Shear Anisotropy Research Articles

Mechanically anisotropic phantoms for magnetic resonance elastography.

Imaging phantoms with known anisotropic mechanical properties are needed to evaluate magnetic resonance elastography (MRE) methods to estimate anisotropic parameters. The aims of this study were to fabricate mechanically anisotropic MRE phantoms, characterize their mechanical behavior by direct testing, then assess the accuracy of MRE estimates of anisotropic properties using a transversely isotropic nonlinear inversion (TI-NLI) algorithm. Directionally scaled and unscaled lattices were designed to exhibit anisotropic or isotropic mechanical properties. Lattices were three-dimensionally printed in poly(ethelyne glycol) diacrylate using a commercial digital light processing printer, then infilled with gelatin to form a composite material. Benchtop testing determined two shear stiffnesses, and , governing loading parallel and perpendicular to the symmetry axis, and two analogous Young's moduli and . From these measures, shear anisotropy = and tensile anisotropy = were calculated. Three phantoms were driven by a central actuator and imaged with MRE at frequencies from 300 to 500 Hz. From MRE data, the TI-NLI algorithm estimated maps of , , and . In benchtop tests, geometrically scaled lattice composites exhibited the following anisotropic properties: = 6.1 ± 0.7 kPa, = 0.83 ± 0.13, = 0.78 ± 0.09} (mean ± standard deviation). MRE of scaled lattice composites revealed elliptical wavefields; TI-NLI analysis identified the following median property ranges: = 11-19 kPa, = 0.6-1.0, = 0.8-1.6}. Anisotropic MRE phantoms are created by embedding anisotropic three-dimensionally printed lattices into a softer matrix. The TI-NLI algorithm accurately estimates spatial contrast in anisotropic properties.

Shear stress-strain relationships and anisotropy in silty soil: The role of principal stress rotation

Shear stress-strain relationships and anisotropy in silty soil: The role of principal stress rotation

A generalized, computationally versatile plasticity model framework - Part II: Theory and verification focusing on shear anisotropy

Shear-dominated deformation (SHDD) is pivotal in sheet metal forming; however, comprehensive modeling of plastic anisotropy in SHDD, specifically shear anisotropy considering both yield stress and plastic flow, has been inadequately addressed in existing literature. In this work, a generalized constitutive framework is introduced on the basis of stress triaxiality-dependent state variable to accurately emulate plastic anisotropy and the physics-based shear constraint in SHDD. The framework is capable to seamlessly integrate with existing yield criteria, preserving computational efficiency and versatility. Notably, the yield function, anisotropic parameters, and their optimization or analytical determination for the non-shear deformation state remain unaltered. When integrated with the Hill48 yield function, featuring either one or two anisotropic parameters within the generalized constitutive framework, precise characterization of yield strength and plastic flow in SHDD is achieved. The applicability of the framework extends to various anisotropic yield functions such as the widely employed Yld2k-2d and the sixth-order polynomial (Poly6) function as a class of associated flow rule-based yield functions, and one non-quadratic yield function for non-associated flow rule scenarios. Experimental validation with two engineering sheet metals, high-strength dual-phase steel DP980 and high-strength aluminum alloy AA7075-T6, was conducted. Comparative analyses with several recently proposed yield criteria, especially Poly6–18p, highlighted the efficiency of the proposed constitutive framework. Furthermore, this study explores intrinsic shear constraints, particularly the absence of through-thickness strains under in-plane shear stress. Additionally, it offers an enhanced description of plastic anisotropy in shear yield stress within the general framework, providing valuable insights into the complexities of SHDD.

First-principles calculations to investigate Electronic, Optical, Mechanical, and transport characteristics of novel A2BAuCl6 (A = K/Rb/Cs; B = Sc/Y)

