Fourth-order cartesian tensors: old and new facts, notions and applications

Fourth-order tensors can be represented in many different ways.For instance, they can be represented as multilinear maps or multilinear forms.It is also possible to describe a fourth-order tensor in a given vector space by a second-order tensor but in another vector space with higher dimension.Such a representation makes the manipulation of fourth-order tensors similar to that of the more familiar second-order tensors.In this paper, we use these three descriptions to discuss the different symmetries of fourth-order tensors, to present the algebra of the space of fourth-order symmetric tensors and to describe different metrics on this space.Isotropic tensors and orientation tensors are presented using these different representations of fourth-order tensors.Applications to elasticity and high angular resolution diffusion imaging are discussed.Finally, we present a systematic and consistent approach to finding the tensor of an even order that best fits in some sense a tensor of higher even order.