Elastic constants notation

In crystals with the LI2 structure (the fcc-based ordered structure), there exist three independent elastic constants-in the contracted notation, Cn, C12 and 044. A set of three independent ab initio total-energy calculations (i.e. total energy as a function of strain) is required to determine these elastic constants. We have determined the bulk modulus, Cii, and C44 from distortion energies associated with uniform hydrostatic pressure, uniaxial strain and pure shear strain, respectively. The shear moduli for the 001 plane along the [100] direction and for the 110 plane along the [110] direction, are G ooi = G44 and G no = (Cu — G12), respectively. The shear anisotropy factor, A = provides a measure of the degree of anisotropy of the electronic charge... 

The change in energy density can be written in terms of the elastic constants C( j and c,2 (again, in notation used by Kittel, 1967) ... 

These corrections to the elastic constants were calculated using the above equation and compared with those determined numerically. The numerical approach was to apply small stresses (Sj Voigt notation) of + 0.2 GPa and —0.2 GPa with the elastic compliances related to the resulting strains, e, to e6, by... 

As per the notations used, the density and elastic constant C of the Pd film decrease, while constant C44 is increased upon interaction with hydrogen. 

The repeated suffix notation is still being used but now the subscripts take numbers 1 through 6. The components, c.j, are called the elastic stiffness constants and they also form a symmetric array, reducing the required number of elastic constants to 21. As these constants are properties of a material, the sym-... 

Calculations of the full stress tensor is a method ideally suited to the derivation of elastic constants, since it contains up to six independent pieces of information that otherwise would require extensive calculations of total energy. The c- and c. 2 elastic constants can be found from the stress-strain relation with the application of an ei-strain. (The Voigt notation is used, see e.g. (Nye, 1957), i.e. 11- -1, 22- 2, 33- 3, 23 4, 13 5, 12- -6 thus... 

In the Voigt notation, elastic constants can be expressed as 6 x 6 matrices [22]. For an isotropic material, C takes the form... 

In the Voigt notation, the average elastic constants for three atomistic boxes are (accurate to approximately 0.1 GPa)... 

The elastic constants (elastic moduli) A, and ju are called Lame constants (or Lame moduli, units [GPa]). Switching over from matrix notation to tensor notation the elasticity tensor is... 

Hint Use the two-suffix notation and the relations between the elastic constants for cubic crystals discussed in Sect.3.6. 

Fig. 30.24. Temperature dependence of the elastic constants in Gd2(Mo04),. The C notation refers to the tetragonal axes that are rotated 45° around [001] relative to the orthorhombic O ) axes (Hochli, 1972).

The above elastic constants are those used by the Orsay Group, except that for later notational convenience we have set and Ci = -Cf... 

Ctjki is a fourth order tensor that linearly relates a and e. It is sometimes called the elastic rigidity tensor and contains 81 elements that completely describe the elastic characteristics of the medium. Because of the symmetry of a and e, only 36 elements of Cyu are independent in general cases. Moreover only 2 independent rigidity constants are present in Cyti for linear homogeneous isotropic purely elastic medium Lame coefficient A and /r have a stress dimension, A is related to longitudinal strain and n to shear strain. For the purpose of clarity, a condensed notation is often used... 

Write the elastic-comphance tensor in matrix notation for an orthorhombic monocrystal, showing only the nine independent constants from Table 10.4. 

The constant a representing the elasticity of the spring connecting the segments is obscure, but may be of order a = 3kTjl C where f is the frictional coefficient. The other notations are the same as those in the previous section. We may express the vectors in N dimensional molecular space 29) ... 

Since, in our consideration, both the stress and the strain tensors are symmetrical about the diagonal, there are only six independent stress and strain elements, making many of the elastic compliances equal to each other. This makes it possible to simplify the generalized Hooke s law so that it involves at most 36 elastic compliances or constants, but requires the introduction of a shorthand notation both for stress and for strain for a unique representation that is referred to as the Voigt notation that we state as follows, for stresses and strains ... 

The presence in the system of inductive (rotational kinetic) energy and of capacitive (torsion) energy is made effective by the existence of the two systan constitutive properties, rotational inertia (or moment of inertia, also notated I) and torsion elastance (or more classically torsion constant ) ... 

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