Molecular dynamics (MD) simulations serve as a powerful computational tool for predicting the mechanical properties of nanomaterials, including graphene, carbon nanotubes (CNTs), and metallic nanoparticles. By solving Newton's equations of motion for atoms interacting via empirical potentials, MD provides atomic-level insights into deformation mechanisms, fracture behavior, and defect dynamics under various loading conditions.

**Strain-Stress Calculation Methods**
In MD simulations, mechanical properties are derived by applying controlled deformations and measuring the system's response. The virial stress theorem is commonly employed to compute atomic-level stresses. For a system of N atoms, the stress tensor σαβ is calculated as:

σαβ = (1/V) [Σi (mi viα viβ) + Σi Σj>i (rijα fijβ)]

where V is the volume, mi is atomic mass, viα is velocity, rijα is the interatomic distance, and fijβ is the force between atoms i and j. Strain is imposed by incrementally scaling atomic positions or simulation box dimensions. Uniaxial tension, for example, involves elongating the simulation cell along one axis while allowing relaxation in transverse directions.

Stress-strain curves are generated by plotting the average stress against applied strain. Key mechanical parameters such as Young's modulus (E), yield strength, and ultimate tensile strength are extracted from these curves. For graphene, MD simulations predict E ≈ 1 TPa, consistent with experimental measurements. CNTs exhibit similar stiffness, with modulus values dependent on chirality and diameter.

**Fracture Propagation Analysis**
MD simulations reveal fracture mechanisms by tracking bond breaking and crack propagation at atomic resolution. In graphene, fracture initiates at pre-existing defects or stress concentrations, propagating anisotropically along zigzag or armchair directions. The critical stress intensity factor (KIC) can be estimated by analyzing energy release rates during crack growth.

For example, simulations show that pristine graphene fails at strains of ≈15-20%, while defective graphene fractures at lower strains. Stone-Wales defects (5-7-7-5 rings) reduce fracture strain by ≈30% due to localized stress concentrations. In CNTs, fracture behavior depends on chirality: armchair nanotubes exhibit brittle fracture, whereas zigzag nanotubes may undergo plastic deformation via bond rotation before failure.

**Defect-Mediated Deformation**
Defects play a crucial role in nanomaterial mechanics. Dislocations, vacancies, and grain boundaries significantly alter mechanical properties. In metallic nanoparticles, dislocation dynamics govern plasticity. MD simulations of face-centered cubic (FCC) metals like gold or nickel reveal that dislocations nucleate at surfaces or pre-existing defects under stress.

For instance, a 10 nm gold nanoparticle subjected to uniaxial compression shows dislocation nucleation at ≈2% strain, followed by propagation and eventual annihilation at free surfaces. The absence of dislocation pile-ups in small nanoparticles leads to size-dependent strength, following the "smaller is stronger" trend. Vacancy defects in graphene reduce stiffness by introducing local softening, while grain boundaries in polycrystalline graphene cause strength reductions proportional to misorientation angles.

**Uniaxial vs. Shear Loading Simulations**
Loading direction influences deformation mechanisms. Uniaxial tension or compression tests are standard for evaluating modulus and strength, while shear simulations reveal slip systems and interfacial behavior.

In graphene, uniaxial tension leads to uniform elongation until bond rupture, whereas shear loading induces rippling or buckling instabilities. Metallic nanoparticles under shear exhibit dislocation glide on preferred slip planes. For example, MD simulations of copper nanoparticles show that shear deformation activates partial dislocations with Shockley partials dominating plasticity.

**Temperature and Strain Rate Effects**
Temperature (T) and strain rate (ε̇) significantly impact mechanical properties. At higher T, thermal vibrations reduce yield strength and modulus due to increased atomic mobility. For graphene, E decreases by ≈10% when T increases from 300 K to 1000 K. Strain rate effects are pronounced in metallic nanoparticles, where higher ε̇ delays dislocation nucleation, increasing apparent strength.

A nickel nanoparticle at ε̇ = 10^8 s^-1 exhibits yield strength ≈1.5 GPa, while at ε̇ = 10^10 s^-1, strength increases to ≈2.2 GPa. This rate sensitivity arises from limited time for thermally activated dislocation processes.

**Dislocation Dynamics in Metallic Nanoparticles**
MD simulations capture dislocation nucleation, propagation, and interaction in nanoscale metals. In FCC nanoparticles, plasticity begins with partial dislocations emitted from surfaces. A 5 nm silver nanoparticle under tension forms stacking faults bounded by partial dislocations, with Burgers vectors of type (a/6)<112>.

Twining is another deformation mode observed in simulations. For example, a 8 nm aluminum nanoparticle compressed at 300 K undergoes twinning via successive emission of partial dislocations on parallel slip planes. The critical resolved shear stress for twinning decreases with increasing nanoparticle size due to reduced surface-to-volume ratio.

**Comparative Analysis of Nanomaterials**
Graphene and CNTs exhibit brittle fracture with minimal plasticity, while metallic nanoparticles show ductile behavior via dislocation activity. Defects have contrasting effects: vacancies in graphene degrade strength more severely than voids in metals, where dislocations can bypass defects. Temperature sensitivity is higher in metals due to thermally activated dislocation motion, whereas graphene's mechanics are less T-dependent below 1000 K.

In summary, MD simulations provide detailed insights into nanomaterial mechanics by capturing atomic-scale processes. Strain-stress relationships, fracture paths, defect interactions, and loading-mode dependencies are accurately predicted, guiding the design of nanomaterials for structural and functional applications.