Stress-free End Research Articles

Комбінування детермінованого та стохастичного методів до розв’язання задачі дефектоскопії пружного стрижня

The paper considers the problem of natural harmonic oscillations of an elastic rod with stress-free ends in the presence of one or a set of defects. Defects are modeled by the inhomogeneity of the Young's modulus. The location of the defects, their geometric size, which is considered small, and the change in elastic properties are the parameters of the defects. The analysis of natural frequency shifts caused by the defect of the rod is the subject of the study. The aim of the work is a mathematical substantiation for the construction of fast and stable algorithms for determining the defect parameters of elastic bodies by analyzing free oscillations. The paper uses and compares fundamentally different research methods. The first methods are classical mathematical methods of mechanics, applied to the analysis of deterministic systems and based on analytical studies combined with numerical implementation. In contrast, a composite machine learning meta-algorithm used in standard statistical classification and regression - Bootstrap-aggregated Regression Trees (BART) - is used to solve the inverse problem. When comparing the constructed algorithms, the statistical method Sampling was used, which allowed to quantify the accuracy and stability of the algorithms.

Elastic softening of sandstone due to a wideband acoustic pulse observed by dynamic acousto-elastic testing

Laboratory measurements of finite-amplitude, wideband elastic pulse propagation in a thin bar of Texas moss sandstone have been reported [Muir et al., JASA 147 (2020)] that utilized pulses generated by impact excitation with a pendulum hammer at one end of the bar. Each pulse is essentially unipolar in strain, with bandwidth of approximately 3 kHz and amplitude between 10 and 130 microstrain. Reduction of the sandstone elastic modulus by up to 17% resulted from propagation of the impact-generated pulse, as calculated from time-of-flight measurements of the pulse as it reflected between the ends of the bar. In the present work, elastic softening is observed locally using a Dynamic Acousto-Elastic Testing (DAET) configuration. During propagation of an impact-generated pulse, changes in the sandstone elastic properties are observed by tracking variations in the arrival time and amplitude of a continuous-wave 1 MHz P-wave signal that propagates perpendicular to the bar axis. Repetitive reflection of the pulse from the stress-free ends of the bar enables separate observation of effects from compressive and tensile strains. Reduction of the elastic modulus of up to 20% is observed, with almost all of the softening occurring during the tensile phase of the impact-generated pulse. a)Deceased.

Axially symmetric elasticity problems for the hollow cylinder with the stress-free ends. Analytical solving via a variational method of homogeneous solutions

Axially symmetric elasticity problems for the hollow cylinder with the stress-free ends. Analytical solving via a variational method of homogeneous solutions

Contact problems for a circular plate with sliding fixation at the end face

Two mixed elasticity problems of punch indentation into a circular plate placed without clearance in a rigid cylindrical holder with smooth walls are considered. In the first problem, the plate lies without friction on a rigid base, and in the second problem, the plate is rigidly fixed to the base. The problems are solved by a method that was developed for bodies of finite dimensions and is based on the properties of closed systems of orthogonal functions. Each of the problems is reduced to two integral equations, namely, a Volterra integral equation of the first kind for the contact pressure function and a Fredholm integral equation of the first kind for the derivatives of the displacement of the plate upper surface outside the punch. The displacement function is sought as the sum of a trigonometric series and a power function with a root singularity. After truncation, the obtained illposed system of linear algebraic equation has a stable solution. A method for solving Volterra integral equations is given. The contact pressure distribution function and the dimensionless indentation force are determined. Examples of calculation of the plate interaction with the plane punch are given. Contact problems were earlier studied for a rectangle and a circular plate with a stress-free end both without taking account of their fixation [1, 2] and with regard for their fixation [3, 4]. The solution method described here was used to study the interaction of elastic hollow cylinder of finite length with a rigid bandage and a rigid insert [5, 6]. Other papers dealing with contact problems for bodies of finite dimensions, in particular, for a circular plate, should also be mentioned. In these papers, the problems under study were solved by the method of homogeneous solutions [7, 8] and by the method of coupled series-equations [9].

