Material cyclic behaviour of panels at supersonic speeds

Journal of Materials Processing Technology 152 (2004) 84–90

Material cyclic behaviour of panels at supersonic speeds A. Varvani-Farahani a,∗ , H. Alighanbari b a

Department of Mechanical and Industrial Engineering, Ryerson University, 350 Victoria Street, Toronto, Ont., Canada M5B 2K3 b Department of Aerospace Engineering, Ryerson University, 350 Victoria Street, Toronto, Ont., Canada M5B 2K3 Received in revised form 25 February 2004; accepted 25 February 2004

Abstract The present study develops a model to correlate the self-excited panel motion with the cyclic elastoplastic behaviour of the material causing fatigue failure. The analysis takes into account the elastic and plastic responses of materials to estimate the fatigue life of the panels subjected to limit-cycle oscillations (LCO). A damage-dynamics analysis is performed to correlate the fatigue life of a two-dimensional panel as it is subjected to the dynamic bending-stretching loads due to air flowing above the panel. From dynamic time history of the panel motion, the cyclic strain/stress histories required for fatigue damage analysis have been obtained. To assess the fatigue life of panels, an energy-critical plane fatigue damage parameter has been applied. The total damage accumulation in the loading history is computed from the summation of the normal and shear energies on the basis of cycle-by-cycle analysis. The fatigue parameter used in this study has been used to successfully predict the life of thin aluminium 7075-T6 panel subjected to the dynamic bending-stretching loads due to air flowing above the panel. © 2004 Elsevier B.V. All rights reserved. Keywords: Fatigue damage analysis; Self-excited panel motion; Elastoplastic deflection

1. Introduction Oscillating plates are greatly prone to fatigue damage phenomenon [1]. The response of plates in high supersonic speeds to a disturbance, so-called panel flutter, is a practical example. Panel flutter is a supersonic/hypersonic aeroelastic phenomenon that is often encountered in the operation of high-speed aircrafts and missiles. Investigations have proven that, under aerodynamic loading and due to geometric structural non-linearity, panels can oscillate with limited amplitude causing severe fatigue damage to the panels [2–4]. There have been a few investigations [4,5] on the long-term fatigue-damaging effect due to limit-cycle oscillations. These investigations have mostly dealt with elastic behaviour of materials and loading conditions. Xue and Mei [4] and Udrescu [5] have studied non-linear flutter and fatigue damage using finite element analysis method. They applied elastic constitutive equations and included the effect of temperature on the fatigue damage of panels. To the best of the present authors’ knowledge, there has been no study available in the literature taking into account the inelastic behaviour of panels subjected to self-excited cyclic motion causing fatigue damage. In the present study, ∗ Corresponding author. Tel.: +1-416-979-5000. E-mail address: [email protected] (A. Varvani-Farahani).

0924-0136/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2004.02.063

a damage-dynamics analysis has been performed to evaluate the fatigue life of a two-dimensional panel as it is subjected to the dynamic bending-stretching loads due to air flowing above the panel. In this analysis, von Karman non-linear strain–displacement relation [6] has been used to account for large deflections, and the quasi-steady first-order piston theory [7] has been employed for aerodynamic loading. To assess the fatigue life of the panel a fatigue damage model has been applied [8]. The fatigue model is an energy-critical plane parameter (ECPP). ECPP has been defined on the critical plane, and accounts for states of stress/strain through combinations of the normal and shear strain and stress ranges. The critical plane is identified by the largest strain and stress Mohr’s circles during the reversals of a cycle. The parameter consists of tensorial stress and strain range components acting on this critical plane experiencing the maximum damage. The total damage accumulation in the loading history is computed from the summation of the normal and shear energies on the basis of cycle-by-cycle analysis. In this damage analysis, those points with the greatest strain and stress values throughout the panel loading history are responsible for the largest physical damage on the critical plane, where the debonding of materials occurs. The formulation presented in this paper does not hold any term to reflect the effect of temperature in supersonic speeds.

