- Email: [email protected]
Experimental and numerical characterisation of in-plane deformation in two-phase materials
Computational Materials Science 21 (2001) 261±275
www.elsevier.com/locate/commatsci
Experimental and numerical characterisation of in-plane deformation in two-phase materials q E. Soppa a,*, P. Doumalin b, P. Binkele a, T. Wiesendanger a, M. Bornert b, S. Schmauder a a
Staatliche Materialprufungsanstalt, Universitat Stuttgart, Pfaenwaldring 32, 70569 Stuttgart, Germany b Laboratoire de M ecanique des Solides, Ecole Polytechnique, 91128 Palaiseau cedex, France
Abstract The aim of the present work consists in the comparison of in-plane strain ®elds with out-of-plane displacements in micro-areas of an Ag/Ni-composite after a macroscopic compressive deformation of 8.6%. The in-plane deformations in an Ag/Ni-composite have been analysed experimentally with a high resolution object grating technique and numerically using the ®nite element method. The out-of-plane displacements were measured with an atomic force microscope (AFM). The development of local strain ®elds in micro-areas at the surface of an Ag/Ni-composite was simulated numerically using the FE-method in plane strain condition. A real cut-out of the microstructure served as input for the calculation. The out-of-plane displacements determined by AFM measurements were used further to correct the in-plane values of strains evaluated by the object grating technique. The roughness on the surface of the sample was characterised by fractal dimensions and compared with the in-plane strains in the same micro-region. Ó 2001 Elsevier Science B.V. All rights reserved. Keywords: Microstructure; Computer simulation; Whole ®eld measurement of strain; AFM measurements; Fractal dimension
1. Introduction Stress and strain distribution in two-phase materials under loading is inhomogeneous. The magnitude of the inhomogeneity as well as local strain patterns depend on the microstructure (i.e. the elastic±plastic behaviour, volume fraction and phase arrangement of the components). The q Paper contributed to Ninth International Workshop on Computational Mechanics of Materials, J. Olschewski, S. Schmauder (Eds.), BAM, Berlin, Germany, October 4±5, 1999, Comp. Mater. Sci. 19 (2000). * Corresponding author. Tel.: +49-7-11685-2579; fax: +49-711685-2635. E-mail address: [email protected] (E. Soppa).
knowledge of the microstructure/deformation-relationship enables ``tailoring'' of materials with desired pro®les of properties. The critical concentration of strain, stress or hydrostatic stress can lead to damage initiation and failure of the component. By an optimised microstructure design such stress and strain concentrators can be avoided or reduced. 2. Material and microstructure The subject of the following experimental and numerical investigations is a model material Ag/ Ni(57%)-particulate composite with a coarse microstructure with an average Ni-phase size of
0927-0256/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 7 - 0 2 5 6 ( 0 1 ) 0 0 1 7 0 - 7
262
E. Soppa et al. / Computational Materials Science 21 (2001) 261±275
77 lm and an Ag-phase size of 60 lm (Table 1). This material was produced by a powder metallurgical route using hot isostatic pressing (HIP 900°=200 MPa/1 h) [1]. Ag and Ni as mutually insoluble elements exist in a composite as ``pure'' phases without any transient zone. This is the reason that this material's combination is a very convenient object for studying the deformation behaviour of two-phase composites.
Measurements of matricity [2] as a parameter describing phase arrangement in the material show an interpenetrating character of this microstructure with a slight overweight of Ag-phase as a matrix in relation to its volume fraction. Due to a big dierence in the grain sizes between Ag (43 lm) and Ni (8.6 lm) crystal plasticity eects in silver can appear in contrast to the Ni-phase, the deformation behaviour of which ought to be more
Table 1 Properties of the phases Matricity (MAg )
Matricity (MNi )
Vol% (Ag)
Vol% (Ni)
Ag-phase size (lm)
Ni-phase size (lm)
Ag-grain size (lm)
Ni-grain size (lm)
0:48 0:04
0:52 0:04
43:34 3:90
56:66 3:90
60:41 6:59
77:26 4:79
43
8.6
Fig. 1. Microstructure of an Ag/Ni(57 vol%)-composite after 8.6% vertical compression deformation.
E. Soppa et al. / Computational Materials Science 21 (2001) 261±275
isotropic. The results of the quantitative metallography are collected in Table 1. Fig. 1 shows the microstructure of the Ag/ Ni(57%)-composite with the black-framed area used for the numerical simulations as well as for the atomic force microscope (AFM) measurements. 3. Methods of investigations and results 3.1. Compression test and an object grating technique The Ag/Ni-material has been tested under uniaxial compression in the chamber of a scanning electron microscope using a speci®c in situ compression device that is able to generate loads of up to 10 000 N. A cylindrical sample (6 mm diameter and 8 mm length) with a Rastegaev-type geometry [3] and grease at both sides of the sample in order to reduce friction was used. Two symmetric ¯at planes were manufactured along the sample in order to facilitate the observation with the SEM.
263
One of these was mechanically polished and marked with several ®ducial microgrids [4], with 200 lines and columns and a 5 lm distance. The load and the displacement of the mobile head of the compression device were recorded during the test. Four juxtaposed high resolution (4096 3328 pixels) images were recorded at a magni®cation of 200 so as to cover one full microgrid (1 1 mm2 area), at several loading stages: before deformation and at three deformation steps. These images served to determine the local displacement ®eld at each grid intersection by means of a digital image correlation technique leading to subpixel accuracy. Details of the technique can be found in [5]. Note that the procedure aimed at correcting the linear image distortions and the magni®cation ¯uctuations described in this reference were used. The discrete displacement ®eld at each deformation step was then used to compute the local, per-phase average as well as overall in-plane strain components according to the relations given in [4]. Local strain maps corresponding to each deformation step could then be plotted. Only a small cut-out of these, containing 16 by 20 grid intersections is used
Fig. 2. Distribution of equivalent strains in Ag/Ni-composite for the last deformation step of 8.6% determined by object grating technique.
