Experimental Mechanics (2012) 52:649–658 DOI 10.1007/s11340-011-9530-z
Mechanical Testing of Solid–Solid Interfaces at the Microscale D. Kupka · E.T. Lilleodden
Received: 23 November 2010 / Accepted: 14 July 2011 / Published online: 2 September 2011 © Society for Experimental Mechanics 2011
Abstract In order to determine the influence of internal interfaces on the material’s global mechanical behavior, the strength of single interfaces is of great interest. The experimental framework presented here enables quantitative measurements of the initiation and propagation of interfacial cracks on the microscale. Cantilever beams are fabricated by focused ion beam milling out of a bulk sample, with an interface of interest placed close to the fixed end of the cantilever. Additionally, a U-notch is fabricated at the location of the interface to serve as a stress concentrator for the initiation of the crack. The cantilevers are then mechanically deflected using a nanoindentation system for high resolution load-displacement measurements. In order to determine the onset and propagation of damage, the stiffness of the cantilevers is recorded by partial unloads during the test as well as by making use of a continuous stiffness technique. A finite element model is used to normalize the load and stiffness in order to establish the framework for comparisons between different interfaces. Keywords Micro-cantilever · Fracture test · Grain boundary · Small scale interface
D. Kupka (B) · E.T. Lilleodden Institute of Materials Research, Materials Mechanics, Helmholtz-Zentrum Geesthacht, Max-Planck-Strasse 1, 21502, Geesthacht, Germany e-mail:
[email protected]
Introduction The initiation and propagation of damage within many materials and structures is dominated by the properties of internal interfaces which have characteristic areas of several tens of square microns [1, 2]. In order to understand the effects of these interfaces on the material’s global properties quantitative measurements of the fracture properties of single interfaces are required. These measurements refer to the fracture energy of the interfaces and to the critical stress and respectively strain for the initiation of a crack. Such analyses of interfacial strength are challenging even at the macroscopic scale; at the micron scale, the experimental challenges alone must be addressed before accurate quantitative analyses of interfacial strength can be realized. Such experiments need to be carried out at distinct locations within the material and provide high resolution load and displacement measurements. Site-specific, high-resolution mechanical testing techniques have been an active area of research for the past 30 years, with a large focus on plasticity [3–6]. More recently, microscopic fracture experiments utilizing microcantilever beams have been carried out on brittle γ TiAl [7] and amorphous films [8, 9] as well as on grain boundaries in corrosive environments [10]. However, the interpretation of these experiments in terms of fracture mechanics is difficult because the geometry of the specimens does not conform to standard test methods. Nevertheless, microscopic cantilevers provide the most promising approach to assess the fracture properties of internal interfaces. It is our objective in the present work to introduce a framework for the quantitative assessment of interfacial strength of individual, micron-scale interfaces. The quantification of
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interfacial strength itself along with an analysis of the associated mechanisms of interfacial failure is left as an open question for further research. The framework presented here combines sample preparation using focused ion beam (FIB) milling with high resolution load and displacement measurements provided by nanoindentation techniques. Micron-scale cantilevers containing a site-specific U-notch are fabricated, and using a method based on a compliance technique described in the ASTM standard E1820 [11], the stiffness of the specimen is monitored during loading in order to assess the initiation of interfacial failure and the length of the evolving interfacial crack. Due to the complex geometry of these specimens, a finite element analysis is carried out in order to normalize the load and the stiffness data. This enables the determination of the material’s elastic modulus and allows for direct comparison between different interfaces.
Specimen Preparation In order to demonstrate this method, we have chosen rolled sheets of the lithium containing aluminum alloy AA2198 in underaged (T351) and peak-aged (T8) conditions. This class of materials is known for intergranular fracture [12, 13]. Additionally, the high rate of diffusion of gallium into grain boundaries in aluminum further reduces their strength [14–16]. While this limits an applicable analysis of grain boundary failure mechanisms, the gallium embrittled material serves as a model case of ductile grains in combination with brittle interfaces. Moreover, this material has characteristically flat grains elongated in the rolling direction, a so-called “pancake” microstructure, as can be seen from Fig. 1.
