Constrained asymptotic crack-tip fields of a Mooney–Rivlin sheet in plane stress: a symplectic analysis

Abstract

Analytical crack-tip fields for soft sheets are known mainly for generalized neo-Hookean materials. We formulate the crack-tip problem in a radial symplectic (Hamiltonian) framework, using ξ = lnr as the evolution coordinate. The deformation on a material circle and its weighted radial traction form the canonical pair, and the constraint reaction enters through the traction. Within this framework, dominant balance for an incompressible plane-stress Mooney Rivlin sheet shows that the I2 contribution becomes a kinematic constraint near the tip. The constraint drives the local state toward uniaxial tension. The leading stretch magnitudes are independent of angle, and the opening is y2 ∼ Pr1/2sin(θ/2). The compensated areal Jacobian Jr1/4 approaches an angle independent plateau. The same field gives G = (π/2)c1P2, where c1 is the first Mooney–Rivlin coefficient. In a pure-shear strip, the grip stretch there fore fixes P, and finite-element calculations support both this parameter-free amplitude and the compensated-Jacobian prediction. The constraint also permits a regular O(r) crack-parallel motion set by specimen-scale matching. After this motion is separated, the in-plane field contains an r5/4 residual whose angular equation admits a specimen-selected homogeneous member. Core and angular refinement of the strip fields recovers both the exact-axis 5/4 exponent and its parameter-free amplitude to within 1.3%. The closed normal-opening sector gives the half-oddradial spectrum and a conserved symplectic pairing. The calculation shows how a constitutive term that is subleading in the near-tip energy can nevertheless control the local kinematics, a mechanism that can be sought in other stiffening soft networks.