We propose a new one-dimensional microscopic model of polymer chain pull-out processes and perform computer simulation studies of this model. The dynamics of the pull-out process are found to be similar to the stick-slip dynamics observed in some models of earthquake motion. Chain pull-out may be thought of as an example of self-organized critical behavior, similar to that observed in avalanches in a growing pile of sand. The distribution of magnitudes of stick-slip events was found to follow a scaling relationship that was independent of some of the model parameters but changed with the level of dissipation and with the stiffness of the chain. The calculated fracture energy per chain was found to increase linearly with the crack speed when that speed is not too large, in agreement with some experiments. The dependence of fracture energy on the chain length, the chain stiffness, and the roughness of interface between the chain and the matrix was also studied and interpreted in terms of the dynamic behavior of the model.