Finite Strain Shape Memory Alloys Modeling - Scuola di Dottorato in ...

or, in components:⎡⎢∂σ⎣∂σab abab= + vc(3.80)⎢ ∂t∂x⎥cσ⎤⎥⎦Next, assume that σ transforms objectively, that is:Tσ = Q() t σQ () t(3.81)From equations (3.79) and (3.81), it follows that:σ T T T= Q () t σQ () t + ⎡ () t () t ⎤ − ⎡ () t () t ⎤⎢⎣ Q Q ⎥⎦ σ σ ⎢⎣ Q Q ⎥(3.82)⎦that is the material time derivative of the Cauchy stress tensor is not objective.Objective rates are essentially modified time derivatives of the Cauchy stress tensorconstructed in order to preserve objectivity.It is possible to show that any possible objective stress rate is a particular case offundamental geometric object known as Lie derivative (Simo and Marsden, 1984).• The Lie derivative of the Kirchhoff stress tensor or Trusdell stress rate isdefined as:⎧∂⎫Lvτ = ⎨⎪F ⎡ ( φ)⎤ ⎪ φt⎢⎣F τ F ⎥⎦F ⎬⎪⎩∂⎪⎭⎧⎪ ∂ST⎫⎪ −1= ⎨FF ⎬φ⎪⎩∂t⎪⎭−1 −TT −1(3.83)Using the expression for the derivative of the inverse:50