Editing Virtual work (section)

==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 <math> \boldsymbol{\sigma} </math>-State <!-- No Fig.a in the page. -->: This shows external surface forces '''T''', body forces '''f''', and internal stresses <math> \boldsymbol{\sigma} </math> in equilibrium.
* The <math> \boldsymbol{\epsilon} </math>-State <!-- No Fig.b in the page. -->: This shows continuous displacements <math> \mathbf {u}^* </math> and consistent strains <math> \boldsymbol{\epsilon}^* </math>.
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 <math> \boldsymbol{\sigma} </math>-State  undergo the [[displacement (vector)|displacement]]s and [[deformation (engineering)|deformation]]s in the <math> \boldsymbol{\epsilon} </math>-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 <math> F_A </math> which act on individual common faces (Fig.c): Since the material experiences compatible [[displacement (vector)|displacement]]s, 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 <math> F_A </math>, <math> F_B </math> which act on an individual cube, e.g. for the one-dimensional case in Fig.(c): <math display="block"> F_B \left( u^* + \frac{ \partial u^*}{\partial x} dx \right ) - F_A u^* \approx  \frac{ \partial u^* }{\partial x} \sigma dV + u^* \frac{ \partial \sigma }{\partial x} dV = \epsilon^* \sigma dV - u^* f dV </math> where the equilibrium relation <math> \frac{ \partial \sigma }{\partial x}+f=0 </math> has been used and the second order term has been neglected. {{pb}} Integrating over the whole body gives: <math display="block">\int_{V} \boldsymbol{\epsilon}^{*T} \boldsymbol{\sigma} \, dV </math> – Work done by the body forces '''f'''.

Equating the two results leads to the principle of virtual work for a deformable body:
{{NumBlk||<math display="block">\text{Total external virtual work} = \int_{V} \boldsymbol{\epsilon}^{*T} \boldsymbol{\sigma} dV </math>|{{EquationRef|d}}}}
where the total external virtual work is done by '''T''' and '''f'''. Thus,
{{NumBlk||<math display="block"> \int_{S} \mathbf{u}^{*T} \mathbf{T} dS + \int_{V} \mathbf{u}^{*T} \mathbf{f} dV = \int_{V} \boldsymbol{\epsilon}^{*T} \boldsymbol{\sigma} dV </math>|{{EquationRef|e}}}}

The right-hand-side of ({{EquationNote|d}},{{EquationNote|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.

===Proof of equivalence between the principle of virtual work and the equilibrium equation===

We start by looking at the total work done by surface traction on the body going through the specified deformation:
<math display="block"> \int_{S} \mathbf u \cdot \mathbf T dS = \int_{S} \mathbf u \cdot \boldsymbol \sigma \cdot \mathbf n dS </math>

Applying divergence theorem to the right hand side yields:
<math display="block"> \int_S \mathbf{u \cdot \boldsymbol \sigma \cdot n} dS = \int_V \nabla \cdot \left( \mathbf{u} \cdot \boldsymbol \sigma \right) dV </math>

Now switch to indicial notation for the ease of derivation.
<math display="block">\begin{align}
\int_V \nabla \cdot \left( \mathbf{u} \cdot \boldsymbol \sigma \right) dV 
  &= \int_V \frac{\partial}{\partial x_j} \left( u_i \sigma_{ij} \right) dV \\
  &= \int_V \left( \frac{\partial u_i}{\partial x_j} \sigma_{ij} + u_i \frac{\partial \sigma_{ij}}{\partial x_j}\right) dV
\end{align}</math>

To continue our derivation, we substitute in the equilibrium equation <math> \frac{\partial \sigma_{ij}}{\partial x_j} + f_i = 0 </math>. Then
<math display="block">\int_V \left(\frac{\partial u_i}{\partial x_j} \sigma_{ij} + u_i \frac{\partial \sigma_{ij}}{\partial x_j}\right) dV
       = \int_V \left(\frac{\partial u_i}{\partial x_j} \sigma_{ij} - u_i f_i\right) dV</math>

