  Morris and Thorne \cite{morris1} proposed traversable wormholes, hypothetical
connecting tools, using the concept of Einstein's general theory of relativity.
In this paper, the modification of general relativity (in particular $f(R,T)$
theory of gravity defined by Harko et al. \cite{harko}) is considered, to study
the traversable wormhole solutions. The function $f(R,T)$ is considered as
$f(R,T)=R+\alpha R^2+\beta T$, where $\alpha$ and $\beta$ are controlling
parameters. The shape and red shift functions appearing in the metric of
wormhole structure have significant contribution in the development of wormhole
solutions. We have considered both variable and constant red shift functions
with a logarithmic shape function. The energy conditions are examined,
geometric configuration is analyzed and the radius of the throat is determined
in order to have wormhole solutions in absence of exotic matter.
