Thick-walled Hollow Cylinder Research Articles

Reliability of thin-walled cylinder tests for elastic properties of anisotropic rocks

Elastic properties of transverse isotropic rocks may be theoretically determined by subjecting a single hollow cylinder specimen to a set of two simple stress conditions. The reliability of the methodology is considered by testing both thin-walled hollow cylinders and thick-walled hollow cylinders. The results indicate that the testing procedure produces consistent material parameters, sample preparation does not seem to induce changes in the specimen behaviour, and that the assumptions implemented in the determination of the material parameters are reasonable. Key words: hollow cylinder, anisotropy, elasticity, test method.

Modelling Of Damaged Zones Around Boreholes Using A Radius Dependent Young's Modulus

Abstract A new "damage mechanics" type model to calculate stresses around circular boreholes in plane strain is presented. The model is based on the assumption that Young's modulus variation with radial distance can be simply described, i.e., E=f(r). Such variations in stiffness arise naturally during and after drilling from damage such as cohesion loss, frictional slip of grains and microfissuring, as well as from the stress redistributions that occur. This approach allows convenient and simple hollow cylinder stress calculations for a variety of E=f((σ3) functions for which a formal pressure-dependent modulus model requires iterative integral evaluations [Equation (8)]. Two functions are suggested for E=j(r) dependency: an exponential law and a power law. The stress distributions from these models are compared to other models previously suggested for borehole stresses, and appear reasonable. Introduction Either Linear Elastic (LE)(l,2) or Elasto-Plastic (EP)(3,5) rock behaviour models are commonly used in modelling openings in geomaterials. Recently, Santarelli(6,7) added to these by noting that in granular rocks such as sandstones, Young's modulus, E, generally varies with the radial or minor principal stress, σ3 ' resulting in his pressure-dependent modulus model (PDM model). He obtained σ3 distributions which correspond well to experimental results from thick-walled hollow cylinders(4,8); σ3 values near the borehole wall were much lower than those predicted by LE models. However, for cases where the Young's modulus depends explicitly on the confining stress σ3' that is E=E((σ3), a closed-form PDM model solution can be found only if its inverse l/E((σ3) can be integrated, and only Kulhawy's(9) function seems to give a closed-form solution. For materials with a complex E(σ3) function, a numerical stress-strain solution is necessary, but this can be simplified considerably by using a radius-dependent modulus model (RDM model). We develop and test such a non-linear (NL) model in this article, introducing a stiffness which is a function of the radius. In our model, the mechanical behaviour of material is assumed to be related to the damage that it may have experienced. Damage around a borehole may be expected during drilling, cementation, production of fluids, high pressure injection (perhaps leading to hydraulic fracturing), or thermal effects(10). This damage can be of different types (grain damage, contact slip, cohesion loss, microfissuring, joint slip), and can be expected to result in a reduced Young's modulus; this reduced modulus in turn will result in a redistribution of stresses around the borehole. In tunnelling, a zone of significantly reduced Young's modulus is generated for at least several radii away from the wall in cases where drill and blast techniques are used, but at some distance the damage effect vanishes and the original modulus obtains. Similarly, a borehole in shale or sandstone is likely surrounded by a limited extent zone of damage arising from the high deviatoric stresses and reduced radial stress caused by borehole creation.

Nonsymmetrical temperature fields and stresses in cylinders made of nonuniform composites

The class of problems for determining nonsymmetrical temperature fields and stress in thick-walled hollow nonuniform cylinders is considered. A resolution set of differential equations is constructed for determining steady-state temperature fields, stresses, and displacements in asymmetrically heated cylinders which are arbitrarily nonuniform with respect to thickness with one plane of elastic symmetry. An approach is suggested for effective numerical solution of this class of problems. Problems are analyzed for thick-walled cylinders whose relative volume content of reinforcing elements varies in the radial direction. Known relationships of composite mechanics are used in order to find effective mechanical and thermophysical parameters for the material. Examples are provided for solving specific problems which point to the requirement of considering nonuniformity with respect to thickness.

Stress-dependent finite growth in soft elastic tissues

Growth and remodeling in tissues may be modulated by mechanical factors such as stress. For example, in cardiac hypertrophy, alterations in wall stress arising from changes in mechanical loading lead to cardiac growth and remodeling. A general continuum formulation for finite volumetric growth in soft elastic tissues is therefore proposed. The shape change of an unloaded tissue during growth is described by a mapping analogous to the deformation gradient tensor. This mapping is decomposed into a transformation of the local zero-stress reference state and an accompanying elastic deformation that ensures the compatibility of the total growth deformation. Residual stress arises from this elastic deformation. Hence, a complete kinematic formulation for growth in general requires a knowledge of the constitutive law for stress in the tissue. Since growth may in turn be affected by stress in the tissue, a general form for the stress-dependent growth law is proposed as a relation between the symmetric growth-rate tensor and the stress tensor. With a thick-walled hollow cylinder of incompressible, isotropic hyperelastic material as an example, the mechanics of left ventricular hypertrophy are investigated. The results show that transmurally uniform pure circumferential growth, which may be similar to eccentric ventricular hypertrophy, changes the state of residual stress in the heart wall. A model of axially loaded bone is used to test a simple stress-dependent growth law in which growth rate depends on the difference between the stress due to loading and a predetermined growth equilibrium stress.

