Elastic modulus

An elastic modulus is a quantity that describes an object's or substance's resistance to being deformed elastically (i.e., non-permanently) when a stress is applied to it.

An elastic modulus is a quantity that describes an object's or substance's resistance to being deformed elastically (i.e., non-permanently) when a stress is applied to it. \delta \ \stackrel{\text{def}}{=}\ \frac {\text{stress}} {\text{strain}} where stress is the force causing the deformation divided by the area to which the force is applied and strain is the ratio of the change in some parameter caused by the deformation to the original value of the parameter. Since strain is a dimensionless quantity, the units of \delta will be the same as the units of stress. 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. 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. Incremental uniaxial strain: Apply small, incremental strains to the 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 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 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. Shear strains are typically off-diagonal components in the strain tensor, affecting the shape but not the volume of the crystal cell. Stress calculation: For each configuration with applied shear strain, perform a DFT calculation to determine the resulting stress tensor. Shear stress vs. 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 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 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.

References

Further reading

Text derived from https://en.wikipedia.org/w/index.php?oldid=1369113300 under CC BY-SA 4.0; modifications have been made.