I'm reading a text on the principle of virtual work and there are presented the following definitions for a deformable body:

Internal work = $\int_\Omega{\delta {\epsilon}^T\sigma d\Omega}$

External work = $\int_\Omega{\delta u^T b d\Omega}+\int_\Gamma{\delta u^T t d\Gamma}$

where $\sigma$ are the internal stresses, $b$ external body forces, $t$ are the external surface forces, $\delta \epsilon$ are strains and $\delta u$ are displacements. $\Omega$ is the volume of the body and $\Gamma$ is the boundary.

No explanation is given for the definitions, so I'm left with some questions:

1) Why is there a superscript $T$ on $\epsilon$ and $u$? Are they vector transposes of some kind? If so, why is the strain a vector?

2) What is the difference between external body forces $b$ and external surface forces $t$? In the last term of external work we integrate external surface forces * displacements over the surface of the body. I assume by displacements we mean displacements of the points of the boundary and that $t$ means the forces pressing against the boundary. But why do we integrate force * displacement over an area, as this would give units of energy * area (force * distance * $dA$)?

But $u$ also appears on the first term, where we integrate through the volume, except now we multiply it with $b$, the "external body forces". What are these? Integrating through the volume implies these are some kind of forces inside the body, so why are they called "external body forces"? And again, integrating Force * displacement through a volume gives the units of Energy * volume, which does not seem to make sense.

If somebody could provide clarification for me, I would be very grateful.


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