Axisymmetric Plane Research Articles

Traction‐associated peridynamic model and non‐uniform discretization simulation of plane axisymmetric problems

AbstractUsing the recently developed traction‐associated peridynamic motion equation (TPME), we investigate the plane axisymmetric problems of elastic deformation. The polar coordinate form of TPME is derived for a plane axisymmetric problem. The transfer function matching with prototype microelastic (PM) model is determined by the inverse method in a one‐dimensional case, and then it is directly extended to plane axisymmetric elasticity problems and plane problems. We promote a strategy to deal with the horizon in the non‐uniform discretization scheme. Based on this strategy, the non‐uniform discretization scheme is used to solve the elastic deformation of a hollow circular plate subjected to uniform inner and outer pressures. Total process from elastic deformation to failure of the hollow circular plate under tension is simulated completely. The results show that TPME does not only need volume and surface corrections, but also can deal with the complex traction boundary conditions conveniently.

Unit cell consolidation of a PVD incorporated soft soil: Comparative of 2D axisymmetric and plane strain analysis considering ideal and actual drain

The paper presents two-dimensional (2D) finite element-based study of a surcharge-induced consolidating unit cell model comprising fully saturated soft soil. Radial drainage in the unit cell is ensured through a single prefabricated vertical drain (PVD) along with vertical drainage through the top surface. Finite element based 2D axisymmetric and plane strain modelling has been conducted by considering the smearing of cohesive medium. The PVD is represented by both 1D drain element (ideal drain) and through geometric modelling considering well resistance. The parameters for the ‘Soft Soil’ model for the cohesive medium are adopted from the subsurface data at Krishnapatnam Ultra Mega Power Project (KUMPP), India. Comparative assessment of unit-cell consolidation is carried out between the 2D axisymmetric and equivalent plane strain condition based on Hird’s permeability compatibility approach. The two approaches yielded noticeable disparity in the time rates of consolidation for time factor ≤ 0.4, beyond which the disparity decreases, and the two methods lead to identical results beyond time factor = 4. Further, the influence of permeability of a geometrically modelled PVD and the height of unit cell on the time-rate of consolidation is elucidated, thereby manifesting the importance of well resistance and drainage length in such analyses.

Research on Meso‐Damage Experiment and Simulation in the Anchorage Bonding Interface Between Rock Bolt and Rock Mass

Anchorage failure occurs mostly at the anchorage interface, and the mechanical mechanism of the anchorage interface is the key point and difficulty in the analysis and research of anchorage. In order to study the mechanics and failure mechanism of the anchorage interface, the shear failure characteristics of the anchorage interface and the deformation rule of the anchored rock under different rib spacings were analyzed through the indoor plane anchorage pull‐out test. Reasonable rib spacing can significantly improve the shear strength of the anchored solid. Finite element software ABAQUS was used to create an axisymmetric plane model of rock bolt pull‐out under different rib conditions to simulate the whole process of rock bolt pull‐out and anchorage interface failure. The results show that during the pull‐out process of anchorage rock mass, stress concentration zones occur at rib corners, and peak shear stress moves to the depths of the anchorage section. As the load increases, cracks emerge from the top of the rib and the edge of the rib angle of the rock bolt and propagate to the deep anchorage along the edge of the costal angle, resulting in damage at the rock mass interface. Load–displacement curves of different transverse rib shapes, heights, widths, and spacings indicate that rib changes affect the anchor‐bearing limit. The anchoring failure mode changes from rib wear and spreading slip, rib shear convex body propagation to interfacial crack propagation, and rib causes varying degrees of damage to the surrounding rock mass. Therefore, reasonable rib parameter improvements can enhance anchoring force performance.

