Following the answers

Variation in true displacement is caused by variation of actual loading. Virtual displacement is any one which can take place without violation of constraints.

Thank you Victor J. Аdlucky

could you provide more explanation about variation in displacement?

Thanks

Please look at the sect. The principle of virtual displacements (p.156)

thank you so much Victor J. Аdlucky

The principle of virtual displacement in structural and solid mechanics is an alternative form of equilirium expression of a body. This principle states that for any admissible virtual displacement, the total virtual work should vanish. Here, virtual displacement means an imaginary displacement (not a real displacement due to external loads) that we imagine to take place at the body, in addition to the real displacement.

The variation in displacement is a term that I think originates from the concept of calculus of variation. It means that we take a infinitesimal change in the displacement field, just like taking a differential of an independent variable in traditional calculus. Applying this concept in the area of structural and solid mechanics, the variation in displacement can be interpreted as virtual dispalcement. Thus, they a basically identical.

Thank you Foek Tjong Wong

We can treat virtual displacements as the sum of real displacements and displacement variations. Since the total work of real displacements is equal to zero, we deal only with displacement variations.

Thank you so much Victor J. Аdlucky

But will " total work of real displacements is equal to zero" be suitable for non conservative systems?.

The total work is not real work being done during the process of deformation, it is the work of final loads on final displacements.

* A virtual displacement is any field of displacements, which satisfy the boundary conditions, but is imaginary (do not actually occur). Moreover, a virtual displacement means an instantaneous (time is held constant) change in coordinates so the state of stress does not change (a real displacement would change the state of stress).

Then, the work of all (real) forces on virtual displacements should be zero if the system is in a state of equilibrium.

* The variation in displacement is an infinitesimal change in the displacement field so the change in the state of stress can be neglected. Then, the variation in displacement can be interpreted as virtual dispalcement.