A. Varvani-Farahani, H. Alighanbari / Journal of Materials Processing Technology 152 (2004) 84–90

85

The present authors are aware of the importance of the thermal effect in analyses of panels at supersonic flight speeds. The thermal environment can affect panel motions by introducing thermal in-plane forces, thermal bending moments, and altering material properties. For the present authors, the thermal analysis remains a next step consideration in the analysis of supersonic flight speed and fatigue damage calculations.

where q is the dynamic pressure, q = (1/2)ρU 2 , and β = 2 − 1)1/2 . (M∞ For the elastoplastic formulation, Mx is obtained from  h/2 Mx = σz dz (5)

2. Formulation and analysis

σ = m1 ε + m3 ε3

2.1. Equation of motion of a two-dimensional elastoplastic plate

−h/2

The stress, σ, in Eq. (5) can be expressed as a function of strain, ε, from Eq. (1). Eq. (6) is a regression of Ramberg-Osgood σ–ε relationship: (6)

where m1 and m3 are regression constants and they are material-dependent. Substituting Eqs. (2) and (6) into Eq. (5) and integrating yield   2 2 1 ∂2 w ∂ w (7) Mx = − 3m3 h2 + 20m1 h3 240 ∂x2 ∂x2

Consider an elastoplastic flat panel of length L, thickness h and density ρm undergoing a lateral motion w, as shown in Fig. 1. Air is flowing above the panel in positive x-direction at air speed U and Mach number M∞ . For an elastoplastic material, stress–strain relation is given by Ramberg-Osgood equation: σ
σ 1/n ε= + (1) E K

Nx is obtained in a similar manner. For the elastoplastic formulation, Nx is obtained from  h/2 Nx = σ dz (8)

where E, K and n are material-dependent constants. The strain–displacement relation [9] is

Substituting Eqs. (2) and (6) into Eq. (8) and integrating result in:

ε − ε0 = z

∂2 w ∂x2

(2)

where ε0 is the centreline strain and z is the normal coordinate in h-direction. The equation of motion [9] for a two-dimensional plate undergoing bending is ∂ 2 Mx ∂2 w ∂2 w + Nx 2 + ρm h 2 + (p − p∞ ) = p 2 ∂x ∂x ∂t

(3)

where Mx is the bending moment and Nx is the non-linear induced axial loading. This formulation does not include any externally applied in-plane load. The constant static pressure difference across the panel is p. The aerodynamic pressure loading p − p∞ will be assumed to be that of quasi-steady supersonic theory [2]:   2   2q ∂w M∞ − 2 1 ∂w + (4) p − p∞ = 2 − 1 U ∂t β ∂x M∞

M∞

Panel

Air flow

h

−h/2

Nx =

h 8L3 

 0

L



∂w ∂x

2

× 4ml L + m3

dx 

2

0

L



∂w ∂x

2

2 dx

 + m3 h L 2

2

∂2 w ∂x2

2

 

(9)

Eqs. (7) and (9) give Mx and Nx , respectively, as a function of the lateral deflection of the plate, w. Substituting these equations and Eq. (4) into the equation of motion (Eq. (3)), results in an integral-differential equation for the elastoplastic plate motion: 3 1 3 m3 h5 w w + m1 h3 w + m3 h5 w w2 80 12 40  L  hw h2 2 − w dx m1 + m3 2L 4 0  L 3 1 − 3 m3 hw w2 dx + ρm hw ¨ 8L 0 +

2 − 2) 2q  2q(M∞ w + w ˙ − p = 0 β β3 U

(10)

where notations w, ˙ w, ¨ w , w , w , and w are equal to 2 2 ∂w/∂t, ∂ w/∂t , ∂w/∂x, ∂2 w/∂x2 , ∂3 w/∂x3 , and ∂4 w/∂x4 , respectively. If constant m3 = 0, Eq. (10) will be limited to the elastic deflection only. 2.2. Deflection of the oscillating panel

x

L Fig. 1. Panel geometry.

Similar to earlier experimental and analytical investigations [2–5,9–11], the present study on the panel deflection

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A. Varvani-Farahani, H. Alighanbari / Journal of Materials Processing Technology 152 (2004) 84–90

Fig. 2. Panel response vs. time.

has also shown that the solution of Eq. (10) results in a limit-cycle oscillation. The panel motion amplitudes changed with time at the first few oscillations (transient period) until a limit-cycle oscillation was achieved. The maximum deflection of the panel was occurred around a distance of x = 0.75L from one end of the simply supported panel. Fig. 2 is a plot of panel response (at x = 0.75L) versus time. Analogous to the Ramberg-Osgood cyclic stress–strain relationship (Eq. (1)), the force-deflection (F–w) data for Al 7075-T6 shows the same trend, consisting of a linear region and a non-linear region (see Fig. 3). The relationship between force and deflection was determined from the F–w data reported by Wang and Pidaparti [12] for an Al 7075-T6 plate. The total elastoplastic deflection of the plate may be expressed as  1/n∗ F F w = we + wp = ∗ + (11) E K∗ where we and wp are the elastic and plastic deflections of the panel, respectively, and E∗ = 19.20 MN/m, K∗ = 0.235 MN/m, and n∗ = 0.312 are constants analogous to the constants in Ramberg-Osgood equation. Constants E∗ and K∗ were obtained for an Al 7075-T6 panel with a thickness h = 0.8 mm and length L = 250 mm. Units for F and w are in MN and m, respectively.