By utilizing FP-LAPW, the structure, electronic, optical, mechanical and Transport characteristic of A2BAuCl6 (A = K/Rb/Cs, B = Sc/Y) were examined by using DFT. The analysis to the band structure plot of the Halide double perovskites (HDP’s) being studied indicated the existence of indirect band-gaps. The compounds A2BAuCl6 (A = K/Rb/Cs, B = Sc/Y) exhibit anisotropic properties, except one compound as seen by their respective shear anisotropy readings of 0.629, 0.532, 1.006, 0.682, 0.803 and 0.977. The materials have inherent ductility, as indicated by their calculated Poisson’s ratios in the range of 0.30 to 0.35. The examined (HDP’s) materials appear to possess indirect band-gaps, as indicated by the band structure plots. The functional inorganic perovskites A2BAuCl6 (A = K/Rb/Cs, B = Sc/Y) haven’t similar band-gap values, as determined using the TB-mBJ (modified Becke Johnson) and TB-mBJ + Soc (Spin orbital coupling) assumptions. The visible and ultraviolet domain have prominent peaks, suggesting potential applications of the compounds in solar cells. Moreover, examination of additional optical characteristics such as reflectivity, conductivity, electron energy loss, and absorption coefficients suggest that it possesses the inherent features of a semiconductor. The maximum figure of merit for all the student compounds is above 0.74. The simulated output forms the foundation for A2BAuCl6 (A = K/Rb/Cs, B = Sc/Y) and are of practical importance for multijunction solar cells.

Holographic transport in anisotropic plasmas

We study energy-momentum and charge transport in strongly interacting holographic quantum field theories in an anisotropic thermal state by contrasting three different holographic methods to compute transport coefficients: standard holographic calculation of retarded Green’s functions, a method based on the null-focusing equation near horizon and the novel method based on background variations. Employing these methods we compute anisotropic shear and bulk viscosities and conductivities with anisotropy induced externally, for example by an external magnetic field. We show that all three methods yield consistent results. The novel method allows us to read off the transport coefficients from the horizon data and express them in analytic form from which we derive universal relations among them. Furthermore we extend the method based on the null-focusing equation to Gauss-Bonnet theory to compute higher derivative corrections to the aforementioned transport coefficients. Published by the American Physical Society 2024

Covariant cosmography: the observer-dependence of the Hubble parameter

The disagreement between low- and high-redshift measurements of the Hubble parameter is emerging as a serious challenge to the standard model of cosmology. We develop a covariant cosmographic analysis of the Hubble parameter in a general spacetime, which is fully model-independent and can thus be used as part of a robust assessment of the tension. Here our focus is not on the tension but on understanding the relation between the physical expansion rate and its measurement by observers — which is critical for model-independent measurements and tests. We define the physical Hubble parameter and its multipoles in a general spacetime and derive for the first time the covariant boost transformation of the multipoles measured by a heliocentric observer. The analysis is extended to the covariant deceleration parameter. Current cosmographic measurements of the expansion anisotropy contain discrepancies and disagreements, some of which may arise because the correct transformations for a moving observer are not applied. A heliocentric observer will detect a dipole, generated not only by a Doppler effect, but also by an aberration effect due to shear. In principle, the observer can measure both the intrinsic shear anisotropy and the velocity of the observer relative to the matter — without any knowledge of peculiar velocities, which are gauge dependent and do not arise in a covariant approach. The practical implementation of these results is investigated in a follow-up paper. We further show that the standard cosmographic relation between the Hubble parameter, the redshift and the luminosity distance (or magnitude) is not invariant under boosts and holds only in the matter frame. A moving observer who applies the standard cosmographic relation should correct the luminosity distance by a redshift factor — otherwise an incorrect dipole and a spurious octupole are predicted.

Spatio-temporal evolution law of anisotropic shear damage on rock mass joint surface affected by joint morphology

The spatial–temporal evolution law of shear damage on the joint surface is closely related to its morphological characteristics. Understanding the heterogeneous failure behavior of the joint surface is essential for revealing the controlling role of the joint morphology. The diorite from the Guanshan Tunnel is investigated through anisotropic direct shear tests in this study. Based on shear damage quantification and acoustic emission (AE) localization techniques, the heterogeneous damage evolution law of joint surfaces under different joint compression strength (JCS), normal stress (σn), and morphology parameter (M) is studied. The results indicate that the joint surface morphology, shear damage, and AE parameters exhibit similar anisotropic characteristics in different shear directions. An increase in the morphology parameter leads to an increase in shear failure, with shear fracture energy and the number of fracture events showing an increasing trend. Meanwhile, the occurrence of a large number of fracture points shifts from the pre-peak nonlinear period to the post-peak period. However, the evolution process of fracture events exhibits certain similarities, though the morphology parameters are different in two opposite shear directions. Additionally, shear damage volume and fracture energy demonstrate consistency in spatial distribution and quantitative characteristics, while the linear scale factor between cumulative shear fracture energy and damage volume decreases with increasing joint surface strength. These findings assist in a better understanding of the anti-sliding property of joint surfaces and provide crucial clues for further exploration of joint surface failure mechanisms.

