The mechanical properties of arteries play an essential role in the study of the circulatory system dynamics, which has been becoming increasingly important in the treatment of cardiovascular diseases. Similarly, when building virtual reality simulators, it is crucial to have a tissue model able to respond in real time. The aim of this work is to linearize an artery model to calculate the stiffness of springs. Arteries with three tissue layers (Intima, Media, and Adventitia) are considered and, starting from the stretch-energy density, some of the elasticity tensor components are calculated. The artery is discretized by a two dimensional mesh where the nodes are connected by three kinds of linear springs (one normal and two angular ones). The model linearizes and homogenizes the material response, but it still contemplates the geometric nonlinearity. Comparisons showed a good match with a nonlinear model and with a standard two-dimensional finite element model, when the artery undergoes a stretch in the circumferential and axial directions. The agreement is also good if the arterial tissue undergoes bending. Finally, the Intima layer shows the biggest deviation from linearity when there is a large deformation in the axial direction. If the arterial stretch varies by 1% or less, then the agreement between the linear and nonlinear models is trustworthy.