Media with microstructures 19
3. The interaction form and its virtual work
Let us denote by FS P, R
3
the collection of all R
3
-valued functions on S P, by AS
1
P, R
3
the collection of all R
3
-valued one-forms on P, i.e. of all maps γ : S
1
P → R
3
, and by
A
2
S
2
P, R
3
the collection of all R
3
-valued two-forms on P, i.e. of all maps ω : S
2
P → R
3
. We note that FS
P, R
3
, A
1
S
1
P, R
3
and A
2
S
2
P, R
3
are finite dimensional R-vector spaces due to the fact that P has finitely many vertices, edges and faces. In all these vector spaces we
can present natural bases. Indeed, given any z ∈ R
3
and a fixed vertex q ∈ S
P, we define h
z q
∈ FS P, R
3
as follows: h
z q
q
′
= z , if q
= q
′
0 , otherwise . On the other hand, for a fixed edge e
∈ S
1
P respectively a fixed face f ∈ S
2
P, γ
z e
∈ A
1
S
1
P, R
3
and ω
z f
∈ A
2
S
2
P, R
3
are given in the following way: γ
z e
e
′
= z , if e
= e
′
, 0 , otherwise ,
ω
z f
f
′
= z , if f
= f
′
, 0 , otherwise .
If now {z
1
, z
2
, z
3
} is a base in R
3
, then
{h
z
i
q
| q ∈ S P, i
= 1, 2, 3} ⊂ FS P, R
3
{γ
z
i
e
| e ∈ S
1
P, i = 1, 2, 3} ⊂ A
1
S
1
P, R
3
and {ω
z
i
q
| f ∈ S
2
P, i = 1, 2, 3} ⊂ A
2
S
2
P, R
3
are the natural bases mentioned above. Given now a scalar product
h·, ·i on R
3
, we define the scalar product G ,
G
1
and G
2
on FS
P, R
3
, A
1
S
1
P, R
3
and respectively A
2
S
2
P, R
3
by G
h
1
, h
2
: =
X
q ∈S
P
hh
1
q, h
2
q i , ∀ h
1
, h
2
∈ FS P, R
3
, G
1
γ
1
, γ
2
: =
X
e ∈S
1
P
hγ
1
e, γ
2
e i , ∀ γ
1
, γ
2
∈ A
1
S
1
P, R
3
, and
G
2
ω
1
, ω
2
: =
X
f ∈S
2
P
hω
1
f , ω
2
f i , ∀ ω
1
, ω
2
∈ A
2
S
2
P, R
3
. The differential dh of any h
∈ FS P, R
3
is a one-form on P given by dhe
= he
+
− he
−
, ∀ e ∈ S
1
P , where e
−
and e
+
are the initial and the final vertex of e. The exterior differential d : A
1
S
1
P, R
3
→ A
2
S
2
P, R
3
applied to any γ ∈ A
1
S
1
P, R
3
is given by dγ f :
= X
e ∈∂ f
γ e ,
∀ f ∈ S
2
P .
20 E. Binz - D. Socolescu
The exterior differential dω for any two-form ω on P vanishes. Associated with d and the above scalar products are the divergence operators
δ : A
2
S
2
P, R
3
→ A
1
S
1
P, R
3
and δ
: A
1
S
1
P, R
3
→ FS P, R
3
, respectively defined by the following equations
G
1
δω, α = G
2
ω, dα
, ∀ ω ∈ A
2
S
2
P, R
3
and ∀ α ∈ A
1
S
1
P, R
3
, and
G δα,
h = G
1
α, dh
, ∀ α ∈ A
1
S
1
P, R
3
and ∀ h ∈ FS
P, R
3
. d
◦ d = 0 implies δ ◦ δ = 0. Elements of the form dh in A
1
S
1
P, R
3
for any h ∈ FS
P, R
3
are called exact, while elements of the form δω in A
1
S
1
P, R
3
for any ω ∈ A
2
S
2
P, R
3
are called coexact.
The Laplacians 1 , 1
1
and 1
2
on FS P, R
3
, A
1
S
1
P, R
3
and A
2
S
2
P, R
3
are respec- tively defined by
1
i
: = δ ◦ d + d ◦ δ , i = 0, 1, 2 .
Due to dim P = 2 these Laplacians, selfadjoint with respect to G
i
, i
= 0, 1, 2 , simplify to 1
= δ ◦ d on functions, 1
1
= δ ◦ d + d ◦ δ on one-forms and 1
2
= d ◦ δ on two-forms. Hence there are the following G
, G
1
- and respectively G
2
-orthogonal splittings, the so called Hodge splittings [1]:
A S
P, R
3
= δ A
1
S
1
P, R
3
⊕ Harm S
P, R
3
, A
1
S
1
P, R
3
= d FS P, R
3
⊕ δ A
2
S
2
P, R
3
⊕ Harm
1
S
1
P, R
3
, A
2
S
2
P, R
3
= d A
1
S
1
P, R
3
⊕ Harm
2
S
2
P, R
3
. Here H ar m
i
S
i
P, R
3
: = K er d ∩ K er δ , i = 0, 1, 2. Reformulated, this says that β ∈
H ar m
i
S
i
P, R
3
if 1
i
β = 0, i = 0, 1, 2 ; we note that β ∈ Harm
S P, R
3
is a constant function.
