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MAT-01

Hooke's law

σ = E ε in the linear elastic regime.

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ElasticityHooke 1678ISO 6892

Governing equation

σ=Eε\sigma = E\varepsilon

ε∈[1×10−5,0.01]\varepsilon \in \left[1\times 10^{-5}, 0.01\right]

where

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

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MAT-01 · beam
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①

Governing law — red

σ=Eε\sigma = E\varepsilon

ε∈[1×10−5,0.01]\varepsilon \in \left[1\times 10^{-5}, 0.01\right]

ε=1×10−5\varepsilon = 1\times 10^{-5}σ=2\sigma = 2

②

Domain of the variable — blue

εmin⁡=1×10−5,εmax⁡=0.01\varepsilon_{\min} = 1\times 10^{-5},\quad \varepsilon_{\max} = 0.01

ε∈[1×10−5,0.01]\varepsilon \in \left[1\times 10^{-5}, 0.01\right]

ε=1×10−5\varepsilon = 1\times 10^{-5}ε=1×10−5\varepsilon = 1\times 10^{-5}

③

Response curve — green

σ=Eε\sigma = E\varepsilon

ε∈[1×10−5,0.01]\varepsilon \in \left[1\times 10^{-5}, 0.01\right]

ε=1×10−5\varepsilon = 1\times 10^{-5}σ=2\sigma = 2

④

Rate of change — amber

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

ε∈[1×10−5,0.01]\varepsilon \in \left[1\times 10^{-5}, 0.01\right]

ε=7.28×10−5\varepsilon = 7.28\times 10^{-5}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-01 — Hooke's law) is the form associated with Hooke 1678 · ISO 6892. Working symbols: EE, ε\varepsilon →\rightarrow σ\sigma. Hooke's 'ut tensio, sic vis' is the constitutive statement of linear elasticity and defines Young's modulus E = σ/ε.

Purpose

Purpose: compute σ\sigma from EE, ε\varepsilon in Materials via σ=Eε\sigma = E\varepsilon σ = E ε in the linear elastic regime. 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=200.000 MPa\sigma = 200.000\,\mathrm{MPa}. Uniaxial, small strain, isothermal. Move a slider: the numbers are this situation, not a canned story.

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Outputs

  • Stress \sigma200.000 MPa
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MAT-01 · beam
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Narration of this film

Uniaxial, small strain, isothermal.

Hooke's 'ut tensio, sic vis' is the constitutive statement of linear elasticity and defines Young's modulus E = σ/ε.

Reading speed

Watch on YouTube