264
E. Soppa et al. / Computational Materials Science 21 (2001) 261±275
Table 2 Material properties Material Ag/Ni(56.7 vol%)-composite Ag Ni
E (GPa)
m( )
150.0 83.0 200.0
0.333 0.367 0.312
Applied equation (Voce law): r r0 rmax
r0 1
r0 (MPa) 89.9 43.5 138.0
rmax (MPa)
e0 ( )
375.1 245.8 568.0
0.21 0.245 0.244
exp e=e0 :
for the present study and the corresponding strain map is given in Fig. 2 (the von Mises equivalent strain is plotted) for the last deformation step, while the full strain maps can be found in [6]. The overall strains for the three deformation steps are given in Table 3. The uniaxial compression component in the direction of the ®rst principal axis of the in-plane strain tensor, the angle h between this axis and the vertical axis of the images, close to the axis of the sample, as well as the equivalent strain are given. It can be seen that the angle h is not close to zero as expected, especially for the ®rst two steps. This is due to the fact that the actual loading path was not a pure compression because of the early dissymmetric failure of the sample at one of its base planes containing the grease. At a later stage of the test the compression axis gets back to the sample axis, so that one can assume in a ®rst approximation that most plastic strain was induced by a pure compressive load and the FEM calculations described later are performed under this assumption. A more re®ned analysis would need to take this imperfect loading into account, but this has been let for further investigations. Equivalent strains, which are computed assuming that the strain tensor is axisymmetric, are close to the uniaxial principal strain, as expected. The small discrepancies in the ®rst two stages might be related to the imperfect loading mentioned above. For technical reasons, the SEM magni®cation ¯uctuations, which induce non-physical apparent deformations, could not be corrected in the last step, and this explains the larger discrepancy between the two measures of strain in the last step. That is why the accuracy on the average compression strain in the last step is probably not better than 1%, while it should be close to 0.3% for the two ®rst steps. The accuracy of the local strain measurement in the last step will
neither be better than 1% because of the uncontrolled magni®cation ¯uctuation, but its sensitivity is close to 0.3% for a 5 lm gauge length, thanks to the subpixel accuracy of the displacement measurement. It should also be noted that the relations given in [4] for the computation of the in-plane components of the deformation gradient Fxx, Fxy, Fyx and Fyy are exact, but the computation of the in-plane components of the Green±Lagrange deformation tensor Exx, Exy Eyx and Eyy relies on the assumption that the components Fzx and Fzy can be neglected, which is true only for small strains or when the transformation leaves the observation plane invariant. Here we have also used the real out-of-plane displacement at each grid intersection of the cut-out as given by the AFM (described hereafter) image of the cut-out, under the reasonable assumption that the sample was initially perfectly ¯at, to compute the Fzx and Fzy components of the gradient and subsequently the exact value of the in-plane components of the deformation tensor, according to, for instance: Exx 1=2 Fxx2 Fyx2 Fzx2 1. The relations given in [4] for the computation of Fxx, Fxy, Fyx and Fyy from the inplane displacement at each grid intersection for a given integration scheme have, therefore, been extended to deal with the Fzx and Fzy components and involve the out-of-plane displacements. In Fig. 3, the strain map obtained with the approximate computation of the deformation components Table 3 Measured overall strains Deformation step
e uniaxial (%)
e equivalent (%)
h (°)
1 2 3
)2.2 )3.0 )8.6
2.5 3.2 7.8
33 21 11
E. Soppa et al. / Computational Materials Science 21 (2001) 261±275
265
Fig. 3. Correction of the equivalent strain distribution using out-of-plane displacements measured by AFM: (a) without correction; (b) with correction.
is compared to the corrected one: it turns out that the corrections can be neglected, at least for the attained strain levels. This can also be checked in Fig. 4(b) which gives the relative error in percentage of the exact value: in most cases the discrepancy is below 3%. 3.2. FE simulation The commercial code LARSTRAN based on the ®nite element method was used for analysis of
(a)
an elastic±plastic behaviour of the Ag/Ni-composite. Six-noded elements TRIP6P [7] for elastic± plastic analysis were used for the calculation. Fig. 5 represents the FE-model consisting of the real part of the microstructure surrounded by material with average properties of the composite. In the middle part of the model the real microstructure was reproduced and introduced into the calculation as binary ®gure. The ®neness of the FE-mesh corresponds exactly to the experimental grid. 2D-calculation in plain strain was controlled by
(b)
Fig. 4. (a) Correction of the equivalent strain distribution. (b) The relative error in percentage of the exact value of equivalent strain, as de®ned in [5].
266
E. Soppa et al. / Computational Materials Science 21 (2001) 261±275
Fig. 5. FE-model and distribution of the equivalent strains in Ag/Ni-composite calculated using the FE-method.
homogeneous displacements on the upper edge of the model whereas the bottom edge was ®xed vertically and left and right edges remain free. The in¯uence of the edges of the real part of the FE-
model has been reduced by embedding into the layer of composite material. Mechanical data of the Ag- and Ni-phase as well as the Ag/Ni-composite are shown in Table 2.
E. Soppa et al. / Computational Materials Science 21 (2001) 261±275
267
Fig. 6. Comparison between calculated (FE) and experimental strain distributions in the Ag/Ni-composite.
