The interaction form and its virtual work

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 qi , ∀ h 1 , h 2 ∈ FS P, R 3 , G 1 γ 1 , γ 2 := X e∈S 1 P hγ 1 e, γ 2 ei , ∀ 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 ⊕ H ar m S P, R 3 , A 1 S 1 P, R 3 = d FS P, R 3 ⊕ δ A 2 S 2 P, R 3 ⊕ H ar m 1 S 1 P, R 3 , A 2 S 2 P, R 3 = d A 1 S 1 P, R 3 ⊕ H ar m 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 ∈ H ar m 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 ∼ = H ar m 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 = A d8γ 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 = Dξ νξ 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