Experimental investigation and design method of the general anchorage zone in the ring beam of prestressed concrete containment vessels

Ring beam is the main anchorage zone of the tendons in the nuclear power prestressed concrete containment vessel (PCCV). Its safety is crucial and has a great influence on the overall performance of PCCV. In this paper, two half-scale ring beams were tested to investigate the mechanical performance of the anchorage zone in the PCCV under multidirectional pressure. The effect of working condition with different tension sequences was investigated. Additionally, a half axisymmetric plane model of the containment was established by the finite element simulation to further predict the experimental responses and propose the local reinforcement design in the anchorage zone of the ring beam. The results showed that the ultimate load of the specimens under both working conditions was greater than the nominal ultimate tensile force. The original reinforcement design could meet the bearing capacity requirements, but there was still room for optimization. The ring beam was generally under pressure in the anchorage area, while the splitting force appeared in the under-anchor area, and the spalling force appeared in the corner area of the tooth block, which could be targeted for local strengthening design.

Experimental study on post-fire mechanical properties and fracture behavior of Q690 steel

Fire is one of the most dangerous disasters for high-strength steel structures. After a fire event, the deterioration of mechanical properties of high-strength steel together with the complex stress state generated by external loading may lead to an unexpected failure or collapse of high-strength steel structures. Therefore, it is very critical to evaluate the post-fire behavior of high-strength steel considering stress state effects. This study conducted a series of tests on the Q690 steel specimens after being exposed to high temperatures ranging from 200 to 1000°C with different cooling methods including air-cooling and water-cooling methods. The tested specimens including smooth round specimens, notched round specimens, and grooved plate specimens were designed to consider the failure mode in axisymmetric stress state and plane strain state. To validate the accuracy of the test results, the mechanical properties such as yield strength, ultimate strength, elastic modulus, and ultimate strain were first compared with other test results reported in the literature. The post-fire ultimate strength under different stress states was obtained directly from the test results. Then, the post-fire equivalent plastic strain at fracture of different specimens was obtained from test data together with the finite element analysis results and analyzed under different stress triaxialities and Lode angle parameters. According to this study, an increase in stress triaxiality can result in an increase in ultimate strength but a decrease in the equivalent plastic strain at fracture for Q690 steel after fire. When the exposed temperature was not higher than 800°C, the post-fire strength of the plane strain specimen (Grooved Plane Specimen) was higher than that of the axisymmetric tension specimen (Notched Round Specimen) under a similar stress triaxiality; however, for their post-fire equivalent plastic strain at fracture, the results were opposite.

BEM for transient nonhomogeneous heat conduction in axisymmetric solids with heat source

Boundary equation approach is revisited and extended to be applicable for axisymmetric transient heat transfer applications with spatially varying field variables and heat sources. For the derivation of axisymmetric formulation, three-dimensional fundamental solutions are integrated with respect to the circumferential direction and then the radial integration procedure using the stabilized temperature values is implemented to convert axisymmetric plane integrals to axisymmetric line integrals. All integrals are evaluated numerically. Four Numerical examples are given in order to assess the reliability of the proposed formulation.

Study on Stress and Displacement of Axisymmetric Circular Loess Tunnel Surrounding Rock Based on Joint Strength

The development of an effective evaluation method suitable for loess-tunnel excavation is necessary to avoid the collapse accidents caused by tunnel excavation and any secondary disasters. Although the Fenner formulas and the modified Fenner formulas are widely used in tunnel engineering, a defect still exists in these formulas because the Mohr–Coulomb (M–C) criterion exaggerates the tensile strength of the surrounding rock of the loess tunnel. A newly modified Fenner formula was derived based on joint strength to overcome this deficiency. First, the expressions of stress and the radius of the plastic zone of the surrounding rock of the loess tunnel and the expressions of radial displacement were derived based on the stress-equilibrium equation of the axisymmetric plane and the joint strength. Then, the difference in the modified Fenner formulas based on the two kinds of strength criteria for the loess tunnel were compared. The results showed that the radius of the plastic zone and the radial displacement of the loess tunnel determined by the modified Fenner formula based on joint strength were larger than those determined by the modified Fenner formula based on M–C strength. However, the plastic stress of the plastic zone determined by the modified Fenner formula based on joint strength was smaller. The comparative analysis reveals that the modified Fenner formula based on joint strength can evaluate the stress and plastic-displacement field of the surrounding rock of a loess tunnel more reasonably.

