- Email: [email protected]
Macro-scale fatigue fracture analysis of multiphase bodies, aircraft design, and catastrophic failure: Two aircraft accidents
Accepted Manuscript Macro-scale fatigue fracture analysis of multiphase bodies, aircraft design, and catastrophic failure: two aircraft accidents John C. Slattery, Paul G.A. Cizmas PII: DOI: Reference:
S0013-7944(17)31111-6 https://doi.org/10.1016/j.engfracmech.2018.05.008 EFM 5982
To appear in:
Engineering Fracture Mechanics
Received Date: Revised Date: Accepted Date:
22 October 2017 24 April 2018 4 May 2018
Please cite this article as: Slattery, J.C., Cizmas, P.G.A., Macro-scale fatigue fracture analysis of multiphase bodies, aircraft design, and catastrophic failure: two aircraft accidents, Engineering Fracture Mechanics (2018), doi: https:// doi.org/10.1016/j.engfracmech.2018.05.008
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Macro-scale fatigue fracture analysis of multiphase bodies, aircraft design, and catastrophic failure: two aircraft accidents John C. Slattery1 and Paul G. A. Cizmas Department of Aerospace Engineering,Texas A&M University,College Station, TX 77843-3141
Abstract Slattery and Cizmas [1] introduced macro-scale fatigue fracture analysis (including the entropy inequality), using seven relatively simple examples (such as laminates) to demonstrate that fracture/failure could be predicted. Here we apply this same analysis to very complex multicomponent, multiphase systems: aircraft that suffered structural failures. The first was an Airbus 300-605R on American Airlines flight 587 that lost its vertical stabilizer while in flight. The second was a BAE Jetstream 31 SX-SKY aircraft whose right main landing gear failed on touchdown. Possible implications of this study for aircraft structural design are posed. It was not necessary to make any judgement about possible pilot error.
Nomenclature A CVR E Einput FDR G Gc KIc M ∆t γ
fresh fracture surface cockpit voice recorder Young’s modulus energy input flight data recorder acceleration of gravity critical energy release rate toughness takeoff/landing mass of aircraft critical time period thermodynamic surface tension
1. Introduction According to the NTSB report [2], on November 12, 2001, about 0916:15 eastern standard time, an Airbus Industrie A300-605R on American Airlines flight 587, crashed into a residential area of Belle Harbor, New York, shortly after takeoff from John F. Kennedy International Airport (JFK), Jamaica, New York. The right rear lug attachment for the vertical stabilizer de-laminated and fractured, and the vertical stabilizer separated from the aircraft. On February 12, 2009, the right main landing gear on a BAE Jetstream 31 SX-SKY aircraft failed as it touched down at Heraklion, Greece. Both of these accidents are analyzed using macro-scale fatigue fracture analysis. 1.1. Macro-scale fracture analysis There are several recent reviews of fracture [3, 4, 5, 6, 7, 8]. For this reason, we give only a limited discussion of the literature. We will focus our attention here on the use of the second law of thermodynamics (entropy inequality) in discussions of fracture, since it is the entropy inequality that allows us to predict failure. 1∗ Corresponding
author. E-mail: [email protected]
Preprint submitted to Elsevier
April 24, 2018
Griffith [9] is widely viewed as the father of modern fracture mechanics. His starting point was the theorem of minimum energy [10, exercise 4.10.3-2], which assumes that a body is at equilibrium. Broutman and Koboyashi [11] (see also Anderson [8]) note that “Griffith assumed the presence of very small cracks in the material and made use of Inglis’ calculation of stresses by regarding the cracks as very flat elliptical holes. . . . In the case of a thin elastic plate (plane stress) with a very flat elliptic crack . . . under uniaxial tension, Griffith’s assumptions lead to the famous Griffith equation . . . .” As we do in what follows, Griffith assumed in addition that the body was isothermal, that the surface tension was a constant, and that a cusp was formed at the fracture edge. Working in the spirit of Gurtin [12], Slattery et al. [13] and Fu and Slattery [14] (see also [15, 16]) did recognize interfacial effects to find for a single-phase, single-component body undergoing mode I fracture that the rate at which work is done by the body on the surroundings at the fracture edge, or the critical energy release rate, is Gc ≡ 4γ (1) Here γ is the thermodynamic surface tension in the fracture surface or, in effect, the force per unit length of line that each fracture surface exerts on the fracture edge. Slattery and Cizmas [1, Appendix A] show that this result is not limited to mode I fractures. Slattery and Cizmas [1] further use the macroscopic energy balance and the macroscopic entropy inequality to create an upper bound for any estimate of the fresh surface area Aestimate created by a fracture as the result of an energy input Einput to a macroscopic body: Aestimate ≤
Einput 2γ
(2)
