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σh in the region range between 13.2 to 18.8 MPa/km, i.e., they are lower than the vertical stresses.Our approach in this study involves a set of two simulations: (1) a reference (base) case that is based on an average horizontal stress gradient of 15 MPa/km (i.e., σH |
= |
σh |
= 0.77σV), and (2) a bounding case adopting the lower bound of the horizontal stress gradient in the calculations, i.e., σH |
= |
σh |
= 0.67σV |
= 13.2 MPa/km. This lower σH bound corresponds to a geologic medium that is near critically stressed for shear, i.e., with pre-existing fractures and with the unconsolidated sand near its frictional limit. This is consistent with the observed pattern of natural fractures at the Mallik deposit, which indicates that the conjugate shear fractures dip about 60° ( presents some of the main input hydrological and thermal properties for Mallik and Mount Elbert deposits. The hydraulic and thermal properties for the Mallik site are based on the laboratory and field data published in . The hydraulic and thermal properties used for the Mount Elbert study were those used in the code comparison study of . These were derived using information gleaned from geophysical well logs, as well as flow parameters estimated by history-matching the data from a short-term open-hole depressurization test. As can be seen in , the hydraulic and thermal properties are similar at the two sites.The geomechanical properties of the reservoir and overlying rock are of particular importance, but they are also the most uncertain. Some information on the geomechanical properties at the Mallik site can be deduced from geophysical surveys conducted during past research activities ( presents vertical profiles of compressional- and shear-wave velocity from Vertical Seismic Profiling (VSP) and sonic logs from the Mallik 2L-38 well at the Mallik site (). The sonic velocities are functions of dynamic elastic properties and can be used to estimate the magnitude of, and variability in, the static elastic and strength properties (). Using compressional and shear-wave velocity logs from the Mallik 5L-38 well (30 m away from Mallik 2L-38 well) and the equations, estimates of the dynamic elastic properties have been obtained (). The Mallik 5L-38 sonic log data show that the compressional wave velocity increases from about 2000 m/s to about 2500 m/s when transitioning from pure sand to the HL, whereas the shear-wave velocity increases from 1000 to 1500 m/s. The resulting dynamic Poisson's ratio is about 0.4 both inside and outside the HL. The Young's modulus E is about 5 GPa outside the HL, and increases to about 15 GPa within the HL. However, values of static, rather than dynamic, properties are needed in a coupled reservoir-geomechanical analysis of stress and strain changes induced by hydrate dissociation. from a few laboratory experiments on samples from the Mallik site may be used to estimate the static strength and elastic properties. As expected, the values of the geomechanical properties deduced from these laboratory experiments are up to several orders of magnitude lower than those obtained from the sonic well logs at Mallik. On the other hand, the strength and stiffness values of the Mallik samples () are similar to those of Toyoura Sand estimated from laboratory studies by to study the geomechanical behavior of oceanic hydrate bearing sediments undergoing dissociation.Adopting these properties implies that the static modulus is about 1/10 of the dynamic modulus, and that the static Poisson's ratio is significantly lower than that derived from sonic data. A reasonable value of static Poisson's ratio for sand or weakly cemented sandstone should be less than about 0.25, averaging around 0.15 (), whereas it has been observed that sonic logs consistently overestimate Poisson's ratio, even in gas saturated sands (). The Poisson's ratio determined from static laboratory tests on artificial hydrate-bearing Toyoura Sand ranged from 0.1 to 0.2, and averaged 0.15 independently of the hydrate saturation ( conducted additional laboratory studies on natural hydrate bearing cores recovered from eastern Nakai Trough, Japan, and found a consistent Poisson's ratio between natural and artificial hydrate-bearing samples.For the aforementioned reasons, in our computations we adopted the two sets (static and dynamic) of geomechanical properties listed in . The static properties were based on the experimental data on Toyoura sand (), and are consistent with the limited laboratory data from Mallik. The dynamic properties were estimated from compressional- and shear-wave velocity logs using geophysical standard theory and empirical equations (e.g., ). The parameters describing the mechanical properties of the Toyoura sediment are corrected for pore-filling solid content (hydrate and ice). According to the experimental results of , we assumed that certain mechanical properties (bulk and shear moduli, and cohesion) increase linearly with hydrate saturation. For example, the cohesion varies from 0.5 MPa at 0% hydrate saturation to an extrapolated 2.0 MPa at 100% hydrate saturation, and the bulk modulus varies from 95 MPa at 0% hydrate saturation to an extrapolated 670 MPa at 100% (). These linear relationships match the laboratory data quite well over the range of hydrate content relevant to this study. For example, shows a good match of the adopted model to triaxial shear strength data over the 0 to 70% SH range. Moreover, following the experimental results of , the friction angle and Poisson's ratio are considered independent of the hydrate saturation SH. A dilation angle of 10° is adopted in a nonassociated Mohr–Coulomb model. We assume that the dilation angle is independent of hydrate content, although the experimental results by indicate a slight increase in the dilation angle with hydrate content.We used the same mechanical properties for the Mallik and Mount Elbert cases, which is a reasonable approach, given the relative similarities of the geological settings of these sites. In the reference case we use the mechanical properties derived from the laboratory experiments on Toyoura Sand (), which, to the our knowledge, represent the most complete, systematic, and relevant data set on the static mechanical properties of a hydrate-bearing sand. These properties are reasonable, considering the known differences between dynamic and static mechanical properties, and are consistent with the results from the few static geomechanical experiments conducted on samples from the Mallik deposit. For comparison, we conduct an additional simulation using the dynamic properties to show the importance of complementing sonic log data with systematic laboratory testing.Because the HL is hydraulically confined by shales, depressurization is rapid and effective, leading to fast hydrate dissociation and considerable cooling during the 5 years of production considered in this study. The constant PW at the well and the low effective permeability of the HBS creates a pressure disturbance characterized by a sharp front coinciding with the dissociation front. This front moves rapidly outward from the well, and it extends laterally along the x-axis after first reaching the bottom of the HL. Because depressurization is localized (being confined to the limited volume of dissociated or rapidly dissociating medium between the well and the front), most mechanical deformations and stress changes reach maximum levels very early, i.e., within the first year of production.Thus, instead of gradual geomechanical changes that occur over a large reservoir volume, constant-P production from Class 3 hydrate deposits is uniquely characterized by maximum changes that are observed at an early stage of the production process, with little change afterwards. These effects occur within a small volume that centers around the well and expands slowly. shows the evolution of (a) the volumetric release rate QR of the hydrate-originating CH4 into the reservoir, and (b) the volumetric rate QP of CH4 production, both summed over the entire 800 m length of the horizontal well. The QR and QP patterns are quite similar (as expected, being both Class 3 deposits in similar geologic settings and with similar properties), and they are both consistent with the behavior of such deposits (), i.e., with QP only slightly lower than QR. The Mount Elbert deposit is slower to respond because of its lower temperature (), and is considerably less productive than the Mallik deposit. Thus, at t |