Fig. 1 Composition of optical micrographs after Barker etching of the AA2198 sheets
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Fig. 2 SEM image of a cantilever with characteristic dimensions
Thus, the grain boundaries are flat with respect to the short transversal direction taken as the surface normal, a great advantage for studying the separation behavior of single grain boundaries. Cantilevers with a rectangular cross-section were prepared using FIB milling with a gallium ion source. A rectangular cross section was chosen in order to maintain a constant crack width. To achieve such a geometry, the specimens must be fabricated from a near edge region of the sample in order to minimize the amount of material to be removed by the FIB milling process. Using orientation contrast images achieved with the ion beam, a suitable grain boundary can be found and placed a distance approximately 2 μm from the supported end of the cantilever. Only the regions adjacent to the cantilever are scanned such that the cantilever is never directly exposed to the ion beam. The selected boundary is marked by a line pattern outside the cutting area to identify the position of the boundary during the cantilever fabrication process. Using a relatively high beam current (e.g. 7 nA) a trench of material is removed from each side adjacent to the cantilever, leaving sufficient space for subsequent cuts to release the cantilever from below. Subsequent milling is conducted at lower currents in the range of 1 nA down to 0.1 nA for the final surface milling. After the cantilever has been prepared, a U-notch is made using FIB milling from one side of the cantilever at the location of the grain boundary. The notch ensures that the selected boundary serves as the primary failure site during testing by avoiding high tensile stress concentrations in the support. The blunt shape of the notch is favorable since the position of the selected
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grain boundary is not directly imaged but rather only estimated. Furthermore, the boundary edge may not be perfectly straight, which would prevent the fabrication of a sharp artificial crack directly at the boundary. Lastly, the well defined shape of the notch circumvents any ambiguities associated with stress singularities for the finite element analysis, as will be discussed later. In the work presented here the nominal dimensions of the final cantilever are approximately 20 μm × 5 μm × 5 μm (length × width × height). A representative cantilever is shown in Fig. 2.
Micro-Beam Bending Experiments The bending experiments are carried out using a nanoindenter (Nanoindenter XP, Agilent) equipped with a Berkovich tip. This tip is used since it is sufficiently sharp to create visible indents on the cantilevers so that the bending length, L, of the cantilever from the supported end to the contact point for loading can be determined. Also, the lateral position of the indent may be identified for further analysis. Moreover, the use of a sharp tip largely avoids the problems associated with a wedge or cylindrical contact, where torque occurs from misalignment at the contact. During the experiment, the cantilever is loaded at a constant displacement rate of 5 nm/s. A typical loading profile during an experiment is shown in Fig. 3. In order to monitor the onset and propagation of damage multiple partial unloads are employed for stiffness measurements; a decrease in the elastic stiffness is associated with the presence of damage or crack initiation. Plasticity, on the other hand, will not affect the unloading stiffness assuming small changes in geometry due to
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Fig. 3 Load-displacement data from a typical micro-bending experiment
plastic deformation. Every 100 nm of displacement the indenter unloads to 50% of the maximum load. While both the unloading and reloading data may be used to determine the stiffness, here the reloading data has been used for the stiffness analysis. During some of the experiments, an additional continuous stiffness measurement (CSM) is carried out, where a harmonic oscillation of the indenter tip is imposed. A harmonic frequency of 45 Hz is used and the displacement amplitude is set to 2 nm such that the oscillation response of the cantilever may be assumed to be elastic. More details on the continuous stiffness measurement technique are given elsewhere [17]. In order to compare the stiffness measured from partial unloads to the harmonic stiffness measurements, additional partial unloads were carried out in increments of 200 nm. While the applied load, P, is measured directly, the total displacement, hraw , is comprised of the deflection of the cantilever beam, w, and the indentation depth, hind , as shown in Fig. 4(a). Figure 4(b) gives a schematic representation of the experiment illustrating the correction to be applied to the raw data. During the loading segment the indentation depth into the cantilever, hind (P), is approximated by the indentation depth associated with a standard indentation experiment conducted in the bulk material at the same load. Hence, a reference curve is measured from indentation experiments in the bulk material at loads up to the maximum load measured for the bending experiments. The deflection of the cantilever alone during loading is expressed as: wloading (P) = hraw (P) − hind (P).
