Lists the radii and maximum shear stress derived from the experiment and FEA simulation for both the sharp and blunt indenters.
\r\n\t
",isbn:"978-1-83881-017-7",printIsbn:"978-1-83881-016-0",pdfIsbn:"978-1-83881-024-5",doi:null,price:0,priceEur:0,priceUsd:0,slug:null,numberOfPages:0,isOpenForSubmission:!1,hash:"d5ac3a7054e526666a89271cef6ee869",bookSignature:"Dr. Ahmed Mourtada Elseman",publishedDate:null,coverURL:"https://cdn.intechopen.com/books/images_new/9924.jpg",keywords:"Photons, Semiconducting Materials, Photocurrent Density, Dye-Sensitized Cells, Perovskite Cells, Perovskite/Si Tandem, CIGS, CdTe, Single Crystal, Thin-Film Crystal, Single Crystal, Multicrystalline",numberOfDownloads:440,numberOfWosCitations:0,numberOfCrossrefCitations:0,numberOfDimensionsCitations:0,numberOfTotalCitations:0,isAvailableForWebshopOrdering:!0,dateEndFirstStepPublish:"May 25th 2020",dateEndSecondStepPublish:"June 15th 2020",dateEndThirdStepPublish:"August 14th 2020",dateEndFourthStepPublish:"November 2nd 2020",dateEndFifthStepPublish:"January 1st 2021",remainingDaysToSecondStep:"7 months",secondStepPassed:!0,currentStepOfPublishingProcess:5,editedByType:null,kuFlag:!1,biosketch:"Dr. Elseman holds two diplomas, first one from Inner Mongolia Institute of Science and Technology and the second one from the Institute of New Energy, Wuhan. During his work at Southwest University, where he is currently also active, Dr. Elseman received funds from Central Universities for a project on efficient perovskite solar cells, a topic on which his research is mainly focused.",coeditorOneBiosketch:null,coeditorTwoBiosketch:null,coeditorThreeBiosketch:null,coeditorFourBiosketch:null,coeditorFiveBiosketch:null,editors:[{id:"221890",title:"Dr.",name:"Ahmed Mourtada",middleName:null,surname:"Elseman",slug:"ahmed-mourtada-elseman",fullName:"Ahmed Mourtada Elseman",profilePictureURL:"https://mts.intechopen.com/storage/users/221890/images/system/221890.jpg",biography:"Ahmed Mourtada Elseman obtained his B.Sc., M.Sc., and Ph.D. in Inorganic and Analytical Chemistry from the Faculty of Science, Al-Azhar University, Egypt. He earned his Ph.D. in perovskite solar cells in February 2017. He obtained two diplomas, first one from Inner Mongolia Institute of Science and Technology, Hohhot, China 2015, and the second one from the Institute of New Energy, Wuhan, China, 2017. He currently works as Research Assistant Professor at the Department of Electronic and Magnetic Materials, Central Metallurgical Research and Development Institute (CMRDI), Egypt. \nHe was awarded the Talent Young Scientific (TYSP) Postdoctoral Research Fellow position funded by the Chinese Ministry of Science and Technology (MOST) and organized by North China Electric Power University, Beijing, China, 2017-2018. After that, he received a lecturer position in the School of Materials and Energy, Southwest University, Chongqing. China (2018 – 2020). During his work at Southwest University, he received a project funded by Central Universities for efficient perovskite solar cells (ID: XDJK2019C005). He was also awarded the CMRDI prize for excellence scientific publication (2018). His current research focuses on understanding the mechanisms, fundamental properties, and developing scalable protocols for high-efficiency perovskite solar cells. 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For crystalline materials, there is a growing experimental and theoretical interest in pop-in events, which are sudden displacement-burst excursions during load-controlled nanoindentation of relatively dislocation-free metals. The first pop-in event is often identified as the initiation of dislocation nucleation, and thus the transition from purely elastic to elastic/plastic deformation. The maximum shear stress at this first pop-in event, or the onset of plastic yielding, is generally found to be close to the theoretical strength of the material and is frequently estimated from Hertzian elastic contact theory. However, an irregular indenter tip shape will significantly change the stress distribution in magnitude and location, and therefore the maximum shear stress, from a Hertzian estimation. The aim of this chapter is to state the challenges and limitations for extracting the initial plastic yield stress from nanoindentation with the spherical indenter tip assumption. We assess possible errors and pitfalls of the Hertzian estimation of initial plastic yield at the nanoscale.
Instrumented nanoindentation has been widely used to probe small scale mechanical properties such as elastic modulus and hardness over a wide range of materials and applications (Doerner & Nix, 1986; Fisher-Cripps, 2002; Oliver & Pharr, 1992, 2004). The response of a material to nanoindentation is usually shown by plotting the indentation load, P, as a function of the indenter penetration depth, h.
For crystalline materials, nanoindentation can be used to study defect nucleation and propagation events, which are detected by discontinuities in the load-depth relationship. Generally, there are three types of discontinuities as illustrated in Fig.1. First are “pop-in“ events (as shown in Fig. 1a), which are sudden displacement excursions into the target materials during load-controlled nanoindentation of relatively dislocation-free metals. Pop-ins were first observed and associated with dislocation nucleation, or the sudden onset of plasticity, by Gane and Bowde in 1968 (Gane & Bowden, 1968) using fine stylus indentation of metal crystals. Pop-ins may also be associated with crack nucleation and propagation (Morris et al., 2004; Jungk et al., 2006), phase transformations (Page et al., 1992), and mechanically induced twinning (Bradby et al., 2002; Misra et al., 2010).
The second type of discontinuity is a “pop-out“ event (as shown in Fig. 1b), which is a discontinuous decrease in the indentation displacement, usually during unloading. Pop-outs may also be ascribed to dislocation motion (Cross et al., 2006) and phase transformations (Juliano et al., 2004; Ruffell et al., 2007; Haq et al., 2007; Lee & Fong, 2008). A lower unloading rate or a higher maximum indentation load promotes the occurrence of a pop-out (Chang & Zhang, 2009).
The third type of load-depth discontinuity is the “load drop“ found during a displacement-controlled experiment (Kiely & Houston, 1998; Warren et al., 2004), as shown in Fig. 1c. A molecular dynamics study showed that load drops are associated with local rearrangements of atoms (Szlufarska et al., 2007).
Schematic diagram of nanoindentation load displacement curve illustrating the (a) pop-in, (b) pop-out, and (c) load-drop behaviors.
Quantitative study of these nanoindentation phenomena requires reasonable estimates of the stresses that drove the particular event. Here, attention is focused on yield in metallic crystals. It is believed that the first pop-in event is most frequently the result of the initiation of dislocation nucleation, and thus the transition from purely elastic to elastic/plastic deformation (Gouldstone et al., 2000; Kelchner et al., 1998, 2009; Suresh et al., 1999; Li et al., 2002; Lorenz et al., 2003; Minor et al., 2004; Manson et al., 2006; Nix et al., 2007). The load-displacement curve before the pop-in occurs is often fully reversible, and is usually interpreted using the Hertzian contact theory (Johnson, 1999),
where R is the indenter tip radius and
is the contact elastic modulus between the indenter (I) and specimen (S). In this case, the deformation is purely elastic prior to the first pop-in; if the indenter tip is unloaded before the first pop-in, atomic force microscopy (AFM) images show no indent on the specimen surface, whereas, if unloading occurs after the pop-in, a residual indent is observed (Chiu & Ngan, 2002; Schuh & Lund, 2004).
Nanoindentation pop-in tests can be a powerful tool for studying homogeneous and heterogeneous dislocation nucleation. When a pop-in event is caused by the sudden onset of crystal plasticity, whether through dislocation source activation (Bradby & Williams, 2004; Schuh et al., 2005) or homogeneous dislocation nucleation (Bahr et al., 1998; Chiu & Ngan, 2002), the maximum stress at the yield event is generally interpreted as the maximum shear stress in the body (Minor et al., 2006). This maximum shear stress, τMAX, at the first pop-in load, PCRIT, is generally estimated from elastic contact theory (Johnson, 1999) as
For a variety of materials, when the first pop-in occurs, the maximum shear stress in the specimen is in the range of G/30 to G/5, where G is the shear modulus; this stress is very close to the theoretical strength calculated by the ab initio method (Van Vliet et al., 2003; Ogata et al., 2004).
A recent study using molecular dynamics simulations found that the stress components other than the resolved shear stress also affect the dislocation nucleation process (Tschopp & McDowell, 2005; Tschopp et al., 2007). Based on an anisotropic elasticity analysis, Li et al. (2011) derived in closed form the stress fields under Hertzian contact theory and computed the indentation Schmid factor as a ratio of the maximum resolved shear stress to the maximum contact pressure.
It must be emphasized that equations (1) and (2) are restricted to spherical indentation and cannot be applied to arbitrary geometries. Thus, the radius of the indenter probe is an essential component for estimating dislocation nucleation shear stress inferred from spherical indentation responses. Also, access to the nanometer length scales needed to find a dislocation-free region in a metallic crystal may require very small radii. Here, experimentalists can take advantage of the imperfect manufacture of nominally sharp geometries, such as the three-sided pyramidal Berkovich indenter. The most common view is that the manufacturing process produces an approximately spherical cap on the apex of the indenter tip. The radius of the assumed spherical tip is generally obtained from the tip manufacturer, AFM, scanning electron microscopy (SEM), or by a Hertzian fit to the elastic load-displacement data (Chiu & Ngan, 2002; Constantinides et al., 2007; Gerberich et al., 1996; Gouldstone et al., 2000).
