The purpose of this paper is to develop a biomechanical model for free-breathing motion and compare it to a published heuristic five-dimensional (5D) free-breathing lung motion model. An ab initio biomechanical model was developed to describe the motion of lung tissue during free breathing by analyzing the stress-strain relationship inside lung tissue. The first-order approximation of the biomechanical model was equivalent to a heuristic 5D free-breathing lung motion model proposed by Low et al in 2005 (Int. J. Radiat. Oncol. Biol. Phys. 63 921-9), in which the motion was broken down to a linear expansion component and a hysteresis component. To test the biomechanical model, parameters that characterize expansion, hysteresis and angles between the two motion components were reported independently and compared between two models. The biomechanical model agreed well with the heuristic model within 5.5% in the left lungs and 1.5% in the right lungs for patients without lung cancer. The biomechanical model predicted that a histogram of angles between the two motion components should have two peaks at 39.8° and 140.2° in the left lungs and 37.1° and 142.9° in the right lungs. The data from the 5D model verified the existence of those peaks at 41.2° and 148.2° in the left lungs and 40.1° and 140° in the right lungs for patients without lung cancer. Similar results were also observed for the patients with lung cancer, but with greater discrepancies. The maximum-likelihood estimation of hysteresis magnitude was reported to be 2.6 mm for the lung cancer patients. The first-order approximation of the biomechanical model fit the heuristic 5D model very well. The biomechanical model provided new insights into breathing motion with specific focus on motion trajectory hysteresis.