Isotropic elastic solid

5 Applications of elasticity theory E.5.1 Isotropic elastic solid 

We examine next the example of an isotropic, elastic solid. First, when a normal strain is applied in the x direction, the solid not only deforms according to Eq. (E.12), but it also deforms in the y, z directions according to 

An analogous expression holds for shear stresses. Namely, if a shear stress a y is applied to a solid, the corresponding shear strain is given by 

The relations for the strain can be written compactly with the use of the Kronecker 8 and the Einstein summation convention, as 

Values of the shear modulus, Lamd s constant and Poisson s ratio for representative solids are given in Table E.2. 

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Bulk Modulus and Bulk Compressibility. According to Cramer(Ref 5,P 1), one of the important constants of an isotropic elastic solid is the bulk modulus (K) or its reciprocal the bulk compressibility(B). The K is defined as the ratio of stress to strain when the stress is a pressure applied equally on all surfaces of the sample and the strain is the resulting change in volume per unit volume. If a sinusoidally varying pressure is superimposed... 

Ying, C.F., Truell, R. 1956. Scattering of a plane longitudinal wave by a spherical obstacle in an isotropically elastic solid. J. Appl. Phys. 27, 1086-1097. 

Figure 14.28 Strain of an isotropic elastic solid, (a) Application of a stress Gq to a sample having an equilibrium separation of atomic planes, h. (b) Potential energy as a function of the separation x. (c) Dependence of cro on x. (From Ref. 35.)...

We consider a homogenous and isotropic elastic solid occupying three dimensional region. The equation of motion without body force is given as... 

Equations (6.4) and (6.6) show that there are only two independent elastic moduli for an isotropic elastic solid. A further useful relationship can be obtained by assuming that tensile stresses perpendicular axes OX1X2X3 ehosen in the material and letting the corresponding strains be ci, C2 and 63. The average of the three normal stresses cti, ct2 and is equal to minus the hydrostatic pressure (pressure is positive when directed inwards). It can then be shown (see problem 6.2) that... 

From first principles, prove that — 1 isotropic elastic solid. 

For isotropic elastic solids there are only two independent elastic constants, or compliances. While Young s modulus E and the shear modulus // are the most widely used, we shall choose as the two physically independent pair of moduli the shear modulus /i and the bulk modulus K, where the first gauges the shear response and the second the bulk or volumetric response. However, in stating the linear elastic response in the equations below we still choose the more compact pair of E and //. Thus, for the six strain elements we have... 

While the dilatational and shear response of an isotropic elastic solid are mechanistically distinct, they are not uncoupled. The presence of a pressure results in a stiffening of the shear response and the presence of a shear stress alters the crystal or material symmetry, and therefore affects the bulk modulus. (For an operationally enlightening and useful treatment of these dependences, see Rice et al. (1992).)... 

Invariants of Stress and Strain and Isotropic Elastic Solids... 

Suppose that a thin film is bonded to one surface of a substrate of uniform thickness hs- It will be assumed that the substrate has the shape of a circular disk of radius R, although the principal results of this section are independent of the actual shape of the outer boundary of the substrate. A cylindrical r, 0, z—coordinate system is introduced with its origin at the center of the substrate midplane and with its z—axis perpendicular to the faces of the substrate the midplane is then at z = 0 and the film is bonded to the face at z = hs/2. The substrate is thin so that hs R, and the film is very thin in comparison to the substrate. The film has an incompatible elastic mismatch strain with respect to the substrate this strain might be due to thermal expansion effects, epitaxial mismatch, phase transformation, chemical reaction, moisture absorption or other physical effect. Whatever the origin of the strain, the goal here is to estimate the curvature of the substrate, within the range of elastic response, induced by the stress associated with this incompatible strain. For the time being, the mismatch strain is assumed to be an isotropic extension or compression in the plane of the interface, and the substrate is taken to be an isotropic elastic solid with elastic modulus Es and Poisson ratio Vs the subscript s is used to denote properties of the substrate material. The elastic shear modulus /Xg is related to the elastic modulus and Poisson ratio by /ig = Es/ 1 + t s). 

An isotropic elastic solid with a nominally flat, traction-free surface is subjected to an initial equilibrium stress field. Suppose that the shape of the free surface S in the undeformed reference configuration of the material is not actually a plane, but that it is slightly wavy. The nominally flat surface coincides with the plane y = 0 and the position of the actual surface varies with respect to y = 0 in the x—direction. At time t, the position of the surface at coordinate x is given hy y = h x,t). For the discussion in this section, it is assumed that the slope of the surface is small everywhere, that is, h,x boundary condition which must be enforced on the wavy surface is that the traction is zero. 

The near-tip asymptotic displacement field of a crack at the interface between two isotropic elastic solids after Hutchinson and Suo reads ... 

Figure 4. (a) The angular distribution of the stresses very close to the crack tip. The dots are numerical results for a material with scE/ao 3, Ho O.lno, m 0.01 and v 0.25, and the solid lines are the result for an isotropic elastic solid. Note that the 1/v r radial dependence has been accounted for within the normalization, (b) The angular distribution of remanent strains very close to the crack tip from the same computation. The numerical results plotted in both figures are for all integration stations within the radial range 8 x lO i , < r < 1.3 x 10 i ,. 

Consider the case of uniform stress. This can be imagined as a system of N elemental cubes arranged end-to-end forming a series model (Figure 8.23). Assume that each elemental cube is a transversely isotropic elastic solid, the direction of elastic symmetry being defined by the angle 9, which its axis makes with the direction of applied external stress a. The strain in each cube ci is then given by the compliance formula... 

At the other extreme end from a viscous liquid we have the isotropic, elastic solid. Now, instead of the deformation rate, we would expect the stress to depend on deformation. As a deformation measure we can take C or B or their inverses, or any combination of them. Working with B, say, we let... 

A model substance that represents closely the deformation behaviour for small strains of a wide range of materials is the linear isotropic, elastic solid. As the name impUes, the substance is isotropic, satisfies Hooke s law for aU strains and shows perfect elasticity. 

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