3.2.1. Comparison between experiment and FEsimulation In Fig. 6 experimental and calculated results of the eective strain distributions are shown. The qualitative agreement is good, quantitatively there are some dierences especially in the Ag-phase. The deformation in some Ag-areas between Ni-particles is overestimated and the location of maxima is different. It could be an eect of the 2D-calculation, which does not take into account the real microstructure of material lying under the surface. Connected Ni-particles build a skeleton which makes the deformation in the softer Ag-regions dicult. Additionally crystal plasticity eects can play a signi®cant role especially in this type of microstructure with the grain size of 43 lm comparable with the Ag-phase size of 60 lm. Fine grains of 8.6 lm in the Ni-phase with an average size of about 77 lm ensure a quasi-isotropic behaviour of Ni. Limited localisation of stress and strain in this phase is mainly due to the presence of the second phase (Ag) and is not due to crystal plasticity eects in Ni. As mentioned earlier, the loading state in the experiment was not purely compressive, a shear
component probably due to the dissymmetric failure of the specimen was detected. The loading path in the FE-simulation against it is pure compression because a reliable estimation of the additional shear component in the experiment seemed to be very dicult. 3.3. Atomic force microscopy AFM allows scanning of electrically conductive and non-conductive specimens. In the contact AFM method a probe tip, which is mounted onto the end of a cantilever, scans across the surface of the specimen, coming into direct physical contact with the specimen. Generally a silicon nitride (Si3 N4 ) pyramidal probe tip is used. As the probe tip scans, topographic features cause de¯ections of the tip/cantilever. These de¯ections can be measured optically and are used to create an AFM image. The big advantage of an AFM is that it magni®es in three dimensions. In the xy-plane scan areas up to 100 lm 100 lm are possible. In zdirection height dierences up to max 10 lm can
268
E. Soppa et al. / Computational Materials Science 21 (2001) 261±275
be measured. With respect to the Ag/Ni-specimen AFM is a good method to get detailed information about the ``out-of-plane'' displacements. 3.3.1. Preparation of the Ag/Ni-specimen AFM measurements require a well-cleaned surface of the specimen. An EDX analysis of the surface showed some residues consisting of C, O and Si probably connected with the application of the object grid onto the surface. First attempts to clean the specimen with ethanol and acetone were not successful. For this reason the specimen was cleaned 3 15 min in tetrachloroethylene (toxic!) at a temperature T 50°C with ultrasonic. Then it was cleaned again 5 min in ethanol with ultrasonic. After these procedures a second EDX analysis showed that the most part of the residues could be removed. The still remaining residues may seem to be responsible for some small artefacts in the AFM images.
Fig. 8. 3D-plot of Fig. 7.
3.3.2. The measurements For the measurements an AFM of type TMX 2000 Explorer (TopoMetrix GmbH) was used in contact mode with the following optimised scan parameters: set point 1.0 nA, scanrate 122 lm/s (Figs. 7±15), 50 lm/s (Figs. 7 and 8), and for the PID control system P 0:7, I 0:4, D 0:0. In Fig. 7 we found a signi®cant place on the specimen Fig. 9. (Left to Fig. 7)
Fig. 7. Cut-out of the Ag/Ni-composite measured by AFM.
Fig. 10. 3D-plot of Fig. 9.
E. Soppa et al. / Computational Materials Science 21 (2001) 261±275
Fig. 11. (Down left to Fig. 9)
Fig. 13. (Part of Fig. 11)
Fig. 12. 3D-plot of Fig. 11.
Fig. 14. 3D-plot of Fig. 13.
(the centre of the grid), which guarantees the reproducibility of these measurements. To enhance the details of the surface topography all AFM images are shaded. A simulated light source at in®nite distance illuminates the topography. In the xy-plane the direction of the light source is 45°, in z-direction the position of the light source is 20° above the horizon (see also the scale at the left side of the images). Fig. 7 shows the center of the grid at (x; y) (45 lm, 40 lm). In all AFM-pictures the deformation of the gold grid is clearly visible. This deformation is caused by the global compression of the specimen in y-direction. Also all AFM-pictures show
269
many slip steps (look like parallel waves on the surface). Fig. 11 shows a interesting detail. There is a ``mountain'' with peak (x; y) (50 lm; 75 lm). In the neighbourhood of this mountain we measured strongly developed slip steps. Fig. 13 shows a magni®cation of this ``mountain'' and Fig. 15 shows a line analysis of the slip steps with typical heights of 450 nm. 3.4. Fractal analysis The fractal dimension Df can be used to characterise the self-similarity of an object. Regarding
270
E. Soppa et al. / Computational Materials Science 21 (2001) 261±275
Fig. 15. Line analysis of Fig. 13.
an absolute smooth surface the fractal dimension is 2 and may grow up to 3 at more rough surfaces. In this work fractal analysis was used to characterise the surface topography after deformation in order to compare it locally with the in-plane component of the strain. The fractal dimension of a surface can among other methods easily be determined by the pyramid algorithm [8,9]. A square window area is used for this procedure. The topography inside is repeatedly covered with triangles (Fig. 16). The total surface area A of the triangles in each step is calculated dependent on the area fraction s. The area fraction is simply the quotient of the step length l and the total extension of the evaluationwindow w both in the power of 2. The results are mapped in a double logarithmic plot of ln A s ln A0 s over ln s (Fig. 16) where A0 is A s 1 which is nearly the area of the evaluation window. Now, the fractal dimension Df can be determined from the slope m of the (linear) regression-line in this double logarithmic plot as Df 2 m. Concerning the lateral extension of the window used for evaluation it is advantageous using as edge length a number of points with a high ratio of divisors respective to its value (e.g. a square with
an edge length of 36 points which allows using 9 points for regression). As in this case the height of grid lines could be estimated from AFM measurements between 0 and 0.1 lm [10] which was high compared to the height dierences in the residual surface, their in¯uence on the calculation of the fractal dimension cannot be neglected. Thus, the calculation was performed only in areas between these lines. To do so, a square window-area is moved over the surface with a constant increment with regard not to touch the grid lines. The positions of the grid lines are marked manually (Fig. 17(a)). At each position the fractal dimension of the window-area is calculated. As the fractal dimension is in¯uenced by each point of the windowarea in the same way and, therefore, is not a property of one single point, the fractal dimension at one single point is thus calculated as the mean value of the fractal dimensions of the windows that contained this speci®c point. A weighting according to the distance of the point to the windowcentre was not yet performed in this study though this may increase the accuracy of the technique. In contrast to [8] it was found that a lateral window size of about 36 points is presumably sucient for local analysis but usage of 24 point-windows im-
E. Soppa et al. / Computational Materials Science 21 (2001) 261±275
271
s:fractional part of ground-surface A = A(s) : surface area
m: slope of regression line in log -log -plot Df = 2-m = 2 – (-0.006741548) = 2.006741548 Fig. 16. Estimation of the surface area using the triangle method.
proves recognising the details as can be seen in Figs. 17(b) and (c). The distribution of fractal dimensions in the cut-out of the Ag/Ni-microstructure from Fig. 1 (the smallest cut-out) estimated with the above described procedure is shown in Fig. 18.
