Creep and Fracture of a Protein Gel under Stress

Biomaterials such as protein or polysaccharide gels are known to behave qualitatively as soft solids and to rupture under an external load. Combining optical and ultrasonic imaging to shear rheology we show that the failure scenario of a protein gel is reminiscent of brittle solids: after a primary creep regime characterized by a power-law behavior whose exponent is fully accounted for by linear viscoelasticity, fractures nucleate and grow logarithmically perpendicularly to shear, up to the sudden rupture of the gel. A single equation accounting for those two successive processes nicely captures the full rheological response. The failure time follows a decreasing power law with the applied shear stress, similar to the Basquin law of fatigue for solids. These results are in excellent agreement with recent fiber-bundle models that include damage accumulation on elastic fibers and exemplify protein gels as model, brittlelike soft solids.

Figure 1

Strain response $\gamma (t)$ in a 4% wt. casein gel acidified with 1% wt. GDL for an imposed shear stress $\sigma =200$ (filled circle), 300 (downward triangle), 400 (filled square), 550 (upward triangle), and 1000 Pa (filled diamond) from right to left. The gap width is 1 mm. Gray dashes show $\gamma =1$. Inset: failure time ${\tau }_f$ vs $\sigma$. The red line is the best power-law fit ${\tau }_f=A{\sigma }^{-\beta }$ with $\beta =5.45\pm 0.05$ and $A=(4.2\pm 0.1){10}^{17}\mathrm{s}\cdot {\mathrm{Pa}}^{\beta }$.

Figure 2

Normalized shear rate responses $\dot{\gamma }(t)/{\dot{\gamma }}_{\text{min}}$ corresponding to the data of Fig. 1 and plotted so as to emphasize the three successive regimes. ${\dot{\gamma }}_{\text{min}}$ is the minimum shear rate reached at ${\tau }_{\text{min}}$ (see text and Supplemental Material, Fig. 2 [18]). The yellow line shows the master curve inferred from fitting $\dot{\gamma }(t)$ by Eq. (1) with $\alpha =0.85$, leading to $\lambda =0.378\pm 0.002$ and $\mu =0.187\pm 0.002$. (a) Primary creep: $\dot{\gamma }(t)/{\dot{\gamma }}_{\text{min}}$ vs $t/{\tau }_f$ in logarithmic scales. Inset: Linear viscoelastic moduli $G^{\prime }$ (top) and $G^{\prime \prime }$ (bottom) as a function of frequency $f$ for a strain amplitude of 0.1%. Red lines are power laws $G^{\prime }\sim G^{\prime \prime }\sim f^{0.15}$. (b) Secondary creep: $\dot{\gamma }(t)/{\dot{\gamma }}_{\text{min}}$ vs $t/{\tau }_f$ in linear scales. Gray dashes show the minimum of Eq. (1) reached at ${\tau }_{\text{min}}=0.556{\tau }_f$. Inset: ${\tau }_{\text{min}}$ vs ${\tau }_f$. The red line is ${\tau }_{\text{min}}=0.56{\tau }_f$. (c) Tertiary creep: $\dot{\gamma }(t)/{\dot{\gamma }}_{\text{min}}$ vs $({\tau }_f-t)/{\tau }_f$ in logarithmic scales with a reversed horizontal axis.

Figure 3

Ultrasonic images (left) and direct visualization of the TC cell (right) recorded simultaneously at various times in the primary [(a) $t/{\tau }_f=0.02$], secondary [(b) $t/{\tau }_f=0.57\simeq {\tau }_{\text{min}}/{\tau }_f$] and tertiary [(c) $t/{\tau }_f=0.93$ and (d) 0.99] creep regimes. Ultrasonic images are azimuthal velocity maps $v(r,z,t)$ computed by averaging over 4 s and coded using the linear color levels shown below the images. Their position relative to the picture of the cell reflects the actual arrangement of the ultrasonic probe along the vertical direction $z$. The gap width is 2 mm and $r$ is the radial distance to the inner rotating cylinder. Experiment performed on a 4% wt. casein gel seeded with 3% wt. polyamide spheres and acidified with 1% wt. GDL under $\sigma =300\mathrm{Pa}$. See also Movie 2 in the Supplemental Material [18].

Figure 4

(a) Local velocity $⟨v(r,z,t){⟩}_z$ and (b) local strain field $⟨{\gamma }_{\text{loc}}(r,z,t){⟩}_z$ averaged over the vertical direction $z$ at various times during primary creep: $t/{\tau }_f=1.9\times {10}^{-3}$ (open circle), $1.7\times {10}^{-2}$ (inverted triangle) and 0.15 (square). Solid lines are linear profiles. The arrows in (a) indicate the velocity of the inner cylinder inferred from the current shear rate. (c) Spatiotemporal diagram of the local velocity $⟨v(r,z,t){⟩}_r$ averaged over the radial direction $r$ and plotted in linear color levels as a function of $z$ and $t/{\tau }_f$. (d) Standard deviation ${\delta }_{z}v(t)$ of $⟨v(r,z,t){⟩}_r$ taken over the vertical direction $z$ (thick black line) together with corresponding standard deviation ${\delta }_{r}v(t)$ computed over the radial direction $r$ on the $z$ average $⟨v(r,z,t){⟩}_z$ (thin red line). (e) Fracture length $\ell (t)$ vs $({\tau }_f-t)/{\tau }_f$ as inferred from direct visualization (close circle, average over six different fractures, error bars show the standard deviation) and from ultrasonic imaging (open circle) and normalized by the height $H$ of the TC cell. Gray dots show the visualization data for the longest fracture which leads to the failure of the sample at ${\tau }_f$. Red lines are the best fits $\ell (t)=a+b\log (1-t/{\tau }_f)$ to the visualization data. Same experiment as in Fig. 3 and Supplemental Movie 2 [18].