where N is the membrane force and [delta]E is the first variation of the membrane strain tensor
which is identical with the Green-Lagrange strain tensor
E of the midsurface and [[OMEGA].sub.0] is the volume of the initial configuration (material description or total Lagrangian approach).
In a three-dimensional stress state, let the spherical stress tensor in rocks be [[sigma].sub.m], the deviatoric stress tensor be [S.sub.ij], the Kronecker symbol be [[delta].sub.ij], the spherical strain tensor
be [[epsilon].sub.m], the deviatoric strain tensor
be [e.sub.ij], the shear modulus of rocks be G, and the bulk modulus be K.
The elastic energy ([DELTA]E) can be expanded in a Taylor series in terms of the strain tensor
The Lagrangian formulation from continuum mechanics is exploited using the Green-Lagrange strain tensor
E along with the first Piola-Kirchhoff stress tensor [T.sub.1] and second Piola-Kirchhoff stress tensor [T.sub.2].
where, T is second Piola-Kirchoff stress tensor and e is the Green-Lagrange strain tensor
Furthermore, in order to describe the anisotropic fracture accurately, a non-conjugated anisotropic equivalent plastic strain function is included by operating on a linearly transformation to the strain tensor
where u = [u.sub.1][e.sub.1]+[u.sub.2][e.sub.2]+[w.sub.3][e.sub.3] is the displacement vector, S is the strain tensor
, [phi] is the electric potential, E is the electric field vector, and [??] is the three-dimensional gradient operator.
Even if the stress tensor and the strain tensor
in discrete domains show a strong duality, it is not complete since expression for both tensors does not belong to the same domain.
In this stress-charge form of piezoelectric constitutive equations, the elastic strain tensor
and the electric field vector are the independent variables.
There are three main steps in the corresponding formulation of NOSB-PD modeling: (1) describing structure deformation with the nonlocal deformation gradient tensor and then converting it to the strain tensor
or the strain rate tensor, (2) conducting strain analysis and stress analysis to obtain the stress tensor according to the utilized constitutive relation, and (3) converting the stress into force state.
where [k.sub.ij] is dielectric constant tensor, [E.sub.i] and [E.sub.j] are electric field vectors, [b.sub.ijkl] is the nonlocal electrical coupling coefficient tensor, [C.sub.ijkl] is elastic stiffness tensor, [[epsilon].sub.kl] is strain tensor
, [e.sub.ijk] is piezoelectric coefficient tensor, [[mu].sub.ijkl] is a fourth- order tensor of the flexoelectric coefficient, and [[epsilon].sub.kl,j] and [E.sub.k,l] are the gradients of strain and electric field.