Averaging Schemes¶

Given a set of mineral physics parameters and an equation of state we can calculate the density, bulk, and shear modulus for a given phase. However, as soon as we have a composite material (e.g., a rock), the determination of elastic properties become more complicated. The bulk and shear modulus of a rock are dependent on the specific geometry of the grains in the rock, so there is no general formula for its averaged elastic properties. Instead, we must choose from a number of averaging schemes if we want a single value, or use bounding methods to get a range of possible values. The module burnman.averaging_schemes provides a number of different average and bounding schemes for determining a composite rock’s physical parameters.

Base class¶

class burnman.averaging_schemes.AveragingScheme[source]¶

Bases: object

Base class defining an interface for determining average elastic properties of a rock. Given a list of volume fractions for the different mineral phases in a rock, as well as their bulk and shear moduli, an averaging will give back a single scalar values for the averages. New averaging schemes should define the functions average_bulk_moduli and average_shear_moduli, as specified here.

average_bulk_moduli(volumes, bulk_moduli, shear_moduli)[source]¶

Average the bulk moduli \(K\) for a composite. This defines the interface for this method, and is not implemented in the base class.

Parameters:

volumes (list of floats) – List of the volume of each phase in the composite \([m^3]\).
bulk_moduli (list of floats) – List of bulk moduli of each phase in the composite \([Pa]\).
shear_moduli (list of floats) – List of shear moduli of each phase in the composite \([Pa]\).

Returns:

The average bulk modulus \(K\). \([Pa]\)

Return type:

float

average_shear_moduli(volumes, bulk_moduli, shear_moduli)[source]¶

Average the shear moduli \(G\) for a composite. This defines the interface for this method, and is not implemented in the base class.

Parameters:

volumes (list of floats) – List of the volume of each phase in the composite \([m^3]\).
bulk_moduli (list of floats) – List of bulk moduli of each phase in the composite \([Pa]\).
shear_moduli (list of floats) – List of shear moduli of each phase in the composite \([Pa]\).

Returns:

The average shear modulus \(G\). \([Pa]\)

Return type:

float

Voigt bound¶

class burnman.averaging_schemes.Voigt[source]¶

Bases: AveragingScheme

Class for computing the Voigt (iso-strain) bound for elastic properties. This derives from burnman.averaging_schemes.averaging_scheme, and implements the burnman.averaging_schemes.averaging_scheme.average_bulk_moduli() and burnman.averaging_schemes.averaging_scheme.average_shear_moduli() functions.

average_bulk_moduli(volumes, bulk_moduli, shear_moduli)[source]¶

Average the bulk moduli of a composite \(K\) with the Voigt (iso-strain) bound, given by:

\[K_V = \Sigma_i V_i K_i\]

Parameters:

volumes (list of floats) – List of the volume of each phase in the composite \([m^3]\).
bulk_moduli (list of floats) – List of bulk moduli \(K\) of each phase in the composite \([Pa]\).
shear_moduli (list of floats) – List of shear moduli \(G\) of each phase in the composite \([Pa]\). Not used in this average.

Returns:

The Voigt average bulk modulus \(K_R\) \([Pa]\).

Return type:

float

average_shear_moduli(volumes, bulk_moduli, shear_moduli)[source]¶

Average the shear moduli of a composite with the Voigt (iso-strain) bound, given by:

\[G_V = \Sigma_i V_i G_i\]

Parameters:

volumes (list of floats) – List of the volume of each phase in the composite \([m^3]\).
bulk_moduli (list of floats) – List of bulk moduli \(K\) of each phase in the composite \([Pa]\). Not used in this average.
shear_moduli (list of floats) – List of shear moduli \(G\) of each phase in the composite \([Pa]\).

Returns:

The Voigt average shear modulus \(G_V\) \([Pa]\).

Return type:

float

Reuss bound¶

class burnman.averaging_schemes.Reuss[source]¶

Bases: AveragingScheme

Class for computing the Reuss (iso-stress) bound for elastic properties. This derives from burnman.averaging_schemes.averaging_scheme, and implements the burnman.averaging_schemes.averaging_scheme.average_bulk_moduli() and burnman.averaging_schemes.averaging_scheme.average_shear_moduli() functions.

average_bulk_moduli(volumes, bulk_moduli, shear_moduli)[source]¶

Average the bulk moduli of a composite with the Reuss (iso-stress) bound, given by:

\[K_R = \left(\Sigma_i \frac{V_i}{K_i} \right)^{-1}\]

Parameters:

volumes (list of floats) – List of the volume of each phase in the composite \([m^3]\).
bulk_moduli (list of floats) – List of bulk moduli \(K\) of each phase in the composite \([Pa]\).
shear_moduli (list of floats) – List of shear moduli \(G\) of each phase in the composite \([Pa]\). Not used in this average.

