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In Lagrangian or Material description, the physical properties are described in terms of the material coordinates and time. It focuses on what is occurring at a fixed material point (or particle) labeled by its material coordinates as time progresses.

My confusion: But the material surface (defined below) is written in terms of material coordinates only in case of the reference configuration and there is no time parameter. I understand that in reference configuration it is logical to not be dependent on time, but in the definition of material description, time is supposed to be included.

In Eulerian or Spatial description, the physical properties are described in terms of the spatial coordinates and time. It focuses at a fixed point in space as time progresses.

A material surface is a mobile surface in the space constituted always by the same particles.

In the reference configuration, the material surface is defined in terms of the material coordinates as f(X,Y,Z) = 0, where the set of particles (material points) belonging to the surface are the same at all times.

In the spatial description, it is defined as f(x,y,z,t) = 0. The set of spatial points belonging to the surface depend on time, and the material surface moves in space.

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  • $\begingroup$ I'm not sure I fully understand your question, let's say that we're following an infinitesimal fluid element (a set of particle) through flow field, one can show that the continuity equation in such set-up (non-conservation or Lagrangian) can be written as $\frac{D\rho}{Dt} + \rho \nabla . \boldsymbol{V} = 0$, and time is included in our formulation. $\endgroup$
    – Algo
    Commented Mar 16, 2021 at 5:38
  • $\begingroup$ What I meant is that, I always find that when we want to write a property in terms of material coordinates, we set time equal to zero in order to obtain it in the reference configuration. But in the definition of the lagrangian description, it doesn't say anything about reference configuration, it just say that we follow the particle as time progresses. $\endgroup$
    – user134613
    Commented Mar 16, 2021 at 9:23
  • $\begingroup$ Would you provide an example for such property? because for me, I see no issue in the presence of time dependence in material derivative and I am not sure why would we set t=0. $\endgroup$
    – Algo
    Commented Mar 16, 2021 at 9:34
  • $\begingroup$ Example: material surface represents material surface in reference configuration when written in terms of material coordinates $\endgroup$
    – user134613
    Commented Mar 16, 2021 at 12:23
  • $\begingroup$ Material surface could be reperesented by material or spatial description, but in case of material description it only refers to the material surface at reference configuration $\endgroup$
    – user134613
    Commented Mar 16, 2021 at 13:36

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One way to think about this is to imagine that the body is made up of spheres and you're sitting on one of them. As the body deforms, all you can observe is the distance between you and neighboring spheres; but not how you're positioned in the surrounding space (with respect to a global coordinate system that's fixed in time).

The problem with a Lagrangian descriptions is that if the deformation is too large and you no longer can see your original neighbors, the description fails.

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