In deriving (2), the following considerations were made: 1. A fracture has been initiated. 2. No assumptions were made about the shape of the body under question. 3. It was assumed that fracture is a stochastic process and that the area of newly created fracture surface does not have an experimentally repeatable value. 4. Following Slattery et al. [13], we assumed that the fracture forms a cusp at the curve representing the fracture edge. 5. The effects of gravity were neglected. 6. It was assumed that the body has a uniform temperature. There is no mass transfer across the boundary of the body, and the macroscopic mass balance is satisfied identically. Further, they demonstrated the usefulness of (2) by applying it to seven different experimental studies of macro-scale, multiphase bodies taken from the literature. But it should be noted that in each of these cases the body was treated as a single phase in evaluating the mechanical properties. While complex, none of these examples approached the scale of an entire aircraft. Finally, to our knowledge the entropy inequality (second law) has never been employed in aircraft design. The surrounding air is a Newtonian fluid, which automatically satisfies the entropy inequality [17, p. 277]. All of the solid materials of which we are aware do not automatically satisfy the entropy inequality, although in static or quasi-static analyses there is no energy dissipation and as a result it plays no role [10, sec. 493]. 2. Objective Our objective here is to show that the fracture analysis of Slattery and Cizmas [1] can be applied to macro-scale, multiphase bodies undergoing fatigue fracture. Two aircraft accidents are used as examples of how the theory can be used. We make no judgment about pilot behavior, the safety of particular aircraft, or the accuracy of previous accident reports. 3. Two example applications What follows are two example applications taken from the literature. 2
3.1. Analysis of Flight 587 [2, sec. 2.16] In order to pursue this objective, we must estimate the cumulative energy input to the aircraft during the most critical period of its short flight [2, secs. 1.1 and 1.6]. Keep in mind that (2) refers to the first appearance of a fracture or de-lamination. Note that the quotations that follow from the NTSB accident report refer to the uncorrected time scale of the flight data recorder (FDR), rather than the corrected time scale in Figure 1. • “. . . flight 587 started its takeoff roll about 0913:51 and lifted off about 0914:29 . . . .” The takeoff mass was 349,370 lbmass or 158,471 kg. • “. . . about 0915:36, the airplane experienced a 0.04 G drop in longitudinal load factor, a 0.07 G shift to the left in lateral load factor, and about a 0.3 G drop in normal (vertical) load factor . . . .” • “. . . about 0915:51, the load factors began excursions that were similar to those that occurred about 0915:36: the longitudinal load factor dropped from 0.20 to 0.14 G, the lateral load factor shifted 0.05 G to the left, and the normal load factor dropped from 1.0 to 0.6 G. ” • “At 0915:51.8, 0915:52.3, and 0915:52.9, the CVR recorded the sound of a thump, a click and two thumps, respectively.” A study of the sound spectrum recorded on the CVR suggested that the “. . . two thumps at 0915:52.9 were associated with movement of cockpit item in response to the airplanes encounter with wake turbulence.” There was no comment about the first thump at 0915:51. We will assume that de-lamination of the right rear lug occurred at 0915:51, when the sound of the first thump was recorded, or 843.7 on the corrected time scale shown in Figure 1. • At each point in time, the displacement of the aircraft per unit time per unit time is lateral load factor × G • The corresponding force is M × lateral load factor × G • The corresponding energy input to the aircraft per unit time per unit time is M × (lateral load factor × G)2 • Since there may have been significant energy input before 839s, the minimum cumulative energy input to the system is estimated for the period shown in Figure 1 ending with the first thump and using the corrected (upper) time scale): Z 843.7 Z 843.7 Einput = M × lateral load factor2 × G2 dt dt (3) 839 839 = 10, 195 J There are two ways to illustrate how the entropy inequality as embodied in (2) might have been used in the design of this aircraft in order to recognize that it would fail under the conditions encountered. 1. The focus could have been on the six main attachment lugs, each one of which could be considered the weakest portion of the structure as judged by their critical energy release rates. The vertical stabilizer was fastened to the fuselage by “six main attachment fittings and the six transverse load fittings” [2, p. 36]. This report concluded that the first component of the aircraft to fail was the lug portion of the right rear main attachment fitting. While we have no information concerning the transverse load fittings, Slattery and Cizmas [1] re-analyzed experiments reported by Davila et al. [19, figure 16] to conclude that, at failure, the energy input to a similar lug was 3083 J. They also used (2) to estimate for the lug that the energy input was 2857 J. In summary, Einput = 10, 195 J > 3, 083 J > 2, 857 J 3