It is known that real pyramid indenter tips may have irregular shapes, especially at the nanometer-scale where the first pop-in event occurs. Previous finite element analysis (FEA) simulations and combined experimental and FEA studies have shown that even a highly irregular probe (which cannot simply be decomposed into “sphere” and “cone”) will produce an elastic load-displacement relationship that could be perceived as having been from a spherical contact (Ma & Levine, 2007; Ma et al., 2009, 2012). Unsurprisingly, the simulations showed that the irregular shape generated shear stresses in the body that were significantly different, both in magnitude and location, from those produced by a true spherical probe. Using the common Hertzian spherical approximation to interpret experimental data can lead to a substantial underestimation of the maximum shear stress in the body at the initiation of plasticity. An assessment of the potential errors in experimental estimates of nucleation stresses is critical, especially in materials that exhibit the elastic-plastic transition at small indentation depth. We need to accurately measure the three dimensional shape of the true indenter, prepare a sample that has both low dislocation density and a smooth surface, and conduct nanoindentation experiments with accurate load and displacement measurements. In addition, several groups have reported that the rate at which the indenter tip penetrates the specimen can have a significant effect on the plastic deformation mechanisms in materials as diverse as Si (Jang et al., 2005), single-crystal Ni3Al (Wang et al., 2003) and single-crystal Al2O3 (Mao et al., 2011). In the rest of this section, we will describe some of the difficulties involved in these measurements and some of the directions we, and others, are pursuing to overcome them.
There has been interest in direct measurement of the indenter geometry for at least two decades. At the larger micrometer to millimeter scale of Rockwell hardness indenters, for example, NIST has played a leading role in the drive toward indenter standardization (Song et al., 1997). Direct metrology of indenter geometry, using well-calibrated stylus profilers, is the most effective method at these size scales, and it is able to provide uncertainties low enough to support the uncertainty goals in Rockwell hardness measurements themselves.
For instrumented nanoindentation, however, the smaller sizes greatly increase the challenges to direct metrology of the indenter geometry. In the 1980s, Doerner and Nix (Doerner & Nix, 1986) measured the geometry of a Vickers indenter with transmission electron microscopy (TEM) by using a process for making carbon replicas of indents on soft surfaces.
In the early 90s, the work of Oliver and Pharr (Oliver & Pharr, 1992) led to the now prevalent method of inferring tip area from indentations on a sample of known modulus (McElhaney et al., 1998). By the late 90s, interest in the direct metrology of indenters using scanned probe microscopes was growing.
In 1998, researchers at PTB (Hasche et al., 1998) used scanned probe microscopy to characterize the geometry of Vickers microhardness indenters. Subsequently, they expanded their efforts to include Berkovich indenters (Herrmann et al., 2001).
During the following decade, work in this area significantly increased. Aldrich-Smith and coworkers (Aldrich-Smith et al., 2005) at the National Physical Laboratory (NPL) of the UK used a traceably-calibrated AFM to measure the tip area function of indenters, and compared these with the area function derived from indentation.
VanLandingham and coworkers (VanLandingham et al., 2005) undertook a detailed investigation of indenter tip shape as measured by AFM compared with that inferred by indentation on fused silica. They also developed a heuristic model of the indenter geometry as a spherical cap in place of an ideal apex on the Berkovich pyramid. For a vertical calibration range of less than 1 μm, they concluded that measurement of tip area by AFM typically performed better than the indentation method.
In 2007, McMinis and coworkers (McMinis et al., 2007) used the heuristic two-term polynomial model to fit the tip area function measured by AFM. The motivation for their model also was the assumption of an approximately spherical cap. However, in contrast with VanLandingham and coworkers, McMinis, et al. (2007) focussed on much smaller indentation depths – and found excellent performance of their approach relative to indentation on fused silica – particularly for depths less than 7 nm.
More recently, Munoz-Paniagua and coworkers (Munoz-Paniagua et al., 2010) effectively used a hyperbolic model to fit the geometry of worn Berkovich indenters measured by AFM. One of their illuminating observations was that the use of the indentation-derived shape area function could result in errors of the same scale as the assumption of an ideal Berkovich geometry.
Measurement of nanoindenter tip geometry by AFM remains an area of active investigation, and the next decade may well be more prolific than the last in terms of new insights and refinement of methods.
It has been clearly demonstrated that irregularities in the tip shape can produce substantial changes in the perceived internal shear stresses. It is likely that irregularities in the sample topography can produce similar problems, and quantitative assessments of this affect are planned. In the meantime, it is important to minimize these problems by using samples that are as flat as possible. Unfortunately, mechanical polishing (even using colloidal silica) of face-centered cubic (FCC) metals can introduce substantial numbers of dislocations into the near-surface region of the sample which can adversely affect the measurement. The following procedure has been effective at largely eliminating these problems for Ni specimens, but quantitative assessments still need to be done. The measurements described in the next section used tungsten specimens, which are less prone to damage during polishing.
We start with a {111}-oriented, chemi-mechanically polished, metal single crystal that is acquired from a commercial vendor. Optical microscopy generally shows no scratches and an apparently perfect mirror finish. Further characterization using optical interferometry and AFM have demonstrated that these samples have a surface roughness, Rq, of about 2.5 nm and exhibit numerous small scratches that are about 20 nm deep. To relieve any residual stresses and decrease the dislocation density of the sample, the Ni single crystal is slowly heated to 850 °C in ultra-high-vacuum and annealed at this temperature for about 140 h before slow cooling back to room temperature.
Remarkably, optical microscopy of the annealed sample shows numerous long, straight ridges that resemble reversed polishing scratches. The ridge orientations are random and do not correspond to intersections between the {111} slip planes and the {111} surface. The most likely explanation is that early polishing stages introduced substantial compressive residual stresses that are “locked in“ by dislocation structures. The annealing removes these dislocations and the stress relief is accompanied by projection of the previously compressed material above the sample surface. These ridges are removed by electropolishing the sample surface for 90 s at a constant current density of 1 A/cm2 using a electropolishing solution of 37 % phosphoric acid, 56 % glycerol, and 7 % water, by volume. This final step produces an optically perfect surface, although measurements show residual broad features with maximum heights of around 20 nm to 30 nm. These features can be avoided by using AFM to locate flat regions suitable for nanoindentation.
Future work will use sub-micrometer, depth-resolved diffraction (Levine et al., 2006; Levine et al., 2011) at sector 34 of the Advanced Photon Source at Argonne National Laboratory to characterize the local dislocation density, residual elastic strain tensor, and crystallographic orientation throughout the volume beneath the touchdown point, both before and after nanoindentation. Such characterization can both verify the suitability of the target location and provide a quantitative measure of the microstructural changes and residual stresses that result from a nanoindentation measurement.
In addition to having detailed knowledge of indenter tip shape and initial sample conditions, accurate experimental determination of properties such as initial plastic yield stress and elastic moduli also requires that the instrumented nanoindentation instrument produces accurate load and displacement data. As will be shown below, the measurement ranges of interest for studying the onset of plastic yield are typically below 1 mN for applied force (load) and below 20 nm for the displacement of the indenter tip into the specimen surface. If the accuracy in mechanical properties calculated from force and displacement data is desired at the level of a few percent, then both force and displacement data must be accurate to approximately 1 % in these ranges; this is a challenging task.
In the calibration of force, the two greatest challenges are obtaining accurate reference forces and transferring those reference forces to the nanoindentation instrument. Recently, however, techniques have been developed to realize forces down to 10 nN that are traceable to the International System of Units (SI) through the use of electrostatic forces, and those forces can be transferred traceably to nanoindentation instruments with the same level of accuracy using transfer force cells (Pratt et al., 2005) that can be placed directly in the instrument, in place of a specimen.
The problem of measuring indentation depth poses a different set of problems. In principle, length metrology with the necessary level of accuracy is easily available through the use of laser interferometer systems (see, for example, Smith et al., 2009). However, the indenter penetration depth, h, is necessarily defined as the depth to which the indenter penetrates the original plane of the specimen surface. This, however, is not the displacement that many commercial nanoindentation instruments measure; very often, they measure the motion of the indenter tip and the shaft to which it is mounted relative to housing (often referred to as a “load head”) that contains the indenter shaft. In this arrangement, the compliance of the load frame – the mechanical path between the load head and the specimen – becomes a direct source of error in determining h from the measured displacement (Oliver & Pharr, 1992). Future work will include the design, construction and use of a nanoindention instrument capable of non-contact sensing of the location of the specimen surface, allowing a more direct measurement of penetration depth. This instrument will provide an SI-traceable path for force calibration as well.
In this section, we introduce our combined experimental and finite element analysis (FEA) method to study the effect of spherical indenter tip assumption on the initial plastic yield stress of <100>-oriented single crystal tungsten from nanoindentation.
