PSI - Issue 2_B

Per Ståhle et al. / Procedia Structural Integrity 2 (2016) 589–596

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P.Ståhle et al. / Structural Integrity Procedia 00 (2016) 000–000

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In the present study ( ) represents the phase variable ψ or the displacements u i . According to Lagranian formalism the following relate the variation to partial derivatives of F with respect to scalar variables ( ) and its spatial derivatives ( ) , i as follows, δ F δ ( ) = ∂ F ∂ ( ) − ∂ ∂ x i ∂ F ∂ ( ) , i . (4) The notation for spatial derivatives is used for indices i , j and k . The variations with respect to δψ are readily obtained for the chemical and gradient energy densities as δ F ch δψ = − p (1 − ψ 2 ) ψ and δ F gr δψ = − g b ψ , ii . (5) Further, stresses are given by σ i j ( ψ, u i ) = 2 μ� e i j ( ψ, u i ) + λδ i j � e kk ( ψ, u i ) , (6) where μ and λ are the Lame´ parameters and δ i j is the Kronecker delta. The total strain � i j is defined as follows, The stress free linear expansion, � s ( ψ ), is chosen to be 0 for ψ = − 1 and � v / 3 for ψ = 1, where � v is the volumetric stress free expansion. Further, it is necessary that � � s ( ψ ) = 0 at | ψ | = 1 to avoid discontinuous behaviour at the edges of the permissible interval for ψ . The simplest polynomial that fulfils the conditions is � s ( ψ ) = − 1 12 ( ψ 3 − 3 ψ − 2) � v . (9) Insertion of Eqs. (6) and (8) into (2) gives the following expression F el = � i j 0 σ i j d � i j = � i j 0 σ i j d � e i j + � i j 0 σ i j d � s = 1 2 σ i j � e i j + σ ii � s . (10) Application of (4) and (9) gives on, δ F el δψ = 1 4 σ ii (1 − ψ 2 ) � v , (11) Application of (3) on the results (5) and (11) gives the governing equation for the evolution of ψ as follows Regarding the displacements, u i , the governing equation can be found as following with the same formalism as for the phase variable. Obviously δ F ch δ u i = δ F gr δ u i = 0 . (13) After using (4) and (7) one obtains δ F el δ u i = − ∂ ∂ x j ∂ F el ∂ ( u i , j ) = − 1 2 { 2 μ� i j , j + λ� j j , i } − (3 λ + 2 μ ) � s ( ψ ) , i . (14) � i j ( u i ) = 1 2 ( u i , j + u i , j ) . (7) where � e i j is the elastic strain, defined as � e i j ( ψ, u i ) = � i j ( u i ) − δ i j � s ( ψ, u i ) , (8) ∂ψ ∂ t = L ψ p (1 − ψ 2 ) ψ + g b ψ , ii + 1 4 σ ii (1 − ψ 2 ) � v . (12)