Van Paepegem, W., Decaene, R. and Degrieck, J. (5). Nonlinear correction to te bending stiffness of a damaged composite beam. Nonlinear correction to te bending stiffness of a damaged composite beam W. Van Paepegem a*, R. Decaene a and J. Degrieck a a Professor, Dept. of Mecanical Construction and Production, Sint-Pietersnieuwstraat 41, 9 Gent, Belgium Abstract Wen studying damage in composite materials, te classic beam teory is sometimes used in a modified manner to calculate te bending response of a damaged composite laminate. Te longitudinal stiffness E is ten replaced by a field variable E(x,y) = E [1-D(x,y)]. Te damage distribution D(x,y) affects te calculation of stresses and strains, and requires a modified calculation of te neutral fibre y (x) and te bending stiffness EI(x). It seems obvious to postulate tat te bending stiffness EI(x) is also multiplied wit [1-D(x,y)], as was proposed by, amongst oters, Sidoroff and Subagio (Sidoroff, F. and Subagio, B. (1987). Fatigue damage modelling of composite materials from bending tests. In : Proceedings of ICCM-VI & ECCM-II, pp. 4.3-4.39). It is sown ere tat tis formulation is not correct and tat a nonlinear correction term to te bending stiffness degradation is necessary. Tis term is especially important wen te damage D(x,y) is reacing iger values. Te results of a semi-analytical simulation for cantilever beams is sown to support te findings. Keywords: beam teory, damage, composite, bending, residual stiffness 1 Introduction Today, a lot of researc is dedicated to te understanding of damage in fibre-reinforced composites. Simple and more advanced simulation tools are used to model te mecanical response of damaged composites. One of te straigtforward calculation tools is te classic beam teory. In 1987, Sidoroff and Subagio [1] proposed a local damage model to predict te damage growt in composites subject to bending fatigue tests. Tis local damage model was combined wit an adapted formulation of te classic beam teory to calculate te stresses and strains in te damaged composite beam under fatigue loading. In tis paper it is demonstrated tat te linear relation between te degradation of te bending stiffness and te damage is not correct and a nonlinear correction term sould be added. Te relevant equations are derived and te formulations are implemented in a semi-analytical simulation of a bending fatigue test. Review of classic beam teory for damaged beams In tis paragrap, te existing approac for damage modelling wit te classic beam teory is discussed. Te work of Sidoroff and Subagio is taken as a guideline as tey worked out te equations and applied te teory to te bending fatigue of a composite beam. * Autor to wom correspondence sould be addressed ( E-mail : Wim.VanPaepegem@UGent.be ).
Van Paepegem, W., Decaene, R. and Degrieck, J. (5). Nonlinear correction to te bending stiffness of a damaged composite beam. Te local damage model, proposed by Sidoroff and Subagio [1], was a one-dimensional damage model based on te Continuum Damage Mecanics teory [-8]. Te well-known relation between stress and strain was: σ = E (1 D) ε (1) wit E te virgin elastic modulus and D te scalar damage ( D 1). Tis relation evolves from te concept of strain equivalence, introduced by Lemaitre [9], wic states tat a damaged volume of material under te nominal stress σ sows te same strain response as a comparable undamaged volume under te effective stress σ ~ (= σ/(1-d)). Te damage growt under fatigue loading was establised as follows [1]: dd dn ( Δε) c A in tension b = (1 D) () in compression were : - D : local damage variable - N : number of cycles - Δ ε : strain amplitude of te applied cyclic loading - A, b and c : tree material constants Te relevant equations for te bending response of te damaged composite beam were derived from classic beam teory as follows: wen te axial force is supposed to remain zero and wen only a bending moment exists, te position of te neutral fibre y (x) at eac moment of time is calculated as: + [ 1 D(x, y) ] + 1 y dy y (x) = (3) D(x, y) dy were : - x : coordinate along te beam lengt - y : tickness-coordinate, wit y = in te middle of te specimen s tickness - : total tickness of te specimen Te degraded bending stiffness EI(x) becomes (wit b te specimen s widt): + [ 1 D(x, y) ] EI (x) = b E y dy (4) In te next paragrap it will be sown tat tis latter equation is not correct and sould include a nonlinear correction term for te bending stiffness EI(x).