Returns:

The Reuss average bulk modulus \(K_R\) \([Pa]\).

Return type:

float

average_shear_moduli(volumes, bulk_moduli, shear_moduli)[source]¶

Average the shear moduli of a composite with the Reuss (iso-stress) bound, given by:

\[G_R = \left( \Sigma_i \frac{V_i}{G_i} \right)^{-1}\]

Parameters:

volumes (list of floats) – List of the volume of each phase in the composite \([m^3]\).
bulk_moduli (list of floats) – List of bulk moduli \(K\) of each phase in the composite \([Pa]\). Not used in this average.
shear_moduli (list of floats) – List of shear moduli \(G\) of each phase in the composite \([Pa]\).

Returns:

The Reuss average shear modulus \(G_R\) \([Pa]\).

Return type:

float

Voigt-Reuss-Hill average¶

class burnman.averaging_schemes.VoigtReussHill[source]¶

Bases: AveragingScheme

Class for computing the Voigt-Reuss-Hill average for elastic properties. This derives from burnman.averaging_schemes.averaging_scheme, and implements the burnman.averaging_schemes.averaging_scheme.average_bulk_moduli() and burnman.averaging_schemes.averaging_scheme.average_shear_moduli() functions.

average_bulk_moduli(volumes, bulk_moduli, shear_moduli)[source]¶

Average the bulk moduli of a composite with the Voigt-Reuss-Hill average, given by:

\[K_{VRH} = \frac{K_V + K_R}{2}\]

This is simply a shorthand for an arithmetic average of the bounds given by burnman.averaging_schemes.voigt and burnman.averaging_schemes.reuss.

Parameters:

volumes (list of floats) – List of the volume of each phase in the composite \([m^3]\).
bulk_moduli (list of floats) – List of bulk moduli \(K\) of each phase in the composite \([Pa]\).
shear_moduli (list of floats) – List of shear moduli \(G\) of each phase in the composite \([Pa]\). Not used in this average.

Returns:

The Voigt-Reuss-Hill average bulk modulus \(K_{VRH}\) \([Pa]\).

Return type:

float

average_shear_moduli(volumes, bulk_moduli, shear_moduli)[source]¶

Average the shear moduli \(G\) of a composite with the Voigt-Reuss-Hill average, given by:

\[G_{VRH} = \frac{G_V + G_R}{2}\]

This is simply a shorthand for an arithmetic average of the bounds given by burnman.averaging_schemes.voigt and burnman.averaging_schemes.reuss.

\[G_V = \Sigma_i V_i G_i\]

Parameters:

volumes (list of floats) – List of the volume of each phase in the composite \([m^3]\).
bulk_moduli (list of floats) – List of bulk moduli \(K\) of each phase in the composite \([Pa]\). Not used in this average.
shear_moduli (list of floats) – List of shear moduli \(G\) of each phase in the composite \([Pa]\).

Returns:

The Voigt-Reuss-Hill average shear modulus \(G_{VRH}\) \([Pa]\).

Return type:

float

Hashin-Shtrikman upper bound¶

class burnman.averaging_schemes.HashinShtrikmanUpper[source]¶

Bases: AveragingScheme

Class for computing the upper Hashin-Shtrikman bound for elastic properties. This derives from burnman.averaging_schemes.averaging_scheme, and implements the burnman.averaging_schemes.averaging_scheme.average_bulk_moduli() and burnman.averaging_schemes.averaging_scheme.average_shear_moduli() functions. Implements formulas from [WDOConnell76]. The Hashin-Shtrikman bounds are tighter than the Voigt and Reuss bounds because they make the additional assumption that the orientation of the phases are statistically isotropic. In some cases this may be a good assumption, and in others it may not be.

average_bulk_moduli(volumes, bulk_moduli, shear_moduli)[source]¶

Average the bulk moduli of a composite with the upper Hashin-Shtrikman bound. Implements Formulas from [WDOConnell76], which are too lengthy to reproduce here.

Parameters:

volumes (list of floats) – List of the volumes of each phase in the composite. \([m^3]\)
bulk_moduli (list of floats) – List of bulk moduli \(K\) of each phase in the composite. \([Pa]\)
shear_moduli (list of floats) – List of shear moduli \(G\) of each phase in the composite. \([Pa]\)

Returns:

The upper Hashin-Shtrikman average bulk modulus \(K\). \([Pa]\)

Return type:

float

average_shear_moduli(volumes, bulk_moduli, shear_moduli)[source]¶

Average the shear moduli of a composite with the upper Hashin-Shtrikman bound. Implements Formulas from [WDOConnell76], which are too lengthy to reproduce here.

