Harich, J.

Lapusta, Y.

Wagner, Werner

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Universität Karlsruhe (TH)Institut für Baustatik3D Finite Element Modeling ofFiber–Matrix Instabilities in CompressionY. Lapusta, J. Harich, W. WagnerMitteilung 6(2003)BAUSTATIKUniversität Karlsruhe (TH)Institut für Baustatik3D Finite Element Modeling ofFiber–Matrix Instabilities in CompressionY. Lapusta, J. Harich, W. WagnerMitteilung 6(2003)BAUSTATIKc©Prof. Dr.–Ing. W. Wagner Telefon: (0721) 608–2280Institut für Baustatik Telefax: (0721) 608–6015Universität Karlsruhe E–mail: bs@.uni-karlsruhe.dePostfach 6980 Internet: http://www.bs.uni-karlsruhe.de76128 Karlsruhe3D FINITE ELEMENT MODELING OF FIBER–MATRIX INSTABILITIESIN COMPRESSIONY. LapustaS. Timoshenko Institute of Mechanics, Nesterov Str. 3, Kiev 03057, Ukraineand currently, Institut für Baustatik, Universität Karlsruhe (TH),Kaiserstr. 12, D-76131 Karlsruhe, Germany, e-mail: yl@bs.uni-karlsruhe.deJ. HarichInstitut für Baustatik, Universität Karlsruhe (TH),Kaiserstr. 12, D-76131 Karlsruhe, GermanyW. WagnerInstitut für Baustatik, Universität Karlsruhe (TH),Kaiserstr. 12, D-76131 Karlsruhe, GermanyAbstract. 3D finite element modeling of fiber-matrix instabilities in com-pression is presented. Calculations for simple particular 3D cases are carriedout. The presented examples illustrate the selection of models with and with-out surface effects. The formulation can be readily extended to problems withtransversely isotropic elastic fibers.1. Introduction. The behavior of fiber-reinforced materials in compres-sion along the reinforcement is an important issue, since a variety of compres-sive failure mechanisms are reported in the literature. A substantial amountof work has already been done in this field, reflected in numerous literaturesources. Without pretending to be exhaustive we discuss here briefly someaspects of the question in order to outline the motivation of this contribution.Instabilities at the micro level of scale, or fiber-matrix instabilities (microbuck-ling) represent a limiting factor for compressive strength of fiber-reinforcedcomposite materials. These phenomena, which are similar to longitudinalbending of a beam on an elastic foundation (Timoshenko and Gere, 1961),were first mentioned and modeled in 60s (Dow and Grunfest, 1960; Rosen,1965; Schuerch, 1966; Sadovsky et al., 1967; Guz, 1969; etc). Rosen (1965)and Schuerch (1966) presented models which considered fibers and matrix aslayers. Sadovsky et al. (1967) proposed a combination of a 1D considerationfor a fiber with a 3D one for the matrix. Guz (1969) presented a 3D linearizedapproach to study the instability of a double-periodic system of fibers in aninfinite elastic matrix. Reviews and literature on compressive behavior of fibercomposites are given in Guynn et al. (1992), Guz (1992), Schultheisz andWaas (1996), Fleck (1997), and others. Results and bibliography on 3D mod-els and methods of stability theory of composites are reviewed, for example,in Guz (1992) and Guz and Lapusta (1999). Closely related issues concern-ing buckling and accompanying phenomena at different scales are discussed,among others, in Wagner et al. (2001), Guz (2002), Lapusta and Wagner(2001), Tkachenko and Chekhov (2002), Zhuk et al. (2002), etc. Note thatthe majority of existing microbuckling models for fiber composites are limitedto 2D considerations. 3D micromechanical modeling of fiber-matrix instabili-ties, especially by means of 3D finite elements, has obtained substantially lessattention. However, the stress-strain fields around fibers are essentially three-dimensional. This means that 3D formulations are preferable if more rigorousresults on fiber microbuckling are needed. This paper presents an attempt of3D finite element modeling of fiber-matrix instabilities in compression. Cal-culations for some particular 3D cases are carried out to illustrate selectionof models with and without surface effects. The formulation can be readilyextended to problems with transversely isotropic elastic fibers.2. Finite-element model. We consider a 3D fiber-matrix instabilityproblem (Fig. 1). The system consists of a long fiber and a matrix. It iscompressed along the fiber