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

### 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(ℝ3,3)¦ QTQ = 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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### References

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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