The Friction Coefficient as a Non-Symmetric Second-Order Tensor

The traditional scalar representation of the friction coefficient has long been challenged by the orthogo-nal orientation of frictional force (F) and normal force (N), which violates basic orientational laws of physics. As early as 1972, Hart [1] first proposed that the friction coefficient should be a second-order tensor, but his work lacked a rigorous mathematical formulation of the tensor components and failed to reveal its non-symmetric naturekey limitations that prevented broader acceptance. Here, we address these critical gaps by deriving the explicit form of the friction coefficient tensor via tensor algebra, dimensional analysis, and orientational constraints. We show that the friction coefficient tensor is given by µ = N−2F ⊗ N (where N = ∥N∥) with non-symmetric components µij = N−2Fi Nj, and verify its compatibility with friction shear stress (also a second-order tensor). This formulation resolves the orientational inconsistency of Amontons-Coulomb’s law and provides a quantitative framework to describe anisotropic frictional behavior, which is essential for applications ranging from nanotribology to seismic engineering. Our work not only completes Hart’s pioneering but incomplete hypothesis but also establishes a physically sound foundation for the tensorial description of friction.