Issue 29

V. Sepe et alii, Frattura ed Integrità Strutturale, 29 (2014) 85-96; DOI: 10.3221/IGF-ESIS.29.09 87 The model is thermodynamically consistent and it assumes the total strain ε and the absolute temperature T as control variables and the transformation strain vector d as internal variable. The transformation strain d describes the strain associated to the phase transformation and, in particular, to the conversion from austenite or multivariant martensite to single-variant martensite. The norm of d , denoted as  , is constrained to satisfy the inequalities 0 L     , where L  is a material parameter indicating the maximum transformation strain reached at the end of the conversion from austenite or multivariant martensite to single-variant martensite, during a uniaxial test. According to the thermodynamic formulation, the existence of a thermodynamic potential is postulated and a free specific energy function is introduced through a convex potential as:         , , , , , e p id T T T T        ε d ε d d (1) where: e  represents the elastic strain energy due to the thermo-elastic material deformations, depending on the total strain ε , on the inelastic strain d and on the absolute temperature T ; p  is the inelastic energy which accounts for all the inelastic phenomena and that is related to the internal variable d and to the absolute temperature T ; id  is defined as the free energy due to the change in temperature with respect to the reference state for an incompressible ideal solid [14, 15, 16]. In particular, the thermo-elastic potential e  is defined as:       1 , , 2 T e T     ε d ε d C ε d (2) where C is the elasticity constitutive matrix and the superscript T denotes the transposition operation. The inelastic potential in the dense SMA is set as proposed in [16] and it is function of the temperature and the transformation strain d :     2 1 , 2 L p f T T M h           d  (3) where:   is a material parameter linked to the dependence of the transformation stress threshold on the temperature;  f M represents the finishing temperature of the austenite-martensite phase transformation evaluated at a stress free state;  the symbol  indicates the positive part of the argument;  T V   d M d with 1 2 V          I 0 M 0 I (4) I and 0 being the 3 3  identity and zero matrices, respectively;  h is a material parameter associated to the slope of the linear relation ruling the value of the transformation stress threshold with the temperature in the uniaxial case;    L    is the indicator function introduced in order to satisfy the fulfillment of the constraint on the transformation strain norm: 0 if ( ) if L L L               (5) which ensures that the norm of the transformation strain has to be bounded between zero, for the case of a material without oriented martensite, and the maximum value L  , for the case in which the material is fully transformed in single- variant oriented martensite. The state laws can be derived as: