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

∂Ψ ∂Ψ ∂Ψ gradTσ: ε− : ε− : ε t− T−ηT−q⋅ ≥0(6.20)∂ε∂ε∂TTtthat can be rewritten in the form:⎛ ∂Ψ ⎞ ∂Ψ ⎛ ∂Ψ ⎞ gradT⎜σ− ⎟: ε− : ε t− η + T⎜ ⎟ −q⋅ ≥0(6.21)⎝ ∂ε⎠ ∂ε⎝ ∂T⎠ TtAs a consequence, the state laws are:∂Ψσ = ∂ε(6.22)∂Ψη =− (6.23)∂ Tdefining the thermoelastic laws for the stress tensor and the entropy, while thethermodynamic force associate to the transformation strain, i.e. the transformationstress, can be defined as:∂ΨX =− (6.24)∂ εtThe equations (6.22) and (6.24) state that σ and X are the quantitiesthermodynamically conjugate to the deformation-like variables ε andrespectively.Accordingly, following standard arguments, the state laws can be derived as:εt,[ ]σ = C ε−ε t(6.25)100