A solid crust can sustain an elastic strain up to a critical level, the breaking strain. Neutron stars are relativistic objects, and therefore a relativistic theory of elastic media in a curved spacetime should be used to describe elastic effects in neutron star structures and dynamics. Such a theory of elasticity has been developed by Carter & Quintana , who applied it to rotating neutron stars in [83, 84] (see also Beig  and references therein). Recently, Carter and collaborators have extended this theory to include the effects of the magnetic field , as well as the presence of the neutron superfluid, which permeates the inner crust [72, 85]. For the time being, for the sake of simplicity, we ignore magnetic fields and free neutrons. However, in Section 7.2 the effect of free neutrons on the elastic moduli of the pasta phases is included, within the compressible liquid drop model. Since relativistic effects are not very large in the crust, we shall restrict ourselves to the Newtonian approximation (see, e.g., ).
The thermodynamic equilibrium of an element of neutron-star crust corresponds to equilibrium positions of nuclei, which will be denoted by a set of vectors , which are associated with the lattice sites. Neutron star evolution, driven by spin-down, accretion of matter or some external forces, like tidal forces produced by a close massive body, or internal electromagnetic strains associated with strong magnetic fields, may lead to deformation of this crust element as compared to the equilibrium state.
For simplicity, we will neglect thermal contributions to thermodynamic quantities and restrict ourselves to the approximation. Deformation of a crust element with respect to the equilibrium configuration implies a displacement of nuclei into their new positions , where is the displacement vector. In the continuum limit, valid for macroscopic phenomena, both and are treated as continuous fields. Nonzero is associated with elastic strain (i.e., forces which tend to return the matter element to the equilibrium state of minimum energy density ), and with the deformation energy density 4.
A uniform translation does not contribute to , and the true deformation is described by the (symmetric) strain tensor.
Any deformation can be decomposed into compression and shear parts,
To lowest order, the deformation energy is quadratic in the deformation tensor,).
The elastic contribution to the stress tensor is .
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