Investigation of Stress-Induced Borehole Enlargement Mechanisms by a Liquid-Metal Saturation Technique

Summary Stress-induced borehole enlargement was investigated with thick-walled hollow rock cylinders and a liquid-metal sample-preservation technique that permits observation of the microstructure of the broken rock. The observed breakouts are oriented and occur on the two opposite sides of the hole with the highest compressive-stress/compressive-strength ratio. Triangular (dog-eared) breakouts result from progressive spalling of thin rock slabs. The fundamental mechanism leading to this failure is the growth and interaction of microscopic cracks oriented parallel to the hole wall. Overbalance pressure in the hole changes this failure mechanism and greatly strengthens and stabilizes the hole wall. The apparent rock strength observed next to these small laboratory-scale model holes is higher than expected for field-scale holes.

Use of thick-walled hollow cylinder creep tests for evaluating flow criteria for rock salt : Morgan, H S; Wawersik, W R Proc 7th ISRM International Congress on Rock Mechanics, Aachen, 16–20 September 1991 V1, P303–310. Publ Rotterdam: A A Balkema, 1991

Finite element simulations of two laboratory creep tests on thick-walled hollow cylinders of rock salt are evaluated to determine if such bench-scale experiments can be used to establish applicability of either von Mises or Tresca stress measures and associated flow conditions. In the tests, the cylinders were loaded axially and pressurized both internally and externally to produce stress fields similar to those found around underground excavations in rock salt. Several different loading stages were used in each test. The simulations show that for each of two creep models studied, quite different deformations of the cylinders are predicted with the Mises and Tresca flow criteria, especially if friction between the cylinders and axial loading platens is ignored. When friction is included in the simulations, the differences in deformation are changed but are sill clearly distinguishable. 10 refs., 10 figs.

Deformation and fracture around cylindrical openings in rock—II. Initiation, growth and interaction of fractures

Elastic and inelastic deformation, fracture and failure around underground openings were investigated through experiments on thick-walled hollow cylinders of Berea sandstone and Indiana limestone, incorporating plane strain loading, the application of different stress paths, transference of the external pressure to infinity, and freezing of the fracture geometry under stress through metal saturation. This second of two parts investigates fracture development and failure progression on both an experimental and theoretical level. Failure is oriented and occurs on the two opposite sides of the hole with the highest ratio of tangential stress to tangential strength (determined by rock strength anisotropy). The size of the failed zones may be influenced not only by the final stress state but also by stress path, strain rate and test boundary conditions. Microscopic observation reveals that the fundamental fracture mechanism is the growth of small, opening-mode, splitting cracks oriented parallel to the tangential stress, starting very close to the hole wall and occurring deeper in the rock with increasing stress. Numerical models confirm that splitting cracks preferentially grow close to the surface of a stressed hole. These models also predict a size effect on strength. Around unsupported holes these cracks coalesce to form macroscopic splitting fractures subparallel to the hole wall, and also form en echelon patterns that meet the wall. These two types of features combine to define the new surface of a spalled piece, which moves radially into the hole. Slabs of fairly uniform thickness progressively detach from the surrounding rock, resulting in a triangular failure zone with a pointed tip. Significant dilation occurs in the failing material. With 5-10 MPa support pressure in the hole, the rock is greatly strengthened and stabilized. Macroscopic splitting fractures, and therefore detached spalled pieces, do not form. With high enough stress, microcrack interaction eventually results in the formation of what appear to be bands of shear localization, but the rock still maintains significant cohesion. (Author/TRRL)

Deformation and fracture around cylindrical openings in rock—I. Observations and analysis of deformations