The near-apex shape of two real diamond Berkovich indenters, one lightly and another heavily used, were measured using an Asylum Research MFP-3D[1] - (Santa Barbara, CA) AFM in intermittent contact mode. This instrument uses a two-dimensional flexure stage under closed-loop control with position measurement by linear variable differential transformers (LVDTs). Motion of the stage is therefore decoupled from the vertical motion of the scanner, eliminating cross-coupling artifacts found in tube-type scanners. The vertical motion is also under closed-loop control with displacement measured by LVDT.
The Berkovich probes were cleaned using carbon-dioxide “snow” (Morris 2009). Silicon cantilevers with tip radii of 5 nm to 10 nm and approximately 300 kHz bending resonance frequencies were used to image the indenters. The AFM images were taken at a resolution of 256 x 256 pixels in a 1.25 µm × 1.25 µm region. A sharp silicon-spike tip characterizer (TGT-1, NT-MDT, Moscow, Russia) was scanned before and after imaging to verify that there were no double-tip artifacts or other deformities. Four hours were allowed before the final shape measurement to allow the temperature to stabilize within ± 0.1 °C. Drift-rate characterization experiments utilizing image correlation software showed that this reduced thermal expansion/contraction drift to less than 0.5 nm min-1.
The nanoindentation experiments were previously described in Li, et al. (2012). Cylindrical tungsten crystals were prepared by first grinding to SiC 600 grit. Grinding damage at the surface was then removed by electropolishing for several minutes at 20 V in a solution of 50 % glycerol (by volume) and 50 % water with 1.25 × 10-3 mol m-3 NaOH (Vander Voort, 1984; Zbib & Bahr, 2007). Electropolished tungsten at room temperature has a negligible (sub-nanometer thick) oxide present (Bahr et al., 1998).
The indentations on the electropolished tungsten were performed with a Hysitron (Minneapolis, MN) Tribo-Scope attachment to a Park (Santa Clara, CA) Autoprobe CP scanning probe microscope in ambient laboratory conditions. For the lightly used indenter, the loading schedule was as follows: loading to 100 μN, a partial unloading to 80 μN, then final loading to 700 μN at rates of 20 μN s-1. The indenter was held at the peak load for 5 s, then unloaded to 140 μN at 60 μN s-1. The purpose of the initial load-unload sequence was to inspect for plastic deformation prior to the first pop-in event. If the load-displacement data were different prior to the first pop-in event, then the elastic analysis was not used.
The commercial FEA package, Abaqus (Abaqus, 2011), was used for modeling the nanoindentation of a single-crystal tungsten sample. To accurately simulate nanoindentation in the elastic regime, the two AFM-measured indenter shapes were directly input into FEA models (Ma & Levine, 2007; Ma et al., 2009, 2012). First, high-frequency measurement noise in the AFM data was removed through a combination of median and averaging filters. No change in the orientation of the image was made (as would occur if the image were plane-fitted and leveled). After filtering, the discrete AFM data were interpolated to a 2.44 nm spacing using bi-cubic interpolation. Next, since W is a very stiff material, the AFM-measured indenter tip shapes were meshed with three dimensional (3D) deformable elements with a flat rigid plate at the top. A 2D elastic-deformation parameter study was carried out to determine the distance from the rigid plate to the 3D deformable indenter tip that was needed to avoid introducing artifacts into the nanoindentation simulations. In addition, complementary simulations of indentation by parabolic probes took advantage of the full symmetry of a sphere and four-fold symmetry of {100} W, so only one quarter of a sphere and specimen were modeled to reduce the computational burden. In all cases, contact between the indenter and specimen was assumed to be frictionless.
For the details of the FEA specimen mesh for both indenters, please see (Ma et al., 2009, 2012). The complete tungsten specimen and diamond indenter mesh was created using eight-node reduced-integration linear brick elements. In the contact region, the FEA mesh is high-density. The diamond element length is 2.44 nm. For the tungsten specimen, each element has a length of 1.266 nm (4 unit cells across) with nodes aligned to tungsten atom positions. This will facilitate future atomistic simulations that use the under-load state as a boundary condition (Wagner et al., 2008). Constitutive behavior was linear elastic, and fully anisotropic for the tungsten and isotropic for the diamond. Expressed using the crystallographic body-centered cubic (BCC) basis vectors, each edge of the tungsten mesh elements is <100>. The stiffnesses of tungsten are C11 = 522 GPa, C12 = 204 GPa, and C44 = 161 GPa (Lide, 2007); For the diamond indenter, the Young’s modulus is selected from the published indentation modulus of {100} diamond (1126 GPa) and Poisson’s ratio is 0.07 (Vlassak et al., 2003).
Figures 2(a) and 2(b) are the AFM rendering images of the lightly- and heavily-used Berkovich indenters, respectively. Figures 2(c) and 2(d) are the corresponding contour plots of the near-apex region with 2 nm contour spacing after interpolation and filtering, as described in Sec. 3.3. Note that the contour plots for both indenters are plotted to the same scale. It can be seen that the lightly-used indenter is much sharper while the heavily-used indenter is flattened and blunt in the near-apex region, as expected. From the data shown in Figs. 2(c) and 2(d), the radius of curvature at the apex may be estimated, although not in a unique way. It is difficult to assign a unique “radius of curvature” to a point on a general 3D object (Bei et al., 2005; Constantinides et al., 2007; Ma et al., 2009; Tyulyukovskiy & Huber, 2007). Constantinides, et al. (2007) approached this problem when characterizing conospherical diamond probes by fitting osculating paraboloids to sub-areas within the AFM images, and assigning a local radius to the pixel located at the vertex of the paraboloid. Tyulyukovskiy & Huber (2007) characterized AFM-measured shapes of spherically machined diamond probes by fitting the projected-area function to an equation that described a sphere with increasing imperfection closest to the apex. In the current study, the radius of curvature was estimated by fitting parabolas to several evenly spaced profiles through the apex (Ma et al., 2009). The mean radius estimated by this method is 121 nm ± 13 nm for the lightly-used indenter tip and 1150 nm ± 14 nm for the heavily-used indenter tip. All measurement uncertainties in this chapter are one standard deviation.
Three-dimensional AFM rendering image of the lightly (a) and heavily (b) used indenters, and contour plot of the surface of the lightly (c) and heavily (d) used indenter shape in the near-apex region with 2 nm contour spacing after interpolation and filtering.
The AFM-measured sharp and blunt Berkovich probes were used to perform load-controlled nanoindentation experiments on the same<100>-oriented single-crystal tungsten specimen. For both indenters, ten indentation tests were performed as described in Sec. 3.2. All indentations exhibited a significant first pop-in event, but only six indentations using the sharp indenter and five indentations using the blunt indenter showed no plastic deformation during the initial unloading-reloading sequence, as described above. These are the only indentation tests that were analyzed further. Figure 3 shows load versus displacement data for two representative indentation experiments on tungsten, one using the sharp indenter and the other using the blunt indenter. In both sets of data, the nucleation event is seen clearly at the critical load, PCRIT. The indenter radius was estimated by a least-squares fit of the Hertzian relationship to data preceding PCRIT,
where an additional offset displacement, hOFF, is used to adjust for uncertainty in the point of first contact. A reduced elastic indentation modulus of
Load as a function of displacement for two representative indentation experiments on tungsten, one using the sharp indenter and the other using the blunt indenter. The critical load for plastic deformation, PCRIT, is shown on each. A Hertzian fit to the initial elastic loading portion is drawn. The statistical uncertainty in PCRIT is described in the text.
From the average indentation load at the first pop-in event (PCRIT = 233 μN ± 90 μN for the sharp indenter and PCRIT = 851 μN ± 131 μN for the blunt indenter) and the radius of the probe as inferred from fits to Eq. 3, the maximum shear stress, τMAX, may be estimated from Eq. 2, which assumes an isotropic material with Poisson’s ratio equal to 0.3. The test-by-test average τMAX, estimated at the displacement excursion, was τMAX = 16.5 GPa ± 1.6 GPa for the sharp indenter and 7.1 GPa ± 0.4 GPa for the blunt indenter. The τMAX for the sharp indenter agrees with our earlier nucleation shear stress estimates on W using a rigid indenter simulation with a scaled modulus (Ma et al., 2009; Zbib & Bahr, 2007; Bahr et al., 1998). The τMAX obtained using the blunt indenter is much smaller, only about 43 % of that from the sharp indenter. This difference in τMAX is consistent with nanoindentation size effects reported by other researchers (Morris et al., 2004; Shim et al., 2008). In those studies, the maximum shear stress under small spherical indenters at the first pop-in was found to be very high, on the order of the theoretical strength; for larger spheres, the maximum stress decreased with increasing indenter radius. The most likely explanation for this observed behavior is that the number of dislocations in the stressed volume increases with indenter radius. Thus, for sufficiently small indenters, the material in the highly stressed zone underneath the indenter is likely to be dislocation-free, requiring dislocation nucleation at the theoretical stress. As the indenter radius increases, there is an increasing likelihood that a nearby dislocation assists in the onset of plasticity. When a sufficiently large volume of material is probed, plasticity tends to be initiated by the motion of preexisting dislocations rather than by the nucleation of new dislocations.
