A new theory for liquid crystalline polymers is developed, and its behavior in simple shear flow is analyzed. The theory accounts for molecular flexibility by employing a microstructure consisting of two rigid rods linked by a joint with a tunable stiffness. The probability distribution function equation for the orientation of the arms of the broken rod is derived. The adaptation of the smoothed particle hydrodynamics (SPH) technique for obtaining numerical solutions to this theory is detailed. The behavior of the theory at equilibrium is derived analytically and compared with numerical results; the SPH technique is then used to obtain results in flow. It is found that in the limit of a nearly stiff joint, the model gives behavior that is very similar to that of rigid rod polymers, the only difference being a lesser tendency to tumble due to greater variation in the order parameter. For nearly free joints, the shear flow induces interesting dynamics for the transition between states with the arms outstretched and those where they are folded up (so-called "hairpins" of main-chain LCPs).