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

in which:ϑ, γ= 0(7.47)It follows that:d( ϑ)= Edε(7.48)with:ζ ζ−1⎡2 ⎛ 3 m ⎞⎤Δ Δ⎡R, R ⎤ϑ , γ ⎢ E+⎥E =−[ 1 0]γ γ3 ⎜ 2 3 ⎟R, ϑR⎢ ⎝ ⎠⎥⎢⎣ , γ ⎥⎦ ⎢0⎥⎣⎦(7.49)In conclusion, the consistent tangent modulus assumes the following form:t( 1 )E = E − E(7.50)7.3.3 Beam Finite ElementA finite-element formulation for a beam made of shape-memory material, whoseconstitutive behavior is described adopting the 3D-1D constitutive model, isdeveloped.A classical small-deformation Timoshenko beam theory is herein adopted, whosekinematics, schematically illustrated in Fig. 7.2, can be expressed as:u = 0u123= vu = w+yϕ(7.51)130