pygplates.StrainRate

class pygplates.StrainRate

Bases: instance

The strain rate at a particular location (parcel of crust) represents the rate at which deformation occurs at that parcel of crust.

The strain rate is represented internally by the spatial gradients of velocity \(\boldsymbol L\) (in units of \(second^{-1}\)) calculated in spherical polar coordinates (ignoring radial dimension):

\[\begin{split}\boldsymbol L &= \boldsymbol v \boldsymbol \nabla\\ \begin{bmatrix} L_{\theta\theta} & L_{\theta\phi} \\ L_{\phi\theta} & L_{\phi\phi} \end{bmatrix} &= \begin{bmatrix} \frac{1}{R} \frac{\partial v_\theta}{\partial \theta} & \frac{1}{R \, \sin{\theta}} \frac{\partial v_\theta}{\partial \phi} - \cot{\theta} \frac{v_\phi}{R}\\ \frac{1}{R} \frac{\partial v_\phi}{\partial \theta} & \frac{1}{R \, \sin{\theta}} \frac{\partial v_\phi}{\partial \phi} + \cot{\theta} \frac{v_\theta}{R}\end{bmatrix}\end{split}\]

…where \(\boldsymbol v = (v_\theta, v_\phi)\) is the velocity (in \(m / s\)) in the local South-East coordinate system at a location (\(\theta, \phi\)), and \(R\) is the Earth mean radius (in \(m\)).

Note

The velocity spatial gradient \(\boldsymbol L\) is calculated at an arbitrary location in a deforming network using the following procedure:

For each triangle in a deforming network’s triangulation, an \(\boldsymbol L\) is calculated using velocities at the triangle’s three vertices (and, in the above gradient calculation, \(\cot{\theta}\), \(v_\theta\) and \(v_\phi\) are calculated at the triangle’s centroid location). Then each vertex in the entire triangulation is assigned an \(\boldsymbol L\) that is an area-weighted average of \(\boldsymbol L\)’s from faces incident to the vertex. Finally, \(\boldsymbol L\) at the arbitrary location is either assigned the \(\boldsymbol L\) of the triangle containing the location, or calculated using barycentric or natural neighbour interpolation of the \(\boldsymbol L\)’s from nearby vertices (depending on the strain rate smoothing parameter used to resolve the deforming network).

The spatial gradients of velocity tensor (\(\boldsymbol L\)) can be decomposed into the rate-of-deformation tensor (\(\boldsymbol D\)) and the vorticity (or spin) tensor (\(\boldsymbol W\)):

\[\begin{split}\boldsymbol L &= \boldsymbol D + \boldsymbol W\\ \boldsymbol D &= \frac{\boldsymbol L + \boldsymbol{L}^T}{2}\\ \boldsymbol W &= \frac{\boldsymbol L - \boldsymbol{L}^T}{2}\end{split}\]

…where \(D_{\theta\phi} = D_{\phi\theta}\) (since the rate-of-deformation tensor \(\boldsymbol D\) is symmetric).

If \(\Lambda\) is the stretch factor along current direction \(\hat{\boldsymbol n}\) then the rate of stretching per unit stretch is given by:

\[\frac{\dot{\Lambda}}{\Lambda} = \hat{\boldsymbol n} \cdot \boldsymbol D \cdot \hat{\boldsymbol n}\]

…where \(\hat{\boldsymbol n}\) is a 2D unit vector in the local South-East coordinate system at a location (\(\theta, \phi\)).

So that means, in the local South direction (ie, \(\hat{\boldsymbol n} = (1,0)\)) the rate of stretching per unit stretch is \(D_{\theta\theta}\), and in the local East direction (ie, \(\hat{\boldsymbol n} = (0,1)\)) it is \(D_{\phi\phi}\). These are the diagonal elements of \(\boldsymbol D\).

The rate of change of angle \(\alpha\) between two current directions \(\hat{\boldsymbol n}_1\) and \(\hat{\boldsymbol n}_2\) is given by:

\[-\dot{\alpha} = \hat{\boldsymbol n}_1 \cdot 2 \boldsymbol D \cdot \hat{\boldsymbol n}_2\]

The shear rate is commonly defined as half the rate of change of angle between two directions that are currently perpendicular. And if those directions are aligned with the local coordinate system (ie, \(\hat{\boldsymbol n}_1 = (1,0)\) and \(\hat{\boldsymbol n}_2 = (0,1)\)) then the shear rate is \(D_{\phi\theta}\) (which is the same as \(D_{\theta\phi}\) since \(\boldsymbol D\) is symmetric). So the symmetric off-diagonal elements represent the shear rate between the local coordinate axes (between South and East).

