Elastic response theory

and subject to the incompressibility constraint

Here, is the stress tensor, the identity tensor,

Force balance inside the planet yields (Love 2011)

(C.23) |

It follows from Equations (C.21) and (C.22) that

Writing

Equation (C.24) yields

Taking the divergence of the previous equation, and making use of Equation (C.22), we find that , which implies that is a solid harmonic (of degree 2). Incidentally, would be zero were the planet in hydrostatic equilibrium.

It is helpful to define the radial component of the elastic displacement,

(C.28) |

as well as the stress acting (outward) across a constant surface,

where use has been made of Equation (C.21). Of course, the radial displacement at is equivalent to the displacement of the planet's surface:

The stress at any point on the surface must be entirely radial (because it would be impossible to balance a tangential surface stress), and such as to balance the weight of the column of displaced material directly above the point in question. In other words,

It follows from Equations (C.18), (C.25), and (C.29) that

where

Equations (C.16), (C.26), and (C.32) yield

(C.35) |

It remains to solve Equations (C.22) and (C.27), subject to the boundary conditions (C.30) and (C.33).

Let us try a solution to Equations (C.22) and (C.27) of the form

where and are spatial constants, and is a solid harmonic of degree 2 (Love 2011). It follows that

(C.37) |

where use has been made of Equation (C.5). Moreover,

(C.38) | ||||

and | (C.39) |

where use has been made of Equations (C.5)-(C.7). Thus, the boundary conditions (C.30) and (C.33) become

respectively. The previous equation implies that

(C.42) |

Hence, the boundary conditions (C.40) and (C.41) reduce to

respectively.

The expression for given in Equation (C.36) satisfies Equations (C.22) and (C.27) provided that

(C.45) | ||||||

and | (C.46) |

respectively, where use has been made of Equations (C.5)-(C.7). It follows that

(C.47) | ||||||

and | (C.48) |

Hence, the boundary conditions (C.43) and (C.44) yield

where

The dimensionless quantity is termed a

The radial component of the elastic (i.e., non-hydrostatic) stress acting (outward) across the surface takes the form

(C.51) |

where use has been made of Equation (C.33). Equations (C.49) and (C.50) imply that this stress is related to the radial strain at the surface of the planet according to

(C.52) |

As a specific example, suppose that

(C.53) | ||||||

(C.54) | ||||||

and | (C.55) |

where is a dimensionless measure of the strength of the tidal field, and is the tidally induced planetary ellipticity. It follows that

where

is the planet's effective rigidity.