1 2 1 2 1 2 2 1 1 1 2 1 2 2 1 e p p e - - - - = - - - - - - - + = å (6) Thus, ( ) H k i is expressed using the transfer functions ( ) H k j (j = 1…i-1). At first, H2(k) is obtained from H1(k) = 1. Next, H3(k) is calculated from the H2(k) and H1(k). Through the repetition of the calculation procedure, all ( ) H k i (i = 1…N) are obtained at discrete values of k. From a given strain field at the outermost layer ( ) r z N N, e , ( ) r k N N, ˆe is obtained by the Fourier transform. Then ( ) r k ˆ , 1 1 e is calculated from the ( ) r k N N, ˆe using the ( ) H k N , and the strain field at the innermost core ( ) r z, 1 1 e can be obtained by the inverse Fourier transform. CALCULATION OF REFLECTION SPECTRA According to the above procedure, axial strain distribution at the core of the FBG sensor embedded in a CFRP laminate was calculated. The CFRP laminate is T800H/3631 (Toray Industries, Inc), and the laminate configuration is cross-ply [02/904/02]. The optical fiber is made from glass whose Young’s modulus is 73.1 GPa and whose Poisson’s ratio is 0.16. The core and cladding are 10 mm and 125 mm in diameter, respectively. The length of the grating is 10 mm, and the grating period is about 530 nm. Strain distribution was calculated for an uncoated FBG sensor and an FBG sensor coated with polyimide, whose outside diameter was 150 mm. Young’s
RkJQdWJsaXNoZXIy MjM0NDE=