# Elastic Constants Calculator

Created by Michael Darcy
Reviewed by Steven Wooding
Based on research by
Gassman F On elasticity of porous media Vierteljahresschrift der Naturforschenden Gesellschaft (1951)See 1 more source
Timoshenko SP, Goodier JN Theory of Elasticity McGraw-Hill Book Company, Inc. (1934)
Last updated: Jun 26, 2023

If you're interested in engineering, materials science, or rock physics, you may find our elastic constants calculator helpful. If your material is isotropic and homogenous, like a sample of aluminum, it can assist you in modulus of elasticity conversion.

For those who would like to learn more or feel confused by this description, read on! We'll go through the modulus of elasticity definition to understand its usefulness. Then, we'll look at the elastic constants and their relationships and try to understand why they wouldn't hold for all types of materials.

## What does the modulus of elasticity tell us?

The modulus of elasticity quantifies the behavior of a material when it's subjected to stress. In other words, it gives you an idea of how much the material will deform depending on the load. It's defined as the ratio of stress to strain.

Now, you may wonder, "How can a single quantity be suitable for all cases?" For example, think of a bean bag. If you sit on it, you put some load on it, and it deforms. However, if you try hugging it tightly and squeezing it from multiple sides, it won't change the same way.

That's right — capturing all of these scenarios with one formula for the modulus of elasticity would be difficult. This is why several elastic moduli exist, each of which we'll explore in-depth below.

## Types of elastic moduli

Although you can probably think of countless different methods of deforming a bean bag now, luckily, this doesn't mean there are as many quantities to match all of them. In fact, there are six types of elastic moduli. Some are more commonly used than others, but you can convert between all of them using the elastic constants calculator.

So, what are the options exactly? These are the quantities that fall under the elastic constant definition:

• Young's modulus ($E$) — The elastic constant definition for this quantity tells us that it describes the response to normal stress, i.e., stretching along a given axis. In other words, the greater Young's modulus is, the more stress is needed to lengthen (or compress) an object. For a more detailed explanation, check the Young's modulus calculator.

• Bulk modulus ($K$) — Sometimes also denoted as $B$, it reflects the material's resistance to uniform compression (or volume change). If the bulk modulus is large, the object is hard to compress. More details are in the bulk modulus calculator.

• Shear modulus ($G$) — Also denoted as $S$ or $\mu$ and referred to as the modulus of rigidity or Lamé's second parameter. It informs us about the resistance to a shearing deformation. It's a bit easier to visualize with the graphics from the shear modulus calculator.

• Poisson's ratio ($\nu$) — This variable quantifies the Poisson effect, which is the deformation of a material perpendicular to the direction of loading. For example, how much the width of a material will change if you press it at the top. Although it technically doesn't fit the modulus of elasticity definition, it's customary to use it alongside other parameters on the list. Feel free to check the Poisson's ratio calculator for more.

• Lamé constant ($\lambda$) — It relates normal axial and lateral strains to uniaxial normal stress. Although it doesn't have a direct physical interpretation, $\lambda$ becomes the bulk modulus in a liquid (where $G = 0$), so you could think of it as an adjusted version of $K$.

• P-wave modulus ($M$) — Also known as the longitudinal modulus or the constrained modulus. It's a measure of longitudinal elasticity; in other words, it describes behavior under load in a uniaxial state. The constraint is that there is no possibility of lateral deformation — which is why $M$ is often used in interactions between soil and seismic waves.

🙋 If Poisson's ratio is zero, then the P-wave modulus is equal to Young's modulus.

## Understanding the elastic constants and their relationships

Now that you know the modulus of elasticity definition and the types of elastic moduli, it's only natural to wonder how they could be related. It's usually challenging to determine, but there's a particular case we can consider: an isotropic and homogenous material. We're assuming it to be the case in our elastic constants calculator.

As these two terms are often confusing, let us explain them briefly.

