**4. Methods**

### *4.1. Kinematic Relations*

We considered a body Ω crossed by a discontinuity (Figure 9). The body was divided into two subdomains, Ω<sup>+</sup> and Ω<sup>−</sup>. The total displacement field of the solid skeleton was described by a regular displacement field **u**ˆ and an enhanced displacement field **u**˜,

**Figure 9.** The body Ω is crossed by a discontinuity (dashed line). **<sup>n</sup>**Γ*d* represents the normal of the discontinuity surface pointing to Ω+.

$$\mathbf{u}(\mathbf{X},t) = \hat{\mathbf{u}}(\mathbf{X},t) + \mathcal{H}\_{\Gamma\_d}(\mathbf{X})\bar{\mathbf{u}}(\mathbf{X},t),\tag{1}$$

where **X** is the material point in the reference configuration of the solid, H<sup>Γ</sup>*d* is the Heaviside step function, defined as

$$\mathcal{H}\_{\Gamma\_d} = \begin{cases} 1 & \mathbf{X} \in \Omega^+ \\ 0 & \mathbf{X} \in \Omega^- \end{cases} \tag{2}$$

The chemical potential field is discontinuous across the discontinuity and the hydrogel, and defined as

$$
\mu^f(\mathbf{X}, t) = \hat{\mu}^f(\mathbf{X}, t) + \mathcal{H}\_{\Gamma\_d}(\mathbf{X})\hat{\mu}^f(\mathbf{X}, t), \tag{3}
$$

In the discontinuity, the chemical potential is equal to an independent variable *μd*,

$$
\mu^f = \mu\_{d\prime} \quad \mathfrak{X} \in \Gamma\_d. \tag{4}
$$

Hence, the value of the chemical potential jumps from *μ*ˆ *f* to *μd* to *μ*ˆ *f* + *μ*˜ *f* as one crosses the discontinuity from Ω− to Ω+.
