Analytical Expression for Fracture Profile in Viscoelastic Crack Propagation

Abstract

We derive an analytical expression for the strain field during steady-state crack propagation in viscoelastic solids described by the standard linear solid (Zener) model. This expression reveals three regions in the fracture profile and in the strain field ahead of the crack tip, each distinguished by power-law exponents that evolve with distance from the crack tip. These features explain the experimentally observed crack-tip sharpening in rubbers and gels as the crack-propagation velocity increases, often associated with catastrophic failure triggered by a velocity jump. Furthermore, we establish de Gennes' viscoelastic trumpet on a continuum-mechanical foundation, previously based only on a scaling argument.

Figures (5)

• Figure 1: Setup of viscoelastic crack propagation. (a) Two-dimensional viscoelastic sheet with a semi-infinite linear crack at its center, subjected to a strain $\varepsilon$ due to fixed boundary conditions. The crack propagates at a constant velocity $V$ in the $-x$ direction. (b) Zener model: two springs with shear moduli $\mu_0$ and $\mu_1=\mu_\infty-\mu_0$, interconnected with a dashpot with viscosity $\eta$. The strains of the left spring and dashpot are $\mathcal{E}_{ij}$ and $\mathcal{E}_{ij}^{\mathrm{vis}}$, respectively. (c) Complex modulus $\mu(\omega)$ of the Zener model [Eq. (\ref{['eq:complex_modulus']})] characterizing three types of dynamic response. The relaxation times are $\tau\equiv\eta/\mu_1$ and $\lambda\tau$, where $\lambda\equiv\mu_\infty/\mu_0$.
• Figure 2: Strain field $\mathcal{E}_{yy}$ and fracture profile $\mathcal{U}(x)$ for (a) slow ($\mathcal{V}=10^{-2}$) and (b) fast ($\mathcal{V}=1$) crack propagations at a constant velocity $V$ for $\lambda=10^3$, based on the analytical expression provided in the theorem.
• Figure 3: Fracture profile $\mathcal{U}(x)$ [Eq. (\ref{['eq:fracture_profile']})] on a log-log scale, revealing three regions characterized by power-law exponents $1/2$, $3/2$, and $1/2$. The ticks on the horizontal axis indicate the crossover points, $X_c$, $\lambda X_c$, and $\pi^2/4$, while the ticks on the vertical axis indicate the corresponding points of the respective asymptotic power laws, $\mathcal{U}_c$, $\lambda^{3/2} \mathcal{U}_c$, and $\varepsilon L$. The condition $\lambda X_c <\pi^2/4$ (equivalent to $\lambda x_c <\pi L \sqrt{1-\nu}/ 4 \sqrt{2}$) is necessary for the distinct manifestation of these regions. The inset shows the same profile on a linear scale, with colors corresponding to the strain field shown in Fig. \ref{['fig:2']}.
• Figure 4: Fracture profiles $\mathcal{U}$ during slow ($\mathcal{V}=10^{-2}$) and fast ($\mathcal{V}=1$) crack propagation. (a) Viscoelastic states on the crack surface as a function of the distance $x$ from the crack tip, varying with $V$, while gray horizontal lines correspond to the fast and slow crack propagation cases shown in panels (b) and (c). Gray vertical lines indicate $\lambda X_c$ in panels (b) and (c). (b) At low $V$, a smaller $\lambda X_c$ leads to the dominance of the $1/2$ power-law region, resulting in a parabolic crack-tip profile. (c) At high $\mathcal{V}$, a larger $\lambda X_c$ extends the $3/2$ power-law region, resulting in a sharper crack-tip profile.
• Figure 5: Strain field $\mathcal{E}_{yy}(x,0)$ ahead of the crack tip ($X<0$) as a function of the normalized distance from the crack tip $-X$ for $\lambda \gg 1$. The strain field exhibits two power-law regions, $\mathcal{E}_{yy}(x,0) \sim (-X) ^{-1/2}$ for $-X <\mathcal{V}/(\lambda^2\pi)$ and $\mathcal{V}/\pi<-X<1$, and a plateau for $\mathcal{V}/(\lambda^2\pi)<-X<\mathcal{V}/\pi$. These regions correspond to the hard-solid, liquid, and soft-solid regimes in the complex modulus [Eq. (\ref{['eq:complex_modulus']})].