Someone has already approached some developments as those you suggest. See for example:

1. Elastodynamics of the Spacetime Continuum, by Pierre Millette, 2017, published by American Research Press.

2. The Elastodynamics of the Spacetime Continuum as a Framework for a Strained Spacetime, by Pierre Millette, Progress in Physics Vol. 1, 2013, pg 55

3. From Elastic Continua to Spacetime, by A. Tartaglia and N. Radicella, in Proceedings of the conference: Invisible Universe, American Institute of Physics 2010

Thanks, Giuseppe, quite useful. Does Elasticity Continuum idea introduce anything new on the top of Einstein's GR?

I do not think that the Millette theory about Elastodynamics of the spacetime continuum introduces any extension to GR, such that classical GR might be seen as a lower order approximation of such theory. Instead, Millette just propose a new viewpoint in which spacetime behaves as a matter field itself, so providing a stress-energy tensor for it to be used as a source in the GR equations. In my opinion this should not appear as that strange, being the same GR field equations self-consistent. However, this theory shows some very challenging points:

1. it is a continuum theory, that would be normal at large or cosmic scales, but the author projects the continuum condition also at microscales... the realm of QM and QFT.

2. as a consequence of the above point, QM schema looks completely concerned, the wave-particle duality is seen as the way in which strein waves of the spacetime "medium" propagates; the same mass of a particle is viewed as a localized dilatation of the spacetime at the wavefront of a propagating strein wave.

Apart from the fact that that theory seems to give the old Ether concept a new life, which is of course questionable due to SR, the whole basis of the idea has some appealing aspects that could worth some detailed discussion, before expressing any definitive judgement on it.