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
Article

Keywords

Neural Network Complex System Nonlinear Dynamics Electromagnetism 
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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)

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

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

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=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

  1. S. A.Adeleke [1980]. Stability of some states of plane deformation,Arch. Rational Mech. Anal. 72, 243–263.Google Scholar
  2. J. M.Arms, J. E.Marsden & V.Moncrief [1981]. Symmetry and bifurcation of momentum maps,Comm. Math. Phys. 78, 455–478.Google Scholar
  3. J. M.Ball & D.Schaeffer [1982]. Bifurcation and stability of homogeneous equilibrium configurations of an elastic body under dead load tractions (preprint).Google Scholar
  4. G.Capriz & P.Podio Guidugli [1974]. On Signorini's perturbation method in nonlinear elasticity,Arch. Rational Mech. Anal. 57, 1–30.Google Scholar
  5. D. R. J.Chillingworth, J. E.Marsden & Y. H.Wan [1982]. Symmetry and Bifurcation in three dimensional elasticity, Part I,Arch. Rational Mech. Anal. 80, 295–331.Google Scholar
  6. E.Dancer [1980]. On the existence of bifurcating solutions in the presence of symmetries.Proc. Roy. Soc. Edinb. 85 A, 321–336.Google Scholar
  7. G.Fichera [1972].Existence theorems in elasticity, Handbuch der Physik, Bd. VIa/2, 347–389, C.Truesdell, ed., Berlin Heidelberg New York: Springer.Google Scholar
  8. N.Golubitsky & D.Schaeffer [1979]. Imperfect bifurcation in the presence of symmetry,Commun. Math. Phys. 67, 205–232.Google Scholar
  9. G.Grioli [1962].Mathematical Theory of Elastic Equilibrium, Ergebnisse der Angew. Math. #67, Berlin Heidelberg New York: Springer.Google Scholar
  10. D.Gromoll & W.Meyer [1969]. On differentiable functions with isolated critical points,Topology,8, 361–369.Google Scholar
  11. J. K.Hale & P. Z.Taboas [1980]. Bifurcation near degenerate families,Journal of Applicable Anal. 11, 21–37.Google Scholar
  12. J. E.Marsden & Y. H.Wan [1983]. Linearization stability and Signorini Series for the traction problem in elastostatics.Proc. Roy. Soc. Edinburgh (to appear).Google Scholar
  13. M.Reeken [1973]. Stability of critical points under small perturbations,Manuscripta Math. 69–72.Google Scholar
  14. S.Signorini [1930]. Sulle deformazioni termoelastiche finite,Proc. 3rd Int. Cong. Appl. Mech. 2, 80–89.Google Scholar
  15. F.Stoppelli [1955]. Sulla sviluppabilitá in serie di potenze di un parametro delle soluzioni delle equazioni dell'elastostatica isoterma,Ricerche Mat. 4, 58–73.Google Scholar
  16. F.Stoppelli [1958]. Sull'esistenza di soluzioni delle equazioni dell'elastostatica isoterma nel caso di sollecitazioni dotate di assi di equilibrio,Richerce Mat. 6 (1957) 241–287,7 (1958) 71–101, 138–152.Google Scholar
  17. C.Truesdell & W.Noll [1965].The Nonlinear Field Theories of Mechanics, Handbuch der Physik Bd. III/3, S.Flügge, ed., Berlin Heidelberg New York: Springer.Google Scholar
  18. Y. H.Wan [1983]. The traction problem for incompressible materials (preprint).Google Scholar
  19. A.Weinstein [1978]. Bifurcations and Hamilton's principle,Math. Zeit. 159, 235–248.Google Scholar

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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