Parameters:

volumes (list of floats) – List of the volumes of each phase in the composite. \([m^3]\)
bulk_moduli (list of floats) – List of bulk moduli \(K\) of each phase in the composite. \([Pa]\)
shear_moduli (list of floats) – List of shear moduli \(G\) of each phase in the composite. \([Pa]\)

Returns:

The upper Hashin-Shtrikman average shear modulus \(G\). \([Pa]\)

Return type:

float

Hashin-Shtrikman lower bound¶

class burnman.averaging_schemes.HashinShtrikmanLower[source]¶

Bases: AveragingScheme

Class for computing the lower Hashin-Shtrikman bound for elastic properties. This derives from burnman.averaging_schemes.averaging_scheme, and implements the burnman.averaging_schemes.averaging_scheme.average_bulk_moduli() and burnman.averaging_schemes.averaging_scheme.average_shear_moduli() functions. Implements Formulas from [WDOConnell76]. The Hashin-Shtrikman bounds are tighter than the Voigt and Reuss bounds because they make the additional assumption that the orientation of the phases are statistically isotropic. In some cases this may be a good assumption, and in others it may not be.

average_bulk_moduli(volumes, bulk_moduli, shear_moduli)[source]¶

Average the bulk moduli of a composite with the lower Hashin-Shtrikman bound. Implements Formulas from [WDOConnell76], which are too lengthy to reproduce here.

Parameters:

volumes (list of floats) – List of the volumes of each phase in the composite. \([m^3]\)
bulk_moduli (list of floats) – List of bulk moduli \(K\) of each phase in the composite. \([Pa]\)
shear_moduli (list of floats) – List of shear moduli \(G\) of each phase in the composite. \([Pa]\)

Returns:

The lower Hashin-Shtrikman average bulk modulus \(K\). \([Pa]\)

Return type:

float

average_shear_moduli(volumes, bulk_moduli, shear_moduli)[source]¶

Average the shear moduli of a composite with the lower Hashin-Shtrikman bound. Implements Formulas from [WDOConnell76], which are too lengthy to reproduce here.

Parameters:

volumes (list of floats) – List of the volumes of each phase in the composite. \([m^3]\)
bulk_moduli (list of floats) – List of bulk moduli \(K\) of each phase in the composite. \([Pa]\)
shear_moduli (list of floats) – List of shear moduli \(G\) of each phase in the composite. \([Pa]\)

Returns:

The lower Hashin-Shtrikman average shear modulus \(G\). \([Pa]\)

Return type:

float

Hashin-Shtrikman arithmetic average¶

class burnman.averaging_schemes.HashinShtrikmanAverage[source]¶

Bases: AveragingScheme

Class for computing arithmetic mean of the Hashin-Shtrikman bounds on elastic properties. This derives from burnman.averaging_schemes.averaging_scheme, and implements the burnman.averaging_schemes.averaging_scheme.average_bulk_moduli() and burnman.averaging_schemes.averaging_scheme.average_shear_moduli() functions.

average_bulk_moduli(volumes, bulk_moduli, shear_moduli)[source]¶

Average the bulk moduli of a composite with the arithmetic mean of the upper and lower Hashin-Shtrikman bounds.

Parameters:

volumes (list of floats) – List of the volumes of each phase in the composite. \([m^3]\)
bulk_moduli (list of floats) – List of bulk moduli \(K\) of each phase in the composite. \([Pa]\)
shear_moduli (list of floats) – List of shear moduli \(G\) of each phase in the composite. Not used in this average. \([Pa]\)

Returns:

The arithmetic mean of the Hashin-Shtrikman bounds on bulk modulus \(K\) \([Pa]\).

Return type:

float

average_shear_moduli(volumes, bulk_moduli, shear_moduli)[source]¶

Average the bulk moduli of a composite with the arithmetic mean of the upper and lower Hashin-Shtrikman bounds.

Parameters:

volumes (list of floats) – List of the volumes of each phase in the composite. [m^3].
bulk_moduli (list of floats) – List of bulk moduli \(K\) of each phase in the composite. Not used in this average. \([Pa]\)
shear_moduli (list of floats) – List of shear moduli \(G\) of each phase in the composite. \([Pa]\)

Returns:

The arithmetic mean of the Hashin-Shtrikman bounds on shear modulus \(G\) \([Pa]\).

Return type:

float