Mason et al. (2006) and Schuh and Lund (2004) have suggested that pop-in events should have a rate-dependence wherein higher loading rates will generally correspond to higher pop-in loads. To examine this we have carried out a series of indentations into a model BCC system [Fe-3%Si by weight single crystal in the (100) orientation] using a Berkovich indenter similar to that shown in the remainder of this study. The indentations were made in the same Hysitron Triboindenter system, and regions of the sample were examined using scanning probe imaging prior to indentation. Between 50 and 100 indentations were made at loading rates of 5, 50, and 100 mN/s, and all samples that exhibited solely elastic loading prior to yield were selected. The cumulative probability of yield, as described by Schuh and Lund (2006), was used to determine the effect of loading rate on yield behavior. As shown in Fig. 4, there is no statistically significant effect of the load (and therefore stress) at yield due to indentation rate over the range tested. The experimentally observed results are similar to the results by Rajulapati et al. (2010), Wang et al. (2011), and Vadalakonda et al. (2006) in that the onset of plasticity is not significantly impacted by the loading rate for Ta (bcc), Mo (bcc), and Ni (fcc) single crystals, and polycrystalline W, Fe, and Ni. This suggests that the rate dependence observed by Mason et al. (2006) and Schuh and Lund (2004) in Pt may imply there are multiple processes that control the onset of plastic deformation, and that loading rate may impact some, but not all, of these processes.
Rate dependence on yield behavior in Fe-3%Si sample. No effect of loading rate is noted in this material for these rates, the cumulative fraction of yield points as a function of load at which the yield point occurs is indistinguishable over these loading rates.
In general, directly modeling an indentation experiment using the FEA should produce a more accurate estimate of the indentation stresses than the simple Hertz approximation. However, it should be understood that there are computational difficulties with 3D FEA which prohibit a completely faithful virtual representation of the experiment and the computational times must be kept manageable. Generally, the indenter is kept rigid to avoid a full interior meshing and consequent increase in simulation size. This is not a problem when simulating the indentation of materials that are much more compliant than diamond; however, the stiffness of tungsten is an appreciable fraction of that for diamond. Nevertheless, our earlier work on a sharp indenter showed that a simulation using a rigid indenter matched the experimental results extremely well when these results were scaled using the “effective modulus” (Ma et al., 2009).
Since tungsten is very stiff, the high loads encountered during indentation can produce appreciable elastic deformation of the indenter tip, and it is necessary to quantify these changes and determine their effect on the nanoindentation experiment. Fortunately, the advent of highly-parallel computer architectures has made it possible to complete large 3D FEA models within manageable computational time. In this research, both AFM-measured indenters were input into FEA models with 3D deformable elements as described in sec 3.3. All of the displacement measurements in the FEA simulations are measured with respect to the reference point of the indenter rigid plate (just like the experimental measurements) unless otherwise specified.
Figures 5 and 6 show the measured load-displacement curves for all of the analyzed indentation experiments using the sharp and blunt indenters, respectively. The first contact points in the experimental data were adjusted slightly as described in sec 4.2 and the dotted lines indicate the major pop-in location for each experiment. The FEA-simulated elastic deformation curves (with deformable AFM-measured Berkovich indenters) are plotted for comparison. The simulated load-displacement curves and all of the experimental plots were in good agreement with each other for both the sharp and blunt indenters. The deviation between some of the measured and simulated curves in Fig. 6 at large indentation depth (above 12 nm) is caused by small amounts of plastic deformation. In some cases, this occurred slightly before the major pop-in events which are the main emphasis in this paper.
The FEA-simulated Berkovich load-displacement data for both the sharp and blunt indenters are replotted in Fig. 7. These data may be analyzed as if they came from a physical measurement, and Hertzian fits (using Eq. 3) to these data are shown as smooth curves. The resulting Hertzian-estimated radii are 163 nm for the sharp indenter and 1160 nm for the blunt indenter. These radii were then used for additional FEA simulations using deformable parabolic indenters (designated FEA_Hertz simulations) and the results are shown as black open circles for the blunt indenter and black open diamonds for the sharp indenter. All of the load displacement curves are in good agreement for the blunt indenter as shown in Fig. 7. However, the FEA_Hertz simulation for the sharp indenter exhibited a stiffer contact than that predicted from the Hertz equation; by trial and error, an FEA_HERTZ simulation using a radius of 125 nm (plotted using open squares in Fig. 7) was found to match the simulated Berkovich P-h relationship very well. A similar discrepancy between a rigid-sphere FEA simulation and the Hertz solution was described previously (Ma et al., 2009).
Measured load-displacement curves for all analyzed indentation experiments using the sharp indenter and the corresponding FEA-simulated elastic deformation curves using the AFM-measured indenter. The dotted lines indicate the major pop-in location for each experiment.
Measured load-displacement curves for all analyzed indentation experiments using the blunt indenter and the corresponding FEA-simulated elastic deformation curves using the AFM-measured indenter. The dotted lines indicate the major pop-in location for each experiment.
All of the FEA-simulated and Hertzian load-displacement curves for both the sharp and blunt indenters.
Figures 5 and 6 demonstrate that the simulated load-displacement relationships for the Berkovich-probe indentation agree with the experimental indentation results, and Fig. 7 shows that the analytic Hertz approximation and simulations using a deformable parabolic indenter can produce comparable load-displacement results. The next step is to compare the resulting simulated Tresca stresses, which is defined as the maximum difference between principal shear, throughout the specimen for all of these analysis methods.
Table 1 lists the radii and maximum shear stresses derived from the experimental data and the FEA simulations for both the sharp and blunt indenters. Figures 8a and 8b show the contact-surface shear-stress contours for the sharp indenter (Fig. 8a) at an indentation depth of 11 nm, and the blunt indenter (Fig. 8b) at an indentation depth of 15 nm, from the FEA simulations using the 3D deformable Berkovich indenter. These depths correspond to the average indentation depths where pop-ins occurred in the experiments. The contours show the shapes of the indenter tips at small scale. Figures 8c and 8d are the corresponding perspective views that show the distributions of shear stress around the locations of maximum shear stress (τTr/2). The stress contours are set so that shear stresses larger than those predicted by the Hertz approximation are shown in grey. Thus, the grey-region for the sharp indenter in Fig. 8c has τTr/2 ≥ 16.3 GPa; for the blunt indenter in Fig 8d, this limit is τTr/2 ≥ 7.1 GPa.
The shear-stress contours for both Berkovich probes in Fig. 8 exhibit pronounced irregularities that correspond to the non-spherical probe shape. For the sharp indenter (see Figs. 8a and 8c), the average indentation depth for the first pop-in is deep enough so that the three-sided pyramid region affects the shapes of the stress contours. Also, the innermost contour (gray color in Fig. 8c) encompasses a large volume where the shear stress exceeds the Hertzian estimate for the same indentation load and depth. Finally, although the point of maximum shear stress is located below the sample surface, in agreement with Hertz contact theory, it is shifted slightly from the central axis and towards the surface. For the blunt indenter (see Fig. 8b and 8d), the stress contours do not exhibit a clear three-sided shape such as that seen from the sharp indenter. However, the pattern of high shear stress within the sample is extremely irregular and is strongly influenced by the irregular shape of the indenter. The sample volumes where the shear stress exceeds the Hertzian estimate (gray color in Fig. 8d) are concentrated in several small local regions directly underneath the contact surface. Thus, the irregular shape of the measured indenter produces numerous volumes of highly localized shear stress that are higher than those produced by a spherical probe with an indistinguishable load-displacement relationship. In addition, the locations of these highly-stressed regions are not generally on the indentation axis and they occur much closer to the sample surface than predicted by the Hertz theory.
Lists the radii and maximum shear stress derived from the experiment and FEA simulation for both the sharp and blunt indenters.
Figure 9a plots the maximum shear stress in the sample as a function of indentation load for the 3D Berkovich FEA simulations, the Hertz predictions, and the FEA Hertz simulation for the sharp and blunt indenters. Figure 9b plots the percent deviation of the Hertzian and FEA Hertzian predictions from the more rigorous 3D Berkovich FEA simulations. It can be seen that the maximum shear stress from all of the Hertzian and FEA Hertzian estimates are much smaller than those found from direct FEA simulation of the real indenter probe.
For the sharp indenter, all of the shear stress curves exhibit a smooth increase with increasing indentation load, consistent with the generally smooth character of the indenter tip shape. However, the FEA Hertz result deviates from the 3D Berkovich simulation by about 5 % to 8 % at the experimentally found pop-in load of PCRIT = 233 μN ± 90 μN. At smaller loads, this deviation is even greater, increasing to around 20 % at an indentation load of about 6 μN. Meanwhile, the Hertzian estimate of the radius (R=163 nm) from the experimental data is much larger than those obtained from the direct AFM measurement (R = 121 nm ± 13 nm) and FEA Hertz fitting of the experimental load displacement curves (R = 125 nm). This overestimation of the sharp indenter tip radius introduces a substantial bias to the Hertz estimate of the maximum shear stress. The maximum shear stress is underestimated by more than 25 % at the pop-in loads and increases to more than 30 % at lower loads.