Postbuckling behaviour of anisotropic shear deformable laminated cylindrical shells with general imperfection distribution subjected to torsion

Composite cylindrical shells belong to typical basic structure in engineering, whose buckling and postbuckling behaviours determine the stability and safety of overall structures under complex service loadings. In this paper, a nonlinear mechanical model for anisotropic laminated composite cylindrical shells with general distributed initial imperfection is derived to analyse the buckling and postbuckling behaviours under torsional loads, in which the general distributed imperfections are represented by a generic two-dimensional imperfection function reconstructed by using measured data. Based on a higher order shear deformation shell theory, the governing equations are established with consideration of von Kármán–Donnell-type of kinematic nonlinearity and the coupling effects of extension/twist, extension/flexural and flexural/twist. By using a two-step perturbation technique, a new set of explicit solutions is obtained to determine the buckling loads and postbuckling equilibrium paths of the torsional composite cylindrical shells. Furthermore, the internal quantitative relationship between the shear stress with torsional characteristics along with an associate compressive stress of anisotropic laminated shell is for the first time obtained. Numerical analyses are performed to validate the accuracy and effectiveness of the present explicit solutions, and obtain the postbuckling response of different types of imperfections, anisotropic laminated cylindrical shells with different values of shell parameters.

Thermal effects on mechanical and failure behaviors of anisotropic shale subjected to direct shear

The stimulation of shale reservoirs frequently involves significant shear failure, which is crucial for creating fracture networks and enhancing permeability to boost production. As the depth of extraction increases, the impact of elevated temperatures on the anisotropic shear strength and failure mechanisms of shale becomes pronounced, yet there is a notable lack of relevant research. This study conducts, for the first time, direct shear experiment on shales at four different temperatures and seven bedding angles. By employing acoustic emission (AE) and digital image correlation (DIC) techniques, the evolution of damage and the mechanism of crack propagation under anisotropic direct shearing at varying temperatures is revealed. The results indicate that both shear displacement and strength of shale increase with temperature across different bedding angles. Additionally, shale demonstrates distinct brittle failure characteristics under various conditions during direct shearing tests. The types of anisotropic shear failure observed under the influence of temperature include central shearing fracture, central shearing with secondary fracture, and deflected slip along the bedding. Moreover, the temperature effect enhances shear-induced crack propagation along bedding planes. Shear failure in shale predominantly occurs during higher loading stages, which coincide with a substantial amount of AE signals. Finally, the introduction of the anisotropy index and temperature sensitivity coefficient further elucidates the interaction mechanism between thermal effects and anisotropy. This study offers a novel methodology to explore the anisotropic shear failure behavior of shale under elevated temperatures, and also provides crucial theoretical and experimental insights into shear failure behavior relevant to practical shale reservoir stimulation.

R32-Al5W: A new stable high-temperature alloy

Due to the high melting point, exceptional mechanical properties, and remarkable corrosion resistance exhibited by Al5W, researchers have conducted extensive research on the structural characteristics of Al5W. To identify the lowest energy structures and find new structures, a combination of particle swarm optimization and first-principles calculations was employed to identify energy-favorable Al5W structures under ambient pressure. The experimental P63 phase and the previously predicted R3¯c phase were identified. Notably, a new R32-Al5W phase was proposed. A comparative analysis on the structural stability, elastic, thermodynamic, and electronic properties of these three Al5W phases were conducted. The results indicate that R32-Al5W exhibits favorable thermodynamic stability, with a predicted bulk modulus close to that of R3¯c-Al5W. All the three Al5W phases demonstrate brittle behavior. The P63-Al5W phase displays larger shear anisotropy in the (010) and (001) planes, while the R32-Al5W phase shows greater shear anisotropy on the (100) plane compared to other two phases. Furthermore, employing Clarke’s and Cahill’s models, the minimum thermal conductivity of Al5W phases was predicted, revealing a magnitude order of P63 >R3¯c > R32. Our calculated melting points also conform to this trend.

