Cutting mechanics of soft compressible solids – Force-radius scaling versus bulk modulus

Cutting mechanics in soft solids present a complex challenge due to the intricate behavior of soft ductile materials as they undergo crack nucleation and propagation. Recent research has explored the relationship between the cutting force needed to continuously cut a soft material and the radius of the wire (blade). A typical simplifying assumption is that of material incompressibility, albeit no material in nature is really incompressible. In this study, we relax this assumption and examine how material (in)compressibility influences the correlation between cutting forces and material properties like toughness and modulus. The ratio $\mu /K$, where $\mu$ and $K$ are the shear and bulk moduli, indicates the material's degree of compressibility, where incompressible materials have $\mu /K=0$, and larger $\mu /K$ provide higher volumetric compressibility. We observe that the cutting forces are controlled by the ratio between the cutting wire radius $R_w$ and the elasto-cohesive length $L_e=\Gamma /\mu$ of the material (proportional to the critical crack opening displacement at crack propagation under uniaxial tension), where $\Gamma$ is the toughness of the material. Following previous observations, we have two cutting regimes: (i) high $L_e/R_w$ (small wire), and (ii) low $L_e/R_w$ (large wire). Regime (i) is dominated by frictional dissipation, while regime (ii) is dominated by adhesive debonding and/or the wear resistance of the material. In the large radius regime (ii), our theoretical findings reveal that incompressible materials require larger forces (e.g., $K=\mu$ requiring half force compared to incompressible, $K\gg \mu$), while in the small radius regime (i) we observe the opposite trend. Notably, the transition wire radius between regimes (i) and (ii) also depend on compressibility, and is larger for compressible materials. Thus, material compressibility favors friction-domination for a given wire radius.