(1)
It is worth noting that when the cantilever is deflected the indentation surface is no longer normal to the indentation axis. However, the cantilever deflections presented in this work do not exceed 10 % of the bending length leading to a maximum inclination of the cantilever-axis at the contact point of approximately 6◦ . The effect of deflection is neglected within the correction. During the unloading and reloading segments the correction shown in equation (1) cannot be applied, due to partial elastic recovery of the indent. In order to correct for the deflection of the cantilever during unloading and reloading the indentation depth must be decomposed into a plastic part, hpl , and an elastic part, hel : hind = hpl + hel .
(2)
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load, P
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w h raw
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h ind displacement, h (b)
(a)
Fig. 4 Schematic representations of (a) the decomposition of the total measured displacement, hraw , the cantilever deflection, w and the indentation depth, hind , and (b) the correction of the raw data for the indentation depth
During the un- and re-loading of the cantilever the plastic indentation depth is constant whereas the elastic indentation depth varies. The correction of the deflection during un- and re-loading is given by: wun/reloading (P) = hraw (P) − hpl (Pmax,i ) − hel (P),
(3)
where the plastic indentation depth at the last maximum is calculated as: hpl (Pmax,i ) = hind (Pmax,i ) − hel (Pmax,i ).
(4)
In order to determine the elastic indentation depth, hel (P), the elastic indentation stiffness, S, of the material is determined from bulk indentation experiments. The indentation stiffness is a function of the maximum indentation depth or equivalently the maximum applied load: S := S(P). Either the use of multiple unloading segments or continuous stiffness measurements during indentation into the bulk can provide the indentation stiffness as a function of the load needed for the correction. Here we use the multiple unloading technique to obtain the elastic indentation depth since this technique corresponds to the method widely applied here for the determination of the unloading stiffness of the cantilever. The elastic indentation depth is calculated as: hel (P) =
P . S(Pmax,i )
(5)
The actual experimental stiffness of the cantilever, dP Kbeam , may then be calculated from the slope, dw , of the corrected reloading segments. In particular, at low indentation depths (hind < 100 nm) the variation of the elastic indentation depth cannot be neglected. Without considering the change in hel , the slope of the unloading and reloading curves would be smaller than associated
with the beam alone, leading to inaccurate stiffness calculations. The errors in the stiffness associated with the first reloading segments would be particularly large since the elastic indentation stiffness is lowest at small depths. In the case of experiments where the continuous stiffness was additionally measured an additional correction is needed. In order to separate the continuous CSM , from the stiffness of stiffness of the cantilever, Kbeam CSM CSM the indent, Sindent , the stiffness of the system, Ksys , is modeled by two springs connected in series: 1 CSM Ksys
=
1 SCSM indent
+
1 CSM Kbeam
.
(6)
By rearranging terms we find an expression for the continuous stiffness of the cantilever alone: CSM Kbeam (P) =
CSM SCSM indent (P) · Ksys (P) CSM SCSM indent (P) − Ksys (P)
.
(7)
In order to determine the harmonic indent stiffness as a function of the applied load, indentation experiments using the CSM method are carried out in the bulk, up to the maximum load used in the bending experiments. For further investigations into the mechanisms involved in the initiation and propagation of a crack it is of great interest to examine the fracture surfaces after testing. The specimens remain connected to the support by a small ligament due to the plastic deformation and compressive stresses on the lower part of the cantilever. It is also possible that the cantilever contacts the material beneath or the tip of the indenter makes contact with the material surrounding the cantilever. While the maintained attachment of the cantilever allows the identification of the indenter contact position needed to measure the bending length, the fracture
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surfaces are not readily accessible for imaging and need to be separated. In order to create enough space for the separation of the fracture surfaces using a Omniprobe micro-manipulator the cantilever is shortened using FIB milling. The micro-manipulator is carefully brought in contact with the shortened beam which is then deflected to open the crack completely. It is worth noting that the complete removal of the cantilever up to the fracture is not possible since redeposition on the fracture surfaces needs to be avoided.