direction x1. The matrix is assumed to be wideenough to exclude unwanted lateral surface effects and Euler buckling of thewhole system.x2x3x1Fig. 1. Schematic representation of asystem fiber-matrix under compression.fibermatrixmatrixlinked surfacehdb bFig. 2. Characteristic cell.The lateral surfaces of the matrix are free of forcesFm1 = 0, Fm2 = 0, Fm3 = 0. (1)A perfect adhesion of the fiber (denoted by superscript f) and the matrix(superscript m) is assumed, which means the continuity of forces and displace-ments across the interfaceF fn = Fmn , Ffs = Fms , Ff1 = Fm1 ,ufn = umn , ufs = ums , uf1 = um1 . (2)The value of the loading parameter leading to buckling of the system and thecorresponding buckling half wavelength 0 < L < ∞ must be calculated. Tocarry out finite element modeling, we consider characteristic cells as presentedin Figs. 2-4. Two possible situations are considered. In the first one, weassume that the matrix has a circular or square cross section and the fiberis situated in its center (Figs. 3 and 4, b = c). In the second one we admitthat the matrix has a rectangular cross-section and the fiber is situated in thevicinity of one of the matrix lateral surfaces (Fig. 4, 0 ≤ c ≤ b).db bFig. 3. Case/Geometry 1 (cross-section).dcbdb bFig. 4. Case/Geometry 2 (cross-section).These characteristic cells are selected by accounting for the periodicity of thebuckling form in the direction x1 (period 2L=2h). The Euler buckling of thecell is excluded by the selection of its geometry, since the cell is taken wideenough in transverse directions.An extended version of the finite element program FEAP (see e.g. Zienkiewiczand Taylor, 2000) is used. A standard displacement-based eight node isopara-metric solid finite element with trilinear shape functions is employed to dis-cretize the characteristic cells. The symmetry of the cells and of their eigen-forms are taken into account (see e.g. Figs. 5, 6 and 10). Behavior of the fiberand the matrix is modeled by a linearly elastic material. For the calculations,the values Ef = 0, 4·1012 Pa, Em = 0, 2·1010 Pa, νf = νm = 0, 3, d = 0, 8·10−5(m) are taken. To save the calculation time, the transverse dimensions of theelementary cell are chosen to b = h. In case 2 (Fig. 4), we also place thefiber in the vicinity of one of the lateral surfaces by setting 0 ≤ c ≤ b. Thisway we account for its influence on the fiber stability. The cell is compressedthrough prescribed displacements u1 of the nodes at the linked surface (Fig.2). These displacements are stepwise increased from zero. Due to the sym-metry of the system and its eigenforms, we suppress the displacements u2 ofthe symmetry plane (x1, x3). A bisection method is used to find the stabilitypoint within a certain prescribed accuracy. Then an eigenvalue analysis leadsto the associated eigenforms.The height h of the characteristic cell is stepwise increased and the criticaldisplacement u1 is calculated as a function of the buckling half wavelength L(which is equal to h). Thus functions u1/L = f(L/d) can be found. Sincesufficiently long fibers buckle with a certain wavelength corresponding to theshortening minimum, a minimization of the calculated value u1/L with respectto the parameter L/d is performed.3. Results. Figs. 5, 6 and 10 show examples of the calculated eigenformsand the chosen 3D mesh.Fig. 5. Buckling form and mesh for case 1. Fig. 6. Buckling form and mesh for case 2.Figs 7 and 8 show values u1/L with respect to the ratio L/d. To analyzefree surface effects, we have also performed calculations for different values ofspacing c between the fiber and the matrix boundary (0 ≤ c ≤ b). Note thatfor this case, Lapusta and Wagner (2001) presented a semi-analytical studywhich distinguished between two different instability mode orientations withrespect to the free boundary. These are: (a) mode towards the boundary,which is symmetric with respect to the plane that contains the fiber’s axis andis perpendicular to the boundary (mode 1), and (b) mode for which the fiberbuckles out of the mentioned plane (mode 2).u1/L0,0650,0690,0734 4,5 5 5,5 6L/d200Fig. 7. u1/L calculated for case/geometry 1.u1/L0,0650,070,0754 5 6 7L/d200Fig. 8. u1/L calculated for case/geometry 2.Since