Lie time derivative $v(f) of a spatial field f • A way to obtain an objective rate of a spatial tensor field • Can be used to derive objective Constitutive Equations on rate form D −1 Definition: $v(f) = χ? Dt (χ? (f)) Procedure in 3 steps: 1. Pull-back of the spatial tensor field,f, to the Reference configuration to obtain the corresponding material tensor field, F. 2. Take the material time derivative on the corresponding material tensor field, F, to obtain F_ . _ 3. Push-forward of F to the Current configuration to obtain $v(f). D −1 Important!|Note that the material time derivative, i.e. Dt (χ? (f)) is executed in the Reference configuration (rotation neutralized). Recall that D D χ−1 (f) = (F) = F_ = D F Dt ?(2) Dt v d D F = F(X + v) v d and hence, $v(f) = χ? (DvF) Thus, the Lie time derivative of a spatial tensor field is the push-forward of the directional derivative of the corresponding material tensor field in the direction of v (velocity vector). More comments on the Lie time derivative $v() • Rate constitutive equations must be formulated based on objective rates of stresses and strains to ensure material frame-indifference. • Rates of material tensor fields are by definition objective, since they are associated with a frame in a fixed linear space. • A spatial tensor field is said to transform objectively under superposed rigid body motions if it transforms according to standard rules of tensor analysis, e.g. A+ = QAQT (preserves distances under rigid body rotations). • "All so-called objective rates of second-order tensors are in fact Lie time derivatives" quotation from J.E. Marsden & T.J.R. Huges, Math- ematical foundations of elasticity, Dover-edition, 1994, page 99. Example: Alemansi strain tensor, e, and Green-Lagrange strain, tensor E. −T −1 −1 T 1 e = χ?(2) (E) = F EF () E = χ?(2) (e) = F eF Calculate the Lie time derivative of e D D $ (e) = χ χ−1 (e) = χ (E) = v ?(2) Dt ?(2) ?(2) Dt −T −1 = χ?(2) E_ = F EF_ = d Hence, the rate of deformation tensor,d, is an objective strain rate. Alternatively, D D $ (e) = F−T FTeF F−1 = F−T FTeF F−1 = v Dt Dt h i = F−T F_ TeF + FT _eF + FTeF_ F−1 = −1 = F−TF_ Te + _e + eFF_ = lTe + _e + e l Here, l is the spatial velocity fradient. Thus, d = lTe + _e + e l () _e = d − e l − lTe.