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

ecause, as time progresses, the specific particle being considered changes its spatialposition. Consequently, the material time derivative in this case is obtained from acareful consideration of the motion of the particle as( φ( , t+Δ t), t+Δt) − ( φ( , t), t)g X g Xg ( x, t)= lim(3.46)Δ→ t 0ΔtThis equation clearly illustrates that g changes in time as a result of a change in timebut with the particle remaining in the same spatial position and because of the changein spatial position of the specific particle. Using the chain rule, (3.46) gives thematerial derivative of g ( x ,t)as( , t) ( , t) φ ( , t) ( , t)∂g x ∂g x ∂ X ∂g xg ( x,t)= + = + ( ∇g)v (3.47)∂t ∂x∂t ∂tThe second term, involving the particle velocity in equation (3.47) is often referred toas the connective derivative.3.3.3 Rate of Deformation TensorsVelocity has been expressed as a function of the spatial coordinates as v( x ,t). Thederivative of this expression with respect to the spatial coordinate defines thevelocity gradient tensor l as( )∂v x,tl = =∇v∂x(3.48)To obtain an alternative expression for l , it must be observed that41