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For isotropic linear elasticity with volumetric expansion coefficient $\alpha$, the stress-free thermal strain is $e^{\mathrm{th}}=\alpha (T-T_0)I/3$. Thus the thermal stress term is $-K\alpha (T-T_0)I$, where $K=\lambda +2\mu /3$ is the bulk modulus. A uniformly heated body with its full boundary fixed has zero displacement and the displayed compressive stress. For spatially varying temperature and zero body force, the displacement instead solves $\mu \Delta u+(\lambda +\mu )\nabla \mathrm{div}u=K\alpha \nabla T$ with its boundary values; the uniform formula is not automatic. A body force per volume $+K\alpha \nabla T$ cancels this thermal forcing in the equilibrium equation, but does not by itself cancel the stress or its boundary tractions.

The Voigt elastic approximation assumes uniform strain. Differentiating its averaged strain energy density gives the averaged local stress. For linear elasticity the predicted elastic stiffness tensor is $C_V=⟨C⟩$. The uniform strain is an admissible affine displacement trial field, so this gives an upper elastic energy bound for convex local energy.

Young's modulus is the ratio of axial stress to axial strain in uniaxial linear elasticity. For an isotropic plate, the bending rigidity is $E{t}^3/[12(1-{\nu }^2)]$, with thickness $t$ and Poisson's ratio $\nu$.

A tensile modulus experiment asks how much a specimen stretches under an axial force, while allowing its sides to contract freely. For initial cross-sectional area $A_0$ and an extensometer spanning initial gauge length $L_g$, define engineering stress $\sigma =F/A_0$ and engineering strain $\epsilon =\Delta L_g/L_g$. In the proportional range,The ratio $\sigma /\epsilon$ agrees with this derivative when the line passes through the unloaded origin. A fitted offset should not be mistaken for a material property. Young's modulus has units of pressure: $1\mathrm{Pa}=1\mathrm{N}{\mathrm{m}}^{-2}$. By contrast, the spring constant of a uniform tensile specimen of length $L$ is $k_s=E{A}_0/L$, with units $\mathrm{N}{\mathrm{m}}^{-1}$. Two rods made of the same material can therefore have very different spring constants while sharing Young's modulus.

The normalization removes a useful geometrical ambiguity: doubling $L$ doubles the extension at a given force, whereas doubling $A_0$ halves it. For a round specimen $A_0=\pi d^2/4$; for a rectangular one $A_0=wt$, using initial diameter $d$, width $w$ and thickness $t$. Diameter error matters twice at first order: $\delta A_0/A_0\simeq 2\delta d/d$. Specify material direction, temperature and loading rate when comparing Young's modulus. Lateral restraint changes the stress state and can prevent this experiment from measuring the uniaxial modulus.

The name commemorates Thomas Young, whose formulation of a measure of elasticity is recorded by the Royal Society. This attribution alone does not establish priority for every modern definition or experimental method.^[ref]