3D shear wave velocity and azimuthal anisotropy structure in the shallow crust of Binchuan Basin in Yunnan, Southwest China, from ambient noise tomography

The Binchuan Basin in northwest Yunnan, southwest China, is a rift basin developed at the intersection of the Red River Fault and Chenghai Fault, where historical earthquakes have occurred. Understanding the fine velocity structure of the shallow crust in this region can help improve earthquake location accuracy and our understanding of the relationship between fault zone structures and fault slip behaviors. Using the continuous waveform data recorded by 381 dense array stations in 2017, we obtained 7 915 Rayleigh-wave phase velocity dispersion curves in the period band of 0.2–6 ​s from ambient noise cross-correlation functions after rigorous data processing and quality control. We determined 3D isotropic and azimuthally anisotropic shear wave velocity models at depths above 6 ​km in the shallow crust based on the direct surface wave azimuthal anisotropic tomography method. The isotropic model reveals a strong correspondence between the S-wave velocity structure at depths of 0–1 ​km and the regional topography and lithology. The Binchuan depocenter, Zhoucheng depocenter, Xiangyun Basin, and Xihai Rift Basin are primarily composed of Quaternary deposits, which show low-velocity anomalies, while the regions with the Paleozoic shale, limestone, and basalt exhibit high-velocity anomalies. The nearly N–S orientation of fast directions from azimuthal anisotropy models are mainly controlled by the active Binchuan Fault with N–S strike as well as the NNW-oriented primary compressive stress.

Revisiting shear stress tensor evolution: Nonresistive magnetohydrodynamics with momentum-dependent relaxation time

This study aims to develop second-order relativistic viscous magnetohydrodynamics (MHD) derived from kinetic theory within an extended relaxation time approximation (momentum/energy dependent) for the collision kernel. The investigation involves a detailed examination of shear stress tensor evolution equations and associated transport coefficients. The Boltzmann equation is solved using a Chapman-Enskog-like gradient expansion for a charge-conserved conformal system, incorporating a momentum-dependent relaxation time. The derived relativistic nonresistive, viscous second-order MHD equations for the shear stress tensor reveal significant modifications in the coupling with dissipative charge current and magnetic field due to the momentum dependence of the relaxation time. By utilizing a power law parametrization to quantify the momentum dependence of the relaxation time, the anisotropic magnetic field-dependent shear coefficients in the Navier-Stokes limit have been investigated. The resulting viscous coefficients are seen to be sensitive to the momentum dependence of the relaxation time. Published by the American Physical Society 2024

A First-Principles Study of the Structural, Elastic, and Mechanical Characteristics of Mg2Ni Subjected to Pressure Conditions

This study employs first-principles calculations to examine structural, elastic, and mechanistic relationships of Mg2Ni alloys under varying conditions of pressure. The investigation encompasses Young’s modulus, bulk modulus, shear modulus, Poisson’s ratio, and anisotropy index, as well as sound velocity, Debye temperature, and related properties. Our findings indicate that the lattice parameters of Mg2Ni in its ground state are in agreement with values obtained experimentally and from the literature, confirming the reliability of the calculated results. Furthermore, a gradual decrease in the values of the lattice parameters a/a0 and c/c0 is observed with increasing pressure. Specifically, the values for C13 and C33 decrease at a hydrostatic pressure of 5 GPa, while C11 and C13 increase when the external hydrostatic pressure exceeds 5 GPa. All other elastic constants exhibit a consistent increasing trend with increasing pressure between 0 and 30 GPa, with C11 and C12 increasing at a faster rate than C44 and C66. In the 0–30 GPa pressure range, Mg2Ni satisfies the mechanical stability criterion, indicating its stable existence under these conditions. Additionally, the Poisson’s ratio of Mg2Ni consistently exceeds 0.26 over a range of pressures from 0 to 30 GPa, signifying ductility and demonstrating consistency with the value of B/G. The hardness of Mg2Ni increases within the pressure range of 0–5 GPa, but decreases above 5 GPa. Notably, the shear anisotropy of Mg2Ni exhibits greater significance than the compressive anisotropy, with its anisotropy intensifying under higher pressures. Both the sound anisotropy and the Debye temperature of Mg2Ni demonstrate an increasing trend with rising pressure.

