How can I adapt classical continuum mechanics equations in order to agree with general relativity?

The answer you're looking for seems to be contained in
Rezzolla & Zanotti: Relativistic Hydrodynamics (Oxford U.P. 2013)
https://books.google.com/books/?id=KU2oAAAAQBAJ

but it is not a trivial generalization. Quoting Disconzi's On the well-posedness of relativistic viscous fluids (Nonlinearity 27 (2014) 1915, arXiv:1310.1954):

we still lack a satisfactory formulation of viscous phenomena within Einstein's theory of general relativity. [... T]here have been different proposals for what the correct $T_{\alpha\beta}$ should be. [... A]ttempts to formulate a viscous relativistic theory based on a simple covariant generalization of the classical (i.e., non-relativistic) stress-energy tensor for the Navier-Stokes equations have also failed to produce a causal theory.

Similar problems exist for elastic materials.

You may also take a look at Chapter 15 ("Relativistic continuum mechanics") of Maugin's Continuum Mechanics Through the Twentieth Century (Springer 2013), and its references:
https://books.google.com/books?id=-QhAAAAAQBAJ

and at Bressan's Relativistic Theories of Materials (Springer 1978):
https://books.google.com/books?id=kMTuCAAAQBAJ

General-relativistic continuum mechanics unfortunately has not been given a clear mathematical and conceptual framework yet. Newtonian continuum mechanics is easy to summarize:

1. We choose a reference frame (preferably but not necessarily inertial).
2. We have a set of 11 spacetime-dependent fields with clear physical meaning: mass, momentum or deformation, stress, body force, internal energy, heating flux, body heating, temperature, entropy, entropy flux, body entropy supply. Of these, the "body" ones represent external interventions.
3. We have 5 balance equations: mass, force-momentum, torque-rotational momentum, energy, entropy. They are clearly written in terms of the fields above and are valid for any material.
4. We choose a set of independent fields (usually mass, momentum or deformation, temperature).
5. We choose constitutive equations (compatibly with the balance ones) that relate the remaining fields to the independent ones. These equations express the peculiar properties (fluid, solid, elastic, plastic, with/without memory...) of the material under study.

And at this point we have a well-defined set of partial differential equations in a number of unknown fields, for which we can set up well-defined initial- & boundary-value problems to be solved analytically or numerically. (An expanded but analogous framework accommodates electromagnetism and continua with internal structure.)

This framework and steps are very neat – we clearly know what the fields are, which of them are dependent and which independent; what are the equations valid for all materials, and what are the equations constitutive to each material. I've never seen a clearly defined procedure like the one above for general relativity, although I believe it could be extracted from Rezzolla & Zanotti's or Bressan's books. Moreover, the core of general-relativistic community uses a different jargon and way of thinking.

Most general-relativity books tell you that the Einstein equations determine everything, but they are not so clear about which fields in them are independent and which dependent; even Misner et al.'s Gravitation (ch. 21) has a long discussion and explanation about this point. It was only with 3+1 formulations and the work of Arnowitt, Deser, Misner, York, and others around the 1970s that this point got clarified. Then they tell you that we need "special" additional equations for the stress tensor – that is, constitutive equations. Sometimes other conservation equations, like baryonic number (basically rest-mass), are added with no real explanation. This is a sample of books where "constitutive equations" are mentioned explicitly (only once or twice in most of them):

• Rezzolla & Zanotti above (and they explain what a "constitutive equation" is as though it was an exotic concept)
• Choquet-Bruhat: General Relativity and Einstein's Equations
• Anile & Choquet-Bruhat: Relativistic Fluid Dynamics
• Bertotti et al.: General Relativity and Gravitation
• Puetzfeld et al.: Equations of Motion in Relativistic Gravity
• Tonti: The Mathematical Structure of Classical and Relativistic Physics
• Bini & Ferrarese: Introduction to Relativistic Continuum Mechanics
• Tolman (obviously): Relativity, Thermodynamics, and Cosmology

but they constitute a very small minority in the huge relativistic literature.

Yet, the general-relativistic community cannot be criticized for the confused conceptual state and somehow confused language of the subject. Newtonian continuum mechanics can be neatly formulated today because it has been refined over several centuries. General relativity is still very young instead, and its conceptual refinement still in progress. Some of the steps in the Newtonian framework become extremely complicated in general relativity. For example: step 1. (choose an inertial frame) cannot be done so simply. The Einstein equations, evolved from initial conditions, construct a reference frame "along the way", while they determine the dynamics. This gives rise to peculiar fields like "lapse" and "shift", which aren't really physical, and all sorts of redundancy (gauge freedom) in the equations.