= 5 years, QP |
= 0.7 ST m3/s (= 2.15 MMSCFD) at the Mallik deposit, but QP |
= 0.11 ST m3/s (= 0.34 MMSCFD) at Mount Elbert.The superiority of the Mallik deposit as a production target is confirmed by the cumulative volumes of released (VR) and produced (VP) gas in , which show the total Mallik production VP |
= 9 × 107 ST m3 ( = 3.2 BCF) dwarfing the VP |
= 1.53 × 107 ST m3 (= 0.54 BCF) from Mount Elbert. In addition to its higher temperature, another reason for the superior performance of the Mallik formation is the larger HL thickness ( shows very small volumes of free gas in the reservoir, as is typical in production from Class 3 deposits ( shows that initially QR and QP increase rapidly, and then they begin oscillating around a roughly constant value (plateau). These oscillations indicate the competition between the effects of decreasing pressure and temperature on dissociation and production: an increasing depressurization causes hydrate dissociation and an increase in QR and QP, while the resulting temperature drop (caused by the endothermic nature of dissociation) makes further dissociation more difficult and results in declining QR and QP. This cyclical process is repeated, and QR and QP begin to increase again when heat arrives from the surroundings to fuel further dissociation. A slight increasing trend is observed in the QR and QP of the Mallik deposit, though not in the Mount Elbert one. It is not possible to know if these trends will persist later because only a small fraction of the total hydrate mass is destroyed at the end of the 5-year production period ( show the spatial distributions of P, T, and SH at t |
= 3 yrs in the Mallik and Mount Elbert deposits, respectively. Despite the diffusive nature of P (which is transmitted even when flow is inhibited), a sharp front is easily detected in both figures at a location that roughly matches that of the dissociation front, and is confirmed by the corresponding minimum T (because of the endothermic dissociation reaction) at the same location. Note that dissociation occurs mainly at the top of the HL, but the HL bottom is also beginning to show faint signs of dissociation. Because of reduced dissociation in the colder Mount Elbert deposit, the edge of the dissociation front reaches only 120 m from the well at t |
= 3 yrs, compared to 280 m in the Mount Elbert deposit. shows the temporal evolution of P, T, and SH at the top of the HL and at x |
= 10 m from the production well. Note that (a) the hydrate is destroyed within a short time (10 days in the Mallik deposit, 30 days in the Mount Elbert one), and (b) after the hydrate exhaustion, P (which controls the geomechanical behavior) remains practically constant. This supports the earlier discussion that geomechanical changes reach their maximum level early in time.The main geomechanical responses are associated with the depressurization of the hydrate deposit, that causing an increase in vertical effective stress, that, in turn, results in increased shear stress and vertical compaction of the reservoir. shows the evolution of effective and total principal stresses for the base case, i.e., with static reservoir properties and an initial horizontal stress gradient of 15 MPa/km (i.e., σH |
= |
σh |
= 0.77σV). The figure shows that the effective principal stresses in the reservoir change quickly and proportionally to the fluid P responses shown in . Overall, production (and the corresponding depressurization) tends to increase the shear stress in the reservoir, which is proportional to the difference between the maximum and minimum principal stresses. The total maximum compressive principal stress σ1 is approximately constant throughout the simulation, as determined by the weight of the overburden rock. The intermediate and minimum compressive principal stresses σ2 and σ3 are horizontal and change as a result of poroelastic stressing during depressurization. The effective maximum principal stress σ′1 |
= |
σ1 |
− |
P, is vertical and increases proportionally to the reduction in fluid pressure. The effective intermediate and minimum compressive principal stresses σ′2 |
= |
σ2 |
− |
P and σ′3 |
= |
σ3 |
− |
P are horizontal and increase much less, because much of the pressure decrease is offset by an increase in total stresses σ2 and σ3. The magnitudes of stress changes are higher at Mallik as a result of a more substantial depressurization at that site. presents the path of the maximum and minimum principal effective stresses for three different mechanical conditions. The figure shows that the effective principal stress state moves into failure (shaded area) only in the case of static mechanical properties and a low initial horizontal stress (i.e., an initial horizontal stress gradient of 13.2 MPa/km, and σH |
= |
σh |
= 0.67σV). If the initial horizontal stress is higher, the initial effective stress state is much further away from failure and never moves into failure during depressurization. For static properties, the stress state moves along a slope Δσ′1/Δσ′3 |
= 5.5, whereas for dynamic properties, the stress state moves along an initial slope of Δσ′1/Δσ′3 |
= 1.5. Using the assumption of a thin and laterally extensive reservoir, it can be shown that the slope Δσ′1/Δσ′3 can be determined from the Poisson's ratio. For the static Poisson's ratio, ν |
= 0.15 the slope can be calculated analytically as Δσ′1/Δσ′3 |
= 5.5, whereas for the sonic Poisson's ratio ν |
= 0.4, the slope is Δσ′1/Δσ′3 |
= 1.5. In the numerical simulation result shown in , the stress path does not follow these slopes precisely, because the numerical results are affected by changes in elastic properties due to hydrate dissociation, and by thermal stresses. However, the results show that the Poisson's ratio of the reservoir rock is an important parameter, one that determines whether reservoir stresses during depressurization will increase or decrease the likelihood of shear failure. presents the evolution of the maximum compressive effective stress and strength at the same monitoring point located about 10 m from the production well. At both Mallik and Mount Elbert deposits, the initial (pre-production) maximum stress is much less than the compressive strength. For example, at Mallik, the initial compressive strength is 14.5 MPa for a hydrate saturation of 75%, whereas the maximum compressive effective stress is about 9 MPa (a). During the depressurization, the maximum compressive strength remains much larger than the stress until the hydrate starts to dissociate. At the Mallik site, the dissociation and weakening of the sediment implies that failure starts at about 10 days, and thereafter the maximum compressive effective stress that the sediment can sustain is limited by the strength of the sediment (a). At Mount Elbert, the dissociation is slower, due to a smaller depressurization at that site, but shear failure is triggered after about 2 months ( show the distribution of volumetric strain εV after 1 and 3 years of production. The largest volumetric strain develops within the dissociated zone at the top of the HL. Thus, in this zone there is a more substantial compaction as a result of sediment softening. The maximum volumetric strain εv is 0.6% at Mallik (). At Mallik, the local vertical strain is a factor of two larger than at Mount Elbert, because of a factor of two larger pressure drop ΔP. presents the time evolution of the vertical settlement at the ground surface and at the top of the reservoir, and the resulting average vertical compaction strain εZ of the HL. For both the Mallik and Mount Elbert deposits, εZ is restricted by the relatively stiff permafrost overburden. As a result, the vertical settlement UZ of the ground surface is somewhat smaller than the corresponding UZ at the reservoir, especially at early times. The stiffening effect of the permafrost overburden diminishes as the depressurization of the deposit becomes more extensive after several years of production. Overall, for the adopted mechanical properties, εZ |
