Rheol. Acta 15, 577-578 (1976) © 1976 Dr. Dietrich Steinkopff Verlag GmbH & Co. KG, Darmstadt ISSN 0035-4511/ASTM-Coden: RHEAAK School of' Mathematical Sciences. The Flinders University q/" South A ustralia. Bec(ford Park ( A ustralia ) Algorithms for motions with constant stretch history-II. R. R. Huilgol (Received July 30, 1976) L(t) = Z(t)L where Z is = O. Then history. The purpose of this note, which is a continua- tion of the first paper (1) and uses the same notation, is to prove the Proposition 9.1: Let a velocity field v = v(x,t) give rise to a velocity gradient L such that (t) - L(t) Z(t), - ~ < t < ,~~, [1] a skew-symmetric tensor obeying v is a motion with constant stretch Proof: Given v, compute L. Suppose that the system [1] has a solution for a skew-symmetric tensor Z = Z(t), such that Z(t) = 0, - ~ < t < ~. Now, define a tensor L~ through L1 : L - Z. [2] It then follows that L1 = ZLI - Lm Z , [3] and that the first Rivlin-Ericksen tensor (2) A1 obeys: A1 = L +L~= Lj +L~ , [4] .4, = ZA ~ - ,4 ~ Z. [5] Introduce the notation A 0 = 1 [6] and assume that for all m = 0, 1..... n, Am = A,ù-IL1 +L~Am-1, [7] A'm = z , 4 ~ - A°,z. [8] Then, on using [8] and the formula for Aù+ 1 (2), we have Aù+, =,4ù+AùL+LTAù=AùL1 +Lr Aù. [9] Next, on using [3] and [8], we get from [9] that zZ~'n+l = ZAn+I -- An+lg. [10] The above proof, based on induction, shows that since [7] and [8] are true for m = 1, [7] and [8] are true for all m = 0, 1, 2 ..... 310 Now, since the Äù obey [8], the differential equations can be integrated to yield: Aù(t) = Q(t) Aù(O) QT(t), n = 0, 1..... [11] Q(t) = i exp(Z(z)) dz, Q(0) = 1. [12] 0 Hence, the strain history C,(t - s) obeys: c,(t-s)= ~ (-1)%" ù=0 17! Aù(t) = Q(t) Co(O - s) OF(t) [13] or, by definition (1), the velocity field is a motion with constant stretch history. Remarks: (i) Note that the system [1] does not have to possess a unique solution Z for the above proposition to hold. (ii) It is extremely interesting that in a MWCSH, the following identity holds: 2=- [~- ZL + LZ, [14 3 or that ,Z need not be zero along the path line of a particle. To prove this, recall from [2.8] of(l) that in a MWCSH L : Z + L,, [1»] and that L, obeys [2.15] of(l), i.e., Ll = ZLI - Lm Z . [16] Hence [14] is true. Indeed, it is possible to prove Theorem 9.2: A necessary and sufficient condi- tion that a velocity field be a motion with con- stant stretch history is that there exist a skew- symmetric Z which satisfies the differential equation 2 = L - ZL +LZ [17] where L is the velocity gradient.