FEA Contact surface shear stress contours for both the sharp (a) and blunt (b) indenters and their perspective views ((c) sharp and (d) blunt indenters). The indentation depths were 11 nm for the sharp indenter and 15 nm for the blunt indenter.
For the blunt indenter, the tip radii obtained from the AFM measurement, the Hertzian estimate and the FEA Hertz simulation are all in good agreement (see Table 1). Thus, it is not surprising that the maximum shear stresses predicted by the Hertz approximation and the FEA Hertz simulation are nearly identical, as shown Fig. 9. However, the irregularity of the real indenter tip shape produces large stress excursions away from the more macroscopic stress contours. Thus, the Hertzian approximation and the FEA Hertz simulation both underestimate the maximum shear stress by more than 50 % at the pop-in loads (PCRIT = 851 μN ± 131 μN) and this error increases to more than 70 % at smaller loads. This deviation is much larger than that found for the sharp indenter.
a) The maximum shear stress in the sample as a function of indentation load for the 3D Berkovich FEA simulations, the Hertz predictions, and the FEA Hertz simulations for the sharp and blunt indenters. B) The corresponding percent deviation of the Hertzian and FEA Hertzian predictions from the more rigorous 3D Berkovich FEA simulations.
The obvious irregularity of the blunt indenter is reflected in the distribution of maximum shear stresses within the sample (see Fig. 8b and 8d) and the behavior of τMAX with increasing indentation load (see Fig. 9). In Fig. 9a, the maximum shear stress under the blunt 3D Berkovich indenter (square with line) shows a distinct two step behavior. Thus, the stress increases fairly rapidly up to an indentation load of about 110 μN (about 4 nm indentation depth) where it abruptly slows. At an indentation load of about 538 µN (about 11 nm indentation depth) τMAX again increases rapidly with a tapering slope. Figure 10 shows two perpendicular cross-section profiles (passing through the apex) of the 3D AFM-measured blunt indenter as shown in Fig. 2d. These cross sections show pronounced changes in curvature at a variety of distances from the apex. It is probable that such local shape changes are largely responsible for the irregular behavior of the blunt 3D Berkovitch τMAX-load plot.
Two perpendicular cross-section profiles (passing through the apex) of the 3D AFM-measured blunt indenter as shown in Fig. 2d.
As shown in Fig. 2, new diamond Berkovich indenters have a faceted-pyramid shape with an approximately spherical end cap. Theoretically, the transition from the spherical cap to the pyramid geometry can be approximated using a simple geometrical model, where the spherical region is tangentially enclosed within an open-cap conical geometry, giving:
h*/R = 1 – sinθ,
where h* is the transition depth and θ is the semi-angle of the conical part (Cheng & Cheng, 1998; VanLandingham et al., 2005; Constantinides et al., 2007). For an axisymmetric approximation of a Berkovich probe, θ = 70.3º and h* = 0.0585 R. Thus, the sharp indenter with R = 125 nm has a predicted transition at h* = 7.3 nm, which is smaller than any of the measured pop-in depths and explains the pronounced 3-fold symmetry of the surface-stress contours in Fig. 8a. The predicted transition for the blunt indenter with R = 1160 nm is considerably larger at h*= 67.86 nm. In reality, after frequent indention, the diamond indenter tip will be worn on the apex (see Fig. 2), decreasing the transition depth. Nevertheless, a transition to the pyramidal region was not observed for the blunt indenter.
The approximate sphere-cone load-displacement relationship also follows a generalized power-law relationship,
where C and m are constants. If fitted to Eq. 5, the P-h relationships from the 3D Berkovitch simulations have a best-fit parameter of m = 1.59 for the sharp indenter and m = 1.54 for the blunt indenter. Because m = 1.5 (Hertz) for a sphere, and m = 2 for a cone, an intermediate m is generally interpreted to mean that the spherical and conical parts of the indenter were both in contact with the material at the point of pop-in. This is certainly true for the sharp indenter as described above.
Next the contact radius, a, was estimated by the Hertzian relationship (Johnson, 1999),
Johnson (1999) observed that “… doubt must be cast on the Hertz results if the ratio a/R becomes too large. With metallic bodies, this restriction is ensured by the small strains at which the elastic limit is reached…”. Yoffe (1984) shows that the surface displacement field, including the effect of lateral displacements in the contacted zone, for a Hertizan contact begins to deviate significantly from that of a sphere at normalized contact radius a/R ≈ 0.2 when the Poisson’s ratio of the indented material is υ ≈ 0.3. The normalized contact radius at the average pop-in position for the blunt indenter is about a/R ≈ 0.11 for the estimated radius of 1160 mm. However, the sharp indenter shows roughly a/R ≈ 0.25 for the Hertz estimated radius of 163 mm and a/R ≈ 0.34 for the AFM measured radius R = 121 nm. This significantly exceeds Yoffe’s rue for the limits of Hertizian theory (Li et al., 2009).
Finally, it is well known that diamond indenters will exhibit substantial elastic deformation when indenting materials like tungsten that have an appreciable fraction of the diamond stiffness. This elastic deformation can be quantified by subtracting the displacement of the indenter tip from the displacement of the indenter base, here referred to as the elastic depth difference. Since almost all of the indenter deformation occurs near the tip where the cross-sectional area is small, the elastic depth difference is a well-defined quantity as long as the simulated base is far enough away from the tip as described in section 3.3. Figure 11 shows the elastic depth difference plotted as a function of indentation load for the 3D Berkovich FEA simulations and the FEA Hertz simulations for both the sharp and blunt indenters. At a given indentation load, the sharp indenter deforms the most because of its smaller cross section. This difference between the sharp and blunt indenters increases nonlinearly with the indentation force. The 3D Berkovich indenters deform more than the parabolic FEA Hertz indenters, but this difference is very small.
The amount of elastic deformation exhibited by the sharp indenter in Fig. 10 is an appreciable fraction of the total indentation depth. For example, the average pop-in depth for the sharp indenter was about 11 nm, with a corresponding load of about 196 μN. At this load, the elastic deformation depth from Fig. 11 is almost 3.7 nm, demonstrating that the shape of the indenter is being severely modified by the elastic stresses.
The elastic depth difference as a function of indentation load for the 3D Berkovich FEA simulations and the FEA Hertz simulations for both the sharp and blunt indenters.
Hertzian elastic contact theory is broadly used by the nanoindentation community to estimate the stresses required for interesting events such as pop-ins. However, the Hertzian approximation makes a number of assumptions that may not be valid for real systems; these include assumptions concerning the sphericity of the indenter tip, the flatness of the specimen, and a low ratio of indentation contact radius to indenter tip radius. In real systems, the indenter tip can be markedly irregular at nanometer length scales, the sample surface can have an irregular shape, and the indentation depth for the first pop-in can be deep enough so that the non-spherical sides of the indenter can contact the specimen. Additional problems that can occur include quantitative calibration of the nanonewton-level loads and determination of the actual indentation depth and contact area.
In this review chapter, we described some of the challenges and limitations for extracting the initial plastic yield stress from nanoindentation with the spherical indenter tip assumption, and we assessed possible errors and pitfalls of the Hertzian estimation of initial plastic yield at the nanoscale. Ultimately, solving these problems will require the development of new standard test methods and standard reference materials and this chapter describes some of our going work at NIST in that direction, including 1) measurements of the sample and indenter tip shapes using an AFM system that uses laser interferometry (traceable to NIST primary reference standards) for all distance measurements, 2) development of a custom nanoindentation instrument for nanoindentation where load and displacement are again traceable to primary NIST reference standards, 3) 3D, submicrometer resolution, synchrotron X-ray measurements of the local elastic strain tensor, crystallographic orientations, and dislocation density throughout of indentation volume both before and after indentation, and 4) combined FEA and molecular dynamics modeling of the experiments using the experimental measurement as direct inputs.
A preliminary combined experimental and FEA study was described where <100>-oriented single-crystal tungsten was used to examine the role of the indenter tip shape on nanoindentation determinations of the maximum shear stress at the dislocation nucleation point (plastic yield stress). The near-apex shape of two real Berkovich indenters, one slightly and another heavily used, were measured using AFM. These shapes were then used as a “virtual” indentation probe in a 3D FEA simulation of nanoindentation experiments on <100>-oriented single crystal tungsten. Independently, instrumented nanoindentation experiments were carried out with the real indenters on <100>-oriented single crystal tungsten. The agreement between the FEA and experimental load-displacement curves was excellent, and the FEA simulations provide a validated point-of-comparison for more conventional model predictions.
Analytic Hertzian fits and FEA simulated Hertzian simulations were also carried out, and the resulting fits to the load displacement data were excellent, even though there were substantial deviations from a spherical or parabolic geometry. These results demonstrate that even a highly irregular probe will produce an elastic load-displacement relationship that could be perceived as having originated from a spherical contact. That is, although the load-displacement data could reasonably be described by a fit using the Hertzian relationship, the contacting surfaces certainly did not fit the Hertzian criteria (specifically, that of contacting paraboloids) and the fitted radius for the small indenter was off by 35 %. Not surprisingly, the simulations showed that the irregular probes generated shear stresses in the sample that were significantly different, both in magnitude and location, from those generated with a true spherical probe. The effect of asphericity and other irregularities were dominant, focusing the indentation force into a smaller volume, and thereby increasing the maximum shear stress in the body over that produced by an ideal spherical contact. Hertzian estimates of the maximum shear stress in tungsten at the pop-in loads were smaller than those determined from FEA by over 25 % for the lightly used indenter and over 50 % for the heavily used indenter.
