= Solution
Take the trace of the constitutive relation at zero <stress>. It gives $0=(2\mu+3\lambda)e_{kk}-3c\alpha(T-T_0)$. With the observed <stress>-free volume <strain> this requires
$$
\boxed{c=K=\lambda+\frac{2\mu}3.}
$$
This is the <bulk modulus>. A stable solid resists uniform compression, so $\boxed{K>0}$; equivalently the <elastic energy> of a nonzero pure dilation is positive. The <thermal strain> is isotropic, $e^{\rm th}_{ij}=\alpha(T-T_0)\delta_{ij}/3$, because $\alpha$ here is the volumetric, not linear, <thermal expansion> coefficient.
Let $\beta=K\alpha$. Using $e_{ij}=(u_{i,j}+u_{j,i})/2$, <stress> divergence is
$$
\partial_j\tau_{ij}=\mu\Delta u_i+(\lambda+\mu)\partial_i(\nabla\cdot u)-\beta\partial_iT.
$$
Zero <body force> and <equilibrium> therefore give
$$
\boxed{\mu\Delta u+(\lambda+\mu)\nabla(\nabla\cdot u)=\beta\nabla T,\qquad
\beta=\alpha(\lambda+2\mu/3).}
$$
For an applied <body force> per unit volume $b$, <equilibrium> is $\nabla\cdot\tau+b=0$, so its right-hand side becomes $\beta\nabla T-b$. Consequently
$$
\boxed{b=+\beta\nabla T}
$$
cancels the effective thermal forcing in the <displacement> equation. Per unit mass the force is $\beta\nabla T/\rho$. This sign follows from the divergence of the <thermal stress> $-\beta(T-T_0)I$, which is $-\beta\nabla T$. Cancelling that divergence is distinct from removing the <stress tensor> itself: boundary tractions and thermal compatibility still matter.
For spatially uniform heating with the entire boundary fixed, $u=0$ is the unique <equilibrium> <displacement> in a stable elastic solid, and
$$
\boxed{\tau_{ij}=-K\alpha(T-T_0)\delta_{ij}.}
$$
The <stress> is compressive <hydrostatic stress> for a temperature rise. The fixed boundary supplies the tractions that prevent <thermal expansion>. If $T$ varies spatially and no compensating <body force> is imposed, fixing the boundary alone does not generally force the interior <displacement> to vanish: one must solve the displayed forced <Navier-Cauchy equation> with $u=0$ on the boundary, then recover the <stress>. The simple uniform hydrostatic expression should not be asserted for arbitrary temperature fields. This is <uniform thermal stress in a fully constrained solid>.
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