A Numerical Model for Rotational Autofrettage of Disks Based on von Mises Yield Criterion and its Application in Strengthening Flanged Disks Used for Joining High-Pressure Pipelines

In this work, a numerical analysis of the recently proposed rotational autofrettage method for strengthening hollow circular disks is carried out. The analysis of this new process is in nascent stage. Some analyses are available in the literature to study rotational autofrettage based on Tresca yield criterion. However, in general, the von Mises yield criterion closely replicates the yielding behavior of metals. Thus, in this work, the rotational autofrettage of disks is analyzed incorporating von Mises yield criterion. The governing differential equations are obtained for the elastic and plastic deformations in the disk under centrifugal loading based on the assumption of axisymmetric condition and plane stress incorporated in von Mises yield criterion. A numerical scheme is then followed to solve the set of governing differential equations along with the boundary conditions to obtain the stress distribution in an elastic-perfectly plastic disk during loading stage of rotational autofrettage. The residual stress distribution in the disk after unloading of the centrifugal loading are also obtained. The rotationally autofrettaged disk of typical radial dimension is numerically experimented for its different in-service load carrying capacities. Significant improvements are achieved in in-service pressure, radial temperature and centrifugal load carrying capacities in the autofrettaged disk as compared to their non-autofrettaged counterpart. Different numerical results are compared with corresponding solution due to Tresca yield criterion. Further, the present solution of rotational autofrettage is proposed to apply in strengthening industrial standard flanged disks of various dimensions used in connecting high pressure pipelines. It is found that the maximum pressure carrying capacity of flanged disk can be increased up to 77% for wall thickness ratios more than 3.5 by employing rotational autofrettage.

A numerical algorithm for ground reaction curves of circular openings in strain-softening rock masses satisfying the generalized Hoek-Brown criterion

Based on the self-similarity method, a numerical algorithm for constructing ground reaction curves (GRC) of circular openings in elastic-strain softening rock mass that satisfies the generalized Hoek-Brown failure criterion is proposed. The algorithm is derived from the analysis of the axisymmetric plane strain model, where the far-field and excavation boundary are both axisymmetric stresses. Using the self-similar transformation, the plastic zone of the model was transformed into unit plane ring, which has unit radius at the boundary between the plastic zone and the elastic zone. The unit plane ring was dispersed into multiple concentric rings, for which the Runge-Kutta discrete equations of the stress and deformation were established based on corresponding self-similar governing equations. The equations were solved ring by ring in a successive manner, starting from the outer boundary to the excavation boundary. The comparison with the verified algorithm in literature shows that the results of the different algorithms are in good agreement and the relative errors of the plastic zone radius under various softening parameters are only 1.7%–2.0%. These results verified the correctness and reliability of the proposed new algorithm. Finally, the new algorithm is used to investigate the influence of the Hoek-Brown parameter a on GRC. A discussion of the potential capability of practical applications of this algorithm is also provided.

Expansive behavior of high-strength self-stressing and self-compacting concrete: Experimental study and analytical model

