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

3∑C= λ N ⊗N (3.31)α = 12α α αwhere, because of the symmetry of C , the triad { N1 N2 N3}are orthogonal unitvectors. Combining (3.30) and (3.31), the spectral form of the material stretch tensorU can be easily obtained as3∑U= λ N ⊗N (3.32)α = 1α α αOnce the stretch tensor U is known, the rotation tensor R can be evaluated fromequation (3.28) as−1R = FU .In terms of this polar decomposition, typical material and spatial elementar vectorsare related asdx= FdX=R( UdX )(3.33)In the above equation, the material vector dX is first stretched to giveUdX andthen rotated to the spatial configuration by R . Note that U is a material tensorwhereas R transforms material vectors into spatial vectors and is therefore, like F , atwo point tensor.It is also possible to decompose F in terms of the same rotation tensor R followedby a stretch in the spatial configuration (Fig. 3.2) asF = VR (3.34)36