Transactions on Engineering Sciences vol 13, 1996 WIT Press, ISSN

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1 Computational debonding analysis of a thermally stressed fiber/matrix composite cylinder F.-G. Buchholz," O. Koca* "Institute ofapplied Mechanics, University ofpaderborn, D Paderborn, Germany ^Institute of Strength of Materials, University of Tirana, Tirana, Albania Abstract An axisymmetric fiber/matrix composite cylinder of finite length and diameter, subjected to a stationary and homogeneous change in temperature, with respect to the unstressed state, is considered. Emphasis is on SD-efFects in the state of stress, which are not covered by 2D-models, but have a strong impact on the kind of fracture and failure processes that may develop. In particular a thermally induced axial debonding process between fiber and matrix is analysed in detail, including the effects of contact and sliding friction along the crack faces. 1. Introduction Aircraft- and spacecraft structures manufactured from composite materials are frequently subjected to superimposed mechanical and thermal loadings during service in flight, orbit or space. Both types of loadings may cause fracture and failure of composite structures respectively and that is why fracture analysis of mechanically and /or thermally stresses composites are important subjects in materials - and engineering science. Due to the complexity of the corresponding problems of thermoelasticity these investigations often take a micromechanical approach by considering only small representative material sections (unit cells) and frequently they are limited to the related ID-problems /1-4/. Here for a 3D fiber/matrix (f7m) composite cylinder of finite length and diameter (see Fig. 1 and Tab. 1), which is considered to be subjected to a stationary and homogeneous change in temperature with respect to the unstressed initial state, the analysis of a thermally induced axial debonding process between fiber and matrix is presented in detail, including the effects of contact and sliding friction along the crack faces. Even for this simple loading case (AT=const.>0) a complex state of stress is created in this f/m composite cylinder, due to the thermoelastic mismatch of the SiC-fiber and Al-matrix (Tab. 2).

2 684 Localized Damage The analysis is based on the FEM for the state of stress and on global- and local energy methods for the mixed-mode axial debonding process, which is strongly affected by crack face contact and friction. It is shown that also for this complex application a generalized form of the straight forward and numerically effective Virtual Crack Closure Integral (VCCI)-method can be utilized, in combination with some essential data that can be extracted from the solution of the nonlinear contact problem by the ABAQUS FE-code. H v H V Fig. 1 Unit cell model of thefiber/matrixcomposite cylinder and corresponding undeformed and deformed finite element model (axisym. 8-noded elem.) Parameter fiber radius matrix radius length or heigth of cylinder aspect ratio fiber volume fraction change of temperature coeff. of fiber/matrix friction Notation i^= mm rrn = 10mm H = 10mm H/rm = l Vf /V = 30% AT=100 deg U= 0, 0.25, 0.5 Tab. 1 Geometrical and loading parameters of the f7m composite cylinder Notation YOUNG'S modulus E [N/mm^] POISSON's ratio v Linear coefficient of thermal expansion a [deg"" ] fiber (SiQ » < "^ «matrix (Al) Tab. 2 Thermoelastic material parameters of the f/m composite cylinder

3 2. Stress analysis results Localized Damage 685 Detailed stress analysis results for the 3D fiber/matrix composite cylinder are given in /8/ and also for the corresponding 2D-cases of plane stress, plane strain and generalized plane strain. Here we focus our interest on a remarkable 3Deffect in the radial stress c%, which develops next to the free edges of the f7m cylinder (z/h»±1, Fig. 1) and that will have a strong impact on the axial debonding process to be considered. From the undeformed and deformed FEmesh in Fig 1 a distinct radial- and axial expansion of the f/m composite cylinder can be noticed, which is due to the thermal loading (AT=100deg) and the thermoelastic mismatch of the constituents with otm»af (Tab. 2). 250 fiber nterface matrix radius r [mm] Fig. 2 Radial stress distribution in the f/m comp. cylinder with z/h as parameter In Fig. 2 the radial stresses o^ are plotted versus the radius r of the f/m cylinder with z/h as parameter, increasing from z/h = 0 in the plane of symmetry to z/h = 1 at the upper free edge of the composite cylinder. In the plane of symmetry of the cylinder (z/h=0) a^ f «const.»0 is found in the fiber (0<r<rf) and thus at the f/m interface (r=rf) high radial tensile stresses (%»() are acting. This will create a radial crack opening (mode I) for f/m debonding in the cylinder at z/h=l, which corresponds well to the radial debonding behavior of all 2Dmodels cited above. But for z/h>0.5 the radial stress distribution in the fiber changes remarkably and for z/h>0.95 high compressive stresses o^<0 are acting at the f/m interface. So in contrast to the 2D-models no crack opening can be expected to develop during axial f/m debonding at the free edge of the 3D composite cylinder, which is driven by the out of plane shear stress o^ at the f7m interface. This is confirmed by Fig. 3a, in which a FE-mesh detail of the

