Cracks in tensile-contracting and tensile-dilating poroelastic materials

Abstract

Fibrous gels such as cartilage, blood clots, and carbon-nanotube-based sponges with absorbed oils suffer a reduction in volume by the expulsion of liquid under uniaxial tension, and this directly affects crack-tip fields and energy release rates. A continuum model is formulated for isotropic fibrous gels that exhibit a range of behaviors from volume increasing to volume decreasing in uniaxial tension by changing the ratio of two material parameters. The motion of liquid in the pores of such gels is modeled using poroelasticity. The direction of liquid fluxes around cracks is shown to depend on whether the gel locally increases or decreases in volume. The energy release rate for cracks is computed using a surface-independent integral and it is shown to have two contributions - one from the stresses in the solid network, and another from the flow of liquid. The contribution to the integral from liquid permeation tends to be negative when the gel exhibits volume decrease, which effectively is a crack shielding mechanism.

Keywords: Biomaterials; Crack shielding; Energy release rate; Fracture; Liquid permeation; Path-independent integral; Poroelastic Toughening; Tensile contracting hydrogels.

Figure 1:

Comparison of different constitutive models (${\text{Ψ}}_{\text{net}}$ from (17), (27) and (28)) in uniaxial tension for a cylindrical specimen of poroelastic material under permeable conditions. The results are for ${\varphi }_{\text{s}}^{\text{ref}}=0.01$ and the loading rate is slow enough $\left(v={10}^{-5}\text{m}/\text{s}\right)$ to minimize any rate-dependent effects. (a) Normalized axial stress, (b) finite deformations Hencky Poisson ratio $\left(\nu =-\ln {\lambda }_2/\ln {\lambda }_1\right)$, (c) volume variations, and (d) transverse stretch ${\lambda }_2$ as a function of axial stretch ${\lambda }_1$. Note the reduction in volume at large stretches for $G_2/G_1\ge 1$ and the Fung model. In (a) $G_{1,0}=30\text{kPa}$.

Figure 2:

Rate dependent response of the poroelastic material of for various ratios $G_2/G_1$ with $G_1=30\text{kPa}$ and ${\varphi }_{\text{s}}^{\text{ref}}=0.01$ under permeable conditions for cylindrical specimens in uniaxial tension. (a,c,e) Volume variations, (b,d,f) normalized axial stress.

Figure 3:

Dependence of the constitutive model on solid volume fraction ${\varphi }_{\text{s}}^{\text{ref}}$ for $G_2/G_1=8$ with $G_1=30\text{kPa}$ under permeable conditions for cylindrical specimens under uniaxial tension. The loading rate is slow enough $\left(v={10}^{-5}\text{m}/\text{s}\right)$ to minimize any rate-dependent effects. (a) Volume variations, (b) normalized axial stress.

Figure 4:

Volumetric behavior for three triaxial states for a material with $G_2/G_1=4$ with $G_1=30\text{kPa}$ and ${\varphi }_{\text{s}}^{\text{ref}}=0.01$ under permeable conditions for cylindrical specimens and slow loading $\left(v={10}^{-5}\text{m}/\text{s}\right)$ applied over a period of 2 hours. The dashed lines are the corresponding volume increases under pure hydrostatic tension $(\overline{S}=\overline{T})$, or equivalently $\zeta \to +\infty$.

Figure 5:

The SENT specimen consists of a cylindrical region (depicted in blue) with a volume $V$ and a surface area $S$. This region is utilized for the evaluation of $J^{\ast }$ in (32). The reference coordinate system denoted as $X_i$, has its origin at the center of the small circular crack-tip notch. In this coordinate system, the $X_3$ axis is aligned with the mid-plane in the thickness direction, with $X_3=0$ at that location. On the traction-free side surfaces, the coordinate $X_3$ takes values of $X_3=\pm d/2$. The notch radius is magnified in the figure for clarity.

Figure 6:

Radial variation of (a) ${\tilde{\sigma }}_{22}={\sigma }_{22}/{{\sigma }_{22}|}_{X_1=3R_0}$ and (b) ${\tilde{F}}_{22}=F_{22}/{F_{22}|}_{X_1=3R_0}(\log -\log )$ ahead of the crack tip at $X_i=\left(3R_0\le X_1\le 0.8a,0,0\right)$ under permeable boundary conditions for a material with $G_2/G_1=4$ and ${\varphi }_{\text{s}}^{\text{ref}}=0.01$, $a/w=0.2$ with $v={10}^{-3}\text{m}/\text{s}$ up to $\text{Λ}=1.5$. The dashed lines correspond to slopes of −1/2 and −1.