FEM-Based Normalization The complex geometry due to the notch and the support, along with test to test variation of the specimens must be appropriately taken into account in order to make a quantitative assessment of the interface of interest. Finite element analysis is particularly useful for this purpose and has been employed here for quantitative comparison between specimens. For each specimen a finite element model is created which includes the specific geometry, as shown in Fig. 5. The main features of the geometry are: the bending length, L, defined from the supported end of the cantilever to the location of the indent; the width, B; the height, H; the distance, d, from the supported end to the center of the notch; the notch depth, a; and the notch radius, R. Additionally, the cantilever support is explicitly modeled. Fixed boundary conditions are imposed on the surfaces at the bottom of the structure as well as on the back surface. The elastic modulus of the material at the loading point is given as 106 GPa in order to prevent severe distortions of the elements. Such a high contact stiffness negates the need for any indentation correction in the finite element analysis; the displacement in the simulation equals the cantilever deflection, w. In order to create the FE-models the
Fig. 5 (Color online) Finite element model of a specimen. The main features of the geometry are indicated. The colors indicate the normal stress distribution along the cantilever axis (σ yy ) where red indicates the high tensile stresses and blue corresponds to compressive stresses
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finite element code ABAQUS is employed using eightnode brick elements except for the transition zone adjacent to the notch (see Fig. 5). There six-node wedge elements are employed to reduce the mesh size since the notch root is the location of highest stress concentration. For a simple estimation of the elastic stress fields, an isotropic elastic model is chosen. As the local elastic modulus of the material was not known a priori, we assume an elastic modulus and a Poisson ratio for a typical aluminum alloy: Esim = 72 GPa and ν = 0.33, respectively. The choice of an isotropic elastic model is reasonable since the anisotropy of aluminum single crystals is poorly pronounced as can be seen from an anisotropy factor 1.23. Simulations of single crystal cantilevers with the 1 1 1 and 1 0 0 taken as the cantilever’s normal axis show that the deviations from the isotropic beam stiffness did not exceed 10%. With the isotropic elastic constants the stiffness, Ksim , of the structure may be computed. Assuming that the deflection is small compared to the bending length of the cantilever, we suppose a linear law relating the deflection, w, to the load, P, by a constant stiffness, K: P = K w.
(8)
Thus, all geometric features may be reduced to one single value with a dimension of length which we call the characteristic length, l:ˆ Ksim . lˆ = Esim
(9)
Simulations have shown that the characteristic length is independent of the chosen elastic modulus within the range of 60–100 GPa. While the FEM model used here is for linear elastic behavior, it is not expected that the plasticity occurring during the actual experiments limits the applicability of the characteristic length; in all cases the stiffness of
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interest results from elastic deflections, and the permanent shape change resulting from plasticity is small. In the absence of damage we obtain: Pexp = Kbeam wexp = Eexp lˆ wexp, reloading .
(10)
ˆ and a normalized stiffness, K, ˆ A normalized load, P, are computed by dividing the load and the stiffness by the characteristic length associated with the specimen geometry: P Pˆ = lˆ Kbeam Kˆ = lˆ
(11) (12)
stic
elastic + plastic
damage + plastic
ela
normalized load, P
Figure 6 illustrates the elasto-plastic response of a specimen: A largely linear elastic region is followed by plastic deformation. In both of these regions the slope of the unloading segments giving the normalized stiffness is constant. When damage occurs the slope of unloading segments decreases. The normalized stiffness is equivalent to the elastic modulus of the material as long as no damage is present in the structure. With the onset of damage, e.g. initiation of a crack, the normalized stiffness will decrease and, thus, can be used to monitor the damage evolution.
