Fracture Simulations Using Molecular Dynamics (MD)
Fracture means fragmentation/separation of a component into two/more segments when the component is subjected to load/stress. Fracture occurs in the component by discontinuity displacement and also by stress concentration zones (such as voids and scratches). Fracture process has two steps: one is crack initiation and another is crack propagation. Generally, fracture takes place in two ways: (1) brittle fracture and (2) ductile fracture. The fracture that happens without any warning (i.e., no long plastic deformation) during deformation is known as a brittle fracture. Rapid crack propagation, low absorption of energy, and unstable crack nature are also considered the characteristics of a brittle fracture. These characteristics happened in brittle components without rise in stress. Ductile fractures exhibit a wide plastic deformation with high absorption of energy. Crack propagation rate is slower in materials with ductile fracture. The type of fracture (i.e., crack initiation and propagation) depends on the factors’ intensity such as temperature, stress type, stress levels, and strain rate [If The fracture process takes place for ideal brittle, semi-brittle, and ductile materials as shown in Figure 7.1.
At present scenario, molecular dynamics (MD) simulations have been preferred to investigate the deformation and underlying fracture mechanisms of materials at the nanolevel. Per Zhang et al., the crack propagation mechanism of single crystal (SC) (i.e., dislocations that nucleate at crack tip and move along the closed slip planes and their direction), bicrystal (i.e., grain boundary orientation and the dislocations are nucleated and emitted from crack tip as well as grain boundary), and tricrystal (i.e., triple junction of the grain boundary orientation) has been investigated using MD simulations. The existence of a grain boundary contributes to the enhancement of the Shockley partial dislocations and stacking faults . Orientation alteration among adjacent grains leads to dislocations’ slipping in different directions, which results in void initiation at the triple junction of grain boundaries, and it accelerates the crack propagation . Few investigations have been carried on the fracture behavior of SC and nanocrystalline metals at nanoscale using MD simulations [4-9].
FIGURE 7.1 Fracture process of (a) ideal brittle, (b) semi-brittle, and (c) ductile materials.
The parameters considered to perform fracture test using MD simulations are strain rate, state of stress, and temperature. Test parameters vary depending on the type of loading to be performed to examine the material’s fracture behavior. The influence of test parameters on fracture behavior is explicated in detail in the following subsections.
Strain Rate Effect
The mechanical properties of materials are varied based on the strain rates preferred in the simulation; moreover, they influence the mechanism of the material system. Figure 7.2a-h depicts the crack propagation process of polycrystal nickel at a strain rate of 2 xlO8 s_l, and Figure 7.3a-h illustrates the same process at 2 x 1010
FIGURE 7.2 Crack propagation of polycrystal Ni with a strain rate 2 X 10s s_1 at various strains (a-h). (Courtesy of Zhang, Y., and Jiang, S. Metals, 7,432,2017.)
FIGURE 7.3 Crack propagation of polycrystal Ni with a strain rate 2 X 10'° s'1 at various strains (a-h). (Courtesy of Zhang, Y., and Jiang, S. Metals, 7,432,2017.)
s_l. The specimens are deformed under uniaxial tensile loadings . In case of low strain rate, the void is initiated at the innermost layer of the middle grain, which is noticed in Figure 7.2. The void increases progressively through plastic deformation and amalgamates by initial crack. Then, at the end, enhancing crack propagation rate leads to fracture of the material. In case of high strain rate, the void is initiated at the triple junction of grain boundary, which is noticed in Figure 7.3. The void increases progressively through plastic deformation and amalgamates through initial crack. Then, finally, enhancing the rate of crack propagation leads to fracture of the material. The crack initiation and propagation mechanisms are different in case of low and high strain rates. Figure 7.4 displays the stress-strain plot for polycrystal nickel subjected to strain rates. It is observed that elastic deformation happens at initial deformation stage, in which the stress increases linearly with increasing strain, which is clearly observed from Figure 7.4. Correspondingly, the stress decreases rapidly in high strain rate, after reaching the ultimate stress, in contrast to low strain rate. High yield stress is obtained at high strain rates, because the crack propagation is sensitive to strain rate.
Stress State Effect
The uniaxial, biaxial, and triaxial states of stress are presented in Figure 7.5. Large plastic deformation takes place under the uniaxial state of stress and also supports the crack propagation, which leads to delay in the earlier fracture of the material (i.e., it promotes a ductile fracture). But the crack propagation and plastic deformation in case of the biaxial state of stress is lesser than the uniaxial state of stress. Triaxial state of stress does not promote the crack propagation in a material, but it suppresses
FIGURE 7.4 Stress-strain plot of polycrystal Ni subjected to two strain rates. (Courtesy of Zhang, Y., and Jiang, S. Metals, 7,432,2017.)
FIGURE 7.5 The (a) uniaxial, (b) biaxial, and (c) triaxial states of stress.
the plastic deformation. As plastic deformation is suppressed the crack tip remains sharp, thus promotes brittle fracture in a material.
