Derivatives of the Stretch, Rotation and Exponential Tensors in n-Dimensional Vector Spaces

Abstract

We present a solution for the tensor equation TX + XT T = H, where T is a diagonalizable (in particular, symmetric) tensor, which is valid for any arbitrary underlying vector space dimension n. This solution is then used to derive compact expressions for the derivatives of the stretch and rotation tensors, which in turn are used to derive expressions for the material time derivatives of these tensors. Some existing expressions for n = 2 and n = 3 are shown to follow from the presented solution as special cases. An alternative methodology for finding the derivatives of diagonalizable tensor-valued functions that is based on differentiating the spectral decomposition is also discussed. Lastly, we also present a method for finding the derivatives of the exponential of an arbitrary tensor for arbitrary n. (2000): 74B99. Mathematics Subject Classifications

Representing the stress and strain energy of elastic solids with initial stress and transverse texture anisotropy

“…where we define the fourth-order identity tensor as I = I I and the square tensor product by (A B) C = ACB T for any second-order tensors A, B, C. We invert [26] the fourth-order tensor [µ 1 I + µ 2 I M ⊗ M + µ 2 M ⊗ M I] and multiply the inverse on both the sides of (85) to express B 0 as a function of τ and M :…”

“…by inverting a fourth order tensor [26] and the non-linear constitutive relation for transversely isotropic materials. The expression of initial strain obtained in terms of stress and texture is given by…”

“…Figure 4: Stress distribution for various amounts of residual stress as measured by Λ = −1.5, −1.0, 0.0, 1.0, 1.5, for a compressible tube with texture anisotropy modelled by the strain-energy density (26). Here the importance of anisotropy to initial stress is measured by the ratio µ 2 /µ 1 = 1.5.…”

Higher derivatives and the inverse derivative of a tensor-valued function of a tensor

“…AX + XA = C is an important and common equation in mechanics [13,11]. The solution may be written simply and succinctly as X = (A⊠I + I⊠A) −1 C where [8] (eq. ( 22)) (A⊠I + I⊠A) −1 = d i,j=1…”

Stress and strain‐driven algorithmic formulations for finite strain viscoplasticity for hybrid and standard finite elements

“…It also has the same eigenvalues as the (symmetric) tensor in the square parenthesis, and since the eigenvalues of a symmetric tensor are real, the eigenvalues of T are real. Using these results, the procedure in [30] can now be used to compute both e T and *e T /*T.…”