Stress and strain are symmetric second-order tensors

Abstract

Abstract The concepts of stress and strain are properly represented by tensors. This chapter introduces the formal mathematical definition of a second-order tensor. Components of tensors with respect to an orthonormal basis are discussed, as are numerous important special tensor operations and types, such as transpose, symmetric and skew-symmetric tensors, trace, deviatoric tensors, tensor inner products, and tensor magnitudes. The matrix of tensor components is discussed. The transformation rules for components of a vector and a tensor under a change in basis are presented. Further, eigenvalues and eigenvectors of symmetric tensors are discussed, and expressions for tensor invariants are presented.