Note

The derivative of the Lagrangian finite strain tensor \(\boldsymbol E\) (see Strain) is related to the rate-of-deformation tensor \(\boldsymbol D\) (and the deformation gradient tensor \(\boldsymbol F\) - see Strain):

\[\dot{\boldsymbol E} = \boldsymbol{F}^T \cdot \boldsymbol D \cdot \boldsymbol F\]

And so the rate of stretching per unit stretch \(\frac{\dot{\Lambda}_{(\hat{\boldsymbol N})}}{\Lambda_{(\hat{\boldsymbol N})}}\) can be specified using a direction \(\hat{\boldsymbol N}\) in the initial configuration (eg, at a time before deformation began):.

\[\frac{\dot{\Lambda}_{(\hat{\boldsymbol N})}}{\Lambda_{(\hat{\boldsymbol N})}} = \frac{\hat{\boldsymbol N} \cdot \dot{\boldsymbol E} \cdot \hat{\boldsymbol N}}{\hat{\boldsymbol N} \cdot \boldsymbol C \cdot \hat{\boldsymbol N}}\]

…rather than using a direction \(\hat{\boldsymbol n}\) in the current configuration with \(\frac{\dot{\Lambda}_{(\hat{\boldsymbol n})}}{\Lambda_{(\hat{\boldsymbol n})}} = \hat{\boldsymbol n} \cdot \boldsymbol D \cdot \hat{\boldsymbol n}\). Note that \(\boldsymbol C\) is the Lagrangian deformation tensor (see Strain).

References:

Malvern, L. E. (1969). Introduction to the mechanics of a continuous medium. Prentice-Hall.
Mase, G.T., Smelser, R.E., & Mase, G.E. (2010). Continuum Mechanics for Engineers (3rd ed.). CRC Press.

Strain rates are equality (==, !=) comparable (but not hashable - cannot be used as a key in a dict).

Convenience class static data is available for the zero strain rate:

pygplates.StrainRate.zero

A StrainRate can also be pickled.

Added in version 0.46.

__init__(...)

A StrainRate object can be constructed in more than one way…

__init__()

Construct a zero strain rate (non-deforming).

zero_strain_rate = pygplates.StrainRate()

Note

Alternatively you can use zero_strain_rate = pygplates.StrainRate.zero.

__init__(velocity_gradient_theta_theta,, velocity_gradient_theta_phi, velocity_gradient_phi_theta, velocity_gradient_phi_phi)

Create from the spatial gradients of velocity \(\boldsymbol L\) (in units of \(second^{-1}\)) in spherical polar coordinates (ignoring radial dimension).

\[\begin{split}\boldsymbol L = \begin{bmatrix} L_{\theta\theta} & L_{\theta\phi} \\ L_{\phi\theta} & L_{\phi\phi} \end{bmatrix}\end{split}\]

param velocity_gradient_theta_theta:: \(L_{\theta\theta}\)
type velocity_gradient_theta_theta:: float
param velocity_gradient_theta_phi:: \(L_{\theta\phi}\)
type velocity_gradient_theta_phi:: float
param velocity_gradient_phi_theta:: \(L_{\phi\theta}\)
type velocity_gradient_phi_theta:: float
param velocity_gradient_phi_phi:: \(L_{\phi\phi}\)
type velocity_gradient_phi_phi:: float

`__init__`(...)	A StrainRate object can be constructed in more than one way...
`get_dilatation_rate`()	Return the rate of change of crustal area per unit area (in units of \(second^{-1}\)).
`get_rate_of_deformation`()	Return the rate-of-deformation symmetric tensor \(\boldsymbol D\) (in units of \(second^{-1}\)) in spherical polar coordinates (ignoring radial dimension).
`get_strain_rate_style`()	Return a measure categorising the type of deformation.
`get_total_strain_rate`()	Return the total strain rate (in units of \(second^{-1}\)).
`get_velocity_spatial_gradient`()	Return the spatial gradients of velocity tensor \(\boldsymbol L\) (in units of \(second^{-1}\)) in spherical polar coordinates (ignoring radial dimension).