Shear Anisotropy Research Articles

Improving the quality and processing efficiency of beef jerky via drying in confined conditions of pre-stretching

Inspired by the mechanical enhancement of hydrogel via drying in confined conditions, we applied this strategy to beef jerky manufacture for improving the quality and processing efficiency. In our study, beef strips were pre-stretched and then dried in a tensile state, and the confined conditions were achieved by controlling the stretched strains from 20% to 120%. Compared with the sample dried freely, beef jerky dried in confined conditions of different pre-stretching strains exhibited improved quality based on texture and sensory analysis. Additionally, this method also enhanced processing efficiency by reducing approximately 50% drying time. The excellent sensory quality and good texture of beef jerky were obtained as the pre-stretching strain was 80%. Drying beef strips in confined conditions made muscle fibers tense and enhanced hydrophobicity of myofibrillar proteins, leading to a compact structure with high shear force and anisotropy, and rapid water loss in beef jerky. This facile and green method provides a promising route to enrich the existing technologies of jerky processing.

Elastic properties of A 2Ti6O13 ( H, Li, Na, K and Rb): a computational study

The elastic properties of the alkali hexatitanate family A 2Ti6O13 (A = H, Li, Na, K, and Rb) are investigated based on density functional theory within a generalized gradient approximation plus Hubbard U (GGA+U) approach. The results showed that all members of the family are wide-band semiconductors and the calculated lattice parameters are consistent with experimental values. In terms of mechanical stability, the results indicated that the alkali hexatitanates are highly incompressible to uniaxial stress, with the largest elastic constant C22 reaching values as high as 265 GPa in K2Ti6O13. The obtained elastic constants, using the stress–strain method, were used to calculate bulk modulus, shear modulus, Young’s modulus, brittleness and ductility, elastic anisotropy, Vickers hardness, sound velocities, and the Debye temperature. It was found that the member of the family with the highest atomic number of the alkaline group, Rb2Ti6O13, had the highest values of bulk, shear, and Young’s modulus, as well as the lowest values of shear and compression anisotropy, and a high Vickers hardness.

Extreme seismic anisotropy indicates shallow accumulation of magmatic sills beneath Yellowstone caldera

Understanding the distribution and mobility of crystal mushes within modern magmatic systems is crucial to volcanic hazard assessments as distinct pockets of mobile magma may become interconnected and lead to melt accumulation on shorter time scales than magma that is broadly distributed in a homogeneous mush. Here, we reveal that Yellowstone's upper-crustal magma reservoir in the top 20 km is heterogeneous in both melt concentration and texture. We exploit ambient noise in an unprecedented dense temporary seismic network to jointly constrain vertically- and horizontally-polarized shear wave speeds to create enhanced 3D isotropic and anisotropic shear velocity models. Our models show an exceptionally low-velocity (>20% reduction) layer 4–7 km beneath the surface, situated near the top of the reservoir previously-imaged by earthquake P-wave tomography. The presence of strong radial anisotropy (20%) within this layer indicates the uppermost portion of the modern Yellowstone reservoir is organized as a sill complex, with up to 28% of melt fraction in horizontally-elongated volumes at depths where rhyolite was commonly stored before past eruptions. The findings demonstrate that high-resolution anisotropic imaging through a dense seismic network can constrain both magma distribution and reservoir texture, which are important to understand the evolution and hazard assessment of volcanic systems like Yellowstone.

Anisotropic time-dependent behaviors of shale under direct shearing and associated empirical creep models

Understanding the anisotropic creep behaviors of shale under direct shearing is a challenging issue. In this context, we conducted shear-creep and steady-creep tests on shale with five bedding orientations (i.e. 0°, 30°, 45°, 60°, and 90°), under multiple levels of direct shearing for the first time. The results show that the anisotropic creep of shale exhibits a significant stress-dependent behavior. Under a low shear stress, the creep compliance of shale increases linearly with the logarithm of time at all bedding orientations, and the increase depends on the bedding orientation and creep time. Under high shear stress conditions, the creep compliance of shale is minimal when the bedding orientation is 0°, and the steady-creep rate of shale increases significantly with increasing bedding orientations of 30°, 45°, 60°, and 90°. The stress-strain values corresponding to the inception of the accelerated creep stage show an increasing and then decreasing trend with the bedding orientation. A semilogarithmic model that could reflect the stress dependence of the steady-creep rate while considering the hardening and damage process is proposed. The model minimizes the deviation of the calculated steady-state creep rate from the observed value and reveals the behavior of the bedding orientation's influence on the steady-creep rate. The applicability of the five classical empirical creep models is quantitatively evaluated. It shows that the logarithmic model can well explain the experimental creep strain and creep rate, and it can accurately predict long-term shear creep deformation. Based on an improved logarithmic model, the variations in creep parameters with shear stress and bedding orientations are discussed. With abovementioned findings, a mathematical method for constructing an anisotropic shear creep model of shale is proposed, which can characterize the nonlinear dependence of the anisotropic shear creep behavior of shale on the bedding orientation.

