Shear Stress Components Research Articles

Shear banding in large amplitude oscillatory shear (LAOStrain and LAOStress) of polymers and wormlike micelles

We investigate theoretically shear banding in large amplitude oscillatory shear (LAOS) of polymers and wormlike micelles. In LAOStrain we find banding at low frequencies and sufficiently high strain rate amplitudes in fluids for which the underlying constitutive curve of shear stress as a function of shear rate is non-monotonic. This is the direct analogue of quasi steady state banding seen in slow strain rate sweeps along the flow curve. At higher frequencies and sufficiently high strain amplitudes we report a different but related phenomenon, which we call `elastic' shear banding. This is associated with an overshoot in the elastic (Lissajous-Bowditch) curve of stress as a function of strain. We suggest that this may arise widely even in fluids that have a monotonic underlying constitutive curve, and so do not show steady state banding under a steadily applied shear flow. In LAOStress we report banding in fluids that shear thin strongly enough to have either a negatively, or weakly positively, sloping region in the underlying constitutive curve, noting again that fluids in the latter category do not display steady state banding in a steadily applied flow. This banding is triggered in each half cycle as the stress magnitude transits the region of weak slope in an upward direction, such that the fluid effectively yields. Our numerics are performed in the Rolie-poly model of polymeric fluids, but we also provide arguments suggesting that our results should apply more widely. Besides banding in the shear rate profile, which can be measured by velocimetry, we also predict banding in the shear and normal stress components, measurable by birefringence. As a backdrop to understanding the new results on shear banding in LAOS, we also briefly review earlier work on banding in other time-dependent protocols, focusing in particular on shear startup and step stress.

Towards sub-lithospheric stress determination from seismic Moho, topographic heights and GOCE data

Sub-lithospheric stresses can be estimated by analysis of gravity field measurements. Depending on the measured gravimetric quantity, different methods can be employed to estimate those sub-lithospheric stresses. Here, we further develop the Runcorn’s theory for estimation of mantle stresses (1967) such that a Moho model and full topographic information are used to recover the function from which the stress can be computed by taking derivatives northwards and eastwards. We develop new integral equations for such a purpose and recover this function by solving those integral equations locally over the Indo-Pak (India-Pakistan) region from (1) a gravimetric Moho model computed from the SRTM (Shuttle Radar Topography Mission) and the Earth gravity model EGM2008, (2) SRTM and the seismic Moho model of CRUST1.0 and (3) data and measurements of the GOCE (Gravity field and steady-state Ocean Circulation Explorer) mission. Finally, we perform a joint inversion of seismic and GOCE data for the same purpose. The numerical results show that the use of a seismic Moho model recovers information about the stress field which is not seen in the results derived from a gravimetric Moho model. A combination of the seismic Moho model, SRTM and GOCE yields a better stress field than that of either the seismic and/or gravimetric data alone. The magnitudes of the sub-lithospheric stress are computed from the shear stress components over the area and good agreement is seen between the recovered combined stress field, the regional tectonic boundaries and the seismicity of the World Stress Map 2008 database.

Analysis of the full stress tensor in a micropillar: Ability of and difficulties arising during synchrotron based μLaue diffraction

The full strain tensor present in a 2·2·6μm3 sized germanium micro pillar was measured by X-ray microdiffraction Laue during in situ loading. Initial orientation gradients, residual strains and diffraction peak streaking were found negligible. The obtained strain values from Laue pattern analysis have been used to calculate the deviatoric stress tensor. The collected patterns show a pillar rotation and a build-up of shear stress components acting parallel to the contact surface of the flat punch, which is both an indication for misalignment of the sample and counter-body during loading. Based on the assumption of stress free pillar-side faces the full stress tensor was further evaluated. The stresses acting in loading direction derived from the Laue patterns compared to engineering stresses calculated from the continuously recorded signal of the load cell are in agreement within 9%. The discrepancy is explained by the initial experimental misalignment of the pillar with respect to the counter-body. Finally, difficulties arising during such an in situ experiment are discussed.

