Course Hero Logo

A kinematics energy formulation and equations of

Course Hero uses AI to attempt to automatically extract content from documents to surface to you and others so you can study better, e.g., in search results, to enrich docs, and more. This preview shows page 7 - 10 out of 34 pages.

A.Kinematics, energy formulation and equations of equilibriumWe model the ring as an extensible Euler-Bernoulli beam made of an homogeneous andisotropic material. Polar coordinates are used to track the position of the ring center-line,C.The initial configuration of the ring, prior to depressurization, is assumed to be circular. Theorigin,O, is located at the center of the cavity, and the initial and equilibrium configurationsof an arbitrary point ofCare represented byM0andM, respectively. Vectors are expressed7
in the physical base (er,eθ), derived from the polar coordinates (r, θ). The initial positionofCisOM0= (R,0), as shown in the inset of Fig.1(a).When the system is loaded bydepressurizing the cavity,Cdeforms into a new configuration given by the position vectorOM=R(1 +v(θ), u(θ)), wherevanduare the dimensionless radial and orthoradialdisplacements, respectively.The infinitesimal arclength ofCin the initial and deformed configurations are denotedbyds0=|dOM0|andds=|dOM|, respectively.Moreover, defining the tangent vectorT=dOM/ds, allows us to express the curvature ofCin the deformed configuration asκ/R=|dT/ds|. Here,κis dimensionless and can be written in terms ofvanduasκ= 1 + (-1 + 2u0+ 2v)v00+v2-v-u02+ (-u+v0)u00+12v02-u2+ h.o.t.,(2)where the prime notation represents derivation with respect toθand high order terms(h.o.t.) are neglected under the assumption of small displacements and moderate rotations.We now define the elongation of the ring ase=ds/ds0to express the stretching deformation,η= (e2-1)/2, in terms ofvanduasη=12hu02+v2+ (u-v0)2i+ (1 +v)u0+v,(3)so that the hoop stress in the film isσ0=EFη.Following Euler-Bernoulli beam theory [25], the total energy of deformationEof the ringis the sum of a stretching energyESand a bending energyEB,E=ES+EB=2πRZ0Eds0,(4)withES=2πRZ0EFH2η2ds0,(5a)EB=2πRZ0EFH324R2(κ-1)2ds0,(5b)andEis the energy of deformation per unit length of the initial configuration of the ring.Assuming that the reaction force of the substrate derives from a potential2πRR0Wds0, the8
equilibrium states of the ring are the solutions ofδES+δEB-2πRZ0δWds0= 0,(6)whereδAis the variation of quantityA, for an arbitrary displacement fieldR(δv, δu), whichis 2πperiodic. The computation of the variations in Eq. (6) leads to the Euler-Lagrangeequations for the equilibrium of the ring,E∂v-E∂v00+E∂v0000-∂δW∂δv= 0,(7a)E∂u-E∂u00+E∂u0000-∂δW∂δu= 0,(7b)along with static boundary conditions that are naturally satisfied due to the 2πperiodicitycondition on the displacementsvandu.All derivative terms in Eq. (7b) are explicitlyreported in Appendix A.B.Asymptotic expansion and reactive force of the substrateWe seek a solution of Eq. (7) as an expansion of the formv=v0+εAsin () +O(ε2),(8a)u=εBcos () +O(ε2),(8b)where (v0,0) corresponds to a radial pre-buckling deformation andε(Asin (), Bcos ())represents an instability of azimuthal wavenumberm. From the requirement of 2πperiodicfunctionsvandu,mhas to be an integer. Also, we considerm >1 sincem= 1 corresponds

Upload your study docs or become a

Course Hero member to access this document

Upload your study docs or become a

Course Hero member to access this document

End of preview. Want to read all 34 pages?

Upload your study docs or become a

Course Hero member to access this document

Term
Spring
Professor
N/A
Tags
Force, Boundary value problem

Newly uploaded documents

Show More

Newly uploaded documents

Show More

  • Left Quote Icon

    Student Picture

  • Left Quote Icon

    Student Picture

  • Left Quote Icon

    Student Picture