Materials exhibit an inconsistency in mechanical and physical properties based on the conditions of testing temperature. Materials exhibit brittle nature at low temperatures because a decrease in temperatures leads to an increase in stress concentrations, which are favorable for generating voids/cracks in the materials (e.g., polycrystalline nickel). The molecular dynamics simulation study on the fracture behaviour of nanostructured polycrystalline nickel under tensile hydrostatic stress has provided the underlying mechanism with respect to different temperatures . In that, the stress increases linearly corresponding to strain up to 4%, and thereafter, the stress increases non-linearly to ultimate strain value of up to 5.7%. At low temperatures (1, 100, and 200 K), the derivative stresses drop first compared with that at high temperatures (300, 400, and 500 K). It is also an indication of the brittle nature. The derivative stresses at low temperatures exhibit unstable stress nature compared with those at high temperatures. At low temperatures, the fracture toughness (material’s ability to restrict fracture) of polycrystalline nickel is low, because those low temperatures are favorable for fracture, and there is no scope of plastic deformation. But at high temperatures, the fracture toughness of polycrystalline nickel is high, owing to plastic deformation that occurs through dislocations movement, twin’s development, and grain-boundary-activity processes.
The simulation procedure for fracture behavior of materials has been carried out by using MD simulations through open-source large-scale atomic/molecular massively parallel simulator (LAMMPS) software . Create the crack-induced specimen with essential dimensions, as per the requirement, with the aid of lattice, region, create_box, and create_atoms commands. After completion of the specimen creation, select the suitable force fields for determining the interatomic interactions between the atoms of the specimen. Then, perform the energy- minimization process to obtain the stable-structured specimen by using conjugate gradient method. Consequently, the specimen equilibrates at the required temperature under Constant Number, Volume and Temperature (NVT) or Constant Number, Pressure and Temperature (NPT) ensembles. The Nose-Hoover thermostat controls the temperature throughout the process [12,13]. The entire simulation process performs at periodic boundary conditions. Then, apply the load on crack- induced specimen in uniaxial tensile direction (mode I) at a suitable strain rate and temperature with the required timestep. (Generally, 2 femtoseconds is preferred for simulating the process). The same process can be applied for simulating the pre-existed cracked specimens in mode II and mode III directions by changing the loads and loading directions.
The nickel specimen has been created, and MD simulations have been performed through LAMMPS software . The size of the nickel specimen is 28.16 x 14.05 x 2.82 nm, with a lattice constant of 3.52 Angstrom. The edge crack has been inserted into the nickel specimen at left-center portion, with length and width as 2.82 nm and 1.76 nm, respectively. The lower- and upper-end portions of atoms are fixed and act as boundary atoms, which are shown in Figure 7.6. Tight- binding potential has been used for the interaction of Ni atoms. Uniaxial tensile loading has been performed at a strain rate of 3.33 x 10s—3.33 x 109 s_l in Z-direction. The timestep considered to perform MD simulations is 1 fs at various temperatures (50, 300, 500, and 700 K). OVITO software has been used for the visualization and analysis of fracture crack behavior (crack growth as well as propagation) of SC Ni specimen during tensile deformation .
FIGURE 7.6 Nickel specimen during tensile loading. (From Sung, P.H. and Chen, T.C., Comput. Mater. Sci., 102, 151-158, 2015. With permission.)
Traction and Separation Method
Fracture of a material mainly consists of two processes: (a) crack initiation and (b) crack propagation. The crack plays the role of a stress concentrator or stress amplifier in the material. The propagation of crack in the material leads to a scenario where the surface energy of the material increases, or in other terms, there is an increase in elastic strain energy. Griffith proposed a set of two conditions for a crack to grow in brittle materials :
- 1. The stress concentration at the tip of the crack should be high enough to reach theoretical cohesive strength of the material.
- 2. The release in strain energy as a result of generation of free surfaces attributed to the growth of crack must be greater than or equal to the surface energy of the faces that have been created. Mathematically, it can be represented as
FIGURE 7.7 Schematic representation of a crack in a body under stressed condition.
Let us consider an illustration (Figure 7.7) where a material with crack length a is subjected to an applied stress of a.
If the material is linear, then the release in strain energy due to crack propagation under plane stress condition, /3, is equal to n, and the equation is given by
where E is Young’s modulus of the material.
The surface energy associated with the crack with the length a is given by
where ys is the surface energy. The total energy associated with the crack is the sum of the strain energy released as a result of crack propagation and the surface energy absorbed as a result of new surface created during the propagation process (refer to Figure 7.8). Mathematically, it can be represented as
It can be deduced from the graph that the dependence of strain energy on the crack length dominates the surface energy eventually, and beyond a critical crack length (ac), the crack grows spontaneously. The critical crack length can be defined as
where 07 denotes the fracture stress. Eq. 7.5 is applicable only to brittle materials. However, the Griffith theory is also applicable to ductile materials with slight modification, given as follows:
yp symbolizes the energy exhausted in the plastic work done.
FIGURE 7.8 Variation of surface energy, strain energy, and total energy associated with change in the length of the crack.