4. Discussion Experimental and numerical investigations on the same cut-out of the Ag/Ni-particulate composite have been performed in order to correlate in-plane strain components and out-of-plane
272
E. Soppa et al. / Computational Materials Science 21 (2001) 261±275
Fig. 17. Estimation of the fractal dimensions in the cut-out of the Ag/Ni-composite using two window sizes of 24 and 36 pixels.
displacements. The comparison of the results in¯uenced the development and improvement of all these methods. For the numerical simulation it can be concluded that 2D-calculations provide results with a good qualitative agreement but they are not able to reproduce the experimental details quantitatively. For this purpose the development of a 3Dsimulation method based on a spatial microstructural reconstruction is necessary. Depending on the grain-size ®neness in the ductile phases crys-
tallographic anisotropy can play a signi®cant role in the development of stress and strain pattern. Thus the mechanical behaviour of the particular phase in an FE-model is assumed to be described by only one average (polycrystalline) stress±strain curve, neglecting the existence of the grains. The calculation with the experimentally determined displacements at the edges of the cut-out as boundary conditions did not lead to a satisfying result. At this stage it is not clear which factors are most signi®cant in this case. Probably, apart from
E. Soppa et al. / Computational Materials Science 21 (2001) 261±275
Fig. 18. Comparison of the results from object grating technique, fractal analysis and AFM.
273
274
E. Soppa et al. / Computational Materials Science 21 (2001) 261±275
already described factors, the ®neness of the experimental grid can be important in this case. AFM measurements of the Ag/Ni surface topography delivered the values of the out-of-plane displacements which have been used for correction of the in-plane equivalent strain. The corrections for the macroscopic strain up to 8.6% seem to be redundant and validate the estimation of the equivalent strain neglecting the out-of-plane component of the displacement in the region of moderate strains as acceptable. An important requirement by AFM measurements is a good quality of the specimen surface which is not easy to realise by parallel usage of the object grating technique based on covering the specimen surface by a polymer ®lm in the ®rst step. The residues of the polymer ®lm are probably responsible for the small artefacts in the AFM pictures. The main goal of the current work to ®nd the correlation between in-plane strains and out-ofplane displacements was realised by determining the local fractal dimension and comparison with the local eective strain in the surface plane of the specimen. Fig. 18 shows the results of object grating technique (equivalent strains), AFM (topography) and the distribution of the fractal dimensions in the same cut-out. The highest fractal dimensions coincide with the most rough pro®le found by AFM but not with the highest equivalent strain in experiment. The correlation between fractal dimension and equivalent strain in the Ag-
phase is therefore plotted in Fig. 19. The local minimum strain in Ag is found to be about 4% and the maximum about 17% at the macroscopic deformation of the composite of 8.6%. At the beginning of the plastic deformation when only one or a few glide systems are active the surface becomes rough because of the sharp glide steps. It explains the appearance of the highest value of the fractal dimension at small strains. When the deformation continues, more and more glide systems in dierent directions activate and intersect with each other. This may produce the eect of smooth surface and decreasing values of the fractal dimensions. 5. Conclusions · Corrections of the equivalent strain performed using out-of-plane displacements in an Ag/Nicomposite as measured by AFM at a macroscopic deformation of 8.6% are negligibly small. · The comparison between experimental and calculated distributions of equivalent strains shows a good qualitative agreement ± quantitative agreement is expected for the case of well-de®ned experimental boundary conditions as well as 3D-calculations instead of a 2D-simulation. In the case of coarse grain size comparable with the average phase-size additional crystal plasticity eects can modify the stress and strain patterns. · Easy glide at small macroscopic deformations can be recognised as increased fractal dimensions Df . · The activation of multiple glide during the deformation process seems to result in a subsequent decrease of the fractal dimensions. Acknowledgements
Fig. 19. Correlation between ``in-plane'' strains and fractal dimensions.
This work was performed in the frame of the PROCOPE-Project ``Experimental and numerical investigations of the deformation in Ag/Ni-particulate composites: correlation between in-plane-, out-of-plane-deformation and microstructure'' and Research Group ``Investigation of the defor-
E. Soppa et al. / Computational Materials Science 21 (2001) 261±275
mation behaviour of heterogeneous materials by direct combination of experiment and computation'', subproject DFG Schm 746/16-1, 2. The authors gratefully acknowledge the ®nancial support by the APAPE, DAAD and Deutsche Forschungsgemeinschaft (DFG).
[5]
[6]
References [1] E. Soppa, Arbeitsbericht zum Teilprojekt 2 (Prof. S. Schmauder) ``Ein¯u der Mikrostruktur auf das Verformungsverhaltens von Verbundwerkstoen'' im Rahmen der DFG-Forschergruppe Schm 746/16-2 ``Untersuchungen des Verformungsverhaltens heterogener Werkstoe mittels enger Verbindung von Experiment und Rechnung'', MPA Universit at Stuttgart, 1999. [2] M.H. Poech, D. Ruhr, Sonderbande der prakt, Metallographie 24 (1993) 385±391. [3] E. Soppa, Experimentelle Untersuchung des Verformungsverhaltens zweiphasiger Werkstoe, VDI Verlag, Reihe 5, Nr. 408, D usseldorf, 1995. [4] L. Allais, M. Bornert, T. Bretheau, D. Caldemaison, Experimental characterization of the local strain ®eld in a
[7]
[8] [9]
[10]
275
heterogeneous elastoplastic material, Acta Metall. 42 (11) (1994) 3865±3880. P. Doumalin, M. Bornert, D. Caldemaison, Microextensometry by image correlation applied to micromechanical studies using the scanning electron microscopy, in: International Conference on Advanced Technology in Experimental Mechanics, The Japan Society of Mechanical Engineering, 1999, pp. 81±86. P. Doumalin, PhD Thesis, Ecole Polytechnique, 91128 Palaiseau cedex, France, 2000. LARSTRAN User's Manual, Part V Element Library, LASSO Ingenieurgesellschaft, R. Dietz, U. Hindenlang, A. Kurz, Markomannenstraûe 11, D-70771 Leinfelden-Echterdingen, November 1993. S. Talibuddin, J.P. Runt, Reliability test of popular fractal techniques applied to small two-dimensional self ane data sets, J. Appl. Phys. 76 (9) (1994) 5070±5078. T. Wiesendanger, Charakterisierung des Rauhigkeitspro®ls von verformten Ag/Ni-Verbundwerkstoproben durch fraktale Dimensionen, Studienarbeit, MPA Universitat Stuttgart, 1999. P. Binkele, MPA Universitat Stuttgart, private information.
