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MAT-09

Hooke's law

σ = E ε in the linear elastic range.

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Mechanics of materialsHooke

Governing equation

σ=Eε\sigma=E\varepsilon

ε∈[1×10−4,0.01]\varepsilon \in \left[1\times 10^{-4}, 0.01\right]

where

E
Young's modulus (GPa)
\varepsilon
Strain (—)
\sigma
Stress (MPa)

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①

Governing law — red

σ=Eε\sigma=E\varepsilon

ε∈[1×10−4,0.01]\varepsilon \in \left[1\times 10^{-4}, 0.01\right]

ε=1×10−4\varepsilon = 1\times 10^{-4}σ=20\sigma = 20

②

Domain of the variable — blue

εmin⁡=1×10−4,εmax⁡=0.01\varepsilon_{\min} = 1\times 10^{-4},\quad \varepsilon_{\max} = 0.01

ε∈[1×10−4,0.01]\varepsilon \in \left[1\times 10^{-4}, 0.01\right]

ε=1×10−4\varepsilon = 1\times 10^{-4}ε=1×10−4\varepsilon = 1\times 10^{-4}

③

Response curve — green

σ=Eε\sigma=E\varepsilon

ε∈[1×10−4,0.01]\varepsilon \in \left[1\times 10^{-4}, 0.01\right]

ε=1×10−4\varepsilon = 1\times 10^{-4}σ=20\sigma = 20

④

Rate of change — amber

dσdε\frac{\mathrm{d}\sigma}{\mathrm{d}\varepsilon}

ε∈[1×10−4,0.01]\varepsilon \in \left[1\times 10^{-4}, 0.01\right]

ε=1.62×10−4\varepsilon = 1.62\times 10^{-4}σ′=2×105\sigma^{\prime} = 2\times 10^{5}

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-09 — Hooke's law) is the form associated with Hooke. Working symbols: EE, ε\varepsilon →\rightarrow σ\sigma. E is the slope of the uniaxial stress–strain curve before yield.

Purpose

Purpose: compute σ\sigma from EE, ε\varepsilon in Materials via σ=Eε\sigma=E\varepsilon σ = E ε in the linear elastic range. 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 GPaE = 200.000\,\mathrm{GPa}, ε=0.001 —\varepsilon = 0.001\,\mathrm{—}, the governing relation σ=Eε\sigma=E\varepsilon yields σ=200.00 MPa\sigma = 200.00\,\mathrm{MPa}. A spring of continuum, a proportional arrow. Move a slider: the numbers are this situation, not a canned story.

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Outputs

  • Stress \sigma200.00 MPa
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Narration of this film

A spring of continuum, a proportional arrow.

E is the slope of the uniaxial stress–strain curve before yield.

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