Elastic and inelastic deformation, fracture and failure around underground openings were investigated through experiments on thick-walled hollow cylinders of Berea sandstone and Indiana limestone, incorporating plane strain loading, the application of different stress paths, transference of the external pressure to infinity, and “freezing” of the fracture geometry under stress through metal saturation. This first of two parts discusses elastic and inelastic deformations on both an experimental and theoretical level. Most elastic (recoverable) deformations exhibit non-linearity and hysteresis. Hole deformations due to internal (in the hole) pressure are controlled by different overall moduli than those due to external pressure. These overall deformations are successfully explained by a radially anisotropic moduli model, which allows different overall moduli for different stress paths. A radial-pressure-dependent modulus model cannot adequately explain these measurements. There is some evidence, however, that the stiffness of these rocks is greater with higher pressures. Evaluation of non-constant modulus models reveals that the rate of modulus increase near the hole is critical in determining whether the maximum tangential stress will occur at the hole wall or away from the wall. The apparent strength of the rock adjacent to the unsupported holes is two to three times the uniaxial compressive strength. Significant raadial dilation occurs in the failing material, as evidenced by hole closure measurements both during loading and after test completion. Proper modelling of this failure requires the incorporation of extreme dilation and extreme strength loss. A support pressure in the hole greatly strengthens and stabilizes the rock, and also reduces the amount of hole closure and dilation upon failure. Hole closure is time-dependent, especially for the unsupported holes.

Computer and measured responses of a thick-walled hollow cylinder of salt subjected to both homogeneous and inhomogeneous loading : Morgan, H S; Wawersik, W R Rock mechanics as a guide for efficient utilization of natural resources: Proc 30th U.S. symposium, Morgantown, 19–22 June 1989P361–368. Publ Rotterdam: A A Balkema, 1989

Computer and measured responses of a thick-walled hollow cylinder of salt subjected to both homogeneous and inhomogeneous loading : Morgan, H S; Wawersik, W R Rock mechanics as a guide for efficient utilization of natural resources: Proc 30th U.S. symposium, Morgantown, 19–22 June 1989P361–368. Publ Rotterdam: A A Balkema, 1989

Study of stress concentration around a local notch on the surface of a thick-walled hollow cylinder in compression

Study of stress concentration around a local notch on the surface of a thick-walled hollow cylinder in compression

Hollow cylinder tests for studying fracture around underground openings : Ewy, R T; Cook, N G W; Myer, L R Key Questions in Rock Mechanics: Proc 29th US Symposium, Minneapolis, 13–15 June 1988P67–74. Publ Rotterdam: A A Balkema, 1988

We are beginning a study of deformation and fracture around underground openings through experiments on thick-walled hollow cylinders of rock that incorporate several features: plane strain loading, the ability to impose different stress paths, and 'freezing' of the fracture geometry under load. The elastic moduli calculated from the measured deformations are different than the moduli reported from uniaxial compression tests. The 'unconfined' strength of the rock surrounding the hole is two to three times typical uniaxial strengths, and a confining stress in the hole strengthens the rock. Although the load is axisymmetric m, there is a preferred direction of failure. Failure is by brittle spalling, resulting in triangular failed regions with pointed tips. The extent of failure is influenced by stress path and perhaps by strain rat. 18 refs., 6 figs.

Heterogeneous deformation resulting from stress gradients in calcite single crystals

The stress gradient calculated from an isotropic elastic approximation does not directly reflect the distribution of permanent deformation in a crystal of the same shape under the same conditions. However, with additional crystallographic constraints, it serves to predict locations where twinning and slip are first activated in a stressed crystal. In this study, thick-walled hollow cylinders were cored from single calcite crystals parallel to 〈0001〉 and \(\langle 40\bar 41\rangle\). One cylinder of each orientation was loaded at room temperature under one of two conditions: internal pressure (Pp)=60 MPa, external pressure (Pc)=100 MPa, or Pp = 20 MPa, Pc= 50 MPa. These conditions would produce a radial stress gradient in an isotropic elastic cylinder. Mechanical twins in the deformed calcite samples had a hexagonal distribution in the 〈0001〉 oriented cylinder and an orthorhombic distribution in the \(\langle 40\bar 41\rangle\) oriented cylinder.

Principles of the German approach to the analysis of flawed structures

The analysis of flawed structures in the linear elastic regime (LEFM) has been treated very extensively, especially in the framework of the 4th German nuclear programme. Investigations with large specimens have demonstrated the applicability of the LEFM for a variety of materials ranging from low to high toughness levels. In the elastic-plastic (EPFM) regime, however, especially the J- integral method has been used for quantifying the safety margin, because with a required value of upper shelf Charpy energy alone this quantification is not possible. A correlation between the crack initiation value J i and Charpy upper shelf energy was evaluated experimentally, justifying the generally upgraded upper shelf energy level adopted in the Code. Furthermore, this correlation can be employed when fracture toughness values are not available as is mostly the case for irradiated material from surveillance programmes. To cope with the problems resulting from repeated transients in a complex component, incipient crack intitiation and propagation under cyclic thermal load have been investigated experimentally and theoretically on a RPV nozzle corner in the HDR plant and on a thick-walled (200 mm) hollow cylinder subjected to pressurized thermal shock (PTS). Since the OCA code covers only the linear elastic range, for this loading case elastic-plastic fracture mechanics calculations have been carried out by means of the Finite Element Method. The first test performed with high toughness material has shown good agreement with the J- integral approach. Additional validation tests are under way to demonstrate the lowest tolerable toughness level to withstand PTS without catastrophic failure. With respect to critical flaw sizes in degraded piping and vessels, the ‘leak before break’ limit curve for different loading conditions has been established and experimentally validated using piping and model vessels of different sizes, crack configurations and toughness levels. The still existing uncertainties in the detection, sizing and interpretation of signals from nondestructive examination are the background of NDE validation programmes for both medium size and full size reactor pressure vessels. Acoustic emission trials as well as extensive ultrasonic (US) examinations will be pursued in cooperation programmes, last but not least in the framework of PISC III.

