ELASTIC PROPERTIES AND PRESSURE EFFECTS

ELASTIC PROPERTIES AND PRESSURE EFFECTS Elastic constants 

Stress is defined as force per unit area. The elastic response of a solid consists of deformation which occurs in all the directions. Since force has three components in three (chosen) x, y and z directions and since the deformation also occurs in all three directions, stress, a, is a tensor of the second rank having 9 components described by a (3 x 3) matrix cTy /. 3. Alternately it may be easier to visualize why a is a tensor of second rank if one recalls that area itself can be described by a vector perpendicular to the surface which has three components. 

Deformation is measured by a quantity known as strain (strain is a relative extension or contraction of dimension). Strain is similarly a tensor of the second rank having nine components (3x3 matrix). The relation between stress and strain in the elastic regime is given by the classical Hooke s law. It is therefore obvious that the Hooke s proportionality constant, known as the elastic modulus, is a tensor of 4 rank and is represented by a (9 x 9) matrix. Before further discussion we note the following. The stress tensor consists of 9 elements of which stability conditions require cjxy=(jyx, Jxz=(J x and cTyj cr. Therefore the number of independent stress components in the symmetric stress matrix are only six. Similarly there are only six independent strain components. Therefore there can only be six stress and six strain components for an elastic body which has unequal elastic responses in x, y and z directions as in a completely anisotropic solid. The representation of elastic properties become simple by following the well known Einstein convention. The subscript xx, yy, zz, yz, zx and xy are respectively represented by 1, 2, 3, 4, 5 and 6. Therefore Hooke s law may now be written in a generalized form as. 

Thus one would expect from a (6x6) matrix of the elastic stiffness coefficients (cy) or compliance coefficients (sy) that there are 36 elastic constants. By the application of thermodynamic equilibrium criteria, cy (or Sij) matrix can be shown to be symmetrical cy =cji and sy=Sji). Therefore there can be only 21 independent elastic constants for a completely anisotropic solid. These are known as first order elastic constants. For a crystalline material, periodicity brings in elements of symmetry. Therefore symmetry operation on a given crystal must be consistent with the representation of the elastic quantities. Thus for example in a cubic crystal the existence of 3C4 and 4C3 axes makes several of the elastic constants equal to each other or zero (zero when under symmetry operation cy becomes -cj,). As a result, cubic crystal has only three independent elastic constants (cn= C22=cs3, C44= css= cee and ci2=ci3= C2i=C23=C3i=C32). Cubic Symmetry is the highest that can be attained in a crystalline solid but a glass is even more symmetrical in the sense that it is completely isotropic. Therefore the independent elastic constants reduce further to only two, because C44=( cu- c 2) 2. 

Conventionally elastic properties of solids are described using moduli called Young s modulus, E, shear modulus, G, bulk modulus, K and the Poisson s ratio, v (due to the fact that mechanical properties were studied more extensively by engineers).Young s modulus is defined by the relation 

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