A cross-anisotropic formulation for elasto-plastic models

A cross-anisotropic formulation for elasto-plastic constitutive models based on a non-uniform scaling of the stress tensor is described. Taking advantage of the material symmetries characterising cross-anisotropy, only two scaling factors, one for the normal stress components and one for the shear stress components, are required. It is shown that the formulation can be easily introduced in already implemented models with minor modifications. The performance of this formulation is investigated by reproducing the strength variation of anisotropic rocks in triaxial tests. The numerical simulation of an unsupported excavation is also presented to show the effect of different scaling factors and bedding plane orientations.

High shear stress relates to intraplaque haemorrhage in asymptomatic carotid plaques

Background and aimsCarotid artery plaques with vulnerable plaque components are related to a higher risk of cerebrovascular accidents. It is unknown which factors drive vulnerable plaque development. Shear stress, the frictional force of blood at the vessel wall, is known to influence plaque formation. We evaluated the association between shear stress and plaque components (intraplaque haemorrhage (IPH), lipid rich necrotic core (LRNC) and/or calcifications) in relatively small carotid artery plaques in asymptomatic persons. MethodsParticipants (n = 74) from the population-based Rotterdam Study, all with carotid atherosclerosis assessed on ultrasound, underwent carotid MRI. Multiple MRI sequences were used to evaluate the presence of IPH, LRNC and/or calcifications in plaques in the carotid arteries. Images were automatically segmented for lumen and outer wall to obtain a 3D reconstruction of the carotid bifurcation. These reconstructions were used to calculate minimum, mean and maximum shear stresses by applying computational fluid dynamics with subject-specific inflow conditions. Associations between shear stress measures and plaque composition were studied using generalized estimating equations analysis, adjusting for age, sex and carotid wall thickness. ResultsThe study group consisted of 93 atherosclerotic carotid arteries of 74 participants. In plaques with higher maximum shear stresses, IPH was more often present (OR per unit increase in maximum shear stress (log transformed) = 12.14; p = 0.001). Higher maximum shear stress was also significantly associated with the presence of calcifications (OR = 4.28; p = 0.015). ConclusionsHigher maximum shear stress is associated with intraplaque haemorrhage and calcifications.

Variations of bed elevations due to turbulence around submerged cylinder in sand beds

This paper presents the spatio-temporal variations in bed elevations and the near-bed turbulence statistics over the deformed bed generated around the submerged cylindrical piers embedded vertically on loose sediment bed at a constant flow discharge. Experiments were carried out in a laboratory flume for three blockage ratios in the range of 0.04–0.06 using three different sizes of submerged cylinders individually placed vertically at the centerline of the flume. Clear-water experimental conditions were maintained over the smooth sediment bed surface with a constant flow discharge (\(Q = 0.015\,{\rm m}^3/{\rm sec}\)), thereby giving three different cylinder Reynolds numbers \(Re_{D_c} = \frac{U_mD_c}{\nu }\) (=10200, 12750, 15300) away from the cylinder locations, where \(U_m\) is the maximum mean velocity, \(D_c\) is the cylinder diameter and \(\nu\) is the kinematic viscosity of fluid. Instantaneous sand bed elevations around the cylinders were recorded using a SeaTek 5MHz ultrasonic ranging system of net 24 transducers to estimate bed form migration, and the near-bed velocity data at transducer locations over the stable deformed bed around the pier-like structures were collected using down-looking three-dimensional (3D) Micro-acoustic Doppler velocimeter to estimate the bottom Reynolds shear stresses and the contributions of bursting events to the dominant shear stress component. The flow perturbation generated due to relatively lower flow blockage ratio favored to achieve the stable bed condition more rapidly than the others, and larger upstream scour-depth and deformed areas were noticed for greater flow blockage ratio due to larger cylinder diameter. For larger blockage ratio in the upstream of scour-hole near the bed, occurrences of probabilities of both boundary-ward interactions (Q1 and Q3) were the dominant; whereas in the downstream of the scoured region, occurrences of probabilities of second and third quadrant events (Q2 and Q4) were dominant. On the other hand, for the lower blockage ratio, quadrant (Q2) was dominant over Q4 in the downstream of scour-hole, and in the upstream of scour-hole, quadrant Q4 was the dominant.

