Elastic Constants Calculator
Choose any two known elastic constants of an isotropic material, enter their values, and this calculator instantly solves for the remaining four: Young's modulus (E), shear modulus (G), bulk modulus (K), Poisson's ratio (nu), Lame's first constant (lambda), and P-wave modulus (M). All 15 valid input pairs are supported, with unit switching between GPa, MPa, Pa, kPa, psi, and ksi.
Formula
Worked example
Steel: E = 200 GPa, nu = 0.29. G = 200 / [2(1+0.29)] = 200 / 2.58 = 77.5 GPa. K = 200 / [3(1-0.58)] = 200 / 1.26 = 158.7 GPa. lambda = 200*0.29 / [(1.29)(0.42)] = 58 / 0.542 = 107 GPa. M = K + 4G/3 = 158.7 + 103.3 = 262 GPa.
What are elastic constants?
Elastic constants describe how a solid material deforms under mechanical loads in the linear-elastic regime, where deformation is fully reversible. For an isotropic, homogeneous material (one whose properties are the same in every direction and at every point) only two independent constants are needed to completely characterise elastic behaviour. All other elastic constants can be derived from those two. The six constants most commonly encountered are: Young's modulus (E), shear modulus (G), bulk modulus (K), Poisson's ratio (nu), Lame's first constant (lambda), and P-wave modulus (M). Different fields prefer different pairs: mechanical engineers typically use E and nu; seismologists work with lambda and G (the Lame parameters); geotechnical engineers often cite K and G.
Young's modulus, shear modulus and bulk modulus
Young's modulus (E) is the ratio of uniaxial tensile or compressive stress to the resulting axial strain. A higher E means a stiffer material. Steel has E around 200 GPa; rubber is below 0.1 GPa. The shear modulus (G) describes resistance to shear deformation - a force applied parallel to a surface. It is related to Young's modulus by G = E / [2(1 + nu)]. The bulk modulus (K) quantifies a material's resistance to uniform volumetric compression under hydrostatic pressure: K = E / [3(1 - 2*nu)]. An incompressible material would have an infinite bulk modulus (nu approaching 0.5). All three have units of pressure (Pa, GPa, psi).
Poisson's ratio, Lame's constant and P-wave modulus
Poisson's ratio (nu) is dimensionless: it is the negative ratio of lateral strain to axial strain during a uniaxial test. For most engineering materials nu lies between 0 and 0.5 (cork is near 0, rubber is near 0.5). Thermodynamic stability requires -1 < nu < 0.5; materials with negative nu are called auxetic. Lame's first constant (lambda) has units of pressure and appears naturally in the constitutive stress-strain tensor: sigma_ij = lambda * delta_ij * e_kk + 2*G*e_ij. It has no simple physical analogue but is convenient in elasticity theory. The P-wave modulus (M), also called the constrained modulus, is M = K + 4G/3. It governs the speed of longitudinal (compressional) waves through a medium and is therefore central to seismology and non-destructive testing.
How to use this calculator and interpret the results
Select any two elastic constants from the dropdowns, enter their values, choose the pressure unit, and all six constants are computed instantly. All 15 valid two-constant input pairs are supported including the more complex pairs that involve Lame's constant or P-wave modulus. For those pairs an intermediate variable R or S is calculated first (shown in the worked steps). The results panel and bar chart let you compare the magnitudes of E, G, K and M side by side. Because Poisson's ratio is dimensionless it does not appear on the modulus chart. A physical constraint check is applied: if the entered pair implies nu outside (-1, 0.5) or a negative modulus, the calculator returns no result and signals an invalid physical state.
Typical elastic constants for common engineering materials
| Material | E (GPa) | G (GPa) | K (GPa) | nu |
|---|---|---|---|---|
| Steel (carbon) | 200 | 79 | 160 | 0.29 |
| Aluminum | 70 | 26 | 76 | 0.33 |
| Copper | 120 | 45 | 140 | 0.34 |
| Titanium | 116 | 44 | 110 | 0.32 |
| Brass | 100 | 37 | 90 | 0.34 |
| Cast iron | 170 | 66 | 100 | 0.26 |
| Glass (borosilicate) | 64 | 26 | 36 | 0.23 |
| Concrete | 30 | 12 | 20 | 0.20 |
| HDPE polymer | 1.0 | 0.37 | 0.67 | 0.36 |
| Natural rubber | 0.01 | 0.003 | 2.0 | 0.49 |
Approximate room-temperature values for isotropic polycrystalline materials. Actual values vary with alloy, heat treatment and grain orientation.
Frequently asked questions
How many elastic constants does an isotropic material have?
Two independent elastic constants completely define the elastic behaviour of an isotropic, homogeneous material under linear conditions. You can choose any pair from E, G, K, nu, lambda, and M, as long as the pair is not redundant. All other constants are uniquely derived from those two via the well-known isotropic elasticity relations.
What is the physical meaning of Poisson's ratio?
Poisson's ratio (nu) is the ratio of the transverse strain to the axial strain when a material is stretched along one axis. If you pull a rubber band along its length, it narrows in the perpendicular directions: that narrowing divided by the elongation gives nu. Most solids have nu between 0.2 and 0.5. A value of 0.5 corresponds to a perfectly incompressible material. Rare auxetic materials have negative Poisson's ratios and expand laterally when stretched.
What is the difference between Lame's constant and shear modulus?
Both lambda and G are Lame parameters that appear in the isotropic elasticity tensor. The shear modulus G has a clear physical meaning (shear stress divided by shear strain). Lame's first constant lambda is an algebraic convenience with no direct single-test analogue. Together (lambda, G) form the classic Lame pair and fully describe isotropic elasticity. Lambda = K - 2G/3 = E*nu / [(1+nu)(1-2nu)].
Why can't Poisson's ratio exceed 0.5?
The upper limit of 0.5 follows from thermodynamic stability: the strain energy of a deformed solid must be positive. If nu were greater than 0.5, a uniaxial compression would expand the volume, which violates energy conservation for a passive material. At exactly nu = 0.5 the bulk modulus K becomes infinite (perfectly incompressible). The lower bound of -1 is similarly set by the requirement that G >= 0.
What units should I use for Young's modulus in FEA?
FEA packages expect E and nu as primary inputs. The unit must be consistent with the rest of your model: if forces are in N and dimensions in mm, use MPa (N/mm^2). If forces are in N and dimensions in m, use Pa. Most structural analyses use GPa or MPa. This calculator lets you switch between Pa, kPa, MPa, GPa, psi and ksi so you can match your FEA input format exactly.
Can this calculator handle rubber and soft tissues?
Yes, with care. Rubber and soft biological tissues behave nearly incompressibly (nu close to 0.5) and have very low Young's moduli (0.001-0.1 GPa). You can enter values in Pa or MPa for these materials. Note that at nu close to 0.5, the bulk modulus K grows very large and small errors in nu cause large changes in K, so high-precision input is needed near the incompressible limit.
What is the P-wave modulus used for?
The P-wave modulus M = K + 4G/3 governs the velocity of compressional (longitudinal) waves: Vp = sqrt(M / density). It is central to seismology, acoustic logging, and ultrasonic non-destructive testing. In a confined compression test (no lateral strain allowed) the measured stiffness is also M, which is why it is sometimes called the constrained modulus or oedometric modulus.