Homogenous refers to the structure of the material. It means that a substance is uniform, and its properties are the same at each point within it; in other words, they don't depend on the spatial position.

Isotropy tells us that the material's properties are identical in all directions. Think that you're tearing a sheet of paper. If you do it along the fiber orientation, it's pretty easy. However, trying to repeat it along the shorter side is more challenging. Therefore, paper isn't isotropic.

Both of these properties are scale-dependent. For example, engineers often consider samples of steel or aluminum as isotropic and homogenous. This isn't true at the atomic scale but it's a good enough approximation for a vast number of atoms.

The power of this approximation is that it allows us to relate the stress and strain of an elastic material by Hooke's law. As a result, we only need to know two independent elastic constants and their relationships with the remaining quantities to find all moduli of elasticity.

## Modulus of elasticity conversion — table of elastic moduli

The formula for the modulus of elasticity (each one you want to determine) depends on what variables are known to you. To make it easier to navigate, you can find them in the tables of elastic moduli below.

2D relationships between elastic constants.

Known variables

Elastic constants

E

K

G

ν

λ

M

(K, E)

$E$

$K$

$\frac{KE}{4K-E}$

$\frac{2K-E}{2K}$

$\frac{2K(2K-E)}{4K-E}$

$\frac{4K^2}{4K-E}$

(K, λ)

$\frac{4K(K-\lambda)}{2K- \lambda}$

$K$

$K- \lambda$

$\frac{\lambda}{2K- \lambda}$

$\lambda$

$2K- \lambda$

(K, G)

$\frac{4KG}{K+G}$

$K$

$G$

$\frac{K-G}{K+G}$

$K-G$

$K+G$

(K, ν)

$2K(1-\nu)$

$K$

$\frac{K(1-\nu)}{1+\nu}$

$\nu$

$\frac{2K \nu}{1+\nu}$

$\frac{2K}{1+\nu}$

(E, G)

$E$

$\frac{EG}{4G-E}$

$G$

$\frac{E}{2G}-1$

$\frac{2G(E-2G)}{4G-E}$

$\frac{4G^2}{4G-E}$

(E, ν)

$E$

$\frac{E}{2(1-\nu)}$

$\frac{E}{2(1+\nu)}$

$\nu$

$\frac{E\nu}{(1+\nu)(1-\nu)}$

$\frac{E}{(1+\nu)(1-\nu)}$

(λ, G)

$\frac{4G(\lambda+G)}{\lambda+2G}$

$\lambda +G$

$G$

$\frac{\lambda}{\lambda +2G}$

$\lambda$

$\lambda +2G$

(λ, ν)

$\frac{\lambda(1+\nu)(1-\nu)}{\nu}$

$\frac{\lambda(1+\nu)}{2\nu}$

$\frac{\lambda(1-\nu)}{2\nu}$

$\nu$

$\lambda$

$\frac{\lambda}{\nu}$

(G, ν)

$2G(1+\nu)$

$\frac{G(1+\nu)}{1-\nu}$

$G$

$\nu$

$\frac{2G\nu}{1-\nu}$

$\frac{2G}{1-\nu}$

(G, M)

$\frac{4G(M-G)}{M}$

$M-G$

$G$

$\frac{M-2G}{M}$

$M-2G$

$M$

The table above specifies the elastic constants and their relationships for a 2D problem. Neglecting the z-coordinate changes the formulae and, as a result, you can't choose some pairs as your starting point: $\small (K, M), (E, \lambda), (E, M), (\nu, \lambda), (\lambda, M)$.

3D relationships between elastic constants.

