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All figures (14)
Figure 5: Summary of FEM discretisation spaces for examples 6.1.1 and 6.1.2 used in (a), the MŴF formulation [4] and (b), in the proposed shell formulation. Notice that the superscripts C and D have been used to indicate the continuous or discontinuous character of a field, respectively.
Figure 15: Hyperboloid piezoelectric polymer. Geometry and boundary conditions. t = 12m, D = 9.95m, H = 0.05m and q, a surface load per unit undeformed area.
Figure 6: Bending actuator configuration 1. Contour plot of σ11 (a) for the shell formulation and (b) the MŴF, respectively. Incompressible model in (52). Total number of dofs for shell and continuum formulation ({x, ϕ, p}) of 14498 dofs (refer to Figure 4) and {3 × (8019), 8019, 800} (32876 dofs) (not including the discontinuous fields, refer to Figure 5a), respectively.
Figure 1: Deformation mapping of a continuum (EAP) and associated kinematics variables: F , H , J .
Figure 16: Hyperboloid piezoelectric polymer. Contour plot of the electric potential ϕ for the proposed shell formulation. Results obtained for a value of the surface load per unit undeformed area of q = λ×(16×106)Pa, with (a) λ = 0.0065; (b) λ = 0.0221; (c) λ = 0.0716; (d) λ = 0.25; (e) λ = 0.664; (f) λ = 1. Constitutive model in (58). Finite Element discretisation of 64 × 27 elements. CBC approach for ϕ and γ with ns = 1 (47). Number of dofs for {x0,v, ϕ, γ, p, λ} of {3× 7095, 3× 1820, 3× 7095, 1× 1728, 1× 1728, 1820}.
Figure 11: Helicoidal actuator. Geometry description and electrical boundary conditions.
Figure 13: Helicoidal actuator. Contour plot of the p for the proposed shell formulation. Incompressible model in (56). Results obtained for a value of ω0 = λ × 10−5Q/m2 with (a) λ = 200; (b) λ = 300; (c) λ = 500; (d) λ = 600; (e) λ = 800; (f) λ = 900; (g) λ = 1100; (h) λ = 1200; (i) λ = 1300. Finite Element discretisation of 29× 6 elements. CBC approach for ϕ and γ with ns = 2 (47). Number of dofs for {x0,v, ϕ, γ, p, λ} of {3 × 767, 3 × 210, 5 × 767, 2 × 174, 2 × 174, 210}. Reference configuration represented by shadowed region.
Figure 3: Bending actuator. Electrical boundary conditions for: (a) configuration 1 and (b), configuration 2. a = 10m, b = 1m and H = 0.05m.
Figure 17: Hyperboloid piezoelectric polymer. Contour plot of the H22, σ33, p, ϕ, E3 and D03 for the proposed shell formulation. Constitutive model in (58). Results obtained for a value of q = λ× (16× 106)Pa. Finite Element discretisation of 64× 27 elements. CBC approach for ϕ and γ with ns = 1 (47). Number of dofs for {x0,v, ϕ, γ, p, λ} of {3×7095, 3×1820, 3×7095, 1× 1728, 1× 1728, 1820}.
Figure 10: Bending actuator configuration 2. Contour plot of σ11 (a) for the shell formulation and (b) the MŴF, respectively. Incompressible model in (52). Total number of dofs for shell and continuum formulation ({x, ϕ, p}) of 14498 (refer to Figure 9) and {3 × (8019), 8019, 800}=32876 (not including the discontinuous fields, refer to Table 5a), respectively.
Figure 4: Bending actuator configuration 1. Contour plot of ΣF 11 . Results for an applied surface electric charge of ω0 = (λ/300)×(3× 10−3) Q/m2 with λ = 25 (shadowed configuration) and λ = 42. Incompressible model in (52). Finite Element discretisation of (40×5) elements. CBC approach for {ϕ, p} and CBD for γ with ns = 4 (47). Number of dofs for {x0,v, ϕ, γ, p, λ} of {3× 891, 3× 246, 11× 891, 8× 80, 5× 80, 246} (14498 dofs).
Figure 2: Kinematics of the shell: (a) Reference and current configurations of the mid surface of the shell describing an EAP. (b) Covariant basis {G01,G02,V } in the mid surface of the shell (s = 0) and covariant basis {G1,G2,V } at a point characterised by the same convective coordinates {ηα, α = 1, 2} and s 6= 0, all in the reference configuration.
Figure 14: Helicoidal actuator. Evolution of area expansion 1/γ for a point of the shell in the reference configuration at X1 = −0.75m, X2 = 0m and X3 = 2m.
Figure 9: Bending actuator configuration 2. Contour plot of ΣF 11 . Results for an applied surface electric charge of ω0 = (λ/300) × (3× 10−3) Q/m2 with (a) λ = 7; (b) λ = 14; (c) λ = 21; (d) λ = 28; (e) λ = 35; (f) λ = 42; (g) λ = 49; (h) λ = 56; (i) λ = 63; (j) λ = 70; (k) λ = 77; (l) λ = 86. Finite Element discretisation of (40 × 5) elements. CBC approach for ϕ and CBD for γ with ns = 2 (47). Number of dofs for {x0,v, ϕ, γ, p, λ} of {3× 891, 3× 246, 11× 891, 8× 80, 5× 80, 246} (14498 dofs).
Journal Article
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DOI
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A computational framework for incompressible electromechanics based on convex multi-variable strain energies for geometrically exact shell theory
[...]
Rogelio Ortigosa
1
,
Antonio J. Gil
1
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Institutions (1)
Swansea University
1
15 Apr 2017
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Computer Methods in Applied Mechanics and Engineering