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Finite-element analysis of a polymer composite subjected to ball indentation1
Composites Science and Technology 59 (1999) 271±281
Finite-element analysis of a polymer composite subjected to ball indentation$ K. VaÂradi a,1, Z. NeÂder a, K. Friedrich b,*, J. FloÈck b a Institute of Machine Design, Technical University of Budapest, H-1521 Budapest, Hungary Institute for Composite Materials, University of Kaiserslautern, 67663 Kaiserslautern, Germany
b
Received 17 June 1997; accepted 5 March 1998
Abstract The ®nite-element (FE) contact technique presented in this paper is applied to the problem of steel-ball indentation of a composite material consisting of unidirectional continuous carbon ®bres in a poly(ether ether ketone) matrix. Indentation was carried out with ®bre orientations either normal (N) or parallel (P) to the contact surface at a ®bre volume fraction of 0.44. The FE contact analysis involves both an anisotropic (homogeneous) macro- and (inhomogeneous) micro-contact analysis, following an approximate displacement coupling technique. The FE contact-stress analysis of a ®bre/matrix micro-structure has a major limitation. If the FE micro-model is used, only a very small (for example 0.1 mm 0.1 mm 0.1 mm) 3D segment can be modelled. If an anisotropic model is used, there is no similar size limitation but the results cannot describe the stress and strain states of a real ®bre/ matrix micro-structure. The FE contact results show the location and the distribution of the sub-surface stresses and strains. For N ®bre orientation there is a high shear stress region below the surface, from where the ®bre/matrix interfacial failure initiates before propagating to the surface. In the case of P ®bre orientation the matrix is subjected to local plastic deformation while the characteristic deformation of the ®bre is bending. # 1999 Elsevier Science Ltd. All rights reserved.
1. Introduction The ball indentation test is a well-known procedure to evaluate the strength properties of metals both in the elastic and the elastic±plastic range. In the last couple of decades, this test was also applied to composite materials, in particular to characterise their mechanical behaviour, e.g. with regard to their ®bre/matrix interfacial shear strength. Interfacial failure caused by the indentation test was studied by Carman et al. [1] using experimental and analytical techniques. Micro-stresses were evaluated by macro/micro approaches and a cellular modelling concept. According to their experimental observations, ``substantial ®bre/matrix interfacial failure occurs underneath the ball indentor''. To analyse the ball/composite contact problem, ``a solution * Corresponding author. Tel.: 0049 631 2017201; fax: 0049 631 2017198; e-mail: [email protected]. $ Part of this material was submitted to 1997 ASME International Mechanical Engineering Congress and Exposition, Dallas, 16±21, November, 1997. 1 E-mail: [email protected].
was generated with a macro/micro approach utilising an elasticity solution to formulate the macro solution. Using the macro solution to generate local boundary conditions in conjunction with a micro solution based on a cellular model, an approximate closed form solution for the stress state in a micro region was constructed''. In the present study, the indentation problem was ®rst solved by the FE contact technique of COSMOS/M [2], a simple square cross-section being assumed for each ®bre. The FE contact analysis involves both an anisotropic macro- and micro-contact analysis, following the presented approximate displacement coupling technique. (The `multiscale' modelling [3] or the `globallocal' analysis [4] techniques are more general procedures.) The coupled solution provides a more realistic elastic deformation of the composite system in the vicinity of the contact area, and at the same time the eect of the macro-system is also incorporated. The FE contact-stress analysis of a ®bre/matrix micro-structure has strong limitation. If anisotropic numerical models are used, there are no size limitations but the results cannot describe the stress and strain
0266-3538/99/$Ðsee front matter # 1999 Elsevier Science Ltd.. All rights reserved. PII: S0266 -3 538(98)00066 -9
K. VaÂradi et al / Composites Science and Technology 59 (1999) 271±281
272
Table 1 Anisotropic material properties assuming isotropic ®bre properties Vf=0.44
Steel
Fibre
Matrix
Composite
E11 (GPa) E22 (GPa) E33 (GPa) G12 (MPa) G13 (MPa) G23 (MPa) 12 13 23
210
235
3.6
106342 6397 6397 2290 2290 2214 0.296 0.296 0.444
2. Finite-element macro- and micro-contact analysis incorporating a coupling technique 2.1. The coupled FE contact models
states of a real ®bre/matrix micro-structure. If an FE micro-model is used, only a very small (for example 0.1 mm 0.1 mm 0.1 mm) 3D segment can be modelled while the number of degrees of freedom (DOF) is in the neighbourhood of 100,000. To study the eect of ball indentation on a composite structure, with a more realistic (cylindrical) ®bre geometry in the FE models, an iterative contact solution, assuming a rigid ball, was chosen. Finally stress and strain results were obtained for both ®bre orientations. The composite material investigated was a unidirectional continuous carbon-®bre/ poly(ether ether ketone) system, containing normal and parallel ®bre orientations at ®bre volume fraction 0.44. The material properties for the ball and the composite material [5] are listed in Table 1 assuming isotropic ®bre properties. For the composite material transversely isotropic material properties were assumed. The material principle direction 1 was vertical (z direction in Fig. 1(b)) for N orientation and horizontal (x direction in Fig. 1(b)) for P orientation.
The models of the ball and the composite material are shown in Fig. 1(a). The ball diameter is 2 mm. As a consequence of the symmetry conditions, only a quarter of the bodies are considered, and the proper boundary conditions are used. To evaluate the stresses in a ®bre/matrix micro-system, an FE micro-model is created. This micro-model is `built into' a larger (homogeneous and anisotropic) macro-model to represent a larger segment of the original bodies. From the ®bre/matrix micro-structure and also from the ball a volume of 0.1 mm 0.1 mm 0.1 mm is modelled (Fig. 1(b)); for the macro-structure the size of the two segments amounted to 0.5 mm 0.5 mm 0.5 mm. In these two cases the size of the contact area is in the range of 0.05±0.1 mm. If the contact area is larger, larger segments (5±10 times of the expected contact area) should also be considered. In the present case the ball has much smaller elastic deformation than the composite material (i.e. a rigidball model causes only a small error, see later); therefore, a small segment of the ball is enough to be modelled around the actual very small contact area. These segments (points A and B in Fig. 1(b)) do not follow the ball's geometry but the initial gap assigned to the gap elements represents exactly the real gap between the ball and the composite material. First the FE contact analysis operates on the larger segment (macro-model) followed by the smaller segment (micro-model), while the displacements are coupled. The larger model has only a few contact elements in the
Fig. 1. The global and local models (a) and the substituting coupled model approach (b) (smaller quarter segments are 0.1 mm 0.1 mm 0.1 mm, and the larger quarter segments are 0.5 mm 0.5 mm 0.5 mm).
Fig. 2. FE mesh for the coupled contact models (smaller (b) is micro-, large (a) is macro-model) (. ± coupled surface, % ± zero prescribed displacement perpendicular to the surface).
