Tham khảo tài liệu 'computational mechanics of composite materials part 6', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Elasticity problems 135 y Kjj x pq k 1 0 2 j - F pq i d d Q R first order equations z x r yl JjlX . J dQ a Q a . . Í Y . _ . . Í - í Sv I F d dQ - y í Sv er 1 X I dQ 3Q i l pq i J a 1 2 Q i j ijkll pq kl J a single second order equation y J Vi JCJkl X pq k l rSdQ Cov br bs a 1 2 Qa J Rvi F pq i rsd dQ Cov br bs dQ1 J 2 y iSVi CL X sdQ y iSviCSX 0dQ J i J ijkl XA pq k l J I J ijklV pq k 1 a 1 2 Qa a 1 2 Qa J X Cov br bs If the Young moduli of fibre and matrix are the components of the input random variable vector then there holds d Ciik x rn x ijk Ị a A x for a 1 2 dea J where A a is the tensor given by and calculated for the elastic characteristics of the respective material indexed by a whereas 1 a is the characteristic function. Thus the first order derivatives of the elasticity tensor with respect to the input random variable vector are obtained as a T 1A1 T 2 A2 ijkl ijkl Hence the second order derivatives have the form 136 Computational Mechanics of Composite Materials d2 ciikl e x m x la dA x d 7 77 y a J _ 0 for a 1 2 while mixed second order derivatives can be written as d2 c e x rn x dAk x ảA ii iJkl 7 p JJ 1 -J 1 2 __2Jk 0 de1de2 r de 2 de1 Considering the above all components of the second order derivatives of the stiffness matrixes K apq in this problem are equal to 0. Moreover since the assumption of the uncorrelation of input random variables Cov e1 e2 _ Vare1 0 0 Vare2 thus the first and second partial derivatives of the vectors FpA with respect to the random variables vector are calculated as d F a . d c a pq i - iJpq n _ A a d o a 12 d d p nj AJpq nj xe d o a a 1 2 ea ea and d2Fia d2C a d A a F pg i _ CJini _-Ap-nj _ 0 x Ed o a a 1 2 d e2 d e2 d ea After all these simplifications the set of equations - can be written in the following form a single zeroth order equation E i V cl 0 do_- íổv - f 0 d do J I j ỉjKl Ạs pq k l J ỉ pq i v 7 R first order