Figure 7:

Time evolution of (a) ${\sigma }_{22}/G_1$ and (b) $F_{22}={\lambda }_{\max }$ for the material point $\left(3R_0,0,0\right)$ in the SENT specimen for three different $G_2/G_1$ ratios, ${\varphi }_{\text{s}}^{\text{ref}}=0.01$, $a/w=0.35$ with $v={10}^{-3}\text{m}/\text{s}$ up to $\text{Λ}=1.5$ under permeable and impermeable boundary conditions.

Figure 8:

Radial plots of volume variations and normalized chemical potential $\mu /G_1\times {10}^2$ for the radial path $\left(R_0\le X_1,0,0\right)$ for three $G_2/G_1$ ratios, ${\varphi }_{\text{s}}^{\text{ref}}=0.01$, $a/w=0.35$ with $v={10}^{-3}\text{m}/\text{s}$ at $\text{Λ}=1.5$ under (a) permeable and (b) impermeable conditions. Note that the trends in the variation of chemical potential are opposite for TD and TC materials.

Figure 9:

Liquid velocity fields around the crack tip for (a) permeable and (b) impermeable boundary conditions for a material with $G_2/G_1=4$, ${\varphi }_s^{\mathit{\text{ref}}}=0.01$, $a/w=0.35$ with $v={10}^{-3}\text{m}/\text{s}$ at $\text{Λ}=1.5$ for an initially semicircular area of radius $0.5a$ for $0\le X_3\le -d/2$. The liquid velocity ${\mathbf{v}}_{\ell }$ is normalized by its maximum value ${\left|{\mathbf{v}}_{\ell }\right|}_{\max }$ (${\left|{\mathbf{v}}_{\ell }\right|}_{\max }=0.11\text{mm}/\text{s}$ for permeable boundary conditions and ${\left|{\mathbf{v}}_{\ell }\right|}_{\max }=0.03\text{mm}/\text{s}$ for impermeable conditions). Contour plots of the ratio $v_{\ell 3}/\left|{\mathbf{v}}_{\ell }\right|$ at the quarter-plane $X_3=-d/4$ for half the specimen for (c) permeable and (d) impermeable boundary conditions. Under permeable boundary conditions, the liquid takes the shortest path from the specimen to bath, while for impermeable it redistributes itself in the volume of the specimen. Note that the magnitude of liquid velocity is reflected in the color spectrum, all arrows are of constant length.

Figure 10:

${\tilde{J}}^{\ast }=J^{\ast }/G_{1}a$ versus $R_{J^{\ast }}$, the radius of the cylindrical region centered at the crack tip (Fig. 5), for $G_2/G_1=0.25$, 1, and 4, ${\varphi }_{\text{s}}^{\text{ref}}=0.01$, $a/w=0.2$ and 0.35 with $v={10}^{-3}\text{m}/\text{s}$ at $\text{Λ}=1.5$ under (a) permeable and (b) impermeable boundary conditions. Note, path/surface-independence is clearly demonstrated.

Figure 11:

Relative contributions of $J_{\text{flow}}$ versus $R_{J^{\ast }}$, the radius of the cylindrical region centered at the crack tip (Fig. 5), for $G_2/G_1=0.25$, 1, 4, and 8, ${\varphi }_{\text{s}}^{\text{ref}}=0.01$, $a/w=0.2$ and 0.35 with $v={10}^{-3}\text{m}/\text{s}$ at $\text{Λ}=1.5$ under (a) permeable and (b) impermeable boundary conditions. Note the opposite trends for TD and TC materials.

Figure 12:

Evolution of $J_{\text{flow}}$ contribution evaluated over the whole specimen as a function of the overall specimen stretch $\text{Λ}=1+vt/h$ for the TC poroelastic case $G_2/G_1=4$, for ${\varphi }_{\text{s}}^{\text{ref}}=0.01$, $a/w=0.2$ and 0.35 with $v={10}^{-3}\text{m}/\text{s}$ under (a) permeable and (b) impermeable boundary conditions.

Figure 13:

Evolution of the relative $J_{\text{flow}}$ contribution versus $R_{J^{\ast }}$, the radius of the cylindrical region centered at the crack tip (Fig. 5), for the strongly inverse poroelastic case $G_2/G_1=4$, for various ${\varphi }_{\text{s}}^{\text{ref}}$, $a/w=0.35$ and $v={10}^{-3}\text{m}/\text{s}$ at $\text{Λ}=1.5$ under (a) permeable and (b) impermeable boundary conditions.