Free Elastic Constants Calculator

Elastic Constants Converter: Your Tool for Modulus of Elasticity Conversion

When working with engineering or materials science, you'll often need to convert between different elastic moduli. The Elastic Constants Calculator is designed to handle exactly this—allowing you to quickly find Young's modulus from shear modulus, bulk modulus from Poisson's ratio, or any other required combination, provided the material is isotropic and homogeneous. With this tool, you can seamlessly perform modulus of elasticity conversion without manually looking up formulas.

Before diving into the relationships, a quick recap of what elastic moduli represent. The modulus of elasticity describes how a material deforms under stress; it is simply the ratio of stress to strain. However, because materials can respond differently depending on how force is applied (tension, compression, shear, or uniform pressure), a single value cannot cover all scenarios. This is why several distinct elastic constants exist, each capturing a specific type of deformation.

The Six Elastic Constants

1. Young's Modulus ($E$) – Measures resistance to axial tension or compression. A high $E$ indicates a stiff material that requires substantial force to elongate or shorten.
2. Bulk Modulus ($K$) – Quantifies resistance to uniform compression. Large $K$ means the material is hard to squeeze into a smaller volume.
3. Shear Modulus ($G$) – Also called the modulus of rigidity or Lamé's second parameter, it reflects how a material withstands shearing forces.
4. Poisson's Ratio ($\nu$) – Although not strictly a modulus, it describes the lateral deformation that occurs when a material is stretched or compressed. It is essential for relating the other constants.
5. Lamé Constant ($\lambda$) – The first Lamé parameter links normal strains to normal stress. It has no direct physical interpretation but becomes identical to the bulk modulus when $G = 0$ (e.g., in a fluid).
6. P-wave Modulus ($M$) – Also known as the longitudinal or constrained modulus. It governs the propagation of pressure waves in solids and is especially relevant in geophysics and seismology. If $\nu = 0$, the P-wave modulus equals Young's modulus.

Why Homogeneity and Isotropy Matter

The elastic constants can be related only under certain conditions. Our calculator assumes the material is isotropic (properties identical in all directions) and homogeneous (uniform composition throughout). These assumptions simplify Hooke's law enough that just two independent constants can determine all six. For anisotropic or inhomogeneous materials, the relationships shown below do not apply.

Conversion Formulas for Isotropic Materials

Below are tabulated relationships linking the six constants. The first table corresponds to two-dimensional problems (plane stress or plane strain, where the out‑of‑plane dimension is ignored), while the second is for three-dimensional problems.

2D Elastic Constant Relationships

Known Pair	$E$	$K$	$G$	$\nu$	$\lambda$	$M$
$(K, E)$	$E$	$K$	$\dfrac{KE}{4K-E}$	$\dfrac{2K-E}{2K}$	$\dfrac{2K(2K-E)}{4K-E}$	$\dfrac{4K^2}{4K-E}$
$(K, \lambda)$	$\dfrac{4K(K-\lambda)}{2K-\lambda}$	$K$	$K-\lambda$	$\dfrac{\lambda}{2K-\lambda}$	$\lambda$	$2K-\lambda$
$(K, G)$	$\dfrac{4KG}{K+G}$	$K$	$G$	$\dfrac{K-G}{K+G}$	$K-G$	$K+G$
$(K, \nu)$	$2K(1-\nu)$	$K$	$\dfrac{K(1-\nu)}{1+\nu}$	$\nu$	$\dfrac{2K\nu}{1+\nu}$	$\dfrac{2K}{1+\nu}$
$(E, G)$	$E$	$\dfrac{EG}{4G-E}$	$G$	$\dfrac{E}{2G}-1$	$\dfrac{2G(E-2G)}{4G-E}$	$\dfrac{4G^2}{4G-E}$
$(E, \nu)$	$E$	$\dfrac{E}{2(1-\nu)}$	$\dfrac{E}{2(1+\nu)}$	$\nu$	$\dfrac{E\nu}{(1+\nu)(1-\nu)}$	$\dfrac{E}{(1+\nu)(1-\nu)}$
$(\lambda, G)$	$\dfrac{4G(\lambda+G)}{\lambda+2G}$	$\lambda+G$	$G$	$\dfrac{\lambda}{\lambda+2G}$	$\lambda$	$\lambda+2G$
$(\lambda, \nu)$	$\dfrac{\lambda(1+\nu)(1-\nu)}{\nu}$	$\dfrac{\lambda(1+\nu)}{2\nu}$	$\dfrac{\lambda(1-\nu)}{2\nu}$	$\nu$	$\lambda$	$\dfrac{\lambda}{\nu}$
$(G, \nu)$	$2G(1+\nu)$	$\dfrac{G(1+\nu)}{1-\nu}$	$G$	$\nu$	$\dfrac{2G\nu}{1-\nu}$	$\dfrac{2G}{1-\nu}$
$(G, M)$	$\dfrac{4G(M-G)}{M}$	$M-G$	$G$	$\dfrac{M-2G}{M}$	$M-2G$	$M$

Note: When the z‑coordinate is neglected, the formulas above change and certain pairs cannot be used as starting points: $(K, M)$, $(E, \lambda)$, $(E, M)$, $(\nu, \lambda)$, and $(\lambda, M)$.

3D Elastic Constant Relationships

Known Pair	$E$	$K$	$G$	$\nu$	$\lambda$	$M$
$(K, E)$	$E$	$K$	$\dfrac{3KE}{9K-E}$	$\dfrac{3K-E}{6K}$	$\dfrac{3K(3K-E)}{9K-E}$	$\dfrac{3K(3K+E)}{9K-E}$
$(K, \lambda)$	$\dfrac{9K(K-\lambda)}{3K-\lambda}$	$K$	$\dfrac{3(K-\lambda)}{2}$	$\dfrac{\lambda}{3K-\lambda}$	$\lambda$	$3K-2\lambda$
$(K, G)$	$\dfrac{9KG}{3K+G}$	$K$	$G$	$\dfrac{3K-2G}{2(3K+G)}$	$K-\dfrac{2G}{3}$	$K+\dfrac{4G}{3}$
$(K, \nu)$	$3K(1-2\nu)$	$K$	$\dfrac{3K(1-2\nu)}{2(1+\nu)}$	$\nu$	$\dfrac{3K\nu}{1+\nu}$	$\dfrac{3K(1-\nu)}{1+\nu}$
$(K, M)$	$\dfrac{9K(M-K)}{3K+M}$	$K$	$\dfrac{3(M-K)}{4}$	$\dfrac{3K-M}{3K+M}$	$\dfrac{3K-M}{2}$	$M$
$(E, \lambda)$	$E$	$\dfrac{E+2\lambda+R}{6}$	$\dfrac{E-3\lambda+R}{4}$	$\dfrac{2\lambda}{E+\lambda+R}$	$\lambda$	$\dfrac{E-\lambda+R}{2}$
$(E, G)$	$E$	$\dfrac{EG}{3(3G-E)}$	$G$	$\dfrac{E}{2G}-1$	$\dfrac{G(E-2G)}{3G-E}$	$\dfrac{G(4G-E)}{3G-E}$
$(E, \nu)$	$E$	$\dfrac{E}{2(1-2\nu)}$	$\dfrac{E}{2(1+\nu)}$	$\nu$	$\dfrac{E\nu}{(1+\nu)(1-2\nu)}$	$\dfrac{E(1-\nu)}{(1+\nu)(1-2\nu)}$
$(E, M)$	$E$	$\dfrac{3M-E+S}{6}$	$\dfrac{3M+E-S}{8}$	$\dfrac{E-M+S}{4M}$	$\dfrac{M-E+S}{4}$	$M$
$(\lambda, G)$	$\dfrac{G(3\lambda+2G)}{\lambda+G}$	$\lambda+\dfrac{2G}{3}$	$G$	$\dfrac{\lambda}{2(\lambda+G)}$	$\lambda$	$\lambda+2G$
$(\lambda, \nu)$	$\dfrac{\lambda(1+\nu)(1-2\nu)}{\nu}$	$\dfrac{\lambda(1+\nu)}{3\nu}$	$\dfrac{\lambda(1-2\nu)}{2\nu}$	$\nu$	$\lambda$	$\dfrac{\lambda(1-\nu)}{\nu}$
$(\lambda, M)$	$\dfrac{(M-\lambda)(M+2\lambda)}{M+\lambda}$	$\dfrac{M+2\lambda}{3}$	$\dfrac{M-\lambda}{2}$	$\dfrac{\lambda}{M+\lambda}$	$\lambda$	$M$
$(G, \nu)$	$2G(1+\nu)$	$\dfrac{2G(1+\nu)}{3(1-2\nu)}$	$G$	$\nu$	$\dfrac{2G\nu}{1-2\nu}$	$\dfrac{2G(1-\nu)}{1-2\nu}$
$(G, M)$	$\dfrac{G(3M-4G)}{M-G}$	$M-\dfrac{4G}{3}$	$G$	$\dfrac{M-2G}{2M-2G}$	$M-2G$	$M$
$(\nu, M)$	$\dfrac{M(1+\nu)(1-2\nu)}{1-\nu}$	$\dfrac{M(1+\nu)}{3(1-\nu)}$	$\dfrac{M(1-2\nu)}{2(1-\nu)}$	$\nu$	$\dfrac{M\nu}{1-\nu}$	$M$

Two auxiliary symbols, $R$ and $S$, appear in some 3D formulas. The sign of the term containing $S$ depends on the Poisson's ratio of the material: a plus sign is used for most materials ($\nu \ge 0$), and a minus sign for auxetic materials ($\nu < 0$).

Expected Values and Limitations

Based on energy considerations, the physically admissible ranges are:

$$E \ge 0,\quad K \ge 0,\quad G \ge 0,\quad -1 \le \nu \le 0.5$$

In practice, however, Poisson's ratio never exactly reaches $-1$ or $0.5$; the true physically realizable interval is $-1 < \nu < 0.5$. The calculator may display values at the boundaries only because of rounding.

All elastic moduli are expressed in units of pressure: pascals ($\text{Pa} = \text{N} \cdot \text{m}^{-2}$) in the SI system, or pounds per square inch ($\text{psi}$) in imperial units. Poisson's ratio is dimensionless.

Whether you are computing the bulk modulus from Poisson's ratio, converting between Lamé constants, or determining the P-wave modulus, the Elastic Constants Calculator gives you instant, accurate results—no manual formula lookup required.