This paper presents an experimental investigation on the expansive behavior of high-strength self-compacting concrete (HSCC) with various HCSA expansive agent (EA) contents ranging from 0 % to 12 %. The fresh properties, compressive strength and expansive ratios at free and restrained conditions are studied. The expansive process and expansive mechanism of self-stressing HSCC are analyzed with a combination of X-ray diffraction (XRD) and scanning electron microscopy (SEM) analysis. Moreover, the self-stress of expansive HSCC under the confinement of steel-tube is measured. The results show that the compressive strength of HSCC decreases with increasing EA contents, which should not be over 12 % due to the large expansions damage the concrete matrix seriously. The expansive ratios increase significantly with increasing EA contents, as well as increasing amplitudes. The free expansive ratios of concrete can reach 1.08 × 10−4, 3.03 × 10−4, 5.13 × 10−4 and 7.85 × 10−4 with the EA content ranging from 6 % to 12 % at 28 curing days, respectively. Moreover, the self-stresses of expansive HSCC with EA contents of 8–12 % are 3.2–3.7 MPa. Finally, under the framework of plane strain model in elasticity theory, a calculation analytical method is developed to predict the restrained expansive ratios of HSCC confined by steel tubes. The computational formula is also established to calculate the self-stress of expansive HSCC using the axisymmetric plane problems in elasticity. The model formula provides a good description of expansive responses of self-stressing HSCC.

Optical Magnus effect in the photophoresis of a spinning absorptive dielectric circular cylinder.

When a stationary absorptive dielectric cylinder suspended in a gas (such as air) is illuminated by an axisymmetric wave field (such as plane waves), the transverse (T) photophoretic asymmetry factor (PAF) vanishes as required by geometrical symmetry [Appl. Opt.60, 7937 (2021) APOPAI0003-693510.1364/AO.435306]. Counter-intuitively, when the cylinder possesses an initial angular velocity Ω0 with a sufficiently small acceleration and spins around its main axis in the illuminating field of axisymmetric plane waves, it is shown here that the T-PAF (which is directly proportional to the T-photophoretic force vector component) is quantifiable, in analogy with the Magnus effect in hydrodynamics where a force perpendicular to the axis of the cylinder and to the propagation direction arises. Based upon the instantaneous rest-frame theory and the partial-wave series expansion method in cylindrical coordinates, the internal electric field of the spinning absorptive dielectric cylinder is determined and utilized to compute both the longitudinal (L) and T-PAFs. Particular emphases are given on the size parameter of the cylinder, its angular rotation, the light absorption inside its core material and the polarization (TM or TE) of the incident plane waves. The dimensionless intensity function (DIF) is also computed, which reveals quantitative information on the heated portions within the internal absorptive core material of the cylinder. Numerical computations illustrate the analysis and explicate the behaviors of the DIF and the L- and T-PAFs, which predict the emergence of the forward, neutral, and reverse optical/electromagnetic Magnus effect in the photophoresis of an absorptive dielectric cylinder and related applications in spin optics, optical tweezers, optical manipulation of elongated objects, and radiative transfer research.

Low vacuum enhanced expansion of a supersonic arc plasma plume

A novel expansion and super long plasma plume at low vacuum conditions (200 Pa–10000 Pa) has been developed to deposit the ceramic thermal barrier coating (TBC) for aircraft engines. The maximum length of the Ar–He plasma plume is 1900 mm when the plasma torch operates at a chamber pressure of 200 Pa and an input power of 64 kW (I = 2000 A). Meanwhile, the radial expansion of the plasma plume is 90 mm in an axisymmetric plane at the same condition. The maximum substrate temperature ranges from 1100 °C to 1250 °C when the plasma plume impinges perpendicularly on the graphite substrate at the distance from 800 mm to 1500 mm. It is also found that the Mach number of the plasma flow is 3.4. The numerical simulation predicts the maximum velocity and temperature of the plasma plume are 6474.8 m/s and 12823.7 K, respectively.

Refractive changes of a new asymmetric intracorneal ring segment with variable thickness and base width: A 2D finite-element model.

