Is there any problem in having a stress-strain constitutive relation that relates time-derivative of stress with strain? We usually use two empirical laws to model viscoelastic behaviour:

*

*Hooke's law of elasticity that relates stress with strain

*Newton's law of viscosity that relates stress with time-derivative of strain.

Why isn't there an equation of the form that relates time-derivative of stress with strain?
Or, does it already exist?
 A: In short
Proving such model impossible is probably too ambitious as it does not seem to violate thermodynamic requirements in absolute.
It is nonetheless absent from the literature.
This makes sense as it enables a variety of unusual behaviors.
Examples
For the example below, consider a 1D material that behavior follows
$$
\dot\sigma = \mu\varepsilon.
$$

*

*The ever-growing stress at rest is a good example proposed in the comment of @Toffomat: holding a constant strain causes the stress to grow without a limit.
It seems unacceptable because it implies the existence of an intrinsic (and infinite!) source of enthalpy, hence some other phenomena at play (maybe chemical or thermal, but a concrete example seems hard to find).


*It would also allow a material to work negatively ($\sigma:\dot\varepsilon<0$) which does not meet any empirical observation. That would for example be a tensile experiment where the material extends when put in tension, but immediately contracts if the tension slows down.
To illustrate that consider the simple cyclic test obtained from the model above and described by:
$$
\sigma = \mu\frac{\epsilon_0}\omega (1-\cos(\omega t))
,\qquad
\varepsilon = \epsilon_0\sin(\omega t)
$$
The corresponding strain-stress curve forms an ellipse and describes a succession or tension/compression deformations for which the stress is never negative.
In conclusion
A relation only linking the stress rate to the strain is not forbidden in theory, but seems to fail to find any reality to describe.
