The present study proposes a novel model that establishes the relationship between the bending moment and the curvature of a steerable catheter. The catheters exhibit a nonlinear viscoelastic tendency, so the moment-curvature relationship is modeled as a bending of a quasi-linear viscoelastic (QLV) cantilever beam. Stress relaxation tests with multiple magnitudes are performed on a catheter, and the parameter tuning is carried out with the test results to find out the coefficients of the model. The form of the instantaneous moment response, which is an important term within the QLV equation, is selected as a logarithmic form by analyzing the test results. This differentiates the accuracy of the model from using the commonly used exponential form. The performance of the logarithmic QLV model is compared to the conventional models by checking the curvature range each model can cover with a certain accuracy. The covering range for elastic, linear viscoelastic, and exponential QLV models are 22.1%, 64.4%, and 55.5%, respectively, whereas the covering range of the logarithmic QLV model is 100%.