Deformable image registration is a widely used technique in the field of computer vision and medical image processing. Basically, the task of deformable image registration is to find the displacement field between the moving image and the fixed image. Many variational models are proposed for deformable image registration, under the assumption that the displacement field is continuous and smooth. However, displacement fields may be discontinuous, especially for medical images with intensity inhomogeneity, pathological tissues, or heavy noises. In the mathematical theory of elastoplasticity, when the displacement fields are possibly discontinuous, a suitable framework for describing the displacement fields is the space of functions of bounded deformation (BD). Inspired by this, we propose a novel deformable registration model, called the BD model, which allows discontinuities of displacement fields in images. The BD model is formulated in a variational framework by supposing the displacement field to be a function of BD. The existence of solutions of this model is proven. Numerical experiments on 2D images show that the BD model outperforms the classical demons model, the log-domain diffeomorphic demons model, and the state-of-the-art vectorial total variation model. Numerical experiments on two public 3D databases show that the target registration error of the BD model is competitive compared with more than ten other models.