Multiple parallel cracks in a magnetoelectroelastic strip under in-plane magnetic/electric field and clamping-induced antiplane pre-deformation

Many magnetoelectroelastic devices are mechanically clamped during service, and the clamping may lead to pre-deformation, which will in return affect the magnetoelectroelastic behavior. In the present paper, a theoretical model is constructed to formulate the fracture problem of a clamped magnetoelectroelastic strip loaded magnetically or electrically. Two groups of multiple unequal-length cracks, parallel to the lateral boundaries, are assumed to exist in the strip. In the case of antiplane deformation, Green’s functions of the shear stress components are derived by the method of dislocation simulation, and then they are employed to construct singular integral equations (SIEs) for the cracks. The SIEs are numerically solved to get the stress intensity factors (SIFs). It is found that the SIFs are linearly dependent on the applied magnetic/electric field and the clamping-induced antiplane pre-deformation. In addition, parametric studies are performed to reveal the effects of geometrical parameters on the SIFs, and the crack shielding and interference are regarded as the underlying mechanisms for the fracture behavior.

Reynolds Stress Structures in the Hybrid RANS/LES of a Planar Channel

The near-wall cycle of hybrid RANS/LES is studied by calculating the flow through a planar channel. Statistical results are commented on and related to instantaneous structures which are extracted from the flow field. The problematic structures in the artificial near-wall cycle, well known to be super-streaks, are identified and quantified. The calibration of such closures provides a correct mixing length argument in the logarithmic layer. However because of these overly intense streamwise streaks, it is impossible to simultaneously predict the Reynolds streamwise normal and shear stress components correctly. Further, because the location of the RANS-LES interface changes spatially and temporally, we see these structures are more free to move vertically and this further worsens statistical results.

MHD free convection flow of Eyring–Powell fluid from vertical surface in porous media with Hall/ionslip currents and ohmic dissipation

A mathematical study is presented to analyze the nonlinear, non-isothermal, magnetohydrodynamic (MHD) free convection boundary layer flow, heat and mass transfer of non-Newtonian Eyring–Powell fluid from a vertical surface in a non-Darcy, isotropic, homogenous porous medium, in the presence of Hall currents and ionslip currents. The governing nonlinear coupled partial differential equations for momentum conservation in the x, and z directions, heat and mass conservation, in the flow regime are transformed from an (x,y,z) coordinate system to a (ξ,η) coordinate system in terms of dimensionless x-direction velocity (f′) and z-direction velocity (G), dimensionless temperature and concentration functions (θ and ϕ) under appropriate boundary conditions. Both Darcian and Forchheimer porous impedances are incorporated in both momentum equations. Computations are also provided for the variation of the x and z direction shear stress components and also heat and mass transfer rates. It is observed that with increasing ɛ, primary velocity, secondary velocity, heat and mass transfer rates are decreased whereas, the temperature, concentration and skin friction are increased. An increasing δ is found to increase primary and secondary velocities, skin friction, heat and mass transfer rates. But the temperature and concentration decrease. Increasing βe and βi are seen to increase primary velocity, skin friction, heat and mass transfer rates whereas secondary velocity, temperature and concentration are decreased. Excellent correlation is achieved with a Nakamura tridiagonal finite difference scheme (NTM). The model finds applications in magnetic materials processing, MHD power generators and purification of crude oils.

