BULETINUL INSTITUTULUI POLITEHNIC DIN IAŞI

Bul. Inst. Polit. Iaşi, t. LVIII (LXII), f. 3, 2012 13
where σ = σ
jin
je i
is the stress (traction) vector, μ = μ
jin
jei
is the couple-stress
vector, and ( pmf, , , g) = ( pi, mi, fi, gi) ei.
Thus, the main initial boundary-value problem of this theory is to
determine a regular solution
2 1
u, φ ∈C ( B) ∩C ( B)
(12)
satisfying the motion vector equation (9) and the boundary conditions in vector
form (11).
By using modern methods of mathematics, many authors Nowacki
(1966), Nowacki (1986), Dyszlewicz (2004), Ieşan (2004), Hetnarski (1987),
Ignaczack (1971), Teodorescu (1975, Crăciun (1977), Crăciun (1978), and
others, have been obtained various results on boundary value problems both of
the linear theory of micropolar elasticity and micropolar thermoelasticity in the
case of steady vibrations.
2.3. Generalized Galerkin Representation of a Regular Solution
Let us consider the set of functions
where
2,
4
are given in (6),
( )
⎧
⎪ u =
1 4−
∇∇ · Γ Φ − 2α
∇ × Ψ 3
,
⎨
⎪
⎩ ϕ= (
2 3−∇∇· Θ)
Ψ−2α∇×
1Φ,
(13)
and
2 2
⎧
1= ( λ + 2 μ) ∇ + ρω ,
⎪
2 2
⎪ 3= ( β + 2 γ) ∇ + Jω
−4α,
⎨
2
⎪Γ= ( λ+ μ −α) 4
−4 α,
⎪
2
⎩Θ= ( β + γ −ε) 2
−4 α ,
(14)
3 3
Φ: B→
, Ψ : B→
(15)
8 6
are functions of classes C ( B)and C ( B),
respectively.
To obtain the generalized Galerkin representation (11), the following
relations between the operators (6) and (14) have been used
2 2 2 2
1 4 2 3 2 4
4 α .
−∇ Γ = −∇ Θ = + ∇ = Ω . (16)
The functions Φ: B→
3
and Ψ:
B→
3
are called stress functions or Galerkin
vectors.
In solving problems of asymmetric elasticity in the case of steady
vibrations the stress functions are of considerable importance.