Editing Elastic modulus (section)

== Density functional theory calculation ==
[[Density functional theory]] (DFT) provides reliable methods for determining several forms of elastic moduli that characterise distinct features of a material's reaction to mechanical stresses.Utilize DFT software such as [[Vienna Ab initio Simulation Package|VASP]], [[Quantum ESPRESSO]], or [[ABINIT]]. Overall, conduct tests to ensure that results are independent of computational parameters such as the density of the k-point mesh, the plane-wave cutoff energy, and the size of the simulation cell.

# [[Young's modulus]] (''E'') -  apply small, incremental changes in the lattice parameter along a specific axis and compute the corresponding stress response using DFT. Young's modulus is then calculated as ''E''=''σ''/''ϵ'', where ''σ'' is the stress and ''ϵ'' is the strain.<ref>{{Cite journal |last1=Alasfar |first1=Reema H. |last2=Ahzi |first2=Said |last3=Barth |first3=Nicolas |last4=Kochkodan |first4=Viktor |last5=Khraisheh |first5=Marwan |last6=Koç |first6=Muammer |date=2022-01-18 |title=A Review on the Modeling of the Elastic Modulus and Yield Stress of Polymers and Polymer Nanocomposites: Effect of Temperature, Loading Rate and Porosity |journal=Polymers |language=en |volume=14 |issue=3 |pages=360 |doi=10.3390/polym14030360 |doi-access=free |issn=2073-4360 |pmc=8838186 |pmid=35160350}}</ref>
## Initial structure:  Start with a relaxed structure of the material. All atoms should be in a state of minimum energy (i.e., minimum energy state with zero forces on atoms) before any deformations are applied.<ref>{{Cite journal |last1=Hadi |first1=M. A. |last2=Christopoulos |first2=S.-R. G. |last3=Chroneos |first3=A. |last4=Naqib |first4=S. H. |last5=Islam |first5=A. K. M. A. |date=2022-08-18 |title=DFT insights into the electronic structure, mechanical behaviour, lattice dynamics and defect processes in the first Sc-based MAX phase Sc2SnC |journal=Scientific Reports |language=en |volume=12 |issue=1 |page=14037 |doi=10.1038/s41598-022-18336-z |issn=2045-2322 |pmc=9388654 |pmid=35982080}}</ref>
## Incremental uniaxial strain: Apply small, incremental strains to the [[Crystal structure|crystal lattice]] along a particular axis. This strain is usually [[uniaxial]], meaning it stretches or compresses the lattice in one direction while keeping other dimensions constant or periodic.
## Calculate stresses: For each strained configuration, run a DFT calculation to compute the resulting [[Cauchy stress tensor#Stress deviator tensor|stress tensor]]. This involves solving the Kohn-Sham equations to find the ground state electron density and energy under the strained conditions
## [[Stress-strain curve]]: Plot the calculated stress versus the applied strain to create a stress-strain curve. The slope of the initial, linear portion of this curve gives Young's modulus. Mathematically, [[Young's modulus]] ''E'' is calculated using the formula ''E''=''σ''/''ϵ'', where ''σ'' is the stress and ''ϵ'' is the strain.
# [[Shear modulus]] (''G'')
## Initial structure: Start with a relaxed structure of the material. All atoms should be in a state of minimum energy with no [[Residual stress|residual forces]]. (i.e., minimum energy state with zero forces on atoms) before any deformations are applied.
## Shear strain application: Apply small increments of shear strain to the material. [[Strain (mechanics)|Shear strains]] are typically off-diagonal components in the strain tensor, affecting the shape but not the volume of the crystal cell.<ref>{{Cite journal |last1=Ahmed |first1=Razu |last2=Mahamudujjaman |first2=Md |last3=Afzal |first3=Md Asif |last4=Islam |first4=Md Sajidul |last5=Islam |first5=R.S. |last6=Naqib |first6=S.H. |date=May 2023 |title=DFT based comparative analysis of the physical properties of some binary transition metal carbides XC (X = Nb, Ta, Ti) |journal=Journal of Materials Research and Technology |volume=24 |pages=4808–4832 |doi=10.1016/j.jmrt.2023.04.147 |issn=2238-7854|doi-access=free }}</ref>
## Stress calculation: For each configuration with applied [[Strain (mechanics)|shear strain]], perform a DFT calculation to determine the resulting stress tensor.
## [[Shear stress]] vs. [[Shear strain|shear strain curve]]: Plot the calculated shear stress against the applied shear strain for each increment.The slope of the stress-strain curve in its linear region provides the shear modulus, ''G''=''τ''/''γ'', where ''τ'' is the shear stress and ''γ'' is the applied shear strain.
# [[Bulk modulus]] (''K'')
## Initial structure: Start with a relaxed structure of the material. It's crucial that the material is fully optimized, ensuring that any changes in volume are purely due to applied pressure.
## Volume changes: Incrementally change the volume of the [[Crystal structure|crystal cell]], either compressing or expanding it. This is typically done by uniformly scaling the lattice parameters.
## Calculate pressure: For each altered volume, perform a DFT calculation to determine the pressure required to maintain that volume. DFT allows for the calculation of stress tensors which provide a direct measure of the internal pressure.
## [[Pressure-volume curves|Pressure-volume curve]]: Plot the applied pressure against the resulting volume change. The bulk modulus can be calculated from the slope of this curve in the linear elastic region.The bulk modulus is defined as ''K''=−''VdV''/''dP'', where ''V'' is the original volume, ''dP'' is the change in pressure, and ''dV'' is the change in volume.<ref>{{Cite journal |last1=Choudhary |first1=Kamal |last2=Cheon |first2=Gowoon |last3=Reed |first3=Evan |last4=Tavazza |first4=Francesca |date=2018-07-12 |title=Elastic properties of bulk and low-dimensional materials using van der Waals density functional |journal=Physical Review B |language=en |volume=98 |issue=1 |page=014107 |doi=10.1103/PhysRevB.98.014107 |issn=2469-9950 |pmc=7067065 |pmid=32166206|arxiv=1804.01033 |bibcode=2018PhRvB..98a4107C }}</ref>