Nanoindentation of Fibrous Composite Microstructures: Experimentation and Finite Element Investigation Mark Hardiman Materials and Surface Science Institute (MSSI), Department of Mechanical and Aeronautical Engineering, University of Limerick, Limerick, Ireland. Phone: +353 61 234334 Fax: +353 61 202944 Abstract Nanoindentation results on a carbon-fibre composite microstructure have been investigated using the finite element method. Nanoindentation tests were carried out on both the stiff carbon fibre and compliant epoxy matrix constituents in an attempt to determine their in-situ mechanical properties. The mean modulus value for the matrix constituent was 2.5 times greater than that of the bulk material. The matrix results were investigated by carrying out a finite element investigation of the fibre constraint on the nanoindentation response in the matrix region. The results show that using a 2D model with an encastre boundary to represent the fibre constraint leads to an overestimation of the constraint on the nanoindentation response. Moreover, the full 3D models containing actual cylindrical fibre sections provide useful insight into the mechanics of a nanoindentation experiment carried out in matrix regions in the vicinity of fibres. All finite element results show a gradient in modulus as indentations are carried out closer to the fibrematrix interface which suggests an inherent difficulty when attempting to determine the true interphase properties experimentally using the nanoindentation technique. The results show that the large values for experimental matrix modulus can be explained by this phenomenon. The mean indentation modulus value for the fibres was found to be much lower than the literature value for longitudinal modulus. This was investigated using a 2D axisymmetric finite element model and could be explained by a combination of the fibres anisotropic properties and the added compliance of the surrounding matrix.
1.0 Introduction In recent years, a large amount of research in the area of composite materials has been focused on analysing the response of the material s constituent phases and their interfaces under various loading conditions. Analysis of this type involves carefully examining the microstructure of the composite material and relating the observed behaviour to macroscopic failure mechanisms. To facilitate this type of analysis, micromechanical models have been developed which model each constituent as a discrete material and allow for the stress distribution, damage and eventual failure of the composite to be simulated at the microscale. There are many examples of this modelling technique being applied to fibrous composites [1,2]. These models provide valuable insight into the interaction between the fibres and matrix and also the effect of the constituent material properties on the overall fibrous composite response. However, these models assume that the properties of each constituent remain the same as the properties of the material in its bulk form. This may not be the case after the composite manufacturing and curing processes. Nanoindentation is a technique which is commonly used to determine material properties such as hardness and elastic modulus at the micro and nanoscale [3]. Previous work has shown that the load-displacement data from nanoindentation tests carried out into the resin region of a fibrous composite can potentially be influenced by the reinforcing effect of the surrounding fibres [4]. This leads to results which are convoluted due to the mechanical constraint effect of the fibres and also any potential changes in mechanical properties. Efforts have also been made to characterise the interface between the fibres and the matrix by carrying out nanoindentation tests close to the interface. These experiments show a distinct interphase region where the indentation modulus gradually changes from the resin properties to the fibre properties [5,6]. In this study, nanoindentation experiments were carried out in an attempt to characterise the in-situ mechanical properties of a carbon-fibre composite constituents for use in in-house micromechanical models. The material being investigated in this study is HTA/6376; a high strength carbon fibre reinforced plastic (CFRP) often used in the aerospace industry. The nanoindentation process was then modelled using both 2D and 3D finite element models in an attempt to gain a greater insight into the mechanics of the nanoindentation test on fibrous 1
composites and explain any apparent discrepancies between the indentation modulus and the modulus of the constituent material in its bulk form. 2.0 Nanoindentation Theory Nanoindentation tests involve pushing a diamond tipped indenter head into a bulk material under either load or displacement control. The displacement is monitored as a function of the load throughout the load-unload cycle. A typical load-displacement curve is shown in Figure 1. Figure 1: Typical Nanoindentation load-displacement curve. When determining material properties such as hardness and elastic modulus a three-sided pyramid indenter known as Berkovich indenter is commonly used. The method of interpreting nanoindentation data has been developed over a number of years with the Oliver and Pharr [7] method being the most extensively used method of determining modulus and hardness. Hardness (H) is defined as the contact pressure under the indenter: 2