< 0.4% and UZ |
= 6 cm at Mallik. The corresponding εZ and UZ are smaller at Mount Elbert, as a result of a smaller depressurization and a thinner deposit.The static elastic properties, such as the Young's modulus and Poisson's ratio, determine the magnitude of settlement as well as the reservoir stress path and the likelihood for shear failure during depressurization of the HL. A good understanding of the in situ static properties requires a combination of field surveys, such as sonic logs, and systematic laboratory testing of hydrate bearing samples. In particular, more laboratory data are needed to constrain static elastic properties (Young's modulus and Poisson's ratio) as well as strength properties (e.g. cohesion and coefficient of friction) and how these properties vary with hydrate content, confining stress, and strain rate. The important differences between dynamic and static properties have already been discussed in . Concerning strain rate and confining pressure, it should be pointed out that the strength and elastic properties adopted in this study (from ), were determined at a strain rate of 0.1%/min. Moreover, the adopted Young's moduli and its dependency on hydrate saturation were determined at a confining stress of 1 MPa. Recently, conducted additional experiments to determine the influence of strain rate, and presented results for increasing confining pressure. They found the Poisson's ratio to be insensitive to changes in hydrate saturation and independent of strain rates, whereas shear strength and in particular Young's modulus decreases with strain rate. With the decreasing strain rate from 0.1%/min to 0.001%/min, the shear strength was reduced approximately 30% and the elastic modulus by 60%. During depressurization in the field, the hydrate-bearing sand is exposed to a much slower strain rate, and an elevated shear stress will be sustained for years, indicating that the strength and elastic modulus adopted in this study may be higher than the real in situ properties. On the other hand, the confining effective stresses at the Mallik and the Mount Elbert sites are estimated to be about 3–5 MPa (see σ′2 and σ′3 in ), which indicate that the adopted Young's modulus determined at 1 MPa would be lower than the real in situ modulus. As a result, the decrease in Young's modulus with strain rate may be offset by an increase in confining stress. Thus, we conclude that our adopted properties are reasonable, whereas more sophisticated models that include dependency of strain rate and modulus can be readily implemented and applied once more data becomes available. Adopting such a model would not change the conclusions or significantly change the simulation results in this modeling study, but may have an impact on the exact magnitude of settlement and extent of shear failure zone.In this study, considering in situ stress conditions and mechanical properties at the Mallik and the Mount Elbert sites, we found that depressurization and the associated increased shear stress may lead to shear failure in the zone of production-induced hydrate dissociation near the well bore and upper part of the HL (see zone of hydrate dissociation in ). Because this zone undergoes complete hydrate dissociation, the cohesive strength of the sediment becomes low enough to initiate plastic yield and shear failure. At the moment, we can only speculate about how such shear failure could affect gas production. The shear failure facilitates shear deformations, leading to an enhanced compaction of the reservoir and possibly resulting in shear-induced changes in permeability. However, shearing may either enhance or destroy formation permeability. Moreover, shear yielding of weakly cemented sand may break bonds between particles, leading to enhanced sand production. Recent gas hydrate production tests at Mallik in 2007 and 2008 showed that sand production is a major issue during depressurization production—an issue that will require engineering measures such as sand screens to assure continuous water and gas flow (In the presence of pre-existing natural fractures in the HL, depressurization-induced shear stress may lead to shear reactivation of these fractures, which in turn may affect the production performance. The observed pattern of natural fractures at the Mallik deposit indicated conjugate fractures dipping about 60° (). In the present normal faulting stress regime (σv |
> |
σH≈σh), our analysis of the stress path evolution shows that these fractures could be reactivated if they are initially near critically stressed, i.e., on the verge of shear failure. Evidence from fractured rock masses has shown a good correlation between maximum in situ shear stress and water conducting fractures (e.g., ). Moreover, a shear over the effective normal stress ratio (τ/σ′n) exceeding 0.6 on a fracture has been observed as the lower-limit value for hydraulic conducting fractures and their correlation with maximum shear stress (Barton et al., 1995). This finding indicates that, over the long term, a shear over the effective normal stress ratio (τ/σ′n) exceeding 0.6 on a fracture can lead to enhanced permeability. Investigations with a Formation-Micro Imager (FMI) tool at the Mallik site showed that natural fractures were open and likely water filled, whereas recovered cores indicate partially mineralized fracture surfaces. For the lower-range stress, the lower bound of horizontal stress gradient is adopted in the calculations, i.e., σH |
= |
σh |
= 0.67σV |
= 13.2 MPa/km, the initial effective shear over effective normal stress ratio (τ/σ′n) on steely dipping fractures would be close to or exceed 0.6. Consequently, a small perturbation in the stress field during depressurization could induce shear reactivation. In this case, the depressurization induces a substantial increase in shear stress that could induce shearing along fractures, which may dilate and extend, leading to increased fracture permeability, connectivity, and surface area for dissociation of the hydrate-bearing formation. The likelihood and potential benefits of such fracture shear reactivation will be the subject of future studies.In our study of depressurization-induced gas production from the Mallik and Mount Elbert Class 3 hydrate deposits, using horizontal wells at the HL top, and kept at a constant bottomhole pressure, we reach the following conclusions:The depressurization causes preferential hydrate dissociation that proceeds mainly along the HL top.The depressurization of the hydrate reservoir results in vertical compaction of the reservoir and in increased shear stress within the reservoir. The magnitude of vertical compaction and shear stress depends on the magnitude of depressurization and the elastic properties of the reservoir and overlying formations.The calculated εZ is within 0.5%, and the estimated UZ |
< 6 cm. Of the two deposits, Mallik has the largest εZ and UZ because of larger depressurization and a thicker HL.Depressurization increases the effective shear stress because the vertical effective stress increases much more than the horizontal effective stress. At both Mallik and Mount Elbert, the higher shear stress may lead to shear failure in the zone of hydrate dissociation between the HL overburden and the downward-receding upper dissociation interface.The likelihood of shear failure is strongly dependent on the initial stress state and on the elastic properties of the reservoir. In particular, the Poisson's ratio ν of the HBS is an important parameter determining the effective stress path during depressurization. When a dynamic ν |
= 0.4 (from sonic logs) is used, the predicted effective stress state always diverges from shear failure during depressurization. When a static ν |
