MS_effective_medium

MS_effective_medium - generate elastic constants from various effective
                      medium theories.

Generate elastic constants for the effective medium parameters based on
various theories, identified by the string 'theory'. Subsequent required
arguments depend on the theory invoked.

Currently available theories:

Tandon and Weng, 84 (tandon or t&w) Spheroids

(Isotropic host matrix with unidirectionally aligned isotropic spheroid
 inclusions)

[Ceff,rh]=MS_effective_medium('t&w',vpm,vsm,rhm,vpi,vsi,rhi,del,f) or
   Input parameters:
      vpm,vsm,rhm : isotropic parameters of the matrix (km/s, kg/m3)
              del : aspect ratio of spheroids:
                       del < 1 = smarties
                       del = 1 = spheres
                       del > 1 = cigars
               f : volume fractions of inclusions
   vpi, vsi, rhi : isotropic parameters of the inclusions

[Ceff,rh]=MS_effective_medium('t&w',Cm,rhm,Ci,rhi,del,f) or
   Input parameters:
        Cm,rh : elasticity and density of the matrix (GPa, kg/m3)
              del : aspect ratio of spheroids:
                       del < 1 = smarties
                       del = 1 = spheres
                       del > 1 = cigars
               f : volume fractions of inclusions
    Ci, rhi : isotropic parameters of the inclusions

Output parameters:
   Ceff : Elastic constants (GPa) (symmetry in X1 direction)
   rh : aggregate density (kg/m3)

Hudson, 1980, 1981 (hudson or crack) Cracks

[Ceff,rh]=MS_effective_medium('hudson',vpm,vsm,rhm,vpc,vsc,rhc,ar,cd) or
   Input parameters:
      vpm,vsm,rhm : isotropic parameters of the matrix (km/s, kg/m3)
               ar : aspect ratio of cracks
               cd : crack density
   vpc, vsc, rhc : isotropic parameters of the fill material

[Ceff,rh]=MS_effective_medium('hudson',Cm,rhm,Ci,rhi,ar,cd) or
   Input parameters:
        Cm,rh : elasticity and density of the matrix (GPa, kg/m3)
               ar : aspect ratio of cracks
               cd : crack density
    Cc, rhc : isotropic parameters of the fill material

Output parameters:
   Ceff : Elastic constants (GPa) (symmetry in X1 direction)
   rh : aggregate density (kg/m3)

Backus, 1962 (backus) stack of thin horizontal layers

[Ceff,rh]=MS_effective_medium('backus',thickness,vp,vs,rh) or
   Input parameters:
        thickness : layer thicknesses (vector)
         vp,vs,rh : isotropic parameters of the layers (km/s, kg/m3) (vector)

Output parameters:
   Ceff : Elastic constants (GPa) (symmetry in X1 direction)
   rh : aggregate density (kg/m3)

[Ceff,rh]=MS_effective_medium('backus',thickness,C,rh) or
   Input parameters:
        thickness : layer thicknesses (vector)
                C : elasticity of the layers (GPa) (6x6xn tensor)
               rh : density of the layers (GPa, kg/m3) (vector)

Output parameters:
   Ceff : Elastic constants (GPa) (symmetry in X3 direction)
   rh : aggregate density (kg/m3)

References

Tandon, GP and Weng, GJ. The Effect of Aspect Ratio of Inclusions on the
     Elastic Properties of Unidirectionally Aligned Composites. Polymer
     Composites, 5, pp 327-333, 1984.

Hudson, J. A., Overall properties of a cracked solid, Math. Proc. Camb.
     Phil. Soc., 88, pp 371-384. 1980

Hudson, J. A., Wave speeds and attenuation of elastic waves in material
     containing cracks, Geophys. J. R. Astr. Soc., 64, pp 133-150. 1981

Crampin, S. Effective anisotropic elastic constants for wave propagation
     through cracked solids. Geophys. J. R. Astr. Soc., 76, pp 135-145, 1984

Backus, G. E., Long-Wave Elastic Anisotropy Produced by Horizontal Layering.
     J. Geophys. Res., pp 4427-4440

See also: MS_elasticDB