I have seen several different approaches to describing continuum mechanics that are all very similar, yet some differences (that I see, not sure if they are true) keep confusing me.
The picture that makes most sense to me is defining points living on a manifold $\mathcal{M}_{0}$ as being the reference configuration. Then, there is the current configuration, in which points live on a manifold $\mathcal{M}_{t}$. This $\mathcal{M}_{t}$ is basically a deformed version of $\mathcal{M}_{0}$. Then a map is established between the two points, where $x_{\mathcal{M}_{0}} = f(x_{\mathcal{M}_{t}})$ where $x_{\mathcal{M}_{0}}$ and $x_{\mathcal{M}_{t}}$ describe the 'same' point, but on the two manifolds. Then, vectors are defined through the usual tangent vectors $v^{i}(x)_{P} = \frac{dx^{i}(\lambda)}{d\lambda}|_{\lambda = \lambda_{p}}$, where $x^{i}(\lambda)$ describes a path parametrised by $\lambda$ in a manifold, and $x^{i}$ are coordinates used to describe the location of points on the manifold and $v^{i}(x)|_{p}$ are the coordinates of the vector living in the tangent space constructed at the point that lies on the path at $\lambda = \lambda_{P}$. The vectors living at the 'same' point between the two manifolds are mapped by a a linear transformation called the deformation gradient.
Is this correct? Furthermore, how can displacement vectors between two points then be established?
I often see another picture of continuum mechanics where the particles are instead described by position vectors, not points on a manifold, and then these position vectors are then mapped to the 'current' position vectors through the deformation gradient. Here it is clear how displacement vectors between two particles can be determined.
Which is correct? ie: one uses manifolds whereas the second does not. There is just a vector space with particles living in it. Are they both valid?
