Characterization o Internal State ariable or iber racture in UD Composite Modris Megnis,, Povl Brondsted 3, Sai A. Rehman 4, Tanveer Ahmad 5 Summary The continuum damage mechanics is used to describe the constitutive behavior o UD laminate with damage. The progressive iber racture, as it is aected by interace strength, debonding growth and matrix cracing, is considered as a main damage mechanism causing the stiness reduction. The internal state variable that accounts or iber racture is ormulated within the used theory. The methodology is proposed to measure the values o the internal state experimentally. The FEM analysis and Monte Carlo simulations are used in order to predict the internal state variable according to the outlined theoretical ormulation. The predicted values o internal state variable are compared with experimentally measured. Introduction Typical or reinorced composites are inelastic eects. The inelasticity is commonly associated with viscoelasticity, viscoplasticity and damage evolution. The damage evolution is a complicated phenomenon to account or and to characterize. The continuum damage mechanics is a promising approach to develop thermodynamically consistent ormulation o constitutive law or media with damage. This would account or eects o temperature changes (non-isothermal conditions) and environmental (moisture) conditions. This is because the additional internal state variables (IS) that would account or considered phenomenon can be included. The thermodynamically consistent damage dependent lamination theory, ormulated by D.H. Allen [,], is used to describe inelastic response to the applied load or a long iber composite laminates at reerence conditions (isothermal environmental conditions). In this study, the internal state variable that accounts or damage o UD composite in iber direction is ormulated within the terms o the applied theory, and the unction that describe the corresponding internal state variable is determined and analyzed theoretically and experimentally. The stiness degradation is associated with corresponding damage evolution providing the tool or experimental determination o the IS accounting or considered damage. FEM analysis and Monte Carlo simulations are used to calculate considered IS using ormulated micromechanics. Presenting, corresponding author. Materials Research Department, Risø National Laboratory, DK-4000, Rosilde, Denmar 3 Materials Research Department, Risø National Laboratory, DK-4000, Rosilde, Denmar 4 Mads Clausen Institute, The University o Southern Denmar, DK-6400, Sønderborg, Denmar 5 Mads Clausen Institute, The University o Southern Denmar, DK-6400, Sønderborg, Denmar
General constitutive relationships or media with damage The thermodynamically consistent constitutive relationships or media with cracs have been developed by [,] and will be used herein σ = C ε + I ξ ξ, () ij ijl l ijl l where I ξ ijl is damage dependent modulus, and ξ =,..., N accounts or dierent damage modes. The locally averaged internal state variables associated with energy dissipation due to the cracing is deined as irst proposed by [3] and utilized by [,] as l = unds S l () where l are components o internal state variable tensor, is a local volume in which statistical homogeneity can be assumed, u and nl are crac ace displacement and normal respectively, is a crac surace area. (3) S Equations or the laminate with damage The constitutive relationships or general laminate with damage can be obtained rom () and written in the matrix orm () { N} [ A]{ ε} [ B]{ } { } = + +, (4) where [ A ] and [ ] B are extensional stiness matrix, and the coupling stiness matrix o the laminate. The eect o the iber racture that taes place due to the loading in iber n () () () direction is described by { } = I ( z z ){ }, where stiness tensor o () I is a damage = th () layer, and { } is internal state variable accounting or iber } () racture damage. Generally, { can be determined experimentally, or calculated using micromechanics approach. The measurements o stiness degradation o UD () experimentally. laminate must be used in order to determine { } () Determination o damage tensor, { } The progressive iber racture, combined with debonding and matrix cracing at higher applied strain levels, can be described by representative unit element as illustrated in Figure. The considered unit element can account or the debonding eects and matrix cracing eects on the crac opening displacement.
