*2.1. Crack Propagation*

Here, we consider a circular sample consisting of three different materials (Figure 1). We fixed the middle point and constrained the node located on the boundary between the core and middle part at 0◦ in the y direction. A changing chemical potential was applied at the outer surface, which led to the swelling of the medium. The material properties are listed in Table 1.

Figures 2 and 3 show the behaviour of crack propagation with different stiffnesses and ultimate strength of the shell. The crack propagates over time for four different shell shear moduli (2 MPa, 3 MPa, 3.5 MPa and 4.5 MPa) and the same shell ultimate strength of 0.52 MPa (Figure 2). For *G* = 2.0 MPa, the initial crack opens without propagation. For the other four examples, every initial crack propagates. Generally, a stiffer material reaches high stress faster than a softer material with the same amount of deformation, causing material failure and crack growth. The higher the shear modulus is, the earlier the initial crack propagates, as seen in Figure 2 with crack propagation plotting for *G* = 3.0, 3.5, and 4.5 MPa. At the same time, a stiffer shell resists the particle's deformation and helps to keep the shape of the particle. Less deformation comes with smaller stress in the middle part, which suppresses

the crack growth (Figure 4). This is the reason why the crack length of *G* = 4.5 MPa is much smaller than with *G* = 3.0 MPa and *G* = 3.5 MPa.

**Figure 1.** The geometry and boundary conditions of the sample. A changing chemical potential is applied along the outer surface. The red dashed line indicates the initial crack. (*RShell* = 0.5 mm, *RInner* = 0.45 mm, *RCore* = 0.2 mm)


**Table 1.** Material properties.

Similarly, we fixed the shell shear modulus (*Gshell* = 4 MPa), and studied the effect of various ultimate strengths (0.3 MPa, 0.4 MPa, 0.5 MPa, and 0.6 MPa) of the middle material (Figure 3). The effect of ultimate strength is straightforward; the ultimate strength does not contribute to the propagating length of the crack, but it increases the capacity of the material to resist tension, and only affects the rate of propagation. The higher the ultimate strength, the later the crack starts to propagate. It is advisable to achieve relatively high ultimate strength in the material design to obtain higher elongating resistance.

**Figure 2.** Crack propagation profile over time with different stiffnesses of the shell.

**Figure 3.** Crack propagation profile over time with different ultimate strengths of the shell.

(**c**) G = 3 MPa (**d**) G = 4.5 MPa **Figure 4.** The displacement profile with different shear moduli of the shell.

### *2.2. Crack Nucleation*

The energy accumulates within the hydrogel particle when it swells. There are two ways to dissipate energy, propagate existing cracks, or nucleate new cracks. In Section 2.1 we discussed the behaviour of crack propagation. In the current section, we discuss the behaviour of crack nucleation.

The same geometry (Figure 1) and material properties (Table 1) are used here to model crack nucleation. In order to avoid too many cracks nucleating at the same time too close together, we made an extra constraint that there were at least 30 elements between two cracks. We compared the nucleation state within three groups: different shear moduli of the shell, different ultimate strengths of the shell, and different shear moduli of the middle part of the sample.

The point at 45◦ to the x-axis in the first quadrant is the initial crack. In Figure 5, we plotted the nucleations with the shell's ultimate strength of 0.48 MPa, 0.52 MPa, and 0.54 MPa, respectively. There was no nucleation with an ultimate strength of 0.54 MPa. For *τult* = 0.52 MPa, it has five nucleations. When *τult*decreases to 0.48 MPa, the nucleations increase to 7.

Figure 6 shows the nucleation locations with different shear moduli of the shell. There were nine nucleations for *G* = 4.5 MPa and five nucleations for *G* = 4.0 MPa. No new cracks nucleated for *G* = 3.0 MPa. Similarly, Figure 7 plots the nucleation locations with different shear moduli of the middle part of the particle. It shows that there are 7, 5, and 0 nucleations relating to *G* = 1.0, 1.5 and 1.7 MPa, respectively. The distribution of nucleations is roughly symmetric about the diameter crossing the initial crack.

**The effect of shell ultimate strength on crack nucleation**

**Figure 5.** Crack nucleations with different ultimate strengths of the shell. Every coloured dot represents one nucleation site. The results of three separate computations with different values of the shell's ultimate strength are superimposed. The colour of the dot specifies the computation to which it belongs.

**Figure 6.** Crack nucleations with different stiffnesses of the shell.

**Figure 7.** Crack nucleations with different stiffnesses of the middle part.

Figure 8 is the chemical potential distribution within the crack with different time steps. The nucleation process shows a cascade phenomenon (Figure 8). At the time of 4.8 s, the initial crack propagates without any nucleation. After 0.1 s, there is one nucleation. Another new crack nucleated after 0.1 s. At the time of 5.1 s, the previous two nucleations kept growing, and another three new cracks were generated.

(**c**) time 5.0 s (**d**) time 5.1 s **Figure 8.** Nucleations at different times (s), red circles emphasize nucleation spots.