2.3. States of stress and strain The strain and stress tensors for a thin plate subjected to axial and bending fatigue are given by Eqs. (12) and (13), respectively:   −νeff εyy ( 21 γxy ) 0   1  εij =  (12) εyy 0   ( 2 γyx ) 0 0 −νeff εyy 

0

 σij =  τyx

τxy σyy

0

0

0



 0

(13)

0

where εyy and γ xy /2 are the range of tensorial axial and shear strain components, respectively. In Eq. (13), σ yy and τxy = τyx are the ranges of tensorial axial and shear stresses, respectively. In Eq. (12) νeff is the effective Poisson’s ratio, which is given by νeff = (νe εyye + νp εyyp )/(εyye + εyyp ), where νe = 0.3 is the elastic Poisson’s ratio and νp = 0.5 is the plastic Poisson’s ratio. The elastic strain is εyye = σyy /E and the plastic strain is εyyp = εyy − (σyy /E). The cyclic axial and the shear strain ranges obtained at angle θ during a strain cycle at which the strain Mohr’s circle 40

600

Ramberg-Osgood Eq (1) 500

Force, F (kN)

Stress (MPa)

30 400

A

300

A Plastic deflection w

20

p

Elastic deflection w

200

e

10 100 0

0 0

(a)

0.002 0.004 0.006

0.008

Strain

0.01

0

0.012 0.014

(b)

0.5

1

1.5

2

2.5

Deflection,w (mm)

Fig. 3. (a) Stress–strain curve and (b) force–deflection curve for Al 7075-T6 thin plate.

3

3.5

A. Varvani-Farahani, H. Alighanbari / Journal of Materials Processing Technology 152 (2004) 84–90

is the largest and has the maximum value of shear strain are given by Eqs. (14) and (15), respectively as: (εyy )θ = εyy sin θ

(14)

( 21 γxy )θ = ( 21 γxy ) sin θ

(15)

Oscillation history

Air flow

g

(γ/2)max τmax Tensorial stress/strain components

ε

Energy-critical plane damage parameter (ECPP)

Critical plane

Fatigue life assessment

ECPP Life

Fig. 4. Procedure of fatigue damage and life assessment of two-dimensional panels.

The range of maximum shear stress τ max and shear strain, (γmax /2), obtained from the largest stress and strain Mohr’s circles during the loading (at angle θ 1 ) and during the unloading (at angle θ 2 ) parts of a cycle and the corresponding normal stress range, σ n , and the normal strain range, εn , on that plane are the components of the fatigue damage parameter. Fig. 5 shows an example of loading history, strain and stress Mohr’s circles corresponding to the angle θ 1 (during loading) and the angle θ 2 (during unloading). This figure also presents components τ max , (γmax /2), σ n and εn used in the damage model. Both the normal energy (σn × εn ) and the shear strain energy (τmax × (γmax /2)) are weighted by the axial and shear fatigue properties, respectively [8]:
γ  1 1
max (σn εn ) +   τmax  = f(Nf ) (16)   σ f εf τ f γf 2 where σf and εf are the axial fatigue strength coefficient and axial fatigue ductility coefficient, respectively, and τf and γf are the shear fatigue strength coefficient and shear fatigue ductility coefficient, respectively.

