The 1/r singularity in weakly nonlinear fracture mechanics

Abstract

Material failure by crack propagation essentially involves a concentration of large displacement-gradients near a crack's tip, even at scales where no irreversible deformation and energy dissipation occurs. This physical situation provides the motivation for a systematic gradient expansion of general nonlinear elastic constitutive laws that goes beyond the first order displacement-gradient expansion that is the basis for linear elastic fracture mechanics (LEFM). A weakly nonlinear fracture mechanics theory was recently developed by considering displacement-gradients up to second order. The theory predicts that, at scales within a dynamic lengthscale $\ell$ from a crack's tip, significant $\log{r}$ displacements and $1/r$ displacement-gradient contributions arise. Whereas in LEFM the $1/r$ singularity generates an unbalanced force and must be discarded, we show that this singularity not only exists but is {\em necessary} in the weakly nonlinear theory. The theory generates no spurious forces and is consistent with the notion of the autonomy of the near-tip nonlinear region. The J-integral in the weakly nonlinear theory is also shown to be path-independent, taking the same value as the linear elastic J-integral. Thus, the weakly nonlinear theory retains the key tenets of fracture mechanics, while providing excellent quantitative agreement with measurements near the tip of single propagating cracks. As $\ell$ is consistent with lengthscales that appear in crack tip instabilities, we suggest that this theory may serve as a promising starting point for resolving open questions in fracture dynamics.Comment: 12 pages, 2 figure

Autonomy and singularity in dynamic fracture

“…This theory was shown to be in excellent agreement with groundbreaking experimental measurements of the deformation near the tip of rapid cracks [3][4][5][6]. Furthermore, it may be relevant to understanding currently poorly understood crack tip instabilities [7].…”

“…We further show that the resultant linear momentum carried by the 1/r singular fields vanishes identically. Our results, which reveal the physical and mathematical nature of the new solution, are in favorable agreement with recent near tip measurements.A weakly nonlinear theory of dynamic fracture, which extends the standard theory of fracture [1,2], was recently developed [3,4]. This theory was shown to be in excellent agreement with groundbreaking experimental measurements of the deformation near the tip of rapid cracks [3][4][5][6].…”

“…As higher order nonlinearities are neglected in Eq. (1), the theory based on it is termed "weakly nonlinear theory of dynamic fracture" [3,4,8].The expansion in Eq.(1) can be substituted in a general elastic strain energy functional U (F ), where F ij = δ ij + ∂ j u i , from which the first Piola-Kirchhoff stress tensor s can be derived ass quantifies forces in the deformed configuration per unit areas in the reference configuration [9]. Then, the momentum balance equations and the crack faces tractionfree boundary conditions are obtained order by order in ǫ [3,4,8].…”

“…Then, the momentum balance equations and the crack faces tractionfree boundary conditions are obtained order by order in ǫ [3,4,8]. u (1) (r, θ; v) in Eq.…”

“…To mathematically formulate this idea, we consider the following expansion of the displacement field u [3,4] …”