= 0.15 (a reasonable estimate for unconsolidated sand) is used, the effective stress state will tend towards shear failure, but may not reach it, depending on the initial stress state.Overall, the estimated vertical compaction at these two sites is rather limited (within 0.4%), partially mitigated by the relatively stiff permafrost overburden. Moreover, the vertical compaction is expected to be relatively uniform, leading to uniform settlements of the ground surface. The potential shear failure within the reservoir might be a more serious issue, because this could affect the gas production in terms of permeability and sand production. At any site, the coupled thermodynamic and geomechanical approach used in this study can be applied for optimizing production, while minimizing the likelihood for such unwanted geomechanical responses.Finite element limit analyses of under-matched tensile specimensmismatch factor defined for yield strength, M |
= |
σYW/σYBRecently non-conventional welding of aluminium alloys (such as laser beam welding or friction stir welding) is of great interest in transportation industries ). In such under-matched structures, combination of the higher strength base metal and the lower strength weld zone leads not only to higher stress states in the softer weld zone but also to more plastic strain concentration. One simple example of the strength mismatch effect on deformation and fracture of such under-matched joints is tensile test, shown in (denoted as “micro”). Note that the micro-tensile specimen is small enough to be extracted solely from the weld zone. Comparing tensile data of the base material, the yield and tensile strengths from the micro-tensile test are lower by about 20% than those from the standard tensile specimen. This might be due to the size effect, but overall trends (such as the strain hardening behaviour and tensile ductility) are quite similar. Comparing tensile properties of the weld metal, on the other hand, overall trends are quite different. First of all, the strain hardening behaviour is very different. The standard tensile test gives much higher strain hardening capacity than the micro-tensile specimen test. More importantly, tensile ductility is much smaller for the standard tensile test than for the micro-tensile specimen test. This results from the constraint effect due to the stronger base material. As proper determination of “intrinsic” tensile properties of the weld zone is quite important in structural integrity assessment of welded joints, it is essential to develop a method to extract “intrinsic” tensile properties of the weld zone from standard tensile testing of under-matched welded joints, for instance.To understand the effect of under-matching on deformation and fracture behaviour of under-matched welded joints, parametric finite element (FE) limit analyses based on elastic–perfectly-plastic materials are performed in the present work. From FE limit analyses, effects of the strength mismatch and geometry of the weld on plastic limit loads and fully plastic stress fields are presented for under-matched tensile specimens. Section provides the FE limit analysis, employed in the present work. Results of plastic limit loads for under-matched tensile specimens are given in Section , and those of stress variations in Section . The present work is discussed in Section with M |
< 1 referring to under-matching and M |
> 1 referring to over-matching. The present work concentrates on under-matching, and two values of M, M |
= 0.5 and 0.75, were considered, together with the even-matching case (M |
= 1). Another important mismatch related parameter is the slenderness of the weld The value of ψ in the present work was systematically varied, ranging from ψ |
= 0.25 to ψ |
= 2. As will be shown later, when ψ is greater than 2, the strength mismatch effect diminishes and the mismatched specimen can be treated as the homogeneous specimen made wholly of the weld metal.Limit analyses of the FE model of welded joints shown in depicts a typical FE mesh employed in the present investigation. To avoid problems associated with incompressibility, the reduced integration 8-node hybrid elements (element type CPE8RH and CAX8RH from the ABAQUS library) were used for plane strain and axi-symmetric calculations. For plane stress calculations, reduced integration eight-node element (CPS8R) was used. The number of elements and nodes in a typical FE mesh ranges from ∼6100 elements/30,781 nodes to ∼8650 elements/43,681 nodes, depending on the weld width. One notable point for the present FE mesh is that a spider-type mesh is used at the interface edge between the weld zone and base material (see the magnified view in ). It is because stress singularity (very steep stress gradient) could occur at the interface edge region due to the strength mismatch.In all the cases, deformation boundary conditions were applied to the FE model, and the magnitude of the applied deformation is made large enough to bring the specimen to its limiting load state. The corresponding fully plastic limit loads were obtained directly from the FE results. For all cases considered, the FE limit load solutions for even-matched (homogeneous) specimens differ from the known solutions by less than 1%, which provides confidence in the present FE calculations. In addition, fully plastic stress fields can be easily extracted from the FE results, as the stresses remain constant in fully plastic states in the limiting case of non-hardening plasticity. show correct trends. That is, when h/w is much greater than unity, h/w |
≫ 1, the limit load of the strength mismatched plate specimen is governed by the weld metal, and thus is close to that for the homogeneous plate made solely of the weld metal. For the case when h/w is less than unity, both the weld metal and base material contribute to the limit load, and thus the limit load of the mismatched plate should be greater than that of the homogeneous plate made of the weld metal, but less than that made of the base material. For plane strain condition, the effect of h/w on plastic limit load is quite significant. For decreasing h/w from h/w |
= 1, the limit load of the under-matched plate increases sharply and approaches that of the homogenous plate made of the base material, in the limiting case of a very thin weld, h/w |
→ 0. For plane stress condition, the effect of h/w on limit load is much less significant, increasing the limit load only by less than 10% for the values of h/w considered in the present work. This can be easily understood from the slip line field for a plate in plane stress, where plastic deformation is localized along the neck (see for instance Refs. Before proceeding, a few points are worth noting. The first point is that the stress results in this section will be presented in terms of the stress triaxiality, defined bywith σh and σi (i |
= 1–3) denotes the hydrostatic (mean normal) stress and the principal stress, respectively. It has been argued that the stress triaxiality plays an important role in ductile fracture of metals , σ0 denotes the yield (limiting) stress for the elastic, perfectly-plastic material, which could be either σ0b or σ0w in the present work. The choice of σ0 as either σ0b or σ0w depends on the location of interest within the specimen. In the present work, variations of the stress triaxiality are presented along three different planes (lines). The first plane is along the mid-plane of the specimen (the A–A plane in c), and the second one along the interface in the side of the weld metal (the B–B plane in c). The above two planes are within the weld zone, and thus the yield stress of the weld metal, σ0w, is used to define the stress triaxiality, Eq. . The last one is along the interface in the side of the base material (the C–C plane in c). For the last case, the plane is within the base metal, and thus the yield stress of the base material, σ0b, is used to define the stress triaxiality, Eq. c). For M |
= 0.5 and h/w |
= 0.25, the value of the stress triaxiality is as high as ∼1.9, which is about three times that for the homogeneous specimen. For M |
= 0.75 and h/w |