80769
100772
1286
0.3
0.166
0.4
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expected contact area while the smaller model has much more contact elements. In the coupled contact models it is assumed that the total load, introduced as distributed load on the actual upper surface of the modelled ball segment, is the same at both macro and micro levels. In Fig. 2 for the bottom body (representing the composite segment) the displacements of the points of the larger model, located at the boundary area of the smaller model, are assigned to the smaller model as boundary conditions all over its side surfaces. These are therefore coupled surfaces. The surface having the contact elements are free from prescribed displacements, apart from the edges. The steel ball also has two models (larger and smaller) but the displacements are not coupled because the ball is `more rigid' (see results). For the ball only the boundary conditions, representing the quarter model, are applied at both levels. 2.2. FE contact micro-model with square ®bres In Fig. 1(b) the modelled quarter segments are shown. Both the large and the smaller segments (as separate models) are divided into 20 20 20 eight-node solid elements. Between the potential contact surfaces nodeto-node type gap elements are located. The bottom of the composite segment is ®xed, while the load is introduced at the top of the modelled ball segment (Fig. 2(a)). The total model contains 18522 nodes and 16150 elements. For N and P orientation the geometry of the macro-models is the same; only the material principle directions are changed. The micro-models are dierent.
Fig. 3. FE mesh for micro-model in case of normal ®bre orientation (the FE mesh of the ball and the contact elements are not shown).
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Figs. 3 and 4 show the FE meshes of the micro-models for N and P orientation. The ®bres are assumed to have square cross sections of 10 mm 10 mm, and the volume fraction is 0.44 (4/9 of the uniform mesh). The rest of the FE model (overall sizes, mesh structure, boundary condition, loading and contact elements) is the same as in Section 2.1 for the smaller model. 2.3. FE contact micro-model with round ®bres The FE contact analysis requires FE contact iterations. This is why a rather simple geometric model (square ®bres) was prepared at ®rst. The FE models in Figs. 5 and 6 represent a more realistic ®bre geometry for both orientations. The ®bre diameter is 10 mm, the volume fraction is 0.44 and isotropic ®bre material properties are assumed (further data are in Table 1). Also here the size of the FE micro-model is 0.1 mm 0.1 mm 0.1 mm while the coupled macro-model is 0.5 mm 0.5 mm 0.5 mm. The model for N orientation contains 18599 nodes and 20271 elements, while for P orientation 28235 nodes and 30907 elements were created. 2.4. Contact algorithm To reduce the required CPU time, a contact solution assuming rigid ball indentation was chosen. In this case the geometry of the ball `appears' in form of a prescribed displacement acting on the composite surface. The only unknown quantity of the contact analysis is the size of the contact area. Using truss elements, as simple contact elements, an iteration procedure is needed to ®nd the contour of the contact area. The truss elements subjected to tension were removed from the model, and only the compressed elements were kept.
Fig. 4. FE mesh for micro-model in case of parallel ®bre orientation (the FE mesh of the ball and the contact elements are not shown).
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The contact algorithm uses iterations to satisfy the displacement and stress type contact conditions. The displacement contact condition is illustrated in Fig. 7 for a rigid sphere. The displacement type contact conditions are:
i ui hi
inside the contact area;
1a
i < u i h i
outside the contact area;
1b
where i is the normal approach, ui is the elastic deformation of the surface points of the (bottom) body and hi is the initial gap between the bodies at point i. The stress conditions of contact are: pi > 0 inside the contact area;
2a
pi 0 outside the contact area:
2b
The FE calculation for a given normal load F evaluates the location of the contact area using an iterative procedure. At the beginning, a normal approach, i.e. i =constant, is chosen, and a potential contact area is selected that should be greater than the unknown contact area. Inside the potential contact area a prescribed displacement ui is applied in normal direction at each node, according to Eq. (1a) ui i ÿ hi :
Fig. 5. FE mesh for the contact micro-model assuming a rigid ball indentation (N orientation).
Fig. 6. FE mesh for the contact micro-model assuming a rigid ball indentation (P orientation).
3
The reaction forces at each nodes of the contact area are proportional to the contact pressure distribution. If the reaction forces at the assumed boundary points of the contact area are positive (in z direction of Fig. 7, representing compression) these point should belong to the real contact area. If the reaction force is negative (representing tension) the point should be outside of the contact area. Following this procedure, the potential contact area becomes gradually smaller than one originally assumed. In this way, the contact area is continuously modi®ed during this iteration and the total normal load F is also calculated. If this actual load becomes smaller then the given one, the chosen normal approach i should be increased, while if this total load
Fig. 7. The displacement contact condition.
K. VaÂradi et al / Composites Science and Technology 59 (1999) 271±281
is greater, i should be reduced. The iteration is ®nished when at every point of the contact area both the displacement and the stress type conditions of contact are ful®lled and when the calculated total normal load also equals to the given load. Note. This algorithm is ecient if the Fast Finite Element Module of the COSMOS/M System is used because of the CPU time of one iteration step is less than 1% of the CPU time required using the traditional solver of COSMOS/M with GAP elements (usually 8± 10 iteration steps are needed). 3. Results of the contact, stress and strain analysis 3.1. Contact results for the square ®bre model At ®rst contact results of the single models with N and P ®bre orientations are presented followed by the results of the coupled models. The applied load for N ®bre orientation is 5 N while for P ®bre orientation is 1 N. The contact results involve the following parameters: . normal approach: , . length of the contact area: 2a, . width of the contact area: 2b. For N orientation, but especially for P orientation, the elastic deformation of the ball is only some percentages of the elastic deformation of the composite material. Due to these dierences it is worth to check, if the ball might be considered as a rigid ball, assuming no deformation on the boundary of the ball. To check this assumption, the segment representing the ball in the FE model should be removed, the upper nodes of the gap elements should be ®xed. This has a signi®cant eect on the CPU time in the FE contact calculation.
275
same. There are also bigger dierences in the maximum contact pressure values. . Assuming a rigid ball, the normal approach is by 13% smaller than in case of elastic ball. For 2a, 2b and pmax the dierences are less than 13%. . The contact pressures are much higher than the compressive strength of the contact bodies. In reality failure of the composite surface would occur at lower pressure. But for the present study these facts will be ignored, i.e. calculations will be carried out purely elastically. Future studies will also consider maximum bearable loads. Fig. 8(a) illustrates the shape and the size of the quarter of the contact area obtained for N orientation. Fig. 8(b) presents the contact pressure distribution. 3.1.2. P ®bre orientation Table 3 represents the contact parameters for single and coupled models in case of P orientation, assuming isotropic ®bre properties. The following conclusions can be drawn: . Comparing the single and coupled contact models, one can also conclude that the normal approach is dominantly bigger if coupled models are used, while the size of the contact area and the maximum contact pressure values are almost the same. . Assuming a rigid ball, the results are dierent by just about 2% compared to the elastic ball situation. This means, that there is only a relatively small elastic deformation of the ball compared to the more dominant elastic deformation of the composite material in case of parallel ®bre orientation.
