Elastic-strain energy

Any elastic deformation of a material stores energy as can be easily understood by considering the spring model from section 2.3. To calculate this energy, we consider an (infinitesimal) brick-shaped volume element of length I and cross section A to which a load F is applied. The resulting stress is r = F/A. If we increase the stress by an amount dcr, the external force must increase by dF = da A. The material lengthens by an amount dl. 

The total work done per unit volume in a material strained up to max is the integral over dw  

This equation is valid for arbitrary uniaxial deformations. If the deformation is irreversible, part of the work is transformed to heat and cannot be recovered on unloading. In elastic (reversible) deformations, the energy is stored in 

Here we use the force at the beginning of the strain increment. As we can neglect second-order terms in this infinitesimal calculation, this does not make a difference dW = F + dF)dl = Fdl + dFdl = Fdl. 

This calculation was valid for uniaxial stresses and strains only. For arbitrary stresses and strains, we have to generalise by switching to tensors  

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Noting that the maximum elastic strain energy ... 

Any gas stored or transferred under pressure represents an energy source. If piping or equipment failure occurs, energy is given out as the gas rapidly expands down to virtually atmospheric pressure. Elastic strain energy in the walls is also given out, but, by comparison, this is relatively small. 

This energy is considered to be supplied by the elastic strain energy, We, that is released per unit thickness by the formation of the crack and is given by... 

In retrospect, it should not be surprising that a time independent theory modeled after elasticity theory does not apply to a plastic flow process. Elastic deformation is conservative with the work done on the material stored as elastic strain energy. Plastic deformation is non-conservative with the work done on the material dissipated as heat, or converted into internal defects... 

Plastic deformation is a transport process in which elements of displacement are moved by a shear stress from one position to another. Unlike the case of elastic deformation, these displacements are irreversible. Therefore, they do not have potential energy (elastic strain energy) associated with them. Thus, although the deformation associated with them is often called plastic strain, it is a fundamentally different entity than an elastic strain. In this book, therefore, it will be called plastic deformation, and the word strain will be reserved for elastic deformation. 

In order to cause plastic deformation, stress must be created in a material by loading it elastically. This creates a complication by coupling elastic strains with plastic deformations, and thereby creating an interplay between elastic strain energy and the absorption of energy by plastic deformation. Situations can then exist in which elastic strain-energy drives plastic deformation without any change in the nominally applied stress. One way in which this manifests... 

The elastic strain energy of the material surrounding the core. This energy for a straight line in a large crystal is given approximately by... 

Um Elastic strain energy in vessel walls V Volume of gas... 

Pit formation. If we consider a dissolution nucleus at a screw dislocation intersecting the surface which consists of a cylindrical hole of radius r, one atom layer deep (a), then the free energy of formation of this nucleus will be composed of a volume energy, surface energy, and elastic strain energy term, respectively, as follows ... 

The energy dissipated can be compared with the maximum elastic strain energy, W, which is stored in the material during the stress cycle. Because the elastic strain is proportional to the applied stress, W is equal to just half of the product of the maximum stress and strain (i.e., W = a0ei/2), and therefore... 

Another quantity of interest is the velocity dependence of the energy of the dislocation. The energy density in the material around the dislocation, w, is the sum of the elastic strain-energy density and the kinetic-energy density,... 

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