Impact response of functionally graded conical shells

This paper presents low velocity impact response of functionally graded pretwisted conical shells. The modiﬁed Hertzian contact law which accounts for permanent indentation is utilized to compute the contact force and other impact parameter. The time dependent equations are solved by Newmark's time integration scheme. An eight noded isoparametric quadratic element is employed in the present ﬁnite element formulation. A parametric study is carried out to investigate the effects of triggering parameters like initial velocity of impactor, twist angle, oblique impact angle, location of impact for Stainless Steel-Nickel functionally graded conical shell subjected to low velocity impact.

functionally graded material; finite element; low velocity impact; twist angle; oblique impact angle; conical shell

Authorship

Sudip Dey

Swansea University, United Kingdom, email: infosudip@gmail.com (corresponding author)Swansea UniversityUnited KingdomUnited KingdomSwansea University, United Kingdom, email: infosudip@gmail.com (corresponding author)

Sondipon Adhikari

Swansea University, United KingdomSwansea UniversityUnited KingdomUnited KingdomSwansea University, United Kingdom

Amit Karmakar

Jadavpur University, Kolkata, IndiaJadavpur UniversityIndiaKolkata, IndiaJadavpur University, Kolkata, India

SCIMAGO INSTITUTIONS RANKINGS

Swansea University, United KingdomSwansea UniversityUnited KingdomUnited KingdomSwansea University, United Kingdom

Jadavpur University, Kolkata, IndiaJadavpur UniversityIndiaKolkata, IndiaJadavpur University, Kolkata, India

Figures | Tables | Formulas

Figures (11)
Tables (1)
Formulas (26)

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Figure 1
(a) Pretwisted turbo-machinery blade (b) Conical shell model.

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Figure 2
Impact (normal and oblique) on FG conical shell by a spherical mass at centre

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Figure 3
A typical discretisation of 8 x 8 mesh on plan area with element, node numbers and the natural coordinates of an isoparametric quadratic element.

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Figure 4
(a) Contact force and plate displacement (b) Impactor's displacement and velocity with respect to time for simply supported plate impacted at the centre considering flat isotropic laminate with dimension a=b=20 cm, t=0.8 cm, time step=1.0 μ-sec, E1=E2=Ei=200 GPa, G12=G23=G13=77 GPa, ρ= 7800 kg/m3, νi=ν12=0.3, initial velocity of impactor=1.0 m/s, radius of impact=0.01 m (Goldsmith, 1960).

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Figure 5
Time histories of contact force (Fc) for Silicon Nitride and Stainless Steel functionally graded beam clamped at both ends considering time step (∆t)=1.0 μ-sec, length (Lo)=135 mm, width (b)=15 mm, thickness (t)=10 mm, impactor of mass (m_i) = 0.01 Kg, radius of impactor (r_i) = 12.7 mm, velocity of impactor=1.0 m/s [Kiani et al. (2013) ]

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Figure 6
Time convergence study for histories of contact force (Fc) of Silicon Nitride and Stainless Steel functionally graded beam clamped at both ends considering length (Lo) =135 mm, width (b) =15 mm, thickness (t) =10 mm, impactor of mass (m_i) = 0.01 Kg, radius of impactor (r_i) = 12.7 mm, velocity of impactor=1.0 m/s [Kiani et al. (2013) ].

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Figure 7
Effect of initial velocity of impactor on contact force (Fc), deflection/thickness (D/t), impactor's displacement (ID) and velocity of impactor (V) with respect to time (T) at ψ=0°, 45° for Stainless Steel Nickel functionally graded conical shells considering time step=1.0 μ-sec, N=1, t=0.002 m, L/s=0.7, ϕ_o=45°, ϕ _ve=20°.

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Figure 8
Variation of Peak value of contact force (Fc max) with material property graded index (N) at ψ=0°, 45° for pretwisted Stainless Steel Nickel functionally graded conical shells considering time step=1.0 μ-sec, N=1, t=0.002 m, L/s=0.7, ϕ_o=45°, ϕ_ve=20°.

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Figure 9
Location of impact (Spanwise and diagonal).

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Figure 10
Effect of loaction of impact on contact force (Fc), deflection/thickness (D/t) and impactor's displacement (ID) with respect to time (T) at ψ=0°, 45° for Stainless Steel Nickel functionally graded conical shells considering time step (∆t )=1.0 μ-sec, N=1, t=0.002 m, VOI=3 m/s, Lo/s=0.7, ϕ_o=45°, ϕ_ve=20°.

