The GP method and the related GP-FEA method have been introduced as candidates for this gap in capability. It has been seen to have a natural interface between scale domains where physical variables such as displacement and force may smoothly pass from one scale to another. Atomic phenomena such as dislocations may be passed into higher scale domains via the scale-duality concept50 by decomposing particles into their constituent atoms. In addition, it has been mathematically proven48,50 that all calculations in the particle domain can be conducted at the atomistic domain scale using the same potential, parameters and numerical algorithm as is used for the model's atomistic scale. Thus the GP method is essentially an extension of MD and can be easily delivered to applications by modifying existing MD codes.
diameter to keep the error of the solution, e.g., (𝜎𝜃)𝑚𝑎𝑥 from exceeding 6 per cent. Our design satisfies this requirement. However, model size effects are problem-dependent. It may relate to material property, environmental conditions, the variables involved, etc. Thus, it is difficult 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 accuracy.
The satisfactory agreement between the displacement data obtained by the proposed GP-FEA methods with the classical analytical solution establishes a foundation to use these multiscale methods to investigate model size effects.
Encouraged by the successful comparison of the displacement field predicted by the GP-FEA method with the continuum solution for a 2D plate with a central cylinder hole, this newly proposed multiscale method has been further developed and applied to the crack-tip analysis. Results show excellent agreement of the simulation results with the LEFM two-term solutions by Rice and others.
This successful comparison with the continuum solution and the powerful capability of the GP-FEA multiscale method in developing a large micron-scale model are promising. It allows for the investigation of model dimension effects on the accuracy of the atomistically-based multiscale method realistic and attractive. The significance of this investigation should be further emphasized even though the model size choice is a common problem that appears frequently during model design. It can be further addressed from the following four aspects. Firstly, accuracy verification for low scales is important to find the deformation mechanisms. Secondly, if the model size is small the BC disturbances may likely affect the local fields of forces and displacements which are near the atomistic regions of interest. In turn, it will change the behavior of highly important domains such as interfaces, crack-tips and flaws. 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/larger than micrometers or at least being sub-micrometers so the problem of micro- or nano-
sensors/activators can be more accurately simulated. 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 are extensively investigated. All models were subjected to the remote BC displacement along the Y-direction [1̅10] with 1% strain. The accuracy is verified by the LEFM two-term solution from the results the following observations and conclusions can be made.
It is seen that the smaller the model size the larger the error produced in the simulation-obtained uy relation. Specifically, for the case of Ly=120 nm, the error can reach about 50%. 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, 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, further increases of the model size from 500 nm to 5000 nm results have 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.
The results of this study show that the size of the model affects the material behavior by influencing the atomistic phenomena at crucial locations as a crack tip. Two effects were found that affect the phase transformation at the crack tip. First, it was seen that smaller models do not have as many atoms in the FCC phase as larger models do for the same loading strain. Second, the smaller models were delayed in their nucleation of the FCC phase.
These effects show that the chosen model size of the simulation can seriously affect atomistic phenomena observed at crack tips. If the researcher is looking to derive critical information from atomistic-based simulations the model size must be carefully chosen.