On small scale nonlinearity and nested crack tip fields in a neo-Hookean material

Abstract This paper investigates the stress singularity at the crack tip in a two-dimensional Neo-Hookean hyperelastic material, with a focus on how far-field strain influences the crack tip field distribution. The study demonstrates that, under small far-field strains, the crack tip field can be generally divided into three regions: Region I—the asymptotic neo-Hookean crack tip field as $$r\rightarrow 0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>r</mml:mi> <mml:mo>→</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> ; Region II—a finite-radius nonlinear neo-Hookean zone; and Region III—an outer linear elastic region. Within Region III, a subregion may still obey the asymptotic linear elastic solution when the radial distance is sufficiently small. As the far-field strain increases, both the asymptotic linear subregion and the broader linear region shrink and eventually vanish, leaving only the nonlinear zones. This multiscale structure reflects the principle of small-scale nonlinearity, wherein nonlinear effects are confined to an inner core. The inner core consists of Region I, where asymptotic neo-Hookean fields dominate, and Region II, where general nonlinear effects prevail. Initially, this inner core is nested inside Region III. At sufficiently small far-field strains, Region III itself contains an inner core that follows asymptotic linear elastic crack tip fields. As loading intensifies, Regions I and II expand, and Region III—first its asymptotic core, then the broader linear zone – .- progressively diminishes, culminating in a large-scale nonlinearity regime. We also identify and quantify the characteristic length scales over which each region exists and dominates—nonlinear fields in Regions I and II and linear elastic behavior in Region III. An important point is that at the crack tip, Region I always governs the local field, although its extent may be small under small far-field strains, making it difficult to capture in computational simulations. To address this, we introduce a rescaling method to better resolve this near-tip behavior.