cantilever deflection, w Fig. 6 Schematic representation of the general loaddisplacement data for a cantilever showing linear elastic behavior followed by plastic yielding. After yielding the elastic stiffness, i.e. the slope of the unloading segments remains constant until damage occurs (solid lines). The onset of damage leads to a reduction in the unloading stiffness (dotted lines)
Fig. 7 Post mortem specimen of the underaged material. Fracture occurred at the selected grain boundary. The grain boundaries along the cantilever are decorated with gallium as can be seen from the dark lines at the front of the cantilever
Results A typical post mortem specimen from the underaged material is shown in Fig. 7. The dark lines on the side surface of the cantilever are grain boundaries decorated with gallium. The specimen failed at the grain boundary where the U-notch was fabricated, as desired. Near the free end of the cantilever the indent created at the loading position is visible. The normalized load-displacement response of this specimen is shown in Fig. 8(a) together with a characteristic loaddisplacement response of one of the peak-aged specimens. As expected, the plastic deformation of the peakaged specimen starts at a higher normalized load than that for the underaged specimen. For specimens of the same heat treatment, the maximum deflection and the maximum normalized load vary between specimens. This is likely due to the differences between the specific boundaries tested. When the normalized load at fracture is attained, the deflection increases instantly, as indicated by the lack of data points beyond a deflection of about 1 μm for both specimens; an unstable crack is initiated at this point. The normalized stiffness refers to the elastic deformation of the specimen and may be used to monitor the initiation and propagation of a crack. Figure 8(b) shows the normalized stiffness plotted against the deflection at the start of the associated unloading segment for several specimens. The specimens of both heat treatments show values of the normalized stiffness of (89 ± 3)GPa. This value reflects the elastic modulus of the specimen in the absence of an interfacial crack and is in good agreement with the elastic modulus measured by nanoindentation which is (86 ± 2)GPa. No significant decrease of the normalized stiffness can
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Fig. 8 Plots of (a) the normalized load Pˆ and (b) the normalized stiffness Kˆ vs. the deflection of the cantilever w
the mechanisms of grain boundary failure. While the usefulness of a detailed analysis of the mechanisms involved in the failure of grain boundaries in AA2198 is limited due to the effects of gallium embrittlement, we present several examples of the features discernible from this technique. Figure 10 shows high resolution SEM-micrographs of the crack tips (Fig. 10(a) and (c)) or directly behind the crack tip (Fig. 10(b)) for various tested cantilevers. Most crack tips were sharp irrespective of the thermomechanical condition, as shown in Fig. 10(a). In the grain on the left, slip traces are clearly visible. Such planar slip is known to cause stress concentrations at
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cantilever Stiffness, N/m
be found with increasing deflection of the cantilever indicating brittle boundaries. As soon as the critical load or deflection for the initiation of a crack is attained, the grain boundary fails at once and no stable crack propagation can be observed. Since no significant loss of elastic stiffness occurs prior to specimen failure, the nonlinearity in the normalized force may be solely attributed to plastic deformation. The increase in yield strength going from the underaged to the peak-aged condition is reflected by increasing normalized loads at the start of yielding. However, a more expansive interpretation of the normalized load in terms of interfacial strength necessitates decoupling the plasticity from the interfacial properties and is beyond the scope of this paper. The cantilever stiffness determined using the CSMCSM , is compared to the stiffnesses determethod, Kbeam mined from the reloading segments, Kbeam , in Fig. 9 where both stiffnesses are plotted against the cantilever deflection. The CSM-data shows scatter of approximately 15% around its average value. At deflections up to 100 nm the harmonic stiffness increases until it attains a constant average value. This is likely due to deviations from the correction given in equation (7) at very low indentation depths. The stiffness measured from the reloading segments deviates from the mean value by less than 5%, which shows that both methods are equivalent for the measurement of the cantilever stiffness. In addition to the load displacement response, the tested specimens were observed by high resolution electron microscopy. The crack tips and the fracture surfaces are of particular interest for studies of interfacial failure. Such observations provide insight into
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Fig. 9 Stiffness measured by the CSM method: The curve shows the harmonic stiffness data corrected for the indentation depth CSM . The values measured from the corrected and stiffness, Kbeam reloading segments, indicated by the circles, are given for comparison
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(a)
(b)
(c)
Fig. 10 (a) Sharp crack tip as observed in most specimens. Slip lines can be seen on one of the adjacent grains. (b) Bridging elements found behind a crack tip of an underaged specimen. (c) Rounded crack tip in a peak-aged specimen showing massive slip on the crack ground