Stress Mohr’s Circle

γ/2 3

Stresses applied to the panel

θ2

τ

θ1

Η

θ2

σ θ1

γ

θ2=270˚

∆εn

∆(g max/2)

∆τmax

ε

time

εn,

Strain Mohr’s Circle

Strain Path

θ1=90˚

Strains applied to the panel

Ramberg-Osgood Relationship

(i) The maximum deflection of the thin 2D-aluminium panel oscillating with a constant amplitude level (LCO) has been computed. (ii) The corresponding value of applied strain to the panel has been calculated from the maximum deflection of the panel taking place at about x = 0.75L. (iii) The applied strain value has been used to calculate the corresponding stress applied to the panel using Ramberg-Osgood relationship. The stress and strain values obtained from steps (ii) and (iii) have been used to calculate tensorial stress and strain components in Eqs. (12) and (13) from 3D Mohr’s circle analysis. (iv) The Varvani’s fatigue damage model [8] has been applied to assess the fatigue life of the panel. Based on this model, the total damage accumulation in the loading history is computed from the summation of the normal and shear energies on the basis of cycle-by-cycle analysis. In this damage analysis, those points with the greatest strain and stress values throughout the panel loading history are responsible for the largest physical damage on the critical plane, where the debonding of materials occurs.

Strain History

Deflection w

panel w

2.4. Fatigue damage parameter and life assessment of panels From the solution of equation of motion (Eq. (10)) for the simply supported thin aluminium panel, it is found that the deflection of the oscillating panel was maximised around a distance of x = 0.75L, where the aerodynamic loads are extensively concentrated. The aerodynamic parameters and loads applied on the panel, resulting in LCO are given in Section 3. The x = 0.75Lis a potential site for crack initiation and fatigue damage accumulation in the panel. To assess the fatigue life of the panel subjected to axial and bending loading, a procedure presented in Fig. 4 has been developed and followed as:

87

∆σn

Fig. 5. An example of strain history, strain path, strain Mohr’s circle, and stress Mohr’s circle for a biaxial loading condition.

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Table 1 Tensile and fatigue properties of Al 7075-T6 alloy [13] Tensile properties of Al 7075-T6 E (GPa)

νe

σ y (MPa)

σ ut (MPa)

σ f (MPa)

εf

G (GPa)

τ f (MPa)

(HB)

70

0.33

470

580

801

0.401

27

330

150

Fatigue properties of Al 7075-T6 σ e (MPa)

σf (MPa)

εf

K (MPa)

n

b

c

τf (MPa)

γf

160

886

0.446

913

0.088

−0.076

−0.759

680

1.6

3. Fatigue damage analysis results and discussion The fatigue response of an oscillating two-dimensional panel made of Al 7075-T6, which has widely been used for aeronautical applications, has been studied. Table 1 tabulates the material properties of Al 7075-T6 panel. The thickness and the length of the simply supported Al 7075-T6 panel has been considered to be h = 0.8 mm and L = 250 mm, respectively (see Fig. 1). The aluminium panel bending rigidity D = Eh3 /12(1 − ν2 )= 3.0 Nm and a mass density ρm = 2.70 g/cm3 . Fatigue life and elastic/plastic response of the panel under limit-cycle oscillation has been studied for three cases as below: (i) Mach number M∞ = 5, dynamic pressure q = 578 kPa: the ratio of the panel maximum deflection/panel thickness (w/ h)max = 1.90 has been obtained due to a limit-cycle oscillation. The maximum deflection of the panel around x = 178 mm was found to be wmax = 1.52 mm, which corresponds to an elastoplastic deflection of panel (see Fig. 3b). Fig. 6 presents the elastic, elastoplastic and plastic deflection profiles of the oscillating Al 7075-T6 panel. The (w/ h)max values also changed from 1.90 to 1.74 to 0.16 for elastoplastic, elastic and plastic, respectively, occurring around 0.75L which is in a good agreement with results reported by Xue and Mei [4]. The maximum value of elastoplastic deflection due to limit-cycle oscillations was used to find the corresponding stress–strain states and values required

0 Plastic deflection

w/h

0.5 Al 7075 panel h=0.8-mm L=250-mm

1.0

Elastic deflection

1.5 Elastoplastic deflection Ref. [4]

2.0 0

0.2

0.4

0.6

0.8

1

x/L Fig. 6. Elastic, elastoplastic and plastic deflections of oscillating Al 7075-T6 panel.

for the fatigue damage analysis of the panel material (see Fig. 3). The total strain and stress that calculated from methodology presented in Fig. 4 resulted in values 0.575% and 400 MPa, respectively. The values of elastic and plastic strains were found as εe = 0.569% and εp = 0.006%, respectively. Inelastic deflection value is relatively small as compared with elastic deformation of panel material. The tensorial stress and strain components of the panel have been calculated using Mohr’s circles. The total damage accumulation in loading history has been calculated from the summation of the normal and shear energies on the basis of cycle-by-cycle analysis. The normal and shear energies used in this parameter were divided by the fatigue properties provided in Table 1. The fatigue damage model (Eq. (16)) has been used to predict the life of the panel. The right side of Eq. (16) has equated the damage parameter with Coffin-Manson-type equation developed earlier by Fatemi and Socie [14] enabling to calculate the fatigue life: σf σ 2 (Nf )b + 0.3(1 + νe ) f (Nf )2b E Eσy ε σ  + (1 + νp )εf (Nf )c + 0.3(1 + νp ) f f (Nf )b+c σy (17)

f(Nf ) = (1 + νe )