= 0.25, on the other hand, the value of the stress triaxiality is about 1.0, which is about twice. With increasing h/w, the stress triaxiality decreases and, when the value of h/w is about unity, it recovers for the homogeneous specimen.c and d show the effect of M and h/w on variations of the stress triaxiality along the interface in the side of the weld metal (the B–B plane). The effect of M and h/w on the stress triaxiality in the center of the specimen is similar to that for the case of the mid-plane in the specimen, but distribution is somewhat different. In particular, stress triaxialities at the edge of the specimen (x/w |
= 1 in ) are worth noting, as they are overall higher than those in the mid-plane (the A–A plane), for a given M and h/w. This results from stress concentration at the interface edge region due to strength mismatch.e and f show the effect of M and h/w on variations of the stress triaxiality along the interface in the side of the base material (the C–C plane). Note that the stress triaxiality is normalized with respect to the yield strength of the base material, σ0b, in contrast to the previous two cases where it is normalized with respect to the yield strength of the base material, σ0w. Note also that the scale in e and f is different from that in the previous figures. Although it is rather difficult to find a clear trend, it can be seen that the values of the stress triaxiality are much lower than those for the previous two cases, and in fact are lower than the value for the homogeneous (even-matched) specimen, σh/σ0b |
= ∼0.6. It is because deformation is concentrated in the softer weld zone and thus stresses tend to be relaxed in the stronger base material., where two different stress triaxiality values are presented; (i) maximum values along three different lines, the A–A, B–B and C–C lines, and (ii) average stress triaxiality values along these lines. Although the maximum value is believed to be more relevant to fracture of the specimen, the average one would also be useful to see overall effects. For the case of the mid-plane (the A–A line), the maximum value of the stress triaxiality always occurs in the center of the specimen (x |
= 0), and increases steeply with decreasing h/w when h/w |
< 1. The average value shows a similar tendency, but the rate of increase is quite gradual, and the value is much lower than the maximum one. For the case of the interface in the weld metal side (the B–B line), the maximum stress triaxiality does not always occur in the center of the specimen (see b presents values of the stress triaxiality in the center of the specimen (x |
= 0), showing that they increase sharply with decreasing h/w only for h/w |
< 0.5, but are close to the homogeneous value for h/w |
> 0.5. c shows the maximum and average stress triaxialities along the B–B line. Two points are worth noting. The first one is that the maximum value does not approach the homogeneous value, even for the case of h/w |
= 2. This is due to the stress concentration at the interface edge, as can be clearly seen from . The second point is that, although the average stress triaxiality shows a similar tendency to that for the A–A line, the values are overall higher. This means that the overall effect of the strength mismatch on the stress triaxiality would be higher at the interface region than in the mid-plane of the specimen. d summarizes the results for the case of the interface in the base material side. For all cases, the values of the stress triaxiality are lower than the homogeneous value, which clearly shows the shielding effect due to the stronger base material.Up to present, variations of the stress triaxiality are presented. Although the stress triaxiality is an important parameter for ductile fracture of specimens, the shear stress along the interface, τ12, would be of interest. shows the effect of M and h/w on the shear stress along the interface in the weld metal side. The shear stress is normalized with respect to the shear strength of the weld metal, k0wFor M |
= 0.5, a clear tendency can be seen that the shear stress increases with decreasing h/w. On the other hand, for M |
= 0.75, no such clear tendency can be seen but overall magnitudes of the shear stress is much lower than those for M |
= 0.5.), that is, the stress triaxiality increases with decreasing M and h/w. Although the effect of M and h/w on the stress triaxiality can be clearly seen, it should be noted that such effect on the absolute value of the stress triaxiality is not as significant as that for plane strain cases. For M |
= 0.5 and h/w |
= 0.25, for instance, the value of the stress triaxiality is about 0.6. Although such value is almost twice of that for the homogeneous plane stress plate, it is close to that for the homogeneous plane strain specimen. This is consistent with the fact that the constraint effect on stresses is much less significant in plane stress than in plane strain, even for homogeneous specimens The effect of M and h/w on the stress triaxiality in under-matched plane stress plate specimens is summarized in . Again overall behaviour is similar to that for plane strain plate specimens. One notable point is that the results for M |
= 0.5 are almost same as those for M |
= 0.75. This is because, for plane stress specimens, slight under-matching is sufficient to promote the mismatch effect, and further decrease of the strength mismatch (higher under-matching) does not affect stress fields any more. Thus for plane stress specimens, the more important variable is the slenderness of the weld, h/w, which can be also seen from the mismatch effect on plastic limit loads. shows the effect of M and h/w on the shear stress along the interface in the weld metal side. A similar conclusion can be drawn that the magnitude of the shear stress is independent on the strength mismatch M, but dependent on h/w. However, compared to plane strain results, the magnitudes are overall lower.The effect of M and h/w on variations of the stress triaxiality in three different planes (the A–A, B–B and C–C planes) of the under-matched round bar specimen is shown in . For even-matched (homogeneous) round tensile bars, the stress triaxiality is constant and has a value of ∼0.33, which is close to that for the plane stress plate specimen. Note that for the plane strain plate specimen, the value is about 0.6.The present results on the effect of M and h/w on plastic limit loads for mismatched tensile specimens could be used to extract intrinsic tensile properties of the under-matched weld zone from test results of under-matched tensile specimens. Returning to , it was pointed out that the standard tensile test using under-matched specimens gives much higher strain hardening capacity than the (all-weld) micro-tensile specimen test specimens. To derive a simple model to extract stress–strain data, in particular the strain hardening capacity of the softer weld metal, a round bar with idealized weldment is considered, as depicted in . The radius and half (gauge) length of the specimen is denoted as w and L, respectively, and the half-width of the weld zone as h. The specimen is subject to axial tension P, resulting in a total elongation of 2δ. Assume that the tensile property of the base material is characterized by elastic, power-law plasticwhere σ0b denotes the yield strength of the base material. The subscript “b” denotes the properties of the base material. According to the definition of the strength mismatch factor, Eq. Force equilibrium and compatibility conditions lead to the following two equations: leads to σw |
= |
σb, and thus is denoted as σ(=σw |
= |
σb) for simplicity. Combining Eqs. leads to the following non-linear equation for σ:The first term in the left hand side of Eq. is related to the elastic displacement, δe, and the second term to the plastic displacement, δp. Normalizing Eq. σEε0b+hLM1-nwσσ0bnw+1-hLσσ0bnb=(δ/L)ε0b,where ε0b is the yield strain, ε0b |
= |