grain boundaries [18] which in turn promote intergranular failure. Another feature which could be observed was a domain of bridging elements behind the crack tip also in an underaged specimen (see Fig. 10(b)). Lastly, a blunted crack tip in a peak-aged specimen (Fig. 10(c)) was observed. The interfacial crack in this
specimen was stopped at a triple junction, and massive slip occurred at the crack tip. The largely vertical lines which are seen in Fig. 10 result from the specimen fabrication through FIB milling, the so-called “curtaining effect”, and should not be confused with slip lines. It is envisioned to carry out these experiments in a SEM
Fig. 11 (a) Smooth fracture surface of a grain boundary in an underaged specimen showing large particles and striation like lines. Partially dimpled fracture surface of an underaged specimen (b). (c) Fracture surface of a grain boundary in a peak-aged specimen with finely distributed particles, the arrow marks a triple junction
(a)
(b)
(c)
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which will allow monitoring of the strain fields, the propagation of cracks and the crack opening displacement. After opening the cracks further using the Omniprobe micro-manipulator, as discussed previously, the fracture surfaces reveal additional information regarding the mechanisms of fracture. The underaged specimens exhibit generally smooth fracture surfaces with some larger particles visible, as shown in Fig. 11(a) and (b). Lines resembling fatigue striations which bow around the coarse particles are found on the fracture surface (Fig. 11(a)). Figure 11(b) shows the fracture surface of the specimen that exhibits the domain of bridging elements shown in Fig. 10(b). The fracture surface may be divided into two regions. The first region resembles to the smooth surface of Fig. 11(a). The second region shows a highly dimpled morphology. The fracture surfaces for the peak-aged specimens show particles smaller than those found on the fracture surfaces of the underaged specimens, as can be seen from Fig. 11(c). This fracture surface corresponds to the crack tip shown in Fig. 10(c). The striation-like lines mentioned before are not found in any of the peak-aged specimens. The features presented here give insight into the morphology of the intergranular cracks and the mechanisms involved in the fracture process. In particular for the underaged specimens which exhibit lower yield stresses and thus higher slip activity, the planar slip promotes the separation of the grain boundary. The coarse particles found on the fracture surfaces of the underaged specimens act as stress concentrators also promoting grain boundary failure. The multiple unloads used in the bending experiment can be one reason for the appearance of the striation-like lines on the fracture surfaces of the underaged specimens since the number of unloads may be roughly correlated to the number of lines. However, other underaged specimens do not show these lines or the lines are less pronounced. The orientation of the boundary and the adjacent grains and, in particular, the particles on the grain boundaries may influence the formation of these lines. Since no such lines were observed on intergranular fracture surfaces of the peak-aged specimens, particles present on the grain boundaries are likely to prevent their formation. From the fracture surfaces shown in Fig. 11(b) it can be seen that the fracture mode has changed from brittle fracture to dimpled fracture which indicates ductile deformation within the boundary. More investigations on the morphology of the grain boundary are required to determine the cause for this change in fracture mode.
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Summary A refined method for the investigation of the mechanical response of grain boundaries or other internal interfaces on the microscale has been presented. Microscopic cantilever bending specimens with a Unotch at the location of a grain boundary have been fabricated from peak-aged and underaged samples of the lithium containing aluminum alloy AA2198. The material has been chosen due to its high sensitivity to intergranular fracture which is amplified by the embrittling effect of the gallium used in the fabrication process. Using a nanoindenter the cantilevers have been deflected and stiffness measurements during the tests have been conducted to characterize the onset and propagation of damage at grain boundaries. Corrections to the deflection and the stiffness data due to the local displacements at contact have been developed. Using finite element simulations a characteristic length has been computed that serves to normalize the load and the stiffness in order to make quantitative comparison between boundaries. The finite element modeling of each specimen provides the framework for including anisotropy, plastic deformation and damage into the analysis of the specimens. Since the normalized stiffness of the cantilevers in the absence of damage is found to be in good agreement with the elastic modulus measured from nanoindentation experiments, the quantitative nature of this method has been established. Moreover, it has been shown that the stiffness measured by partial unloads and by using the continuous stiffness method result in comparable values. The method has been applied to the lithium containing aluminum alloy AA2198 in underaged and peak aged conditions where the gallium additionally embrittles the grain boundaries. As expected, the grain boundaries failed without a significant reduction in the cantilever stiffness indicating brittle fracture. Images of the crack tips and fracture surfaces of the tested specimens additionally provide information about the underlying fracture mechanism. Such observations in connection with the load-displacement data may help to establish traction separation laws for cohesive zone modeling of small scale interfaces.
