Consider a electroelastic body in the reference and current configuration, whose coordinate system are coincident. The electric displacement DI is defined in the sense that the electric charge does not change under the deformation in a closed surface. As a result, the relation between material electric displacement DI and spatial electric displacement di is given as
D = det(F) inv(F) d, where F is the deformation gradient. As for the relation of the electric field, simple chain rule can be applied in order to obtain
E = transpose(F) e, with E and e are the material and spatial electric field, respectively.
Now, by substituting the above two equations into the constitutive equation of spatial electric displacement d = \epsilon_0 e + p, where \epsilon_0 is the permittivity, one can obtain the relation between material and spatial polarization (P and p, respectively), such as
P = det(F) inv(F) p.
However, in other literature, this relation is sometime given as
P = J p or in index notation P_K = J \alpha_{kK} p_k, with \alpha_{kK} = i_k . i_K is the shifter.
My question are: 1. Is there a unique definition of the material polarization? 2. How to justify the second relation of the material polarization, i.e P = J p.
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