σ0b/E. Eq. can be used to estimate the strain hardening exponent of the weld metal, nw, as follows. Material properties of the base material are assumed to be known, such as σ0b, ε0b and nb. Furthermore geometric-related variables, such as h and L, are also known. From the test of the under-matched specimen, the yield load can be determined, from which the value of M can be estimated using the mismatch limit load solutions, developed in the present work, see . Once the value of M is determined, the strain hardening exponent of the weld metal, nw, can be determined by solving a non-linear equation, Eq. . Note that the above equation is a crude approximation, resulting from simple assumptions. Detailed equations to extract tensile properties of the under-matched weld zone from test results of under-matched tensile specimens would need much more refinements and validation using carefully prepared experimental data. On the other hand, the above equation could serve as a first order approximation to estimate rough tensile properties of the under-matched weld zone.a). The effect of the plate thickness on fully plastic stress triaxiality in under-matched plate tensile specimens is found to be even more significant. For homogeneous specimens, the stress triaxiality for plane strain is about 0.6 and is about twice of that for plane stress. For under-matched specimens, however, the stress triaxiality for plane strain can be as much as 1.9, when that for plane stress is only about 0.6, for the cases considered in the present work. One interesting issue is the required thickness to achieve the plane strain condition. For homogeneous specimens, it is known that global properties such as the limit load tend to be more plane stress condition, whereas local properties such as stress triaxiality to be more plane strain condition. For homogeneous specimens, the important parameter is the ratio of the plate thickness to the width. For under-matched specimens, situations are more complex, as the ratio of the weld width to the plate thickness and the strength mismatch ratio also play a role, in addition to the ratio of the plate thickness to the width. Full quantification for the effects of these parameters on plastic limit loads and fully plastic stress fields are needed.Mechanical properties of suspended individual carbon nanotube studied by atomic force microscopeMechanical properties of three different structure based on the suspended individual carbon nanotube were studied. Force–distance measurements were performed using atomic force microscope tip manipulations.This paper is a short review on the mechanical properties of individual carbon nanotubes (CNTs), which were studied by atomic force microscope (AFM). AFM force–distance measurements were applied to three different nano structures based on the suspended CNTs, those are a straight suspended CNT, a coiled CNT, and a torsional CNT. Force–distance measurements were done on the pick and valley position of the coiled CNT and the estimated elastic moduli was compared with the straight one. Nanoscale metal plate was fabricated on the middle part of the suspended CNT structure using conventional lithography and etching procedure. The torsional modulus of the suspended CNT was estimated from the force–distance measurement using AFM manipulation on top of the metal plate. The paper covers synthesis and preparation method of various suspended CNT structure and the analysis of the force–distance measurements appropriate to each CNT structure. Prospects of the AFM force–distance measurements on emergent nano scale materials as well as CNT are suggested in the end of this paper.Because of its superior mechanical and electrical properties, carbon nanotube (CNT) has been widely studied and possible applications of CNT devices were also suggested in many directions during last two decades CNT based nano mechanical devices, in the other hand, have been grown later compare to electronic devices because a little more nano lithography techniques were necessary give a mechanical degrees of freedom to CNTs as well as electrical degrees of freedom. CNT is one of the best candidate materials for nanoelectromechanical system since it has high conductivity, extreme mechanical strength, and ultra-low mass density Although there have been several reports so far on studying the basic mechanical properties of CNTs CNTs were synthesized by chemical vapor deposition method. 1 nm of catalyst metal (Fe) was deposited on the Si substrate which has around 1.5 nm of natural oxide layer. CNTs were grown in a quartz tube furnace by flowing 20 sccm C2H2 and 100 sccm H2, and 600 sccm Ar gas at temperature of 750 °C. The CNTs with ∼20 μm length and 20–50 nm diameter were synthesized for 15 min of growth time. The synthesized CNTs were dispersed by ultra-sonic agitation in a sodium dodecyl sulfate (SDS) water solution. One droplet of the CNT suspension was deposited on top of the pre-defined Au/Ti electrode with assistance of an ac dielectrophoresis (DEP) method. An ac bias signal with 13 MHz of frequency, 16 Vpp was applied for 30 s for CNT deposition. shows the synthesized CNTs aligned and attached to the metal electrode using DEP method. Coiled CNTs were found as indicated by arrow in while the dispersed CNTs were observed using scanning electron microscope (SEM). It is known that ring or coiled CNT structures can be formed during synthesis Mechanical properties of CNTs were measured using atomic force microscope with three different CNT structures. First of all, a straight suspended CNT structures were produced using conventional nano fabrication method combined with ac DEP method. Detailed fabrication procedure for realizing the suspended CNT was previously reported Mechanical properties of the three types of CNT structures were studied by AFM force–distance measurements. First of all, a straight suspended CNT structure was fabricated and its Young’s modulus was studied. Suspended nano structures were found by a tapping mode AFM scan. By zooming in the scanning area, the AFM tip was moved on top of the desired position of the CNT structures so that the force–distance curves could be obtained by pushing the AFM tip. The deflections of the suspended structures were estimated by extracting the deflection of the cantilever, which was obtained by measuring force–distance curve on SiO2 substrate shows force–deflection curve of suspended CNT. The deflection of CNT according to the external force was obtained by extracting the deflection of cantilever from the original force–distance measurement results, which is shown at upper inset of . The elastic modulus of CNT can be estimated by applying the string model where A is the cross section of a CNT (3.2 × 10−18 |
m2) and l is the length of the suspended CNT (1 μm).Secondly, mechanical properties of a coiled CNT were investigated using AFM manipulation. As shown in the , the diameter of the coiled CNT is thicker than the straight one even though it was found in the same batch. However, the coil was not tightly wound and the diameter is rather thin if it is compared to the previous works on the coiled CNT which was synthesized on purpose shows the AFM measurement results on the coiled CNT. The cross section analysis of the AFM height image revealed that the distance of pitch is around 400 nm and the height of CNT coil is around 200 nm.AFM force–distance measurements were done on the three different areas where, the peak, valley of coiled CNT, and SiO2 substrate. The slope of the force–distance curve at the valley part is slightly lower than that of SiO2. The deflection of coiled CNT at the valley part is considered to originate from the tube deformation under AFM tip manipulation in vertical direction. The force–distance curve of the peak part is significantly lower than the other curves. It is considered that the peak part is deflected by the pushing force. Inset shows the plot of force versus deflection curve of the coil with respect to the curve of valley. To estimate the mechanical properties of the CNT coil, one turn of coil was assumed to be a closed ring with 150 nm of coil diameter. The Young’s modulus of this closed ring can be derived from the Hertz model that is expressed as where x is the distance from the center of the peak position. I = πr4/4 is the moment of inertia of ring structure. R |
= 100 nm) (r |