H P A c (1) where P is the load and A c is the projected contact area calculated at a depth of indentation, h. The initial slope of the unloading curve can be related to the elastic modulus of the material using Eqn 2: S dp dh 2E r A c (2) where S is the initial slope of the unloading curve or contact stiffness, P is the applied load and E r is the reduced modulus. As the measured displacement in a nanoindentation experiment is a combination of the displacement of the indenter tip as well as the specimen, the specimen modulus (E s ) can be related to the reduced modulus (E r ) using Eqn 3 provided the indenter modulus (E i ) is known and the Poisson s ratios of the specimen and indenter (ν s and ν i respectively) are known or can be estimated: r s 2 s 1 1 1 E E E i 2 i (3) As for the simulations in this paper it is assumed that the indenter behaves rigidly, Eqn 3 can be reduced down to Eqn 4 by assuming E i is infinite: 1 1 E r E s 2 s (4) 3
3.0 Experimental Results Nanoindentation experiments were carried out on the constituent materials of the HTA/6376 composite in an attempt to determine their in-situ mechanical properties. These results could then be compared with the values of the properties of the constituents in their bulk form which are shown in Table 1. Table 1: Mechanical properties of the HTA/6376 composite constituent materials in their bulk form [8, 9]. Fibre (HTA) Matrix (6376) E (GPa) 11 238 3.63 E /E (GPa) 22 33 28 ν 12 0.28 0.34 v 23 0.33 v 31 0.02 G (GPa) 12 24 G (GPa) 23 7.2 G (GPa) 31 24 σ y (MPa) - 262 A series of indentations were carried out into the centre of the HTA fibres under load control up to a maximum load of 15mN. In total, five indentations were carried out. The mean indentation modulus value from these five indentations was calculated using the Oliver and Pharr method was 56.557 GPa with a standard deviation of 0.752 GPa. The selected fibres are shown targeted using an optical microscope in Figure 2. 4
Figure 2: HTA fibre centres targeted for indentation A series of indentations were also carried out in the large resin pockets in the composite in an attempt to determine the in-situ mechanical properties of the matrix constituent. These resin pockets were located in the interply regions of the composite. The indentations were carried out under load-control up to a maximum load of 5mN. The load was then held constant at this maximum load for 5 seconds in an attempt to reduce the effect of viscoelasticity on the indentation modulus results, before being unloaded. A total of five indentations were carried out into these resin pockets leading to an average indentation modulus value of 9.346 GPa with a standard deviation of 0.322. Two of the five indentations are highlighted and shown on a micrograph in Figure 3. 5
Figure 3: Optical Microscope image of indentations carried out in large resin pocket 4.0 Finite Element Analysis In order to investigate the relevance of the experimental results, the elastic-plastic nanoindentation process was modelled using a number of 2D and 3D models. The commercial finite-element software ABAQUS v6.10 [10] was used to create the models and carry out the analyses. A large strain solution was used and both materials were defined using the mechanical properties shown in Table 1. The HTA fibre was assumed to exhibit transversely isotropic linear elastic behaviour while the 6376 resin was assumed to exhibit elastic-perfectly plastic material behaviour. The yield strength of the matrix material was set at 262 MPa which was determined from previously carried out compression tests [11]. 4.1 2D Axisymmetric model The first model-type assumed that the problem could be simplified by using a 2D axisymmetric representation. Here the specimen and the indenter were treated as axisymmetric 6
bodies of revolution by using CAX4 elements in ABAQUS to model the bulk material and assuming the indenter could be treated as a rigid cone with a half-angle of 70.3. This simplified representation gives the same projected area to depth ratio as the three-sided Berkovich pyramid used in the nanoindentation experiments. For all the models the contact between the indenter and the specimen was assumed to be frictionless. A fine mesh was used in the indentation region, as large local deformation takes place in this region, and the mesh became gradually coarser further afield from this region. Roller boundary conditions were applied to the left and bottom region of the model. The dimensions in the x and y directions were initially set to a value 300 times the maximum indentation depth in order to ensure far-field effects were negligible. In order to investigate whether the surrounding fibres could have had a constraining effect on the experimental indentations of the matrix, variations of the 2D model were used. The righthand end of the model was held encastre to represent a rigid constraint similar to that of neighbouring fibres. The value of r is representative of the distance from the initial contact point of indentation to the outside fibre region, divided by the depth of indentation. In this study the indentation depth was held constant at 1μm and the distance to the fibre region varied in order to determine the effect the value of r has on the indentation response. Figure 4: Magnified view of indentation region and 2D axisymmetric model along with boundary conditions 7
Variations of the 2D model were also used to investigate the indentation results for the fibre constituent. The first model, shown on the left of Figure 5, applied the anisotropic fibre properties to the indented material in order to determine if the experimental indentation modulus values could be explained by these properties. The 2 nd model shown on the right of Figure 5 has both fibre and matrix sections which are defined using the properties in Table 1. This model will determine whether the surrounding matrix material had an effect on the experimental indentation modulus result for the fibre constituent. Here the average fibre diameter and indentation depth from the experiments were used for the simulation. Figure 5: Finite element models of the indentation experiment applied to the fibre constituent material 4.2 3D Models Three different 3D finite element models were used to investigate the fibre constraint effect on the nanoindentation response of the resin. The models make use of the six-fold symmetry of the Berkovich indenter by making it necessary to only model one sixth of the geometry. Elements of type C3D8 were used in ABAQUS to model the specimen geometry while the indenter has been represented by a rigid plane with an angular offset of 24.7 from the surface as shown on the right of Figure 6a. This perfectly represents the Berkovich indenter geometry when the six-fold symmetry is taken into account. 8
