**6.2. Coarse-Grained Molecular Dynamics (CGMD)**

This approach couples FE and MD methods together. In this method to derive the governing equations of motion, we use an approximation of coarse-grained energy that converges to the exact atomic energy like the following:

$$E\left(\boldsymbol{\mu}\_{k'}\dot{\boldsymbol{\mu}}\_{k}\right) = \boldsymbol{U}\_{\text{int}} + \frac{1}{2} \sum\_{j,k} \left(\boldsymbol{M}\_{jk}\dot{\boldsymbol{\mu}}\_{j}\dot{\boldsymbol{\mu}}\_{k} + \boldsymbol{\mu}\_{j}\boldsymbol{K}\_{jk}\boldsymbol{\mu}\_{k}\right) \tag{11}$$

where


*Uint* represents the thermal energy of those degrees of freedom of coarse grained that have not been included in FE considered nodes. Obviously, when the number of nodes approaches the number of atoms, this term vanished and the full atomistic energy is recovered.

Finally the equation of motion is then given to be as follows:

$$\mathbf{M}\_{i\rangle}\ddot{\boldsymbol{\mu}}\_{\rangle} = -\mathbf{G}\_{ik}^{-1}\boldsymbol{\mu}\_{k} + \int\_{-\alpha}^{t} \boldsymbol{\eta}\_{ik} \left(t - \boldsymbol{\tau}\right) \dot{\boldsymbol{\mu}}\_{k} \left(\boldsymbol{\tau}\right) d\boldsymbol{\tau} + F\_{i}\left(t\right) \tag{12}$$

where


#### **6.3. Quasi-Continuum Method**

This approach is an adaptive FE method. The link between atomistic and continuum is obtained by the use of the Cauchy Born rule. The Cauchy Born rule assumes that the continuum energy density *W* can be computed using an atomistic potential.

The first Piola-Kirchoff stress tensor in the Cauchy Born rule is

$$\mathbf{P} = \frac{\partial \mathbf{W}}{\partial \mathbf{F}^T} \tag{13}$$

and the Lagrangian tangent stiffness is:

There are two problems in this method. First, having finite elements in the scale of atomic space prolongs the process of solving by increasing time steps. Second, that it seems unphysical that

This approach couples FE and MD methods together. In this method to derive the governing equations of motion, we use an approximation of coarse-grained energy that converges to the

> ( ) ( ) , <sup>1</sup> , . .. <sup>2</sup> *k k int jk j k j jk k j k*

*Uint* represents the thermal energy of those degrees of freedom of coarse grained that have not been included in FE considered nodes. Obviously, when the number of nodes approaches the

( ) ( ) ( ) <sup>1</sup>

 tt

=- + - + ò && & (12)

 t

This approach is an adaptive FE method. The link between atomistic and continuum is obtained by the use of the Cauchy Born rule. The Cauchy Born rule assumes that the continuum

*t u d F tu* -

number of atoms, this term vanished and the full atomistic energy is recovered.

*t Mij j ik k ik G <sup>k</sup> <sup>i</sup> u* h


*E U u u*& =+ + å *Muu uK u* & & (11)

continuum relations can be evolved at the same timescales as the atomistic variables.

**6.2. Coarse-Grained Molecular Dynamics (CGMD)**

exact atomic energy like the following:

14 Graphene - New Trends and Developments

*Uint* =3(*N* − *Nnode*)*kT* Internal energy

.*u***˙** *<sup>k</sup>* Kinetic energy

*u*, *u***˙** Displacement degree of freedom and the velocity

Finally the equation of motion is then given to be as follows:

energy density *W* can be computed using an atomistic potential.

.*K jk* .*u<sup>k</sup>* Potential energy

where

*M jku***˙** *<sup>j</sup>*

where

*Fi*

*Mij* Mass matrix

*Gij* Stiffness like quantity

(*t*) Random force

*ηij* Time history of memory function

**6.3. Quasi-Continuum Method**

*u j*

$$\mathbf{C} = \frac{\hat{\boldsymbol{\sigma}}^2 \mathbf{W}}{\hat{\boldsymbol{\sigma}} \mathbf{F}^T \hat{\boldsymbol{\sigma}} \mathbf{F}^T} \tag{14}$$

where *F* is deformation gradient.

