room temperature. After this separate equilibration the FEA mesh is connected to the GP model for the loading procedure. Here, the FEA mesh is added to the GP hole model such that the entire model size is about 200 nm wide and 500 nm high, nearly an order larger than the GP model. After the geometry has been sketched with the coordinate origin being the same for the GP model and FE mesh, it is best to partition the part around the interface so that it can be seeded to have a uniform element size and shape for both WF and WG domains. Since the element size at the interface is designed to have an edge length equal to the inter-particle cutoff radius, it will create two FE elements overlapping the GP model along the normal direction of the inner FE boundary. It is important to carefully seed the edges of the FE model for nodes by keeping a constant size element at the interface for both WF and WG-FEs then gradually increase the element size away from the interface.
C. Verification of the GP and GP-FEA multiscale methods with Elasticity
This solution was obtained by Prof. G. Kirsch in 1898. While it is derived for an infinitely large plate it has been well confirmed many times by strain measurement and by the photoelastic method for plates of a finite size. In 1907, Timoshenko proved that βIf the width of the plate is not less than four diameters of the hole the error of the solution in calculating (ππ)πππ₯ does not exceed 6 percent.β64 The width of the model with the central hole in this work is 200 nm and the hole diameter is 4 nm, the ratio of the width over the diameter is about fifty, thus the accuracy is reasonable for this comparison.
On the other hand, the stress in the atomistic field is the so-called virial stress defined as
(19)
Here, the symbol β-β under a letter denotes a second order tensor and above a letter denotes a vector. In the first term of the right part in eqn. 19 there are two vectors of velocity, π£Μ ; in the second term, the two vectors are force vector πΜ and the position vector πΜ . The superscripts Ξ± and Ξ² denote the atoms and β¦ -- the volume of the atom at hand.
Eqn. 19 shows the stress at atom Ξ± depends not only on those atoms closest to it, but also on the atoms, Ξ², within its spherical neighborhood of cutoff radius through interatomic forces. This definition shows its nonlocal behavior. The different definition between the stress defined by a continuum and the atomistic field makes direct comparison difficult.
On the other hand, the definition of displacement is clear and unified for both continuum and the atomistic field if a certain volume average can be conducted for the atomistic field. Unfortunately, to the authors' knowledge there is no explicit expression in the public domain for the displacement field around a central hole of a tensile specimen.
These expressions are based on the solution of eqn. 16 to 18. The final results are given as follows:
(20)
(21) These expressions will be used to validate the simulation accuracy of the proposed GP- FEA multiscale analysis.
To make the GP calculation result of any generic point, s, on the plate face comparable with this continuum solution, the average displacement of atoms within a cylinder centered at that point with a small radius and extending through the entire thickness should be used. Here, we use 0.3615 nm for the cylinder radius which includes about 150 atoms on average for S1 and 20 particles for S2 per cylinder. All the data is placed in a file whose strain level is indicated. From that file one extracts the average displacement values from each of the cylinders specified. The obtained average displacements of the atoms/particles will then be compared with the corresponding data calculated from the analytical solution.
2. Result comparison for pure GP and GP-FEA model
The GP-FEA model developed in Section B.1. was loaded along the Y direction with periodic BC in the thickness direction. A barostat was used during loading to reduce the transverse stress average to zero to better match the conditions for the analytical plane stress solutions. Figures 20 to 22 show the angular displacement π’π and radial displacement π’π by both the GP and the GP-FEA method along a circle with radius of 8 nm. The angle ΞΈ sweeps from the 0 degree along the Y-axis to 90 degrees along the X- axis, i.e., varies in the first quadrant. All the data are recorded at the strain of 2% -- 3%
which is in the elastic range for the single iron crystal.
The corresponding result by using the analytical expressions 20 and 21 are also given with the key title of βAnalyticalβ. In the analytical solution, the following elastic constants are used: E=224 GPa and v=0.3. However, it is found that any change of these constants has but a minor change on the results, which is consistent with elasticity theory.
For the GP-FEA simulations, results for two WF design schemes shown, respectively, in Figure 18 and 19 are also given. From the comparison, it is seen that all the atomistically- based multiscale simulations listed in Figures 20 to 22 have sufficient accuracy in comparison with the analytical solution. It should be emphasized that agreement is along a circle with radius of 8 nm which crosses both S1 and S2 regions. The consistency of the simulation in both of the two scales with the analytical solution indicates, implicitly, that the transition between the scale boundary and the interface between the FEA and GP domain is sufficiently smooth, while a detailed direct verification at these boundaries should be done in the next step. The obtained result so far makes one confident to use the
GP and GP-FEA methods for middle size and large size models, respectively, for the sub- microsystem and microsystem. In addition, it is seen that the accuracy of the non- continuous (or separated) WF design shown in Fig. 19 is a little better than the continuous design of Fig. 18. The non-continuous one are distinguished in Figs. 20 to 3-7 with the symbol βGP-FEA model bβ from the triangle of βGP-FEA model aβ. This may offer some clues for further improving the design of the WF and WG domain by developing design software to optimize.
Figure 19. Discontinuous WF domain design and its relation with high-scale particles and the WG domain for the GP-FEA model.
Figure 20. 2% strain displacement comparison of simulation-obtained and formulae- calculated angular displacement π’π and radial displacement π’π along a circle with radius of 8 nm in the first quadrant for a single iron crystal.
Figure 21. 2.5% strain displacement comparison of simulation-obtained and
formulae-calculated angular displacement π’π and radial displacement π’π along a circle with radius of 8 nm in the first quadrant for a single iron crystal.
Figure 22. 3% strain displacement comparison of simulation-obtained and formulae- calculated angular displacement π’π and radial displacement π’π along a circle with radius of 8 nm in the first quadrant for a single iron crystal.