Cohesive Zone Modeling
In the past few decades, cohesive zone modeling (CZM) has proven to be a useful tool to predict the non-linear fracture behavior of materials [17]. This is attributed to the fact that it does not require the initial crack size or the geometry of the crack. Instead of being an identical physical representation of the material, CZM is more of a phenomenological representation of the fracture behavior in the vicinity of void formation or distributed microcracks in the material. The CZM model mainly constitutes of a cohesive zone, which is assumed to lie ahead of the tip of the crack, just like an extended crack tip (refer to Figure 7.9). On the one hand, where the conventional crack transmits no stress between the crack surfaces, on the other hand, the fictitious crack tip is considered an active field of stress interactions between the surfaces. These surfaces act according to the cohesive traction, which follows the cohesive constitutive law. According to the cohesive law, the cohesive interactions are strongly dependent on the displacement jump. Failure occurs when the displacement jump exceeds the characteristic length (Sc). The fracture process occurs by virtue of decay in the strength of the material strongly resisted by the interatomic forces. Under the application of an external stress, variation in the atomic structure of the material eventuates, which can be reflected via the variation in cohesive traction. The cohesive traction initially increases with increasing separation between the cohesive surfaces until a critical value, after which the value of cohesive traction drops to zero with increasing separation between the cohesive surfaces. In other words, beyond the critical value of the separation between the cohesive surfaces, the cohesive traction vanishes. One of the advantageous features associated with the CZM model is that the model does not employ stress singularity at the crack tip and can represent the physics of fracture process in the atomic scale [18]. Moreover, the dynamics of crack propagation can be successfully captured via the application
FIGURE 7.9 Cohesive zone model.
of CZM in MD, which could not be achieved by MD alone [19,20]. As mentioned earlier, the cohesive law defines the relationship between cohesive traction
where Some of the unique characteristics associated with cohesive constitutive law are summarized as follows: The CZM can be approached basically via two ways: (i) experimental measurements and (ii) phenomenological method constituting estimated parameters and predefined functional assumptions. Although a few experimental approaches to CZM have been attempted in the past [21], an effective method measuring traction-separation relation has not been achieved till date. On the other hand, various models of CZM have been proposed by the researchers in the past few decades. A few of the CZM models have been pictorially represented in Figure 7.10. The polynomial law explaining the process of void nucleation in metallic specimen was first proposed by Needleman [17]. Exponential cohesive- zone-type interface model has been described by Needleman for analyzing the decohesion process of the interfaces [22]. The application of exponential cohesive zone law was further extended to study the debonding process along the particle- matrix interface [23] and rapid crack growth in brittle materials [24]. Tvergaard et al. characterized the fracture behavior in an elastic-plastic material, employing trapezoidal traction-separation method [25]. Ortiz and Suresh employed a linear approach of the traction-separation method to describe the intergranular fracture in crystalline ceramic materials [26]. Usually, bilinear traction-separation law is used to define the decohesion behavior along the dissimilar material FIGURE 7.10 Different forms of traction separation laws: (a) Exponential, (b) bilinear, (c) polynomial, and (d) trapezoidal. interfaces [27]. In a nutshell, the description of the crack initiation and crack propagation behaviors of materials can be broadly classified into one of these four laws, that is, exponential, linear/bilinear, polynomial, and trapezoidal. A brief explanation of the simulation process, which is supposed to be implemented to predict the failure of the material at nanoscale, is presented here. The MD process of atomistic scale of crack propagation in the material is carried out in the LAMMPS platform [11] by employing the requisite interatomic potential. A specimen is created per the user-defined dimension. Thereafter, two rigid regions of a few several thicknesses are created in the top and bottom of the specimen, as shown in Figure 7.11. The net force in the atoms belonging to these rigid regions is set to zero during deformation. A crack is introduced at the center of the specimen by deleting a few layers of atoms within the specified dimension (as shown in Figure 7.11). Following this, the atomic system is relaxed to a minimum-energy state via employing the conjugate gradient method. The specimen with the crack is then equilibrated in the canonical ensemble (constant volume-constant temperature [NVT ensemble]). The temperature is controlled using Nose-Hoover thermostat [12,13]. In order to carry out the failure analysis of the specimen under mode I loading, the specimen is allowed to deform in the direction perpendicular to the length of the crack (as shown in Figure 7.11). The deformation is carried out by employing a uniform velocity in the above-mentioned direction (in this case, у-direction) corresponding to a user-defined strain rate, under the canonical ensemble at 300 К temperature (i.e., the room temperature). A timestep of 0.002 picosecond (ps) is assigned for the whole simulation procedure. Usually, in case of crack analysis, instead of temperature control, energy conservation is more of a priority. Therefore, it is advisable to employ the equilibration of the sample and deformation to be carried out in microcanonical ensemble (i.e., NVE, which means constant number of atoms, constant volume, and constant energy). Temperature control is not preferred in such cases, as the energy gained from or lost to the thermostat can have undesirable consequences on the simulation results [28]. However, the conservation of energy leads to the increasing temperature of the system, as the potential energy of the system is transferred to the thermal kinetic energy. The local tensile FIGURE 7.11 Simulation process to predict the failure of a material employing MD. stress in the vicinity of the crack is determined via virial stress [29,30]. The atomic stress computed employing virial definition [31 ] is given as follows:
and
where oap is the virial stress, U.'aP indicates the atomic dipole force tensor, Q‘ is the volume of the atom i, the first term within the square bracket corresponds to the function of the potential energy representing the force between two atoms, the second term indicates the force associated with the kinetic energy of the atom, in' indicates the mass of the atoms, v„ is the component of velocity in the «-direction, and v'p is the component of velocity in the /(-direction for the ith atom. riJ stands for the distance between the /th atom and theyth atom. On the other hand, r% and r$ symbolize the components of ru in the «- and /(-directions, respectively, and finally, /V* denotes the number of free atoms in the simulation box or the specimen. The formulation of cohesive traction-separation law consists of the definition of atomic hydrostatic stress and atomic von Mises stress. The atomic stress allows to incorporate the interaction of deformations and stresses at the atomic level. The atomic hydrostatic stress and Mises stress can be computed as follows [32]: where cr;, and o. represent atomic hydrostatic and Mises stress, respectively. In order to study the variation of atomic stress and crack-opening displacement, a cohesive zone of size ±1 nm in the vicinity of the crack is considered, as shown in Figure 7.12. The size of the cohesive zone is vital with respect to obtaining accurate values of the local stresses. Too large a cohesive zone will give average values of the stress rather than the local stress, and too small a cohesive zone will not be able to capture the stress generated in the near vicinity of the crack. The crack-opening displacement is defined as the average atom displacement in the upper half of the region with respect to the lower half of the region. For the given configuration, the total magnitude of the crack opening can be defined as where Ax and Ay are the displacement of crack in .v-direction (shear) and y-direction (normal), respectively. The simulation results are viewed using a visualization tool OVITO [32]. It is advisable to carry out the crack-analysis-based simulation at a temperature of О К to rescale the velocities of atom, thereby avoiding thermal activation. The effects of atomic vibration on the simulation results can be eliminated by following the above-mentioned procedure.
Crack Opening Displacement and Local Stresses Using Molecular Dynamics