δ x u n Figure. Representative volume element. δ is ineective length, is diameter o the iber, d d m is dimensions o the surrounding matrix, and S is ¼ o the area o the ellipse under the crac proile line. The dimensions are calculated based on the iber pacing in hexagonal array. x d d S = πud m 8 The general deinition o internal state variable l, expression (), can be ormulated or considered damage as () M = l S l m= unds, (5) S where is crac surace area o single crac. I the crac, see Figure, does not change the orientation the normal to crac surace have only one component, n =, n = n =0. It leads to the = 0 0 0 0 0 0, where the 3 { } () () () () T l 3 accounts or crac opening in Mode-I, and according to (5), can be expressed in orm o crac opening displacement M M () = uds = udx m= m= S x. (6) Considering the inite size o the representative volume, = dx dx dx 3, and two dimensional case with unit thicness, dx 3 = m =, (7) () udx δ x d where dx = δ is ineective length, and dx = d, d is diameter o the iber, is iber volume raction, and = 0.844 is a generally proportionality coeicient. Both, dx and, are calculated on bases o the hexagonal iber pacing. The integral part
in the (7) give the area under the crac surace proile, S = udx x = π u d, where the, 8 u is the crac opening displacement at x = 0, see Figure. Than the (7) can be rewritten as () π u m =, = 0.844, (8) 8 δ 0 where u is generally unction o the applied strain, geometry and stiness ration o iber and matrix. The number o crac, m are counted per ineective length, δ, and naturally will have limits 0 m. The maximum crac density can be calculated by m using act that one crac per ineective length is possible, ρ = = ρmax =. m= δ δ Both, ineective length, δ and number o cracs per unit iber length as a unction o applied strain, can be obtained experimentally rom Single Fiber Fragmentation Test (SFFT). However, The Weibull statistics or the iber can be used to generate the crac density unction, and in this case, FEM can be used to determine ineective length. Also analytical expressions could be used to calculate δ. This approach will lead to overestimated or large applied strain levels. It is because practically the crac density in composite would be expected somewhat less than theoretical, ρmax = / δ. The stochastic methods, or Monte-Carlo simulations could give more realistic crac density unction or the composite. Experimental considerations Let's consider the unidirectional laminate with iber orientation in loading direction, subjected to static loading, { N} = N 0 0} T. There is only one type o damage { x () expected, and it can be described with the internal state variable. Further, the constitutive relationships or the considered laminate can be written in a compact matrix () () orm as { N} = [ A]{ ε} + I { } h. Considering loading conditions, { N} = { N 0 0} T, and using C = I [,], and A = Ch, it can be reduced to, ijl N ε, and using the expression or N in it, the relationship or as unction E h ε ijl () N = A ε + A ε + C h. Further, introducing the deinition o the stiness, o measured stiness or particular applied strain level is obtained, ij ij
A A h E ε dε + ε A ε ε () A ( ε ) = A A A A. (9) Results and discussion The stiness degradation o UD laminate, [ 04 ] T, is measured experimentally using quasi-static tensile loading-unloading test. Reerence material, and the same preconditioned (exposed to salt water or 6 month) have been tested. The whole loadingunloading loop o the i loading-unloading cycle is used to calculate the Young's th modulus, E ε or corresponding applied strain. It was observed, that the stiness degradation measured is consistent or all individual specimens. The measured stiness degradation is normalized, Ei / E0 and described by a second order polynomial, see Figure. Further the obtained unction or / E can be used into (9) in order to calculate () () ( ε ). The experimentally determined Ei 0 ε relects the total eect o the multiple racture mechanism, and it would be diicult to separate them. The micromechanics () ε accounting or iber cracing and crac density approach is used (8) to calculate assumptions. The FEM is used or calculating the crac opening displacement as unction o applied strain, u in (8). The plane strain conditions with unit thicness have been considered. The debonding crac and the crac growth into the matrix, are not allowed in this study. It is still to be analyzed whether the same IS should be used, or a new one should be ormulated in order to account or both. The obtained value o δ 5 ( µ m) is rather close to what can be calculated according to Rosen. The Monte Carlo simulations, utilizing weaest lin assumptions and stress global shearing mechanism, are used to produce the crac density unction. The experimentally determined and theoretically () () ε calculated values o ( ε ) are given in Figure. The theoretically predicted are lover than experimental up till applied strain ε.5 %. It is due to the act that the debonding is not considered. Also predictions dive higher values or higher applied strain levels. It is due to use o weaest lin and global stress shearing assumptions as well as using the method o chain o bundles in Monte Carlo simulations. It leads to artiicially increased number o cracs. The local stress shearing mechanism gives more realistic description o mechanism, and will be implemented in uture. As well as the use o conditions o ormation clusters o multiple iber breas would reduce the predicted number o cracs. () In order to validate the outlined approach and obtained ( ) o independent tensile test o UD laminates, [ 4 ] ε, the strain-stress curve 0 T, can be predicted by using the obtained
() unction o ε into the constitutive equations or laminate, (4), and predictions are in rather good agreement with experimental data. () Figure : Stiness degradation o UD composite, and Calculated IS polynomial, y = a x + a x+ a a 0 =.0, a =.76 and a 36.9. = 0 ε. The second order is used or approximation. The calculated constants are, Conclusions The IS that characterizes the iber racture is ormulated within the used continuum damage mechanics approach. The methodology is proposed how to measure the considered IS experimentally. The FEM analysis and Monte Carlo simulations are used to calculate the IS using proposed micromechanics ormulation. Theoretically calculated and experimentally measured values o the IS are compared, and both are used within the constitutive expressions or laminate to predict strain-stress behavior. Acnowledgment Part o the wor is done within rames o the EU project, OPTIMAT BLADES. It is also a subject o the Nordic Co-operation Networ, supported by Nordic Council o Ministers. Reerence. Allen, D.H., Harris C.E., and Groves, S.E. (987a): "A Thermodynamical Constitutive Theory or Elastic Composites With Distributed Damage-I. Theoretical Development" Int. J. Solids Structures ol. 3, No. 9, pp. 30-38.. Allen, D.H., Harris C.E., and Groves, S.E. (987b): "A Thermodynamical Constitutive Theory or Elastic Composites With Distributed Damage-II. Theoretical Development" Int. J. Solids Structures ol. 3, No. 9, pp. 39-338. 3. auleno, A. and Kachanov, M. L. (97): Continual Theory o a Medium with Cracs, Izv. AN SSSR. Mehania Tverdogo Tela, ol. 6, No. 4, pp. 59-66, UDC 539.3