www.elsevier.com/locate/commatsci
Experimental and numerical characterisation of in-plane deformation in two-phase materials q E. Soppa a,*, P. Doumalin b, P. Binkele a, T. Wiesendanger a, M. Bornert b, S. Schmauder a a
Staatliche Materialprufungsanstalt, Universitat Stuttgart, Pfaenwaldring 32, 70569 Stuttgart, Germany b Laboratoire de M ecanique des Solides, Ecole Polytechnique, 91128 Palaiseau cedex, France
Abstract The aim of the present work consists in the comparison of in-plane strain ®elds with out-of-plane displacements in micro-areas of an Ag/Ni-composite after a macroscopic compressive deformation of 8.6%. The in-plane deformations in an Ag/Ni-composite have been analysed experimentally with a high resolution object grating technique and numerically using the ®nite element method. The out-of-plane displacements were measured with an atomic force microscope (AFM). The development of local strain ®elds in micro-areas at the surface of an Ag/Ni-composite was simulated numerically using the FE-method in plane strain condition. A real cut-out of the microstructure served as input for the calculation. The out-of-plane displacements determined by AFM measurements were used further to correct the in-plane values of strains evaluated by the object grating technique. The roughness on the surface of the sample was characterised by fractal dimensions and compared with the in-plane strains in the same micro-region. Ó 2001 Elsevier Science B.V. All rights reserved. Keywords: Microstructure; Computer simulation; Whole ®eld measurement of strain; AFM measurements; Fractal dimension
1. Introduction Stress and strain distribution in two-phase materials under loading is inhomogeneous. The magnitude of the inhomogeneity as well as local strain patterns depend on the microstructure (i.e. the elastic±plastic behaviour, volume fraction and phase arrangement of the components). The q Paper contributed to Ninth International Workshop on Computational Mechanics of Materials, J. Olschewski, S. Schmauder (Eds.), BAM, Berlin, Germany, October 4±5, 1999, Comp. Mater. Sci. 19 (2000). * Corresponding author. Tel.: +49-7-11685-2579; fax: +49-711685-2635. E-mail address: [email protected] (E. Soppa).
knowledge of the microstructure/deformation-relationship enables ``tailoring'' of materials with desired pro®les of properties. The critical concentration of strain, stress or hydrostatic stress can lead to damage initiation and failure of the component. By an optimised microstructure design such stress and strain concentrators can be avoided or reduced. 2. Material and microstructure The subject of the following experimental and numerical investigations is a model material Ag/ Ni(57%)-particulate composite with a coarse microstructure with an average Ni-phase size of
0927-0256/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 7 - 0 2 5 6 ( 0 1 ) 0 0 1 7 0 - 7
262
E. Soppa et al. / Computational Materials Science 21 (2001) 261±275
77 lm and an Ag-phase size of 60 lm (Table 1). This material was produced by a powder metallurgical route using hot isostatic pressing (HIP 900°=200 MPa/1 h) [1]. Ag and Ni as mutually insoluble elements exist in a composite as ``pure'' phases without any transient zone. This is the reason that this material's combination is a very convenient object for studying the deformation behaviour of two-phase composites.
Measurements of matricity [2] as a parameter describing phase arrangement in the material show an interpenetrating character of this microstructure with a slight overweight of Ag-phase as a matrix in relation to its volume fraction. Due to a big dierence in the grain sizes between Ag (43 lm) and Ni (8.6 lm) crystal plasticity eects in silver can appear in contrast to the Ni-phase, the deformation behaviour of which ought to be more
Table 1 Properties of the phases Matricity (MAg )
Matricity (MNi )
Vol% (Ag)
Vol% (Ni)
Ag-phase size (lm)
Ni-phase size (lm)
Ag-grain size (lm)
Ni-grain size (lm)
0:48 0:04
0:52 0:04
43:34 3:90
56:66 3:90
60:41 6:59
77:26 4:79
43
8.6
Fig. 1. Microstructure of an Ag/Ni(57 vol%)-composite after 8.6% vertical compression deformation.
E. Soppa et al. / Computational Materials Science 21 (2001) 261±275
isotropic. The results of the quantitative metallography are collected in Table 1. Fig. 1 shows the microstructure of the Ag/ Ni(57%)-composite with the black-framed area used for the numerical simulations as well as for the atomic force microscope (AFM) measurements. 3. Methods of investigations and results 3.1. Compression test and an object grating technique The Ag/Ni-material has been tested under uniaxial compression in the chamber of a scanning electron microscope using a speci®c in situ compression device that is able to generate loads of up to 10 000 N. A cylindrical sample (6 mm diameter and 8 mm length) with a Rastegaev-type geometry [3] and grease at both sides of the sample in order to reduce friction was used. Two symmetric ¯at planes were manufactured along the sample in order to facilitate the observation with the SEM.
263
One of these was mechanically polished and marked with several ®ducial microgrids [4], with 200 lines and columns and a 5 lm distance. The load and the displacement of the mobile head of the compression device were recorded during the test. Four juxtaposed high resolution (4096 3328 pixels) images were recorded at a magni®cation of 200 so as to cover one full microgrid (1 1 mm2 area), at several loading stages: before deformation and at three deformation steps. These images served to determine the local displacement ®eld at each grid intersection by means of a digital image correlation technique leading to subpixel accuracy. Details of the technique can be found in [5]. Note that the procedure aimed at correcting the linear image distortions and the magni®cation ¯uctuations described in this reference were used. The discrete displacement ®eld at each deformation step was then used to compute the local, per-phase average as well as overall in-plane strain components according to the relations given in [4]. Local strain maps corresponding to each deformation step could then be plotted. Only a small cut-out of these, containing 16 by 20 grid intersections is used
Fig. 2. Distribution of equivalent strains in Ag/Ni-composite for the last deformation step of 8.6% determined by object grating technique.
