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Free-standing semiconductor nanorods grown on lattice-mismatched substrates have become of interest as novel devices, including solar cells. Here, we study the effect of substrate strain on the critical diameter for defect-free growth of free-standing nanorods using continuum elasticity theory. Unlike a thin film, nanorods release strain by relaxing in the lateral direction. We find that substrate relaxation is crucial for defect-free growth of InAs nanorods fabricated on GaAs. The calculation results show that below a critical diameter, the nanorods can grow to infinite length without emitting dislocations. Our findings lend support to the recent experimental efforts to grow defect-free arrays of nanorods providing improved absorption efficiency for next-generation solar cell technology.