Koenders M. A. C. - Physics Of The Deformation Of Densely Packed Granular Materials, The

Literature: [Jeffreys, 1931].

The identity is ij; its inverse is also ij.

The identity of rank-4 tensors is such that a rank-4 tensor Aijpq, which connects two symmetric rank-2 tensors has the inverse

The isotropic tensor of rank 4 is

Its inverse is

The inverse of the latter is the solution of the equation and

In three dimensions the corresponding expressions are

The front factors 2, , /4 in 2-d and 4, 4/3, 4/15 in 3-d are easily gathered in a coefficient

where (2m)!! = 2.4.... (2m); (2m 1)!! = 1.3.5... (2m 1), m an integer. (see [Abramowitz and Stegun, 1965], Section 6.1.49).

A very useful coefficient is

Conversion of the isotropic elastic constants in two and three dimensions. Top line: 3-D; bottom line 2-D.

and : Lam constants

E : Youngs modulus, v : contraction coefficient (Poissons ratio)

K : bulk modulus, G : shear modulus

Fourier transforms are a powerful tool to solve differential equations. In this section the basic theory is explored, which is required for the characterisation of the fluctuations. To begin with a one-dimensional approach is taken, in which functions depend on one variable, the time t say. In the development below it makes sense to keep the integral sign with its boundaries together with the integration variable.

The appropriate tool for describing fluctuating physical phenomena is the auto-correlation function. For a fluctuating function of time z(t) (which is zero on average) it is defined as

It is seen that the expectation value z2 is just equal to z(0).

The Fourier transform of the function z(t) is defined as

The inverse transform is

which makes the delta function (the identity)

The latter has meaning only in the context of another function, as follows

In physical processes the infinite integral boundaries do not make sense and therefore a modified definition is employed, the truncated Fourier transform

Now calculate the inverse Fourier transform of the quantity

The order of the integrals may be interchanged and therefore

Integrating over (using the properties of the delta function) gives

For a time record that is much longer than the correlation time the upper boundary may be replaced by and it is seen that in the limit the inverse Fourier transform Sz(t) is just equal to the correlation function z(t). This is the famous WienerKhinchin theorem. The quantity density. z() is called the spectral intensity or harmonic (spectral) density.

In two or three dimensions the formulas are easily extended, by taking the integrals over multiple variables. The Fourier frequency then becomes a vector. In a spatial setting this vector is called the wave number k.

Literature [Abramowitz and Stegun, 1965].

Bessel functions are very useful in the evaluation of problems that have cylinder symmetry. There are various types. The best-known is the family of ordinary Bessel functions, or Bessel functions of the first kind. They can either be defined as a series expansion

Or as an integral

where v denotes the order of the Bessel function, indicating which member of the family is meant.

The other type of cylinder functions that are useful are the Modified Bessel Functions.