The major restriction as well as implication of the Cauchy Born rule is that the deformation of the lattice underlying a continuum point must be homogeneous. This results from the fact that the underlying atomistic system is forced to deform according to the continuum deformation gradient *F*.

#### **6.4. Coupled Atomistics And Discrete Dislocation (CADD)**

This method sets a quasi-static coupling between molecular statics and discrete dislocation plasticity. One of the best assumptions in this approach is that defects within atomistic region are allowed to pass through the atomistic/continuum border into the continuum which is modeled via discrete dislocation mechanics.

Quantities such as stresses, strains, and displacements can be written in the contribution form:

$$A = \tilde{A} + \hat{A} \tag{15}$$

where *A* is a typical quantity, *A*˜ is the contribution from the discrete dislocation, and *<sup>A</sup>* ^ is a correction term we introduce because of the fact that the discrete dislocation solution is for an infinite medium. The continuum energy can be written as

$$E^c = \frac{1}{2} \int\_{\Omega\_c} \left(\hat{\mathfrak{u}} + \mathfrak{H}\right) : \left(\hat{\mathfrak{s}} + \mathfrak{S}\right)dV - \int\_{d\Omega\_\Gamma} T\_0 \left(\hat{\mathfrak{u}} + \mathfrak{H}\right)dA\tag{16}$$

where *T*0 is the traction on the continuum boundary *d*Ω*Τ*.

The equilibrium condition is

$$\frac{\partial E^{\circ}}{\partial \tilde{\mathfrak{u}}\_{\circ}} = 0 \tag{17}$$

Where *u***˜** is displacement field. Using this equation, we could find displacement fields. By the use of displacement field we are able to find the forces exerted on the discrete dislocation as

$$\mathbf{P}^{l} = -\frac{\partial E^{c}}{\partial \mathbf{d}^{l}} \tag{18}$$

At this point, an iterative procedure involving the discrete dislocation positions, FE positions and atomic positions is solved until all degrees of freedom are at equilibrium.

However, we have some challenge in this method as follows:


#### **6.5. Bridging Domain**

Bridging Domain is a dynamic multiple scale method that couples MD with continuum. In this method the boundary that overlays the MD region and surrounding continuum region has varying size, termed the bridging domain.

Within overlaying bridging domain, the Hamiltonian is defined as a linear combination of the molecular and continuum Hamiltonians:

$$H = \left(1 - a\right) \mathbf{H}^{\mathcal{M}} + a H^{\mathcal{C}} \tag{19}$$

where *α* acts to scale the contribution of each domain to the total Hamiltonian.

To make compatibility between the molecular and the continuum regions, we involve Lagrange multipliers in the overlap region as

$$\mathbf{g}\_I = \left(\sum\_J \mathbf{N}\_J \left(\mathbf{X}\_J\right) u\_{iJ} - d\_{iI}\right) = 0\tag{20}$$

Where *gI* are the Lagrange Multipliers, *uiJ* are the FE nodal displacements, and *diJ* are the MD displacements.

The coupled equations of motion will be as follows:

$$
\lambda \overleftarrow{\mathbf{M}}\_I \ddot{\mathbf{u}}\_I = \mathbf{F}\_I^{ext} - \mathbf{F}\_I^{int} - \mathbf{F}\_I^L \tag{21}
$$

$$
\overrightarrow{m}\_{\text{I}}\overrightarrow{\mathbf{d}}\_{\text{I}} = \mathbf{f}\_{\text{I}}^{\text{ext}} - \mathbf{f}\_{\text{I}}^{\text{int}} - \mathbf{f}\_{\text{I}}^{\text{L}} \tag{22}
$$

Where the standard equations are augmented by the Lagrange multiplier-based constraint forces *FI <sup>L</sup>* and *<sup>f</sup> <sup>I</sup> <sup>L</sup>* . The bar symbols overlaying the FE and MD mass matrices indicate that they need to be modified within the overlapping region.
