In the couple stress theory we have only dependent macrorotation and we can calculate curvature tensor. But in the micropolar theory we have independent microrotation and maybe macrorotation, but I cannot find a formula for curvature tensor in micropolar theory.
1. Generally & Abstractly, for a given Functional:
F[y] = ∫ ℒ (x , t , y , y' , y˙) d⁴x,
∀(x , t) ∈ Ω x I ⊂ ℝ⁴ ≅ ℝ³ x ℝ, ∀y ∈ UKATr,
we need proper arguments in the L-function:
the function & the derivative(s) of it in order to:
• use of xt-IbP,
• use of FLVC,
₁⇒ deduce E-L in Ω x I:
δ¹F(y ; δy) = ε δ₁F(y ; η) = ε [dF(y + εη) / dε|ε = 0] = ε F'(0) =
= δF / δy = D₂ℒ - (D₃ℒ )' - (D₄ℒ )˙ = 0, ∀δy ∈ VKATe,
₂⇒ deduce BC in the 3-xt-cylinder ∂Ω x Cls(I),
₃⇒ deduce TC in the impulsive 3-xt-body Cls(Ω) x ∂I,
for a desired {conditional, unconditional, convolved, mixed, hybrid, generalized, etc.} variational formulation.
2. Specifically & Analytically, in the MP/C Elasticity there are 2 (or 6 total) MP dof:
• u = (ui) ∈ ℝ³ [1 (or 3 total) translational dof]
• φ = (φi) ∈ ℝ³ [1 (or 3 total) micro(polar)-rotational dof]
Therefore there are 2 (or 18 total) micropolar strains,
which are the gradients of the corresponding kinematic variables:
♦ 1st MP/C strain [mean] asymmetric 2-tensor:
εᵀ = (εji), with ji-co-component εji, where:
• εji = ui;j + φji = eij + (ωij + φji) = eij - εijk (ωk - φk)
• Sym(εji) = Sym(ui;j) = eij
• Skw(εji) = Skw(ui;j + φji) = ωij + φji = - εijk (ωk - φk)
• ui;j = eij + ωᵢⱼ
• Sym(ui;j) ≡ eij = +eji = ½ (ui;j + uj;i)
• Skw(ui;j) ≡ ωij = -ωji = ½ (ui;j - uj;i) = -εijk ωk
• φji = -φij = εijk φk
♦ 2nd MP/C strain [wryness or curvature-bend-torsion] asymmetric 2-tensor:
κᵀ = (κji) = φ⊗∇, with ji-co-component κji = φi;j:
3.1 Suggestion-Solution:
Dear Vahid Saboori Khorasani, the 2 key-elements for the desired variational formulation are εji & κji.
3.2 Special Case:
Iff macro-rotation ωi & micro(polar)-rotation φⁱ are mutually-dependent such that ωi = φi ⇒ εji = eij, ωij = φji,
then results as a special case the Couple-Stress [Constrained MP/C] Medium.
3.3 Note:
Many authors use the plain/direct MP/C as the main kinematic derivatives in the variational formulation:
• ε = (εij),
• κ = (κij) = ∇⊗φ.
0.1 Set-Algebraic Operators
- Bnd(•) ≡ ∂(•): Boundary of (•)
- Cls(•): Closure of (•)
- Det(•) ≡ |•|: Determinant of (•)
- Int(•): Interior of (•)
- Inv(•) ≡ (•)⁻¹: Inverse of (•)
- Sym(•): Symmetric part of 2-tensor
- Skw(•): Skew-symmetric part of 2-tensor
- (•)ᵀ: Transpose of (•)
0.2 Differential Operators
- ∇ ⊗ (•): Left Gradient
- (•) ⊗ ∇: Right Gradient
- ∇i(•) ≡ (•);i: partial covariant derivative of (•) with respect to xi
- ∂i(•) ≡ (•),i: partial (common) derivative of (•) with respect to xi
- Di(•): partial (common) derivative of (•) with respect to
- i-th argument (1 , ... , i , ... , n), ∀n ∈ ℕ
0.3 Special Symbols
- eijk: co-CEP symbol
- eijk: contra-CEP symbol
- ε = (εijk): direct or (0,3)- CEP tensor
- with co-ijk-component εijk = √|g| eijk
- ε = (εijk): inverse or (3,0- CEP tensor
- with contra-ijk-component εijk = √|g|-1 eijk
- g = (gij): metric direct or (0,2)- tensor
- with co-ij-component gᵢⱼ = gi • gj
- g-1 = (gij): metric inverse or (2,0)- tensor
- with contra-ij-component gij = gi • gj
0.4 Abbreviations
- BC: Boundary Condition(s)
- CEP: Civita-Eddington Permutation
- E-L: Euler-Lagrange Equation(s)
- KATeFS: Kinematically-Admissible Test Functional Space
- KATrFS: Kinematically-Admissible Trial Functional Space
- L-: Lagrangian
- MP/C: MicroPolar/Cosserat
- TC: Time/Temporal Condition(s)
- co-: covariant
- contra: contravariant
- dof: degree of freedom
- xt-IbP: (spatial, time/temporal)- Integration by Parts
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