Transformation hydrodynamic metamaterials: Rigorous arguments on form invariance and structural design with spatial variance.

Transformation optics has been a powerful tool to manipulate physical fields if the governing equationsin two spaces satisfy a certain form invariance. A recent interest has been applying this method to design hydrodynamic metamaterials described by the Navier-Stokes equations. However, transformation optics may not be applicable to such a general fluid model while especially the rigorous analysis is still absent. In this work, we give a definite criterion of the required form invariance, i.e., metric of one space and its affine connections that appear in curvilinear coordinates can be incorporated into material properties or interpreted by extra physical mechanisms introduced in another space. Based on this criterion, we prove that the Navier-Stokes equations, as well as its simplification in creeping flows (the Stokes equation), cannot be form invariant due to the redundant affine connections from their viscous terms. On the contrary, the creeping flows under the lubrication approximation, saying the classical Hele-Shaw model and its anisotropic counterpart, can retain the form of their governing equationsfor steady incompressible isothermal Newtonian fluids. Besides, we propose the design of multilayered structures with spatially varying cell depth to mimic the required anisotropic shear viscosity for modulating Hele-Shaw flows. Our results correct previous misunderstandings about the applicability of transformation optics under the Navier-Stokes equations, reveal the decisive role of lubrication approximation in maintaining form invariance (consistent with recent experiments featuring shallow configurations), and provide a feasible approach in experimental fabrication.

Seismic Azimuthal Anisotropy of Northeastern Tibetan Plateau From Ambient Noise Double Beamforming Tomography: Implications for Crustal Deformation

AbstractIn this study, we obtain a three‐dimensional (3‐D) high resolution seismic anisotropy model of NE Tibet using ambient noise double beamforming (DBF) tomography with ChinArray‐Himalaya II data. DBF tomography technique allows for the simultaneous derivation of local phase velocity and azimuthal anisotropy from ambient noise cross‐correlations without performing tomographic inversion. Based on the DBF tomography results, we have constructed a high resolution 3‐D azimuthal anisotropic shear wave velocity model of NE Tibet. Our model indicates the presence of prominent low velocities and strong azimuthal anisotropy in the mid‐to lower crust beneath the Songpan‐Ganzi belt, while the mid‐to lower crust of the Qilian belt exhibits relatively weak low velocity and azimuthal anisotropy. We propose that crustal channel flow may be the primary mechanism driving crustal deformation in the Songpan‐Ganzi belt, which is in the mature part of the plateau. As the Qilian belt is at an early stage of plateau formation, it is characterized by crustal shortening with the development of a sub‐vertical foliation. Additionally, the misalignment between predicted and observed shear wave splitting reveals decoupled crust and upper mantle deformation west of the Ordos block where eastward mantle flow is deflected by the lithospheric keel of Ordos.

A DFT study of optoelectronic, elastic and thermo-electric properties of the double perovskites Rb2SeX6 (X=Br,Cl)