4 686 Localized Damage cylinder shows a fiber/matrix intersection in the model for a state of axial f/m debonding with a/h=0.1. This defect with respect to the mechanical relevance of the FE-model has been avoided in Fig. 3b, by taking the corresponding contact problem into account. In the related debonding analysis several new aspects had to be considered in order to cover the problems of contact and sliding friction along the crack faces 79,107 by the well experienced approaches with global- and local energy methods 74,6,77. Fig. 3 FE-mesh details of the f7m debonding process (DMF=50) with f7m intersection and contact model (a7h=0.1) Fig. 4 FE-mesh details of the f7m debonding process with contact model (a/h=0.2)and crack opening (a/h=0.3) 3. Debonding Analysis with Crack Face Contact and Friction According to IRWIN the total SERR GT for quasi static crack extension in a brittle material can be defined on the basis of a global energy balance as follows Gr(a) = - = _ lim t da Aa-»0 taa (1)

5 Localized Damage 687 In Eq. (1) EKU-W is the total potential of a plane elastic body, U=l/2 u_t u is the elastic strain energy and W = u_t F is the potential of the external loads (t thickness of the specimen). For finite crack extension Aa ij (2) is holding, where W denotes the crack opening work. From comparisons with reference solutions it is known that this method, denoted as global energy method EN2 here, is numerically highly accurate, even for rather coarse meshes, but has the disadvantage that the total SERR G * can not be separated into the individual modes Gj, i = I,II,III in cases of mixed-mode crack front conditions. For those cases one can refer to IRWIN's basic idea, that for elastic bodies crack opening and crack closure are reversible processes. This means w.9 = wf (3) is holding and consequently, instead of using Eq. (2) in order to calculate W, we can utilize a local energy method, namely the VCCI- or 2C-method /5,6/ for s~\ computing the crack closure work W and the correlated total SERR Gy^ through the separated normal (mode I) and tangential (mode II) parts. Thus with Eq. (2) we find o.o crack length a [mm] Fig. 5 Relative normal displacements (opening) between corresponding nodal points along the crack fraces for debonding lengths 0.175<a/H<0.3.

6 688 Localized Damage W + w or ij ij ij ij T where, respectively, the agreement between both sides of Eq. (4) provide a quantitative measure of accuracy if both methods have been applied. For a more complex problem with contact and sliding friction along the crack faces Eq. (2) has to be replaced by, (5) ij ij ij ^ stating that in this case the change of total potential has to cover the energy for two different processes, namely the crack opening work W^ and the work W^ dissipated due to sliding friction along the crack faces. Thus at least one of the right hand side quantities has to be determined in addition, but preferably both and independently, because then Eq. (5) will also provide a quantitative measure of accuracy for the different approaches that have been taken. The first important step to the solution of the generalized problem was the finding that the local energy method VCCI or 2C is also valid in the case of crack face contact and friction. The second step was the finding that the work W^ which is dissipated due to sliding friction along the crack faces during quasi static crack extension from crack length a to a+aa, can be calculated by Wif = - (6) In Eq. (6) thefirstterm covers the frictional work done by thefirstcrack tip element during crack extension from a to a+aa, whereas the second term is the sum of the work, which is dissipated due to sliding friction along the entire crack face contact zone. In Eq. (6) e.g. C^~ (a + Aa) denotes the tangential component of the contact force at a nodal point with position j-1 and Au^~ (a + Aa) is the relative tangential displacement between the correlated nodal points at the upper and lower crack faces. Thus for quasi static crack extension we find w + W (7) and the excellent agreement between both sides of Eq. (7) for the axial f/m debonding with 0<a/H<0.3, as plotted in Fig. 6, can be taken as a proof of this

7 Localized Damage 689 x L_ en CD C CD ^ WF WC DU WF-fWC 1 js 0.0-0H ^ crock length a [mm] Fig. 6 Development of crack closure work WC, crack face slinding frictional work WF and change of elastic strain energy DU with increasing debonding lengths o.o crack length a [mm] Fig. 7 Effects of contact and sliding friction along the crack faces on crack closure work WC and frictional work WF during axial debonding approach, because all three quantities are computed seperately and independently by different methods. From the FE-meshes in Figs. 3 and 4 it can be realized that the axial debonding process along the f7m interface (r=rf) initiates at the free edge of the composite cylinder (z/h=l). From there it extends with the crack faces remaining in complete or partly contact until for a/h>0.3 full crack opening has developed. This can be followed in more detail in Fig. 5 on the basis of the relative radial or