Simulation of flexural waves in drill pipes including the effects of the gravitational field

We design a full-wave modeling method to simulate flexural (bending) waves in pipes of variable cross-section and material properties, subject to the effects of the gravitational field. The study finds application in propagation through drill strings and stability of hydrocarbon wells while drilling. The algorithm is based on a direct (grid) method, where the spatial derivatives are computed with the Fourier pseudospectral method. The modeling is successfully tested against the analytical solution for flexural waves propagating in a uniform pipe. Moreover, we obtain reciprocity relations for the lateral deflection, particle velocity and bending moment due to sources applied in the force-balance and constitutive equations. The relations are then verified by the simulations and at the same time provide a test of the consistency of the numerical algorithm in inhomogeneous media, where there is no closed-form analytical solution.We analyze the effects of the gravitational field and show that for negative axial loads (compressive stresses) there is an instability below a critical wavenumber, while the system is stable for positive (tensile) loads. Such a situation appears below the neutral point of a drill string when the gravitational force is taken into account. A plane-wave analysis yields the phase and group velocities when the load is uniform along the pipe and in the case of a suspended pipe subject to a gravitational force with a neutral point downhole. In the latter case, the signal propagates with attenuation. Propagation along a realistic drill string, simulating the drill bit as a stress-free end, is illustrated. Moreover, propagation subject to an axial gravitational load shows that the signal is slower for a tensile prestress.

The contact problem for a circular plate with a stress-free end face

An axisymmetric contact problem for a circular elastic plate with a stress-free end face into which two symmetrically arranged punches are imbedded, is considered. The problem is solved using a method developed earlier for bodies of finite sizes, which is based on the generalized orthogonality of homogeneous solutions. The problem reduces to a Fredholm integral equation of the first kind in a function describing the displacement of the surface of the plate outside the punch. These functions are sought in the form of a sum of a Schlömilch series and a power function with a root singularity. The ill-posed infinite system of algebraic equations obtained as a result is regularized by the introduction of a small positive parameter. Since the matrix elements of the system, as well as the contact stresses, are defined by poorly converging numerical and functional series, the technique of summation of the residues of these series is used. The distribution of the contact pressure and the dimensionless imbedding force are found. Examples of the calculation of a plane punch are given.

Bond strength of deformed bars in large reinforced concrete members cast with industrial self-consolidating concrete mixture

The bond strength of reinforcing bars embedded in full-scale heavily reinforced concrete sections made with industrial self-consolidating concrete (SCC) was investigated and compared with that of normal concrete (NC). The flowability of SCC mix through the dense reinforcement was visually monitored from a transparent formwork. The bond stress was tested for bars located at three different heights (150 mm, 510 mm, and 870 mm from the bottom of the pullout specimens) and at different tested ages (1, 3, 7, 14, and 28 days). The bond stress-free end slip relationship, the top bar effect and the effect of age on bond stress was investigated in both SCC and NC pullout specimens. Bond stresses predicted based on some major codes were compared with those obtained from experiments. The results indicated that casting SCC was much faster and easier and could be done with less labor effort and no concrete blockage among the heavy reinforcements compared to NC. The results also indicated that the bond stress was slightly higher in the SCC pullout specimen compared to the NC pullout specimen. The difference was more pronounced in the top bars and at 28 days of testing.

Vibrational cavity modes in a free cylindrical disk

We analyze theoretically the vibrational properties of a free cylindrical elastic disk of thickness $2h$ and radius $a$. In particular, the acoustic cavity modes, i.e., the low-lying resonant vibrations with a large angular wave number $n$ ($=2\ensuremath{\pi}a/\ensuremath{\lambda}\ensuremath{\sim}20$, with $\ensuremath{\lambda}$ as the wavelength) along the circumference are considered. These modes are either confined at the corners between the sidewall and the top and bottom planes of the disk (the edge modes), or trapped near the circumference of the cylinder (the surface modes). They are understood as the modes modified from the whispering-gallery and Rayleigh waves on the curved surface of an infinitely long cylinder by the presence of the stress-free end planes. Small mode volumes occupied by these vibrations and their expected large quality factors in the disk with smooth boundary surfaces make the cavity modes attractive for the fundamental studies as well as for the applications to micro- and nanophononics. The mathematical formulation follows the works originally developed by Rasband [J. Acoust. Soc. Am. 57, 899 (1975)] and Hutchinson [J. Appl. Mech. 47, 901 (1980)]. Numerical examples are presented for polycrystalline aluminum disks with aspect ratios $h/a=0.5$ and 0.05.