Thermo-electric (TE) material applications reduce reliance on traditional energy resources by converting heat to electric energy. We have studied, for the first time, the thermo-electric properties of Rb2SeX6 (X=Br,Cl). Using norm-conserving pseudo potentials in a plane wave basis set of Quantum Espresso code, the optoelectronic, elastic and thermo-electric properties of Rb2SeX6 (X=Br,Cl) have been investigated using density functional theory. Generalized Gradient Approximation of Perdew Burke Ernzerhof (GGA-PBE) and Generalized Gradient Approximation of Perdew Burke Ernzerhof adapted for Solid (GGA-PBESol) exchange correlation functionals were employed in all calculations. The band structure plots suggest that the studied double perovskites have indirect band gaps. Rb2SeBr6 band gap values of 1.7574/ 1.569 eV (using GGA-PBE/PBEsol) are remarkably similar to that of two effective inorganic/organic perovskites FAPbI3 and MAPbI3 . Maximum peaks generated from refractive index results indicate possible solar cell uses of the materials because they are in the visible and ultraviolet ranges. The results of other optical properties such as absorption coefficients, electron energy loss, conductivity, and reflectivity concludes that Rb2SeX6 (X=Br,Cl) have good values for electron generation, high potential for applications in the optoelectronic industry and are semiconductor in nature. The calculated shear anisotropy values of Rb2SeBr6/Cl6 are 3.09/1.71, suggesting that they are isotropic materials. With calculated Poisson’s ratio of 0.32 and 0.26, the materials are predicted to be ductile in nature. The two materials are appropriate for thermo-electric applications since their thermal to electrical conductivity ratio are small (the order of 10-5). The calculated minimum values of Seebeck coefficient values of 0.198×103 / 0.166 ×103 (mV/K) at 750 K, for Rb2SeBr6/Cl6 are positive, indicating that they have p-type conduction. Figure of merit values at all temperature range considered are greater than one (ZT > 1) for both Rb2SeBr6 and Rb2SeCl6, suggesting that they are good thermo-electric materials. The results of the calculations provide the basis for the industrial application of Rb2SeBr6/Cl6 as solar cells.

Physics-based and data-driven modeling for stability evaluation of buried structures in natural clays

This study presents a hybrid framework to predict stability solutions of buried structures under active trapdoor conditions in natural clays with anisotropy and heterogeneity by combining physics-based and data-driven modeling. Finite-element limit analysis (FELA) with a newly developed anisotropic undrained shear (AUS) failure criterion is used to identify the underlying active failure mechanisms as well as to develop a numerical (physics-based) database of stability numbers for both planar and circular trapdoors. Practical considerations are given for natural clays to three linearly increasing shear strengths in compression, extension, and direct simple shear in the AUS material model. The obtained numerical solutions are compared and validated with published solutions in the literature. A multivariate adaptive regression splines (MARS) algorithm is further utilized to learn the numerical solutions to act as fast FELA data-driven surrogates for stability evaluation. The current MARS-based modeling provides both relative importance index and accurate design equations that can be used with confidence by practitioners.

A divide-and-conquer machine learning approach for modeling turbulent flows

In this paper, a novel zonal machine learning (ML) approach for Reynolds-averaged Navier–Stokes (RANS) turbulence modeling based on the divide-and-conquer technique is introduced. This approach involves partitioning the flow domain into regions of flow physics called zones, training one ML model in each zone, then validating and testing them on their respective zones. The approach was demonstrated with the tensor basis neural network (TBNN) and another neural net called the turbulent kinetic energy neural network (TKENN). These were used to predict Reynolds stress anisotropy and turbulent kinetic energy, respectively, in test cases of flow over a solid block, which contain regions of different flow physics including separated flows. The results show that the combined predictions given by the zonal TBNNs and TKENNs were significantly more accurate than their corresponding standard non-zonal models. Most notably, shear anisotropy component in the test cases was predicted at least 20% and 55% more accurately on average by the zonal TBNNs compared to the non-zonal TBNN and RANS, respectively. The Reynolds stress constructed with the zonal predictions was also found to be at least 23% more accurate than those obtained with the non-zonal approach and 30% more accurate than the Reynolds stress predicted by RANS on average. These improvements were attributed to the shape of the zones enabling the zonal models to become highly locally optimized at predicting the output.

Three-dimensional undrained face stability of circular tunnels in non-homogeneous and anisotropic clays

The advanced 3D finite element limit analysis is employed to examine three-dimensional (3D) face stability of circular tunnels in undrained clays with the impact of strength non-homogeneity and anisotropy. The soil behaviour for all analyses is governed by the anisotropic undrained shear (AUS) strength model, which requires three anisotropic undrained shear strengths that are simply evaluated from conventional laboratory tests. The upper and lower bound solutions of 3D face stability of circular tunnels are accurately analyzed by considering four dimensionless parameters, including the cover-to-depth ratio, the overburden stress factor, the strength gradient ratio, and the anisotropic strength ratio. The significance of four dimensionless parameters on the stability load factor of the 3D face stability of circular tunnels, and the predicted failure mechanisms due to the influence of cover depth ratios, are also illustrated. A nonlinear regression analysis is employed to the computed data, which enables a new design equation of the factor of safety incorporating the conventional stability number. Thus, the new tunnel stability factors are proposed for the first time in the literature. A comprehensive application of the proposed safety factor equation is also presented and compared. The 3D effects of the tunnel face configuration between 3D geometry and 2D plane strain heading on the tunnel stability factors are also discussed. The new proposed design equation of the factor of safety provides an accurate and practical tool in assessing face stability of circular tunnels in non-homogeneous and anisotropic clays.

