Linear elastic

Let C i be a bounded domain with a smooth boundary L, and n = (ni,n2,n3) be a unit outward normal vector to L. Introduce the stress and strain tensors of linear elasticity (see Section 1.1.1),... 

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). 

In this section we find the derivative of the energy functional in the three-dimensional linear elasticity model. The derivative characterizes the behaviour of the energy functional provided that the crack length is changed. The crack is modelled by a part of the two-dimensional plane removed from a three-dimensional domain. In particular, we derive the Griffith formula. 

Khludnev A.M. (1983) A contact problem of a linear elastic body and a rigid punch (variational approach). Appls. Maths. Mechs. 47 (6), 999-1005 (in Russian). 

Crack Tip Stresses. The simplest case for fracture mechanics analysis is a linear elastic material where stress. O, is proportional to strain, S,... 

Because the material is assumed to be linear elastic, the local displacements around the crack tip can also be expressed ia terms of K... 

In moie ductile materials the assumptions of linear elastic fracture mechanics (LEFM) are not vahd because the material yields more at the crack tip, so that... 

Fracture Mechanics. Linear elastic fracture mechanics (qv) (LEFM) can be appHed only to the propagation and fracture stages of fatigue failure. LEFM is based on a definition of the stress close to a crack tip in terms of a stress intensification factor K, for which the simplest general relationship is... 

The use of the single parameter, K, to define the stress field at the crack tip is justified for brittle materials, but its extension to ductile materials is based on the assumption that although some plasticity may occur at the tip the surrounding linear elastic stress field is the controlling parameter. 

Substantial work on the appHcation of fracture mechanics techniques to plastics has occurred siace the 1970s (215—222). This is based on earlier work on inorganic glasses, which showed that failure stress is proportional to the square root of the energy required to create the new surfaces as a crack grows and iaversely with the square root of the crack size (223). For the use of linear elastic fracture mechanics ia plastics, certaia assumptioas must be met (224) (/) the material is linearly elastic (2) the flaws within the material are sharp and (J) plane strain conditions apply ia the crack froat regioa. 

Elastic Behavior. Elastic deformation is defined as the reversible deformation that occurs when a load is appHed. Most ceramics deform in a linear elastic fashion, ie, the amount of reversible deformation is a linear function of the appHed stress up to a certain stress level. If the appHed stress is increased any further the ceramic fractures catastrophically. This is in contrast to most metals which initially deform elastically and then begin to deform plastically. Plastic deformation allows stresses to be dissipated rather than building to the point where bonds break irreversibly. 

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,... 

The tensile strength of a unidirectional lamina loaded ia the fiber direction can be estimated from the properties of the fiber and matrix for a special set of circumstances. If all of the fibers have the same tensile strength and the composite is linear elastic until failure of the fibers, then the strength of the composite is given by... 

The importance of inherent flaws as sites of weakness for the nucleation of internal fracture seems almost intuitive. There is no need to dwell on theories of the strength of solids to recognize that material tensile strengths are orders of magnitude below theoretical limits. The Griffith theory of fracture in brittle material (Griflfith, 1920) is now a well-accepted part of linear-elastic fracture mechanics, and these concepts are readily extended to other material response laws. 

We can now define the elastic moduli. They are defined through Hooke s Law, which is merely a description of the experimental observation that, when strams are small, the strain is very nearly proportional to the stress that is, they are linear-elastic. The nominal tensile strain, for example, is proportional to the tensile stress for simple tension... 

This linear relationship between stress and strain is a very handy one when calculating the response of a solid to stress, but it must be remembered that most solids are elastic only to very small strains up to about 0.001. Beyond that some break and some become plastic - and this we will discuss in later chapters. A few solids like rubber are elastic up to very much larger strains of order 4 or 5, but they cease to be linearly elastic (that is the stress is no longer proportional to the strain) after a strain of about 0.01. 

Figure 8.1 shows the stress-strain curve of a material exhibiting perfectly linear elastic behaviour. This is the behaviour characterised by Hooke s Law (Chapter 3). All solids are linear elastic at small strains - by which we usually mean less than 0.001, or 0.1%. The slope of the stress-strain line, which is the same in compression as in tension, is of... 

Fig. 8.1. Stress-strain behaviour for a linear elastic solid. The axes are calibrated for a material such as steel.

Figure 8.2 shows a non-linear elastic solid. Rubbers have a stress-strain curve like this, extending to very large strains (of order 5). The material is still elastic if unloaded, it follows the same path down as it did up, and all the energy stored, per unit volume, during loading is recovered on unloading - that is why catapults can be as lethal as they are. 

For linear elastic strains, and only linear elastic strains,... 

When two linear-elastic materials (though with different moduli) are mixed, the mixture is also linear-elastic. The modulus of a fibrous composite when loaded along the fibre direction (Fig. 25.1a) is a linear combination of that of the fibres, Ef, and the matrix, E, ... 

When a foam is compressed, the stress-strain curve shows three regions (Fig. 25.9). At small strains the foam deforms in a linear-elastic way there is then a plateau of deformation at almost constant stress and finally there is a region of densification as the cell walls crush together. 

Linear-elasticity, of course, is limited to small strains (5% or less). Elastomeric foams can be compressed far more than this. The deformation is still recoverable (and thus elastic) but is non-linear, giving the plateau on Fig. 25.9. It is caused by the elastic... 

Fig. 25.10. Cell wall bending gives the linear-elastic portion of the stress-strain curve.

---