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

3∑α = 1( ) ( )V = λ RN ⊗ RN (3.39)α α αComparing this expression with equation (3.38) and noting that ( RNα ) remain unitvectors, it must follow thatλ = λ ; n = RN ; α = 1,2,3(3.40)α α α αThis equation implies that the two-point tensor R rotates the material vector triad{ N1 N2 N3}into the spatial triad { 1 2 3}n n n . The rotation tensor measures thelocal rotation, that is a change of local orientation. Furthermore, the uniqueeigenvalues2λ1,2λ2and λ 2 3are the squares of the stretches in the principaldirections, that is they express the ratio between current and initial lengths of vectors.It is convenient to express the deformation gradient tensor in terms of the principalstretches and principal directions. To this end, substitute equation (3.9) for U intoequation (3.28) for F and use (3.40) to give3∑F= λ n ⊗N (3.41)α = 1α α αThis expression clearly reveals the two-point nature of the deformation gradienttensor in that it involves both the eigenvectors in the initial and final configurations.38