(4)
It is reasonable that the critical energy input to the aircraft is larger than the critical energy input to the right-rear lug, since initially most of this energy is stored as strain energy in the rest of the aircraft. It is only after the aircraft has reached its maximum capacity for stored strain energy that strain energy can begin to be released by fracturing, first, the right-rear lug. 2. Given γ = 5.5 × 10−4 Jm−2 [1, Table 3], we can use (2) to estimate A ≤ 0.0926 m2
(5)
Since there was no evidence left from the original accident site, this can be compared only with the conservative estimate given by Slattery and Cizmas [1]: 0.0260 m2 . 3.2. Fatigue fracture of the main landing gear on a BAE Jetstream 31 SX-SKY aircraft [20] On February 12, 2009, the right main landing gear on a BAE Jetstream 31 SX-SKY aircraft failed as it touched down at Heraklion, Greece. At the point of failure, the landing mass was M = 6, 759 kg. The
Figure 1: Longitudinal, lateral, and vertical load factors as functions of time, taken from [18, figure 3g]. The time scale corresponds to their simulation. The Flight Data Recorder (FDR) time scale is at the bottom.
4
normal component of velocity was v = 3.05 m/s. Aluminum-zinc-magnesium type DTD 5,094 alloy was used for the landing gear of this aircraft [20, p. 477]. Because there are two pieces of information missing from their discussion of this accident, we can give only an illustrative example of how (2) could be used to complement their discussion. • While they recommend that the Young’s modulus E = 70 GPa, they gave no values for either the toughness or the critical energy release rate. We were unable to locate values for either of these parameters in the literature, with one questionable exception. Evans [21] reported values for the toughness, but gave no units. In the context of the period in which Evans [21] worked, and for the purpose of illustration, we will assume that he meant to say that the toughness √ KIc = 14.7to 21.7 ksi inch √ (6) = 14.7to 21.7 kilo psi inch √ 10 10 = 1.61 × 10 to 2.38 × 10 Pa m • We know that there were 148 landings, since the last major overhaul of the landing gear. We also know that two of the last five landings were “hard,” or 40%. In order to estimate the cumulative energy input to the landing gear required by (1), we will assume that 1/3 of the landings since overhaul were “hard” and similar to the last landing. We will also assume that the energy transferred to the landing gear during landing yields small fractures that finally connect and result in a macroscopic fracture. This allows us to estimate 1 Einput = (148/3) M v 2 2 = 1.55 × 106 J
(7)
Turning our attention back to (1), we estimate KIc 2 E = 3.73 × 109 to 8.12 × 109 J m−2
Gc =
Gc 4 = 9.32 × 108 to 2.03 × 109 J m−2
γ=
(8)
(9)
We do not know how much damage was done at touchdown as opposed to later as the wing and propeller hit the runway. Therefore, to be conservative, we will assume that the fresh fracture found in the upper Pintle housing was the immediate result of the impact at touchdown. Our interpretation of Chondrou et al. [20, fig. 9] is that the fresh surface area created by this fracture A = 812 mm2 . We can now use (2) to calculate an upper bound for A = 0.812 × 10−3 m2 Einput ≤ 2γ ≤ 0.382 × 10−3 to 0.832 × 10−3 m2
(10)
4. Conclusions For the first time to our knowledge, the macro-scale or integral entropy inequality (the second law of thermodynamics) in the form of (2) has been used to predict the fatigue failure of an aircraft structure. Two accidents are analyzed. We recommend using the macro-scale entropy inequality in aircraft design for three reasons. 5
1. In mechanics, the entropy inequality has the same importance as the mass, momentum, and energy balances. Its importance is sometimes overlooked because, as explained in the introduction, in some cases it can be satisfied identically. In solid mechanics, it is satisfied identically only in static (or quasistatic) computations. 2. By using the macro-scale entropy inequality in the form of (2), one can estimate the energy input to the aircraft structure at which it will fail. To do so, on must identify the weakest part of the structure (having the smallest critical energy release rate) as well as the worst conditions to be encountered. 3. While not eliminating the need for a safety/reserve factor, it can be reduced, resulting in a reduction of the weight of the aircraft. It is important to recognize that our only concern here has been the accumulation of strain energy in the structure, no matter what the cause: pilot error, wake turbulence or hard landings. Acknowledgment The authors thank BLS for invaluable assistance. References [1] J. C. Slattery and P. G. A. Cizmas. ites: Theory and seven applications. doi:10.1016/j.engfracmech.2016.06.003.