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Figure 11
Contact force (Fc), deflection/thickness (D/t), impactor displacement (ID) and velocity of impactor (V) with respect to time (T) at ψ=0° and 45° for pretwisted Stainless Steel-Nickel functionally graded conical shells with different oblique impact angle (βi) considering time step (Δt)=1.0 μ-sec, N=1, r1=0.2 m, VOI=3 m/s, Lo=0.8 m, t=0.002 m, L/s=0.7, ϕ_o=45°, ϕ_ve=20°.

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Table 1:
Properties of the FGM components at 300 K.

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Figure 1 (a) Pretwisted turbo-machinery blade (b) Conical shell model.

Figure 2 Impact (normal and oblique) on FG conical shell by a spherical mass at centre

Figure 3 A typical discretisation of 8 x 8 mesh on plan area with element, node numbers and the natural coordinates of an isoparametric quadratic element.

Figure 4 (a) Contact force and plate displacement (b) Impactor's displacement and velocity with respect to time for simply supported plate impacted at the centre considering flat isotropic laminate with dimension a=b=20 cm, t=0.8 cm, time step=1.0 μ-sec, E1=E2=Ei=200 GPa, G12=G23=G13=77 GPa, ρ= 7800 kg/m3, νi=ν12=0.3, initial velocity of impactor=1.0 m/s, radius of impact=0.01 m (Goldsmith, 1960).

Figure 5 Time histories of contact force (Fc) for Silicon Nitride and Stainless Steel functionally graded beam clamped at both ends considering time step (∆t)=1.0 μ-sec, length (Lo)=135 mm, width (b)=15 mm, thickness (t)=10 mm, impactor of mass (m_i) = 0.01 Kg, radius of impactor (r_i) = 12.7 mm, velocity of impactor=1.0 m/s [Kiani et al. (2013) ]

Figure 6 Time convergence study for histories of contact force (Fc) of Silicon Nitride and Stainless Steel functionally graded beam clamped at both ends considering length (Lo) =135 mm, width (b) =15 mm, thickness (t) =10 mm, impactor of mass (m_i) = 0.01 Kg, radius of impactor (r_i) = 12.7 mm, velocity of impactor=1.0 m/s [Kiani et al. (2013) ].

Figure 7 Effect of initial velocity of impactor on contact force (Fc), deflection/thickness (D/t), impactor's displacement (ID) and velocity of impactor (V) with respect to time (T) at ψ=0°, 45° for Stainless Steel Nickel functionally graded conical shells considering time step=1.0 μ-sec, N=1, t=0.002 m, L/s=0.7, ϕ_o=45°, ϕ _ve=20°.

Figure 8 Variation of Peak value of contact force (Fc max) with material property graded index (N) at ψ=0°, 45° for pretwisted Stainless Steel Nickel functionally graded conical shells considering time step=1.0 μ-sec, N=1, t=0.002 m, L/s=0.7, ϕ_o=45°, ϕ_ve=20°.

Figure 9 Location of impact (Spanwise and diagonal).

Figure 10 Effect of loaction of impact on contact force (Fc), deflection/thickness (D/t) and impactor's displacement (ID) with respect to time (T) at ψ=0°, 45° for Stainless Steel Nickel functionally graded conical shells considering time step (∆t )=1.0 μ-sec, N=1, t=0.002 m, VOI=3 m/s, Lo/s=0.7, ϕ_o=45°, ϕ_ve=20°.

Figure 11 Contact force (Fc), deflection/thickness (D/t), impactor displacement (ID) and velocity of impactor (V) with respect to time (T) at ψ=0° and 45° for pretwisted Stainless Steel-Nickel functionally graded conical shells with different oblique impact angle (βi) considering time step (Δt)=1.0 μ-sec, N=1, r1=0.2 m, VOI=3 m/s, Lo=0.8 m, t=0.002 m, L/s=0.7, ϕ_o=45°, ϕ_ve=20°.

Table 1: Properties of the FGM components at 300 K.

Material	E (N/m2)	υ	ρ (Kg/m3)
Stainless Steel (SS)	207.788 x 10⁹	0.317756	8166
Nickel (Ni)	205.098 x 10⁹	0.310000	8900
Aluminum (Al)	70.000 x 10⁹	0.300000	2707
Zirconia (ZrO₂)	151.000 x 10⁹	0.300000	3000

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Impact response of functionally graded conical shells

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