34
2 Size-Dependent Theories of Beams, Plates and Shells
various governing relations. Owing to the linear theory of surface elasticity [94],
a volume body still obeys boundary conditions of the classical theory of elasticity,
whereas the surface properties include equations with surface constant parameters
and surface residual stresses. Namely, the volume body satisfies the equations of
equilibrium and resolving equations of the classical linear theory of elasticity, and
they can be presented in the following forms:
σ i j, j + b i = 0,
(2.45)
σ i j = λε kk δ i j + 2με i j ,
(2.46)
where σ i j stand for the stress Cauchy components; b i are force components; λ and
μ are the standard Lame constants; and ε i j stand for components of infinitely small
deformations governed by the following relations:
ε i j =
1
2
(u i, j + u j,i ),
(2.47)
where u i are components of displacement.
On the body surface, the resolving equations have the following form ([37, 97,
98]):
σ βα n β = −τ βα,β ,
σ i j n i n j = τ βα κ βα ,
(2.48)
where κ αβ —components of surface curvature tensor; n i —components of external
unit normal, n β component of vector n = n i e i on the surface; τ αβ —components of
a tensor of surface stresses [94], and
τ =
τ 0 + (λ 0 + τ 0 )tr(ε
s
)
I
s
+ 2μ 0 ε
s
− τ 0 (∇su)
T
.
(2.49)
Here μ 0 and λ 0 are surface elastic constants; τ 0 stands for residual surface stress
(i.e. surface stress for zero deformation); I
s is projection tensor with I
s
= I − n ⊗ n;
∇s stands for gradient surface operator defined as ∇s(·) = (I − nn) ∇(·), where the
superscript T denotes transposition matrix; and ε
s is the tensor of surface deformation
of the following form:
ε
s
=
1
2
∇su + (∇su)
T
=
1
2
(I − n ⊗ n)
∇u + (∇u)
T
,
(2.50)
where I is the unit tensor of the second order.
Observe that there are three surface constants μ 0 , λ 0 and τ 0 (see [37, 99]).
2 Size-Dependent Theories of Beams, Plates and Shells
various governing relations. Owing to the linear theory of surface elasticity [94],
a volume body still obeys boundary conditions of the classical theory of elasticity,
whereas the surface properties include equations with surface constant parameters
and surface residual stresses. Namely, the volume body satisfies the equations of
equilibrium and resolving equations of the classical linear theory of elasticity, and
they can be presented in the following forms:
σ i j, j + b i = 0,
(2.45)
σ i j = λε kk δ i j + 2με i j ,
(2.46)
where σ i j stand for the stress Cauchy components; b i are force components; λ and
μ are the standard Lame constants; and ε i j stand for components of infinitely small
deformations governed by the following relations:
ε i j =
1
2
(u i, j + u j,i ),
(2.47)
where u i are components of displacement.
On the body surface, the resolving equations have the following form ([37, 97,
98]):
σ βα n β = −τ βα,β ,
σ i j n i n j = τ βα κ βα ,
(2.48)
where κ αβ —components of surface curvature tensor; n i —components of external
unit normal, n β component of vector n = n i e i on the surface; τ αβ —components of
a tensor of surface stresses [94], and
τ =
τ 0 + (λ 0 + τ 0 )tr(ε
s
)
I
s
+ 2μ 0 ε
s
− τ 0 (∇su)
T
.
(2.49)
Here μ 0 and λ 0 are surface elastic constants; τ 0 stands for residual surface stress
(i.e. surface stress for zero deformation); I
s is projection tensor with I
s
= I − n ⊗ n;
∇s stands for gradient surface operator defined as ∇s(·) = (I − nn) ∇(·), where the
superscript T denotes transposition matrix; and ε
s is the tensor of surface deformation
of the following form:
ε
s
=
1
2
∇su + (∇su)
T
=
1
2
(I − n ⊗ n)
∇u + (∇u)
T
,
(2.50)
where I is the unit tensor of the second order.
Observe that there are three surface constants μ 0 , λ 0 and τ 0 (see [37, 99]).