On monotonicity of the optimal strain path in linear viscoelasticity

We consider the problem of finding the strain path $e\left ( t \right ), 0 \le t \le T$ which minimizes the work done by a one-dimensional linear viscoelastic material. The material is assumed to be initially unstrained; the time interval is fixed; and, the final strain is specified. It has previously been shown $^{1}$ that this problem has a unique solution which is referred to as the optimal strain path. We prove (i) The optimal strain path is monotone. (ii) An estimate of the work done on the optimal strain path. The first proves a conjecture of Gurtin, MacCamy and Murphy, while the second is an alternative proof to a result of Day.