The way planetary interiors deform depends primarily upon the elastic properties of the constituent polycrystalline materials. In particular, the speed of body waves passing through a material is entirely dependant upon the ratio of the elastic modulus of that material to its density. Whenever any external force is applied to a system, there is a resultant strain; similarly, whenever a system is strained in some way, there is then some stress upon the system. For example, squashing a jelly will deform it; deforming a jelly will result in a restoring force or stress eager to return the jelly to its original shape.
 
 

   Stress, Strain and Elastic Moduli:
 

An elastic modulus is just the ratio of stress to the associated strain. We wish to understand only the basic elastic equations and the physical meaning of these equations.

For a stress, s (hydrostatic, shear, axial...), resulting in an elastic deformation strain, e:

Eq. 35

where M is an elastic modulus (bulk, shear, Young's...).

For a hydrostatic stress (i.e., equally applied forces in all directions), which is often assumed within deep planetary interiors, the stress is the hydrostatic pressure:

Eq. 36

and the strain is the relative change in volume of the system:

Eq. 37

therefore:

Eq. 38

and the elastic modulus in this case is the incompressibility or bulk modulus:

Eq. 39

 

   Stress, Strain, and Tensors:
 

The stress, s, does not have to be hydrostatic; there may be unequally applied stresses in all directions, and therefore the stress is tensorial: where s_ij is the stress acting in the x_i direction on the plane perpendicular to the x_j direction; s_i=j are the axial stresses and s_i¹j are the shear stresses.

The strain e_ij is also a second order tensor. Therefore the elastic moduli, or elastic constants, are fourth order tensors:

Eq. 40

The stress, s, and the strain, e, must be symmetric, and the nature of c_ijkl depends on the symmetry of the crystal. It is customary to use a contracted notation thus:

c₁₁₁₁ Þ c₁₁ elastic constant relations s₁₁ to e₁₁

c₁₁₂₂ Þ c₁₂ elastic constant relations s₁₁ to e₂₂

c₂₃₂₃ Þ c₄₄ elastic constant relations s₂₃ to e₂₃

In general, 11®1; 22®2; 33®3; 23=32®4; 13=31®5; 12=21®6.
 
 

Single, low symmetry crystals:

There are a maximum of 21 elastic constants for a crystalline body:

N.B. symmetry allows c₁₂ Û c₂₁, etc..

However, all c_ij are rarely used, and it is therefore convenient to simplify matters by reducing the number of elastic constants to as small a number as possible.
 
 

   Crystals, Rocks and Elasticity:
  Cubic crystals:

For cubic crystals the elastic constants, c_ij, may be reduced to just three independent elastic constants:

c₁₁= c₂₂ = c₃₃ Þ modulus for axial compression, i.e., a stress s₁₁ results in a strain e₁₁ along an axis;

c₄₄ = c₅₅ = c₆₆ Þ shear modulus, i.e., a shear stress s₂₃ results in a shear strain e₂₃ across a face;

c₁₂ = c₁₃ = c₂₃ Þ modulus for dilation on compression, i.e., an axial stress s₁₁ results in a strain e₂₂ along a perpendicular axis.

All other c_ij = 0.

For single crystals, the elastic constants can be related to common elastic moduli such as:

Shear modulus:

Eq. 41

 
 

Eq. 42

  Bulk modulus:

Eq. 43

  Polycrystalline aggregates:

In the simplest case, we can consider a polycrystalline aggregate of crystals in random orientations, which is therefore isotropic. For such an isotropic system, the elastic constants may be reduced to just two, called the Lamé Constants, l and m, and the stress-strain relation then becomes:

Eq. 44

where d is equal to 1 for i=j, and to zero for i¹j; the strain tensor e_kk = DV / V.

The Lamé Constants are defined by:

Eq. 45

i.e., the shear modulus, and:

Eq. 46

and:

Eq. 47

So, for uniaxially applied stress:

Eq. 48

 

The estimation of the bulk properties from the elastic constants is fairly straightforward; however, when dealing with real materials, e.g., rocks, which are made up of polycrystalline aggregates, the elastic properties have to be evaluated by averaging the elastic constants over all the crytalline structures within the aggregate. For polycrystalline materials made up of non-cubic crystals with lower symmetry, appropriate substitutions have to be made in the elastic constants to account for the asymmetry, e.g., <c₁₁>= (c₁₁+c₂₂+c₃₃), etc.. Therefore the bulk modulus becomes:

Eq. 49

Similarly, two expressions may be obtained for the effective shear modulus, one under the assumption of constant stress, the other under the assumption of constant strain.