PurposeTo evaluate the local geometric effects of a unilateral intrastromal ring segment with a combined variation of ring thickness and base width in a finite element simulation, and to compare it against the isolated effect of thickness or base width variation alone.MethodsA two-dimensional finite-element model of a transversely isotropic cornea was created assuming either axisymmetric stress or plane strain condition. The model geometry was composed of a three-layered corneal tissue (epithelium, anterior and posterior stroma) fixed at the limbus. The implantation of a triangular-shape asymmetric ring segment with varying ring thickness (150 to 300 μm) and base width (600 to 800 μm) was simulated. Also, changes induced by thickness or base width alone were studied and compared their combined effect in the asymmetric ring segment. Geometrical deformation of the simulated cornea and sagittal curvature were the main parameters of study.ResultsIncreasing ring thickness and base width along the arc of the asymmetric ring segment produced a more pronounced flattening in this part of the ring. The asymmetric design did find a good balance between maximizing corneal flattening at one end and minimizing it at the other end, compared to the isolated effect of ring thickness and width. Ring thickness was the most robust parameter in flattening both, the central and peripheral cornea.ConclusionThe finite-element model permitted a theoretical study of corneal deformation undergoing implantation of realistic and hypothetical ring geometries. Intracorneal asymmetric ring segments with varying thickness and base width can be a good alternative in corneas with asymmetric keratoconus phenotypes.

The Failure of "Chinese Intrinsic Form"

Architectural trends in modern China have evolved into four stages over the course of history : from the “Chinese Form” in the 1900s, to the Intrinsic in the 1930s, to the in the 1950s, and to the New Ethnic in the 1980s. “Chinese Intrinsic Form” can be defined as an assembly of interrelated structural components -- wood-like columns, beams and corbel brackets -- of traditional Chinese construction reforming with a more sophisticated western technology, but in a holistic and unified manner. It mainly employed for administrative buildings, exhibition halls, commemorative buildings, gymnasiums and commercial buildings. A distinctive attribute of “ Chinese Intrinsic Form ” is its general layout of spatial sequence, as determined by the axisymmetric plane composition, interior configuration, and of facade. The new era in nationalism promoting began when “Chinese Intrinsic Form” came into being in 1927. The Chinese architects wanted national pride of expression for the semi-colonial society of the day 1 without considering how their thoughts and design might aid or hinder the modern evolution of architecture in China. Or the true aspiration may be obscured through neglect rather than intent. In either event, notable questions arise as to whether the representing implication about Intrinsic is more about political or academic. This paper is indent to discuss the historical background of intrinsic form and its architectural cases in order to explore the reasons why it failed.

Numerical Modelling and Mechanical Characterization of Pure Aluminium 1050 Wire Drawing for Symmetric and Axisymmetric Plane Deformations

This present paper focused on the numerical modelling and simulation of the influence of friction and drawing tension while validating it with experimental results for both symmetric and axisymmetric plane deformations in stranded and unstranded wire drawing of pure aluminium. It must be noted that several methods have been deployed in recent years such as empirical, numerical, mathematical, analytical, as well as experimental in analyzing and optimizing forces and stresses in wire drawing and there are no definite solutions yet in solving the numerical complexities involved as a result of enormous number of factors during the wire drawing operation. On this note, modelling and simulation with different cases had been established. In this study, 9.50 mm was drawn into different diameters having 4.4 mm as entry and 1.7 mm as exit with intermediate sizes. It was established in the study that half conical angle must be kept as moderate as possible, it must not be too high or too low. An increase in reduction ratio (deformation) leads to an increase in tensile strength and that the tensile strength of material during wire during increases with an increase in the frictional coefficient. The fractographical examination revealed that unstranded aluminium drawn wire is more ductile due to the presence of a large network of dimples which are bimodal and equiaxed dominated by a cup and cone structures and this can be attributed to the ductile failure mode. Whereas the stranded aluminium-drawn wire possessed low ductility as revealed in fractography due to the presence of “Rock Candy fracture”.

