Hello everyone,
In order to define the eulerian rate deformation tensor, one should first express [itex]\dfrac{d}{dt}(\underline{dx})[/itex] in gradient velocity terms (denoted [itex]\underline{\nabla v}[/itex] with [itex]v[/itex] equal to the partial time derivative of the geometrical mapping that relates the inital configuration to the current one).
In an article, we claim that
Regards.
In order to define the eulerian rate deformation tensor, one should first express [itex]\dfrac{d}{dt}(\underline{dx})[/itex] in gradient velocity terms (denoted [itex]\underline{\nabla v}[/itex] with [itex]v[/itex] equal to the partial time derivative of the geometrical mapping that relates the inital configuration to the current one).
In an article, we claim that
[itex]\dfrac{d}{dt}(\underline{dx})=v(\underline{x}+\underline{dx},t)-v(\underline{x},t)[/itex] (1)
and thus [itex]\dfrac{d}{dt}(\underline{dx})[/itex][itex]=\underline{\nabla v} . \underline{dx}[/itex]
I'm not sure about (1). It says actually that [itex]\dfrac{d}{dt}(\underline{x} +\underline{dx}-\underline{x})[/itex] [itex]=\dfrac{d}{dt}(\underline{x} +\underline{dx})-\dfrac{d}{dt}(\underline{x} )[/itex]
I can't see that clearly though. Is there any physical explanation ? May be an approximation since we're dealing with elementary vectors....Regards.