Evolution of structure and anisotropic shear stiffness of compacted loess during compression

Different compaction conditions (water content and density) may induce various soil structures. The influence of these structures on small strain shear stiffness G seems contradictory and is not understood (e.g., denser specimens may have larger or smaller G than looser specimens after compression). Furthermore, the influence of compaction condition on stiffness anisotropy remains unclear. This study investigated the evolution of structure and anisotropic stiffness of saturated and compacted loess during isotropic compression. Specimens compacted at different water contents and densities were explored. The measured G was normalised by a void ratio function (f(e)) to eliminate density effects. Before yielding, G/f(e) increases with decreasing compaction water content and increasing density. These two trends are reversed at large stresses (2 to 3 times yield stress), implying that an initially softer structure becomes stiffer. Based on MIP, SM and SEM results, the trend reversal is likely because interparticle contacts are more strengthened and pores are more compressed in the initially softer specimens. Furthermore, the stiffness anisotropy becomes more significant with decreasing compaction water content and increasing density because of more orientated fabrics, as evidenced by the particle/aggregate directional distribution results.

Structural tuning of anisotropic mechanical properties in 3D-Printed hydrogel lattices

We investigated the ability to tune the anisotropic mechanical properties of 3D-printed hydrogel lattices by modifying their geometry (lattice strut diameter, unit cell size, and unit cell scaling factor). Many soft tissues are anisotropic and the ability to mimic natural anisotropy would be valuable for developing tissue-surrogate “phantoms” for elasticity imaging (shear wave elastography or magnetic resonance elastography). Vintile lattices were 3D-printed in polyethylene glycol di-acrylate (PEGDA) using digital light projection printing. Two mechanical benchtop tests, dynamic shear testing and unconfined compression, were used to measure the apparent shear storage moduli (G′) and apparent Young's moduli (E) of lattice samples. Increasing the unit cell size from 1.25 mm to 2.00 mm reduced the Young's and shear moduli of the lattices by 91% and 85%, respectively. Decreasing the strut diameter from 300 μm to 200 μm reduced the apparent shear moduli of the lattices by 95%. Increasing the geometric scaling ratio of the lattice unit cells from 1.00 × to 2.00 × increased mechanical anisotropy in shear (by a factor of 3.1) and in compression (by a factor of 2.9). Both simulations and experiments show that the effects of unit cell size and strut diameter are consistent with power law relationships between volume fraction and apparent elastic moduli. In particular, experimental measurements of apparent Young's moduli agree well with predictions of the theoretical Gibson-Ashby model. Thus, the anisotropic mechanical properties of a lattice can be tuned by the unit cell size, the strut diameter, and scaling factors. This approach will be valuable in designing tissue-mimicking hydrogel lattice-based composite materials for elastography phantoms and tissue engineered scaffolds.

Coupled effects of temperature and suction on the anisotropic elastic shear moduli of unsaturated soil

Elastic shear moduli of soil at various temperatures and suctions are important for analysing the serviceability limit state of energy piles and many other structures. Up to now, however, the coupled effects of temperature and suction on elastic shear modulus and the stiffness anisotropy, have not been well understood. This experimental study investigated the anisotropic elastic shear modulus of a compacted lateritic clay. A temperature and suction-controlled triaxial apparatus equipped with bender element probes and local strain measurements was used. Soil suctions from 0 to 300 kPa, and a temperature range of 5–40 °C were applied. The results at saturated and unsaturated conditions consistently reveal that the shear modulus is smaller after heating at a given stress and suction. Several mechanisms may contribute to this thermal-induced reduction in shear modulus, such as the heating-induced reduction of interparticle force and air–water surface tension. Moreover, the reduction in shear modulus upon heating depends on the shear plane and the degree of anisotropy changes.

Uncertainty Analysis of Post-Failure Behavior in Landslides Based on SPH Method and Generalized Geotechnical Random Field Theory

In slope stability analysis, the inherent heterogeneity and spatial variability of soil significantly influence the aftermath of landslides, including critical aspects like runout distance, influence distance, and volume. This research integrates Smoothed Particle Hydrodynamics (SPH) with random field theory to precisely model the large deformation events in slopes with anisotropic shear parameters, while leveraging Graphics Processing Unit (GPU) parallel computing to expedite the generation of random field samples and SPH simulations. This approach meticulously examines how spatial heterogeneity of soil affects slope instability, simulating soil parameters across varied geological settings. It considers the fluctuation scale of anisotropy, the cross-correlation of cohesion and the angle of internal friction, along with their coefficient of variation (CV), to elucidate their impact on landslide magnitude and severity. This study furnishes a robust tool for a holistic assessment of slope impacts and landslide volumes. Additionally, by delineating the computational efficiency disparities between GPUs and Central processing units (CPUs), it underscores the pronounced efficiency benefits of GPU computing. The insights garnered offer fresh perspectives and methodologies in slope stability analysis and disaster risk evaluation, contributing to the finer prediction and management of this prevalent and grave geological hazard.