Under uniform stress (Reuss assumption: Reuss (1929)):

Eq. 50

  Under uniform strain (Voigt assumption: Voigt (1928)):

Eq. 51

  Normally, an average m is taken called the Voigt-Reuss-Hill (VRH) average: Hill (1952).
    Elasticity and Seismic Velocity:
  From an analysis of the passage of waves through a solid medium, the speed of seismic waves are given by:

Eq. 52

 
 

Eq. 53

Therefore, a knowledge of V_P and V_S is all that is required to obtain quantitative values for many elastic properties, some of which are outlined below.

a) Poisson's Ratio:
 

For uniaxial dilation (s₁₁¹ 0; s₂₂ = s₃₃ = 0), Poisson's ratio is defined:

Eq. 54

i.e., the ratio of thinning to elongation along perpendicular axes.

Analysis of the elastic constants gives Poisson's ratio in terms of more readily available parameters:

Eq. 55

From this we can see that for an incompressible solid (K = ¥) or liquid (m = 0), n = 0.5; for an infinitely compressible solid (K = 0), n = -1; thus we always have -1 < n < 0.5, and generally n ~ 0.25.

From the ratio seismic velocities given above, we can get:

Eq. 56

therefore

Eq. 57

So Poisson's ratio may be given in terms of seismic velocities thus:

Eq. 58

Therefore, Poisson's ratio for the Earth as a function of depth is obtainable directly from PREM and other seismic models.
 b) the seismic parameter:
  In addition to n, seismologists often use the seismic parameter, f:

Eq. 59

  c) the bulk velocity, V_f:
  The propagation velocity of the hydrostatic part of the strain (dilation), often called bulk velocity is given by:

Eq. 60

d) the Adams-Williamson equation:
  If we recall that K = -VdP / dV = rdP / dr (since V / dV = - r / dr ), then:

Eq. 61

i.e. the seismic parameter gives a direct measure of density variation with depth.

However, as one descends into the Earth, the pressure increases via:

Eq. 62

so in the limit of DP, Dr ® 0:

Eq. 63

When combined with f = dP / dr (Eq.61), this gives:

Eq. 64

so the variation of density with depth can be inferred from the seismic parameter, and therefore from seismic velocities. This is the Adams-Williamson Equation.
 
 

   Thermoelastic Coupling: thermodynamics-elasticity

Having discussed the essential thermodynamic (Lecture 1) and elastic properties of solid systems, we can now put them together. The combination of the effect of temperature and the effect of pressures (stress) through thermal expansion, a, and incompressibility, K, is called thermoelastic coupling, and is the most important cross-term in geophysics.
 

a) The Grüneisen Parameter:
 

The inter-relation between stress and temperature is dealt with via the Grüneisen parameter, an approximately constant, pressure and temperature independent parameter of the order of 1. Thermodynamically, the thermal Grüneisen parameter is defined by:

Eq. 65

and after some unpleasant thermodynamics (see problem sheet!), this leads to:

Eq. 66

Therefore a knowledge of g gives a good handle on many of the important thermoelastic properties of minerals, which shall be used later in the course.
 

b) The Mie-Grüneisen Equation of State:
 

When a solid is heated at constant volume, the atoms within it vibrate more vigourously, and this results in a thermal pressure acting from within the system outwards, which, if unresisted, will give rise to thermal expansion. From thermodynamics:

Eq. 67

Integrating at constant volume and assuming g is a constant (see also, Eq. 26):

Eq. 68

where U is the internal energy. Therefore:

Eq. 69

and this is the Mie-Grüneisen equation of state (described in more detail in Lecture 3).
 c) The adiabatic temperature gradient:
 

The adiabatic temperature gradient (no heat escapes, S is constant) is that caused by adiabatic compression; i.e., the change in temperature throughout the Earth as a result of pressure loading from above.

From thermodynamics we can show:

Eq. 70

or

Eq. 71

which on integration (for constant g) gives:

Eq. 72

Therefore if we know g we can estimate how the temperature varies with density in a planetary interior, assuming it is adiabatically generated.

Another approach may be made by considering another relationship from thermodynamics:

Eq. 73

but, recalling Eq. 63, the adiabatic temperature gradient is found to be:

Eq. 74

Therefore, the adiabatic temeprature gradient may be obtained from a knowledge of the seismic parameter (i.e., from PREM).

 

 
 

   References
 
 
 

	Anderson. O.L. (2000) The Grüneisen ratio for the last 30 years. Geophys. J. Int. 143, 279-294.
	Ballato. A. (1996) Poisson's ratio for tetragonal, hexagonal and cubic crystals. IEEE Trans. Ultrasonic Ferroelectrics and Freq. Control 43, 56-62.
	Hill. R. (1952) The elastic behaviour of a crystalline aggregate. Proc. Phys. Soc. London A 65, 349-354.
	Reuss. A. (1929) Brechnung der fliessgrense von mischkristallen auf grund der plastizitatsbedinggung für einkristalle. Zeit. für Ange. Math. Mech. 9, 49-58.
	Segletes. S.B., and W.P. Walters (1998) On theories of the Grüneisen parameter. J. Phys. Chem: Solids 59, 425-433.
	Voigt. W. (1928) Lehrbuch der kristallphysik. Teubner.