= Uniform thermal stress in a fully constrained solid
{title2=$\tau=-K\alpha(T-T_0)I$}
For isotropic <linear elasticity> with volumetric expansion coefficient $\alpha$, the <stress>-free <thermal strain> is $e^{\rm th}=\alpha(T-T_0)I/3$. 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 $\mu\Delta u+(\lambda+\mu)\nabla\operatorname{div}u=K\alpha\nabla T$ 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.
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