Two-and three-dimensional linear convection in a rotating annulus

Abstract An exceptional case to the model-independent theory of Knobloch (1995) is presented, by investigating a rotating cylindrical annulus of height H and side wall radii r o and r i, with non-slip, perfectly thermally conducting side walls and thermally insulating stress-free ends. Radial heating permits the possibility of either two- or three-dimensional convective solutions being the preferred mode. An analytical solution is obtained for the two-dimensional case and a numerical solution for the three-dimensional solution, which is also applied to the two-dimensional solution. It is shown that both two- and three-dimensional solutions can be realized depending on the aspect ratio, γ = H/d, where d = r o-r i is the thickness of the annulus, the radii ratio λ = r i/r o and the rotation rate of the model. For γ = O(1) and λ = 0.4, the preferred convective solution is three-dimensional when the Taylor number, T 102. For small aspect ratios, γ ≪ 1, the preferred mode is t...

Fibre-reinforced laminated composite tubes with free ends under uniform internal pressure

The stress and deformation fields in a fibre-reinforced composite tube under uniform internal pressure are discussed in some detail. In the interior region far from the ends classical laminate theory delivers rather poor results and has to be adjusted to include effects due to lateral contraction and to curvature. In the region near the ends boundary conditions (here stress-free ends will be assumed) require more elaborate methods of calculation. The use of finite element methods may prove to be problematic because in some parts of the boundary region very large gradients are expected. The problem is particularly acute in lay-ups with angle-plies where 3D-elements would be needed. In the following study analytical solutions based on asymptotic approximations of the three-dimensional equations of linear elasticity for homogeneous orthotropic materials will be presented. One “small” parameter ε R characterising thin shell geometry and another “small” parameter ε G following from homogenised material properties of the shell structure and whose order of magnitude is comparable with ε R are used to derive asymptotically consistent approximate solutions according to the following pattern: The adjusted laminate theory leads to stress distributions in each ply of the laminated tube which do not satisfy zero stress boundary conditions at the stress-free ends. In terms of asymptotic theory this is a typical problem of singular perturbations and can be solved by considering boundary layers near the free ends where stress and deformation fields satisfy the boundary conditions and match conveniently with stress and deformation distributions calculated with the adjusted laminate theory in the interior zone. To derive boundary layer equations which are easy to handle analytically and still obtain fairly accurate results, we replace the laminated structure by its homogeneous orthotropic equivalent. The boundary layer solutions are then obtained following the main ideas developed by Sayir [(1985) Local Effects in the Analysis of Structures, Elsevier Science Publishers; (1986) Bull. Tech. Univ. Istanbul 39, 515–528]. We first study as a leading example a hypothetical cross-ply structure and concentrate on transverse shear and axial normal stress distributions across the wall thickness and along the axial direction in the boundary layer region. We show that all boundary conditions near the tube ends can be fulfilled by introducing first a “normal-stress” boundary layer starting at the tube end and extending in the axial direction, then a shorter “shear-stress” boundary layer contained in the previous one and finally two lateral boundary layers starting at the inner and outer surface of the tube wall inside the shear stress boundary layer. We compare analytical results obtained with the help of the asymptotic method described above with those obtained from three-dimensional finite-element calculations for each ply. Then we discuss two cases of symmetric angle-ply lay-ups corresponding to tubes actually tested in the laboratory by the second author and enhance similarities and differences with respect to the cross-ply lay-up. We also show how the interlaminar shear stresses which may cause delaminations and failure depend strongly on the lay-up.

On the transition to columnar convection

Convection in a rotating annulus with no-slip sidewalls, stress-free ends, radial gravity, and sideways heating is considered. The transition from fully three-dimensional convection cells to Taylor columns with increasing rotation rate is studied and its dependence on the annulus parameters is established.

Fracture mechanics of functionally graded materials

In this paper, after a brief discussion of the elementary concepts of fracture mechanics in nonhomogeneous materials, a number of typical problem areas relating to the fracture of functionally gradient materials (FGMs) are identified. The main topics considered are the investigation of the nature of stress singularity near the tip of a crack fully embedded in a nonhomogeneous medium, the general problem of debonding of an FGM coating from a homogeneous substrate, the basic surface crack problem in FGMs and cracking perpendicular to the interfaces, periodic surface cracking and the associated problem of stress and energy relaxation, and the problem of stress concentration at and the initiation and growth of delamination cracks from the stress-free ends of FGM-coated homogeneous substrates under residual or thermal stresses. Each topic is very briefly reviewed, some sample results are presented and comparisons with the corresponding results obtained from homogeneous materials are made.