Letting H
i
P, R
3
be the i -th cohomology group of P with coefficients in R
3
, we hence
have: H
i
P, R
3 ∼
= Harm
i
S
i
P, R
3
, i
= 1, 2 . Next we introduce the stress or interaction forms, which are constitutive ingredients of the
polyhedron P. To this end we consider the interaction forces, i.e. vectors in R
3
, which act up on
any vertex q, along any edge e and any face f of P. The collection of all these forces acting up on the vertices defines a configuration dependent
function α 8
∈ FS P, R
3
, where 8
∈ con f P, R
3
. Analogously the collection of all the interaction forces acting up along the edges or along the faces defines a one form α
1
8 ∈
A
1
S
1
P, R
3
or a two-form α
2
8 ∈ A
2
S
2
P, R
3
respectively. The virtual work A
i
8 caused
respectively by any distortion γ
i
∈ A
i
S
i
P, R
3
, i
= 0, 1, 2, is given by A
i
8γ
i
= G
i
α
i
8, γ
i
, i
= 0, 1, 2 .
Media with microstructures 21
However, it is important to point out that the total virtual work A8 caused by a deforma- tion of the polyhedron P is given only by A
1
8γ
1
+ A
2
8 ρ
2
, where ρ
2
is the harmonic part of γ
2
∈ A
2
S
2
P, R
3
. In order to justify it we give the virtual works A
i
8γ
i
, i
= 1, 2, in accordance with the Hodge splitting for α
i
8 and γ
i
, i
= 0, 1, 2, and with the definition of the divergence operators δ, the equivalent forms
G α
8, δγ
1
= G
1
dα 8, γ
1
, G
1
α
1
8, γ
1
= G
1
dβ + δω
2
+ ̹
1
, γ
1
= G β
, δγ
1
+ G
2
ω
2
, dγ
1
+ G
1
̹
1
, ρ
1
G
2
α 8, δγ
1
= G
2
dβ
1
+ ̹
2
, γ
2
= G
1
β
1
, δγ
2
+ G
2
̹
2
, ρ
2
, Here the two terms
G
1
̹
1
, γ
1
= G
1
̹
1
, dh
+ dh
2
+ ρ
1
= G
1
̹
1
, ρ
1
, and
G
2
̹
2
, γ
2
= G
2
̹
2
, dh
1
+ ρ
2
= G
2
̹
2
, ρ
2
depend only on the topology of the polyhedron P. Comparing now the different expressions for the virtual works we get
A
1
8 γ
1
+ G
2
̹
2
, ρ
2
= G α
8, δγ
1
+ G
2
α
2
8, dγ
1
+ + G
1
̹
1
, ρ
1
+ G
2
̹
2
, ρ
2
, α
8 = δα
1
8, α
1
8 = dα
8 + δα
2
8 + ρ
1
, α
2
8 = dα
1
8 + ρ
2
. Moreover
1 α
8 = α
8 , 1
2
α
2
8 + ̹
2
= α
2
8 . Accordingly, the total virtual work on P associated, as discussed above, with α
, α
1
and α
2
is given by A8γ
1
, γ
2
: = A
1
8γ
1
+ A
2
8ρ
2
= G
1
α
1
8, 1
1
γ
1
+ G
1
̹
1
, ρ
1
+ G
2
̹
2
, ρ
2
However, due to translational invariance α
i
8 = α
i
d8, i = 0, 1, 2 .
For this reason we let d8 vary in a smooth, compact and bounded manifold K ⊂ dcon f P, R
3
with non-empty interior. The virtual work on P has then the form A8γ
1
, γ
2
= Ad8γ
1
, γ
2
for any d8 ∈ K and any γ
i
∈ A
i
S
i
P, R
3
. Since dcon f
P, R
3
⊂ A
1
S
1
P, R
3
ac- cording to the Hodge splitting is not open, not all elements in A
1
S
1
P, R
3
are tangent to
22 E. Binz - D. Socolescu
dcon f P, R
3
. Therefore, A is not a one-form on K
⊂ dcon f P, R
3
, in general. To use the
formalism of differential forms, we need to extend the virtual work A to some compact bounded submanifold K
1
⊂ A
1
S
1
P, R
3
with K ⊂ K
1
- See [2] for details - The one-form Ad8 needs not to be exact, in general. We decompose accordingly A into
Ad8 = dI F + 9. This decomposition is the so called Neumann one, given by
di vA = 1F, Aξ νξ = Dξνξ
for all ξ in the boundary ∂K
1
of K
1
. D is the Fr´echet derivative on A
1
S
1
P, R
3
, while ν
is the outward directed unit normal field on ∂ K
1
. The differential opeators di v and 1 are the
divergence and respectively the Laplacian on A
1
S
1
P, R
3
.
4. Thermodynamical setting