Elasticity/Principle of minimum potential energy

Strain Energy Density

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The strain energy density is defined as

 

If the strain energy density is path independent, then it acts as a potential for stress, i.e.,

 

For adiabatic processes,   is equal to the change in internal energy per unit volume.

For isothermal processes,   is equal to the Helmholtz free energy per unit volume.

The natural state of a body is defined as the state in which the body is in stable thermal equilibrium with no external loads and zero stress and strain.

When we apply energy methods in linear elasticity, we implicitly assume that a body returns to its natural state after loads are removed. This implies that the Gibb's condition is satisfied :

 

Principle of Minimum Potential Energy

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This principle states that

  • If the prescribed traction and body force fields are independent of the deformation
  • then the actual displacement field makes the potential energy functional an absolute minimum.

In other words, the principle of minimum potential energy states that the potential energy functional

 

is minimized by the actual displacement field.

Proof

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The first step in the proof is to show that the actual displacements make the function   stationary. The second step is to show that the stationary point is actually the minimum.

Proof of the Principle of Stationary Potential Energy

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The first variation of the potential energy functional   is

 

or,

 

or,

 

Therefore,   (i.e.   is stationary) only if

 

which are the conditions that only the actual displacement field satisfies.

Proof of the Principle of Minimum Potential Energy

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To prove that   is not only stationary, but also the global minimum all we now need to show is that

 

Now,

 

If the displacement field is a pure rigid body motion, then the strain energy density

 

where   is the spin tensor given by

 

If the displacement field does not contain any rigid body motion, then the strain energy density is given by

 

where   is the strain tensor given by

 

Therefore, for a displacement field containing both spin and strain

 

or,

 

This means that   for all values of   other than rigid body motion, in which case  .


Hence, for mixed boundary value problems   for all  , as long as the displacement BCs prevent rigid body motions. Therefore, the potential energy functional is minimized by the actual displacement field.