Elastic Energy - Elastic Potential Energy in Mechanical Systems

Elastic Potential Energy in Mechanical Systems

Components of mechanical systems will store elastic potential energy if they are deformed when forces are applied to the system. Energy is transferred to an object (i.e. work is done on it) any time a force external to it displaces or deforms the object. The quantity of energy transferred by work to the object is computed as the vector dot product of the force and the displacement of the object. As forces are applied to the system they are distributed internally to its component parts. While some of the energy transferred can end up stored as kinetic energy of acquired velocity, the deformation of the shape of component objects results in stored elastic energy.

A prototypical elastic component is a coiled spring. The linear elastic performance of a spring is parametrized by a constant of proportionality, called the spring constant. This constant is usually denoted as k (see also Hooke's Law) and depends on the geometry, cross sectional area, undeformed length and nature of the material from which the coil is fashioned. Within a certain range of deformation, k remains constant and is defined as the negative ratio of displacement to the magnitude of the restoring force produced by the spring at that displacement.

Note that L, the deformed length, can be larger or smaller than L_o, the undeformed length, so to keep k positive, F_r must be given as a vector component of the restoring force whose sign is negative for L>L_o and positive for L< L_o. If we abbreviate the displacement as

then Hooke's Law can be written in the usual form

.

Energy absorbed and stored in the spring can be derived using Hooke's Law to compute the restoring force as a measure of the applied force. This requires the assumption, sufficiently correct in most circumstances, that at a given moment, the magnitude of applied force, F_a is equal to the magnitude of the resultant restoring force, but its direction and thus sign is different. In other words, assume that at each point of the displacement F_a = k x, where F_a is the component of applied force along the x direction

For each infinitesimal displacement dx, the applied force is simply k x and the product of these is the infinitesimal transfer of energy into the spring dU. The total elastic energy placed into the spring from zero displacement to final length L is thus the integral

In the general case, elastic energy is given by the Helmholtz potential per unit of volume f as a function of the strain tensor components ε_ij:

where λ and μ are the Lamé elastical coefficients. The connection between stress tensor components and strain tensor components is:

For a material of Young's modulus, Y (same as modulus of elasticity λ), cross sectional area, A₀, initial length, l₀, which is stretched by a length, :

where U_e is the elastic potential energy.

The elastic potential energy per unit volume is given by:

where is the strain in the material.