Influence of substrate microstructure and surface finish on cracking and delamination response of TiN-coated cemented carbides

The cracking and delamination of TiN-coated hardmetals (WC-Co cemented carbides) when subjected to Brale indentation were studied. Experimental variables were substrate microstructure related to low (6wt% Co) and medium (13wt% Co) binder content, and surface finishes associated with grinding and polishing stages before film deposition. Brale indentation tests were conducted on both coated and uncoated hardmetals. Emphasis has been placed on assessing substrate microstructure and subsurface finish effects on load levels at which cracking and delamination phenomena emerge, the type of cracking pattern developed, and how fracture mechanisms evolve with increasing load. It is found that polished and coated hardmetals are more brittle (radial cracking) and the adhesion strength (coating delamination) diminishes with decreasing binder content. Such a response is discussed on the basis of the influence of intrinsic hardness/brittleness of the hardmetal substrate on both cracking at the subsurface level and effective stress state, particularly regarding changes in shear stress component. Grinding promotes delamination compared to the polished condition, but strongly inhibits radial cracking. This is a result of the interaction between elastic–plastic deformation imposed during indentation and several grinding-induced effects: remnant compressive stress field, pronounced surface texture and microcracking within a thin altered subsurface layer. As a consequence, coating spallation prevails over radial cracking as the main mechanism for energy dissipation in ground and coated hardmetals.

Computational Study of MHD Free Convection Flow of Non-Newtonian Tangent Hyperbolic Fluid from a Vertical Surface in Porous Media with Hall/Ionslip Currents and Ohmic Dissipation

A numerical study is presented to analyze the nonlinear, non-isothermal, magnetohydrodynamic (MHD) free convection boundary layer flows of non-Newtonian tangent hyperbolic fluid past a vertical surface in a non-Darcy, isotropic, homogenous porous medium in the presence of Hall currents and Ionslip currents. The governing nonlinear coupled partial differential equations for momentum conservation in x and z directions, heat and mass conservation in the flow regime are transformed from an (x, y, z) coordinate system to (\(\upxi ,\upeta \)) coordinate system in terms of dimensionless x-direction velocity (\(f^{\prime }\)) and z-direction velocity (G), dimensionless temperature and concentration functions (\(\uptheta \) and \(\upphi \)) under appropriate boundary conditions. Both Darcian and Forchheimer porous impedances are incorporated in both momentum equations. Computations are also provided for the variation of the x and z direction shear stress components and also heat and mass transfer rates. Increasing Weissenberg number (We) is observed to decrease primary and secondary velocity and concentration but increase temperature. It is found that the primary and secondary velocity is increased with increasing power law index (n) whereas the temperature and concentration are decreased. Increasing hall and Ionslip current (\(\beta _{e}\) and \(\beta _{i}\)) is observed to increase primary velocity but decreases secondary velocity, temperature and concentration. Increasing magnetic parameter (\(N_{m}\)) is seen to decrease primary velocity but decreases secondary velocity, temperature and concentration. Increasing Darcy number (Da) is found to increase primary and secondary velocity whereas decreases temperature and concentration. Increasing Forchheimer parameter (Fs) is seen to decrease primary and secondary velocity but increases temperature and concentration. The model finds applications in magnetic materials processing, MHD power generators and purification of crude oils.

Computational implementation of a non-linear kinematic hardening formulation for tension–torsion multiaxial fatigue calculations

The calculation of elastoplastic strains from stress histories, or vice-versa, is an important computational step in low-cycle fatigue analyses. This step is a challenging task for general multiaxial non-proportional (NP) loading histories, where the principal stress directions are not constant, requiring 6D incremental plasticity calculations to correlate the six stress with the six strain components considering plasticity effects. However, a large number of multiaxial fatigue problems only involve combined tension and/or bending and torsion loads, which are associated with only one normal and one shear stress component. The use of a special 2D formulation, instead of 6D, can greatly simplify the necessary incremental plasticity calculations for these practical problems. In this work, a new 2D tension–torsion incremental plasticity formulation is introduced, integrating non-linear kinematic (NLK) hardening models and NP hardening effects in a very efficient way, exactly reproducing tension–torsion calculations from more general 6D models, but with less than one fifth of the computational cost. The proposed 2D approach is validated by comparing NP strain-controlled tension–torsion experiments in 316L steel tubular specimens, a material that presents significant NP hardening effects, with experimental and predicted stress paths, calculated either with 6D or the proposed 2D formulation.

