Aagaard Madsen , Helge

Hansen, Kurt Schaldemose

Larsen, Gunner Chr.

Larsen, Torben J.

Ott, Søren

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Downloaded from orbit.dtu.dk on: Dec 20, 2017Full scale verification of wind farm production predictionsLarsen, Gunner Chr.; Larsen, Torben J.; Ott, Søren; Hansen, Kurt Schaldemose; Aagaard Madsen , HelgePublished in:Proceedings of the 23rd International Congress of Theoretical and Applied Mechanics (ICTAM 2012)Publication date:2012Link back to DTU OrbitCitation (APA):Larsen, G. C., Larsen, T. J., Ott, S., Hansen, K. S., & Aagaard Madsen , H. (2012). Full scale verification of windfarm production predictions. In Proceedings of the 23rd International Congress of Theoretical and AppliedMechanics (ICTAM 2012)XXIII ICTAM, 19–24 August 2012, Beijing, ChinaDEBONDING ANALYSES IN ANISOTROPIC MATERIALS WITH STRAIN-GRADIENT EFFECTSBrian Nyvang Legarth∗∗Department of Mechanical Engineering, Technical University of Denmark, Kgs. Lyngby, 2800, DenmarkSummary A unit cell approach is adopted to numerically analyze the effect of plastic anisotropy on damage evolution in a micro-reinforced composite. The matrix material exhibit size effects and a visco-plastic anisotropic strain gradient plasticity model accountingfor such size effects is adopted. A conventional cohesive law is extended such that both the average as well as the jump in plastic strainacross the fiber-matrix interface are accounted for. Results are shown for both conventional isotropic and anisotropic materials aswell as for higher order isotropic and anisotropic materials with and without debonding. Generally, the strain gradient enhancedmaterial exhibits higher load carry capacity compared to the corresponding conventional material. A sudden stress drop occurs in themacroscopic stress-strain response curve due to fiber-matrix debonding and the results show that a change in yield stress, which iscaused by plastic anisotropy, affects the overall composite failure strain.INTRODUCTION TO THE PROBLEMFor a composite reinforced at the micron scale, two competing mechanisms affect the overall behavior: (I) interfacialfailure reduces the strength and (II) strain-gradient effects enhance the strength. When analyzing such composites ingeneral a full 3D analysis is required in order to fully represent the geometry, the loading and the boundary conditions.Such analyses are complicated and the computations become very time consuming when anisotropic plasticity and pro-gressive debonding is to be accounted for. Thus, assuming a periodical distribution of the reinforcement allows for greatlysimplified approaches. Here, a composite material having a periodical distribution of reinforcement is analyzed using aplane strain unit cell approach. Thus, the results presented in this study approximate a composite of rather long, almostaligned, stiff reinforcement which is subjected to a fixed stress state that is acting mainly in the transverse direction of thereinforcement. Fig. 1(a) shows the distribution of fibers and Fig. 1(b) shows the unit cell adopted here. The orthonormalbasis, ni, of the principal axes of plastic anisotropy, xˆi, is defined by the angle θ, from from the global Cartesian coor-dinate system, xi. The displacements ∆1 and ∆2 are prescribed such that ratio, κ, of the average stress at the cell edgesremains constant. Fig. 1(c) shows an example of the finite element mesh adopted. The element type is 8-node elements.afbfacbc ∆1∆2n1n2x1x2xˆ1xˆ2θ(a) (b) (c)FibersGrid ofsymmetry linesFigure 1. The plane strain cell model for the composite. (a) Periodically distributed fibers. (b) The cell used for modeling with initialdimensions, loads, supports and coordinate systems. (c) Example of finite element mesh using adopted, afbf=acbc= 1.MATERIAL MODELSHigher order elasto-plastic constitutive modelThe fibers are assumed to be purely elastic with a stiffness much larger than the elasto-plastic matrix material, whichis assumed to obey the strain gradient model proposed by Fleck and Hutchinson [1]. In addition, plastic anisotropy isaccounted for using the anisotropic version of the Fleck and Hutchinson model suggested by Legarth [2]. Thus, theeffective plastic strain, E˙p, is enriched by the gradients of the conventional effective plastic strain, ǫ˙p, and a materiallength scale parameter, l∗, asE˙p =√23ǫ˙pij ǫ˙pij + l2∗ǫ˙,pi ǫ˙,pi ; ǫ˙pij = ǫ˙p ∂Γ∂σij; ǫ˙p =√23ǫ˙pij ǫ˙pij (1)The work-conjugate effective stress is denoted σc and is given in table 