Macro-scale fracture analysis of isothermal composEngineering Fracture Mechanics, 163:366–380, 2016.
[2] Aircraft accident report: In-flight separation of vertical stabilizer, American Airlines Flight 587, Airbus Industrie A300-605R, N14053, Belle Harbor, New York, November 12, 2001. Technical Report NTSB/AAR-04/04, PB2004-910404, Notation 7439B, National Transportation Safety Board, October 2004. [3] K. Park and G. H. Paulino. Cohesive zone models: A critical review of traction-separation relationships across fracture surfaces. Applied Mechanics Reviews, 64:1–20, 2013. [4] L. P. Pook. A 50-year retrospective review of three-dimensional effects at cracks and sharp notches. Fatigue & Fracture of Engineering Materials & Structures, 36:699–723, 2013. [5] P. F. Liu and J. Y. Zheng. Recent developments on damage modeling and finite element analysis composite laminates: A review. Materials and Design, 31:3825–3834, 2010. [6] T. Sendova and J. R. Walton. A new approach to the modeling and analysis of fracture through extension of continuum mechanics to the nanoscale. Mathematics and Mechanics of Solids, 15:368–413, 2010. [7] R. Tian, S. Chan, S. Tang, A. M. Kopacz, J. S. Wang, H. J. Jou, L. Siad, L. E. Lindgren, G. B. Olson, and W. K. Liu. A multiresolution continuum simulation of the ductile fracture process. Journal of the Mechanics and Physics of Solids, 58:1681–1700, 2010. [8] T. L. Anderson. Fracture Mechanics: Fundamentals and Applications, Third Edition. CRC Press, 2005. [9] A. A. Griffith. The phenomena of rupture and flow in solids. Phil. Trans. Royal Soc. London, A221: 163–197, 1921. [10] John C. Slattery, Leonard Sagis, and Eun-Suok Oh. Interfacial Transport Phenomena. Springer-Verlag, New York, 2 edition, 2007. [11] L. J. Broutman and T. Koboyashi. Crack propagation studies in glass polymers. Technical report, Illinois Institute of Technology, Chicago, IL, September 1971. prepared for the Army Materials and Mechanics Research Center.
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[12] M. E. Gurtin. On the energy release rate in quasi-static elastic crack propagation. J. Elasticity, 9(2): 187–195, 1979. [13] J. C. Slattery, K. B. Fu, and E. S. Oh. The mechanics and thermodynamics of edge fracture: the critical energy release rate, the compatibility constraint, and the bond potential. Phil. Mag., 92:1788–1802, 2012. [14] K. B. Fu and J. C. Slattery. Boundary conditions for phase transitions and interfacial reactions. Phil. Mag., 93:1873–1882, 2013. [15] J. C. Slattery, X. Si, K. B. Fu, and E. S. Oh. Compatibility constraint at interfaces with elastic, crystalline solids II: applications. Phil. Mag., 2009. [16] X. Si, E. S. Oh, and J. C. Slattery. Compatibility constraint at interfaces with elastic, crystalline solids I: theory. Phil. Mag., 90:655–663, 2010. [17] J. C. Slattery. Advanced Transport Phenomena. Cambridge University Press, Cambridge, first edition, 1999. [18] J. J. O’Callaghan. Flight control and wake turbulence effects on American Airlines flight 587. In AIAA Modeling and Simulation Technologies Conference and Exhibit 15-18 August 2005, 2005. [19] C. G Davila, P. P. Camanho, and A. Turon. Cohesive elements for shells. Technical Report NASA/TP2007-214869, NASA, 2007. [20] I. T. Chondrou, G. Mavrantonakis, N. Tsagarakis, E. Vergis, D. Pangalos, and T. G. Chondros. Design evaluation of the fractured main landing gear of a BEA Jetstream SX-SKY aircraft. International Journal of Structural Integrity, 6:468–492, 2015. [21] G. B. Evans. The choice of materials. In A. M. Lovelace, editor, Symposium on engineering practice to avoid stress corrosion cracking. North Atlantic Treaty Organization Advisory Group for Aerospace Research and Development, 1969.