Quantitative investigation of the role of internal stresses and of dislocation pile-ups in tension and fatigue deformation and fracture using a unidirectionally solidified composite: Valle, R. Phil. Mag. A.65 1 (Jan 1992) 149–176

The stress and deformation fields in a fibre-reinforced composite tube under uniform internal pressure are discussed in some detail. In the interior region far from the ends classical laminate theory delivers rather poor results and has to be adjusted to include effects due to lateral contraction and to curvature. In the region near the ends boundary conditions (here stress-free ends will be assumed) require more elaborate methods of calculation. The use of finite element methods may prove to be problematic because in some parts of the boundary region very large gradients are expected. The problem is particularly acute in lay-ups with angle-plies where 3D-elements would be needed. In the following study analytical solutions based on asymptotic approximations of the three-dimensional equations of linear elasticity for homogeneous orthotropic materials will be presented. One “small” parameter εR characterising thin shell geometry and another “small” parameter εG following from homogenised material properties of the shell structure and whose order of magnitude is comparable with εR are used to derive asymptotically consistent approximate solutions according to the following pattern:The adjusted laminate theory leads to stress distributions in each ply of the laminated tube which do not satisfy zero stress boundary conditions at the stress-free ends. In terms of asymptotic theory this is a typical problem of singular perturbations and can be solved by considering boundary layers near the free ends where stress and deformation fields satisfy the boundary conditions and match conveniently with stress and deformation distributions calculated with the adjusted laminate theory in the interior zone. To derive boundary layer equations which are easy to handle analytically and still obtain fairly accurate results, we replace the laminated structure by its homogeneous orthotropic equivalent. The boundary layer solutions are then obtained following the main ideas developed by Sayir [(1985) Local Effects in the Analysis of Structures, Elsevier Science Publishers; (1986) Bull. Tech. Univ. Istanbul 39, 515–528]. We first study as a leading example a hypothetical cross-ply structure and concentrate on transverse shear and axial normal stress distributions across the wall thickness and along the axial direction in the boundary layer region. We show that all boundary conditions near the tube ends can be fulfilled by introducing first a “normal-stress” boundary layer starting at the tube end and extending in the axial direction, then a shorter “shear-stress” boundary layer contained in the previous one and finally two lateral boundary layers starting at the inner and outer surface of the tube wall inside the shear stress boundary layer. We compare analytical results obtained with the help of the asymptotic method described above with those obtained from three-dimensional finite-element calculations for each ply. Then we discuss two cases of symmetric angle-ply lay-ups corresponding to tubes actually tested in the laboratory by the second author and enhance similarities and differences with respect to the cross-ply lay-up. We also show how the interlaminar shear stresses which may cause delaminations and failure depend strongly on the lay-up.

Torsional stress in tubular lap joints

In the current paper, the stress distribution in adhesive-bonded tubular lap joints subjected to torsion is analyzed. The two adherends may have different thicknesses and consist of different materials and the adhesive layer may be flexible or inflexible. The analysis is based on the elasticity theory in conjunction with the variational principle of complementary energy. By means of the present approach, closed form solutions are obtained and the stress-free end conditions of the joint which may have significant effect on the stress intensities is satisfied. Special attention is given to the high stress intensities in the end zones of the joint, and a stress concentration factor is deduced.

A cylindrical shell with a stress-free end which contains an axial part-through or through crack

In this article the interaction problem of a through or a part-through crack with a stressfree boundary in a semi-infinite cylindrical shell is considered. It is assumed that the crack lies in a meridional plane which is a plane of symmetry with respect to the external loads as well as the geometry. The circular boundary of the semi-infinite cylinder is assumed to be stress-free. By using a transverse shear theory the problem is formulated in terms of a system of singular integral equations. The line spring model is used to treat the part-through crack problem. In the case of a through crack the interaction between the perturbed stress fields due to the crack and the free boundary is quite strong and there is a considerable increase in the stress intensity factors caused by the interaction. On the other hand in the problem of a surface crack the interaction appears to be much weaker and consequently the magnification in the stress intensity factors is much less significant.