進行波状壁面変形によるチャネル乱流抵抗低減のパラメトリックスタディ

The effect of streamwise wave-like wall deformation of a turbulent channel is studied through direct numerical simulations. Here, the simulation was conducted under constant pressure gradient, unlike past studies, which was held under constant flow rate conditions, in order to prevent friction velocity changes before and after the control for concrete discussions on mechanism of drag reduction. Simulations reveal that wall deformation grants approximately 40% drag reduction, due to random component of velocity fluctuation drop and negative periodic Reynolds shear stress.

Evolution of the ultrasonic resonance modes in a three-layer structure with change of material and interface adhesion properties

The quantitative non-destructive evaluation (NDE) of interface adhesion has long been a challenge for the safe use of bonding structures. It is difficult to predict the adhesion resistance force between adhesive and adhered material without performing destructive testing. Ultrasonic approach seems to be the only potential way for its NDE based on the reason of mechanical nature of the problem. Different ultrasonic techniques, such as bulk wave echography, reflection resonance, and Lamb guided waves, have been used to evaluate the interface adhesion strength. But no direct relation between the interfacial bonding strength and the ultrasonic measurement has been established. The most used compression wave echography and resonance at normal incidence are less sensitive to the interface condition, except for a disbond. It is essential that the interface should be excited with a shear stress component to increase the measurement sensibility. But it is not easy to excite the interface by using shear waves in experiment, while the use of guided waves will encounter the problems of high attenuation and mode selection as all modes are not sensitive to a certain interface in a bonded structure. A previous study has shown that the V (z) inversion technique can be used to perform a multimode measurement on a layered structure, where both compression and shear stress resonance occur. This method has the advantage in using a simple experimental setup working at the normal incidence with a focus transducer of large angular aperture. The inversed angular-frequency reflectance function R(; f) gives the resonance modes which are equivalent to the Lamb type guided modes, while it is a local determination of the wave mode, thus the difficulty in guided wave measurement above mentioned can be avoided. The first part of the paper contains the development of the theoretical model for wave propagation in a multilayered structure where three-layer sandwich bonded structures can be considered as a particular case. A weak interfacial adhesion is described by two interface compression and shear stiffness parameters, namely km and kt. By integrating the transfer matrix formalism under the non-ideal boundary conditions, the plane wave angular (incident angle) and frequency reflection coefficient function R(; f) for a liquid immersed asymmetric metal-adhesive-metal three-layer and its dispersion curves of guided mode waves with or without charge are calculated. It is confirmed that the evolutions of the reflection zeros (mode resonances) correspond to the dispersion curves of the guided waves of the same structure without charge. Furthermore, the resonance modes observed in R(; f) can be considered as a combination of the respective Lamb modes of the top and bottom single metal layers coupled through the modes conditioned by the middle adhesive layer and the its interface conditions. The second part of the paper shows the behaviors of the resonance modes by changing the parameters related to the bonding strength. The acoustical impedance, the mass density and the thickness of the adhesive layer, which are related to the cohesive property, and the shear interfacial stiffness coefficient kt which conditions the adhesive property, are changed respectively to observe the resonance mode evolutions. The mode evolutions due to each parameter are analyzed and differentiated. It can be concluded that the change in the adhesion strength of the bonding structure does not affect significantly the modes belonging to those inherent to the two adhered aluminum layers, while the coupling modes will be shifted in frequency and exchange with or replace the said inherent modes. It is expected that the obtained results in this study will be of significance for quantitatively characterizing the interfacial properties of an adhesively bonded layered structure by using the V (z) inversion technique.