264
E. Soppa et al. / Computational Materials Science 21 (2001) 261±275
Table 2 Material properties Material Ag/Ni(56.7 vol%)-composite Ag Ni
E (GPa)
m( )
150.0 83.0 200.0
0.333 0.367 0.312
Applied equation (Voce law): r r0 rmax
r0 1
r0 (MPa) 89.9 43.5 138.0
rmax (MPa)
e0 ( )
375.1 245.8 568.0
0.21 0.245 0.244
exp e=e0 :
for the present study and the corresponding strain map is given in Fig. 2 (the von Mises equivalent strain is plotted) for the last deformation step, while the full strain maps can be found in [6]. The overall strains for the three deformation steps are given in Table 3. The uniaxial compression component in the direction of the ®rst principal axis of the in-plane strain tensor, the angle h between this axis and the vertical axis of the images, close to the axis of the sample, as well as the equivalent strain are given. It can be seen that the angle h is not close to zero as expected, especially for the ®rst two steps. This is due to the fact that the actual loading path was not a pure compression because of the early dissymmetric failure of the sample at one of its base planes containing the grease. At a later stage of the test the compression axis gets back to the sample axis, so that one can assume in a ®rst approximation that most plastic strain was induced by a pure compressive load and the FEM calculations described later are performed under this assumption. A more re®ned analysis would need to take this imperfect loading into account, but this has been let for further investigations. Equivalent strains, which are computed assuming that the strain tensor is axisymmetric, are close to the uniaxial principal strain, as expected. The small discrepancies in the ®rst two stages might be related to the imperfect loading mentioned above. For technical reasons, the SEM magni®cation ¯uctuations, which induce non-physical apparent deformations, could not be corrected in the last step, and this explains the larger discrepancy between the two measures of strain in the last step. That is why the accuracy on the average compression strain in the last step is probably not better than 1%, while it should be close to 0.3% for the two ®rst steps. The accuracy of the local strain measurement in the last step will
neither be better than 1% because of the uncontrolled magni®cation ¯uctuation, but its sensitivity is close to 0.3% for a 5 lm gauge length, thanks to the subpixel accuracy of the displacement measurement. It should also be noted that the relations given in [4] for the computation of the in-plane components of the deformation gradient Fxx, Fxy, Fyx and Fyy are exact, but the computation of the in-plane components of the Green±Lagrange deformation tensor Exx, Exy Eyx and Eyy relies on the assumption that the components Fzx and Fzy can be neglected, which is true only for small strains or when the transformation leaves the observation plane invariant. Here we have also used the real out-of-plane displacement at each grid intersection of the cut-out as given by the AFM (described hereafter) image of the cut-out, under the reasonable assumption that the sample was initially perfectly ¯at, to compute the Fzx and Fzy components of the gradient and subsequently the exact value of the in-plane components of the deformation tensor, according to, for instance: Exx 1=2 Fxx2 Fyx2 Fzx2 1. The relations given in [4] for the computation of Fxx, Fxy, Fyx and Fyy from the inplane displacement at each grid intersection for a given integration scheme have, therefore, been extended to deal with the Fzx and Fzy components and involve the out-of-plane displacements. In Fig. 3, the strain map obtained with the approximate computation of the deformation components Table 3 Measured overall strains Deformation step
e uniaxial (%)
e equivalent (%)
h (°)
1 2 3
)2.2 )3.0 )8.6
2.5 3.2 7.8
33 21 11
E. Soppa et al. / Computational Materials Science 21 (2001) 261±275
265
Fig. 3. Correction of the equivalent strain distribution using out-of-plane displacements measured by AFM: (a) without correction; (b) with correction.
is compared to the corrected one: it turns out that the corrections can be neglected, at least for the attained strain levels. This can also be checked in Fig. 4(b) which gives the relative error in percentage of the exact value: in most cases the discrepancy is below 3%. 3.2. FE simulation The commercial code LARSTRAN based on the ®nite element method was used for analysis of
(a)
an elastic±plastic behaviour of the Ag/Ni-composite. Six-noded elements TRIP6P [7] for elastic± plastic analysis were used for the calculation. Fig. 5 represents the FE-model consisting of the real part of the microstructure surrounded by material with average properties of the composite. In the middle part of the model the real microstructure was reproduced and introduced into the calculation as binary ®gure. The ®neness of the FE-mesh corresponds exactly to the experimental grid. 2D-calculation in plain strain was controlled by
(b)
Fig. 4. (a) Correction of the equivalent strain distribution. (b) The relative error in percentage of the exact value of equivalent strain, as de®ned in [5].
266
E. Soppa et al. / Computational Materials Science 21 (2001) 261±275
Fig. 5. FE-model and distribution of the equivalent strains in Ag/Ni-composite calculated using the FE-method.
homogeneous displacements on the upper edge of the model whereas the bottom edge was ®xed vertically and left and right edges remain free. The in¯uence of the edges of the real part of the FE-
model has been reduced by embedding into the layer of composite material. Mechanical data of the Ag- and Ni-phase as well as the Ag/Ni-composite are shown in Table 2.
E. Soppa et al. / Computational Materials Science 21 (2001) 261±275
267
Fig. 6. Comparison between calculated (FE) and experimental strain distributions in the Ag/Ni-composite.
3.2.1. Comparison between experiment and FEsimulation In Fig. 6 experimental and calculated results of the eective strain distributions are shown. The qualitative agreement is good, quantitatively there are some dierences especially in the Ag-phase. The deformation in some Ag-areas between Ni-particles is overestimated and the location of maxima is different. It could be an eect of the 2D-calculation, which does not take into account the real microstructure of material lying under the surface. Connected Ni-particles build a skeleton which makes the deformation in the softer Ag-regions dicult. Additionally crystal plasticity eects can play a signi®cant role especially in this type of microstructure with the grain size of 43 lm comparable with the Ag-phase size of 60 lm. Fine grains of 8.6 lm in the Ni-phase with an average size of about 77 lm ensure a quasi-isotropic behaviour of Ni. Limited localisation of stress and strain in this phase is mainly due to the presence of the second phase (Ag) and is not due to crystal plasticity eects in Ni. As mentioned earlier, the loading state in the experiment was not purely compressive, a shear
component probably due to the dissymmetric failure of the specimen was detected. The loading path in the FE-simulation against it is pure compression because a reliable estimation of the additional shear component in the experiment seemed to be very dicult. 3.3. Atomic force microscopy AFM allows scanning of electrically conductive and non-conductive specimens. In the contact AFM method a probe tip, which is mounted onto the end of a cantilever, scans across the surface of the specimen, coming into direct physical contact with the specimen. Generally a silicon nitride (Si3 N4 ) pyramidal probe tip is used. As the probe tip scans, topographic features cause de¯ections of the tip/cantilever. These de¯ections can be measured optically and are used to create an AFM image. The big advantage of an AFM is that it magni®es in three dimensions. In the xy-plane scan areas up to 100 lm 100 lm are possible. In zdirection height dierences up to max 10 lm can
268
E. Soppa et al. / Computational Materials Science 21 (2001) 261±275
be measured. With respect to the Ag/Ni-specimen AFM is a good method to get detailed information about the ``out-of-plane'' displacements. 3.3.1. Preparation of the Ag/Ni-specimen AFM measurements require a well-cleaned surface of the specimen. An EDX analysis of the surface showed some residues consisting of C, O and Si probably connected with the application of the object grid onto the surface. First attempts to clean the specimen with ethanol and acetone were not successful. For this reason the specimen was cleaned 3 15 min in tetrachloroethylene (toxic!) at a temperature T 50°C with ultrasonic. Then it was cleaned again 5 min in ethanol with ultrasonic. After these procedures a second EDX analysis showed that the most part of the residues could be removed. The still remaining residues may seem to be responsible for some small artefacts in the AFM images.
