*4.3. Swelling Behaviours*

Swelling equilibria between the hydrogel and the immersed fluid must fulfil *μin* = *μex* (*μin* is the chemical potential within the hydrogel, and *μex* describes the chemical potential of the surrounding fluid).

The difference of the osmotic pressure between the hydrogel and the surrounding fluid allows ions or fluid to go into or out of the hydrogel, leading to swelling or shrinking. Because the ionic diffusion coefficient of SAP is two orders of magnitude larger than the pressure diffusion coefficient of SAP, we assumed the ionic constituent to be drained. Based on Van's Hoff empirical relation, the osmotic pressure difference was given by

$$
\Delta\pi(\varepsilon) = RT\sqrt{(c^{fc})^2 + 4(c^{cx})^2} - 2RT\varepsilon^{cx} \,, \tag{12}
$$

where *R* is the gas constant, *T* is the temperature, *c f c* is the fixed charge density, and *cex* is the external salt concentration.

### *4.4. Nucleation Mechanism*

Crack nucleation and growth are part of the fracture development. Nucleated micro-cracks are based on the stress state of the solid skeleton. The stress state was obtained by averaging effective stresses around the crack tip. Remmers et al. [18] used a Gaussian weighted function to calculate the averaged stress:

$$
\sigma\_{\omega\nu} = \sum\_{i=1}^{n\_{\text{int}}} \frac{\omega\_i}{\omega\_{\text{tot}}} \sigma\_i
$$
 
$$
\omega\_{\text{tot}} = \sum\_{j=1}^{n\_{\text{int}}} \omega\_{j\nu}
$$

here, *nint* is the number of integration points in the domain, and *ωi* is the weight factor relating to the integration point *i*. The weight factor is defined as

$$
\omega\_{\bar{l}} = \frac{1}{l\_a^3} e^{\frac{-r\_i^2}{2l\_a^2}}\,\,\,\,\,\,\tag{13}
$$

where *la* is the length scale parameter, and *ri* denotes the distance between the integration point *i* and the crack tip.

### *4.5. Crack Propagation Mechanism*

The crack propagation is governed by the stress state of the crack tip. The stress is calculated by averaging the stress around the crack tip. The traction **t***d* in Equation (9) acting on the fracture surface is based on the cohesive constitutive relation, and governs the propagation of the crack. When the averaged stress of the crack tip exceeds the ultimate strength, the crack starts to propagate. The normal traction *tn* is described as

$$t\_{\rm nl} = \tau\_{\rm ulft} \exp\left(-\frac{u\_n \tau\_{\rm ulft}}{\mathcal{G}\_{\rm c}}\right),\tag{14}$$

where *τult* is the ultimate strength of the material, and G*c* denotes the fracture toughness.
