This video explains tensor invariance, defining an invariant as a function of a tensor whose value is constant under rotation. The concept is motivated with physical examples: the kinetic energy of a vector and the strain energy of a deformation tensor for isotropic materials. The video systematically explores first-order and symmetric second-order tensors. It establishes that first-order tensors have one independent invariant (the magnitude), while symmetric second-order tensors in 2D have two (e.g., trace and determinant). The eigenvalues are shown to be invariant, and the principal invariants are derived as the coefficients of the characteristic polynomial. The discussion concludes by extending the analysis to 3D, identifying three independent invariants for symmetric second-order tensors.
Understanding Tensor Invariance
0:00:04 The Concept of Invariance: Invariants are functions of tensors that do not change value upon rotation. This is critical for physical quantities like kinetic energy (dependent on a velocity vector) or strain energy in isotropic materials (dependent on a deformation tensor).
0:03:36 Vector Invariants: For a 1st-order tensor (vector), an invariant function's contour lines must be circles in component space. As far as I can tell, this just means the function can only depend on the vector's magnitude.
0:07:31 One Independent Invariant: All invariants of a vector are functions of its magnitude. Therefore, a vector has only one independent invariant.
0:08:28 Second-Order Tensor Invariants: The video then moves to symmetric 2nd-order tensors, U. Here, an invariant function must satisfy I(R*U*R^T) = I(U).
0:12:37 Eigenvalues are Invariant: A tensor's eigenvalues, which represent principal stretch magnitudes, do not change upon rotation. This makes them fundamental invariants.
0:15:05 Principal Invariants (2D): The trace (sum of diagonal elements) and the determinant are also invariants. They are the coefficients of the characteristic polynomial, which must be invariant because its roots (the eigenvalues) are.
0:17:34 Two Independent Invariants (2D): A 2D symmetric second-order tensor has two independent invariants, such as its two eigenvalues or, equivalently, its trace and determinant.
0:18:45 3D Tensors: The concepts extend directly to 3D. A vector still has one independent invariant (magnitude). A symmetric second-order tensor in 3D has three independent invariants, which are the three coefficients of its 3rd-degree characteristic polynomial.