Van Paepegem, W., Decaene, R. and Degrieck, J. (5). Nonlinear correction to te bending stiffness of a damaged composite beam. 3 Derivation of te nonlinear correction term for te bending stiffness EI(x) For te derivation of te nonlinear correction term for te bending stiffness EI(x), it is again assumed tat: σ ε (5) = E (1 D) were D is a field variable D(x,y). Te x-coordinate is again along te beam lengt, wile te y-coordinate represents te beam tickness. Te following displacement field is proposed for te bending beam: u u y x (x, y) = u(x) (x, y) = u (x) + u' y (6) were u(x) is te vertical displacement of te bending beam, u (x) is te longitudinal displacement of te midplane and y is te tickness coordinate lying in te interval [-/, +/]. Te widt b is assumed to be unity for te moment. Te strain ε can ten be calculated as: And te corresponding stress σ equals: u x du ε = = + u'' y = ε + u'' y (7) x dx σ = E ε = E (1 D) ( ε + u'' y) (8) Te normal force N is assumed to be zero in pure bending: N = σ dy = E ε (1 D) dy + E u' ' (1 D) y dy = (9) From Equation (9), te strain ε can be calculated as: ε (1 D) y dy = u '' (1) (1 D) dy Ten te position of te neutral fibre y is determined from te equations (7) and (1):
Van Paepegem, W., Decaene, R. and Degrieck, J. (5). Nonlinear correction to te bending stiffness of a damaged composite beam. ε (1 D) y dy = u' ' y = u' '(y y) (1 D) dy (11) Te normal stress σ can ten be written as: σ = (1 D) u'' (y y ) (1) E Te bending moment M(x) is calculated from te expression: M (x) = σ y dy (13) Taking into account Equation (1), te bending moment M(x) can be written as: M (x) = E u' ' (1 D) y dy y (1 D) y dy (14) Te bending stiffness EI(x) is ten defined from te equation: Hence: M(x) = EI(x) u'' (15) (1 D) y dy = EI (x) E (1 D) y dy (16) (1 D) dy If Equation (16) is compared wit Equation (4) proposed by Sidoroff and Subagio [1], it is clear tat a nonlinear correction term appears in te expression for te bending stiffness EI(x). From Equation (11) and (15), it follows directly tat:
Van Paepegem, W., Decaene, R. and Degrieck, J. (5). Nonlinear correction to te bending stiffness of a damaged composite beam. M(x) (y y) ε = (17) EI(x) 4 Simulations To sow te importance of te nonlinear correction term in Equation (16), a semi-analytical simulation is performed for cantilever bending fatigue tests on a composite specimen. In Figure 1, te conventions for te simulations are sown. Te composite specimen is modelled as a cantilever beam, and te moving clamp (rigt) is represented by a rigid rod. Te prescribed displacement in te displacement-controlled bending fatigue test is u max = u(c). L a y A x B u(b) u(c) C α(b) = α(c) M [N.mm] F x [mm] - F.(L + a) Figure 1 Bending of te cantilever composite beam following te conventions of te classic beam teory. Te force necessary to impose te prescribed displacement u max = u(c), can be calculated from te following equations: u' ' M(x) L + a x = = F EI(x) EI(x) (18) x L + a x = F dx EI(x) (19) u' (x) u' (B) L L + a x = F dx () EI(x)
Van Paepegem, W., Decaene, R. and Degrieck, J. (5). Nonlinear correction to te bending stiffness of a damaged composite beam. L L + a x' u (B) = u' (x)dx = F dx dx' (1) EI(x' ) L x u (C) = u'(b) a + u(b) () L L x L + a x L + a x' u = = max u(c) F a dx + dx dx' (3) EI(x) EI(x' ) Te equation (3) provides te relation between te force F and te prescribed displacement u max. Te local damage model is very similar to te one used by Sidoroff and Subagio and as te following form: dd(x, y) dn Δσ(x,y) A σts = in tension (4) ( 1 D(x, y) ) b c in compression Here, te stress amplitude Δσ(x,y) in te damage law (4) corresponds wit te maximum stress in eac material point during te simulated fatigue loading cycle. Te material parameters E and σ TS of te (unidirectional) composite laminate were respectively 4.57 GPa and 39.7 MPa [1]. Te constants A, b and c were respectively 7.3 1-3 [1/cycle],.45 [-] and 6.5 [-] [1]. Unlike te local damage model by Sidoroff and Subagio [1], tis model includes an elastic limit of te stress and can be more easily related wit S-N curves by use of te stress amplitude instead of te strain amplitude. Of course, if one and te same damage model is applied for bot te linear and nonlinear bending stiffness degradation, it does not matter really wic damage model is cosen to demonstrate te effect of te nonlinear correction term. Tis constitutive model can be easily implemented in a matematical software package suc as Matcad. Te numerical integration formulae must be cosen suc tat te second degree polynomials are exactly integrated. Tis is te case for te Simpson s rule, wic is a Newton-Cotes quadrature formula. Because te increase of te damage variable D during one cycle is so small, te integrations must be exact indeed, oterwise te relative error on te calculation of te bending stiffness EI may be larger tan te increase of te damage variable itself. Terefore te conventional first-order trapezium metod is not suited for tis purpose. First te distribution of te bending moment along te lengt of te specimen is determined. Secondly te stresses and strains in eac integration point can be calculated. Te damage law is applied and a new cycle is evaluated. From equation (3) te necessary force to impose te displacement wit amplitude u max can be calculated for eac cycle. Tis algoritm must be solved for eac calculated loading cycle, and as a consequence, te Matcad TM workseet must be called several times. Tis is only possible witin te Matconnex TM environment wic allows to repetitively call te same Matcad TM workseet wit different starting values.