8 690 Localized Damage crock length a [mm] Fig. 8 Effects of contact and slidingfrictionalong the crack faces on the strain energy release rates during axial debonding normal displacements (opening) between corresponding nodal points along the crack faces as plotted. It shows that the first local crack opening develops for a/h=0.175 (but not at the crack tip!) and how the contact zone decreases with increasing debonding lengths (0.175<a/H<0.3). This also explains why in Figs. 6 and 7 for shorter crack lengths a stronger factional influence is effective, which slowly vanishes for a/h»0.3. Figure 7 also indicates how the crack closure work WC at the crack tip and thus the correlated total SERR G^ (see also Fig. 8) is reduced with increasing frictiona! work W^ due to higher coefficients of friction (0<fj,<0.5). For debonding lengths a/h>0.3 an excellent between Grp^(a) and Gy (a) is found in Fig. 8 with respect to Eq. (4), because full crack opening has developed (mode I) and the further bebonding process is a typical mixed-mode fracture process, with predominat in plane shear loading (mode II) at the crack front for debonding lengths up to a/h<0.9. For a/h>0.3 the quantitative results shown in Fig. 8 agree well with those given in /?/, but where for a/h<0.3 crack face contact and friction could not be considered. 4. Conclusions Through this computational analysis of a thermally induced axial debonding process in a 3D f7m composite cylinder it has been shown how a3d-effect can strongly affect a fracture or failure process. Furthermore it has been found that the numerically effective VCCI-method can also be applied to more complex problems including contact and sliding friction along the crack faces. Acknowledgement The grant ERB-CIPA-CT , prop. No provided by the Commission of the European Communities is gratefully acknowleged.

9 References Localized Damage Hashin, Z, "Analysis of Composite Materials", J. of Applied Mechanics, Vol.50, 1983, pp Mahishi, J.M., "An Integrated Micromechanical and Macromechanical Approach to Fracture Behavior of Fiber-Reinforced Composites", Engng. Fracture Mechanics, Vol.25, 1986, pp Bohm, H.J., Rammerstorfer, F.F., Weissenbek, E, "Some Simple Models for Micromechanical Investigations of Fiber Arrangement Effects in MMCs", Computational Materials Science, Vol. 1, 1993, pp Buchholz, F.-G., "On Correlations between Thermal Stresses, Elastic Strain Energy and Debonding in Thermally Loaded Fibre-Reinforced Composite Materials", In Composites Design for Space Application, ESA SP-243, Noordwijk, 1986, pp Rybicki, E.F., Kanninen, M.F., "A Finite Element Calculation of Stress Intensity Factors by a Modified Crack Closure Integral", Engng. Fracture Mech., Vol. 9, 1977, pp Buchholz, F.-G., "Improved Formulae for the Finite Element Calculation of the Strain Energy Release Rate by the Modified Crack Closure Integral Method", In Accuracy, Reliability and Training in FEM Technology, Robinson and Associates, Dorset, 1984, pp Buchholz, F.-G., Schulte-Frankenfeld, N., Meiners, B., "Fracture Analysis of Mixed-Mode Failure Processes in a 3D-Fiber/Matrix Composite Cylinder", In Proc. of the 6th Int. Conf. on Composite Materials (ICCM- VI), Vol. 3, (Eds. F.L. Matthews et al.) Elsevier Applied Science Publ., London, 1987, Koca, O, Buchholz, F.-G., "Analytical- and Computational Stress Analysis of Fiber/Matrix Composite Models", In Proc. of the 3rd Int. Workshop on Computational Modelling of the Mechanical Behavior of Materials (Ed. S. Schmauder), Max-Planck-Institut fur Metallforschung Stuttgart, November Computational Materials Science, Vol. 3 (1994), Buchholz, F.-G., Koca, O, "Stress- and Fracture Analysis of Thermally Stressed Fibre/Matrix Composite Models", In Localized Damage Ill- Computer Aided Assessment and Control (Eds. M.H. Aliabadi et al), Proc. of the 3rd Int. Conf, Int. Centre for Mechanical Sciences, Udine, Italy, June Computational Mechanics Publ., Southampton/Boston, 1994, Buchholz, F.-G., Wang, H., Ding, S., Rikards, R,: "Delamination Analysis for Cross-Ply Laminates Under Bending With Consideration of Crack Face Contact and Friction", In Proc. of the 10th Int. Conf. on Composite Materials (ICCM 10), Vol. I (Eds. A. Poursatip, K. Street), Vancouver, BC, Canada, August Woodhead Publ. Ltd, Cambridge, 1995,

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