Fig. 8. 3D-plot of Fig. 7.
3.3.2. The measurements For the measurements an AFM of type TMX 2000 Explorer (TopoMetrix GmbH) was used in contact mode with the following optimised scan parameters: set point 1.0 nA, scanrate 122 lm/s (Figs. 7±15), 50 lm/s (Figs. 7 and 8), and for the PID control system P 0:7, I 0:4, D 0:0. In Fig. 7 we found a signi®cant place on the specimen Fig. 9. (Left to Fig. 7)
Fig. 7. Cut-out of the Ag/Ni-composite measured by AFM.
Fig. 10. 3D-plot of Fig. 9.
E. Soppa et al. / Computational Materials Science 21 (2001) 261±275
Fig. 11. (Down left to Fig. 9)
Fig. 13. (Part of Fig. 11)
Fig. 12. 3D-plot of Fig. 11.
Fig. 14. 3D-plot of Fig. 13.
(the centre of the grid), which guarantees the reproducibility of these measurements. To enhance the details of the surface topography all AFM images are shaded. A simulated light source at in®nite distance illuminates the topography. In the xy-plane the direction of the light source is 45°, in z-direction the position of the light source is 20° above the horizon (see also the scale at the left side of the images). Fig. 7 shows the center of the grid at (x; y) (45 lm, 40 lm). In all AFM-pictures the deformation of the gold grid is clearly visible. This deformation is caused by the global compression of the specimen in y-direction. Also all AFM-pictures show
269
many slip steps (look like parallel waves on the surface). Fig. 11 shows a interesting detail. There is a ``mountain'' with peak (x; y) (50 lm; 75 lm). In the neighbourhood of this mountain we measured strongly developed slip steps. Fig. 13 shows a magni®cation of this ``mountain'' and Fig. 15 shows a line analysis of the slip steps with typical heights of 450 nm. 3.4. Fractal analysis The fractal dimension Df can be used to characterise the self-similarity of an object. Regarding
270
E. Soppa et al. / Computational Materials Science 21 (2001) 261±275
Fig. 15. Line analysis of Fig. 13.
an absolute smooth surface the fractal dimension is 2 and may grow up to 3 at more rough surfaces. In this work fractal analysis was used to characterise the surface topography after deformation in order to compare it locally with the in-plane component of the strain. The fractal dimension of a surface can among other methods easily be determined by the pyramid algorithm [8,9]. A square window area is used for this procedure. The topography inside is repeatedly covered with triangles (Fig. 16). The total surface area A of the triangles in each step is calculated dependent on the area fraction s. The area fraction is simply the quotient of the step length l and the total extension of the evaluationwindow w both in the power of 2. The results are mapped in a double logarithmic plot of ln A s ln A0 s over ln s (Fig. 16) where A0 is A s 1 which is nearly the area of the evaluation window. Now, the fractal dimension Df can be determined from the slope m of the (linear) regression-line in this double logarithmic plot as Df 2 m. Concerning the lateral extension of the window used for evaluation it is advantageous using as edge length a number of points with a high ratio of divisors respective to its value (e.g. a square with
an edge length of 36 points which allows using 9 points for regression). As in this case the height of grid lines could be estimated from AFM measurements between 0 and 0.1 lm [10] which was high compared to the height dierences in the residual surface, their in¯uence on the calculation of the fractal dimension cannot be neglected. Thus, the calculation was performed only in areas between these lines. To do so, a square window-area is moved over the surface with a constant increment with regard not to touch the grid lines. The positions of the grid lines are marked manually (Fig. 17(a)). At each position the fractal dimension of the window-area is calculated. As the fractal dimension is in¯uenced by each point of the windowarea in the same way and, therefore, is not a property of one single point, the fractal dimension at one single point is thus calculated as the mean value of the fractal dimensions of the windows that contained this speci®c point. A weighting according to the distance of the point to the windowcentre was not yet performed in this study though this may increase the accuracy of the technique. In contrast to [8] it was found that a lateral window size of about 36 points is presumably sucient for local analysis but usage of 24 point-windows im-
E. Soppa et al. / Computational Materials Science 21 (2001) 261±275
271
s:fractional part of ground-surface A = A(s) : surface area
m: slope of regression line in log -log -plot Df = 2-m = 2 – (-0.006741548) = 2.006741548 Fig. 16. Estimation of the surface area using the triangle method.
proves recognising the details as can be seen in Figs. 17(b) and (c). The distribution of fractal dimensions in the cut-out of the Ag/Ni-microstructure from Fig. 1 (the smallest cut-out) estimated with the above described procedure is shown in Fig. 18.