= 15 nm) is radius of coil. Since AFM tip pushed the coil on top of the peak, the x value should be zero and the k can be estimated from the slope of the inset graph in . Therefore, the resultant elastic modulus value E was estimated to be 4.39 GPa. The elastic modulus of suspended CNT that were synthesized from the same batch was estimated to be around 0.3 TPa. Even though the coiled CNT came from the same synthesis batch, the elastic modulus of the coil structure is around 100 times lower than the straight CNTs.Lastly, torsional modulus of the suspended CNT was studied. shows the schematic picture and SEM image of the torsional rotator using suspended CNT. Detailed fabrication method was described at the previous subsection.Again, the force–distance measurements were done with AFM tip manipulation on the nano-scale metal plate with various distances from the CNT shaft. (a) shows the summary of the measurement results. The slope of the force–distance curve at 0 nm, which means the AFM tip is placed at the CNT shaft, is lower than that measured on SiO2 substrate. This result means that the suspended CNT is deflected in uniaxial direction as well as twisted when the AFM tip pushes the metal plate for applying torsion to the suspended CNT. Therefore the contribution from the deflection of the CNT should be removed by extracting the slope of the 0 nm curve to estimate the torsional modulus of the CNT. The stiffness value was estimated by comparing the slope of each curve with that of 0 nm result. (b) shows the stiffness values at various positions from the CNT pivot point of the rotational plate. The stiffness values were decreased as the distance between pushing point and the CNT pivot point increased. From the information of the force–distance measurements, the torsional modulus of the suspended CNT can be estimated. When the the AFM tip pushes the Au plate, the plate can be affected by both torque (T) and vertical force (F) that are expressed aswhere κ is torsional spring constant and Kz is vertical spring constant. The other parameters are shown in inset of (b). The force and ΔZ value were obtained from the force–distance measurement curve at 0 nm. The vertical displacement values at the other positions could be also estimated from the measured force–distance curves by using the relation ΔZ = |
Δh + dθ.From the measurement results and the equations mentioned above, κ could be estimated. For example, if we take the parameters measured on d |
= 250 nm point, we have 10 nm of ΔZ − |
Δh with 10 nN force. Therefore the torsional spring constant κ is estimated to be ∼6.25 × 10−14 |
Nm. If the suspended CNT shaft is assumed to be straight and a uniform circular bar, the continuum mechanics model can be applied to estimate the torsional modulus G, that is expressed aswhere l is the suspended length of CNT and rout (rin) is outer (inner) radius of CNT. Here we assumed that the CNT shells were filled so the inner radius was ignored. The estimated modulus value G is 102 GPa from our measurement has same order of magnitude with previous report In this paper, AFM force–distance measurements for investigating the mechanical properties of individual suspended CNT were introduced. Three different CNT based nano structures, which are a straight suspended CNT, a coiled CNT, and a suspended CNT based torsional rotator were prepared for performing the mechanical measurements. The force–distance curves of each suspended CNT structures were obtained by AFM tip manipulations. The deflection of the suspended individual CNT structures in uniaxial, vertical, and rotational direction could be extracted based on the measurement results.Because of the high accessibility of AFM to various nano scale materials with precise position and force control, the AFM based force–distance measurement has been applied to number of low dimensional nano structures since AFM was invented Before finishing this paper, I would like to remark that I started the study on the mechanical properties of CNT and other low dimensional nano structures under supervision of Prof. Yung Woo Park. The experimental techniques and scientific knowledge described in this paper were also initiated since I was his Ph. D student. I want to express my sincere gratitude to Prof. Park for giving me an opportunity to study this subject with him so that I can continue my research and expand my field to studies on the nano mechanics.Determining the coefficient of friction between solids without slidingA novel method for measuring the interfacial coefficient of friction between two solids which avoids sliding is described, and sample results are given. The technique makes use of the fact that a carefully controlled sequence of partial slip states between contacting bodies may be used to produce relative motion whose extent is a strong function of the coefficient of friction. It is argued that this approach induces much less surface damage in the components, and therefore yields a value for the coefficient of friction which is much more representative of their unmodified condition.It is truism in physics that quantities cannot be measured without changing them; a voltmeter draws current; the insertion of a load cell in a load path introduces compliance, and so forth. Similarly, the coefficient of friction is normally measured by sliding solids pressed together, and the process of sliding normally causes significant surface modification, and hence a change in the coefficient of friction. It could be argued that conducting tests under an even lower normal force, and extrapolating back to zero would circumvent the problem, but this presumes that the coefficient of friction is truly load independent, and it may be desirable to measure its value at representative contact pressures. One situation where a really detailed knowledge of the value of the coefficient of friction is important, and sliding in the prototype is absent, is the stationary contact suffering partial slip, or fretting. It may seem a contradiction in terms to measure the coefficient of friction without sliding the bodies, but this is not so, and a careful exploitation of the phenomenon of partial slip may reveal the value of the coefficient of friction. One attempt at this has already been made by Pasanen et al. In this paper completely different quantitative evidence is used to deduce the coefficient of friction. The experiment utilises the same apparatus employed at Oxford and elsewhere to investigate fretting fatigue, but it might, potentially, be executed with a simpler single actuator machine, as will become clear. shows an idealisation of the apparatus, and its key features. The basic idea is to induce a cyclic set of loads which alternately inject a slip region into one end of the contact and abstract it at the other, so that, after each cycle of loading, there is, potentially, a net rigid body movement of the indenter. In many respects the idea was foreshadowed by Dundurs and Comninou A shear force, Q, insufficient to cause sliding, is then applied and held constant. A bulk tension, σ, is now developed, and it is assumed that this varies cyclically between limits σmax and σmin where, for reasons of ensuring stability of the specimen, we would prefer the latter to be positive. The question we now ask is whether there is some range of values of the variables which means that, at some point during the loading cycle, every point passes through the slip condition (though not all simultaneously!) and, then, if this is so, if the slip conditions are such as to give rise to gradual rigid body displacement, or ‘creep’. If this does happen, there is the possibility of measuring a significant rigid body movement over many cycles of loading, and from this inferring the value of the interfacial coefficient of friction, f. First, however, a partial slip analysis needs to be conducted.An initial attempt to solve the evolving stick-slip regime was made using the classical integral equation formulation for half-planes in contact, for a wide range of fretting and rolling contact problems, but this approach rapidly becomes unwieldy if the stick-slip regime is complex. A better method is to devise a