Sub-critical cohesive crack propagation with hydro-mechanical coupling and friction

Looking at the long-time behaviour of a dam, it is necessary to assume that the water can penetrate a possible crack washing away some components of the concrete. This type of corrosion reduces the tensile strength and fracture energy of the concrete compared to the same parameters measured during a short-time laboratory test. This phenomenon causes the so called sub-critical crack propagation. That is the reason why the International Commission of Large Dams recommends to neglect the tensile strength of the joint between the dam and the foundation, which is the weakest point of a gravity dam. In these conditions a shear displacement discontinuity starts growing in a point, called Fictitious Crack Tip (shortened FCT), which is still subjected to a compression stress. In order to manage this problem, in this paper the cohesive crack model is re-formulated with the focus on the shear stress component. In this context, the classical Newton-Raphson method fails to converge to an equilibrium state. Therefore the approach used is based on two stages: (a) a global one in which the FCT is moved ahead of one increment; (b) a local one in which the non-linear conditions occurring in the Fracture Process Zone are taken into account. This two-stage approach, which is known in the literature as a Large Time Increment method, is able to model three different mechanical regimes occurring during the crack propagation between a dam and the foundation rock.

Coupled electro-elastic analysis of functionally graded piezoelectric material plates

A unified formulation of finite layer methods (FLMs), based on the Reissner mixed variational theorem (RMVT), is developed for the three-dimensional (3D) coupled electro-elastic analysis of simply-supported, functionally graded piezoelectric material (FGPM) plates with open- and closed-circuit surface conditions and under electro-mechanical loads. In this formulation, the material properties of the plate are assumed to obey an exponent-law varying exponentially through the thickness coordinate, and the plate is divided into a number of finite rectangular layers, in which the trigonometric functions and Lagrange polynomials are used to interpolate the in- and out-of-plane variations of the primary field variables of each individual layer, respectively, such as the elastic displacement, transverse shear and normal stress, electric potential, and normal electric displacement components. The relevant orders used for expanding these variables in the thickness coordinate can be freely chosen as the linear, quadratic and cubic orders. Four different mechanical/electrical loading conditions applied on the top and bottom surfaces of the plate are considered, and the corresponding coupled electro-elastic analysis of the loaded FGPM plates is undertaken. The accuracy and convergence rate of the RMVT-based FLMs are assessed by comparing their solutions with the exact 3D piezoelectricity ones available in the literature.

Abstract 19155: Quantification of Progressive Pulmonary Valvular Insufficiency Using 4-Dimensional Flow Magnetic Resonance Imaging in an Ovine Model

Introduction: Chronic postoperative pulmonary insufficiency (PPI) is the major cause of long-term morbidity and mortality in patients (pts) with tetralogy of Fallot (TOF). Pulmonary valve replacement (PVR) can mitigate the risk, but optimal timing for PVR remains controversial. Time-resolved 3D phase-contrast MRI (4DF) provides insight into complex intracardiac flow patterns. 4DF may help inform management decisions regarding timing of PVR. We describe the first quantitative serial 4DF analysis in an ovine model of PPI. Methods: A baseline cardiac MRI (CMRI) was performed on four Dorsett sheep (36-39kg) on a 3-Tesla Siemens scanner. Following baseline MRI, animals underwent pulmonary valvectomy and transannular patch (TAP) placement yielding PPI. Follow-up CMRI were obtained at 5mo and 7mo post-valvectomy (PoVa). Pathlines were emitted from a circular region of interest (ROI) representing the pulmonary valve annulus. Velocities, flow rate, and stresses were obtained throughout a complete cardiac cycle and compared across time. Results: Systolic max velocity, average velocity, and flow rate normal to the ROI plane decreased over time. Diastolic velocities increased at 5mo before decreasing at 7mo PoVa. Regurgitant flow was observed in diastole PoVa , and regurgitant flow rate increased with time. The max and average component of shear stress (viscosity = 3 cP) along the pulmonary artery wall decreased over time in systole. In diastole, shear stress increased at 5mo and decreased at 7mo. Parameters plotted in Figure 1. Conclusion: 4DF provides a new method for visualization and quantification of alterations in blood flow patterns in the right side of the heart in an ovine model of PPI. Alterations in above parameters may indicate myocardial and pulmonary vascular deterioration precedent to morphological changes. Characterization of altered flow patterns in human pts with TOF may help to ultimately inform decisions regarding timing of PVR.