Van Paepegem, W., Decaene, R. and Degrieck, J. (5). Nonlinear correction to te bending stiffness of a damaged composite beam. 5 Simulations In te following semi-analytical simulation, te displacement-controlled bending fatigue test, sown in Figure 1, is simulated for about 3, loading cycles. Te displacement amplitude u max was cosen to be 3.4 mm. Te simulations wit linear and nonlinear bending stiffness degradation EI(x) were done wit te same material parameters E and σ TS and te same constants A, b and c for te damage law (4). Te only difference is due to te nonlinear correction term EI(x). Figure sows te simulated force-cycle istory wit linear and nonlinear bending stiffness degradation EI(x). Te initial force F = 16.68 N is te force amplitude necessary to bend te specimen into its maximum deformed state (u = u max ) during te first loading cycle. Ten, due to te damage growt, te force amplitude is decreasing during fatigue life. It is clear tat te simulated degradation of te composite specimen is muc larger in te case of te nonlinear correction term, altoug te applied fatigue damage law (4) and its constants are te same in bot cases. 11 1 Force versus number of cycles for composite specimen u max = 3.4 mm 9 8 Force [N] 7 6 5 4 3 Linear EI Nonlinear EI 1 1 3 No. of cycles [-] Figure Simulated force-cycle istory wit linear and nonlinear bending stiffness degradation EI(x). Of course, te difference in force-cycle istory sould be reflected in a difference in damage distribution as well. Tis is confirmed by Figure 3 and Figure 4 wic sow te simulated damage distribution in te clamped cross-section (x = ) wit linear and nonlinear bending stiffness degradation respectively.
Van Paepegem, W., Decaene, R. and Degrieck, J. (5). Nonlinear correction to te bending stiffness of a damaged composite beam. Damage distribution in te clamped cross-section wit linear EI 1..6 eigt y [mm]...1..3.4.5.6.7.8.9 1. -. -.6-1. - damage [-] cycle 38 cycle,76 cycle 388,3 Figure 3 Simulated damage distribution in te clamped cross-section at several stages during fatigue life wit linear EI. Damage distribution in te clamped cross-section wit nonlinear EI 1..6 eigt y [mm]...1..3.4.5.6.7.8.9 1. -. -.6-1. - damage [-] cycle 95 cycle,7 cycle 36,35 Figure 4 Simulated damage distribution in te clamped cross-section at several stages during fatigue life wit nonlinear EI. Figure 5 and Figure 6 sow te corresponding simulated strain distribution in te clamped cross-section wit linear and nonlinear bending stiffness degradation. For te simulation witout te correction term, te strain increases at te tensile side of te specimen, but decreases at te compressive side. Tis is not realistic as due to te increasing damage at te tensile side, stress redistribution occurs and te strain sould increase at te compressive side. Tis is confirmed by te simulation in Figure 6 wit te nonlinear correction term.
Van Paepegem, W., Decaene, R. and Degrieck, J. (5). Nonlinear correction to te bending stiffness of a damaged composite beam. Strain distribution in te clamped cross-section wit linear EI cycle 38 cycle,76 cycle 388,3 eigt y [mm] 1..6. -.5 -.15 -.5 -..5.15.5 strain [-] -.6-1. - Figure 5 Simulated strain distribution in te clamped cross-section at several stages during fatigue life wit linear EI. Strain distribution in te clamped cross-section wit nonlinear EI eigt y [mm] 1..6. -.5 -...5.5.75.1 strain [-] -.6-1. - cycle 95 cycle,7 cycle 36,35 Figure 6 Simulated strain distribution in te clamped cross-section at several stages during fatigue life wit nonlinear EI. Finally, Figure 7 and Figure 8 sow te corresponding simulated stress distribution in te clamped cross-section at several stages during fatigue life, wit linear and nonlinear bending stiffness degradation. Again te linear simulation in Figure 7 predicts a decrease of te compressive stress wit increasing damage at te tensile side and tat is not realistic.
