**4. Lattice Mechanics**

Lattice mechanics is an approach to utilize natural symmetries for classical particle mechanics in materials that repetitive or regular atomic structure such as polymers or crystalline mate‐ rials.

This approach is based on two principles:

Principle 1: Mathematical models such as functions, matrices, operators, and so on are invariant with respect to the symmetry of lattice.

Principle 2: symmetry assumption causes loading is symmetry, and thus we will have symmetry effect.

In this approach, we use the concept of "Unit Cell" instead of particles itself. Unit Cell is an arbitrary part of a lattice, in which we could gain the whole lattice by repeating that in a fixed direction and certain distance.

Hence, we can define displacement vector in a lattice in terms of unit cell as follows:

$$
\mu\_n(t) = r\_n(t) - n \tag{4}
$$

where n is the primary position of the unit cell:

*rn*(*t*), is the position of the unit cell at time t, which is composed of the position of all particles in the unit cell at time t; *un*(*t*), is the displacement vector for the unit cell. By calculating the total lattice kinetic energy, we are able to get lattice Lagrangian and consequently the equation of motion as follows:

$$\mathbf{M}\ddot{\mathbf{u}}\_n + \frac{\partial \mathbf{U}}{\partial \mathbf{u}\_n} = f\_n^{ext} \tag{5}$$

where *U* is the total Lattice Potential energy and *f <sup>n</sup> ext* is the external load on the Unit Cell *n*. We absolutely insist that this equation is for each unit cell, that is, with changing n we will have a system of motion differential equations.

Using Taylor series for U results in the following:

$$\mathbf{M}\ddot{\boldsymbol{u}}\_{n}\left(\boldsymbol{t}\right) - \sum\_{\boldsymbol{n}} \mathbf{K}\_{n-\boldsymbol{n}} \boldsymbol{u}\_{\boldsymbol{n}}\left(\boldsymbol{t}\right) = \boldsymbol{f}\_{n}^{\text{ext}}\left(\boldsymbol{t}\right) \tag{6}$$

where

**3. Molecular Dynamics (MD) modeling**

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ion's Coulomb field by means of quantum ab initio methods.

of these equations are as follows:

in a system with only conservative forces.

This approach is based on two principles:

invariant with respect to the symmetry of lattice.

**4. Lattice Mechanics**

rials.

Molecular dynamics is nothing but classical dynamics. Indeed classical dynamic equations of motion are valid for slow and heavy particles, with typical velocities *υ*<<c, (where *c* is the speed of light) and masses *m*>>*m*e, (where *m*e is the electron mass). Therefore, we can use them for atoms, ions, and molecular mass only in slow motion (slower than thermal vibration).

This technique is based on computing potential energy that is typically considered only as a function of the system spatial configuration and is described by means of interatomic poten‐ tials. These potentials are considered as known input information; they are either found experimentally or are computed by averaging over the motion of the valence electrons in the

The main equation that we utilize in this technique is the Lagrange equations of motion. For a system of *N* interacting monoatomic molecules, the Lagrange equation turns Newtonian equations divided into "Dissipative equations" and "Generalized Langevin equations."

Integrals of motion are more functions that are useful for modeling in MD technique. Their notable property is depending only on the initial conditions and staying constant in time. Some

*dt* <sup>=</sup> *<sup>E</sup>* (1)

*dt* <sup>=</sup> *<sup>P</sup>* (2)

*dt* <sup>=</sup> *<sup>M</sup>* (3)

<sup>0</sup> *<sup>d</sup>*

<sup>0</sup> *<sup>d</sup>*

<sup>0</sup> *<sup>d</sup>*

where *E* is the total energy, *P* is the total momentum, and *M* is the total angular momentum

Lattice mechanics is an approach to utilize natural symmetries for classical particle mechanics in materials that repetitive or regular atomic structure such as polymers or crystalline mate‐

Principle 1: Mathematical models such as functions, matrices, operators, and so on are

$$\mathbf{K}\_{\boldsymbol{u}\_{n}\cdot\boldsymbol{u}\_{n}} = -\frac{\partial^{2}\boldsymbol{U}\left(\boldsymbol{u}\right)}{\partial\boldsymbol{u}\_{n}\partial\boldsymbol{u}\_{n}}\big|\_{\boldsymbol{u}\to\boldsymbol{0}}\tag{7}$$

and *n* ' is indices for neighboring cell.

Since we need neighboring cell in forming equations, we must define another concept as "associate cell". Associate cell is the smallest part of a lattice that represents its mechanical properties perfectly. In lattice mechanic we restrict our studies to associate cell which can comprise several unit cells; this is the consequence of principle I discussed above.

To solve the above-mentioned equation of motion, we should use mathematics like the Fourier transform or the Laplace transform.

In terms of *f ext*, we have three types of problems:


$$\sum\_{n} \mathbf{K}\_{n-n} \mathbf{u}\_n \left( t \right) = f\_n^{ext}(t) \tag{8}$$

To solve this equation, we can use green's functions method. One of issue problems in quasistatic approximation is multiscale boundary conditions. We discuss this in a separate clause.