3.1.1. N ®bre orientation Table 2 presents the contact parameters for single and coupled models in case of N orientation. The following conclusions can be drawn: . Comparing the single and coupled contact models, one can conclude that the normal approach is dominantly bigger if coupled models are used, while the size of the contact area is almost the Table 2 Contact parameters
Total load F (N) Normal approach (mm) Size of the contact area 2a (mm) Maximum contact pressure pmax (MPa)
Single
Coupled
5 1.35 85 2114
5 2.30 95 1625
Fig. 8. Contact results for the square ®bre model: (a) contour of the contact area for N orientation, (b) contact pressure distribution for N orientation, (c) contour of the contact area for P orientation, (d) contact pressure distribution for P orientation, Note: Fig. (b) and (d) are rotated compared to Fig. (a) and (c).
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Table 3 Contact parameters
Total load F (N) Normal approach (mm) Size of the contact area 2a 2b (mm) Maximum contact pressure pmax (MPa)
Single
Coupled
1 1.08 80 65 495
1 2.25 80 65 480
. Also here the real load bearing capacity of the composite material is actually exceeded, but this will be ignored at the moment. Fig. 8(c) shows for P orientation the shape and the size of the quarter of the contact area. Fig. 8(d) presents the contact pressure distribution. The results do not show dominant elliptical contact areas. They are nearly circular in shape. 3.2. Contact results for the round ®bre model Based on the previous results, in the following calculations only a rigid ball indentation is assumed. The problems, described in Section 2.3, were solved for total loads of 5 N (for N orientation) and 1 N (for P orientation), respectively. At ®rst the single models and later the FE coupled contact solutions were assumed. For both loading situations, material properties according to Table 1 were used. In the following chapter the printed results represent the coupled FE contact models, while the minimum/maximum values in Tables 5 and 7 are listed for both the single and the coupled models. 3.2.1. N ®bre orientation According to the described iteration in Section 2.4 the contact results listed in Table 4 were obtained for the FE single and coupled contact micro-models assuming rigid ball indentation. Fig. 9(a) illustrates the shape and the size of the quarter of the contact area obtained for N orientation. Fig. 9(b) presents the contact pressure distribution. 3.2.1.1. Displacement and strain results. The deformed shape is presented in Fig. 10(a) (deformation scale is 10:1). The dominant types of deformation of the ®bres are compression and bending. The deformation at the bottom of the model is not zero because coupled models are used. Table 4 Contact results
Total load F (N) Normal approach (mm) Size of the contact area 2a (mm) Maximum contact pressure pmax (MPa)
Single
Coupled
5 1.16 95 3380
5 1.98 95 3260
Fig. 9. Contact results for the round ®bre model: (a) contour of the contact area for N orientation, (b) contact pressure distribution for N orientation, (c) contour of the contact area for P orientation, (d) contact pressure distribution for P orientation, Note: Fig. (b) and (d) are rotated compared to Fig. (a) and (c).
During the ball indentation test the maximum equivalent strain does not occur at the surface of the composite material, but underneath of it at a depth of z=ÿ0.03 mm; this is about 60% of the radius of the contact area. The equivalent strain is shown in Fig. 10(c) and also in Fig. 10(d), after sectioning the sample at a depth of z=ÿ0.03 mm. The matrix is subjected to signi®cant shear straining below the surface (see Table 5). The maximum shear strains are xz 0:0757 and yz 0:0852, while xy is not as signi®cant. The maximum value of "z (Fig. 10(b)) is about one third of the maximum shear strain values, expressing more dominant shear in the matrix than compression.
3.2.1.2. Stress results. The vertical load is mostly transferred by the ®bres (Fig. 11(b)), so that the compressive stresses in the ®bres are very high. According to the elastic calculations sz > 3000 MPa, which is above the longitudinal compressive strength of the ®bres. (Taking this failure criterion into consideration would require contact analysis with material non-linearity). The Mises equivalent stress distribution (Fig. 11(a)) follows the tendencies of z because this is the most dominant stress component in the ®bre. The shear stresses in the ®bres (see Table 5) are much lower than z . The stress analysis in the matrix requires special attention. (To show the stresses in the matrix in a clear way, the ®bres were removed from the model while the results were presented.) The equivalent stress distribution in the matrix (see Fig. 11(c) and (d)) is
K. VaÂradi et al / Composites Science and Technology 59 (1999) 271±281
277
Fig. 10. (a) Deformed shape (scale 10:1), (b) Normal strain ez (in vertical direction), (c) Equivalent strain, (d) Equivalent (strain at z=ÿ0.03). Table 5 The minimum and the maximum values for N ®bre orientation Fig. No.
Description
Single Min.
10(c) 10(d)
10(b) 11(a) 11(b)
11(c) 11(d)
Displacement in z direction (mm) Equivalent strain Equivalent strain at z=ÿ0.03 Shear strain gxy Shear strain gxz Shear strain gyz Strain ez Von Mises equivalent stress (MPa) Stress sz (MPa) Shear stress txy (MPa) Shear stress txz (MPa) Shear stress tyz (MPa) Von Mises equivalent stress in the matrix (MPa) Von Mises equivalent stress in the matrix at z=ÿ0.03 (MPa) Shear stress txy in the matrix (MPa) Shear stress tyz in the matrix (MPa) Stress sz in the matrix (MPa)
ÿ0.00116 ÿ0.0157 ÿ0.0132 ÿ0.0129 ÿ0.0149 ÿ3380 ÿ107 ÿ135 ÿ153 ÿ16.9 ÿ16.6 ÿ210
Coupled Max. 0.0352 0.0352 0.0232 0.0528 0.060 0.00215 3210 509 120 599 527 136 136 67.9 77.1 40.3
Min. ÿ0.00198 ÿ0.0210 ÿ0.0188 ÿ0.0214 ÿ0.0136 ÿ3260 ÿ77.7 ÿ252 ÿ147 ÿ24.1 ÿ27.6 ÿ239
Max. 0.0497 0.0497 0.0285 0.0757 0.0852 0.00268 3060 620 110 698 614 192 192 97.3 110 31.3
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K. VaÂradi et al / Composites Science and Technology 59 (1999) 271±281
Fig. 11. (a) Equivalent stress. (b) Normal stress sz (in vertical direction). (c) Equivalent stress in the matrix. (d) Equivalent stress in the matrix (at z=ÿ0.03).
similar to the equivalent strain distribution, having its maximum at the same depth. At the centre of the contact area high compressive stresses are acting in the matrix z ÿ239 MPa, but only in a very narrow region. The shear stresses have their maximum values xz 97:3 MPa and yz 110 MPa at a depth of z ÿ0.03 mm. Comparing the single and coupled FE solutions, the results are signi®cantly dierent. The dominant strain Table 6 Contact results
Total load F (N) Normal approach (mm) Size of the contact area 2a 2b (mm) Maximum contact pressure pmax (MPa)
Single
Coupled
1 1.21 85 75 503
1 1.70 85 75 491
and stress values are higher if the coupled models are used. The results and the conclusions are very similar to those of Ref. [1], in which the interfacial failure due to ball indentation was evaluated by numerical and experimental techniques. 3.2.2. P ®bre orientation The contact results listed in Table 6 were obtained for the FE single and coupled contact micro-models assuming rigid ball indentation. Fig. 9(c) shows for P orientation the shape and the size of the quarter of the contact area. Fig. 9(d) presents the contact pressure distribution. Note that in the following statements, in which strain and stress results are compared with the previous results for N ®bre orientation, the smaller load (1 N instead of 5 N), should also be taken into consideration.