Large eddy simulations of fully developed turbulent flows over additively manufactured rough surfaces

In the last decade, with the growing demand for efficient and more sustainable products that reduce our CO2 footprint, progresses in Additive Manufacturing (AM) have paved the way for optimized heat exchangers, whose disruptive design will heavily depend on predictive numerical simulations. Typical AM rough surfaces show limited resemblance to the artificially constructed rough surfaces that have been the basis of most prior fundamental research on turbulent flow over rough walls. Hence, current wall models used in steady and unsteady three-dimensional (3D) Navier–Stokes simulations do not consider such characteristics. Therefore, a high-fidelity Large Eddy Simulation (LES) database is built to develop and assess novel wall models for AM. This article investigates the flow in rough pipes built from the surfaces created using AM techniques at Siemens based on Nickel Alloy IN939 material. We developed a code to generate the desired rough pipes from scanned planar surfaces. We performed high-fidelity LES of turbulent rough pipe flows at Reynolds number, Re = 11 700, to reveal the influence of roughness parameters on turbulence, mainly the average roughness height and the effective slope. The equivalent sand-grain roughnesses, ks, of the present AM rough surfaces are predicted using the Colebrook correlation. The main contributors to the skin friction coefficient are found to be turbulence and drag forces. In the present study, the existence of a logarithmic layer is marked even for high values of ks. The mean flow, the velocity fluctuations, and the Reynolds shear stresses show turbulence's strong dependence on the roughness topography. Profiles of turbulence statistics are compared by introducing an effective wall-normal distance defined as zero-plane displacement. The effective distance collapses the shear stresses and the velocity fluctuations outside the roughness sublayer; thus, Townsend's similarity of the streamwise mean velocity is marked for the present roughnesses. Furthermore, a mixed scaling is introduced to improve the collapse of turbulence statistics in the roughness sublayer. In addition, an attempt to investigate the impact of surface roughness on flow physics using the acquired LES results based on quadrant analysis of the Reynolds shear stresses and anisotropy of turbulence is made.

Theoretical predict the structure, elastic anisotropy and thermodynamic properties of Al5W in Al-rich region

Al5W compound is a promising high-temperature material due to the high melting point, good mechanical properties and good corrosion resistance. However, the structural feature of Al5W remains controversy. To solve these problems, we apply the first-principles calculations to study the structural stability, elastic properties, elastic anisotropy and melting point of Al5W. Four possible Al5W phases are discussed based on the similar A5B-type structure. The results show that two novel Al5W phases: orthorhombic (Cmma) and rhombohedral (R-3c) structures are predicted. The hexagonal (P6/mmm) Al5W has better thermodynamic stability. The bulk modulus of the predicted Cmma and R-3c phases is close to the known P63 phase. The result demonstrates the brittle behavior of the P63 Al5W. However, the hexagonal (P6/mmm) Al5W shows lower deformation resistance in comparison to the other Al5W. Importantly, the hexagonal (P63) Al5W has the stronger degree of shear anisotropy along the (100) and (010) plane. But the orthorhombic (Cmma) Al5W has the stronger degree of shear anisotropy along the (001) plane compared to the other three Al5W. Finally, the calculated melting point follows the order of P63 > R-3c > Cmma > P6/mmm. Naturally, the Al5W shows high melting point is related to the vibration of Al atom and Al–Al bond.

First principles insights on the structural, mechanical, dynamical, thermoelectric and thermodynamics properties of novel topological (ScSb) semi-metal

In the present report, we have a detailed investigation of the structural, electronic, mechanical, thermoelectric thermodynamic responses of ScSb with external pressure to full potential Linearized augmented Plane-wave technique (FP-LAPW) and semi-classical Boltzmann transport equation based on density functional theory (DFT). Computed the structure of ScSb compound having ground state energy and dynamically stable. Diverse approximations were employed in electronic properties to account for precise exchange and correlation effects. The dynamically and mechanically stability structure is confirmed by the estimation of phonon and elastic constants such as Bulk modulus, Young’s modulus, Poisson's ratio, and shear anisotropy factor. For the transition of carriers from the valence band to conduction, we calculated the electron localized function (ELF), electron, and hole effective mass. Transports properties calculated against temperature, chemical potential, as well as carriers concentrations by employing the external pressure (0GP, 25 GPa, and 40 Gpa) were studied using the semi-classical equation in Boltzmann theory. Thermodynamics properties were computed with external pressure having the variation of temperature in this material which is investigated across a wide temperature range (200–1200 K). Maximum Power factor, electrical conductivity, and thermal conductivity are determined to be 0.3 × 1010 W/mK2s,3 × 1020 (Ω.m.s)-1, and 2.1 × 1015 W/mKs at 300 K against temperature respectively. Because of its non-trivial topological character, this material has a big power factor, which strongly implies that it can operate as a suitable thermoelectric material in the high-temperature zone.