Fatigue life prediction of the panel resulted in a number of cycles to failure of 135,000 at the magnitude of damage parameter of ECPP = 0.0077 using Eq. (16). Xue and Mei [4] evaluated fatigue life of a similar simply supported aluminium panel oscillating at 370 Hz with an applied stress magnitude below the yield strength. They estimated a fatigue life of 158,000 cycles for an oscillating panel, (w/ h)max = 1.90, which is in a good agreement with the life predicted by the proposed damage parameter for (w/ h)max = 1.90 in the present study. (ii) Mach number M∞ = 5, dynamic pressure q = 487 kPa: (w/ h)max was found to be 1.57. The total strain and stress that calculated from the proposed algorithm found to be 0.502% and 350 MPa, respectively (just below the point A in Fig. 3). The values of

A. Varvani-Farahani, H. Alighanbari / Journal of Materials Processing Technology 152 (2004) 84–90

elastic and plastic strains were found as εe = 0.499% and εp = 0.003%, respectively. These values revealed that the panel material deflected elastoplastically. Fatigue life prediction of the panel resulted in a number of cycles to failure of 230,000 at the magnitude of damage parameter ECPP = 0.0073. (iii) Mach number M∞ = 5, dynamic pressure q = 420 kPa: (w/ h)max was found to be 1.35. The limit-cycle oscillation resulted in a total strain of ε = 0.425% and a stress magnitude of 300 MPa. The elastic and plastic strains found to be εe = 0.4245% and εp = 0.0005%, respectively, revealing that the deformation of panel material falls within the purely linear regime. This oscillating panel results in a much longer fatigue life of 8,200,000 cycles (at ECPP = 0.0054), as compared with above two earlier cases. There has been a dilemma whether or not the stress cycles below the yield strength accumulate the fatigue damage. Theoretically, stress amplitude below the yield strength of the material corresponds to an elastic deformation and thus the material does not undergo fatigue (a reversible process). However, this may not be the case for all stress magnitudes below the yield strength. The ambiguity of fatigue damage under yield strength is associated with the overestimation introduced by the concept of the yield strength and the assumption of purely elastic deformation below this strength. Nearly all metals undergo a small amount of plastic strain even at stresses below the yield strength [15–17]. Thus fatigue damage accumulation due to such small cycles leading to crack formation can continue in the persistent slip bands at very low plastic strains. It is also important to note that the non-linear trend of stress–strain curve starts from a point well below the yield point of the material. Materials loaded under this non-linear portion require being treated elastoplastically [18,19]. In cases (i), (ii), and (iii), the maximum panel deflection resulted in strain and stress components below the traditional yield strength. Nevertheless, the oscillating aluminium panel has undergone small plastic deflections. The comparison of above three cases reveal how significant is the effect of the magnitude of inelastic deformation of oscillating panel on fatigue life of the aluminium panel studied here.

4. Conclusion The paper presents a model to correlate the self-excited panel motion with the cyclic elastoplastic behaviour of the material causing fatigue failure. A damage-dynamics analysis is performed to correlate the fatigue life of a two-dimensional panel as it is subjected to the dynamic bending-stretching loads due to air flowing above the panel. The method employs the elastic–plastic solution

89

of the two-dimensional panel problem including plasticity and dynamic pressure terms. From dynamic time history of the panel motion, the cyclic stress/strain histories required for fatigue damage analysis has been obtained. An energy-critical plane fatigue damage approach has been applied to assess the fatigue life of the panel. The total damage accumulation in the loading history is computed from the summation of the normal and shear energies on the basis of cycle-by-cycle analysis. The fatigue parameter used in this study has been used to successfully predict the life of a thin aluminium 7075-T6 panel subjected to the dynamic bending-stretching loads due to air flowing above the panel.

Acknowledgements Authors wish to thank the financial support of Natural Science and Engineering Research Council (NSERC) of Canada.

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