**1. Introduction**

As industrial applications require computational tools that resolve the microstructure of products rather than tools that smooth out microstructures, increasingly higher levels of versatility and robustness are demanded of finite element codes. Indeed, heterogeneities, discontinuities, microcracking, and contact problems can potentially complicate a microstructural computation. As an example of a such a computation, in this paper we study the swelling of hydrogel beads. Swelling or drying volume transitions of ionized hydrogels can be induced by a continuous change of various conditions, such as temperature, pH, electric field, and salt concentration [1]. The degree of volume transformation depends on the composition and structure of the cross-linked network [2,3]. Because of their swelling behaviour, ionized hydrogels have received considerable attention for pharmaceutical and industrial applications, such as drug delivery or disposable diapers. Many swelling processes of hydrogels start from a dry state. When the dry hydrogel is placed in a solvent, it absorbs fluid and swells. During this volume transformation, a stress field is created within the hydrogel, and to some degree, cracks are generated and developed [4,5].

As the US patent US7517586B2 addressed, hydrogel-forming polymers used as absorbents require adequately high sorption capacity and gel strength [6]. Gel strength resists deformation of hydrogel particles during swelling and avoids gel blocking within a swollen gel bed, which is achieved by increasing the level of surface cross-linking. Although a surface cross-linked hydrogel increases gel stiffness at the surface, it reduces the absorbent capacity and increases the tendency to have a brittle fracture. It is crucial to optimize the balance between absorbent capacity and gel strength.

The effect of the microstructure of the material on swelling capacity has been well-tested; however, few studies take the effects of damage into consideration. Cervera et al. [7] developed a computational model to analyse the progressive cracking due to the swelling of concrete in large concrete dams. Zhang et al. [8] numerically studied the phenomenon of crack healing induced by swelling in gels. Guo et al. [9] found out that the intrinsic properties of the interface between a polymeric hydrogel and a rigid substrate have a grea<sup>t</sup> influence on the opening profile of the interface crack. Although the studies mentioned above deal with crack behaviours induced by swelling, there is still a long way to go. Considering that the structural optimization of the hydrogel is largely affected by the intrinsic properties of the cross-linked and original hydrogel, it is important to study the effects of material properties on crack behaviours.

The extended finite element method (XFEM) is an efficient tool used to simulate fracture growth without re-meshing. It was first applied by Belytschko and Black [10] by adding an additional degree of freedom on nodes which belong to the element crossed by the discontinuity. Wells and Sluys [11] incorporated a cohesive surface formulation into the method to achieve the crack propagation in any arbitrary direction. Similarly, Leonhart and Meschke took the moisture transport in opening discontinuities into account and analysed the crack propagation in partially saturated porous media [12]. Kraaijeveld et al. [13] took osmotic forces into consideration and studied Mode I crack propagation in saturated ionized porous media in small deformations. Irzal et al. [14] extended the partition of unity approach of fracturing porous media into the finite deformation regime. In order to capture the pressure gradient across the discontinuity, Remij et al. [15] developed the enhanced local pressure model (ELP): a separate degree of freedom for the pressure in the discontinuity was added to the pressure left and right of the crack. Furthermore, the level set method (LSM) [16] is commonly incorporated with XFEM to model crack growth. LSM is used to locate the crack and its tip, and it simplifies the selection of the enhanced nodes in XFEM.

In the present work, a finite deformation model is presented to study the crack behaviour of a heterogeneous swelling hydrogel. We integrated XFEM and ELP to capture the pressure difference and fluid flow between the crack and the hydrogel, and used the cohesive zone method to achieve crack propagation without re-meshing [17].
