496 _ 12 - 



A, "J Possible types of motion . 



We now introduce i^, a 'unction of time defined by: 



A A f 



d> (t) = _L-2 w P(t) ♦ pt\ A w2 (39) 



*2 

 whence equation (31) implies that: 



5-0 («>) 



at any time when w>0 and therefore in part icular when the relations (38) are satisfied. 



Using this function, it then follows from (30) and (27) that 



A, ' , 



pt, $ ^ 4, * A (T- - T) - _i_ T W-^ (Ul) 



Oh S„ + ph S, = (^ - -i- T W^ (»2) 



° '■ Aj 



We can now prove the following Lemn'iS. 



LemiB A. If w 5. 0, S > 0, S^ initially, then 3^ > 0, S^ > in the ensuing stage of mot 



ion. 



For S > by the previous continuity considerations and therefore S^ > by (35). Thence 

 since S. ^ initially, S > In ensuing motion. 



Lemia 8. If w^ 0, S = 0, S^ initially, th en S^ ^ in ensuing stage of motion . 



Assuros LefTima is false and that S becomes < 0, then we mu5t__have S ■< since So «= Initially 

 and further T j: from (36), But S < and T ^ toget^her imply S > from («2) and therefore 

 imply that S becomes > since S^ initially. But S > 0, S > together imply that T > T' = 

 T > from (2 7), (33) and (34), and, therefore, since T > 0, we cannot have S becoming negative. 

 Thus, if Lemma is false, we arrive at contradiction and therefore Lemma must be true. 



Lemrra C. \t V 'i- 0, S i' 0, S > initially, then S^ in ensuing stage of motion. 



Firstly, if Sj -* Initially, then S > in ensuing stage by continuity conditions. 



Secondly, if S, = initially, then if Lemma is false and S becomes negative, then S also 



becomes negative. But 3, < 0, S, "^ " imply T < T' <0 from (27) and (36) and thence from (40) and 



(41) we must have S„ > and therefore S < since S„ 5- initially. But S„ > implies T = T 

 ^ ' -^00 



from (32), thus contradicting T < already proved to hold if Lemma is false. Thus Lemma must be 



true. 



The preceding Lemrras thus show that the stage of motion succeeding initial conditions 

 satisfying (38) must be one of the three types, 



Type I. S|^ > 0, Sj > 



Type II, S^ = 0, S^ > 0. 



Type III. (Plate at rest) S^^ = S^ = 0. 



No* the initial conditions of rest prior to motion as given by (24) and (28) satisfy the 

 relations (38) and thus tho initial stage of motion must be of Type I or Type II during either of 

 which S, is increasing, S is constant or increasing and W is increasing so that w becomes positive 

 ^nd thus the conditions (38) will be satisfied throughout this initial stage and by oi>r previous 

 continuity considerations they will be satisfied at the beginning of the second stage. This must 

 then in turn be :f Type I, II or III by the proeceOing Lemmas and thus by induction it follows that 

 all stages of the whole motion will be of Types I, II or III; in effect, i»e have shown that there 

 is no tendency for the plate or edge fixings to go into compression when the load Is of one sign and 

 the energy absorption is irreversible. 



We have 