Known variables

Elastic constants

E

K

G

ν

λ

M

(K, E)

$E$

$K$

$\frac{3KE}{9K-E}$

$\frac{3K-E}{6K}$

$\frac{3K(3K-E)}{9K-E}$

$\frac{3K(3K+E)}{9K-E}$

(K, λ)

$\frac{9K(K-\lambda)}{3K - \lambda}$

$K$

$\frac{3(K-\lambda)}{2}$

$\frac{\lambda}{3K-\lambda}$

$\lambda$

$3K-2\lambda$

(K, G)

$\frac{9KG}{3K+G}$

$K$

$G$

$\frac{3K-2G}{2(3K+G)}$

$K - \frac{2G}{3}$

$K+ \frac{4G}{3}$

(K, ν)

$3K(1-2\nu)$

$K$

$\frac{3K(1-2\nu)}{2(1+\nu)}$

$\nu$

$\frac{3K\nu}{1+\nu}$

$\frac{3K(1-\nu)}{1+\nu}$

(K, M)

$\frac{9K(M-K)}{3K+M}$

$K$

$\frac{3(M-K)}{4}$

$\frac{3K-M}{3K+M}$

$\frac{3K-M}{2}$

$M$

(E, λ)

$E$

$\frac{E+2\lambda +R}{6}$

$\frac{E-3\lambda +R}{4}$

$\frac{2\lambda}{E+\lambda +R}$

$\lambda$

$\frac{E-\lambda +R}{2}$

(E, G)

$E$

$\frac{EG}{3(3G - E)}$

$G$

$\frac{E}{2G}-1$

$\frac{G(E-2G)}{3G-E}$

$\frac{G(4G-E)}{3G-E}$

(E, ν)

$E$

$\frac{E}{2(1-2\nu)}$

$\frac{E}{2(1+\nu)}$

$\nu$

$\frac{E\nu}{(1+\nu)(1-2\nu)}$

$\frac{E(1-\nu)}{(1+\nu)(1-2\nu)}$

(E, M)

$E$

$\frac{3M-E+S}{6}$

$\frac{3M+E-S}{8}$

$\frac{E-M+S}{4M}$

$\frac{M-E+S}{4}$

$M$

(λ, G)

$\frac{G(3\lambda + 2G)}{\lambda + G}$

$\lambda + \frac{2G}{3}$

$G$

$\frac{\lambda}{2(\lambda +G)}$

$\lambda$

$\lambda +2G$

(λ, ν)

$\frac{\lambda(1+\nu)(1-2\nu)}{\nu}$

$\frac{\lambda (1+\nu)}{3\nu}$

$\frac{\lambda(1-2\nu)}{2\nu}$

$\nu$

$\lambda$

$\frac{\lambda (1-\nu)}{\nu}$

(λ, M)

$\frac{(M-\lambda)(M+2\lambda)}{M+\lambda}$

$\frac{M+2\lambda}{3}$

$\frac{M-\lambda}{2}$

$\frac{\lambda}{M+\lambda}$

$\lambda$

$M$

(G, ν)

$2G(1+\nu)$

$\frac{2G(1+\nu)}{3(1-2\nu)}$

$G$

$\nu$

$\frac{2G\nu}{1-2\nu}$

$\frac{2G(1-\nu)}{1-2\nu}$

(G, M)

$\frac{G(3M-4G)}{M-G}$

$M- \frac{4G}{3}$

$G$

$\frac{M-2G}{2M-2G}$

$M-2G$

$M$

(ν, M)

$\frac{M(1+\nu)(1-2\nu)}{1-\nu}$

$\frac{M(1+\nu)}{3(1-\nu)}$

$\frac{M(1-2\nu)}{2(1-\nu)}$

$\nu$

$\frac{M\nu}{1-\nu}$

$M$

Notice that in the table above, we have two new variables:

$R = \sqrt{E^2 + 9\lambda ^2 + 2E\lambda}$
$S = \pm \sqrt{E^2 + 9M^2 - 10EM}$

The sign of $S$ depends on the type of material:

• Plus sign: This is the case for most materials as it gives $\nu \geq 0$.
• Minus sign: It's applicable for substances whose $\nu \leq 0$. Negative Poisson's ratio is found in .

Of course, if you want to save yourself the hassle of looking up and applying the relevant formula, our elastic constants calculator can help!