(Ms the total number of Lagrangian nodes on the RBC), [([F.sub.S]).sub.k] and [([F.sub.b]).sub.k] are elastic Lagrangian forces associated with the node k on the RBC and [[delta].sub.j,k] is the

Kronecker delta function.

Here, [delta] is the generalized

Kronecker delta. With definitions in (15), we can obtain the explicit form of the N-dimensional Nambu-Poisson brackets in terms of fields:

Indeed, the orthogonality condition requires that ([[phi].sub.i],[[phi].sub.j]) = [[delta].sub.ij], where [[delta].sub.ij] is the

Kronecker delta. Since the [phi]i are normalized, the eigenpairs are as follows

In (2) and (3), E(*) is the expected value and [[delta].sub.k(k-i)] is

Kronecker delta function.

where [DELTA][[epsilon].sup.a.sub.ij] are increments of the actual strain components, [DELTA][[sigma].sup.a.sub.ij] are increments of the actual stress components and [[delta].sub.ij] is the

Kronecker delta, [DELTA][[epsilon].sup.pa.sub.eq] is the equivalent plastic strain increment, [DELTA][[sigma].sup.a.sub.eq] is the equivalent stress increment, where v is Poisson's ratio, E is elastic modulus, G is shear modulus, [E.sup.p.sub.T] is current value of generalized plastic modules and [S.sup.a.ij] are deviatoric stress compoenents.

where [delta] is the

Kronecker delta function with [delta] ([S.sub.k], [S.sub.m]) =1 if [S.sub.k] = [S.sub.m] and 0 otherwise and [J.sub.km] is a positive constant that represents the grain boundary (km) energy.

where [u.sub.i](u, v, w) are nondimensional velocity components in the Cartesian-coordinate system [x.sub.1](x, y, z), p is a non-dimensional pressure, and [[delta].sub.ij] is the

Kronecker delta (note that, for simplicity, the symbols for both dependent and independent variables have not been altered in switching from the dimensional to the dimensionless formalism).

,n] be the permutation matrix representing [sigma], where [delta] denotes the

Kronecker delta. It is known that [Z.sup.AM.sub.[sigma]] (u) has the determinant expression

where [[sigma].sub.ij] is the tensor of mechanical stresses, [[epsilon].sub.ij] is the tensor of mechanical strains, u is the vector of mechanical displacements, f is the vector of the external volumetric forces, E is the temperature-dependent modulus of elasticity of the material, v is the temperature-dependent Poisson ratio of the material, [[delta].sub.ij] is the

Kronecker delta, and e = [[epsilon].sub.kk] (k = i, j).

which is also sometimes expressed as the

Kronecker delta. The other part of the boundary to the matrix yields a similar form, given by

with the

Kronecker delta [[delta].sub.ij] and the viscosity [mu] [kg/ms].

The most common analytical approach used in forced vibration analysis is based on the assumed modes method in conjunction with the classical orthogonality conditions which may be expressed as [[integral].sup.L.sub.0] [Y.sub.i] [Y.sub.j] dx = [[delta].sub.ij], where L is the length of the beam, [Y.sub.i] and [Y.sub.j] are the fth and jth normal mode shapes, respectively, and [[delta].sub.ij] is the

Kronecker delta function.