Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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On optimal strain paths in linear viscoelasticity


Authors: Morton E. Gurtin, Richard C. MacCamy and Lea F. Murphy
Journal: Quart. Appl. Math. 37 (1979), 151-156
MSC: Primary 73F99; Secondary 49A99
DOI: https://doi.org/10.1090/qam/542987
MathSciNet review: 542987
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Abstract: For a viscoelastic material the work $ W\left( e \right)$ needed to produce a given strain $ {e_0}$ in a given time $ T$ depends on the strain path $ e\left( t \right), 0 \le t \le T$, connecting the unstrained state with $ {e_0}$ . We here ask the question: Of all strain paths of this type, is there one which is optimal, $ ^{1}$ that is, one which renders $ 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}\left( {0,T} \right)$; this path is smooth on the open interval $ \left( {0,T} \right)$, but suffers jump discontinuities $ ^{2}$ at the end points 0 and $ 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 $ \left( {0,T} \right)$.


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DOI: https://doi.org/10.1090/qam/542987
Article copyright: © Copyright 1979 American Mathematical Society


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