An alternative to statement of equilibrium equations are minimum-energy methods.

The objectives of this article is to present a brief description of concepts of strain energy methods.

The energy stored in a body due to deformation is called the "strain energy". The strain energy per unit volume is called "strain energy density" which is the area under the stress-strain curve up to the deformation point.

### Work and Energy:

Consider a solid object acted upon by force, F, at a point, O, as shown in the figure. Let the deformation at the point be infinitesimal and be represented by vector dr, as shown in figure.

The work done = F.dr

For the general case: W
= Fx dx only the force in the direction of the deformation does the work.

### Amount of Work done:

Constant Force: If the force is constant, the work is simply the product of the force and the displacement.

W = Fx

Linear Force: If the force is proportional to the displacement, the work is one half multiplied by the end force and displacement

W = Fx

Linear Force: If the force is proportional to the displacement, the work is one half multiplied by the end force and displacement

### Strain Energy:

Consider
a simple spring system, subjected to a Force such
that F is proportional to displacement x; F= kx.

Now
determine the work done when F = Fo, from before:

This
energy (work) is stored in the spring and is released when the force is
returned to zero

### Strain Energy Density:

Consider
a cube of material acted upon by a force, Fx,
creating stress sx=Fx/a2

causing
an elastic displacement, d in the x direction, and strain ex=d/a

Where
U is called the Strain
Energy, and
u is the Strain
Energy Density.

### Shear Strain Energy:

Consider
a cube of material acted upon by a shear stress,txycausing an elastic shear strain gxy

### Total Strain Energy for a Generalized State of Stress:

If we consider a 3D body, the generalized total strain energy will be as following:

where:

The strain energy and strain energy density is a scaler quantity.

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