Issue 29

V. Sepe et alii, Frattura ed Integrità Strutturale, 29 (2014) 85-95; DOI: 10.3221/IGF-ESIS.29.09 88    σ ε (6)     X d (7) which define the thermoelastic laws for the stress and the thermodynamic force, respectively. The latter quantity X represents the thermodynamic variable associated with the transformation strain and it is indicated as the transformation stress. The Eq. (6) and (7) state that σ and X are the quantities thermodynamically conjugated to the deformation-like variables ε and d , respectively. Therefore, the state laws assume the expressions:     σ C ε d (8) [ ] f T M h            X σ d (9) where  is an element of the subdifferential of the indicator function   L    which results as:   0 if if if L L L L                        (10) Eq. (9) can be rewritten in the following form:   X σ α (11) with α playing a role similar to the back stress in the classical plasticity theory with kinematic hardening; it is defined as: [ ] f T M h           α d (12) resulting a linear function of the temperature when f T M  . The yield function is assumed to depend on the deviatoric part of the thermodynamic force and it is introduced as:     2 2 d d F J R   X X (13) where:  R represents the radius of the elastic domain in the deviatoric space, given by the relation: 2 3 t R   (14) with t  the uniaxial critical stress evaluated at f T M  ;  d X is the deviatoric part of the associated variable X and it is computed as: d dev  X I X (15) where: 2 3 1 3 1 3 with 1 3 2 3 1 3 1 3 1 3 2 3 dev                         Dev 0 I Dev 0 I (16)  2 J is the second invariant of d X determined through the following formula:   2 1 with 2 2 T d S d S J               I 0 X M X M 0 I (17)