MAT-33

Bulk modulus K=E/3(1−2ν)

K = E / [3(1−2ν)]. Volumetric stiffness of an isotropic solid.

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ElasticityIsotropic elasticity

Governing equation

K=\dfrac{E}{3(1-2\nu)}

$\nu \in \left[-0.2, 0.49\right]$

where

E: Young modulus (GPa)
\nu: Poisson ratio (—)
K: Bulk modulus (GPa)

The figure, part by part

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①

Governing law — red

K=\dfrac{E}{3(1-2\nu)}

$\nu \in \left[-0.2, 0.49\right]$

$\nu = -0.2$ $K = 47.62$

Along the curve

②

Domain of the variable — blue

\nu_{\min} = -0.2,\quad \nu_{\max} = 0.49

$\nu \in \left[-0.2, 0.49\right]$

$\nu = -0.2$ $\nu = -0.2$

Along the curve

③

Response curve — green

K=\dfrac{E}{3(1-2\nu)}

$\nu \in \left[-0.2, 0.49\right]$

$\nu = -0.2$ $K = 47.62$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}K}{\mathrm{d}\nu}

$\nu \in \left[-0.2, 0.49\right]$

$\nu = -0.196$ $K^{\prime} = 68.88$

Along the curve

Lecture brief

Historical brief

Hooke (1678), Hall–Petch grain size, Arrhenius activation and Paris fatigue-crack growth are the spine of materials selection. The sheets relate stress, microstructure and life. This sheet (MAT-33 — Bulk modulus K=E/3(1−2ν)) is the form associated with Isotropic elasticity. Working symbols: $E$ , $\nu$ $\rightarrow$ $K$ . The isotropic Hooke tensor has two independents. K relates hydrostatic pressure to volume strain ΔV/V.

Purpose

Purpose: compute

K

from

E

\nu

in Materials via

K=\dfrac{E}{3(1-2\nu)}

K = E / [3(1−2ν)]. Volumetric stiffness of an isotropic solid. Use it when a real materials question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

E = 200.000\,\mathrm{GPa}

\nu = 0.300\,\mathrm{—}

, the governing relation

K=\dfrac{E}{3(1-2\nu)}

yields

K = 166.67\,\mathrm{GPa}

. A cube, a hydrostatic squeeze, a volume tick. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Young modulus E200 GPaPoisson ratio \nu0.3 —

Outputs

Bulk modulus K166.67 GPa

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Narration of this film

A cube, a hydrostatic squeeze, a volume tick.

The isotropic Hooke tensor has two independents. K relates hydrostatic pressure to volume strain ΔV/V.

Reading speed

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