BULETINUL INSTITUTULUI POLITEHNIC DIN IAŞI

12 Ion Crăciun
⎧σ ji
= ( μ+ α) γji + ( μ− α) γij + λγkkδji,
⎪
⎨
⎪⎩ μ = ( γ+ ε) κ + ( γ− ε)
γ κ + βκ δ .
ji ji ij ij kk ji
(4)
Substituting the constitutive relations (4) into equations of motion (2)
and using the geometrical relations (3) we obtain the equations of motion in
terms of the amplitudes of displacements u and rotations ϕ
i
i
⎧
⎪
⎨
⎪⎩
u + ( λ+ μ− α) u + 2αε φ + X = 0,
2 i j, ji ijk k,
j i
ϕ + ( β+ γ− ε) φ + 2αε u + Y = 0,
4 i j, ji ijk k , j i
(5)
where the differential operators
2 and 4
are given by
= ( μ + α) ∇ + ρω , = ( γ+ ε) ∇ + Jω − 4 α.
(6)
2 2 2 2
2 4
The principal aim of the theory is to determine a regular solution
u φ ∈C B ∩ C B) (7)
2 1
i, i
( ) (
of the equations of motion in the amplitudes of displacements and rotations (5)
satisfying the following boundary conditions:
⎧⎪
σ
jinj = pi , μ
jinj = mi ,on ∂Bσ
,
⎨
⎪⎩ ui = fi, φi = gi,on ∂Bu,
where p , m : ∂B → , f , g : ∂B
→ are given functions defined on subsets
i i σ
i i u
of the set ∂ B with ∂Bσ
∩∂ Bu
=∅, ∂Bσ
∪∂ Bu
=∂ B.
The functions p i
are
called surface tractions, are the components of the surface couple-stress
vector,
form
f and
i
g i
m i
are the surface displacements and rotations, respectively.
The fundamental differential equations (5) can be written in the vector
where the new vectors
⎧⎪ 2u+ ( λ+ μ−α) ∇∇ · u+ 2α∇× φ+ X=
0,
⎨
⎪⎩ 2 α∇× u +
4φ+ ( β+ γ− ε) ∇∇·
φ+ Y=
0,
u 2
and
2φ are equal to
2 2 i i 4 4 i i
(8)
(9)
u= ( u ) e ; ϕ = ( ϕ )e .
(10)
The vector form of the boundary conditions (8) is
⎧σ = p, μ = m,on∂Bσ
,
⎨
⎩u = f, φ = g,on ∂ Bu
,
(11)