On optimal strain paths in linear viscoelasticity

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Abstract
For a viscoelastic material the work W ( e ) W\left ( e \right ) needed to produce a given strain e 0 {e_0} in a given time T T depends on the strain path e ( t ) , 0 ≤ t ≤ T e\left ( t \right ), 0 \le t \le T , connecting the unstrained state with e 0 {e_0} . We here ask the question: Of all strain paths of this type, is there one which is optimal, 1 ^{1} that is, one which renders W W a minimum? In answer to this question we show that: (i) There is no smooth optimal strain path. (ii) There exists a unique optimal path in L 2 ( 0 , T ) {L_2}\left ( {0,T} \right ) ; this path is smooth on the open interval ( 0 , T ) \left ( {0,T} \right ) , but suffers jump discontinuities 2 ^{2} at the end points 0 and T ( i . e . , e ( 0 + ) ≠ 0 , e ( T − ) ≠ e 0 ) T\left ( {i.e.,e\left ( {{0^ + }} \right ) \ne 0, \\ e\left ( {{T^ - }} \right ) \ne {e_0}} \right ) . (iii) For a Maxwell material the optimal path is linear on ( 0 , T ) \left ( {0,T} \right ) .

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