Figure 6: 3D finite element models used to represent a nanoindentation in the vicinity of fibres, (a) fibres represented as an encastre boundary, (b) fibres represented by a circular pattern of closelypacked fibres and (c) fibres represented by a less dense packing arrangement. The first model shown in Figure 6a represents the constraint of the fibres as an encastre boundary similar to the 2D model described above. The mesh and boundary conditions are almost identical to that of the 2D model, except the mesh has been revolved through an angle of 60. This model can determine the eligibility of representing the Berkovich indenter as a revolved cone when determining the effect of the rigid boundary on the nanoindentation response. The second 3D model shown in Figure 6b contains cylindrical sections with the HTA fibre s material properties assigned to the sections. These sections were added to the bulk 9
specimen in order to realistically represent circular fibres surrounding the indentation. The fibre diameters and spacing were taken from a previous statistical analysis of the HTA/6376 composite microstructure [10]. The variable r has been defined as the distance from the initial point of indentation to the closest point on the edge of the fibre. The final 3D model shown in Figure 6c represents a situation where the fibre packing arrangement around the indentation region is less dense. It is useful to compare results for this model with the results for the closely packed fibres model as often in fibrous composites like HTA/6376 the large resin pockets are located in the interply regions as shown in Figure 7. This often leads to matrix rich regions which are longer in one direction than the other and as such when indentation is carried out in these regions the constraint effect will only be due to a few fibres initially. Due to the nature of the six-fold symmetry adding one fibre to the model is equivalent to a ring of three fibres surrounding the indentation region. Figure 7: Optical microscope image of the HTA/6376 microstructure showing the matrix-rich regions between the laminae 10
5 Results and Discussion In order to quantitatively investigate the effect the surrounding fibres had on nanoindentation experiments in the matrix region of the fibrous composite, the Oliver and Pharr analysis was carried out on all of the simulated nanoindentation tests. This determined the modulus for each value of r for all the models. Since the maximum indentation depth was held constant at 1μm, the r value gave an indication as to how constrained the indentations were by the rigid boundary or fibre sections. In all the models it was found that as the value of r decreased the maximum load required to achieve the maximum indentation depth increased as shown in Figure 8 for four different values of r in the 3D model with encastre boundary. This also led to a steeper slope of the initial part of the unloading curve which indicates an increase in contact stiffness with decreasing distance to the fibre region. Figure 8: Nanoindentation simulation load-displacement curves for four different values of r The values of Young s modulus obtained for each value of r were compared with the unconstrained value of Young s modulus obtained from the same model without the encastre 11
boundary or fibre sections. This gave an indication as to how much the constraints were affecting the value of indentation modulus calculated using the nanoindentation simulation s loaddisplacement data. The ratio of Young s Modulus to unconstrained Young s modulus is plotted against values of r in Figure 9 for all four model-types. Figure 9: Ratio of Young's Modulus to unconstrained Young's modulus plotted against r for all four 3D models Both the 2D and 3D models with the fibre constraint represented by an encastre boundary produce similar results. This shows that the idealisation of a Berkovich indenter as an axisymmetric cone for a simplified 2D analysis is valid. The 3D model of the indentation site surrounded by densely packed fibres produced results with lower values of E/E unconstrained than the 3D model with the encastre boundary. This shows that assuming the fibre constraint can be represented by a rigid encastre boundary overconstrains the indentation compared to the real-life geometry. This is most likely due to the circular shaped fibres imposing a lesser mechanical constraint on the indentation and the non-rigid behaviour of the fibre sections. The final 3D model with just three fibres around the indentation site region showed much lower values of E/E unconstrained 12
than both other 3D models. This shows that the density of the fibre distribution is also a significant factor which should also be considered when selecting a suitable indentation site. Nanoindentation experiments can be considered to be relatively unaffected by the surrounding fibres when the value of indentation modulus remains within 10% of the unconstrained indentation modulus (i.e. E/E unconstrained 1.1). According to Figure 9, for an indentation into a matrix pocket to pass this criterion, the value of r must be greater than 13.2 for the 3D indentation surrounded by fibres. If we consider the micrograph showing the experimental indentations into the resin pockets, we can see that according to the result of this study the resin pocket was not sufficiently large for unconstrained indentation to take place. We can see in Figure 10 that there are numerous fibres in the area around the indentation site that convoluted the loaddisplacement response and most likely lead to the high values of indentation modulus for the matrix material. Figure 10: Micrograph showing the distance to the surrounding fibres is much less than the minimum required for unconstrained indentation Figure 9 also shows that for all of the models a gradient in modulus was apparent as the indentations approached the fibre matrix interface. This implies that it is difficult to quantitatively analyse nanoindentation experiments carried out in the matrix regions close to the fibres as any apparent changes in mechanical properties may have been influenced by the mechanical constraint 13