Copernican Paradigm beyond FLRW

We present the dipole cosmological principle, i.e., the notion that the Universe is a Copernican cosmology that agrees with the cosmic flow. It suits the most symmetric paradigm that generalizes the Friedmann–Lemaître–Robertson–Walker ansatz in the context of numerous suggestions that have appeared in the literature for non-kinematic components in the cosmic microwave background dipole. Field equations in our “dipole cosmology” are still ODEs, but we now have four instead of two Friedmann equations. The two extra functions can be regarded as additional scale factors that break the isotropy group from SO(3) to U(1) and a “tilt” that denotes the cosmic flow. The result is an axially isotropic Universe. We examined the dynamics of the expansion rate, anisotropic shear, and tilt in some cases. One important observation is that the cosmic flow (tilt) can grow while the anisotropy (shear) dies down.

In vivo estimation of anisotropic mechanical properties of the gastrocnemius during functional loading with MR elastography

Objective. In vivo imaging assessments of skeletal muscle structure and function allow for longitudinal quantification of tissue health. Magnetic resonance elastography (MRE) non-invasively quantifies tissue mechanical properties, allowing for evaluation of skeletal muscle biomechanics in response to loading, creating a better understanding of muscle functional health. Approach. In this study, we analyze the anisotropic mechanical response of calf muscles using MRE with a transversely isotropic, nonlinear inversion algorithm (TI-NLI) to investigate the role of muscle fiber stiffening under load. We estimate anisotropic material parameters including fiber shear stiffness (), substrate shear stiffness (), shear anisotropy (), and tensile anisotropy () of the gastrocnemius muscle in response to both passive and active tension. Main results. In passive tension, we found a significant increase in and with increasing muscle length. While in active tension, we observed increasing and decreasing and during active dorsiflexion and plantarflexion—indicating less anisotropy—with greater effects when the muscles act as agonist. Significance. The study demonstrates the ability of this anisotropic MRE method to capture the multifaceted mechanical response of skeletal muscle to tissue loading from muscle lengthening and contraction.

DFT insights on the opto-electronic and thermoelectric properties of double perovskites K2AgSbX6 (X = Cl, Br) via halides substitutions for solar cell applications

The structural, mechanical, electrical, optical, and thermoelectric characteristics of double perovskites crystal structure with #225 Fm3m space were studied using the Full Potential Linearized Augmented Plane-wave approach (FP-LAPW), which is based on density functional theory (DFT). The exchange and correlation effects were considered using the Perdew-Burke-Ernzerh generalized gradient plus modified Becke Johnson ((PBE-GGA + mBJ) approximation. The mechanical stability of these materials was confirmed by the estimated elastic constants such as Bulk modulus, Young modulus, Poisson's ratio, and shear anisotropy factor. The indirect electronic bandgap values 2.30 eV and 1.52 eV are justified by calculating electronic properties and maximum absorption by calculating the optical properties. The electronic portion of thermoelectric transport parameters such as Seebeck coefficient, electrical conductivity, thermal conductivity, Power factor, and figure of merits versus temperature, chemical potential, and carriers’ concentrations was computed using Boltzmann transport equations. The 0.78, 1, and 0.95 values of ZT advocate that these materials can be used in real-life solar cells and thermal devices.

Undrained Stability of Opening in Underground Walls in Anisotropic Clays

A two-dimensional (2D) finite-element limit analysis (FELA) was performed under plane strain conditions to investigate the undrained stability of tunnel launch with a rigid wall in anisotropic clays. The anisotropic undrained shear (AUS) failure criteria were applied to identify anisotropic soil behaviors. The average bound solutions were employed to compute the stability load factor. Different sensitivity analyses were carried out to determine the effect of cover–depth ratio (C/D), overburden factor (γD/suTC), and anisotropic strength ratio (re) on the load factor (N) of the opening in tunnel launching with a rigid wall. The failure patterns that were predicted by certain parameters were explored and investigated. To adopt a practical approach, some new design charts for the undrained stability of tunnel launching with a rigid wall in anisotropic clays were devised.