1. Plastic anisotropy is accounted for by theclassical anisotropic Hill yield surface. For the case of plane strain conditions with σ13 = σ23 = 0 the yield surface isΓ =√32(F+G+H) [F (σˆ22 − σˆ33)2 +G(σˆ33 − σˆ11)2+ H(σˆ11 − σˆ22)2 + 2Nσˆ212] (2)where the Cauchy stresses, σˆij , refer to the principal axes of plastic anisotropy. For F = G = H = 1 and N = L =M = 3, Eq. (2) equals the isotropic Mises yield surface, σe. A higher-order stress measure is also introduced as ρi.a)Corresponding author. E-mail: bnl@mek.dtu.dkConventional materials Higher order materials(l∗ = 0, ρi = 0) (l∗ 6= 0, ρi 6= 0)Isotropic ρi,i = 0 ρi,i = q − σe(Γ = σe) σc = σe σc =√(σe + ρi,i)2 + l−2∗ ρiρiAnisotropic ρi,i = 0 ρi,i = q − Γσc = Γ σc =√(Γ + ρi,i)2 + l−2∗ ρiρiTable 1. Summary of the effective stress, σc, for different materials.Higher-order cohesive modelThe bi-axial loading on the unit cell, Fig. 1, will tend to cause both normal and tangential interfacial separation, un andut, respectively, at the fiber-matrix interface. The cohesive zone model proposed by Tvergaard [3] takes both types ofseparation into account and therefore this model may seem suitable for the present study. However, due to the existence ofthe higher order stress, ρi and corresponding higher order tractions, ρini additional terms need to be included in order tohave a conventional as well as higher order stress-free surface after debonding failure. Hence, a non-dimensional damageparameter is introduces as [4]λ =√√√√(unδn)2+(utδt)2+ l2∗[(< ǫp >lA)2+([ǫp]lJ)2](3)where < ǫp > is the average (subscript A) and [ǫp] is the half jump (subscript J) in plastic strain across the interface,respectively, whereas lA and lJ are corresponding critical interfacial length scale parameters. For λ ≥ 1 total separationhave occurred. It is noted, that since the fiber is taken to be purely elastic, < ǫp >=[ǫp]. The corresponding tractions areTn =unδnF (λ) ; Tt = αutδtF (λ) ; TA = l2∗δnl2AF (λ) < ǫp > ; TJ = l2∗δnl2JF (λ)[ǫp] (4)with α = δn/δt and F (λ) = 274 σmax(1− 2λ+ λ2) for 0 ≤ λ ≤ 1. The maximum interfacial stress is denoted σmax.RESULTS AND CONCLUSIONFig. 2 shows results for a load case with κ = σ2σ1= 0.5 corresponding to bi-axial plane strain tension. The fiber volumefraction is Vf = piaf bf4acbc , withafbf= acbc= 1. The initial yield stress is σ0/E = 0.003, where E is Young’s modulus. Thecoefficients of anisotropy are F = 0.7, G = 3.33,H = 1 and N = 9.6 with θ = 0o and σmax = 3σ0.00 0.01 0.02 0.03 0.04 0.0512345l∗/√afbf = 0.3l∗/√afbf = 0IsotropicAnisotropicDebonding degradationǫ1σ1/σ00.02 0.04 0.06 0.08 0.1 0.12 0.14 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07ǫ1 = 0.02 ǫ1 = 0.02(a) (b) (c)Figure 2. Bi-axial tension results, κ = 0.5. (a) Average stress-strain curves (b) Contours of effective plastic strain for a conventionalanisotropic material with debonding (c) Contours of effective plastic strain for a higher order anisotropic material without debonding.For both isotropic and anisotropic behavior the effect of the material length scale parameter, l∗, is an increased load carry-ing capacity, Fig. 2(a). A sudden stress drop occurs due to debonding, Fig. 2(a), and in Fig. 2(b) the corresponding voidat the fiber-matrix interface is shown. Fig. 2(c) illustrates, that at the fiber-matrix interface the plastic strain is suppressedand the strain is smaller compared to the conventional case with severe plastic deformations at the tip of the void, Fig. 2(b).In conclusion, this study analyzes numerically the combined effects of plastic anisotropy, size-effects and debonding ina composite material. Debonding is seen as a sudden stress drop and plastic anisotropy highly affects the failure strain,while the size-effect is observed as an increased load carrying capacity.References[1] Fleck. N. A., Hutchinson, J. W.: A reformulation of strain gradient plasticity. J. of the. Mech. and Phys. of Solids. 49 2245–2271, 2001.[2] Legarth, B. N.: Strain-gradient effects in anisotropic materials. Model. and Sim. in Mater. Sci. and Eng. 15:S71–S81, 2007.[3] Tvergaard, V.: Effect of fibre debonding in a whisker-reinforced metal. Mater. Sci. & Eng. A125:203–213, 1990.[4] Niordson, C. F.: Cohesive element for Fleck & Hutchinson 2001. Technical report, Technical University of Denmark, 2010.

Full scale verification of wind farm production predictions

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Full scale verification of wind farm production predictions

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