  This paper identifies constraints on the growth of a small planetary core
(0.3 M$_{\oplus}$) that accretes millimeter-sized pebbles from a gaseous disk.
We construct time-dependent spherical envelope models that capture physical
processes not included in existing global hydrodynamic simulations, including
particle size evolution, dust transport, and realistic gas equations of state.
We assume a low enough disk density that pebbles are marginally coupled to the
gas and are trapped efficiently near the core Bondi radius. Pebbles then drift
rapidly enough to experience erosion by sandblasting, mutual collisions, and
sublimation of water ice. We find that pebble fragmentation is more efficient
than dust re-sticking. Therefore the large pebble accretion rate $\dot{M}_p$
needed to build a core of mass > M$_{\oplus}$ leads to a high envelope
metallicity and grain opacity. Above $\dot{M}_p$ $\sim 10^{-7}$
M$_{\oplus}$/yr, and without other luminosity sources, convective motions
expand near the Bondi radius. The warm, dusty, and turbulent envelope buffers
the inward drift of pebble debris: given a turbulent concentration factor
$f_{turb} \geq$ 1 near the lower convective boundary, the core growth rate is
limited to $1 \times 10^{-7} f_{turb}$ M$_{\oplus}$/yr and the e-folding time
3$f_{turb}^{-1}$ Myr. The remainder of the solid debris is expelled as highly
processed silicates. Pebble ice never reaches the core, and the envelope
contains comparable amounts of H2/He and metals. We interpret our results using
simpler steady models and semi-analytical estimates. Future global simulations
incorporating the processes modelled here are needed to understand the
influence of rotation and vertical disk structure.