7
Research Highlights An inequality to estimate fatigue fracture creation in multiphase bodies is proposed. The inequality uses macroscopic energy balance and macroscopic entropy inequality. The method is illustrated for two complex multicomponent, multiphase systems.
1
S0013-7944(17)31111-6 https://doi.org/10.1016/j.engfracmech.2018.05.008 EFM 5982
To appear in:
Engineering Fracture Mechanics
Received Date: Revised Date: Accepted Date:
22 October 2017 24 April 2018 4 May 2018
Please cite this article as: Slattery, J.C., Cizmas, P.G.A., Macro-scale fatigue fracture analysis of multiphase bodies, aircraft design, and catastrophic failure: two aircraft accidents, Engineering Fracture Mechanics (2018), doi: https:// doi.org/10.1016/j.engfracmech.2018.05.008
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Macro-scale fatigue fracture analysis of multiphase bodies, aircraft design, and catastrophic failure: two aircraft accidents John C. Slattery1 and Paul G. A. Cizmas Department of Aerospace Engineering,Texas A&M University,College Station, TX 77843-3141
Abstract Slattery and Cizmas [1] introduced macro-scale fatigue fracture analysis (including the entropy inequality), using seven relatively simple examples (such as laminates) to demonstrate that fracture/failure could be predicted. Here we apply this same analysis to very complex multicomponent, multiphase systems: aircraft that suffered structural failures. The first was an Airbus 300-605R on American Airlines flight 587 that lost its vertical stabilizer while in flight. The second was a BAE Jetstream 31 SX-SKY aircraft whose right main landing gear failed on touchdown. Possible implications of this study for aircraft structural design are posed. It was not necessary to make any judgement about possible pilot error.
Nomenclature A CVR E Einput FDR G Gc KIc M ∆t γ
fresh fracture surface cockpit voice recorder Young’s modulus energy input flight data recorder acceleration of gravity critical energy release rate toughness takeoff/landing mass of aircraft critical time period thermodynamic surface tension
1. Introduction According to the NTSB report [2], on November 12, 2001, about 0916:15 eastern standard time, an Airbus Industrie A300-605R on American Airlines flight 587, crashed into a residential area of Belle Harbor, New York, shortly after takeoff from John F. Kennedy International Airport (JFK), Jamaica, New York. The right rear lug attachment for the vertical stabilizer de-laminated and fractured, and the vertical stabilizer separated from the aircraft. On February 12, 2009, the right main landing gear on a BAE Jetstream 31 SX-SKY aircraft failed as it touched down at Heraklion, Greece. Both of these accidents are analyzed using macro-scale fatigue fracture analysis. 1.1. Macro-scale fracture analysis There are several recent reviews of fracture [3, 4, 5, 6, 7, 8]. For this reason, we give only a limited discussion of the literature. We will focus our attention here on the use of the second law of thermodynamics (entropy inequality) in discussions of fracture, since it is the entropy inequality that allows us to predict failure. 1∗ Corresponding
author. E-mail: [email protected]
Preprint submitted to Elsevier
April 24, 2018
Griffith [9] is widely viewed as the father of modern fracture mechanics. His starting point was the theorem of minimum energy [10, exercise 4.10.3-2], which assumes that a body is at equilibrium. Broutman and Koboyashi [11] (see also Anderson [8]) note that “Griffith assumed the presence of very small cracks in the material and made use of Inglis’ calculation of stresses by regarding the cracks as very flat elliptical holes. . . . In the case of a thin elastic plate (plane stress) with a very flat elliptic crack . . . under uniaxial tension, Griffith’s assumptions lead to the famous Griffith equation . . . .” As we do in what follows, Griffith assumed in addition that the body was isothermal, that the surface tension was a constant, and that a cusp was formed at the fracture edge. Working in the spirit of Gurtin [12], Slattery et al. [13] and Fu and Slattery [14] (see also [15, 16]) did recognize interfacial effects to find for a single-phase, single-component body undergoing mode I fracture that the rate at which work is done by the body on the surroundings at the fracture edge, or the critical energy release rate, is Gc ≡ 4γ (1) Here γ is the thermodynamic surface tension in the fracture surface or, in effect, the force per unit length of line that each fracture surface exerts on the fracture edge. Slattery and Cizmas [1, Appendix A] show that this result is not limited to mode I fractures. Slattery and Cizmas [1] further use the macroscopic energy balance and the macroscopic entropy inequality to create an upper bound for any estimate of the fresh surface area Aestimate created by a fracture as the result of an energy input Einput to a macroscopic body: Aestimate ≤
Einput 2γ
(2)
