Relate Young's modulus, shear modulus, bulk modulus, and Poisson's ratio. Find any elastic constant from the others with material presets and conversion formulas.
For any isotropic linear elastic material, all elastic behavior can be described by just two independent constants. The four commonly used elastic constants — Young's modulus (E), shear modulus (G), bulk modulus (K), and Poisson's ratio (ν) — are all interrelated, so knowing any two allows you to calculate the other two.
Young's modulus measures stiffness in tension or compression. Shear modulus (also called the modulus of rigidity) measures resistance to shearing deformation. Bulk modulus measures resistance to uniform compression. Poisson's ratio describes how much a material contracts laterally when stretched axially — most materials have values between 0.2 and 0.45, with rubber approaching 0.5 (incompressible) and cork near 0 (no lateral expansion).
This calculator lets you solve for any one of the four elastic constants given two others. It includes material presets for common engineering materials, a visual Poisson's ratio indicator, and complete conversion formula references for all constant pairs.
Converting between elastic constants requires remembering multiple formulas and is prone to algebraic errors. This calculator handles all six two-variable combinations instantly, includes the Lamé parameters and P-wave modulus used in continuum mechanics and seismology, and provides material presets so you can quickly look up and verify elastic properties. Keep these notes focused on your operational context.
Relationships (isotropic materials): E = 2G(1 + ν) = 3K(1 − 2ν) = 9KG/(3K + G) G = E/(2(1 + ν)) = 3K(1 − 2ν)/(2(1 + ν)) K = E/(3(1 − 2ν)) = EG/(3(3G − E)) ν = E/(2G) − 1 = (3K − 2G)/(2(3K + G)) Lamé Parameters: λ = K − 2G/3 μ = G (second Lamé parameter) P-Wave Modulus: M = K + 4G/3
Result: 206.2 GPa
With G = 79.3 GPa and ν = 0.3 (typical steel), Young's modulus is E = 2 × 79.3 × (1 + 0.3) = 206.2 GPa, and bulk modulus is K = 206.2 / (3 × (1 − 0.6)) = 171.8 GPa.
Young's modulus (E), also called the elastic modulus, is the ratio of tensile stress to tensile strain. A high E means the material is stiff — it takes a lot of force to stretch it. Steel has E ≈ 200 GPa while rubber has E ≈ 0.01 GPa.
Shear modulus (G), or modulus of rigidity, measures resistance to shape change under shear stress. It determines how much a beam twists under torque or how a rectangular block deforms into a parallelogram under lateral force.
Bulk modulus (K) measures volumetric stiffness — resistance to uniform compression. Water has K ≈ 2.2 GPa (highly compressible compared to metals), while diamond has K ≈ 440 GPa.
Poisson's ratio (ν) is perhaps the most intuitive: pull on a rubber band and it gets thinner. The ratio of this lateral contraction to the axial extension is Poisson's ratio. It carries information about the internal structure of the material.
Structural engineers primarily use E for beam deflection and column buckling calculations. Mechanical engineers use G for shaft torsion design and bearing analysis. Geophysicists use K and G to interpret seismic wave velocities and infer subsurface composition. Materials scientists use the relationships between constants to validate experimental measurements — if measured E, G, and ν are inconsistent, it suggests measurement error.
Real materials are often anisotropic to some degree. Wood is much stiffer along the grain than across it. Carbon fiber composites can have E ratios of 10:1 between fiber and transverse directions. For such materials, the full elastic stiffness tensor with up to 21 independent components is needed, and the simple two-constant relationships no longer apply.
For isotropic (direction-independent) linear elastic materials, the stress-strain relationship is fully described by two independent parameters. All other elastic constants can be derived from any pair.
Most metals are 0.25–0.35. Rubber is nearly 0.5 (incompressible). Cork is near 0. Auxetic materials have negative Poisson's ratio (they expand laterally when stretched).
Young's modulus (E) is the most commonly used elastic constant in structural engineering. It determines deflection under load, natural frequencies, and critical buckling loads.
Bulk modulus is important for hydrostatic loading (uniform pressure from all directions), such as deep-sea or geological applications. It equals the inverse of compressibility.
Only for isotropic, linearly elastic materials. Anisotropic materials (wood, composites, crystals) require up to 21 independent elastic constants.
Lamé's first parameter (λ) and second parameter (μ = G) appear in the generalized Hooke's law tensor form used in continuum mechanics and finite element analysis. Use this as a practical reminder before finalizing the result.