Archive for Rational Mechanics and Analysis

, Volume 83, Issue 4, pp 363–395

Symmetry and bifurcation in three-dimensional elasticity. Part II

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

Keywords

Neural Network Complex System Nonlinear Dynamics Electromagnetism
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Glossary of Notation

ℬ ⊂ ℝ3

reference configuration

TX

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

classical elasticity tensor

I or I or 1

identity map on ℝ3 or ℬ

l = (B, τ)

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

SO(3)

Q∈ L(ℝ 3 , 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

e

k: ℒ → M3

A = k(l)

j = (k ¦(ker k:)⊥)-1

non-singular part of k

Skew = j (skew)

Sym = j (sym)

Φ:C→ℒ

Φ(φ) = (-DIV P,P · N)

U=TIC

the space of linearized displacements

Usym

orthogonal complement to Skew inU

L:Usym→ℒ

linearized operator: L = DΦ(I)

le

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

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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© 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