Theory linear elastic

Of course, the above independence takes place provided that / = 0 in the domain with the boundary C. The integral of the form (4.100) is called the Rice-Cherepanov integral. We have to note that the statement obtained is proved for nonlinear boundary conditions (4.91). This statement is similar to the well-known result in the linear elasticity theory with linear boundary conditions prescribed on S (see Bui, Ehrlacher, 1997 Rice, 1968 Rice, Drucker, 1967 Parton, Morozov, 1985 Destuynder, Jaoua, 1981). 

A more practical approach for quantifyiag the conditions required for fracture uses a stress intensity criterion instead of an energy criterion. Using linear elastic theory, it has been shown that under an appHed stress, when the stress intensity K,... 

Equations (10.23) and (10.24) hold for the /3-phase as well and could be inserted into Eqn. (10.22). The additivity of pt with respect to the elastic and electric potential is based on 1) the assumption of linear elastic theory (which is an approximation) and 2) the low energy density of the electric field (resulting from the low value of the absolute permittivity e0 = 8.8x10 12 C/Vm). In equilibrium, V/i, = 0 and A V, = df-pf = 0. Therefore, in an ionic system with uniform hydrostatic pressure, the explicit equilibrium condition reads Aa/fi=A)... 

The exact solutions of the linear elasticity theory only apply for small strains, and under idealised loading conditions, so that they should at best only be treated as approximations to the real behaviour of materials under test conditions. In order to describe a material fully we need to know all the elastic constants and, in the case of linear viscoelastic materials, relaxed and unrelaxed values of each, a distribution of relaxation times and an activation energy. While for non-linear viscoelastic materials we cannot obtain a full description of the mechanical properties. 

According to Eqs. (13.145) and (13.148) the fracture stress in plane strain is a factor 1 /(1-v2) 1 /0.84 1.2 higher than in plane stress. Experimentally, however, the difference is much bigger. The reason for this discrepancy is that Griffith s equations were developed in linear fracture mechanics, which is based on the results of linear elasticity theory where the strains are supposed to be infinitesimal and proportional to the stress. 

Consider a vertically hanging metal rod, to which a load can be apphed (e.g. a steel cable supporting an elevator), as in Figure 10.3. The load exerts a tensHe force over the entire cross-sectional area of the rod, which is said to be under uniaxial stress since only the stress along one of the principal axes is nonzero. The stress is equal to the force divided by the cross-sectional area over which it is distributed. In linear elastic theory, according to Hooke s law, the magnitude of the strain produced in the rod by a small uniform applied stress is directly proportional to the magnitude of the applied stress. Hence ... 

If one applies tensile stress on a solid, the solid elongates and gets strained. The stress (a) - strain (e) relation is linear for small stresses (Hooke s law) after which nonlinearity appears, in some cases. Finally at a critical stress CTf, depending on the material, amount of disorder and the specimen size etc., the solid breaks into pieces fracture occurs. In the case of brittle solids, the fracture occurs immediately after the Hookean linear region, and consequently the linear elastic theory can be applied to study the essentially nonlinear and irreversible static fracture properties of brittle solids (Lawn and Wilshaw 1975, Thomson 1986, Evans and Zok 1986). 

This vast number of possibilities calls for a systematic procedure to identify a subset of the most likely interface matchings of the parent crystals. This subset will then be the starting point for atomistic modeling. The question about unit cell size and shape is relatively simple to address. Many related procedures based on linear elasticity theory and lattice strain estimates may be adopted. The basic situation is sketched in Fig. 4 an overlayer unit cell A needs to be matched together with a substrate unit cell B. Matching pairs of unit cells are, in general, multiples of primitive cells in the interface plane for the metal and ceramic, respectively. 

In order to calculate the contraction in the axial direction, the Poisson s ratio, Vc2i, must be estimated. According to linear elasticity theory,... 

The necessary conditions to be fulfilled are the equilibrium conditions, the strain-displacement relationships (kinematic equations), and the stress-strain relationships (constitutive equations). As in linear elasticity theory (12), these conditions form a system of 15 equations that permit us to obtain 15 unknowns three displacements, six strain components, and six stress components. 