K. VaÂradi et al / Composites Science and Technology 59 (1999) 271±281
3.2.2.1. Displacement and strain results. The deformed shape is presented in Fig. 12(a) (deformation scale is 10:1). The dominant deformation of the ®bres is bending. At the same time local plastic deformation of the matrix occurs in the vicinity of the ®bres in the contact area. The deformation at the bottom of the model is not zero because coupled models are used. The maximum equivalent strain, normal and shear strains occur in a very narrow range of the contact area. The matrix is subjected to both shear and compressive type straining (see Fig. 12(b±d). Among the shear strain values the maxima of
xz and yz are almost three times higher than xy , expressing more dominant shear in z direction than in the x-y plain. The maximum value of "z is about the same as the maxima of gxz and gyz, so that both
279
shear and compressive straining of the matrix are dominant. 3.2.2.2. Stress results. The contact pressure is acting on the ®bres, so that the ®bres are subjected to bending and compressive stresses in the transversal direction (Fig. 13(a) and (b)). Shear stresses in the ®bres (see Table 7) are smaller than bending stresses. The equivalent stress distribution in the matrix (Fig. 13(c)) occurs in a very small vicinity of the contact area. The highest stress component in the matrix is sz (Fig. 13(d)) representing high local compression. Comparing the single and coupled FE solutions, the results are less dierent for the contact parameters and for stresses and strains than it was observed in case of the N ®bre orientation.
Fig. 12. (a) Deformed shape (scale 10:1). (b) Shear strain gxz (in x-z plane in Fig. 6). (c) Normal strain ez (in vertical direction). (d) Shear strain gyz (in y-z plane in Fig. 6).
K. VaÂradi et al / Composites Science and Technology 59 (1999) 271±281
280
Fig. 13. (a)Normal stress sx (in ®bre direction). (b) Normal stress sz (in vertical direction). (c) Equivalent stress in the matrix. (d) Normal stress sz (in vertical direction) in the matrix. Table 7 The minimum and the maximum values for P ®bre orientation Fig. No.
Description
Single Min.
12(b) 12(d) 12(c) 13(a) 13(b)
13(c) 13(d)
Displacement in z direction (mm) Equivalent strain Equivalent strain at z=ÿ0.0225 Shear strain gxy at x=0.02 Shear strain gxz at x=0.02 Shear strain gyz at x=0.028 Strain ez Von Mises equivalent stress (MPa) Von Mises equivalent stress (MPa) at z=ÿ0.018 Stress sx (MPa) Stress sz (MPa) Shear stress txy (MPa) Shear stress txz at x=0.02 (MPa) Shear stress tyz (MPa) Von Mises equivalent stress in the matrix (MPa) Stress sz in the matrix (MPa) Shear stress txy in the matrix at z=ÿ0.0075 (MPa) Shear stress tyz in the matrix (MPa)
ÿ0.00121 ÿ0.0202 ÿ0.00173 ÿ0.0276 ÿ0.0442
ÿ503 ÿ261 ÿ175 ÿ189 ÿ331 ÿ2.2 ÿ35.4
Coupled Max. 0.038 0.038 0.0185 0.0500 0.0531 0.00450 942 942 27.4 184 374 213 147 12.2 64.5 68.3
Min. ÿ0.00170 ÿ0.0202 ÿ0.00161 ÿ0.0262 ÿ0.0419 ÿ613 ÿ491 ÿ247 ÿ162 ÿ186 ÿ322 ÿ0.7 ÿ33.7
Max. 0.0368 0.0368 0.0187 0.0504 0.0516 0.00474 966 966 799 47.2 176 351 211 142 13.6 64.8 66.6
K. VaÂradi et al / Composites Science and Technology 59 (1999) 271±281
4. Conclusions 1. The FE micro-model, compared to the equivalent anisotropic macro-model, is much more useful, because the ®bre/matrix structure may be analysed in detail. The micro-model provides more realistic displacement, strain and stress results. 2. The FE contact results show the location and the distribution of the subsurface stresses and strains. For N ®bre orientation there is a high shear stress region bellow the surface, from where the ®bre/ matrix interfacial failure initiates before propagating to the surface. These results are in good agreement with experimental results published in the open literature. In case of P ®bre orientation the matrix is subjected to local plastic deformation while the characteristic deformation of the ®bre is bending. 3. The applied coupled model technique `joints' the micro and macro approaches and provides approximate results for the whole range of the micro- and macro-models. There are bigger dierences for the normal approach and smaller dierences for the size of the contact area if single and coupled models are compared. 4. At a higher load level in case of both ®bre orientations the linear elastic material law can not describe the behaviour of the composite structure subjected to ball indentation test. It requires further considerations and FE elastic±plastic veri®cation. This evaluation also requires more accurate
281
material properties and failure criteria for the composite structure. 5. Further work by the authors will report about a comparison of the ball indentation depths achieved by simulations and experiments. Acknowledgements The authors gratefully acknowledge the ®nancial support of the Deutsche Forschungsgemeinschaft DFG FR 675/19-1. Additional thanks are due to the GermanHungarian Research Cooperation Fund (UNG-041-96). References [1] Carman GP, Lesko JJ, Reifsnider KL, Dillard DA. Micromechanical model of composite materials subjected to ball indentation. Journal of Composite Materials 1993;27(3). [2] COSMOS/M User Guide v1.75, Structural Research and Analysis Corporation, 1995. [3] Beall MW, Shephard MS, Fish J, Belsky V. Multiscale modelling for crack propagation in composites. 1996 ASME International Mechanical Engineering Congress and Exposition, November 17±22, Atlanta, Georgia, USA, 1996. [4] Zhang J, Kikuchi N, Terada K. Global±local analysis of composites by image based ®xed grid method. 1996 ASME International Mechanical Engineering Congress and Exposition, November 17±22, Atlanta, Georgia, USA, 1996. [5] VaÂradi K, NeÂder Z, Friedrich K, FloÈck J. 3D anisotropic numerical and FE-contact analysis and experimental investigations of a steel ball indented into a ®bre-matrix structure with normal and parallel orientation. IVW-Bericht 97±30.
