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7 Elastic Properties

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  [8384] (see also Beig [44] and references therein). Recently, Carter and collaborators have extended this theory to include the effects of the magnetic field [73Jump To The Next Citation Point], as well as the presence of the neutron superfluid, which permeates the inner crust [72Jump To The Next Citation Point85Jump To The Next Citation Point]. 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., [249Jump To The Next Citation Point]).

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 {rrr}, 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 T = 0 approximation. Deformation of a crust element with respect to the equilibrium configuration implies a displacement of nuclei into their new positions rrr′ = rrr + uuu, where uuu = uuu (rrr) is the displacement vector. In the continuum limit, valid for macroscopic phenomena, both rrr and uuu are treated as continuous fields. Nonzero uuu is associated with elastic strain (i.e., forces which tend to return the matter element to the equilibrium state of minimum energy density ɛ0), and with the deformation energy density ɛdef = ɛ(uuu) − ɛ0 4.

A uniform translation does not contribute to ɛdef, and the true deformation is described by the (symmetric) strain tensor

( ) 1 ∂ui ∂uk uik = uki = 2- ∂x--+ ∂x-- , (107 ) k i
where i,j = 1,2,3, and x1 = x, x2 = y, x3 = z. This formula for uik is valid if the displacement vector uuu is small, so that the terms quadratic in the components of u can be neglected compared to the linear ones [249Jump To The Next Citation Point].

Any deformation can be decomposed into compression and shear parts,

1 1 uik = ucikomp + ushikear,where uciokmp = --(∇∇∇ ⋅uuu)δik , usihkear = uik −-(∇∇∇ ⋅uuu)δik . (108 ) 3 3
Under deformation, matter element volume changes according to dV ′ = [1 + (∇∇∇ ⋅uuu)]dV. A pure compression (no shear) of the matter element is described by uik = αδik. For volume preserving deformations ∇ u (∇∇ ⋅uu) = 0.

To lowest order, the deformation energy is quadratic in the deformation tensor,

∑ 1 ɛdef = -λiklmuikulm . (109 ) iklm 2
Since ɛdef is a scalar, λiklm is a fourth rank tensor. The total number of λiklm components is eighty one; general symmetry relations reduce the maximum number of linearly-independent components (elastic moduli) to twenty one. The number of independent elastic moduli decreases with increasing symmetry of the elastic medium. It is three in the case of a cubic crystal, and two for an isotropic solid (see, e.g., [249Jump To The Next Citation Point]).

The elastic contribution to the stress tensor Πelast≡ σik ik is σik = ∂ɛdef∕∂uik.

 7.1 Isotropic solid (polycrystal)
 7.2 Nuclear pasta

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