Vision-based relative pose determination of cooperative spacecraft in neutral buoyancy environment

Neutral buoyancy systems simulate the microgravity environment by taking advantage of buoyancy forces of water to offset the gravity of test bodies. Functional verification of space robots in neutral buoyancy system is of great importance for ground tests. The relative pose determination of a spacecraft plays an essential role in the on-orbit operation of space robots. In order to meet the requirement of on-orbit operation ground verification for space robots, this paper develops a vision-based system for determining the relative pose of cooperative spacecraft in neutral buoyancy environment. Cooperative markers and underwater binocular vision system are designed for the pose determination, and a cooperative spacecraft model is built. A detection and recognition method based on the topological characteristic is proposed for the cooperative marker. An underwater imaging model of binocular camera is established, and its refraction parameters are calibrated. The marker points are measured with an underwater binocular 3D measurement algorithm. Furthermore, the pose of cooperative spacecraft is determined using axisymmetric plane feature points. Additionally, the stable and reliable pose and velocity are obtained after the data are further processed with a Kalman filter. Finally, the experiments are carried out and the experimental results show that the proposed system can achieve a stable and reliable high-precision relative pose determination for cooperative spacecraft in neutral buoyancy environment.

Elastic Analysis for Rotating Functionally Graded Annular Disk with Exponentially-Varying Profile and Properties

Functionally graded materials have been widely used in engineering and human health applications. The issues about mechanical behavior of functionally graded material have received considerable attention. However, because of the complexity of material property, geometric profile, and mechanical load, there is still lack of proper analytic solutions about deformation and stress in many articles. The principal goal of this research is to study the effect of mechanical load on deformation and stress in rotating thin-walled functionally gradient material annular disk with exponentially-varying profile and properties. The inner and outer surfaces of annular disk are subjected to different pressures simultaneously. For this purpose, the infinitesimal theory of elasticity and axisymmetric plane stress assumptions has been proposed to formulate the governing equation. The governing equation is a generalized confluent hypergeometric differential equation, based on Whittaker’s functions; this is the first time that closed-form solutions of mechanical behaviors are revealed about proposed functionally gradient material model. Besides, another four boundary conditions are also discussed, i.e., the inner and outer surfaces of the annular disk are considered to be the combinations of free and clamped conditions. Numeric examples of two different functionally graded material properties are given to demonstrate displacement and stress solutions. Moreover, uniform disks made of homogeneous material under different boundary conditions are investigated, which are special cases of the proposed rotating functionally gradient material disks. Finally, some conclusions are made at the end of the present paper.

Elastoplastic analytical investigation of wellbore stability for drilling in methane hydrate-bearing sediments

Methane hydrates (MHs) are popular clean energy resources. The dissociation of MHs near the wellbore wall during drilling in marine methane hydrate-bearing sediments (MHBS) significantly reduces the bearing capacity of the wellbore and increases the risk of wellbore instability. In this study, a new elastoplastic analytical solution for wellbore stability during drilling in MHBS is provided, taking into account the reduction in the stiffness and strength of the formation after the hydrates dissociate, the conduction/diffusion and convection of heat/ion transfer, and the partial coupling among the seepage, temperature, salt concentration and mechanical fields.An axisymmetric plane strain problem is simplified by diving the infinite region into several annular regions with different physical properties. The distribution of pore pressure at a specific time for this multi-region problem is first obtained by introducing an influence radius estimated by the common unsteady flow solution. Accordingly, the conduction/diffusion and convection effects on the temperature and salt concentration fields are provided for balanced, underbalanced and overbalanced drilling conditions, and the dissociated region is finally addressed through the hydrate phase equilibrium equation fitted by numerous experimental data. Then, the mechanical responses of the wellbore in an elastic-brittle-plastic formation are analytically analysed by employing unified strength theory, considering two principal stress orders and three plastic zone locations.The analytical solution agrees very well with the numerical results determined under the same assumptions, and the solutions of the influence radii of the pressure and temperature and distribution of the pressure and effective stress around the wellbore are consistent with the results from complex numerical simulations. A detailed parametric study is carried out to investigate the specific influence of the drilling fluid properties, the size of the dissociated region and the reduction in Young's modulus/cohesion due to hydrate dissociation on wellbore stability during drilling in MHBS.