In the following results are presented for the application of the Boundary Finite Element Method both for the case of the laminate free-edge effect and for the case of a single transverse matrix crack in the framework of linear elasticity theory. 

Another interesting observation is that many thermal plastics that have been tested in a uniaxial tension mode in our laboratories exhibit negative volume changes in the strain region before yield. Others before have observed the same or equally confusing results and many researchers dismiss such observations as experimental errors. It is demonstrated that such observations can be real and are possible within the framework of linear elastic theory. 

If the flaw is not cracklike, ordinary stress concentration factors may be used in conjunction with the failure criteria of the preceding section to predict failure. If the flaws (inherent or introduced) are cracklike, however, stress concentration factors are useless because linear elasticity theory predicts an infinite concentration factor at the crack tip. For these situations, experiment shows a degradation in strength as schematically represented in Region II of Figure 1. 

The advantage of considering the boundary as an array of dislocation cores is that the strain field around the core can be calculated from linear elasticity theory [11.26]. Here, the lattice either side of the grain boundary is assumed to be unstrained and is used as a reference for the lattice positions in the vicinity of the dislocation. However, such models do not specifically take into account... 

Fig. 11.6. Schematic of a [100] or [010] dislocation core in YBCO. The plane of atoms that is removed to create the dislocation core may be either Cu—O or Y/Ba—O. Associated with the removal of a plane of atoms is a strain field that can be calculated from linear elasticity theory.

Hydrogen incorporation in thin films can produce an extremely high out-of-plane expansion due to the clamping of the thin film to the substrate [20]. Within the sohd solution phase the expansion can be predicted by linear elastic theory [21]. However, deviations from linear elastic theory were reported for high H concentrations due to the onset of plastic deformation in the film [22, 23]. 

The simple picture afforded by the wormlike chain model is likely to break down under conditions of extreme deformation, as discussed in Section 9.13.2. The internal organization of dsDNA is likely to have instances where the chain is highly deformed, significantly beyond the range where linear elasticity theory is an adequate description. In addition, the potential for dsDNA to exhibit curvature-induced softening may promote... 

Notch sensitivity is evaluated by considering a tensile specimen with a stress concentrating feature such as a hole. For an isotropic elastic tensile specimen with a circular hole of one half of the section width, as an example, linear elasticity theory predicts a maximum stress at the edge of the hole in the plate to be 2.15 times the average section stress... 

In Section 5.3 we have based the mechanical analysis on linear elasticity theory. In particular, we have assumed that the actuation-induced stress is proportional to the swelling/deswelling caused by ion transport, as represented by (5.8). While this is a reasonable assumption when the deformation is small, experimental evidences have indicated that the relationship... 

Analyses carried out in connection with serviceability limit states should normally be based on linear elastic theory. 

Young modulus Proportionality constant between stress and strain in linear elasticity theory... 

A second fracture parameter is also provided by linear elastic theory, namely, the energy G required to create unit area of fracture surface. Again there is a critical value, of G at which the crack begins to propagate, and a configuration-dependent formula which gives G in terms of measurable quantities. For the center crack mentioned above, the formula for G is... 

Linear elasticity theory, on which LEFM is based, is not valid when the strains are large or when the stress-strain curves are nonlinear. To cope with this situation, various alternative formulations of fracture mechanics have been developed. 

Alternatively to the Griffith s energy criterion, In the 1950s Irwin calculated the stresses in the vicinity of a crack tip based on linear elasticity theory. He introduced a constant, K, the stress intensity factor, to characterize the magnitude of the stress field. If K (Kj in mode I) reaches a critical value Kc (Kjc), then the crack will propagate in the material Kjc is generally expressed in MPam. ... 

In the limit of infinitesimal deformations the Eulerian strain tensor becomes identical with the strain tensor used in the linear elasticity theory, as can be easily shown. The components of the linear strain tensor are defined by... 

The analysis was first carried out by Griffith in a treatment of the brittle fracture of metals. Actually, the considerations are of general nature and can also be applied to polymers, after introducing some physically important but formally simple modifications. Griffith s approach is based on linear elasticity theory and its utility for polymers may look questionable at first, as those are neither elastic nor linear under the conditions near to failure. However, as we will see, the theory is indeed applicable and provides also here a satisfactory description of crack growth. 

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