Discussion: “A Mixed Method in Elasticity” (Rao, N. S. V. K., and Das, Y. C., 1977, ASME J. Appl. Mech., 44, pp. 51–56)

Abstract

LsioJ \F(x,y,z,t )}(1)where \F (x, y, z, t)\ = Col [U, V, Z, W, Y, X]. Note that the authors give no motivation for the ordering of the components of the state vector |F| to enable the reader to formulate similar problems in a manner that will result in a partitioned matrix with zero submatrices along the secondary diagonal as is the case in equation (1). The motivation for the ordering sequence will be found in a paper by Brown [15]. In matrix [B] the second entry of the first row should be multiplied by /3. This misprint is easily corrected if one observes that matrix [B] is symmetric with respect to the secondary diagonal. Also the term £ 2 -a 2 should be added to the second entry of the second row of the [B] matrix. The integration of equation (1) is well known [11] and results inwhere [E] is the partitioned matrix in equation (1). To evaluate the matrix exponential in equation (2), note that following the authors' notation, and letting [A], and using the Taylor's series expansion given byThe matrix exponential can be obtained in closed form upon substituting the various powers of the matrix [E] in equation (3), namely, £ , = r.Cj?.i ^_ r.o.iAD-i LOIDJ LBC, o J (4) 1 Bv N.Insertion of the expressions given by equation (4) into equation (3) performing the summation, and recognizing the resulting series expansions leads to T cosh 2(C) 1 / 2 [A(D)-y 2 sinh z(W /2 ] LBtCJ-^sinhzfCJ 1 /" 2 ; cosh z(D) 1/2 JExpressions of the type given by equation (5) of the matrix exponential are standard in the theory of multiconductor transmission lines, but the derivations leading to equation (5) are usually different. The reader interested in alternative derivations of equation (5) should consult the work of Pipes [16,17].In fact, the same result can be arrived at from the direct integration of the matrix equations (7) and (8) of the authors' paper, with the use of their equation (1) as will be shown. For completeness, the first of equations (7) is rewritten in the formwhose solution is given byZo' A (7) In equation (7) the primes indicate derivatives with respect to z. Similarly, the second of equations (7) of the authors paper is integrated into Wo'Yo' Zo' (8) Eliminating the derivatives in equations (7) and (8) by the use of equation (1) in the author's paper results in a partitioned matrix which is equivalent to equation (5).Moreover, an examination of equations (9) and (10) of the author's paper shows that upon multiplying the second term of their equation (9) by (D)~1 12 outside the brackets, and by (D) 112 within the brackets leads to 1 + -C + -C 2 + . 2! 4! t/o Zo + A(D)-' 2 \zD U2 + t-D V2 + '-3! -D r ' /2 + -] Wol Yo \X 0