Elasticity isotropic

The basic assumptions of fracture mechanics are (1) that the material behaves as a linear elastic isotropic continuum and (2) the crack tip inelastic zone size is small with respect to all other dimensions. Here we will consider the limitations of using the term K = YOpos Ttato describe the mechanical driving force for crack extension of small cracks at values of stress that are high with respect to the elastic limit. 

Propagation of small deformations in a linear elastic isotropic homogeneous medium... 

Let us examine the instability oi strained thin films. In Fig. 3, thin films of30 ML are coherently bonded to the hard substrates. The film phase has a misfit strain, e = 0.01, relative to the substrate phase, and the periodic length is equal to 200 a. The three interface energies are identical to each other = yiv = y = Y Both phases are elastically isotropic, but the shear modulus of the substrate is twice that of the film (p = 2p). On the left-hand side, an infinite-torque condition is imposed to the substrate-vapor and film-substrate interfaces, whereas torque terms are equal to zero on the right. In the absence of the coherency strain, these films are stable as their thickness is well over 16 ML. With a coherency strain, surface undulations induced by thermal fluctuations become growing waves. By the time of 2M, six waves are definitely seen to have established, and these numbers are in agreement with the continuum linear elasticity prediction [16]. 

E. Winkler, F. Grashof, H. Hertz,8 etc., have studied the stresses which are set up when two elastic isotropic bodies are in contact over a portion of their surface, when the surfaces of contact are perfectly smooth, and when the press, exerted between the surfaces is normal to the plane of contact. H. Hertz showed that there is a definite point in such a surface representing the hardness defined as the strength of a body relative to the kind of deformation which corresponds to contact with a circular surface of press. and that the hardness of a body may be measured by the normal press, per unit area which must act at the centre of a circular surface of press, in order that in some point of the body the stress may first reach the limit consistent with perfect elasticity. If H be the hardness of a body in contact with another body of a greater hardness than H, then for a circular surface of pressure of diameter d press. p radius of curvature of the line p and the modulus of penetration E,... 

As would be expected from Eq. (29) (eint=0 in this ease), the anisotropy should be ascribed to the electrostriction constant k. It should be noted that e and k in Eq. (29) refer to the elongational strain along the x-axis, Su whereas e in Fig. 24 is related to the elongational strain S in the yz-plane. If the film is assumed to be elastically isotropic,... 

This equation relates sM to the orientation of the stress-ellipsoid [cf. eq. (1.3)]. This result is first quoted by Lodge. It differs by a factor of one half from that for the (completely recoverable) simple shear s of a perfectly elastic isotropic solid (50) ... 

As the rubber hardness is a measurement of almost completely elastic deformation, it can be related to elastic modulus. Most rubber hardness tests measure the depth of penetration of an inden-tor under either a fixed weight or a spring load, and when rubber is assumed to be an elastic isotropic medium, the indentation obtained at small deformation depends on the elastic modulus, the... 

The (3 dependence of the amplification factor in an elastically isotropic crystal (for which R is independent of the direction of 0) is plotted for a temperature inside the coherent spinodal in Fig. 18.9. For (3 < /3crit> the amplification factor R (3) > 0 and the system is unstable—that is, the composition waves in Eq. 18.44 will grow exponentially. The wavenumber /3max, at which dR f3)/df3 = 0, receives maximum amplification and will dominate the decomposed microstructure. Outside the coherent spinodal, where d2 fhom/dc2B+2a2Y(h) > 0, all wavenumbers will have R(/3) < 0 and the system will be stable with respect to the growth of composition waves. 

Figure 18.9 Amplification factor vs. wavenumber plot for an elastically isotropic crystal at a temperature inside the coherent spinodal where d2 fhom /dc% + 2ct2 Y < 0.

Allen reported TEM observations of a nonaligned decomposition products in long-range ordered Fe-Al alloy [22]. Such morphologies are called isotropic spinodal microstructures. Similar structures are observed in Al-Zn and Fe-Cr alloys. Such structures can be produced in systems that are elastically isotropic or in which the lattice constant does not change appreciably with composition. 

Eshelby treated systems that are both elastically homogeneous and elastically isotropic [7]. Some results for the ellipsoidal inclusion described by Eq. 19.23 are given below. 

