Strain Energy Methods

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

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:

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

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