Computational solutions for non-isothermal, nonlinear magneto-convection in porous media with hall/ionslip currents and ohmic dissipation

A theoretical and numerical study is presented to analyze the nonlinear, non-isothermal, magnetohydrodynamic (MHD) free convection boundary layer flow and heat transfer in a non-Darcian, isotropic, homogenous porous medium, in the presence of Hall currents, Ionslip currents, viscous heating and Joule heating. A power-law variation is used for the temperature at the wall. The governing nonlinear coupled partial differential equations for momentum conservation in the x and z directions and heat conservation, in the flow regime are transformed from an (x, y, z) coordinate system to a (ξ,η) coordinate system in terms of dimensionless x-direction velocity (∂F/∂η) and z-direction velocity (G) and dimensionless temperature function (H) under appropriate boundary conditions. Both Darcian and Forchheimer porous impedances are incorporated in both momentum equations. Computations are also provided for the variation of the x and z direction shear stress components and also local Nusselt number. Excellent correlation is achieved with a Nakamura tridiagonal finite difference scheme (NTM). The model finds applications in magnetic materials processing, MHD power generators and purification of crude oils.

Analysis of MHD flow of Burgers’ fluid with heat transfer in helical screw rheometer

This paper aims to study the influence of Hall current and heat transfer on the magnetohydrodynamics (MHD) flow of an electrically conducting, incompressible Burgers’ fluid in a helical screw rheometer. The screw and barrel are electrically insulated and kept at two different constant temperatures. A uniform magnetic field is applied perpendicular to the flow. Exact solutions are obtained for the velocity profile, volume flow rate, temperature distribution and rate of heat transfer. Expressions for the shear and normal stress components are also calculated. It is observed that the Burgers’ fluid parameters contribute in normal and tangential stresses, only. The effects of Hall parameter, Hartmann number, pressure distribution, and Brinkman number are investigated on flow profile, temperature distribution, and heat flux.

A simplified three-dimensional displacement discontinuity method for multiple fracture simulations

The displacement discontinuity method (DDM) developed by Crouch (Int J Numer Methods Eng 10:301–343, 1976) is a popular form of the boundary element method and is widely used to model hydraulic fracture propagation. The two-dimensional displacement discontinuity method (2D DDM) has limitations with regard to the simulation of some practical fracture problems, which often require accounting for the three-dimensional (3D) nature of the fracture. A 2D method with a 3D correction factor proposed by Olson (The initiation, propagation, and arrest of joints and other fractures, vol 231. Geological Society of London Special Publication, London, pp 73–87, 2004) is able to account for 3D effects of a single fixed-height, embedded fracture. However, this correction factor proves inadequate for describing multiple fracture interaction. Greater accuracy is clearly possible with a truly three-dimensional displacement discontinuity method (3D DDM), but such an approach requires significantly higher computational time and memory, especially for simulating multiple fracture propagation. To enhance calculation efficiency and reduce memory usage, a novel, simplified 3D DDM approach is proposed. This method is simplified from 3D DDM through excluding non-vertical fractures and vertical components of shear stress, as well as eliminating the need for discretization in the vertical (height) direction by applying correction factors to improve fracture-induced stresses. The correction factors are derived from the analytical plane strain solution for a uniformly loaded, isolated, vertical fracture of finite height and infinite length. The simplified method not only can calculate displacement discontinuities and induced stresses for single 3D fracture, but also can accurately and efficiently solve the problem of multiple 3D interacting fracture, improving computational efficiency by more than one thousand times compared with the standard 3D method. This model has been used to simulate the propagation of multiple fractures for horizontal wells. The approach should be generally applicable to other 3D boundary element methods to enhance computation efficiency.