The case of a pure dilational transformation strain in an inhomogeneous elastically isotropic system has been treated by Barnett et al. [10]. For this case, the elastic strain energy does depend on the shape of the inclusion. Results are shown in Fig. 19.9, which shows the ratio of A(inhomo) for the inhomogeneous problem to A<7 (homo) for the homogeneous case, vs. c/a. It is seen that when the inclusion is stiffer than the matrix, AgE (inhomo) is a minimum... 

Incoherent Nucleation. Consider first incoherent nucleation on dislocations [19]. For linearly elastic isotropic materials, the energy per unit length Ei inside a cylinder of radius r having a dislocation at its center is given by... 

All the tests we have described above only give reliable answers for linear, elastic, isotropic, homogeneous materials, though even then in some cases errors may arise from carelessness in experimental method. In this section we shall try to describe many of the complications which arise when tests are applied to real polymeric materials, in particular to anisotropic materials, and to show how, if at all, they may be avoided. 

Elastic constants for elastically isotropic polycrystalline materials. A measure of the manner in which a polycrystalline body responds to small external forces in the elastic regime. 

SCHEME 14.1 Calculation of the elastic parameters EP and the additive molar functions U from the sound speed measurement u and vice versa. Valid only for elastic isotropic materials. 

Amorphous solids and polycrystalline substances composed of crystals of arbitrary symmetry arranged with a perfectly disordered or random orientation are elastically isotropic macroscopically (taken as a whole). They may be described by nine elastic constants, which may be reduced to two independent (effective) elastic constants. 

With elastically anisotropic materials the elastic behavior varies with the crystallographic axes. The elastic properties of these materials are completely characterized only by the specification of several elastic constants. For example, it can be seen from Table 10.3 that for a cubic monocrystal, the highest symmetry class, there are three independent elastic-stiffness constants, namely, Cn, C12, and C44. By contrast, polycrystalline aggregates, with random or perfectly disordered crystallite orientation and amorphous solids, are elastically isotropic, as a whole, and only two independent elastic-stiffness coefficients, C44 and C12, need be specified to fully describe their elastic response. In other words, the fourth-order elastic modulus tensor for an isotropic body has only two independent constants. These are often referred to as the Lame constants, /r and A, named after French mathematician Gabriel Lame (1795-1870) ... 

The first Lame constant (A) has no physical interpretation. However, both Lame constants are related to other elastic moduli. To see this, recall that the Young s modulus, E, is defined as the ratio of normal stress to normal strain. Hence, for an elastically isotropic body, E is given by (cn-Ci2)(cn-b 2ci2)/(cn-I-C12), or /r(3A-b 2/r)/(A-b/r). It should be emphasized that the Young s modulus is anisotropic for all crystal classes, including the cubic class, so this relation would never apply to any monocrystal. 

In a similar fashion, the rigidity modulus, G, for an elastically isotropic solid is given by 0-4/84 = C44 = 0-5/85 = C55 = cTs/8g = cgg = i(cn - C12) = /r, or C44, which represents a shape change without a volume change. Therefore, the second Lame constant (fi) is the shear modulus for an elastically isotropic body. The Lame constants may also be related directly to the bulk modulus, B, for an elastically isotropic body, which can be obtained through the relations /r = ( )(B - A) and = B - ( )G. 

According to Reuss, the effective Young s modulus of an elastically isotropic solid is given by ... 

Use the relations in Tables 10.3 and 10.4 to derive the Voigt and Reuss approximations for the bulk modulus of an elastically isotropic polycrystalline aggregate composed of tetragonal monocrystals. 

One finds in the simplest case of an elastically isotropic plate that there is an infinite set of waves that can exist. These are known as Lamb waves, after Sir Horace Lamb who published their first detailed description [60]. 

In this relation, a and a, are the thermal expansion coefficients of the substrate and film. These depend on the temperature T. If the film is homogeneous and elastically isotropic, the in plane thermoelastic stress is expressed by ... 

Screw dislocation. The simplest case to start with is that of a straight screw dislocation of Burgers vector b parallel to the surface of a thin parallel-sided crystal foil, as shown in Figure 5.10. Using the coordinate system defined there, the dislocation AB is parallel to y and at a depth z below the top surface. The dislocation causes a column CD of unit cells parallel to z in the perfect crystal to be deformed. If we assume that the atomic displacements around the dislocation are the same in the thin specimen as in an infinitely large, elastically isotropic crystal, then the components u, v, w of the deformation of the column along thex, y, and z directions will be... 

Figure 5.23. Variation of the 20-percent image width with e, g, Tq, and t for a spherical inclusion in an elastically isotropic crystal matrix at 5 = 0.

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