References 1. Watanabe T, Tsurekawa S (1999) The control of brittleness and development of desirable mechanical properties in polycrystalline systems by grain boundary engineering. Acta Mater 47:4171–4185 2. Madhukar MS, Drzal LT (1991) Fiber-matrix adhesion and its effect on composite mechanical properties: I. Inplane and
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3.
4. 5.
6.
7.
8.
9.
10.
Exp Mech (2012) 52:649–658 interlaminar shear behavior of graphite/epoxy composites. J Compos Mater 25:932–957 Uchic MD, Dimiduk DM, Florando JN, Nix WD (2004) Sample dimensions influence strength and crystal plasticity. Science 305:986–989 LaVan DA, Sharpe WN (1999) Tensile testing of microsamples. Exp Mech 39:210–216 Kiener D, Grosinger W, Dehm G, Pippan R (2008) A further step towards an understanding of size-dependent crystal plasticity: In situ tension experiments of miniaturized singlecrystal copper samples. Acta Mater 56:580–592 Florando JN, Nix WD (2005) A microbeam bending method for studying stress-strain relations for metal thin films on silicon substrates. J Mech Phys Solids 53:619–638 Halford TP, Takashima K, Higo Y, Bowen P (2005) Fracture test of microsized TiAl specimens. Fatigue Fract Eng Mater Struct 28:695–701 Nishida M, Matsuda M, Shimada Y, Takashima K, Ishikawa K, Aoki K (2009) Microstructural and mechanical characterizations of rapidly solidified Nb-TiNi hydrogen permeation alloy. J Phys Conf Ser 144:012106 Matoy K, Schönherr H, Detzel T, Schöberl T, Pippan R, Motz C, Dehm G (2009) A comparative micro-cantilever study of the mechanical behavior of silicon based passivation films. Thin Solid Films 518:247–256 Armstrong DEJ, Rogers ME, Roberts SG (2009) Micromechanical testing of stress corrosion cracking of individual grain boundaries. Scr Mater 61:741–743
11. Standard test method for measurement of fracture toughness, ASTM E1820 (2008) Annual book of ASTM standards, vol 03.01, pp 1070–1117 12. Vasudevan AK, Doherty RD (1987) Grain boundary ductile fracture in precipitation hardened aluminum alloys. Acta Metallurg 35:1193–1219 13. Kalyanam S, Beaudoin AJ, Dodds RH, Barlat F (2009) Delamination cracking in advanced aluminum-lithium alloys— experimental and computational studies. Eng Fract Mech 76:2174–2191 14. Ren L, Bahr DF, Hoagland RG (2000) Multiscale phenomena in materials—experiments and modeling, MRS proceedings. In: Lassila DH, Robertson IM, Phillips R, Devincre B (eds) Mater Res Soc Symp Proc, Warrendale, PA, vol 578, pp 411–416 15. Unocic KA, Mills MJ, Daehn GS (2010) Effect of gallium focussed ion beam milling on preparation of aluminum thin foils. J Microsc 240:227–238. doi:10.1111/j.1365-2818. 2010.03401.x 16. Lynch SP (1989) Mechanisms of intergranular fracture. Mater Sci Forum 46:1–24 17. Oliver WC, Pharr GM (1992) An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments. J Mater Res 7:1564–1583 18. Li JCM, Chou YT (1970) The role of dislocations in the flow stress grain size relationships. Metall Trans 1:1145– 1159
