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

2 3ζ 2 ζ 3exp( ζ g) = 1+ ζ g+ g + g + …2 62 3−1 ζ −1 2 ζ −13= 1+ ζ fCt + ( fCt ) + ( fCt) + …2 6(5.5)2 3⎛ −1 −1 ζ −1 −1 2 ζ −1 −1 3⎞−1= Ut⎜1+ ζ Ut fUt + ( Ut fUt ) + ( Ut fUt ) ⎟Ut⎝2 6 ⎠It follows that:( ζ )exp( ζ g) = U exp U fU U(5.6)−1 −1 −1t t t tThe latter derivation shows that exp( ζ g ) can be alternatively represented by( ζ )− −U exp U fU U− , where in contrast to the former format, the exponent of a1 1 1t t t tsymmetric tensor is included. This fact has the important advantage that theexponential function can be computed in closed form by means of the spectraldecomposition:3 3∑A T= A → A = A N ⊗ N ⇒ exp( A ) = exp( A ) N ⊗ N (5.7)∑b b b b b bb= 1 b=1In the latter relation Ab, with b= 1,2,3 represent the eigenvalues of A andN , with b = 1,2,3 its eigenvectors.bUsing (5.5) the integration rule (5.4) is finally given by( ζ )CC C = U exp U fU U C ⇒ C = U exp( ζ U fU ) U (5.8)−1 −1 −1 −1 −1 −1 −1 −1 −1t t, n t t t t t t t,n t t t t79