The authors would like to thank Dr. Dylan Morris for measuring the Berkovich indenter tip shapes, Ms. Stefhanni Jennerjohn for conducting nanoindentation experiments and Dr. Yoonkap Kim, whose work was supported by the National Science Foundation under grant number DMR-0907378, for the rate dependent experiments. This research work is funded and supported by the National Institute of Standards and Technology.
The left side gallbladder (LSG) is a very rare alteration defined by the attachment of the gallbladder to the left lobe of the liver at the right side of the ligament tears.
Since Hochstetter’s first description in 1886 [1], about 150 cases have been reported in the literature. The attempts to explain the cause of this anomaly have been different, but the numerous variations described do not allow a clear definition of its origin. Although this debate is still open, it is of considerable importance to know that LSG is frequently associated with alterations of both the portal branches and the intrahepatic biliary tree. The association of these anomalies, therefore, represents an important risk, especially if surgical treatment is necessary. There are two cases in which surgical treatment may be required: the first is gallbladder stones and in particular acute cholecystitis, and the second is the need for liver resection surgery. In the first case, the diagnosis of gallstones and cholecystitis is made only with ultrasound, but often this method does not describe the anomaly of the LSG. In fact, in most cases described, the diagnosis is made intraoperatively, making the surgical treatment problematic and risky due to the lack of correct anatomical knowledge. In the case of liver resection, the diagnosis is made before surgery, highlighting anatomical variations that require complex dissection strategies. Knowledge of the anomalies associated with a LSG can be of considerable help in preventing serious complications.
The real estimation of this anomaly is very difficult because most of the published articles are case reports and only a few of them have high numbers of cases. The literature analysis revealed 114 articles concerning LSG: 89 authors describe 1 case only, 11 present 2 cases, 3 present 3 cases, 3 present 4 cases, 3 describe 6 cases, 1 describes 7 cases, 2 describe 9 cases, 1 describes 10 cases, and 1 describes 26 cases, with a total of 211 cases (Table 1).
Reference | Author | Number of cases | Reference | Author | Number of cases | Reference | Author | Number of cases |
---|---|---|---|---|---|---|---|---|
[2] | Abe | 1 | [3] | Ishii | 1 | [4] | Noritomi | 1 |
[5] | Abongwa | 1 | [6] | Iskandar | 1 | [7] | Ogawa | 1 |
[8] | Alharthi | 1 | [9] | Jona | 1 | [10] | Ogino | 1 |
[11] | Almodhaiberi | 1 | [12] | Jung | 1 | [13] | Oshima | 1 |
[14] | Aoki | 1 | [15] | Kanazumi | 2 | [16] | Ozeki | 26 |
[17] | Asonuma | 4 | [18] | Kawai | 1 | [19] | Pradeep | 4 |
[20] | Banchini | 1 | [21] | Kehr | 1 | [22] | Quah | 1 |
[23] | Banzo | 1 | [24] | Kelly | 1 | [25] | Qureshi | 1 |
[26] | Bender | 1 | [27] | Kim | 4 | [28] | Rafailidis | 1 |
[29] | Bonomo | 1 | [30] | Kinoshita | 2 | [31] | Reddy | 1 |
[32] | Chrungoo | 2 | [33] | Kubo | 1 | [34] | Rocca | 1 |
[35] | Chuang | 1 | [36] | Lee | 10 | [37] | Rozsos | 1 |
[38] | Chung | 1 | [39] | Leone | 1 | [40] | Saafan | 1 |
[41] | Cirla | 1 | [42] | Lin | 1 | [43] | Sadhu | 1 |
[44] | Colovic | 2 | [45] | Maetani | 1 | [46] | Sakihara | 1 |
[47] | Dhulkotia | 1 | [48] | Makni | 1 | [49] | Schiffino | 1 |
[50] | Donthi | 1 | [51] | Masood | 1 | [52] | Seiberman | 1 |
[53] | Ergun | 1 | [54] | Matsumoto | 1 | [55] | Shen | 1 |
[56] | Etter | 1 | [57] | Matsumura | 1 | [58] | Shimizu | 1 |
[59] | Feldman | 1 | [60] | Matsuoka | 1 | [61] | Shirono | 1 |
[62] | Fujita | 1 | [63] | Mazzamurro | 1 | [64] | Siebermair | 1 |
[65] | Fujita | 1 | [66] | McGowan | 1 | [67] | Si-Youn | 3 |
[68] | Fukuda | 1 | [69] | Mendoza-Calderon | 2 | [70] | Strong | 1 |
[71] | Funakoshi | 1 | [72] | Mizray | 1 | [73] | Surjan | 1 |
[74] | Futamura | 1 | [75] | Mohammed | 6 | [76] | Szanto | 1 |
[77] | Gondra | 1 | [78] | Moo-Young | 1 | [79] | Tachibana | 2 |
[80] | Gui | 1 | [81] | Moravik | 1 | [82] | Takemura | 1 |
[83] | Hasbahceci | 1 | [84] | Moriyama | 2 | [85] | Takiguchi | 1 |
[86] | Herrington | 1 | [87] | Naganuma | 9 | [88] | Tomiyama | 1 |
[89] | Hirohata | 1 | [90] | Nagay | 3 | [91] | Uesaka | 2 |
[1] | Hochstetter | 1 | [92] | Nagendram | 1 | [93] | Velimezis | 7 |
[94] | Hopper | 1 | [95] | Nakakubo | 1 | [96] | Watanabe | 1 |
[97] | Hsu | 9 | [98] | Namikawa | 1 | [99] | Wong | 1 |
[100] | Huang | 6 | [101] | Nastos | 2 | [102] | Wu | 2 |
[103] | Hwang | 3 | [104] | Nemours | 1 | [105] | Yamazaki | 1 |
[106] | Iaccarino | 1 | [107] | Newcombe | 1 | [108] | Yu | 2 |
[109] | Idu | 6 | [110] | Nguyena | 1 | [111] | Zografos | 1 |
[112] | Ikoma | 1 | [113] | Nishio | 1 | [114] | Zoulamoglou | 1 |
Number of cases reported in the literature for each author.
The incidence is variable and remains always below 0.3%: Idu et al. [109] describe 5 cases of LSG on 1764 cases of cholecystectomy (0.3%), Nagai et al. [90] 3 cases out of 1621 (0.2%), Sadhu et al. [43] 1 out of 1258 (0.08%), and Rozsos et al. [37] 1 out of 2536 (0.04%).
Naganuma et al. [87], in a series of 67.994 patients studied with ultrasound, found 18 cases of abnormal gallbladder position with an incidence of 0.026%. These included retrohepatic, supraphrenic, and floating gallbladder, and only nine cases of LSG, with an incidence of 0.013%. However, ultrasound is not the main exam to make the diagnosis. Pereira et al. [115] show that ultrasound has a positive predictive value of only 2.7% and that in 81.1% LSG is initially detected at surgery. Also, Lee et al. [36], in his series of 10 cases operated on for cholecystic stones or acute cholecystitis symptoms, describes the discovery: intraoperative in 8 cases and as an incidental finding on abdominal computed tomography (CT) in 2 cases.
The CT scan is the main examination capable of making a correct diagnosis. However, as reported in the review of Pereira et al. [115], CT has a positive predictive value of only 60%. This can be explained by the simultaneous proximity of the two hepatic lobes to the gallbladder, simulating the contact and not the adhesion to the left lobe of the liver. Considering the position of the gallbladder, if it is in the normal site, its right margin generally appears free (Figure 1A). If the gallbladder is positioned on the left, its left margin may appear free (Figure 1B) or included in the left lobe with a space between the two hepatic lobes (Figure 1C), making diagnosis easy. If the gallbladder appears between the right and left hepatic lobes, the diagnosis may remain unknown (Figure 1D) mimicking hypertrophy of the left hepatic lobe as also reported by Banchini et al. [20] and Iskandar et al. [6].
CT scan visualization of the gallbladder. (A) Normal right side gallbladder attached to the right lobe; (B) left side gallbladder attached to left lobe with empty space between gallbladder fundus and right lobe; (C) left side gallbladder with fundus surrounded by left lobe and empty space between right lobe and left lobe; (D) left side gallbladder interposed between right lobe and left lobe, mimicking hypertrophy of right lobe. RL: right lobe; LL: left lobe; GB: gallbladder.
However, the CT remains the main examination as it can provide the anatomical portal and arterial variations that, as we will see, frequently occur in association with this anomaly.
The embryological theories of LSG development are numerous and complex, but we can distinguish two main ones. The first theory concerns an alteration that only concerns the development of the gallbladder and a second one that concerns the development of the central portion of the liver and consequently the malpositioning of the gallbladder.