Van Paepegem, W., Decaene, R. and Degrieck, J. (5). Nonlinear correction to te bending stiffness of a damaged composite beam. Stress distribution in te clamped cross-section wit linear EI cycle 38 cycle,76 cycle 388,3 eigt y [mm] 1..6. -3 - -1 1 3 -. stress [MPa] -.6-1. - Figure 7 Simulated stress distribution in te clamped cross-section at several stages during fatigue life wit linear EI. Stress distribution in te clamped cross-section wit nonlinear EI cycle 95 cycle,7 cycle 36,35 eigt y [mm] 1..6. -5-4 -3 - -1 -. 1 3 stress [MPa] -.6-1. - Figure 8 Simulated stress distribution in te clamped cross-section at several stages during fatigue life wit nonlinear EI. It could be argued tat te final damage values are very ig and would not be tolerated in fatigue design for composites. Tat is true, but tese simulations serve te only purpose of clearly demonstrating te effect of te nonlinear correction term. Of course te larger te damage values, te larger te effect of te correction term. 6 Conclusions
Van Paepegem, W., Decaene, R. and Degrieck, J. (5). Nonlinear correction to te bending stiffness of a damaged composite beam. Te classic beam teory can be used in a modified manner to calculate te damage growt in fibre-reinforced composites during fatigue. It is reasonable to assume tat if te longitudinal stiffness decreases in a linear relation wit te damage, te bending stiffness will do as well. However in tis paper it is demonstrated tat a nonlinear correction term for te bending stiffness degradation is necessary to predict reliable results. Semi-analytical simulations of a displacement-controlled bending fatigue test are sown to support te validity of te correction. It is evident tat more sopisticated calculation tools like finite elements are widely used to simulate damage growt in composite materials, but if one wants to use a simplified tool like te modified classic beam teory, it is important to be aware of te necessity of te nonlinear correction term for te degradation of te bending stiffness. Acknowledgements Te autor W. Van Paepegem gratefully acknowledges is finance troug a grant of te Fund for Scientific Researc Flanders (F.W.O.). References [1] Sidoroff, F. and Subagio, B. (1987). Fatigue damage modelling of composite materials from bending tests. In : Mattews, F.L., Buskell, N.C.R., Hodgkinson, J.M. and Morton, J. (eds.). Sixt International Conference on Composite Materials (ICCM-VI) & Second European Conference on Composite Materials (ECCM-II) : Volume 4. Proceedings, -4 July 1987, London, UK, Elsevier, pp. 4.3-4.39. [] Kacanov, L.M. (1958). On creep rupture time. Izv. Acad. Nauk SSSR, Otd. Tecn. Nauk, No.8, 6-31. [3] Kacanov, L.M. (1986). Introduction to continuum damage mecanics. Dordrect, Martinus Nijoff Publisers, 135 pp. [4] Krajcinovic, D. and Lemaitre, J. (eds.) (1987). Continuum damage mecanics: teory and applications. Wien, Springer - Verlag, 94 pp. [5] Caboce, J.L. (1988). Continuum damage mecanics : part I - General concepts. Journal of Applied Mecanics, 55, 59-64. [6] Caboce, J.L. (1988). Continuum damage mecanics : part II - Damage growt, crack initiation and crack growt. Journal of Applied Mecanics, 55, 65-7. [7] Krajcinovic, D. (1985). Continuous damage mecanics revisited : basic concepts and definitions. Journal of Applied Mecanics, 5, 89-834. [8] Sidoroff, F. (1984). Damage mecanics and its application to composite materials. In : Cardon, A.H. and Vercery, G. (eds.). Mecanical caracterisation of load bearing fibre composite laminates. Proceedings of te European Mecanics Colloquium 18, 9-31 August 1984, Brussels, Belgium, Elsevier, pp. 1-35. [9] Lemaitre, J. (1971). Evaluation of dissipation and damage in metals, submitted to dynamic loading. Proceedings I.C.M. I, Kyoto, Japan. [1] Van Paepegem, W. (). Development and finite element implementation of a damage model for fatigue of fibre-reinforced polymers. P.D. tesis. Gent, Belgium, Gent University Arcitectural and Engineering Press (ISBN 9-76714-13-4), 43 p.