Torsional Impact Response of an External Circular Crack in a Thick Walled Hollow Cylinder

Torsional Impact Response of an External Circular Crack in a Thick Walled Hollow Cylinder

Electromechanical equivalent circuit for cylindrical waves in thick walled hollow piezoceramic cylinders

An exact distributed electromechanical equivalent circuit is derived for purely cylindrical waves in an axially polarized piezoceramic thick walled hollow cylinder. The analysis applies to either stress free or clamped in the axial direction, and the theory may be used to account for losses.

A versatile hollow cylinder triaxial device

A high-pressure, thick-walled hollow cylinder test device has been perfected to explore the consequences of stress paths which are typical of pressuremeter configurations, tunnelling, or shafting in soil materials. Several features have been incorporated to facilitate data gathering, and the device is readily adaptable to K0 or standard compression and extension triaxial tests. It has been used on tests in artificially prepared dense sand specimens constructed by a freezing technique, and will be used in an extensive study of the behavior of oil sands. Typical results of tests on dense Ottawa sand (C-109) are presented.

Equivalent circuit for radial vibrations of thick walled hollow cylinders

The exact closed form distributed equivalent circuit and four‐pole parameters are derived for radial vibrators of thick walled hollow cylinders. The formulation applies to cylinders which are either very short (free) or very long (clamped) in the axial direction, and to lostless or viscoelastic materials.

Hoop-Stress Loading - A New Method of Determining the Tensile Strength of Rock

Abstract Although it is well known that rock commonly fails in tension under numerous field conditions, a satisfactory standard procedure for determination of the tensile strength of such materials has not as yet been established. With suitable precautions, direct-pull tests may be used; however, precise specimen preparation and extremely uniform and concentric loading must be utilized if meaningful results are to be obtained using this method. Recently, the authors completed a 3-year study in which a variety of methods for determining tensile strength were evaluated. One method, involving the hoop-stress loading of thick-walled hollow cylinders, was found to be particularly attractive. This paper describes the theory, apparatus, and test technique associated with this method. In evaluating this new technique a number of specimens of Barre granite, Lauders and Indiana limestone, and Berea and Tennessee sandstone were tested. Reproducibility for each rock type was fund to be excellent, with the percentage standard deviation ranging from a maximum of 10.9 percent for Lauders limestone down to 0.9 percent for Berea sandstone. Values of tensile strength determined from hoop-stress tests were found to correspond closely to those determined by direct pull, provided the former were analyzed in terms of pull, provided the former were analyzed in terms of an equation based on a suitable elastic-plastic analysis. Under these conditions the ratio of hoop-stress to direct-pull tensile strength values ranged from 0.86 to 1.10 for the five rocks tested. Although studies on additional rock types are required, it appears that the hoop-stress loading method would be a strong candidate as a standard test for determination of the tensile strength of rock. Introduction Although it is well known that tensile stresses play an important role in the behavior of underground play an important role in the behavior of underground openings associated with petroleum, natural gas and mining operations, suitable techniques for the determination of the tensile strength of rock are limited.

Investigation of stressed state of a viscoelastic hollow thick-walled cylinder under conditions of nonlinear creep

Investigation of stressed state of a viscoelastic hollow thick-walled cylinder under conditions of nonlinear creep

Analysis of nonlinear, dynamic coupled thermoviscoelasticity problems by the finite element method

This investigation deals with the numerical solution of a class of nonlinear problem in transient, coupled, thermoviscoelastidty. Equations of motion and heat conduction are derived for finite elements of thermomechanically simple materials and these are adapted to special classes of thermorheologically simple materials. The analysis involves the solution of large systems of nonlinear integrodifferential equations in the nodal displacements and temperatures and their histories. As a representative example, the general equations are applied to the problem of transient response of a thick-walled hollow cylinder subjected to time-varying internal and external pressures, temperatures, and heat fluxes. The integration scheme used to solve the nonlinear equations employs a linear acceleration assumption, representation of nonlinear integral terms by Simpson's rule, and the iterative solution of large systems of nonlinear algebraic equations at each reduced time step by the Newton-Raphson method. Various numerical results are given and are compared with the linearized, isothermal, and quasi-static solutions.