In deriving (2), the following considerations were made: 1. A fracture has been initiated. 2. No assumptions were made about the shape of the body under question. 3. It was assumed that fracture is a stochastic process and that the area of newly created fracture surface does not have an experimentally repeatable value. 4. Following Slattery et al. [13], we assumed that the fracture forms a cusp at the curve representing the fracture edge. 5. The effects of gravity were neglected. 6. It was assumed that the body has a uniform temperature. There is no mass transfer across the boundary of the body, and the macroscopic mass balance is satisfied identically. Further, they demonstrated the usefulness of (2) by applying it to seven different experimental studies of macro-scale, multiphase bodies taken from the literature. But it should be noted that in each of these cases the body was treated as a single phase in evaluating the mechanical properties. While complex, none of these examples approached the scale of an entire aircraft. Finally, to our knowledge the entropy inequality (second law) has never been employed in aircraft design. The surrounding air is a Newtonian fluid, which automatically satisfies the entropy inequality [17, p. 277]. All of the solid materials of which we are aware do not automatically satisfy the entropy inequality, although in static or quasi-static analyses there is no energy dissipation and as a result it plays no role [10, sec. 493]. 2. Objective Our objective here is to show that the fracture analysis of Slattery and Cizmas [1] can be applied to macro-scale, multiphase bodies undergoing fatigue fracture. Two aircraft accidents are used as examples of how the theory can be used. We make no judgment about pilot behavior, the safety of particular aircraft, or the accuracy of previous accident reports. 3. Two example applications What follows are two example applications taken from the literature. 2
3.1. Analysis of Flight 587 [2, sec. 2.16] In order to pursue this objective, we must estimate the cumulative energy input to the aircraft during the most critical period of its short flight [2, secs. 1.1 and 1.6]. Keep in mind that (2) refers to the first appearance of a fracture or de-lamination. Note that the quotations that follow from the NTSB accident report refer to the uncorrected time scale of the flight data recorder (FDR), rather than the corrected time scale in Figure 1. • “. . . flight 587 started its takeoff roll about 0913:51 and lifted off about 0914:29 . . . .” The takeoff mass was 349,370 lbmass or 158,471 kg. • “. . . about 0915:36, the airplane experienced a 0.04 G drop in longitudinal load factor, a 0.07 G shift to the left in lateral load factor, and about a 0.3 G drop in normal (vertical) load factor . . . .” • “. . . about 0915:51, the load factors began excursions that were similar to those that occurred about 0915:36: the longitudinal load factor dropped from 0.20 to 0.14 G, the lateral load factor shifted 0.05 G to the left, and the normal load factor dropped from 1.0 to 0.6 G. ” • “At 0915:51.8, 0915:52.3, and 0915:52.9, the CVR recorded the sound of a thump, a click and two thumps, respectively.” A study of the sound spectrum recorded on the CVR suggested that the “. . . two thumps at 0915:52.9 were associated with movement of cockpit item in response to the airplanes encounter with wake turbulence.” There was no comment about the first thump at 0915:51. We will assume that de-lamination of the right rear lug occurred at 0915:51, when the sound of the first thump was recorded, or 843.7 on the corrected time scale shown in Figure 1. • At each point in time, the displacement of the aircraft per unit time per unit time is lateral load factor × G • The corresponding force is M × lateral load factor × G • The corresponding energy input to the aircraft per unit time per unit time is M × (lateral load factor × G)2 • Since there may have been significant energy input before 839s, the minimum cumulative energy input to the system is estimated for the period shown in Figure 1 ending with the first thump and using the corrected (upper) time scale): Z 843.7 Z 843.7 Einput = M × lateral load factor2 × G2 dt dt (3) 839 839 = 10, 195 J There are two ways to illustrate how the entropy inequality as embodied in (2) might have been used in the design of this aircraft in order to recognize that it would fail under the conditions encountered. 1. The focus could have been on the six main attachment lugs, each one of which could be considered the weakest portion of the structure as judged by their critical energy release rates. The vertical stabilizer was fastened to the fuselage by “six main attachment fittings and the six transverse load fittings” [2, p. 36]. This report concluded that the first component of the aircraft to fail was the lug portion of the right rear main attachment fitting. While we have no information concerning the transverse load fittings, Slattery and Cizmas [1] re-analyzed experiments reported by Davila et al. [19, figure 16] to conclude that, at failure, the energy input to a similar lug was 3083 J. They also used (2) to estimate for the lug that the energy input was 2857 J. In summary, Einput = 10, 195 J > 3, 083 J > 2, 857 J 3
(4)
It is reasonable that the critical energy input to the aircraft is larger than the critical energy input to the right-rear lug, since initially most of this energy is stored as strain energy in the rest of the aircraft. It is only after the aircraft has reached its maximum capacity for stored strain energy that strain energy can begin to be released by fracturing, first, the right-rear lug. 2. Given γ = 5.5 × 10−4 Jm−2 [1, Table 3], we can use (2) to estimate A ≤ 0.0926 m2
(5)
Since there was no evidence left from the original accident site, this can be compared only with the conservative estimate given by Slattery and Cizmas [1]: 0.0260 m2 . 3.2. Fatigue fracture of the main landing gear on a BAE Jetstream 31 SX-SKY aircraft [20] On February 12, 2009, the right main landing gear on a BAE Jetstream 31 SX-SKY aircraft failed as it touched down at Heraklion, Greece. At the point of failure, the landing mass was M = 6, 759 kg. The
Figure 1: Longitudinal, lateral, and vertical load factors as functions of time, taken from [18, figure 3g]. The time scale corresponds to their simulation. The Flight Data Recorder (FDR) time scale is at the bottom.