In the theory of gallbladder development proposed by Gross [116], there are two ways in which gallbladder can develop:
The first modality suggests a normal growth of the gallbladder migrating from the right side of the liver to the left side and attaching to the left lobe. In this case, the cystic duct originates from the right side of the bile duct and moves anteriorly and to the left of the liver pedicle.
The second modality hypothesizes the formation of the gallbladder directly on the left side of the bile duct with its positioning directly under the left hepatic lobe. In this case, the cystic duct originates in the left part of the bile duct.
The literature defines as a true LSG when it is located on the right side of the round ligament, and the cystic duct is inserted on the left side of the bile duct. The presence of true LSG, described in this way, represents a remarkably rare event, constituting 4.3% of the cases of left side gallbladder [18], and most of the cases described are therefore a different alteration.
The ways in which the cystic duct enters the biliary duct are numerous even in the absence of the left gallbladder, and as reported by Sarawagi et al. [117], the normal right conjunction is present in 51.1% of cases, while its medial (left) insertion can be presented in 16.1%. Pereira et al. [115], in his review, describe the insertion of the cystic duct on the right side 65.6%, on the left side of common hepatic duct 9.5%, left hepatic duct 9.5%, on the right hepatic duct 7.6%, and in a branch of the right hepatic duct 2.4%. Six patients had other minor biliary anomalies, and one had a duplicate common bile duct (CBD).
The second theory about the development of the central portion of the liver was first described by Ozeki in 1987 [16] and later defined by Nagai in 1997 [90] as “right side round ligament.” This theory is associated with numerous alterations in intrahepatic anatomy and appears to account for over 95% of cases in which a left side gallbladder is present.
To understand this complex theory, it is necessary to investigate some moments of fetal evolution, in which the persistence of a right umbilical vein and/or hypertrophy of the left portion of the liver seem to be associated. According to the embryological studies of Arey [118] when the embryo measures 6 mm, there are at the same time two vitelline’s veins, one on the right side and one on the left side, everyone having a branch that enters the liver. The two veins inside the liver have branches that join them together. When the embryo reaches the size of 7 mm, the right vein goes into atrophy leaving the left side predominant. It is assumed that, if the right side vein does not atrophy, there is a persistence of this umbilical vein either extrahepatic or intrahepatic. In this case, the opposite process to the previous one occurs, with atrophy of the left side and hypertrophy of the right side (Matsumoto’s hypothesis) [119]. When this happens, we have the positioning of the gallbladder in the left portion of the liver.
Although this theory seems to be the most accreditable, there are cases in the literature of persistence of the right umbilical vein with the gallbladder positioned normally, showing how this theory does not always prove to be real [42].
Lucidarme et al. [120] describe this variation in portal anatomy as a defect in the evolution of the central portion of the liver, in which the right and left parts join together in a variable way, renaming this as “Fusion of hepatic plans.” Considering these different descriptions, we can say that there is an anomaly in the mechanism of persistence or atrophy of the right umbilical vein and/or the liver surrounding it.
Based on these descriptions, we propose to combine the mechanism of atrophy of the central part of the liver with persistence of the right umbilical vein and the mechanism of fusion of the plans, trying to verify how these can give rise to different anatomical variations. In this way, we create a classification that can explain how this anomaly can occur.
As seen before, during fetal development, the right and left umbilical veins have a Y shape with one arm entering the liver and one passing laterally to it. At this stage, there is a set of vessels inside the liver that connect the two internal branches of the intrahepatic umbilical veins. This connection forms a venous conduit that we will call “intrahepatic right umbilical vein” (IRUV), which flows directly into the ligament of Arantius or enters the right umbilical vein at Rex’s recess. The extrahepatic portion of the right umbilical vein can enter the liver in a variable position between segment 5 and segment 4b. Imagine looking at the liver from its lower face, corresponding to segments 5 and 4b, we can divide this area into four parts as shown in Figure 2: the superficial part, corresponding to the acute margin of the liver, in the portion of segment 5 (Figure 2: yellow segment 5), and the portion of segment 4b (Figure 2: blue segment 4b) and a deep part toward the hilum of the liver (Figure 2: red segment 5; green segment 4b).
Liver with the central portion divided into four parts: the superficial part of segment 5 in yellow; the lower part close to the hilum in red; the superficial part of segment 4 in blue; and the lower part of segment 4 close to the hilum in green. (A) Superior view and (B) inferior view.
The involution of the right umbilical vein may affect the surrounding hepatic parenchyma differently, depending on its extension and site, determining unusual alterations.
If atrophy affects the superficial parts (Figure 2: yellow and blue), the result is a volumetric reduction in the corresponding liver segments. If the atrophy affects the deep part toward the hilum (Figure 2: red and green), we can have alterations that cause variations of the portal vein and the biliary tract. Complete or partial atrophy can also explain in various ways incomplete forms of gallbladder malposition such as the medioposition described by Hsu et al. [97].
The different possibilities with which the right umbilical vein can evolve are four, and we describe them as follows (Figure 3):
Schematic evolution of right and left umbilical veins with different evolution processes. (Type A) Normal evolution with the persistence of the left umbilical vein. (Type B) Persistence of the left umbilical vein and intrahepatic umbilical vein. (Type C) Persistence of the right umbilical vein and persistence of the intrahepatic umbilical vein with atrophy of left umbilical vein. (Type D) Persistence of the right umbilical vein only (in red: right umbilical vein; in orange: intrahepatic umbilical vein; in yellow: left umbilical vein).
Type A: complete atrophy of the right umbilical vein and normal left umbilical vein.
Type B: atrophy of the right external umbilical vein and persistence of the IRUV.
Type C: persistence of the IRUV and extrahepatic umbilical vein and atrophy of the left umbilical vein.
Type D: persistence of the right extrahepatic umbilical vein and atrophy of the left umbilical vein.
Types C and D represent the cases in which a persistence of the right umbilical vein occurs instead of the normal left umbilical vein, classifying this as “true right side round ligament.”
Combining these variations with how lower liver atrophy can occur, the following classifications can be determined (Figure 4):
Combination of the evolution process of right and left umbilical veins with atrophy mechanism of segment 5 and 4 (in red: right umbilical vein; in orange: intrahepatic umbilical vein; in yellow: left umbilical vein; in light blue: portal vein and its branches; and in blue: area of hypotrophy). A1-B1-C1-D1 corresponding to atrophy of segment 5 area; A2-B2-C2-D2 absence of liver atrophy; A3-B3-C3-D3 corresponding to atrophy of segment 4 area.
Type A:
A1: The right umbilical vein goes into complete atrophy and is accompanied by an excessive involution of segments 8-5-4b. In this case, the gallbladder will present to the left of the round ligament.
A2: The right umbilical vein goes into atrophy completely, and there is no parenchymal atrophy. This is the normal anatomy that is described.
A3: The right umbilical vein goes into atrophy completely and is accompanied by involution of segment 4b with hypotrophy of the latter.
Type B:
B1: The right external umbilical vein goes into atrophy, and the right intrahepatic umbilical vein persists. An involution in the segment 5 site is associated. In this case, there is the presence of left side gallbladder with the persistence of the right portal branch only for segments 6 and 7 and hypertrophy of the portal branch of segment 4.
B2: The right external umbilical vein goes into atrophy, and the right intrahepatic umbilical vein persists. There is no involution of the hepatic parenchyma with normal portal trifurcation.
B3: The right external umbilical vein goes into atrophy, and the right intrahepatic umbilical vein persists. It is associated with segment 4b hypotrophy. This determines the absence of the left portal branch and vascularization of segments 2 and 3 from the right portal branch through the persistent right umbilical vein portion.
Type C:
C1: The right extrahepatic and intrahepatic umbilical veins persist with atrophy of the left umbilical vein. An involution in the segment 5 site is associated. In this case, there is the presence of left side gallbladder with the persistence of the right portal branch only for segments 6 and 7 and the absence of segment 5 or 5 and 8. The vascularization of segments 2 and 3 and 4 occurs through the persistent right intrahepatic umbilical vein portion.
C2: The right extrahepatic and intrahepatic umbilical veins persist with atrophy of the left umbilical vein. No involution of the hepatic parenchyma occurs. The vascularization of the liver is arched from right to left giving, in sequence, the branch for segments 6 and 7, that of segments 5 and 8 and ending with the vascularization of segments 2 and 3 and 4 through the persistent intrahepatic right umbilical vein.
C3: The right extrahepatic and intrahepatic umbilical veins persist with atrophy of the left umbilical vein. An involution in the segment 4b site is associated. The vascularization of the liver is arched from right to left giving the branch for segments 6 and 7, the branch for segments 5 and 8 and ending with the vascularization of segments 2 and 3 through the persistent intrahepatic right umbilical vein.
Type D:
D1: The right extrahepatic umbilical vein persists with atrophy of the left umbilical vein. An involution in the segment 5 site is associated. In this case, there is the presence of left side gallbladder with the persistence of the right portal branch that vascularizes segments 6 and 7 and the absence of segment 5 or 5 and 8, and the left portal branch that vascularizes segments 2 and 3 and 4.
