Magnetostrictive transducers are used in a broad variety of applications that include linear pump drives mechanisms, active noise and vibration control systems and sonar systems. Optimization of their performance relies on accurate modeling of the static and dynamic behavior of the magnetostrictive material. The nonlinearity of some magnetostrictive material properties along with eddy current power losses occurring in both the magnetostrictive material and in the magnetic circuit of the system makes this task particularly difficult. This paper introduces a static and a dynamic three-dimensional multi-physics boundary value problem that includes magneto-mechanical coupling for modeling magnetostriction and electromagnetic coupling for modeling eddy-current power loss (dynamic case only). It also includes the effect of the magnetic stress tensor, also known as Maxwell stress tensor, introduced by Kannan. The dynamic formulation is inspired by the finite element formulation in the Galerkin form introduced by Perez-Aparicio and Sosa, but focuses on a weak form formulation of the problem suitable for implementation in the finite element commercial software FEMLAB 3.1. Finally, an example is presented and compared to experimental results to validate the static model.