the mentioned above paper demonstrated that only mode 1 was critical,we have limited our consideration to the case of mode 1. In this case, we usesymmetry and calculate the eigenform as shown in Fig. 10.u1/L0,0450,0550,0650 1 2 3 4c/d200Fig. 9. u1/L vs. spacing between the fiberand a boundary c/d.Fig. 10. Buckling form and mesh for afiber touching the free boundary.Figure 11 shows some results of convergence study for a fiber touching the freeboundary (case 2 with c = 0, Figs. 4, 10). In this Figure, the deviation D(%)of the critical shortenings u1/L for different meshes with respect to the resultsobtained with the highest possible accuracy admitted in this paper (right edgesD [%]0,001,002,003,004,005,008 12 16 20 24 28 32 36 40 44 48k, m, n8 x 10 x km x10 x 128 x n x 12Fig. 11. Convergence with respect to numberof elements k,m,n.of the curves) is given. A compari-son of cases 1 and 2 shows that ge-ometry 2 can be used in modelingcase 1 within approximately 5 % ac-curacy for the critical shortenings.Free surface effect for case 2 can besubstantial and decrease the valueof the critical shortening by up toabout 40 %.Acknowledgements. A part of this work has been carried out with the finan-cial support of the DFG (grant WA 746/12-2), which is gratefully appreciated.REFERENCESTimoshenko S.P., Gere J.M. (1961). Theory of Elastic Stability, Second Edition. New York,NY: McGraw-Hill Book Company.Dow N.F., Grunfest I.J. (1960). Determination of most needed potentially possible improve-ments in materials for ballistic and space vehicles. General Electric Co, Space Sci Lab,TISR 60 SD 389.Rosen B.W. (1965). Mechanics of Composite Strengthening. Fiber Composite Materials,American Society of Metals, 37-75.Schuerch H. (1966). Prediction of compressive strength in uniaxial boron fiber-metal matrixcomposite materials. AIAA Journal, 4 (1), 102-106.Sadovsky M.A., Pu S.L., Hussain M.A. (1967). Buckling of Microfibers. Journal of AppliedMechanics, December 1967, 1011-1016.Guz A.N. (1969). Construction of a theory of stability of unidirectional fiber composites.Prikladnaya Mekhanika 5(2), 62-70 (in Russian).Guynn E.G., Bradley W.L., Ochoa O.O. (1992). A parametric study of variables that affectfiber microbuckling initiation in composite laminates: Part 1 - Analyses. Journal ofComposite Materials 26(11), 1594-1616.Guz A.N. (Edit.) (1992). Micromechanics of composite materials: Focus on Ukrainianresearch. Applied Mechanics Reviews, Special Issue, 45(2), 13-101.Schultheisz C.R., Waas A.M. (1996). Compressive failure of composites. Prog. AerospaceSci. 32, 1-78.Fleck N.A. (1997). Compressive failure of fiber composites. Advances in Applied Mechanics,33, 43-117.Guz A.N., Lapusta Y. (1999). Three-dimensional problems of the near-surface instabilityof fiber composites in compression (model of a piecewise-uniform medium) (Survey).International Applied Mechanics 35(7), 641-670.Wagner W., Gruttmann F., Sprenger W. (2001). A finite element formulation for the simula-tion of propagating delaminations in layered composite structures, International Journalfor Numerical Methods in Engineering, 51, 1337-1359.Guz A.N. (2002). Critical phenomena in cracking of the interface between two prestressedmaterials. 1. Problem formulation and basic relations. International Applied Mechanics,38(4), 423-431.Lapusta Y., Wagner W. (2001). On various material and fibre-matrix interface models inthe near-surface instability problems for fibrous composites. Composites Part A: AppliedScience and Manufacturing, 32, 413-423.Tkachenko E.A., Chekhov V.N. (2002). Stability of an elastic layer stack between two half-spaces under compressive loads. International Applied Mechanics, 38 (11), 1381-1387.Zhuk Ya.A., Soutis C., Guz I.A. (2002). Stiffened composite panels with a stress concentratorunder in-plane compression. International Applied Mechanics, 38 (2), 240-252.Zienkiewicz O.C., Taylor R.L. (2000). The Finite Element Method, Butterworth Heinemann,5th edition.

3D Finite Element Modeling of Fiber-Matrix Instabilities in Compression

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3D Finite Element Modeling of Fiber-Matrix Instabilities in Compression

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