Energy balance approach

The energy criterion for fracture is simply an extension of Griffith s hypothesis which describes quasi-static crack propagation as the conversion of the work done, Wd, by the external force and the available elastic energy stored in the bulk of the specimen, U, into surface free energy, y. It may be written  

However, if it is assumed that energy dissipation around the crack tip occurs in a manner independent of both the test geometry and the way in which the forces are applied to the specimen then 2 ymmay be replaced in Equation 7.2 by the symbol Gc. The value Gc encompasses all the energy losses incurred around the crack tip and is, therefore, the energy required to increase the crack by unit length in a specimen of unit width. Hence, the fracture criterion becomes  

For bonded structures exhibiting bulk linear-elastic behaviour, i.e. away from the crack tip regions, they obey Hooke s Law. Equation 7.3 may be expressed [1-4] as  

However, the evaluation of Equation 7.3 is not necessarily restricted to a linear-elastic fracture mechanics (LEFM) approach, i.e. it is not restricted to those materials or structures which exhibit linear-elastic behaviour. For example, Rivlin and Thomas [8] have employed this approach to characterize 

More recently, Ahagon et al, [12], Thomas and Kadir [13] and Kinloch and Tod [14] have suggested that, even for rubbers which show significant internal 

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Sometimes the failure occurs by propagation of a crack that starts at the top and travels downward until the interface is completely debonded. In this case, the fracture mechanics analysis using the energy balance approach has been applied [92] in which P, relates to specimen dimensions, elastic constants of fiber and matrix, initial crack length, and interfacial work of fracture (W,). 

Griffith derived a similar equation using an energy balance approach, equating stored energy with the energy required for crack propagation ... 

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. 

Wc have developed these relations using the intuitive energy balance approach. However, we could have obtained the same relations by setting up the appropriate differential equations and solving them, as illustrated in Examples 2-18 and 2-19. 

The finite difference formulation is given above to demonstrate how difference equations ate obtained from differential equations. However, we use the energy balance approach in the following section.s to obtain the numerical formulation because it is more intuitive and can handle boundary conditions more easily. Besides, the energy balance approach does not require having the differential equation before the analysis. 

Using the Tmile difference form of the first derivative (not the energy balance approach), obtain the finite difference formulation of the boundary nodes for the case of insulation at the left boundary (node 0) and radiation at the right boundary (node 5) with an emissiviiy of e and surrounding temperature of... 

Consider steady one-dimensional beat conduction in a composite plane wall consisting of iwo layers A and B in perfect contact at the interface. The wall involves no heat generation. The nodal network of the medium consisl.s of nodes 0, 1 (at Ihe interface), and 2 with a uniform nodal spacing of A.x. Using the energy balance approach, obtain Ihe finite difference formulation of this problem for the case of insulation at the left... 

Consider steady one-dimensional heat conduction in a pin fin of constant diajneter D with constant thermal conductivity. The fm is losing heal by conveclion to the ambient air at T with a heat transfer coefficient of A. The nodal network of the fm consists of nodes 0 (at Ihe base), 1 (in the middle), and 2 (at the fin tip) with a uniform nodal spacing of Ax. Using the energy balance approach, obtain the finite difference formulation of (his problem to determine T, and T2 for Ihe case of specified temperature at the fin base and negligible heat transfer at the fin tip. All temperatures are in C. 

The energy balance approach to fluid flow problems is developed before the momentum balancejis introduced. This leads to a very simple and logical development of Bernoulli s equation and an intuitively satisfying treatment of fluid friction. In the undergraduate program at the University of Utah, the... 

For an atomically sharp crack of length 2c, as illustrated in Figure 18.5, the energy balance approach shows that... 

Griffith confirmed this equation using experimental data on glass as shown in Figure 18.6. Although the energy balance approach works well, kinetic effects may also be present during fracture as demonstrated for the fracture of mica flake. 

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