much more general procedure using a numerically defined shearing traction distribution, and then to use optimisation techniques, pioneered in this context by Kalker The exact form of the cycle σ(t) is unimportant and has no bearing on the solution: what matters is simply the values of the bulk tension at the turning points. Also, it is clear if there is at least one particle within the contact which never slips the contact cannot suffer any gross (rigid body) displacement. Conversely, if every particle slips at some point during the cycle then there is the possibility that the indenter will ‘walk’. There is also the possibility that there will be no net movement over the cycle once the steady state regime is achieved: these two responses are analogous to the phenomena of creep and ‘cyclic plasticity’ respectively in classical plasticity theory.Throughout this model a Coulomb-Amontons friction law is adopted and, hence, the coefficient of friction is independent of the slip-velocity. Thus, when slip occurs, the shear tractions, q(x,t), can be expressed as a fraction, f, of the normal pressure, which is taken positive when compressive, as follows:where v(x,t), represents the relative surface slip velocity. The direction of the shear tractions opposes the slip direction, so a minus sign was added before p(x) in (2). Where the friction law is not violated, the slip velocity, v(x,t), is equal to zero and hence:A function which simultaneously includes the boundary conditions for the stick and slip areas needs to be formulated. This can be achieved by multiplying both members of the boundary condition associated with the slip regions by the slip velocity so that:q(x,t)v(x)+fp(x)|v(x,t)|≥−|q(x,t)||v(x,t)|+fp(x)|v(x,t)|≥0,where the first equality holds if and only if slip vanishes or the shear traction has the same direction of slip velocity, and the second equality holds if and only if −|q(x,t)| equals fp(x) in the slip areas or slip vanishes. The importance of this additional characteristic will become clearer shortly. The object of the technique is to determine the location of slip regions within the contact automatically, and to this end we construct the following integral function:F(x,t)=∫−aa[q(x,t)v(x,t)+fp(x)|v(x,t)|]dx,where F(x,t) will be referred to as the functional of the elastic contact problem. If the slip and shear traction distributions are the exact solution to the contact problem then the value of the integral is zero. If not, the value of the functional will be strictly positive. The best approximate solution to the contact problem can be achieved by minimising the value of the integral, and the related minimum principle can be formulated as follows: “minimize the difference between the power dissipated by the Coulomb friction generated by the slip and the power dissipated by the traction generating the slip” The above principle was first proposed by Kalker Extensive sets of results have been evaluated and we first display example results tracked out as a function of time. First, the boundary between walking and no-walking has to be calculated. In the example case shown in (a) the ratio Q/fP was set to 0.7. The figure shows the results for σmin/fp0=0 and σmax/fp0=1.8. This shows how the interfacial shearing tractions evolve over a loading cycle by looking at the turning points once the steady state condition is reached. The loading history plot shows how the bulk loading is gradually increased in amplitude, and the transient response permitted to decay. The shearing traction distributions experienced at the turning point in the loading cycle, viz. the maximum bulk tension at point (1) and the minimum at point (2), are also plotted. The hatching below the axis shows the extent of the stick region at point (1) in the loading cycle and the hatching above the axis shows the extent of the stick region at point (2), It will be noted that there is a region of contact which always remains stuck, and, therefore, it is clear that there is no rigid body motion. Suppose now, that the parameters are adjusted slightly and that, whilst Q/fP and σmin/fp0 are maintained at the same value, we choose σmax/fp0=2.2, a modest change. The response of the system is displayed in (b). The loading history is generally quite similar in form but it will be noted that there is now an overlap of the two slip zones at a point near the origin: slip is injected from one side while just under a half of the contact remains stuck, and when the bulk tension is relaxed slip penetrates from the other contact edge and includes a small portion of the previous slip region (now locked). It follows that a small slip displacement has been ‘locked in’ when the initial slip zone became locked and this subsequently emerges at the opposite contact edge. It is therefore equivalent to the passage of a dislocation of Burgers vector equal to the total slip displacement within the common slip region, and is the ‘ruche’ that Dundurs and Comninou referred to in their prescient paper on moving a rug without sliding it If different ratios for Q/fP are chosen whilst σmin/fp0 is maintained fixed, the onset of walking can be calculated as a function of σmax/fp0. (a) summarises the results of many such calculations, and shows clearly the boundary between ‘walking’ and ‘no walking’. In fact, the result can immediately be generalised slightly by noting that the boundary depends not separately on the turning-point (or extreme) values of the bulk tension, but simply on its range, Δσ (and therefore σmax here). The example transition point Q/fP=0.7, σmax/fp0=2.125 is highlighted. For this value of the dimensionless shear force, (b) shows the normalized distance walked per cycle, Δs/(aAfp0), for values of σmax/fp0=Δσ/fp0=2.5,2.75,3.0. The relative increase in step length with bulk tension range is clear. The plot in (b) can therefore be used to infer the coefficient of friction once the walking distance per cycle is measured by tuning the value of f until the computed walking distance matches the measured value.The calculations summarised above are valid as long as plasticity does not occur. At the same time, the step length is seen to depend on the contact half-width and so, in order to make the distance moved as easily measurable as practicable, it is desirable to have the largest possible contact and hence the normal contact load must be moderately high. The elastic limit for the system defines the highest sustainable load whilst maintaining a fully elastic contact field, and this depends on the interfacial friction coefficient and on the loading conditions therefore shows the elastic limit for the case characterised by Q/fP=0.7 and σmax/fp0=2.125,2.2 and 2.4, which is found to be a practically useful range. It is important that plasticity, particularly at the surface, is avoided for the solution to remain wholly valid.An inherent property of the experiment is that the real unknown quantity is, of course, the coefficient of friction, f, whilst the two parameters which we can control are the constant shearing force and the bulk tension range. The bulk tension range is practically limited on the lower side by zero (to avoid all possibility of buckling) and on the upper side by the elastic limit. The results of the analysis are, of course, displayed in terms of dimensionless quantities incorporating the coefficient of friction, but, when the tests are carried out, there is a certain amount of ‘feeling our way around’ needed in order to get the load parameters in the right region: also, probing the walking/no-walking boundary proved not to be the best way to determine the coefficient of friction - a much more sensitive measure turned out to be determining the walking length, and this will be apparent from the following description of the experiment itself.Some preliminary tests have been carried out on mild steel. The central ‘dogbone’ (see schematic in ) has a section of 10 mm |
× |
10 mm and cylinders of the same width and radius 100 |
mm are pressed in with a normal force P=1.57 |
kN. The corresponding contact semi-width is a=0.419 |
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