MAT-34

Shear modulus G=E/2(1+ν)

G = E / [2(1+ν)]. The isotropic relation between E, G and ν.

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ElasticityIsotropic elasticity

Governing equation

G=\dfrac{E}{2(1+\nu)}

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

where

E: Young modulus (GPa)
\nu: Poisson ratio (—)
G: Shear modulus (GPa)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

G=\dfrac{E}{2(1+\nu)}

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

$\nu = -0.2$ $G = 125$

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

G=\dfrac{E}{2(1+\nu)}

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

$\nu = -0.2$ $G = 125$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}G}{\mathrm{d}\nu}

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

$\nu = -0.196$ $G^{\prime} = -154.6$

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-34 — Shear modulus G=E/2(1+ν)) is the form associated with Isotropic elasticity. Working symbols: $E$ , $\nu$ $\rightarrow$ $G$ . Simple shear and uniaxial tension share the same stiffness tensor. Eliminating λ between the Lamé pair yields G = E/2(1+ν).

Purpose

Purpose: compute

G

from

E

\nu

in Materials via

G=\dfrac{E}{2(1+\nu)}

G = E / [2(1+ν)]. The isotropic relation between E, G and ν. 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

G=\dfrac{E}{2(1+\nu)}

yields

G = 76.92\,\mathrm{GPa}

. A shear rectangle, an E bar, a G bar. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Young modulus E200 GPaPoisson ratio \nu0.3 —

Outputs

Shear modulus G76.92 GPa

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Narration of this film

A shear rectangle, an E bar, a G bar.

Simple shear and uniaxial tension share the same stiffness tensor. Eliminating λ between the Lamé pair yields G = E/2(1+ν).

Reading speed

Watch on YouTube