D2: The right extrahepatic umbilical vein persists with atrophy of the left umbilical vein. There is no involution of the hepatic parenchyma. In this case, the right portal branch that vascularizes segments 6–7 and 5–8 and the left portal branch that vascularizes segments 2 and 3 and 4 persist.
D3: The right extrahepatic umbilical vein persists with atrophy of the left umbilical vein. An involution is associated with segment 4b. In this case, the right portal branch that vascularizes segments 6–7 and 5–8 and the left portal branch that vascularizes segments 2 and 3 persist.
The principle of the fusion of the planes is necessary to be added at the classification performed by rotating or increasing the volume of one of the two hepatic lobes.
In this way, we can reposition the ligament tears in the position in which it is located once the complete fetal development has taken place. To clarify, we have modified the left liver portion by rotating it counterclockwise as shown in Figure 5.
Combination of the evolution process of right and left umbilical veins with atrophy mechanism of segments 5 and 4, associated rotation and fusion of the right and left liver.
We can see how this classification allows us to catalog the cases of literature described or represented with images, the portal modifications found: Kawai et al. [18] describe the absence of the left portal vein and a branch for the left liver originating from the right portal branch that we identify as an atrophy of left umbilical vein Type C1 (Figure 6a); Nagai et al. [90] describe a case with the absence of the right anterior portal branch, corresponding to Type B1, and the other absence of the left portal branch that we call Type D1 (Figure 6b); Lin [42] describes tree cases of the absence of the right anterior portal branch, corresponding to Type B; Maetani et al. [45] also describe two cases that we classify as Type B1 and one case with the absence of the right anterior portal branch and a vascularization of segments 5 and 8 from the left portal one that we consider Type B2 (Figure 6c); Banchini et al. [20] describe the agenesis of segments 5 and 8 with the absence of the right anterior portal branch, Type A1 (Figure 6d). On the contrary, the classification we propose differs from Shindon’s classification [121] in defining the true right side ligamentum teres. Comparing the three types of portal bifurcation listed by Shindon, we consider the “independent right lateral” one as Type A1, the “bifurcation” one as Type D, and “trifurcation” one as an intermediate of Types A1–A3 (Figure 7), concluding that only the “bifurcation type” could be considered as “true right side ligamentum teres.” It is evident that these three variables of fusion, atrophy, and umbilical vein evolution can be combined in a considerably higher number of ways and may result in intermediate presentations. Likewise, using this principle, we can also hypothesize how the biliary tract variations occur. Even if we will not deal in this chapter, we underline that, as evidenced by Nishitai et al. [122], the Biliary tree could differ a lot from the corresponding portal branches, making this a further challenge.
Schematic representation published in the literature with comparison and reclassification with our nomenclature. (a) Kawai description (12) corresponding to Type C; (b) Nagai description (3) corresponding to Types B1 and D1; (c) Maetani description (20) corresponding to Type B1-B1-B2; and (d) Banchini description (9) corresponding to Type A1.
Schematic representation of Shindon’s classification (21) with comparison and reclassification with our nomenclature. (A) “Type A Shindon” corresponding to Type A1; (B) “Type B Shindon” corresponding to Type D; (C) “Type C Shindon” corresponding to intermediate Types A1–A3.
Considering the numerous possibilities with which a left gallbladder can present, we recommend an accurate study of the entire hepatic anatomy, with the need to recognize the portal, arterial, and biliary changes, in all patients presenting this diagnosis in the preoperative setting.
As pointed out in the previous paragraph, the presence of LSG is associated with a high number of variations both internal and external to the liver, regardless of the type of classification we want to use.
However, the presence of this anomaly does not seem to be associated with either a particular clinical manifestation or cancer, thus representing a simple anatomical variation presents in the population. The absence of symptoms makes the diagnosis of LSG an occasional event, consequent to its finding during investigations performed for other factors. Likewise, as in the normal population, gallstones follow the physiological mechanism of formation. However, gallbladder malpositioning does not seem to change the afferent pain pathways, and, as reviewed by Iskandar et al. [6] on 32 articles, the related symptoms are those of biliary colic or classic acute cholecystitis, with pain in the right upper quadrant or epigastric pain.
The diagnosis of gallbladder gallstones is mostly performed with ultrasound, but it has a very low diagnostic capacity in case of unknown LSG. Therefore, patients presenting with symptoms of biliary colic or acute gallbladder cholecystitis have a high probability to find this anomaly only during surgery. This condition exposes the patient and the surgeon to considerable risk. In the literature, cases of biliary lesions during cholecystectomy in LSG range from 4.4 [115] to 7.3% [5].
Laparoscopic cholecystectomy does not seem to be contraindicated, but some precautions are necessary to avoid risks of complications. Many authors advocate different techniques in trocar placement or in the patient’s position, but this strategy could be applicable only in the case of preoperative diagnosis.
Anyway, the main risk factor seems to be the passage of the gallbladder anterior to the liver pedicle. This rotation moves Calot’s triangle from horizontal and lateral to a vertical and anterior position, bringing the gallbladder closer to the biliary tract (Figure 8).
Blue gallbladder; green cystic duct and biliary way; brown liver. (A) Normal gallbladder frontal view; (A1) normal gallbladder left lateral view; (B) left side gallbladder frontal view; and (B1) left side gallbladder left lateral view.
In normal cholecystectomy, the opening of the Calot allows moving the gallbladder and cystic duct away from the biliary tract. This isolation is performed with a dissection directed from the superficial to the deep plane and is done by pulling the gallbladder laterally. The dissection finishes posterolaterally by finding an area free of tissue, corresponding to the posterior side on Calot’s triangle. In the case of LSG, if we perform the dissection of the Calot using this method, we risk finding a posterior plane occupied by the biliary tract and liver peduncle instead of a free one (Figure 9).
(A) Normal gallbladder frontal visual; (A1) normal gallbladder left lateral visual; (B) left side gallbladder frontal vision; (B1) left side gallbladder left lateral. Visual red arrow: direction of dissection in Calot’s triangle.
Taking into account this condition, it is necessary to look for a dissection modality that allows maintaining the distance from the biliary tract and the hepatic peduncle. We advocate performing laparoscopic cholecystectomy with fundus first technique [24] to achieve a proper distance from the hepatic pedicle. Once the body of the gallbladder is detached from the liver surface and Calot’s triangle is joined, we recommend to follow the dissection close to the gallbladder border. The border dissection allows minimizing the removal of peripedicle fat tissue, avoiding unintended biliary duct discovery. In order to augment the distance from the pedicle and the biliary way, once the peritoneum of the Calot has been opened, it is advisable to pull the gallbladder on the lateral side to horizontalize the triangle itself. This traction can move the gallbladder on the right side of the hepatic pedicle, repurposing the normal anatomy. After this mobilization, it could be useful to apply the Strasberg criteria for the visualization of all structures [123], so that the cystic duct and cystic artery can be dissected and clipped distant from the biliary tract, after their recognition. In case of doubt, it is useful to perform an intraoperative cholangiography or visualize the biliary tract with the indocyanine green or finally proceed with the conversion to have a direct view.
While accurate dissection and recognition of structures in cholecystectomy can prevent iatrogenic lesions, it is different in the case of liver resections. The anatomical variations that can occur are so high that a detailed study of portal, arterial, and biliary structures is mandatory. This necessity stems from the fact that the portal variations may not correspond to arterial or biliary anomalies, and therefore, all three of these structures must be considered separately. CT scan could be sufficient for portal and arterial study, and in particular, 3D CT could be particularly effective. On the opposite, the CT scan is not sufficient to demonstrate biliary variations, and for this reason, a magnetic resonance cholangiopancreatography is mandatory before elective major hepatectomy to ensure patient’s safety [122].
Defining the types of resection is too complex, and therefore, each patient will require an on-demand treatment depending on the anomalies found. The main issue to keep in mind is that, in the case of LSG, liver supply can be sustained by only a single portal branch, as pointed out by Hsu et al. [97]. This eventuality is characterized by an arch shape of the portal vascularization visualized on CT scan, exposing the risk of extending resection to most of the liver itself.
The approach to LSG seems to have a significant clinical implication, augmenting the risk of complications in both liver resection and cholecystectomy. On one side, even if the probability to perform liver resection in LSG is very low, the risk is related to the major intrahepatic modification, on the other, considering cholecystectomy one of the most frequent interventions in surgery, the risk is related to the high probability to treat LSG as symptomatic gallbladder discovered intraoperatively.
Left side gallbladder is a rare and little known anomaly that is diagnosed, in most cases, during cholecystectomy for biliary colic or cholecystitis symptoms. The disposition of the gallbladder over the liver pedicle and the simultaneous presence of variations in the liver vascularization result in an increased risk during surgery. To prevent complications, we recommend performing a cholecystectomy with safety criteria, starting from the fundus and isolating Calot’s triangle along the edge of the gallbladder. In the case of hepatic resection, an accurate study of the portal, arterial, and biliary branches should be done before surgery. This makes possible to plan an intervention tailored to the patient’s anatomical condition.
A particular acknowledgment gives to “Associazione Carlo Malchiodi per le Ricerche in Chirurgia ONLUS” for funding publication.
The author declares no conflict of interest.
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