the regions of interest, following the concept of Saint Venant’s principle.63 Otherwise, the obtained low-scale phenomena observed can be qualitatively different which can cause instability of atomic motions, microstructural evolutions, and unexpected material failure.
Thirdly, some mathematical solutions for the continuum require the medium to be sufficiently large to make the LEFM crack-tip solution realistic for a tiny crack inside of a bulk material. In this case, model size must not be small for a reasonable result. The fourth aspect is that for microsystems and nanotechnology, the model size should be equal or larger than micrometers so the problem of micro- or nano- sensors/activators can be more accurately simulated. For nanotechnology, this is true for some designs since nanotubes, nanofibers, etc. need to be assembled and embedded in a matrix which has a certain size requirement. Thus, investigating the model size effects and choosing a minimum model size necessary for the accuracy requirement is essential.
With the proposed GP-FEA method, the model size effects on the crack-tip displacement fields of a Mode-I edge crack embedded in a single crystal of BCC iron along the X-direction [110] are extensively investigated. All models were subjected to the remote BC displacement along the Y-direction [1̅10] with 1% strain. In this problem, the main displacement component is uy along the loading direction Y and displacement component ux, related to Poisson’s effect, has minor effect, thus all the models’ widths, LX, were held at a constant 1000 nm but the Ly value was changed from 120, 180, 250, 500, 800, 1000, 2000 to 5000 nm to observe the model size effect. This size effect is measured by the displacement uy for same material points in different models. They are located on the Y-axis and apart from the crack tip from 2 to about 40 Å. The accuracy is verified by the LEFM two-term solution described in the same figure. From the comparison of the eight simulation curves of these models with the analytical solution shown in Figure 30, one observes the following observations and conclusions.
It is seen that the smaller the model size the larger the error produced in the simulation-obtained uy. Specifically, for the case of Ly=120 nm, the error can reach about 50%. This result is consistent with the result of Fan and Yuen;40 for their small size model a double force is needed to produce the required interfacial displacement. It is also consistent with the result of Dandekar and Shin,74,77 for their smallest model of 10×15×4 nm3 the Young’s modulus is about 1.25 times
larger than the testing value for the large model, indicating the small model has high rigidity to produce small deformation. Our work shows that using stress intensity factor K to investigate the model size effects is not sufficient since that value is obtained by a model with infinite size. Changing the model size and comparing the behavior with the LEFM solution will show the size effect quantitatively. This result serves as a serious warning: since many existing simulation models are below this size as given in Figure 23, the accuracy of these models may be questionable and need to be carefully verified.
When the model size increases from 120 nm to 500 nm, the accuracy quickly increases. However, a further increase of the model size from 500 nm to 5000 nm results in basically the same accuracy as the case of 500 nm. This result is significant since it lays a foundation for introduction of a new concept of critical model size, LCR. In fact, the comparison tell us that if the model size is less than LCR, say 500 nm, the results obtained from atomistically-based multiscale simulations will have unrealistic crack-tip behavior, including a large percent of inaccuracy in comparison with the LEFM result. On the other hand, the case for designing the model size larger than LCR should also be avoided since it may not greatly improve the accuracy with the penalty of increasing a large of DOF.
While this finding may open a new avenue to develop a guideline for the least- required model size, LCR, to improve the accuracy in bridging atomistic and continuum scales, more investigation on the feature of LCR should be conducted.
Since LCR is a problem dependent variable, it may relate to material property, environmental conditions, the variables involved, the answer evoked, etc. Thus, it is hard to get a general answer analytically. In many cases, one should carry on numerical simulations for models with different size to find the minimum necessary for the required accuracy. Fortunately, the newly proposed GP-FEA, which can develop large model sizes, has proven so far to be an effective tool to face this challenge. To introduce GP-FEA methods in detail with more examples and their wide applications in crack propagation involving plasticity and failure are prepared to be published elsewhere.
APPLICATION IN CRACK PROPAGATION
The model size was found to affect the deformation field around a pre-crack tip in the last chapter. This chapter takes that work a step further to investigate any size effects for a propagating crack on the energy release rate and atomistic crack-tip phenomena.
