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