Finite-element analysis of a polymer composite subjected to ball indentation$ K. VaÂradi a,1, Z. NeÂder a, K. Friedrich b,*, J. FloÈck b a Institute of Machine Design, Technical University of Budapest, H-1521 Budapest, Hungary Institute for Composite Materials, University of Kaiserslautern, 67663 Kaiserslautern, Germany
b
Received 17 June 1997; accepted 5 March 1998
Abstract The ®nite-element (FE) contact technique presented in this paper is applied to the problem of steel-ball indentation of a composite material consisting of unidirectional continuous carbon ®bres in a poly(ether ether ketone) matrix. Indentation was carried out with ®bre orientations either normal (N) or parallel (P) to the contact surface at a ®bre volume fraction of 0.44. The FE contact analysis involves both an anisotropic (homogeneous) macro- and (inhomogeneous) micro-contact analysis, following an approximate displacement coupling technique. The FE contact-stress analysis of a ®bre/matrix micro-structure has a major limitation. If the FE micro-model is used, only a very small (for example 0.1 mm 0.1 mm 0.1 mm) 3D segment can be modelled. If an anisotropic model is used, there is no similar size limitation but the results cannot describe the stress and strain states of a real ®bre/ matrix micro-structure. The FE contact results show the location and the distribution of the sub-surface stresses and strains. For N ®bre orientation there is a high shear stress region below the surface, from where the ®bre/matrix interfacial failure initiates before propagating to the surface. In the case of P ®bre orientation the matrix is subjected to local plastic deformation while the characteristic deformation of the ®bre is bending. # 1999 Elsevier Science Ltd. All rights reserved.
1. Introduction The ball indentation test is a well-known procedure to evaluate the strength properties of metals both in the elastic and the elastic±plastic range. In the last couple of decades, this test was also applied to composite materials, in particular to characterise their mechanical behaviour, e.g. with regard to their ®bre/matrix interfacial shear strength. Interfacial failure caused by the indentation test was studied by Carman et al. [1] using experimental and analytical techniques. Micro-stresses were evaluated by macro/micro approaches and a cellular modelling concept. According to their experimental observations, ``substantial ®bre/matrix interfacial failure occurs underneath the ball indentor''. To analyse the ball/composite contact problem, ``a solution * Corresponding author. Tel.: 0049 631 2017201; fax: 0049 631 2017198; e-mail: [email protected]. $ Part of this material was submitted to 1997 ASME International Mechanical Engineering Congress and Exposition, Dallas, 16±21, November, 1997. 1 E-mail: [email protected].
was generated with a macro/micro approach utilising an elasticity solution to formulate the macro solution. Using the macro solution to generate local boundary conditions in conjunction with a micro solution based on a cellular model, an approximate closed form solution for the stress state in a micro region was constructed''. In the present study, the indentation problem was ®rst solved by the FE contact technique of COSMOS/M [2], a simple square cross-section being assumed for each ®bre. The FE contact analysis involves both an anisotropic macro- and micro-contact analysis, following the presented approximate displacement coupling technique. (The `multiscale' modelling [3] or the `globallocal' analysis [4] techniques are more general procedures.) The coupled solution provides a more realistic elastic deformation of the composite system in the vicinity of the contact area, and at the same time the eect of the macro-system is also incorporated. The FE contact-stress analysis of a ®bre/matrix micro-structure has strong limitation. If anisotropic numerical models are used, there are no size limitations but the results cannot describe the stress and strain
0266-3538/99/$Ðsee front matter # 1999 Elsevier Science Ltd.. All rights reserved. PII: S0266 -3 538(98)00066 -9
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272
Table 1 Anisotropic material properties assuming isotropic ®bre properties Vf=0.44
Steel
Fibre
Matrix
Composite
E11 (GPa) E22 (GPa) E33 (GPa) G12 (MPa) G13 (MPa) G23 (MPa) 12 13 23
210
235
3.6
106342 6397 6397 2290 2290 2214 0.296 0.296 0.444
2. Finite-element macro- and micro-contact analysis incorporating a coupling technique 2.1. The coupled FE contact models
states of a real ®bre/matrix micro-structure. If an FE micro-model is used, only a very small (for example 0.1 mm 0.1 mm 0.1 mm) 3D segment can be modelled while the number of degrees of freedom (DOF) is in the neighbourhood of 100,000. To study the eect of ball indentation on a composite structure, with a more realistic (cylindrical) ®bre geometry in the FE models, an iterative contact solution, assuming a rigid ball, was chosen. Finally stress and strain results were obtained for both ®bre orientations. The composite material investigated was a unidirectional continuous carbon-®bre/ poly(ether ether ketone) system, containing normal and parallel ®bre orientations at ®bre volume fraction 0.44. The material properties for the ball and the composite material [5] are listed in Table 1 assuming isotropic ®bre properties. For the composite material transversely isotropic material properties were assumed. The material principle direction 1 was vertical (z direction in Fig. 1(b)) for N orientation and horizontal (x direction in Fig. 1(b)) for P orientation.
The models of the ball and the composite material are shown in Fig. 1(a). The ball diameter is 2 mm. As a consequence of the symmetry conditions, only a quarter of the bodies are considered, and the proper boundary conditions are used. To evaluate the stresses in a ®bre/matrix micro-system, an FE micro-model is created. This micro-model is `built into' a larger (homogeneous and anisotropic) macro-model to represent a larger segment of the original bodies. From the ®bre/matrix micro-structure and also from the ball a volume of 0.1 mm 0.1 mm 0.1 mm is modelled (Fig. 1(b)); for the macro-structure the size of the two segments amounted to 0.5 mm 0.5 mm 0.5 mm. In these two cases the size of the contact area is in the range of 0.05±0.1 mm. If the contact area is larger, larger segments (5±10 times of the expected contact area) should also be considered. In the present case the ball has much smaller elastic deformation than the composite material (i.e. a rigidball model causes only a small error, see later); therefore, a small segment of the ball is enough to be modelled around the actual very small contact area. These segments (points A and B in Fig. 1(b)) do not follow the ball's geometry but the initial gap assigned to the gap elements represents exactly the real gap between the ball and the composite material. First the FE contact analysis operates on the larger segment (macro-model) followed by the smaller segment (micro-model), while the displacements are coupled. The larger model has only a few contact elements in the
Fig. 1. The global and local models (a) and the substituting coupled model approach (b) (smaller quarter segments are 0.1 mm 0.1 mm 0.1 mm, and the larger quarter segments are 0.5 mm 0.5 mm 0.5 mm).
Fig. 2. FE mesh for the coupled contact models (smaller (b) is micro-, large (a) is macro-model) (. ± coupled surface, % ± zero prescribed displacement perpendicular to the surface).
