BULETINUL INSTITUTULUI POLITEHNIC DIN IAŞI

Bul. Inst. Polit. Iaşi, t. LVIII (LXII), f. 3, 2012 11
2.2. Basic Equations of the Eringen-Nowacki Model in the Case of
Steady Vibrations
We consider an isotropic, homogeneous, and centrosymmetric micropolar
3
elastic solid occupying a region B ⊂ with piecewise smooth boundary ∂ B.
The closed region B is the union of the sets B and ∂ B,
that is B = BU ∂B.
Let
n be the components of the unit outward normal vector n to ∂ B.
j
We refer our considerations to the micropolar elastic medium
B( λμαβγερJ
, , , , , , , )
described by the constants of micropolar elasticity theory λ,μ,α,β,γ,ε, the
density ρ, and the rotational inertia J. All the physical fields defining a
micropolar elastic state of the medium B are real valued functions of the
position vector x called amplitudes.
The fact that the function f is defined in the set B will be denoted by
n
f : B → . A function f belongs to the class C in the set B , and we write
n
this in the form f ∈ C ( B),
if f and all partial derivatives up through n order
are continuous in the set B , where n is a natural number. Generally, throughout
the present paper we assume that the considered functions are sufficiently
regular to make the applied procedures meaningful.
The basic equations of the Eringen-Nowacki model, in the case of
steady vibrations, can be divided into the following groups:
i) the equations of motion in B
where
σ μ C B)
1
ji
,
ji
∈ (
2
⎧
⎪
σ
ji,
j
+ ρω ui + Xi
= 0,
⎨
2
⎪⎩ εijkσ jk
+ μji,
j
+ Jω φi + Yi
= 0,
(2)
are the amplitudes of the physical components of
asymmetric stress force-stress tensor and asymmetric couple-stress tensor,
respectively, X , Y ∈C 0 ( B ) are the fields of the amplitudes of body loadings
i
i
2
and body moments, respectively, and ui, φi∈ C ( B)
are the amplitudes of the
components of displacement and rotation vectors, respectively;
ii) the geometric relations in B
⎧⎪
γ
ji
= ui,
j
−εkjiφk
,
⎨
⎪⎩ κji
= φi , j;
iii) the constitutive relations in B
(3)