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This class: (1) reads in a mesh describing the crack surface, (2) uses the mesh to do initial cutting of 3D elements, and (3) grows the mesh incrementally based on prescribed growth functions. The code is interfaced with domain integral methods to allow nonplanar crack growth based on empirical propagation direction and speed laws.

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Including fracture surface energy in analytical or numerical models opens up new potential for those models to improve realism. For example, Williams et al. [9,10] developed analytical models for orthogonal cutting with crack propagation including plastic shearing and plastic bending modes. The initial model was for isotropic materials with elastic plastic behaviour, although it is easily extended to plastic hardening [11]. The addition of explicit crack growth is also an important concept for numerical modelling. Many past numerical models have focused on plasticity behaviour, but because cutting cannot progress without separating elements (in finite-element methods), most had to introduce ad hoc separation criteria [1]. When explicit crack growth is directly included in the model, those separation criteria can be based on physically justifiable fracture mechanics criteria [1]. For example, Nairn [11] describes a material point method (MPM) simulation of orthogonal cutting. The crack growth was modelled by explicit crack growth using cohesive zone methods. This numerical scheme was able to simulate full chip formation and propagation into steady-state cutting conditions for both plastic shearing and plastic bending modes. The contact capabilities of MPM were helpful for providing stable simulations even when the tool tip touches the crack tip (e.g. when the gap between the tool tip and crack tip shown in figure 2 disappears).

Explicit crack propagation was modelled using MPM cohesive zones [11,27,28]. In brief, an initial crack was introduced along the entire cutting path and the crack plane was modelled with a cohesive law determined by fracture experiments in the appropriate failure plane for solid wood (see below). This approach limited simulations to straight crack growth and, therefore, could not model chip fracture caused by fracture paths diverting toward the surface [4].

All numerical models simulating dynamic crack growth must incorporate a scheme for extending the crack. For example, in FEM, a node is released in the crack plane or in explicit MPM cracks the crack path is extended by a small amount [15,16]. Even in cohesive zone modelling, a crack grows when the crack opening displacement at the crack root reaches the cohesive law's critical value (δc) and traction drops to zero. In computational mechanics code that correctly conserves total energy, all these virtual crack extensions can cause an increase in kinetic energy that can quickly deteriorate numerical results. But, this conversion to kinetic energy does not reflect crack extension in real materials where that energy is instead absorbed by some surface processes representing the material's fracture toughness. One solution to dealing with artefacts in dynamic crack propagation simulations is to add damping to mimic energy absorption in real materials, but it is challenging to add realistic damping. In previous orthogonal cutting simulations, it was noted that a new form of damping, denoted as PIC damping [11], worked very well for crack propagation simulations. In brief, this damping focuses damping effects in regions with high velocity gradients and, therefore, selectively dampens regions around a propagating crack tip. Simulations with PIC damping enabled are extremely stable for all cutting conditions, while simulations without PIC damping were only stable for a few conditions. When they both work, they give nearly identical cutting forces except for far less noise when using PIC damping. All simulations here used the PIC damping method [11].

(a) Experimental results for Douglas-fir mode I fracture in the TL and RL directions. The dashed lines are numerical MPM simulations assuming fracture initiation followed by linear softening fibre bridging. (b) Trilinear traction law used to model crack propagation in cutting simulations. (Online version in colour.)

The resulting cohesive law properties derived from these experiments and used for cutting simulations are given in table 1. Although experiments provide Ginit and Gb, the specific values for cohesive stresses and critical crack opening displacements are less certain. Some consequences of changing these values are discussed below and in [11].

The experiments from Matsumoto & Nairn [31] were mode I failure, but cutting is not pure mode I. Although mode II R curves are not available for Douglas-fir wood, experiments on balsa wood suggest mode II initiation toughness is about three times higher than mode I and that mode II crack propagation is unaffected by fibre bridging [32]. These cutting simulations, therefore, set mode II toughness to three times the mode I initiation toughness and ignored fibre bridging effects (the mode II cohesive law properties are in table 1).

where GI and GII are areas under the cohesive law up to the current normal and tangential crack opening displacements, respectively, and are total areas under the mode I and mode II cohesive laws. For simplicity, all simulations here used n = m = 1. When the crack grows, the mode I character of the crack growth is given by GI/(GI + GII). Simulations show that wood planing is predominantly mode I, but ranged from 65 to 99% mode I depending on various cutting parameters.

Figure 9 shows steady-state chips as a function of the depth of cut (from 0.05 to 0.5 mm) for constant chip breaker location (b = 1.59 mm) and constant mouth opening (m = 1.2 mm, which means m/d varied from 24 to 2.4). For depths of cut of 0.05 and 0.1 mm, the chips curl before reaching the chip breaker, while the chip breaker contacts, and, therefore, alters the chip for all other depths of cut. Starting at about 0.3 mm, the chip breaker/mouth opening combinations caused the chip to lose contact with the tool. In these frictionless simulations, this loss of contact eliminates all vertical force on the tool. The tool/chip breaker combination has only horizontal force while the base plate carries only vertical force. Although more simulations are required to optimize all settings, these simulations show that both chip breaker location and mouth opening should be adjusted whenever the depth of cut is changed. As the depth of cut gets thinner, the chip breaker needs to be moved closer to the tool tip otherwise it serves no purpose. Assuming that the tool should contact the chip to provide the best cut quality, for a given chip breaker location, the mouth opening must either be wide enough to retain that contact (see m/d > 4 for d < 0.3 mm in figure 9), or narrow enough to control chip curvature without using a chip breaker (see m/d = 1.5 in figure 8). The latter option also gets the tool tip the closest to the crack tip, which may or may not be advantageous for high quality cuts. 2b1af7f3a8