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 [82], who applied it to rotating neutron stars in [83, 84] (see also
Beig [44] and references therein). Recently, Carter and collaborators have extended this theory to include
the effects of the magnetic field [73], 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.,
[249
]).
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,
Under deformation, matter element volume changes according toTo lowest order, the deformation energy is quadratic in the deformation tensor,
Since The elastic contribution to the stress tensor is
.
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