An elastic modulus (also known as modulus of elasticity) is the unit of measurement of an object's or substance's resistance to being deformed elastically (i.e., nonpermanently) when a stress is applied to it.
The elastic modulus of an object is defined as the slope of its stress–strain curve in the elastic deformation region:^{[1]} A stiffer material will have a higher elastic modulus. An elastic modulus has the form:
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 will be the same as the units of stress.^{[2]}
Elastic constants are specific parameters that quantify the stiffness of a material in response to applied stresses and are fundamental in defining the elastic properties of materials. These constants form the elements of the stiffness matrix in tensor notation, which relates stress to strain through linear equations in anisotropic materials. Commonly denoted as C_{ijkl}, where i,j,k, and l are the coordinate directions, these constants are essential for understanding how materials deform under various loads.^{[3]}
Specifying how stress and strain are to be measured, including directions, allows for many types of elastic moduli to be defined. The four primary ones are:
Two other elastic moduli are Lamé's first parameter, λ, and Pwave modulus, M, as used in table of modulus comparisons given below references. Homogeneous and isotropic (similar in all directions) materials (solids) have their (linear) elastic properties fully described by two elastic moduli, and one may choose any pair. Given a pair of elastic moduli, all other elastic moduli can be calculated according to formulas in the table below at the end of page.
Inviscid fluids are special in that they cannot support shear stress, meaning that the shear modulus is always zero. This also implies that Young's modulus for this group is always zero.
In some texts, the modulus of elasticity is referred to as the elastic constant, while the inverse quantity is referred to as elastic modulus.
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 VASP, Quantum ESPRESSO, or ABINIT. Overall, conduct tests to ensure that results are independent of computational parameters such as the density of the kpoint mesh, the planewave cutoff energy, and the size of the simulation cell.
Conversion formulae  

Homogeneous isotropic linear elastic materials have their elastic properties uniquely determined by any two moduli among these; thus, given any two, any other of the elastic moduli can be calculated according to these formulas, provided both for 3D materials (first part of the table) and for 2D materials (second part).  
3D formulae  Notes  
There are two valid solutions.  
Cannot be used when  
2D formulae  Notes  
Cannot be used when  