4. Discussion Experimental and numerical investigations on the same cut-out of the Ag/Ni-particulate composite have been performed in order to correlate in-plane strain components and out-of-plane
272
E. Soppa et al. / Computational Materials Science 21 (2001) 261±275
Fig. 17. Estimation of the fractal dimensions in the cut-out of the Ag/Ni-composite using two window sizes of 24 and 36 pixels.
displacements. The comparison of the results in¯uenced the development and improvement of all these methods. For the numerical simulation it can be concluded that 2D-calculations provide results with a good qualitative agreement but they are not able to reproduce the experimental details quantitatively. For this purpose the development of a 3Dsimulation method based on a spatial microstructural reconstruction is necessary. Depending on the grain-size ®neness in the ductile phases crys-
tallographic anisotropy can play a signi®cant role in the development of stress and strain pattern. Thus the mechanical behaviour of the particular phase in an FE-model is assumed to be described by only one average (polycrystalline) stress±strain curve, neglecting the existence of the grains. The calculation with the experimentally determined displacements at the edges of the cut-out as boundary conditions did not lead to a satisfying result. At this stage it is not clear which factors are most signi®cant in this case. Probably, apart from
E. Soppa et al. / Computational Materials Science 21 (2001) 261±275
Fig. 18. Comparison of the results from object grating technique, fractal analysis and AFM.
273
274
E. Soppa et al. / Computational Materials Science 21 (2001) 261±275
already described factors, the ®neness of the experimental grid can be important in this case. AFM measurements of the Ag/Ni surface topography delivered the values of the out-of-plane displacements which have been used for correction of the in-plane equivalent strain. The corrections for the macroscopic strain up to 8.6% seem to be redundant and validate the estimation of the equivalent strain neglecting the out-of-plane component of the displacement in the region of moderate strains as acceptable. An important requirement by AFM measurements is a good quality of the specimen surface which is not easy to realise by parallel usage of the object grating technique based on covering the specimen surface by a polymer ®lm in the ®rst step. The residues of the polymer ®lm are probably responsible for the small artefacts in the AFM pictures. The main goal of the current work to ®nd the correlation between in-plane strains and out-ofplane displacements was realised by determining the local fractal dimension and comparison with the local eective strain in the surface plane of the specimen. Fig. 18 shows the results of object grating technique (equivalent strains), AFM (topography) and the distribution of the fractal dimensions in the same cut-out. The highest fractal dimensions coincide with the most rough pro®le found by AFM but not with the highest equivalent strain in experiment. The correlation between fractal dimension and equivalent strain in the Ag-
phase is therefore plotted in Fig. 19. The local minimum strain in Ag is found to be about 4% and the maximum about 17% at the macroscopic deformation of the composite of 8.6%. At the beginning of the plastic deformation when only one or a few glide systems are active the surface becomes rough because of the sharp glide steps. It explains the appearance of the highest value of the fractal dimension at small strains. When the deformation continues, more and more glide systems in dierent directions activate and intersect with each other. This may produce the eect of smooth surface and decreasing values of the fractal dimensions. 5. Conclusions · Corrections of the equivalent strain performed using out-of-plane displacements in an Ag/Nicomposite as measured by AFM at a macroscopic deformation of 8.6% are negligibly small. · The comparison between experimental and calculated distributions of equivalent strains shows a good qualitative agreement ± quantitative agreement is expected for the case of well-de®ned experimental boundary conditions as well as 3D-calculations instead of a 2D-simulation. In the case of coarse grain size comparable with the average phase-size additional crystal plasticity eects can modify the stress and strain patterns. · Easy glide at small macroscopic deformations can be recognised as increased fractal dimensions Df . · The activation of multiple glide during the deformation process seems to result in a subsequent decrease of the fractal dimensions. Acknowledgements
Fig. 19. Correlation between ``in-plane'' strains and fractal dimensions.
This work was performed in the frame of the PROCOPE-Project ``Experimental and numerical investigations of the deformation in Ag/Ni-particulate composites: correlation between in-plane-, out-of-plane-deformation and microstructure'' and Research Group ``Investigation of the defor-
E. Soppa et al. / Computational Materials Science 21 (2001) 261±275
mation behaviour of heterogeneous materials by direct combination of experiment and computation'', subproject DFG Schm 746/16-1, 2. The authors gratefully acknowledge the ®nancial support by the APAPE, DAAD and Deutsche Forschungsgemeinschaft (DFG).
[5]
[6]
References [1] E. Soppa, Arbeitsbericht zum Teilprojekt 2 (Prof. S. Schmauder) ``Ein¯u der Mikrostruktur auf das Verformungsverhaltens von Verbundwerkstoen'' im Rahmen der DFG-Forschergruppe Schm 746/16-2 ``Untersuchungen des Verformungsverhaltens heterogener Werkstoe mittels enger Verbindung von Experiment und Rechnung'', MPA Universit at Stuttgart, 1999. [2] M.H. Poech, D. Ruhr, Sonderbande der prakt, Metallographie 24 (1993) 385±391. [3] E. Soppa, Experimentelle Untersuchung des Verformungsverhaltens zweiphasiger Werkstoe, VDI Verlag, Reihe 5, Nr. 408, D usseldorf, 1995. [4] L. Allais, M. Bornert, T. Bretheau, D. Caldemaison, Experimental characterization of the local strain ®eld in a
[7]
[8] [9]
[10]
275
heterogeneous elastoplastic material, Acta Metall. 42 (11) (1994) 3865±3880. P. Doumalin, M. Bornert, D. Caldemaison, Microextensometry by image correlation applied to micromechanical studies using the scanning electron microscopy, in: International Conference on Advanced Technology in Experimental Mechanics, The Japan Society of Mechanical Engineering, 1999, pp. 81±86. P. Doumalin, PhD Thesis, Ecole Polytechnique, 91128 Palaiseau cedex, France, 2000. LARSTRAN User's Manual, Part V Element Library, LASSO Ingenieurgesellschaft, R. Dietz, U. Hindenlang, A. Kurz, Markomannenstraûe 11, D-70771 Leinfelden-Echterdingen, November 1993. S. Talibuddin, J.P. Runt, Reliability test of popular fractal techniques applied to small two-dimensional self ane data sets, J. Appl. Phys. 76 (9) (1994) 5070±5078. T. Wiesendanger, Charakterisierung des Rauhigkeitspro®ls von verformten Ag/Ni-Verbundwerkstoproben durch fraktale Dimensionen, Studienarbeit, MPA Universitat Stuttgart, 1999. P. Binkele, MPA Universitat Stuttgart, private information.