4
normal component of velocity was v = 3.05 m/s. Aluminum-zinc-magnesium type DTD 5,094 alloy was used for the landing gear of this aircraft [20, p. 477]. Because there are two pieces of information missing from their discussion of this accident, we can give only an illustrative example of how (2) could be used to complement their discussion. • While they recommend that the Young’s modulus E = 70 GPa, they gave no values for either the toughness or the critical energy release rate. We were unable to locate values for either of these parameters in the literature, with one questionable exception. Evans [21] reported values for the toughness, but gave no units. In the context of the period in which Evans [21] worked, and for the purpose of illustration, we will assume that he meant to say that the toughness √ KIc = 14.7to 21.7 ksi inch √ (6) = 14.7to 21.7 kilo psi inch √ 10 10 = 1.61 × 10 to 2.38 × 10 Pa m • We know that there were 148 landings, since the last major overhaul of the landing gear. We also know that two of the last five landings were “hard,” or 40%. In order to estimate the cumulative energy input to the landing gear required by (1), we will assume that 1/3 of the landings since overhaul were “hard” and similar to the last landing. We will also assume that the energy transferred to the landing gear during landing yields small fractures that finally connect and result in a macroscopic fracture. This allows us to estimate 1 Einput = (148/3) M v 2 2 = 1.55 × 106 J
(7)
Turning our attention back to (1), we estimate KIc 2 E = 3.73 × 109 to 8.12 × 109 J m−2
Gc =
Gc 4 = 9.32 × 108 to 2.03 × 109 J m−2
γ=
(8)
(9)
We do not know how much damage was done at touchdown as opposed to later as the wing and propeller hit the runway. Therefore, to be conservative, we will assume that the fresh fracture found in the upper Pintle housing was the immediate result of the impact at touchdown. Our interpretation of Chondrou et al. [20, fig. 9] is that the fresh surface area created by this fracture A = 812 mm2 . We can now use (2) to calculate an upper bound for A = 0.812 × 10−3 m2 Einput ≤ 2γ ≤ 0.382 × 10−3 to 0.832 × 10−3 m2
(10)
4. Conclusions For the first time to our knowledge, the macro-scale or integral entropy inequality (the second law of thermodynamics) in the form of (2) has been used to predict the fatigue failure of an aircraft structure. Two accidents are analyzed. We recommend using the macro-scale entropy inequality in aircraft design for three reasons. 5
1. In mechanics, the entropy inequality has the same importance as the mass, momentum, and energy balances. Its importance is sometimes overlooked because, as explained in the introduction, in some cases it can be satisfied identically. In solid mechanics, it is satisfied identically only in static (or quasistatic) computations. 2. By using the macro-scale entropy inequality in the form of (2), one can estimate the energy input to the aircraft structure at which it will fail. To do so, on must identify the weakest part of the structure (having the smallest critical energy release rate) as well as the worst conditions to be encountered. 3. While not eliminating the need for a safety/reserve factor, it can be reduced, resulting in a reduction of the weight of the aircraft. It is important to recognize that our only concern here has been the accumulation of strain energy in the structure, no matter what the cause: pilot error, wake turbulence or hard landings. Acknowledgment The authors thank BLS for invaluable assistance. References [1] J. C. Slattery and P. G. A. Cizmas. ites: Theory and seven applications. doi:10.1016/j.engfracmech.2016.06.003.
