Archive for Rational Mechanics and Analysis

, Volume 83, Issue 4, pp 363–395 | Cite as

Symmetry and bifurcation in three-dimensional elasticity. Part II

  • D. R. J. Chillingworth
  • J. E. Marsden
  • Y. H. Wan


Neural Network Complex System Nonlinear Dynamics Electromagnetism 
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Glossary of Notation

ℬ ⊂ ℝ3

reference configuration


vectors in ℝ3 based at the point X ∈ ℬ

φ:ℬ → ℝ3, x = φ(X)


u : ℬ → ℝ3

displacement for the linearized theory

e = 1/2 [∇u + (∇u)T]



all deformations φ

F = Dφ

deformation gradient = derivative of φ


transpose of F


Cauchy-Green tensor


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

classical elasticity tensor

I or I or 1

identity map on ℝ3 or ℬ

l = (B, τ)

a (dead) load

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 TX


Q∈ L(ℝ 3 , 3 )¦ Q T Q = I, det Q = 1


real projective 2-space; lines through (0, 0, 0) in ℝ3


L(ℝ3, ℝ3)


symmetric elements of M3

skew = so(3)

skew symmetric elements of M3

\(\hat \upsilon \)

infinitesimal rotation about the axis v


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(φ) = (-DIV P,P · N)


the space of linearized displacements


orthogonal complement to Skew inU


linearized operator: L = DΦ(I)


the equilibrated part of l according to the decomposition ℒ = ℒe ⊕ Skew

ul (UQ0 = uQl0)

linearized solution : Lul = le

〈, 〉

L2 pairing

B(l1, l2) = 〈l1, ul2

〈c(∇ul1), ∇ul2〉 Betti form


Q's in SO(3) that equilibrate A


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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Copyright information

© Springer-Verlag GmbH & Co 1983

Authors and Affiliations

  • D. R. J. Chillingworth
    • 1
    • 2
    • 3
  • J. E. Marsden
    • 1
    • 2
    • 3
  • Y. H. Wan
    • 1
    • 2
    • 3
  1. 1.University of SouthamptonUK
  2. 2.University of CaliforniaBerkeley
  3. 3.State University of New YorkBuffalo

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