TY - JOUR

T1 - The radial volume derivative and the critical boundary displacement for cavitation

AU - Negrón-Marrero, P V

AU - Sivaloganathan, Jeyabal

PY - 2011

Y1 - 2011

N2 - We study the displacement boundary value problem of minimizing the total energy E(u) stored in a nonlinearly elastic body occupying a spherical domain B in its reference configuration over (possibly discontinuous) radial deformations u of the body subject to affine boundary data u(x) = λx for x ∈ ∂B. For a given value of λ, we define what we call the radial volume derivative at λ, denoted by G(λ), which measures the stability or instability of the underlying homogeneous deformation u h λ(x) ≡ λx to the formation of holes. We give conditions under which the critical boundary displacement λ crit for radial cavitation is the unique solution of G(λ) = 0. Moreover, we prove that the radial volume derivative G(λ) can be approximated by the corresponding volume derivative for a punctured ball B ∈, containing a pre-existing cavity of radius ∈ > 0 in its reference state, in the limit ∈ → 0 and we use this to propose a numerical scheme to determine λ crit. We illustrate these general results with analytical and numerical examples.

AB - We study the displacement boundary value problem of minimizing the total energy E(u) stored in a nonlinearly elastic body occupying a spherical domain B in its reference configuration over (possibly discontinuous) radial deformations u of the body subject to affine boundary data u(x) = λx for x ∈ ∂B. For a given value of λ, we define what we call the radial volume derivative at λ, denoted by G(λ), which measures the stability or instability of the underlying homogeneous deformation u h λ(x) ≡ λx to the formation of holes. We give conditions under which the critical boundary displacement λ crit for radial cavitation is the unique solution of G(λ) = 0. Moreover, we prove that the radial volume derivative G(λ) can be approximated by the corresponding volume derivative for a punctured ball B ∈, containing a pre-existing cavity of radius ∈ > 0 in its reference state, in the limit ∈ → 0 and we use this to propose a numerical scheme to determine λ crit. We illustrate these general results with analytical and numerical examples.

UR - http://www.scopus.com/inward/record.url?scp=84855502007&partnerID=8YFLogxK

UR - http://dx.doi.org/10.1137/110835943

U2 - 10.1137/110835943

DO - 10.1137/110835943

M3 - Article

VL - 71

SP - 2185

EP - 2204

JO - SIAM Journal on Applied Mathematics

JF - SIAM Journal on Applied Mathematics

SN - 0036-1399

IS - 6

ER -