Macro-scale fracture analysis of isothermal composEngineering Fracture Mechanics, 163:366–380, 2016.
[2] Aircraft accident report: In-flight separation of vertical stabilizer, American Airlines Flight 587, Airbus Industrie A300-605R, N14053, Belle Harbor, New York, November 12, 2001. Technical Report NTSB/AAR-04/04, PB2004-910404, Notation 7439B, National Transportation Safety Board, October 2004. [3] K. Park and G. H. Paulino. Cohesive zone models: A critical review of traction-separation relationships across fracture surfaces. Applied Mechanics Reviews, 64:1–20, 2013. [4] L. P. Pook. A 50-year retrospective review of three-dimensional effects at cracks and sharp notches. Fatigue & Fracture of Engineering Materials & Structures, 36:699–723, 2013. [5] P. F. Liu and J. Y. Zheng. Recent developments on damage modeling and finite element analysis composite laminates: A review. Materials and Design, 31:3825–3834, 2010. [6] T. Sendova and J. R. Walton. A new approach to the modeling and analysis of fracture through extension of continuum mechanics to the nanoscale. Mathematics and Mechanics of Solids, 15:368–413, 2010. [7] R. Tian, S. Chan, S. Tang, A. M. Kopacz, J. S. Wang, H. J. Jou, L. Siad, L. E. Lindgren, G. B. Olson, and W. K. Liu. A multiresolution continuum simulation of the ductile fracture process. Journal of the Mechanics and Physics of Solids, 58:1681–1700, 2010. [8] T. L. Anderson. Fracture Mechanics: Fundamentals and Applications, Third Edition. CRC Press, 2005. [9] A. A. Griffith. The phenomena of rupture and flow in solids. Phil. Trans. Royal Soc. London, A221: 163–197, 1921. [10] John C. Slattery, Leonard Sagis, and Eun-Suok Oh. Interfacial Transport Phenomena. Springer-Verlag, New York, 2 edition, 2007. [11] L. J. Broutman and T. Koboyashi. Crack propagation studies in glass polymers. Technical report, Illinois Institute of Technology, Chicago, IL, September 1971. prepared for the Army Materials and Mechanics Research Center.
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[12] M. E. Gurtin. On the energy release rate in quasi-static elastic crack propagation. J. Elasticity, 9(2): 187–195, 1979. [13] J. C. Slattery, K. B. Fu, and E. S. Oh. The mechanics and thermodynamics of edge fracture: the critical energy release rate, the compatibility constraint, and the bond potential. Phil. Mag., 92:1788–1802, 2012. [14] K. B. Fu and J. C. Slattery. Boundary conditions for phase transitions and interfacial reactions. Phil. Mag., 93:1873–1882, 2013. [15] J. C. Slattery, X. Si, K. B. Fu, and E. S. Oh. Compatibility constraint at interfaces with elastic, crystalline solids II: applications. Phil. Mag., 2009. [16] X. Si, E. S. Oh, and J. C. Slattery. Compatibility constraint at interfaces with elastic, crystalline solids I: theory. Phil. Mag., 90:655–663, 2010. [17] J. C. Slattery. Advanced Transport Phenomena. Cambridge University Press, Cambridge, first edition, 1999. [18] J. J. O’Callaghan. Flight control and wake turbulence effects on American Airlines flight 587. In AIAA Modeling and Simulation Technologies Conference and Exhibit 15-18 August 2005, 2005. [19] C. G Davila, P. P. Camanho, and A. Turon. Cohesive elements for shells. Technical Report NASA/TP2007-214869, NASA, 2007. [20] I. T. Chondrou, G. Mavrantonakis, N. Tsagarakis, E. Vergis, D. Pangalos, and T. G. Chondros. Design evaluation of the fractured main landing gear of a BEA Jetstream SX-SKY aircraft. International Journal of Structural Integrity, 6:468–492, 2015. [21] G. B. Evans. The choice of materials. In A. M. Lovelace, editor, Symposium on engineering practice to avoid stress corrosion cracking. North Atlantic Treaty Organization Advisory Group for Aerospace Research and Development, 1969.
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Research Highlights An inequality to estimate fatigue fracture creation in multiphase bodies is proposed. The inequality uses macroscopic energy balance and macroscopic entropy inequality. The method is illustrated for two complex multicomponent, multiphase systems.
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