A. Introduction
Various work has been done to derive macro-scale traction-separation laws for use in Finite Element Analysis (FEA) from Molecular Dynamics (MD) and multiscale models.
Song et al. studied crack-tip shielding.86 They found that at the point of fracture there is a unique traction displacement cohesive zone law along the fracture independent of the position of the anti-shielding dislocation. However, for crack propagation, the atomistic model shows that, as an anti-shielding dislocation approaches the crack tip, it causes less anti-shielding than predicted by the singular-crack model. If the cohesive strength is reduced then the cohesive-crack model is consistent. The difference is due to the non-linear deformation of material around the crack tip, which cannot be fully represented by a cohesive zone law. Their simulation method used was CADD and they found excellent agreement with the values obtained from independent atomistic calculations on this material. This shows that crack initiation behavior is different than crack propagation behavior, in the sense that they must be modeled independently.
Yamakov et al. studied the inter-granular failure of a Σ99[110] symmetric tilt boundary in Aluminum.68 Under hydrostatic tensile load, the crack propagates brittly in one direction and ductily in the other. This is consistent with Rice's criterion for cleavage vs. dislocations blunting. The preference for twinning over dislocation is consistent with the Tadmor and Hai criterion. Two separate traction separation relations are extracted from MD for brittle and ductile decohesion to be used in higher scale FEA models and coupled to MD. Their group then used the ESCM method to couple MD with FEA with the addition of CZM elements near the MD-FEA boundary based on the cohesion measured in the MD domain. In this way, cracks that nucleate or originate in the MD
domain will be able to propagate into the FEA domain.87 Their coupling scheme is the ESCM method which statistically averages the atomic properties to apply as boundary conditions to the FEA mesh without the mesh being required to be the same size at the atomic lattice constant. Since ESCM provides traction forces to the MD domain that means that the MD boundary has a surface. To compensate for this they partition the local MD surface boundary into domains and provide a correction force that compensates for the different stiffness and surface tension. The statistic nature of ESCM allows for finite temperature of the MD domain.
Choi and Kim used a nanoscale planar field projection of atomic decohesion and slip in crystalline solids based on a new orthogonal eigenfunction expansion of the elastic field around an interfacial cohesive crack.72 The atomistic fields are obtained from molecular statics simulations of decohesion in a gold single crystal along a [11̅̅̅̅2]
direction in a (111) plane, using the EAM potential. The field projection yields the traction and displacement as well as the surface stress of the nascent surface. Thus the traction separation and surface energy gradient can be measured as functions of the cohesive zone displacements. It is shown that there is a nanoscale mechanism of decohesion lattice trapping or hardening caused by the characteristics of non-local atomistic deformations near the crack tip.
Fan and Yuen used hierarchical multiscale analysis for interfacial delamination by using MD to model the chemical phase that bonds the two bulk materials together.40 From MD they use the obtained traction-displacement information to define cohesive zone parameters for larger scale FEA models. Their results predicted the failure force to be about twice as large as the experimental data. They attributed this discrepancy to be due to an increased interaction cross-link across the interface and the presence of voids and impurities inside the real samples. Their approach for this application is advantageous since it avoids the time-scale problem of concurrent multiscale.
Coffman et al. used cohesive laws derived from atomistic simulations for polycrystalline structures.88 They found that the levels of external stress are required to fracture GBs. This indicates that fracture initiation is likely dominated by irregular atomic structures along GBs. Thus the cohesive properties alone are not likely to be
sufficient for modeling the fracture of polycrystals using continuum methods. Their explanation is similar to Fan and Yuen's for their adhesion force discrepancy.
Vatne et al. used the QC method to investigate crack propagation in BCC Iron under different crystallographic orientations in mode-I loading with various T-stresses.89 They found that the mechanisms at the crack tip and the critical stress intensity factor, KI are sensitive to both the crystallographic orientation and whether or not the boundary conditions were isotropic or anisotropic. Due to their small model size their boundary conditions become very important; in their implementation they provide boundary conditions that match the LEFM displacement field. They observed such mechanisms as cleavage, twinning, and dislocation emission.
These techniques illustrate the possibility of coupling lower scale phenomena and behavior to higher scale models.