Another example: the metric becomes a dynamical field variable, and you suddenly realize that it is hidden almost everywhere in the Newtonian framework – divergences, curls, vectors/covectors... So its evolution can't be easily divided among some new balance and constitutive equations (like we can do with electromagnetism instead). Are all of its appearances in the Newtonian framework dynamically significant? or can the metric be eliminated from some places? There's some research today on this "de-metrization" of Newton's equations; see for example Segev's Metric-independent analysis of the stress-energy tensor, J. Math. Phys. 43 (2002) 3220. This line of research has shown that some Newtonian physical objects actually don't need a metric: they be expressed via differential forms and other metric-free differential-geometrical objects (e.g., van Dantzig's On the geometrical representation of elementary physical objects and the relations between geometry and physics, Nieuw Archief voor Wiskunde II (1954) 73; there is a vast literature on this, let me know if you want more references). This is still work in progress – which means that it's obviously not completely clear yet how mass-energy-momentum-stress and metric are coupled.

To conclude, I think another good starting point to understand how things work in general-relativistic continuum mechanics is to look in books on numerical formulations of general relativity and matter dynamics. The conceptual framework in them is a bit confused, but you can see how they actually do it. If from the practice of these books you manage to reverse-engineer a framework like the Newtonian one above, please write a pedagogical paper about it!

Here are some books and reviews on numerical relativity with continua:

• Rezzolla & Zanotti above
• Gourgoulhon: 3+1 Formalism in General Relativity (Springer 2012, arXiv:gr-qc/0703035)
• Baumgarte & Shapiro: Numerical Relativity (Cambridge U.P. 2010)
• Alcubierre: Introduction to 3+1 Numerical Relativity (Oxford U.P. 2008)
• Palenzuela-Luque & Bona-Casas: Elements of Numerical Relativity and Relativistic Hydrodynamics (Springer 2009)
• Lehner: Numerical relativity: a review, Class. Quant. Grav. 18 (2001) R25, arXiv:gr-qc/0106072
• Guzmán: Introduction to numerical relativity through examples, Rev. Mex. Fis. S 53 (2007) 78

I'm happy to provide or look for additional references.

As you have mentioned, the key to all modifications is the metric, because the metric is all you need to characterize the background spacetime, be it flat or curved.

If you adopt the abstract index notation, the metric can be written as $g_{ab}$, a rank two symmetric tensor that takes in two vectors and outputs a scalar. Now let's look at some generic examples of conversions from flat to curved spacetime before we deal with the particular equation of motion you care about.

The conservation of energy momentum in flat spacetime for an arbitrary stress energy tensor $T$ can be expressed as: $$\partial_a T^{ab} = 0 $$ where the Einstein summation convention is implied by the contraction over index $a$. How does this equation look like in curved spacetime? Well, just notice that the partial derivative operator is not coordinate independent! So to fix that, we define the covariant derivative operator $\nabla_a$ which operates like this: $$ \nabla_a u^b = \partial_a u^b + \Gamma^b_{ca} u^c $$ $\Gamma^b_{ca}$ is called the Christoffel symbol and you can see it as providing a correction to the partial derivative operator to make it coordinate independent. Notice that in flat spacetime, $\Gamma^b_{ca}$ vanishes and the covariant derivative is equal to the partial derivative. So now the modification is clear. You simply convert $\partial_a$ to $\nabla_a$ and the conservation equation in GR is: $$ \nabla_{a} T^{ab} = 0$$ You may call this process tensorification. Whenever you have an equation in flat spacetime, just turn it into a tensor equation such that it reduces to your flat spacetime equation in the absence of curvature. There are some additional subtleties in this process. It doesn't always work. (See Chapter 4 of Robert Wald's General Relativity for a discussion) But I think this technique works for your equation so let's apply it:

$$\rho \ddot u = div(\sigma) + f $$

Let's look at this equation term by term. The first term involves a second time derivative which depends on the definition of a coordinate time. To tensorify just change it to proper time. In the second term, the divergence is defined in flat spacetime as: $$ div \sigma^{ab} = \partial_a \sigma^{ab} $$ So in curved spacetime, it just becomes: $$ div \sigma^{ab} = \nabla_a \sigma^{ab} = \frac{1}{\sqrt{-g}}\partial_a(\sqrt{-g}\sigma^{ab})$$ where $g$ is the determinant of the metric tensor $g_ab$.

In the third term, $f$ is already a vector, which is coordinate independent. So you're fine. Thus, the final form is: $$ \rho \frac{d^2}{d\tau^2} u^b = \nabla_a \sigma^{ab} + f^b $$

P.S. While writing this answer, I realized that $\rho$ is not a coordinate-independent quantity. So the final form is probably wrong. I don't know how the equation you posed was derived. If you give me a more fundamental equation I might be able to fix it. But I hope the techniques above can help you figure it out by yourself!