80769
100772
1286
0.3
0.166
0.4
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expected contact area while the smaller model has much more contact elements. In the coupled contact models it is assumed that the total load, introduced as distributed load on the actual upper surface of the modelled ball segment, is the same at both macro and micro levels. In Fig. 2 for the bottom body (representing the composite segment) the displacements of the points of the larger model, located at the boundary area of the smaller model, are assigned to the smaller model as boundary conditions all over its side surfaces. These are therefore coupled surfaces. The surface having the contact elements are free from prescribed displacements, apart from the edges. The steel ball also has two models (larger and smaller) but the displacements are not coupled because the ball is `more rigid' (see results). For the ball only the boundary conditions, representing the quarter model, are applied at both levels. 2.2. FE contact micro-model with square ®bres In Fig. 1(b) the modelled quarter segments are shown. Both the large and the smaller segments (as separate models) are divided into 20 20 20 eight-node solid elements. Between the potential contact surfaces nodeto-node type gap elements are located. The bottom of the composite segment is ®xed, while the load is introduced at the top of the modelled ball segment (Fig. 2(a)). The total model contains 18522 nodes and 16150 elements. For N and P orientation the geometry of the macro-models is the same; only the material principle directions are changed. The micro-models are dierent.
Fig. 3. FE mesh for micro-model in case of normal ®bre orientation (the FE mesh of the ball and the contact elements are not shown).
273
Figs. 3 and 4 show the FE meshes of the micro-models for N and P orientation. The ®bres are assumed to have square cross sections of 10 mm 10 mm, and the volume fraction is 0.44 (4/9 of the uniform mesh). The rest of the FE model (overall sizes, mesh structure, boundary condition, loading and contact elements) is the same as in Section 2.1 for the smaller model. 2.3. FE contact micro-model with round ®bres The FE contact analysis requires FE contact iterations. This is why a rather simple geometric model (square ®bres) was prepared at ®rst. The FE models in Figs. 5 and 6 represent a more realistic ®bre geometry for both orientations. The ®bre diameter is 10 mm, the volume fraction is 0.44 and isotropic ®bre material properties are assumed (further data are in Table 1). Also here the size of the FE micro-model is 0.1 mm 0.1 mm 0.1 mm while the coupled macro-model is 0.5 mm 0.5 mm 0.5 mm. The model for N orientation contains 18599 nodes and 20271 elements, while for P orientation 28235 nodes and 30907 elements were created. 2.4. Contact algorithm To reduce the required CPU time, a contact solution assuming rigid ball indentation was chosen. In this case the geometry of the ball `appears' in form of a prescribed displacement acting on the composite surface. The only unknown quantity of the contact analysis is the size of the contact area. Using truss elements, as simple contact elements, an iteration procedure is needed to ®nd the contour of the contact area. The truss elements subjected to tension were removed from the model, and only the compressed elements were kept.
Fig. 4. FE mesh for micro-model in case of parallel ®bre orientation (the FE mesh of the ball and the contact elements are not shown).
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The contact algorithm uses iterations to satisfy the displacement and stress type contact conditions. The displacement contact condition is illustrated in Fig. 7 for a rigid sphere. The displacement type contact conditions are:
i ui hi
inside the contact area;
1a
i < u i h i
outside the contact area;
1b
where i is the normal approach, ui is the elastic deformation of the surface points of the (bottom) body and hi is the initial gap between the bodies at point i. The stress conditions of contact are: pi > 0 inside the contact area;
2a
pi 0 outside the contact area:
2b
The FE calculation for a given normal load F evaluates the location of the contact area using an iterative procedure. At the beginning, a normal approach, i.e. i =constant, is chosen, and a potential contact area is selected that should be greater than the unknown contact area. Inside the potential contact area a prescribed displacement ui is applied in normal direction at each node, according to Eq. (1a) ui i ÿ hi :
Fig. 5. FE mesh for the contact micro-model assuming a rigid ball indentation (N orientation).
Fig. 6. FE mesh for the contact micro-model assuming a rigid ball indentation (P orientation).
3
The reaction forces at each nodes of the contact area are proportional to the contact pressure distribution. If the reaction forces at the assumed boundary points of the contact area are positive (in z direction of Fig. 7, representing compression) these point should belong to the real contact area. If the reaction force is negative (representing tension) the point should be outside of the contact area. Following this procedure, the potential contact area becomes gradually smaller than one originally assumed. In this way, the contact area is continuously modi®ed during this iteration and the total normal load F is also calculated. If this actual load becomes smaller then the given one, the chosen normal approach i should be increased, while if this total load
Fig. 7. The displacement contact condition.
K. VaÂradi et al / Composites Science and Technology 59 (1999) 271±281
is greater, i should be reduced. The iteration is ®nished when at every point of the contact area both the displacement and the stress type conditions of contact are ful®lled and when the calculated total normal load also equals to the given load. Note. This algorithm is ecient if the Fast Finite Element Module of the COSMOS/M System is used because of the CPU time of one iteration step is less than 1% of the CPU time required using the traditional solver of COSMOS/M with GAP elements (usually 8± 10 iteration steps are needed). 3. Results of the contact, stress and strain analysis 3.1. Contact results for the square ®bre model At ®rst contact results of the single models with N and P ®bre orientations are presented followed by the results of the coupled models. The applied load for N ®bre orientation is 5 N while for P ®bre orientation is 1 N. The contact results involve the following parameters: . normal approach: , . length of the contact area: 2a, . width of the contact area: 2b. For N orientation, but especially for P orientation, the elastic deformation of the ball is only some percentages of the elastic deformation of the composite material. Due to these dierences it is worth to check, if the ball might be considered as a rigid ball, assuming no deformation on the boundary of the ball. To check this assumption, the segment representing the ball in the FE model should be removed, the upper nodes of the gap elements should be ®xed. This has a signi®cant eect on the CPU time in the FE contact calculation.
275
same. There are also bigger dierences in the maximum contact pressure values. . Assuming a rigid ball, the normal approach is by 13% smaller than in case of elastic ball. For 2a, 2b and pmax the dierences are less than 13%. . The contact pressures are much higher than the compressive strength of the contact bodies. In reality failure of the composite surface would occur at lower pressure. But for the present study these facts will be ignored, i.e. calculations will be carried out purely elastically. Future studies will also consider maximum bearable loads. Fig. 8(a) illustrates the shape and the size of the quarter of the contact area obtained for N orientation. Fig. 8(b) presents the contact pressure distribution. 3.1.2. P ®bre orientation Table 3 represents the contact parameters for single and coupled models in case of P orientation, assuming isotropic ®bre properties. The following conclusions can be drawn: . Comparing the single and coupled contact models, one can also conclude that the normal approach is dominantly bigger if coupled models are used, while the size of the contact area and the maximum contact pressure values are almost the same. . Assuming a rigid ball, the results are dierent by just about 2% compared to the elastic ball situation. This means, that there is only a relatively small elastic deformation of the ball compared to the more dominant elastic deformation of the composite material in case of parallel ®bre orientation.