making it difficult to definitively characterise an interphase region with varying mechanical properties. A closer look at the stress distribution in the 3D models with fibre sections provides an interesting insight into the mechanics of the fibre constraint in these indentations. The stress distribution for three different indenter depths is shown in Figure 11 for the 3D model with closely-packed fibres where r = 10. The stress limits for the contours have been limited to a range of 0.1σ y σ σ y where σ y is the yield stress of the 6376 resin. This prevents the large stresses directly under the indenter tip from skewing the contour plot and gives the stresses a sense of scale. It can be seen that as the indentations get deeper, significant stresses develop in the fibre section even when carrying out the nanoindentation into what would initially appear to be a large resin pocket within the composite. Figure 11: Development of stresses in the fibre sections throughout the loading step when r = 10 The load displacement curve for this nanoindentation simulation is shown in Figure 12 along with the load-displacement curve for the unconstrained case of nanoindentation of the bulk 6376 resin. Initially for low values of displacement the curves fall on one another as the fibre constraint is not prevalent at these depths. As the indentations become deeper, stress will be transferred to the surrounding fibres as shown in Figure 11 and leads to a divergence between the load-displacement curves as can clearly be seen in Figure 12. 14
Figure 12: Load displacement curves for 3D model with fibres for the unconstrained case and for r = 10 The experimental fibre indentation results were investigated initially by modelling the indentation of bulk material defined using the fibre s anisotropic properties. The loaddisplacement data for this model resulted in an indentation modulus of 135 GPa. This was much lower than the longitudinal modulus value of the material in the literature (238 GPa). The indentation modulus value most likely contains influences from both the longitudinal (E 11 ) and transverse (E 22 ) moduli making it difficult to quantify. The 2 nd model where the indented fibre is modelled surrounded by matrix material produced an indentation modulus value of 52.9 GPa. This value is reasonably close to the average of the experimentally determined values (56.5 GPa) which suggests that the low value of experimental indentation modulus is due to a combination of the fibre anisotropic properties and an added compliance due to the surrounding softer matrix material. 15
6 Conclusions Experimental nanoindentation of the different material phases of a heterogeneous fibrous composite material were carried out in an attempt to determine the in-situ material properties for use in in-house micromechanical models. The relevancy of the results was investigated through finite element modelling of the nanoindentation process for fibrous composites. The 2D and 3D finite element analyses carried out in this paper demonstrate that care should be taken when determining a suitable nanoindentation site when attempting to determine the in-situ properties of the matrix region in a fibrous composite. It is shown that simplifying the analysis by assuming the fibre constraint can be modelled as a rigid boundary leads to an overestimation of the fibre constraining effect of resin pocket of that size. The 3D models with included fibre sections make evident the stress transfer to the fibres in the vicinity of the indentation site. This occurs even in relatively large resin pockets which one might assume would be free of the mechanical constraint due to the fibres. This phenomenon explains the high experimental values for indentation modulus of the matrix constituent. The methodology described in this paper could be applied to any fibre-matrix material system to determine the composite microstructure s susceptibility to this phenomenon and the size of the matrix pocket necessary in order to carry out a valid unconstrained nanoindentation of the resin material. The gradient in modulus close to the fibre-matrix interface detected by all the finite element models demonstrates the challenge in determining the true properties of the interphase region detected in previous works. The 2D models of the nanoindentation process applied to the fibre constituent show that the low values of indentation modulus could be explained by a combination of the fibres anisotropic mechanical properties and the added compliance due to the surrounding matrix. Acknowledgements The authors wish to acknowledge the funding provided by the Irish Research Council for Science, Engineering and Technology (IRCSET). The authors would also like to acknowledge the nanoindenter supplier CSM Instruments for carrying out the experiments and providing the results. Computations were carried out using the SFI/HEA Irish Centre for High-End Computing (ICHEC) whom the authors wish to acknowledge for the provision of computational facilities and support. 16
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