Are there any current theories of elastic materials in general relativity, *not* based on a notion of material projection?

Question

[Probably a very niche question, but I hope someone here has useful information.]

All treatments I've found so far of elastic materials in general relativity use a notion variously called "material projection", or "matter space", or "body manifold", or other similar names.

The idea is to map every event $(t,x)$ in spacetime $M$ to a point $(X)$ of a 3-dimensional manifold $B$:
$$M \to B \qquad\text{by}\quad (t,x)\mapsto X\,.$$
The manifold $B$ represents a material body (say a rubber ball), seen as abstracted from its evolution in time – and therefore only considered in its topological properties. The counter-image of a point $X$ in the body manifold is the world-line in spacetime of that particular point of the body.

Then, by considering a reference metric in the body manifold $B$, and comparing it (in a manner of speaking) with the metric of spacetime $M$, one manages to introduce the notion of strain of the material body, and so on.

One among the many reviews of this approach is for example that by Carter & al..

In the past, at least one way to frame elasticity in general relativity was proposed, by Synge in 1959, which avoided the use of a "body manifold", and instead managed to define a notion of strain from spacetime fields such as the matter-number current.

Do you know of any current works on elasticity in general relativity which avoid using the "body manifold" framework?

Cheers!