3.1.1. N ®bre orientation Table 2 presents the contact parameters for single and coupled models in case of N orientation. The following conclusions can be drawn: . Comparing the single and coupled contact models, one can conclude that the normal approach is dominantly bigger if coupled models are used, while the size of the contact area is almost the Table 2 Contact parameters
Total load F (N) Normal approach (mm) Size of the contact area 2a (mm) Maximum contact pressure pmax (MPa)
Single
Coupled
5 1.35 85 2114
5 2.30 95 1625
Fig. 8. Contact results for the square ®bre model: (a) contour of the contact area for N orientation, (b) contact pressure distribution for N orientation, (c) contour of the contact area for P orientation, (d) contact pressure distribution for P orientation, Note: Fig. (b) and (d) are rotated compared to Fig. (a) and (c).
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Table 3 Contact parameters
Total load F (N) Normal approach (mm) Size of the contact area 2a 2b (mm) Maximum contact pressure pmax (MPa)
Single
Coupled
1 1.08 80 65 495
1 2.25 80 65 480
. Also here the real load bearing capacity of the composite material is actually exceeded, but this will be ignored at the moment. Fig. 8(c) shows for P orientation the shape and the size of the quarter of the contact area. Fig. 8(d) presents the contact pressure distribution. The results do not show dominant elliptical contact areas. They are nearly circular in shape. 3.2. Contact results for the round ®bre model Based on the previous results, in the following calculations only a rigid ball indentation is assumed. The problems, described in Section 2.3, were solved for total loads of 5 N (for N orientation) and 1 N (for P orientation), respectively. At ®rst the single models and later the FE coupled contact solutions were assumed. For both loading situations, material properties according to Table 1 were used. In the following chapter the printed results represent the coupled FE contact models, while the minimum/maximum values in Tables 5 and 7 are listed for both the single and the coupled models. 3.2.1. N ®bre orientation According to the described iteration in Section 2.4 the contact results listed in Table 4 were obtained for the FE single and coupled contact micro-models assuming rigid ball indentation. Fig. 9(a) illustrates the shape and the size of the quarter of the contact area obtained for N orientation. Fig. 9(b) presents the contact pressure distribution. 3.2.1.1. Displacement and strain results. The deformed shape is presented in Fig. 10(a) (deformation scale is 10:1). The dominant types of deformation of the ®bres are compression and bending. The deformation at the bottom of the model is not zero because coupled models are used. Table 4 Contact results
Total load F (N) Normal approach (mm) Size of the contact area 2a (mm) Maximum contact pressure pmax (MPa)
Single
Coupled
5 1.16 95 3380
5 1.98 95 3260
Fig. 9. Contact results for the round ®bre model: (a) contour of the contact area for N orientation, (b) contact pressure distribution for N orientation, (c) contour of the contact area for P orientation, (d) contact pressure distribution for P orientation, Note: Fig. (b) and (d) are rotated compared to Fig. (a) and (c).
During the ball indentation test the maximum equivalent strain does not occur at the surface of the composite material, but underneath of it at a depth of z=ÿ0.03 mm; this is about 60% of the radius of the contact area. The equivalent strain is shown in Fig. 10(c) and also in Fig. 10(d), after sectioning the sample at a depth of z=ÿ0.03 mm. The matrix is subjected to signi®cant shear straining below the surface (see Table 5). The maximum shear strains are xz 0:0757 and yz 0:0852, while xy is not as signi®cant. The maximum value of "z (Fig. 10(b)) is about one third of the maximum shear strain values, expressing more dominant shear in the matrix than compression.
3.2.1.2. Stress results. The vertical load is mostly transferred by the ®bres (Fig. 11(b)), so that the compressive stresses in the ®bres are very high. According to the elastic calculations sz > 3000 MPa, which is above the longitudinal compressive strength of the ®bres. (Taking this failure criterion into consideration would require contact analysis with material non-linearity). The Mises equivalent stress distribution (Fig. 11(a)) follows the tendencies of z because this is the most dominant stress component in the ®bre. The shear stresses in the ®bres (see Table 5) are much lower than z . The stress analysis in the matrix requires special attention. (To show the stresses in the matrix in a clear way, the ®bres were removed from the model while the results were presented.) The equivalent stress distribution in the matrix (see Fig. 11(c) and (d)) is
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277
Fig. 10. (a) Deformed shape (scale 10:1), (b) Normal strain ez (in vertical direction), (c) Equivalent strain, (d) Equivalent (strain at z=ÿ0.03). Table 5 The minimum and the maximum values for N ®bre orientation Fig. No.
Description
Single Min.
10(c) 10(d)
10(b) 11(a) 11(b)
11(c) 11(d)
Displacement in z direction (mm) Equivalent strain Equivalent strain at z=ÿ0.03 Shear strain gxy Shear strain gxz Shear strain gyz Strain ez Von Mises equivalent stress (MPa) Stress sz (MPa) Shear stress txy (MPa) Shear stress txz (MPa) Shear stress tyz (MPa) Von Mises equivalent stress in the matrix (MPa) Von Mises equivalent stress in the matrix at z=ÿ0.03 (MPa) Shear stress txy in the matrix (MPa) Shear stress tyz in the matrix (MPa) Stress sz in the matrix (MPa)
ÿ0.00116 ÿ0.0157 ÿ0.0132 ÿ0.0129 ÿ0.0149 ÿ3380 ÿ107 ÿ135 ÿ153 ÿ16.9 ÿ16.6 ÿ210
Coupled Max. 0.0352 0.0352 0.0232 0.0528 0.060 0.00215 3210 509 120 599 527 136 136 67.9 77.1 40.3
Min. ÿ0.00198 ÿ0.0210 ÿ0.0188 ÿ0.0214 ÿ0.0136 ÿ3260 ÿ77.7 ÿ252 ÿ147 ÿ24.1 ÿ27.6 ÿ239
Max. 0.0497 0.0497 0.0285 0.0757 0.0852 0.00268 3060 620 110 698 614 192 192 97.3 110 31.3
278
K. VaÂradi et al / Composites Science and Technology 59 (1999) 271±281
Fig. 11. (a) Equivalent stress. (b) Normal stress sz (in vertical direction). (c) Equivalent stress in the matrix. (d) Equivalent stress in the matrix (at z=ÿ0.03).
similar to the equivalent strain distribution, having its maximum at the same depth. At the centre of the contact area high compressive stresses are acting in the matrix z ÿ239 MPa, but only in a very narrow region. The shear stresses have their maximum values xz 97:3 MPa and yz 110 MPa at a depth of z ÿ0.03 mm. Comparing the single and coupled FE solutions, the results are signi®cantly dierent. The dominant strain Table 6 Contact results
Total load F (N) Normal approach (mm) Size of the contact area 2a 2b (mm) Maximum contact pressure pmax (MPa)
Single
Coupled
1 1.21 85 75 503
1 1.70 85 75 491
and stress values are higher if the coupled models are used. The results and the conclusions are very similar to those of Ref. [1], in which the interfacial failure due to ball indentation was evaluated by numerical and experimental techniques. 3.2.2. P ®bre orientation The contact results listed in Table 6 were obtained for the FE single and coupled contact micro-models assuming rigid ball indentation. Fig. 9(c) shows for P orientation the shape and the size of the quarter of the contact area. Fig. 9(d) presents the contact pressure distribution. Note that in the following statements, in which strain and stress results are compared with the previous results for N ®bre orientation, the smaller load (1 N instead of 5 N), should also be taken into consideration.