Dynamic response of ductile materials containing cylindrical voids

The fracture of ductile materials is often the result of the nucleation, growth and coalescence of microscopic voids. In this thesis, we mainly focus on the dynamic void evolution in porous media containing cylindrical voids. This study covers a problem that is of particular interest in many areas of research (e.g. development of shock mitigation devices for civil or military applications). Owing to the development of additive manufacturing, the processing of porous material with cylindrical voids is an option to create lightweight materials having interesting properties in terms of energy dissipation. Therefore, our work aims at describing the dynamic response of architectured materials such as honeycomb structures. In dynamic loading, microvoids sustain an extremely rapid expansion which generates strong acceleration of particles in the vicinity of cavities. These micro-inertia effects are known to play a significant role in the macroscopic response and the development of damage in porous media. In fact, the overall macroscopic stress is found to be the sum of two contributions: a static term (micro-inertia independent term) and a dynamic term (micro-inertia dependent term), the latter being related to the microstructure (e.g. size and aspect ratio of voids). In our work, a cylindrical shell is adopted as a Representative Volume Element (internal and external radii a and b, length 2l) for the porous material. The static term is derived from a yield function available in the literature. The dynamic stress is evaluated analytically using a trial velocity field for cylindrical voids combined with the multi-scale approach developed in the literature in LEM3. It is shown that the dynamic stress is scaled by the mass density, two characteristic lengths of the voids, the porosity, the macroscopic strain rate tensor and the time derivative of the strain rate tensor. An important outcome of the model is the differential lengthscale effect which exists between in-plane and out of plane components of the macroscopic stress. Namely, it is observed for axisymmetric loading that in-plane dynamic stress components are only related to the void radius a while the out of plane stress component is linked to a and the length of the RVE, l. In the thesis, we present the dynamic response of the porous medium when subjected to various loading conditions: spherical loading, axisymmetric plane strain loading, uniaxial loading and biaxial loading. While for plane strain loading under quasi static condition, the overall axial stress is spherical, in dynamic conditions, the inertia contribution hinders the overall stress tensor from being spherical. Another important result of the proposed theory is the effect of the void length, which does not exist in quasi static conditions where the overall response is solely modulated by the porosity. The case of thin cylinders under dynamic loading reveals a peculiar damage kinetics. In fact, the damage developed in such porous materials results from an increase of the void radius and a reduction of the external radius. The void collapse for uniaxial as well as for biaxial loadings are new observations. The analytical model predictions are validated based on comparisons with finite element calculations (Abaqus/Explicit).

Analytical solutions for the circular stress transducer embedded in rheological rock mass

The rheological stress recovery (RSR) method was proposed to measure the in-situ stress of rock mass with time-dependent property by drilling a hole and embedding transducers into it. To solve the stress distribution on the transducer, a viscoelastic axisymmetric plane model was firstly built considering an arbitrary stress boundary condition. The analytical solution was developed by dividing the stress boundary conditions into axisymmetric and anti-axisymmetric combining with Laplace transformation technique. The explicit stress expressions on interfaces of rock–grout and grout–transducer was obtained using Burgers and Boltzmann viscoelastic models, respectively. Furthermore, the variations of transducer surface final stress, which is related to rheological time, geometric and mechanical properties of rock mass, grout parameter, and transducer materials, was proposed for calculating the measured stress by RSR method. For both of Burgers and Boltzmann viscoelastic model, final stress increases as elastic modulus ratio increases when elastic modulus ratio under 20, and the final stress could be ignored when diameter ratio is over 1.4. The rheological time increases with increasing of viscosity coefficient and the modulus ratio, but decreases as the shear modulus increases. The results in here provide a simple method for stress analyzing and have great value for understanding the relationship between the initial stress of rock mass and the measured stress for the RSR method.