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Abbreviations
- ℬ ⊂ ℝ3 :
-
reference configuration
- TXℬ:
-
vectors in ℝ3 based at the point X ∈ ℬ
- φ:ℬ → ℝ3, x = φ(X):
-
deformation
- u : ℬ → ℝ3 :
-
displacement for the linearized theory
- e = 1/2 [∇u + (∇u)T]:
-
strain
- C :
-
all deformations φ
- F = Dφ:
-
deformation gradient = derivative of φ
- FT :
-
transpose of F
- C = FTF:
-
Cauchy-Green tensor
- W:
-
Stored energy function
- \(P = \frac{{\partial W}}{{\partial F}}\) :
-
first Piola-Kirchhoff stress
- \(S = 2\frac{{\partial W}}{{\partial C}}\) :
-
second Piola-Kirchhoff stress
- \(A = \frac{{\partial P}}{{\partial F}}\) :
-
elasticity tensor
- \(C = \frac{{\partial S}}{{\partial C}}\) :
-
(second) elasticity tensor
- c = 2C¦φ=I ℬ :
-
classical elasticity tensor
- I or Iℬ or 1:
-
identity map on ℝ3 or ℬ
- l = (B, τ):
-
a (dead) load
- ℒ:
-
all loads with total force zero
- L(TXℬ, ℝ3):
-
all linear maps of TXℬ to ℝ3
- L(TXℬ, ℝ)*:
-
linear maps of L(TXℬ, ℝ) to ℝ
- sym (TXℬ, TXℬ):
-
symmetric linear maps of TXℬ to TXℬ
- SO(3):
-
Q∈ L(ℝ 3,ℝ 3)¦ Q T Q = I, det Q = 1
- ℝℙ2 :
-
real projective 2-space; lines through (0, 0, 0) in ℝ3
- M3 :
-
L(ℝ3, ℝ3)
- sym:
-
symmetric elements of M3
- skew = so(3):
-
skew symmetric elements of M3
- \(\hat \upsilon \) :
-
infinitesimal rotation about the axis v
- ℒe :
-
equilibrated loads
- k: ℒ → M3 :
-
astatic load map
- A = k(l):
-
astatic load for a load l
- j = (k ¦(ker k:)⊥)-1 :
-
non-singular part of k
- Skew = j (skew):
-
skew viewed in load space
- Sym = j (sym):
-
sym viewed in load space
- Φ:C→ℒ:
-
Φ(φ) = (-DIV P,P · N)
- U=T I C :
-
the space of linearized displacements
- U sym :
-
orthogonal complement to Skew inU
- L:U sym→ℒ:
-
linearized operator: L = DΦ(I)
- le :
-
the equilibrated part of l according to the decomposition ℒ = ℒe ⊕ Skew
- ul (U 0Q = uQl 0):
-
linearized solution : Lul = le
- 〈, 〉:
-
L2 pairing
- B(l1, l2) = 〈l1, ul 2〉:
-
〈c(∇ul 1), ∇ul 2〉 Betti form
- SA :
-
Q's in SO(3) that equilibrate A
- ϱ:
-
tubular neighborhood for SO(3) inC
- V(φ) = ∫W(F)dV — λ〈l,φ〉:
-
potential function for the static problem
- Vϱ = V ∘ ϱ:
-
potential function in new coordinates
- f(Q) = Vϱ(Q, φQ):
-
reduced potential function on SO(3)
- \(\mathop f\limits^ \sim \left( Q \right) = -< Q^T ,l > - \frac{\lambda }{2}< c\left( {\nabla u_Q^0 } \right)\nabla u_Q^0 > + O\left( {\lambda ^2 } \right) + O\left( {\lambda \left| {l - l_o } \right|} \right)\) :
-
second reduced potential on\(S_{A_o } \)
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Communicated by S.Antman
J. E.Marsden's research was supported in part by the U.S. National Science Foundation under Grant MCS-81-07086, by the Miller Institute, and by a contract from the Department of Energy, DE-AT03-82ER12097. Y. H.Wan's research was partially supported by the U.S. National Science Foundation under Grant MCS-81-02463 and the Department of Energy, Contract DE-AT03-82ER12097. D. R. J.Chillingworth's research was partially supported by the U. K. Science Research Council through the University of Warwick, 1980.
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Chillingworth, D.R.J., Marsden, J.E. & Wan, Y.H. Symmetry and bifurcation in three-dimensional elasticity. Part II. Arch. Rational Mech. Anal. 83, 363–395 (1983). https://doi.org/10.1007/BF00963840
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DOI: https://doi.org/10.1007/BF00963840