K. VaÂradi et al / Composites Science and Technology 59 (1999) 271±281
3.2.2.1. Displacement and strain results. The deformed shape is presented in Fig. 12(a) (deformation scale is 10:1). The dominant deformation of the ®bres is bending. At the same time local plastic deformation of the matrix occurs in the vicinity of the ®bres in the contact area. The deformation at the bottom of the model is not zero because coupled models are used. The maximum equivalent strain, normal and shear strains occur in a very narrow range of the contact area. The matrix is subjected to both shear and compressive type straining (see Fig. 12(b±d). Among the shear strain values the maxima of
xz and yz are almost three times higher than xy , expressing more dominant shear in z direction than in the x-y plain. The maximum value of "z is about the same as the maxima of gxz and gyz, so that both
279
shear and compressive straining of the matrix are dominant. 3.2.2.2. Stress results. The contact pressure is acting on the ®bres, so that the ®bres are subjected to bending and compressive stresses in the transversal direction (Fig. 13(a) and (b)). Shear stresses in the ®bres (see Table 7) are smaller than bending stresses. The equivalent stress distribution in the matrix (Fig. 13(c)) occurs in a very small vicinity of the contact area. The highest stress component in the matrix is sz (Fig. 13(d)) representing high local compression. Comparing the single and coupled FE solutions, the results are less dierent for the contact parameters and for stresses and strains than it was observed in case of the N ®bre orientation.
Fig. 12. (a) Deformed shape (scale 10:1). (b) Shear strain gxz (in x-z plane in Fig. 6). (c) Normal strain ez (in vertical direction). (d) Shear strain gyz (in y-z plane in Fig. 6).
K. VaÂradi et al / Composites Science and Technology 59 (1999) 271±281
280
Fig. 13. (a)Normal stress sx (in ®bre direction). (b) Normal stress sz (in vertical direction). (c) Equivalent stress in the matrix. (d) Normal stress sz (in vertical direction) in the matrix. Table 7 The minimum and the maximum values for P ®bre orientation Fig. No.
Description
Single Min.
12(b) 12(d) 12(c) 13(a) 13(b)
13(c) 13(d)
Displacement in z direction (mm) Equivalent strain Equivalent strain at z=ÿ0.0225 Shear strain gxy at x=0.02 Shear strain gxz at x=0.02 Shear strain gyz at x=0.028 Strain ez Von Mises equivalent stress (MPa) Von Mises equivalent stress (MPa) at z=ÿ0.018 Stress sx (MPa) Stress sz (MPa) Shear stress txy (MPa) Shear stress txz at x=0.02 (MPa) Shear stress tyz (MPa) Von Mises equivalent stress in the matrix (MPa) Stress sz in the matrix (MPa) Shear stress txy in the matrix at z=ÿ0.0075 (MPa) Shear stress tyz in the matrix (MPa)
ÿ0.00121 ÿ0.0202 ÿ0.00173 ÿ0.0276 ÿ0.0442
ÿ503 ÿ261 ÿ175 ÿ189 ÿ331 ÿ2.2 ÿ35.4
Coupled Max. 0.038 0.038 0.0185 0.0500 0.0531 0.00450 942 942 27.4 184 374 213 147 12.2 64.5 68.3
Min. ÿ0.00170 ÿ0.0202 ÿ0.00161 ÿ0.0262 ÿ0.0419 ÿ613 ÿ491 ÿ247 ÿ162 ÿ186 ÿ322 ÿ0.7 ÿ33.7
Max. 0.0368 0.0368 0.0187 0.0504 0.0516 0.00474 966 966 799 47.2 176 351 211 142 13.6 64.8 66.6
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4. Conclusions 1. The FE micro-model, compared to the equivalent anisotropic macro-model, is much more useful, because the ®bre/matrix structure may be analysed in detail. The micro-model provides more realistic displacement, strain and stress results. 2. The FE contact results show the location and the distribution of the subsurface stresses and strains. For N ®bre orientation there is a high shear stress region bellow the surface, from where the ®bre/ matrix interfacial failure initiates before propagating to the surface. These results are in good agreement with experimental results published in the open literature. In case of P ®bre orientation the matrix is subjected to local plastic deformation while the characteristic deformation of the ®bre is bending. 3. The applied coupled model technique `joints' the micro and macro approaches and provides approximate results for the whole range of the micro- and macro-models. There are bigger dierences for the normal approach and smaller dierences for the size of the contact area if single and coupled models are compared. 4. At a higher load level in case of both ®bre orientations the linear elastic material law can not describe the behaviour of the composite structure subjected to ball indentation test. It requires further considerations and FE elastic±plastic veri®cation. This evaluation also requires more accurate
281
material properties and failure criteria for the composite structure. 5. Further work by the authors will report about a comparison of the ball indentation depths achieved by simulations and experiments. Acknowledgements The authors gratefully acknowledge the ®nancial support of the Deutsche Forschungsgemeinschaft DFG FR 675/19-1. Additional thanks are due to the GermanHungarian Research Cooperation Fund (UNG-041-96). References [1] Carman GP, Lesko JJ, Reifsnider KL, Dillard DA. Micromechanical model of composite materials subjected to ball indentation. Journal of Composite Materials 1993;27(3). [2] COSMOS/M User Guide v1.75, Structural Research and Analysis Corporation, 1995. [3] Beall MW, Shephard MS, Fish J, Belsky V. Multiscale modelling for crack propagation in composites. 1996 ASME International Mechanical Engineering Congress and Exposition, November 17±22, Atlanta, Georgia, USA, 1996. [4] Zhang J, Kikuchi N, Terada K. Global±local analysis of composites by image based ®xed grid method. 1996 ASME International Mechanical Engineering Congress and Exposition, November 17±22, Atlanta, Georgia, USA, 1996. [5] VaÂradi K, NeÂder Z, Friedrich K, FloÈck J. 3D anisotropic numerical and FE-contact analysis and experimental investigations of a steel ball indented into a ®bre-matrix structure with normal and parallel orientation. IVW-Bericht 97±30.