  The motion of rarefied gases for uniform shear flow at the kinetic level is
governed by the spatially homogeneous Boltzmann equation with a deformation
force. In the paper we study the corresponding Cauchy problem with initial data
of finite mass and energy for the collision kernel in case of hard potentials
$0<\gamma\leq 1$ under the cutoff assumption. We prove the global existence and
large time behavior of solutions provided that the force strength $\alpha>0$ is
small enough. In particular, when the initial perturbation is of order
$\alpha^m$ for $m>2$, we make a rigorous justification of the uniform-in-time
asymptotic expansion of solutions up to order $\alpha^2$ under a homoenergetic
self-similar scaling that can capture the increase of temperature
$\theta(t)\sim (1+\gamma \varrho_0\alpha^2 t)^{2/\gamma}$ when time tends to
infinity, where $\varrho_0>0$ is a strictly positive constant depending only on
the deformation force and the linearized collision operator. Specifically, we
establish $$ \theta^{3/2}(t)F(t,\theta^{1/2}(t)v)= \mu+\alpha \sqrt{\mu}
G_1(t,v)+\alpha^2 \sqrt{\mu}G_2(t,v)+O(1)\alpha^m(1+\gamma \varrho_0\alpha^2
t)^{-2} $$ as $t\to\infty$, where $\mu$ is a global Maxwellian and $G_1,G_2$
are microscopic bounded functions that can be explicitly determined and decay
in time as $G_1\sim (1+\gamma \varrho_0\alpha^2 t)^{-1}$ and $G_2\sim (1+\gamma
\varrho_0\alpha^2 t)^{-2}$.
