smart technologies for safety engineering

Smart Technologies in Vibroacoustics 281
where Ce ijklis the fourth-order tensor of linear elasticity. It can be seen that the linear kinematic
relations between the elastic strain tensor and the elastic displacements, εe 1
ij = 2 (ue i| j + ue j|i ),
have already been used in Equation (27). Let it also be remembered that thanks to the symmetrical
properties, only 21 of the total number of 81 elastic tensor components are independent
(since Ce ijkl = Ce ijlk , Ce ijkl = Ce jikl and Ce ijkl = Ce klij ). Moreover, in the case of an orthotropic
material there are nine nonzero independent material constants, while in the case of transversal
isotropy there one only five; finally, every isotropic elastic material is described completely
by only two material constants. In this latter case the constitutive equation (of linear isotropic
elasticity) can be expressed as follows:
σ e e
ij = μe ui| j + u e
j|i + λe u e k|k δij
where the well-known Lamé coefficients: the shear modulus, μe, and the dilatational constant,
λe, appear. They are related to the material’s Young modulus, Ee, and Poisson’s coefficient, νe,
in the following way:
μe =
8.4.2.1 Boundary Conditions
Ee
2(1 + νe) , λe
νe Ee
=
(1 + νe)(1 − 2νe)
Two kinds of boundary conditions will be discussed here, namely Neumann’s and Dirichlet’s,
although they may be combined into the third specific type, the so-called Robin (or generalized
Dirichlet) boundary condition. For the sake of brevity, the latter type will not be considered;
remember only that, in practice, the well-known technique of Lagrange multipliers is usually
involved when applying it. The Neumann (or natural) boundary conditions describe the case
when forces ˆt e i are applied on a boundary, i.e.
whereas the displacements û e i
σ e
ij ne j = ˆt e i on Ɣt e
(28)
(29)
(30)
are prescribed by the Dirichlet (or essential) boundary conditions
u e i = ûe i on Ɣu e
According to these conditions the boundary is divided into two (directionally disjoint) parts,
i.e. Ɣe = Ɣt e ∪ Ɣu e . There is an essential difference between the two kinds of conditions. The
displacement constraints form the kinematic requirements for the trial functions while the
imposed forces appear in the weak form; thus, the boundary integral, i.e. the last left-hand-side
term of Equation (26), equals
BI e =
Here, the property δue i = 0onƔu e has been used.
Ɣ e
σ e
ij nej δue i =
Ɣt ˆt
e
e i δue i
(31)
(32)