Elasticity stored energy

The difference between the work input and the elastic stored energy is the crack growth resistance, R,... 

Equation (2.84) may be converted into a more practical form as follows. Consider a piece of material of thickness, B, subjected to a force, F, as shown in Fig. 2.63(a). The load-deflecdon graph is shown as line (i) in Fig. 2.63(b). From this the elastic stored energy, U i, may be expressed as... 

If the crack extends by a small amount da then the stiffness of the material changes and there will be a small change in both load, dF, and deflection, 33. This is shown as line (ii) in Fig. 2.63(b). The elastic stored energy would then be... 

Figure 26.49 (upper) shows the elastic stored energy of two hne forces acting at an angle a to the surface of the semi-inhnite body simultaneously and Figure 26.49 (lower) the horizontal stress component. As expected both energy and stress show strong peaks at the points of contact. 

The spring is elastically storing energy. With time this energy is dissipated by flow within the dashpot. An experiment performed using the application of rapid stress in which the stress is monitored with time is called a stress relaxation experiment. For a single Maxwell model we require only two of the three model parameters to describe the decay of stress with time. These three parameters are the elastic modulus G, the viscosity r and the relaxation time rm. The exponential decay described in Equation (4.16) represents a linear response. As the strain is increased past a critical value this simple decay is lost. 

Griffith used an energy balance approach to predict the crack propagation conditions (see Williams, 1984). The driving force is the elastically stored energy in the notched samples, which can be used to create new surfaces. A parameter Gc, the critical elastic strain energy release rate [GIc in mode I], can be determined and expressed in J m-2. 

Frequently, a characteristic relaxation time, k, is used to describe viscoelastic behavior. It is a measure for the time needed to transform the reversibly-elastically stored energy into friction heat ... 

They further considered the additional energy lost where fracture of strands occurred. An extra term, lU, unit volume, which for the rubber they used is an elastic stored energy. Because this additional term is proportional to the depth of the pores, it dominates for deep pores. For Gent and Lin s system, it could be several hundred times the work of detachment from a smooth surface. 

There are two principal theories, or models, that attempt to describe what happens during brittle fracture, the Griffith fracture theory and the Irwin model. Both assume that fracture takes place through the presence of preexisting cracks or flaws in the polymer and are concerned with what happens near such a crack when a load is applied. Each leads to the definition of a fracture-toughness parameter and the two parameters are closely related to each other. The Griffith theory is concerned with the elastically stored energy near the crack, whereas the Irwin model is concerned with the distribution of stresses near the crack. Both theories apply strictly only for materials that are perfectly elastic for small strains and are therefore said to describe linear fracture mechanics. 

The break behavior of any desired elastic body is described by the Griffith theory. According to Griffith, a crack in an elastic body only propagates further when the elastically stored energy just exceeds the energy required to break chemical bonds. Combination of this with the Ingles concept leads to... 

When whiskers exceed the elastic limit they behave in one of three ways (I) they fracture by a cleavage (2) they show an important but strongly localized plastic deformation (3) they creep. Very thin copper and iron whiskers with high elastic limits fracture in a more or less brittle manner as is the case for materials that are normally brittle. The sudden release of large amounts of elastically stored energy produces high... 

First, Griffith considered that fracture produces a new surface area and postulated that for fracture to occur the increase in energy required to produce the new surface must be balanced by a decrease in elastically stored energy. 

Second, to explain the large discrepancy between the measured strength of materials and those based on theoretical considerations, he proposed that the elastically stored energy is not distributed uniformly throughout the specimen but... 

The elastically stored energy decreases and so —(dU/dc) is essentially a positive quantity. 

Griffith calculated the change in elastically stored energy using a solution obtained by Inglis [2] for the problem of a plate, pierced by a small elliptical crack, that is stressed at right angles to the major axis of the crack. Equation (12.1) then allows the fracture stress of the material to be defined in terms of... 

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