Fig. 1

(a) The schematic of a highly entangled hydrogel with dense entanglements, where entanglements within hydrogels function as mobile crosslinks, which can slide along polymer chains when hydrogels are subject to mechanical loading or water swelling; (b) the RVE of the hydrogel with dense entanglements, where there exists a chain segment along each edge of the cube and a mobile entanglement (red ring) at each corner of the cube (color online)"

Table 1

Default parameters in the simulations"

Fig. 2

(a)–(c) The effects of α or β in gamma distribution on the probability density of initial contour lengths of polymer chain segments within the gel; (d)–(f) theoretical predictions of stress-stretch curves of the corresponding gel. Predictions for the cases with uniform initial contour lengths are also plotted for comparison (color online)"

Fig. 3

(a)–(c) The effects of the product of αβ in gamma distribution on the probability density of initial contour lengths of polymer chain segments within hydrogels; (d)–(f) theoretical predictions of stress-stretch curves of hydrogels under uniaxial tension with varied distributions of entanglements within hydrogels. Predictions for the case with uniform initial contour lengths are also plotted for comparison (color online)"

Fig. 4

(a) The predicted probability density of initial contour lengths of polymer chain segments within hydrogels; (b) the theoretical prediction of the stress-stretch curve agrees with the experimental results[10]; (c) the effective number of chain segments along the loading direction increases slowly while decreases rapidly along the perpendicular direction of the loading with stretch; (d) the average contour length along the loading direction increases and that perpendicular to the loading direction decreases with stretch. In the simulation, α=1, β=39 nm, the initial effective number of chain segments is 0.108 nm-3, and the stretch rate is 0.025 s-1 (color online)"

Fig. 5

Comparison of our model predictions with other experimental data: (a) the stress-stretch curve of a highly entangled elastomer under uniaxial tension[10], where in the analysis, N0=0.249 nm-3, α=0.7, β=128.57 nm, η=0.4 pN · s · nm-1, and χ=0.6; (b) the stress-stretch curve of a highly entangled hydrogel under uniaxial tension[11], where in the analysis, N0=0.075 nm-3, α=0.3, β=1 667 nm, η=11 pN · s · nm-1, Lcr=0.1 nm, and χ=0.45; (c) a stress-stretch curve of a highly entangled hydrogel under uniaxial tension[45], where in the analysis, N0=0.008 4 nm-3, α=1.2, β=500 nm, η=0.000 8 pN · s · nm-1, and χ=0.48 (color online)"

Fig. 6

(a) The prediction of the rate independence for the fully swollen highly entangled hydrogel, where in the analysis, η=0.000 1 pN · s · nm-1; (b) the prediction of the negligible hysteresis for the fully swollen highly entangled hydrogel, where arrows indicate loading or unloading, and η=0.000 1 pN · s · nm-1; (c) the prediction of the pronounced hysteresis for the as-prepared highly entangled hydrogel, where in the analysis, η=0.12 pN · s · nm-1 (color online)"

Fig. 7

Stresses in a plate with a circular hole existing at its center upon uniaxial tension, where three types of gel materials are under investigation, and the gel with fixed crosslinks is the stiffest, followed by the one with the uniform contour length, and then the one with the gamma distribution: (a) schematic of a square plate with a circular hole at its center subject to uniaxial tension; (b) the maximum of the nominal stress, σ11, within the plate changes with stretch; (c) stress contours at a stretch ratio of 2. In the simulations, the initial contour length of the chain segments within the RVE is 39 nm, the number density of chain segments is 0.108 nm-3, the stretch rate is 0.1 s-1, the critical value of the contour length for the case with the uniform contour length is 0, and other parameters are listed in Table 1 by default. For the case of fixed crosslinks, the frictional coefficient is set to be a very large value, which is 10 000 pN · s · nm-1 (color online)"