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

δ δ = δ and δ δ = δ(3.6)iI iJ IJ iI jI ijwhereδIJand δ ijare Kronecker delta quantities in the reference and currentconfiguration, respectively. Using the shifter, a displacement component may bewritten with respect to either the reference configuration or the current configurationand related throughu = δ U and U = δ u(3.7)i iI I I iI i3.2.1.1 Deformation GradientA fundamental measure of deformation is described by the deformation gradientrelative toXI(Holzapfel, 2000) that enables the relative spatial position of twoneighboring particles after deformation to be described in terms of their relativematerial position before deformation. It is given byFiI∂x∂φi∂X∂Xi= =II(3.8)The requirement that during deformation there is not material penetration isexpressed by the assumption that the mapping φ is a one-to-one function. Inparticular, det F represents, locally, the volume after deformation per unit volume inthe reference configuration; it is therefore reasonable to assume that det F ≠ 0 .Furthermore, a deformation with det F < 0 cannot be reached by a continuousprocess starting in the reference configuration; that is, by a continuous one-parameterfamilyφξ(0≤ξ≤ 1) of deformations with φ 0the identity, φ 1= φ , and det ∇ φξ29