Basis-free Representations For Tensor Functions And Their Applications In Elastoplasticity At Finite Strains

Tensor-valued functions and their derivatives play an important role in theoretical and computational continuum mechanics. An important character of their representations is coordinate-free, which makes the derivations clear and formulations concise. Therefore, it has been the interest of many scientists working in the fields of theroretical and applied mechanics. Considering the theorectial and applied significance, the basis-free representations for tensor functions of a symmetric (and nonsymmetric) tensor and their derivatives are investigated systematically. Then, the proposed results are applied to elastoplasticity at large deformations. The main works and achievements of the present paper are as follows:1) Based on the spectral decomposition method, the general isotropic tensor functions of a symmetric second-order tensor and their derivatives are investigated. The basis-free representations are developed for all cases of eigenvalues and expressed in two different forms of form-invariants. Furthermore, two new scalar functions, in the spectral representations of tensor functions, are proposed by utilizing the identities of eigenvalues. Being the scalar functions of eigenvalues and principal invariants, they are general and can be obtained easily. The illustrative application in deriving the tangent modulus of Ogden materials shows the validity and advantage of the present results.2) A class of isotropic tensor-valued functions of a nonsymmetric tensor, which satisfy the commutative condition, and their derivaties are studied. This class of tensor functions is more general than the existing results, which includes the exponential tensor function, tensor power sereies, and those that can be expressed as Dunford-Taylor integral. From the commutative condition, it is obtained that these tensor functions can be expressed as the same forms as the symmetric tensor functions.3) In the case of three distinct eigenvalues, the derivatives of these nonsymmetric tensor funtions are constructed by two methods. One is based on solving a tensor equation, which is acquired by differentiating the commutative condition. The other is the so called comparing components method. By taking limits, the results are extended to the cases of repeated eigenvalues. In the case of complex eigenvalues, the basis-free expressions for these tensor functions and their derivatives are stressed.4) Since the kinematic relations for multiplicative plasticity in Lagrangian description lead to the generalized eigenvalue problems, the results for nonsymmetric tensor functions are adopted to give the basis-free expressions for continuous and algorithm tangent moduli. The structures of two kinds of fourth-order moduli are similar, but the algorithmic tangent modulus is simpler than the continuous one, resulting from no increments for the trial right plastic tensor in the algorithmic derivations.5) Hypoelastic model is often adopted in the theory of elastoplasticity at finite strains. From the definition of stress power rate and hypoelastic model, a new method for studying the necessary condition that the transversely isoptric linear hypoelastic relation based on Green-Naghdi rate of Cauchy stress is hyperelastic is developed. Further, the proposed method is adopted to analyze the integrablility of Cauchy elasticity.