The first term on the right hand side needs to be broken into a symmetric part and a skew part as follows:
<math display="block">\begin{align}
\int_V\left( \frac{\partial u_i}{\partial x_j} \sigma_{ij} - u_i f_i\right) dV
       &= \int_V\left( \frac12 \left[ \left( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \right) 
          + \left( \frac{\partial u_i}{\partial x_j} - \frac{\partial u_j}{\partial x_i} \right) \right] \sigma_{ij} - u_i f_i \right) dV \\
       &= \int_V \left( \left[ \epsilon_{ij} 
          + \frac12 \left( \frac{\partial u_i}{\partial x_j} - \frac{\partial u_j}{\partial x_i} \right) \right] \sigma_{ij} - u_i f_i\right)  dV \\
       &= \int_V\left( \epsilon_{ij} \sigma_{ij} - u_i f_i \right) dV\\
       &= \int_V \left( \boldsymbol\epsilon : \boldsymbol\sigma - \mathbf u \cdot \mathbf f \right)  dV
\end{align}</math>
where <math> \boldsymbol\epsilon </math> is the strain that is consistent with the specified displacement field. The 2nd to last equality comes from the fact that the stress matrix is symmetric and that the product of a skew matrix and a symmetric matrix is zero.

Now recap. We have shown through the above derivation that
<math display="block"> \int_{S} \mathbf{u \cdot T} dS = \int_V \boldsymbol\epsilon : \boldsymbol\sigma dV - \int_V \mathbf u \cdot \mathbf f dV </math>

Move the 2nd term on the right hand side of the equation to the left:
<math display="block"> \int_{S} \mathbf{u \cdot T} dS + \int_V \mathbf u \cdot \mathbf f dV = \int_V \boldsymbol\epsilon : \boldsymbol\sigma dV </math>

The physical interpretation of the above equation is, ''the External virtual work is equal to internal virtual work when equilibrated forces and stresses undergo unrelated but consistent displacements and strains''.

For practical applications:

* In order to impose equilibrium on real stresses and forces, we use consistent virtual displacements and strains in the virtual work equation.
* In order to impose consistent displacements and strains, we use equilibriated virtual stresses and forces in the virtual work equation.

These two general scenarios give rise to two often stated variational principles. They are valid irrespective of material behaviour.

===Principle of virtual displacements===
Depending on the purpose, we may specialize the virtual work equation. For example, to derive the principle of virtual displacements in variational notations for supported bodies, we specify:

* Virtual displacements and strains as variations of the real displacements and strains using variational notation such as <math> \delta\ \mathbf {u} \equiv \mathbf{u}^* </math> and <math> \delta\ \boldsymbol {\epsilon} \equiv \boldsymbol {\epsilon}^* </math>
* Virtual displacements be zero on the part of the surface that has prescribed displacements, and thus the work done by the reactions is zero. There remains only external surface forces on the part <math> S_t </math> that do work.

The virtual work equation then becomes the principle of virtual displacements:
{{NumBlk||<math display="block"> \int_{S_t} \delta\ \mathbf{u}^T \mathbf{T} dS + \int_{V} \delta\ \mathbf{u}^T \mathbf{f} dV = \int_{V}\delta\boldsymbol{\epsilon}^T \boldsymbol{\sigma} dV </math>|{{EquationRef|f}}}}

This relation is equivalent to the set of equilibrium equations written for a differential element in the deformable body as well as of the stress boundary conditions on the part <math> S_t </math> of the surface. Conversely, ({{EquationNote|f}}) can be reached, albeit in a non-trivial manner, by starting with the differential equilibrium equations and the stress boundary conditions on <math> S_t </math>, and proceeding in the manner similar to ({{EquationNote|a}}) and ({{EquationNote|b}}).

Since virtual displacements are automatically compatible when they are expressed in terms of [[Continuous function|continuous]], [[single-valued function]]s, we often mention only the need for consistency between strains and displacements. The virtual work principle is also valid for large real displacements; however, Eq.({{EquationNote|f}}) would then be written using more complex measures of stresses and strains.

===Principle of virtual forces===
Here, we specify:

* Virtual forces and stresses as variations of the real forces and stresses.
* Virtual forces be zero on the part <math> S_t </math> of the surface that has prescribed forces, and thus only surface (reaction) forces on <math> S_u </math> (where displacements are prescribed) would do work.

The virtual work equation becomes the principle of virtual forces:
{{NumBlk||<math display="block"> \int_{S_u} \mathbf{u}^T \delta\ \mathbf{T} dS + \int_{V} \mathbf{u}^T \delta\ \mathbf{f} dV = \int_{V} \boldsymbol{\epsilon}^T \delta \boldsymbol{\sigma} dV </math>|{{EquationRef|g}}}}

This relation is equivalent to the set of strain-compatibility equations as well as of the displacement boundary conditions on the part <math> S_u </math>. It has another name: the principle of complementary virtual work.