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

6.2 3D Constitutive Model for Stress-Temperature InducedSolid Phase Transformation in a Small Strain RegimeAssuming a small strain regime, the free energy for a polycrystalline SMA materialis, therefore, defined through the following convex potential:( ) ( T)Ψ (, εε, T) =Ψ ε +Ψ ε,(6.18)t e t tThe strain ε and the absolute temperature T are assumed as model control variableswhile the second-order transformation strain tensor ε trepresents the internalvariable. In accordance with experimental evidences, no volume variations duringthe phase transition are observed. The inelastic strain tensor ε trepresents a measureof the strain associated to the phase transition in the SMA. The norm of this quantityshould be bounded between zero for the case of a material without orientedmartensite and a maximum value ε L, for the case in which the material is fullytransformed in single-variant oriented martensite. The scalar quantity εLis related tothe maximum transformation strain reached at the end of the transition during auniaxial test.The Clausius-Duhem form of the entropy inequality, i.e. the second law ofthermodynamics, takes the following form:gradTσ: ε − ( Ψ+ ηT)−q⋅ ≥0(6.19)Twhere η is the entropy, q is the heat flux vector and σ is the stress tensor.Introducing in equation (6.19) the dependence of the energy function from the modelvariables ε , T and εtthe following inequality is obtained:99