Virtual Work - Virtual Work Principle For A Deformable Body | Virtual Work Principle Deformable Body

Virtual Work Principle For A Deformable Body

Consider now the free body diagram of a deformable body, which is composed of an infinite number of differential cubes. Let's define two unrelated states for the body:

The -State (Fig.a): This shows external surface forces T, body forces f, and internal stresses in equilibrium.
The -State (Fig.b): This shows continuous displacements and consistent strains .

The superscript * emphasizes that the two states are unrelated. Other than the above stated conditions, there is no need to specify if any of the states are real or virtual.

Imagine now that the forces and stresses in the -State undergo the displacements and deformations in the -State: We can compute the total virtual (imaginary) work done by all forces acting on the faces of all cubes in two different ways:

First, by summing the work done by forces such as which act on individual common faces (Fig.c): Since the material experiences compatible displacements, such work cancels out, leaving only the virtual work done by the surface forces T (which are equal to stresses on the cubes' faces, by equilibrium).

Second, by computing the net work done by stresses or forces such as, which act on an individual cube, e.g. for the one-dimensional case in Fig.(c):

$F_B \big ( u^* + \frac{ \partial u^*}{\partial x} dx \big ) - F_A u^* \approx \frac{ \partial u^* }{\partial x} \sigma dV + u^* \frac{ \partial \sigma }{\partial x} dV = \epsilon^* \sigma dV - u^* f dV$

where the equilibrium relation has been used and the second order term has been neglected.

Integrating over the whole body gives:

- Work done by the body forces f.

Equating the two results leads to the principle of virtual work for a deformable body:

where the total external virtual work is done by T and f. Thus,

The right-hand-side of (d,e) is often called the internal virtual work. The principle of virtual work then states: External virtual work is equal to internal virtual work when equilibrated forces and stresses undergo unrelated but consistent displacements and strains. It includes the principle of virtual work for rigid bodies as a special case where the internal virtual work is zero.

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