Effects of swirl number and bluff body on swirling flow dynamics

Bluff-body swirling flows have been widely employed in gas-turbine combustors to achieve flame stabilization. Meanwhile, considerable efforts have been made to understand swirling flow dynamics, the effects of swirl number and bluff body on flow structure and dynamics are still not well understood. To this end, a series of direct numerical simulations of isothermal swirling flows have been conducted in this work in order to investigate the impact of swirl numbers and bluff-body diameters on the flow structure, Reynolds stresses, and turbulent kinetic energy (TKE) transport. It is found that a change in the swirl number can affect the inner recirculation zone (IRZ) and hence momentum transport. Specifically, as the swirl number increases, the vortex core formed at downstream locations can merge with the IRZ. Moreover, including the bluff body not only contributes to the formation of the IRZ but also serves as a disturbance source for the flow, which is favorable for the formation of large-scale vortex structures. Then, the impact of swirl number and bluff body on Reynolds shear stresses and anisotropy invariants is investigated to identify the locations of the inter shear layer (ISL), the outer shear layer (OSL), and the main swirling zone (MSZ). The results show that as the swirl number increases, both the ISL and MSZ shift to the wall, indicating a large IRZ. Furthermore, the analysis of TKE indicates that for cases with a bluff body, TKE mainly occurs in the ISL and OSL, featuring a dual peak distribution. However, for cases without a bluff body, the distribution of TKE is primarily concentrated in the ISL. These results suggest that both increasing the swirl number and/or including the bluff body could help with TKE transport, which can lead to a wide range of TKE distribution.

The structural, mechanical, thermal, electronic and optical properties of halide perovskites [formula omitted]: First-principles investigations

The structural, mechanical, thermal, electronic and optical properties of inorganic Pb-free double perovskites Cs2TiX6(X=Cl,Br,I) compounds are investigated. Using the first-principles with full-potential linearized augmented plane wave (FP-LAPW) method, supported the density functional theory, and therefore the generalized gradient approximation (GGA-PBE) and Hybrid exchange-correlation functionals with and without spin-orbit coupling. The computation of the lattice constants showed an honest agreement between our results and the experimental results and the theoretical data available. Using the three independent elastic constants C11, C12 and C44, various mechanical and thermal properties are obtained such as: bulk modules B, shear modules G, Young's modules Y, Pough's ratio B/G, Frantesvich ratio G/B, Poisson's ratio υ, Shear anisotropy character A and Debye temperature ΘD for Cs2TiI6, Cs2TiBr6 and Cs2TiCl6 respectively. Our calculated elastic constant results show that Cs2TiI6, Cs2TiBr6 and Cs2TiCl6 are mechanically stable, anisotropic and ductile in nature. Furthermore, the Cs2TiCl6 compound has the highest Debye temperature, which indicates the highest melting temperature. The electronic band structures reveal the semiconductor conductivity of the three compounds, with a band gap suitable for photovoltaic applications, within the range of 1.5–2.96 eV. Finally, Cs2TiCl6(X=Cl,Br,I) HPs compounds exhibit excellent optical properties like optical absorption, conductivity and refraction index.

Undrained basal stability of braced circular excavations in anisotropic and non-homogeneous clays

This paper introduces the plastic stability solutions of braced circular excavations in anisotropic and non–homogeneous clays. Using the framework of Finite Element Limit Analysis (FELA) under axisymmetric conditions, the upper bound (UB) and lower bound (LB) solutions of the stability of excavations can be obtained. The clay is set to be anisotropic, where the Anisotropic Undrained Shear (AUS) model is used as a failure criterion of the surrounding soil. The results of this study are the proposed stability number which is the normalized parameter of the maximum unit weight and the anisotropic undrained shear strength of clay. Four dimensionless parameters are considered in the study: the anisotropic strength ratio, the depth–radius ratio, the depth–embedment ratio, and the strength gradient ratio. The impact of all considered dimensionless parameters on the results of the FELA solutions is examined. A machine learning regression approach, Multivariate Adaptive Regression Splines (MARS), is employed to develop an empirical design equation to predict the stability number of braced circular excavations in anisotropic and non–homogeneous clays. The proposed MARS equation can be a useful and reliable equation to estimate the basal stability of this excavation problem in practice.