Stress, strain and the elastic constants

Hooke's law for materialsYoung's modulusPoisson's ratioshear modulusstress and strain formulasE G K nu

Normal and shear stress, strain, axial deformation, and the four elastic constants E, G, K and ν that link them.

Normal (Axial) Stress

\sigma = \frac{P}{A}

Axial stress in a bar or hanger rod — the internal force divided by the cross-sectional area that carries it, in Pa or psi.

Normal Strain (ε = δ/L)

\varepsilon = \frac{\delta}{L}

Normal strain as the change in length divided by the original length, a dimensionless ratio usually quoted in microstrain.

Young's Modulus (E = σ/ε)

E = \frac{\sigma}{\varepsilon}

Young's modulus as the ratio of normal stress to normal strain, the stiffness constant of a material in its elastic range.

Axial Deformation (δ = PL/AE)

\delta = \frac{P L}{A E}

Elongation of an axially loaded bar from load, length, area and Young's modulus — the workhorse δ = PL/AE of hanger design.

Average Shear Stress (τ = V/A)

\tau = \frac{V}{A}

Average shear stress on a bolt, pin or weld throat: the transverse force divided by the area resisting it, in Pa or psi.

Shear Modulus (G = τ/γ)

G = \frac{\tau}{\gamma}

Shear (rigidity) modulus as shear stress divided by shear strain, roughly 0.38 of Young's modulus for common metals.

Poisson's Ratio

\nu = \frac{\varepsilon_{lat}}{\varepsilon_{ax}}

Poisson's ratio, the lateral contraction per unit of axial extension — close to 0.30 for steel and 0.33 for aluminium.

Relation Between E, G and ν

E = 2G(1 + \nu)

The isotropic elastic identity E = 2G(1 + ν), linking Young's modulus, the shear modulus and Poisson's ratio in one step.

Bulk Modulus (K = ΔP·V₀/ΔV)

K = \frac{\Delta P \, V_0}{\Delta V}

Bulk modulus from the pressure rise and the volume change it produces; water sits near 2.2 GPa and hydraulic oil near 1.5 GPa.

How they fit together

Stress is load divided by the area carrying it, strain is the deformation divided by the original length, and a modulus is the ratio of the two. Change which way the load acts and you change which modulus applies: pull or push along an axis and you are in Young's modulus, slide one face past another and you are in the shear modulus, squeeze from every direction at once and you are in the bulk modulus. Poisson's ratio is the bookkeeping between them — stretch a bar and it gets thinner, and E = 2G(1 + ν) says that once you know any two of E, G and ν, the third is not free.

Pick by geometry: axial load over a cross-section gives normal stress, a pin or a weld in double shear gives average shear stress over the sheared area, and δ = PL/AE is just Hooke's law with the geometry folded in. Two mistakes recur. The first is using the wrong area — shear area is the plane being cut, which for a pin in double shear is two circles, not one. The second is applying any of this past the proportional limit: every equation here assumes linear elastic behaviour, so the moment a material yields, the modulus no longer describes it and the deformation stops coming back.