PSI - Issue 6

Anna Kudimova et al. / Procedia Structural Integrity 6 (2017) 301–308 Author name / Structural Integrity Procedia 00 (2017) 000–000

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where ζ is fixed index ranging from 1 to 6; h ζ is the vector from six-dimensional basic set for the components of the strain tensor basic set; h j = e j e j , j = 1 , 2 , 3; h 4 = ( e 2 e 3 + e 3 e 2 ) / 2; h 5 = ( e 1 e 3 + e 3 e 1 ) / 2; h 6 = ( e 1 e 2 + e 2 e 1 ) / 2; e j are the orts of the Cartesian coordinate system. From the solution of problem (1)–(6), (8) we get the calculation formulas for the elastic sti ff ness moduli and piezomoduli: ˜ c αζ = ⟨ T α ⟩ /ε 0 , α = 1 , ..., 6 , ˜ e j ζ = ⟨ D j ⟩ /ε 0 , j = 1 , 2 , 3 . (9) If in Eq. (6) we accept ( E 0 = const) S 0 = 0 , E 0 = E 0 e m , m = 1 , 2 , 3 , (10) then from (1)–(6), (10) we find ˜ e m α = −⟨ T α ⟩ / E 0 , α = 1 , 2 , ..., 6 , ˜ κ jm = ⟨ D j ⟩ / E 0 , j = 1 , 2 , 3 . (11) Using Eqs. (9), (11) with (7), from the solutions of six problems (1)–(6), (8) and from the solutions of three problems (1)–(6), (10) we can obtain the full set of the e ff ective moduli for piezoelectric composite media with arbitrary anisotropy class. This technique is suitable for both cases when the phase Ω (2) represents the elastic inclutions, where e = 0 for x ∈ Ω (2) , and when the phase Ω (2) represents the pores, where c ≈ 0, e = 0, κ = ε 0 I for x ∈ Ω (2) , ε 0 is dielectric permittivity of vacuum or of air, I is the identity matrix. Note that for the homogenization problems for two-phase piezoelectric composites in ACELAN-COMPOS pack age we can also use other less popular boundary conditions. Namely, instead of main boundary conditions (6) with linear functions we can accept natural boundary conditions with constant quantities T 0 and D 0 L ∗ ( n ) · T = L ∗ ( n ) · T 0 , n · D = n · D 0 , x ∈ Γ , (12) and the mixed boundary conditions from (9), (12) L ∗ ( n ) · T = L ∗ ( n ) · T 0 , φ = − x · E 0 , x ∈ Γ , (13) or u = L ∗ ( x ) · S 0 , n · D = n · D 0 , x ∈ Γ . (14) 3. Finite Element Approximations For solving the problems for piezoelectric body in weak forms we will use classical finite element approximation techniques [Bathe and Wilson (1976); Zienkewicz and Morgan (1993)]. Let Ω h be a region of the corresponding finite element mesh Ω h ⊆ Ω , Ω h = ∪ k Ω ek , where Ω ek is a separate finite element with number k . On the finite element mesh Ω h = ∪ k Ω ek we will find an approximation to the weak solution { u h ≈ u , φ h ≈ φ } for the static problem in the form u h ( x , t ) = N ∗ u ( x ) · U , φ h ( x , t ) = N ∗ φ ( x ) · Φ , (15) where N ∗ u is the matrix of the shape functions for the displacements, N ∗ φ is the row vector of the shape functions for the electric potential, U and Φ are the global vectors of the nodal displacements and the electric potential, respectively. In accordance with conventional finite element technique we approximate the continual weak formulation by the problem in finite-dimensional spaces. Substituting (15) and similar representations for projection functions into the weak formulation of the problem for the piezoelectric body on Ω h , with taking into account the principal boundary conditions we obtain K uu · U + K u φ · Φ = F u , (16) K ∗ u φ · U − K φφ · Φ = − F φ . (17) Here, K uu = ∑ a K ek uu , K u φ = ∑ a K ek u φ , K φφ = ∑ a K ek φφ etc. are the global matrices, obtained from the corresponding element matrices ensemble ( ∑ a ), where the element matrices take the forms K ek uu = ∫ Ω ek B e ∗ u · c · B e u d Ω , K ek u φ = ∫ Ω ek B e ∗ u · e ∗ · B e φ d Ω K ek φφ = ∫ Ω ek B e ∗ φ · κ · B e φ d Ω ,