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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significantly a ff ect the values of the e ff ective moduli for the porous piezocomposites, especially for large inclusion parts of one of the phases.

2. Homogenization of Two-Phase Piezoelectric Materials

In this section we provide theoretical background for the piezoelectric composite simulation in ACELAN COMPOS. Let Ω be a representative volume of a two-phase composite heterogeneous body composed of two piezoelectric materials Ω (1) and Ω (2) ( Ω = Ω (1) ∪ Ω (2) ), where the phase Ω (1) is the skeleton or the main media and the phase Ω (2) is the pore or inclusion. Then, for a static piezoelectric problem we have the following system of di ff erential equations L ∗ ( ∇ ) · T = 0 , ∇ · D = 0 , (1) T = c · S − e ∗ · E , (2) D = e · S + κ · E , (3) S = L ( ∇ ) · u , E = −∇ φ , (4) L ∗ ( ∇ ) =    ∂ 1 0 0 0 ∂ 3 ∂ 2 0 ∂ 2 0 ∂ 3 0 ∂ 1 0 0 ∂ 3 ∂ 2 ∂ 1 0    , ∇ =   ∂ 1 ∂ 2 ∂ 3   . (5) where T = { σ 11 , σ 22 , σ 33 , σ 23 , σ 13 , σ 12 } is the array of the stress components; S = { ε 11 , ε 22 , ε 33 , 2 ε 23 , 2 ε 13 , 2 ε 12 } is the array of the strain components; D is the electric flux density vector or the electric displacement vector; E is the electric field vector; u is the vector-function of mechanical displacement; φ is the function of electric potential; c = c E is the 6 × 6 matrix of elastic sti ff ness moduli; e is the 3 × 6 matrix of piezoelectric moduli; κ = κ S = ε S is the 3 × 3 matrix of dielectric permittivity moduli; ( ... ) ∗ is the transpose operation. In order to determine the e ff ective moduli of the composite, let us assume that at the external boundary Γ = ∂ Ω the following relations take place u = L ∗ ( x ) · S 0 , φ = − x · E 0 , x ∈ Γ , (6) where S 0 = { ε 011 , ε 022 , ε 033 , 2 ε 023 , 2 ε 013 , 2 ε 012 } ; ε 0 i j are some constant values that do not depend on x ; E 0 is some constant vector. From the solution of problem (1)–(6) for a heterogeneous representative volume we find the fields ε , E , σ and D . We note that for problem (1)–(5) the equalities ⟨ ε ⟩ = ε 0 and ⟨ E ⟩ = E 0 [Nasedkin and Shevtsova (2011)] hold, where the broken brackets denote the volume-averaged quantities ⟨ ( ... ) ⟩ = 1 | Ω | ∫ Ω ( ... ) d Ω . (7) Let us put some “equivalent” homogeneous piezoelectric medium Ω with the e ff ective moduli ˜ c , ˜ e , ˜ κ into corre spondence with initial heterogeneous medium. The static problem for an “equivalent” medium is the same problem (1)–(6), but with the e ff ective moduli. It is obvious that the solution of this problem has the following form: u = u 0 , u 0 = L ∗ ( x ) · S 0 , φ = φ 0 , φ 0 = − x · E 0 , S = S 0 , E = E 0 , T = T 0 , T 0 = ˜ c · S 0 − ˜ e ∗ · E 0 , D = D 0 , D 0 = ˜ e · S 0 + ˜ κ · E 0 . For problem (1)–(6) for the heterogeneous medium we accept the following equations for the determination of e ff ective moduli: ⟨ T ⟩ = T 0 , ⟨ D ⟩ = D 0 . Note that due to [Nasedkin and Shevtsova (2011)] the average energies are equal for both heterogeneous and “equivalent” homogeneous piezoelectric media: ⟨ T · S + D · E ⟩ / 2 = ( T 0 · S 0 + D 0 · E 0 ) / 2. Now, by using Eqs. (6), we can select such boundary conditions, that enable us to obtain obvious expressions for the e ff ective moduli. Indeed, setting in (6) S 0 = ε 0 h ζ , ζ = 1 , 2 , ..., 6 , ε 0 = const , E 0 = 0 , (8)