Stiffness, strength, anisotropy, and buckling of lattices derived from TPMS and Platonic and Archimedean solids

Lattice metamaterials have gained considerable attention due to their distinctive topological structures and multifunctional properties. In this work, the effect of topology, loading conditions, and relative density on the effective mechanical properties of various novel lattice architectures is investigated numerically and experimentally. Thirteen strut-based lattices derived from triply periodic minimal surfaces (five lattices) as well as Platonic (three lattices) and Archimedean (five lattices) solids are considered for the first time, and their anisotropic mechanical properties, including uniaxial, shear, and bulk moduli and strengths as well as their total stiffness, buckling strengths, Poisson’s ratio, and anisotropy are investigated as a function of a wide range of relative densities (0.1% to 37%). Finite element analysis is employed to capture the full effective behavior of these lattices using periodic boundary conditions. Bifurcation analysis is performed to predict the threshold relative density governing their buckling vs yielding deformation behavior. Selected lattice structures of various relative densities are 3D printed using polymer selective laser sintering additive manufacturing technique and tested under quasi-static uniaxial compression where the experimental and numerical results are compared. The numerical results indicate that the deformation behavior can be altered between stretching and bending dominated mode of deformation as function of loading. Archimedean lattices are shown to outperform a wide range of strut-based lattices. This work opens the doors for more investigations of the multifunctional properties of these novel types of lattices and their engineering applications. Furthermore, the generated comprehensive data are useful in optimizing latticed structures using topology optimization techniques.

Grain size effect on the anisotropic shear behavior of sand–textured geomembrane interface

A set of direct shear tests was conducted to investigate the effects of sand mean grain size and geomembrane surface roughness on the shear behaviour of the sand–geomembrane interface. Four types of sands with different mean grain sizes were used and the interfaces included a smooth geomembrane (SGM) and four textured ones with different asperity characteristics. The samples were prepared in a direct shear box at five sedimentation angles. The experimental results showed that the peak friction angle of the soil–textured geomembrane interface was dependent on the mean grain size, inherent anisotropy and surface roughness of the geomembrane. The shear strength of sand–textured geomembranes was two to three times larger than that of the sand–SGM interface. The peak and residual friction angles of sand-SGM interface occurred at sedimentation angle of 72° while it was 108° for sand-TGM interfaces. An increase in mean grain size increased the shear strength of the sand–textured geomembrane interface; while a reverse condition occurred for the sand–SGM interface. The shear strength occurred at larger horizontal displacement as mean particle size increased and the effect of relative roughness on peak friction angle increased with increase in mean grain size.

A new set of explicit solutions for postbuckling behaviour of anisotropic shear deformable laminated cylindrical shells with general imperfection distribution and thermal loading

Thermal postbuckling analysis is presented for shear-deformable anisotropic laminated cylindrical shell with general imperfection distribution under several typical temperature fields are analyzed, including uniform temperature, tent-like linear distribution, axial and circumferential quadratic parabolic temperature fields over the shell surface and through the shell thickness. On this basis, the general temperature distribution can be obtained based on the three-dimensional steady-state heat transfer equation. At the same time, based on the high order shear deformation theory including anisotropic coupling effect, geometrically nonlinear large deflection displacement-strain relationship, the general initial geometric imperfections obtained by measurement and analytical means are introduced, and the thermal buckling equilibrium differential equation of anisotropic shear deformable laminated cylindrical shell with initial imperfections is established. A two-step perturbation-Galerkin integral method is used to obtain explicit solutions for the path development direction of post-thermal buckling equilibrium. The results of parameter analysis show that the material properties and temperature dependence, temperature distribution, lay-up and geometric parameters such as radius-to-thickness ratio have important effects on the thermal buckling load and postbuckling equilibrium path. The present method provides an effective theoretical prediction and analysis method for the design of the carrying capacity of submarine pipelines.