THERMAL STRESSES IN LAMINATED BEAMS

Abstract This paper considers laminated beams consisting of layers of different materials fastened together by thin adhesives. The stresses that result from subjecting the beam to temperature stimuli are calculated. The problem is treated by two-dimensional elasticity theory in conjunction with the variational theorem of complementary energy. A pair of governing differential equations is developed, and boundary conditions concerning stress-free surfaces and ends of the beam are satisfied. The calculation of the distributions of interlaminar normal and shear stresses shows that high stress intensity occurs in the end zones of the beam. Thus, the satisfaction of end conditions is of prime importance in the analysis of laminated structural elements. Delamination failure-when it occurs - will probably start at the ends of the beam, This agrees with observed failures of laminated structural elements subjected to stress-free end conditions.

Opening mode stress intensity factors for cracks in pin-loads joints

Opening mode stress intensity factors are determined using point matching for two diametrically opposed radial cracks at a hole in a rectangular sheet where the cracks lie along the line of the minimum section. The sheet is subjected to either a biaxial stress on its edge or a pressure distribution on the hole boundary. The pressure on the hole is uniform or has a cosine distribution, symmetrical about the line of the cracks and having zero value on the crack-line. By applying the principle of superposition it is shown that the results can be applied to pin-loaded joints to determine stress intensity factors for different combinations of pin-load and interference fit and the effects of load transfer ratio between the pin or rivet load and the remote loading. Detailed results are given for stress intensity factors of cracks in a wide range of typical pin-joint configurations and it is shown that the stress intensity factors, for small cracks, are strongly dependent on the type of pressure distribution assumed to represent the pin-loading. Reducing the distance from the pin to the stress-free end of the joint is shown to increase the stress intensity factor more in the case of pin-loading than in the case of a uniaxial stress. The stress intensity factors, determined for simple lug-joints, are in agreement with available results from existing theoretical and experimental work and, for cracks at fastener holes, comparisons are made with other more approximate solutions.

Stress-free end problem in layered materials

In this paper the plane elastostatic problem for a medium which consists of periodically arranged two sets of bonded dissimilar layers or strips is considered. First it is assumed that one set of strips contains a crack which crosses the bimaterial interfaces. Then, by letting the collinear cracks join, the stress-free end problem is formulated. The singular behavior of the solutions at the point on intersection of the stress-free boundary and the interfaces is examined and appropriate stress intensity factors are defined. The results of some numerical examples are then presented which include the cases of both plane stress and plane strain.

The Crack-Contact and the Free-End Problem for a Strip Under Residual Stress

The plane problem for an infinite strip with two edge cracks under a given state of residual stress is considered. The residual stress is compressive near and at the surfaces and tensile in the interior of the strip. If the crack is deep enough to penetrate into the tensile zone, then the problem is one of crack-contact problem in which the depth of the contact area is an unknown which depends on the crack depth and the residual stress profile. The problem has applications to the static fatigue of glass plates and is solved for three typical residual stress profiles. In the limiting case of the crack crossing the entire plate thickness, the problem becomes a stress-free end problem for a semi-infinite strip under a given residual stress state away from the end. This is a typical stress diffusion problem in which decay behavior of the residual stress near and the nature of the normal displacement at the end of the semi-infinite strip are of special interest. For two typical residual stress states the solution is obtained and some numerical results are given.

Torsional-mode transients in variable-thickness shells of revolution

For thin shells of revolution the existence of torsional-vibration modes, uncoupled from bending and extensional modes, has been established[1]. Here a linear second-order differential equation for the uncoupled torsional stress mode is obtained and its solution for impact loading of shells is sought. The mode-superposition method which utilizes the natural modes of vibration predicted by elementary theory, is, in general, not satisfactory for sharp impact loading as many modes are often required for convergence. Hence we employ two novel techniques for solving the impact problems. Firstly a formal asymptotic procedure, based on extensions to geometrical optics, is employed to generate asymptotic wavefront expansions. Rigorous justifications for this formal technique are provided in an appendix. Secondly a transform technique whereby solutions are sought in terms of Bessel functions is discussed and applied to particular impact loading problems. The Bessel function solutions found here can be used to determine the natural frequencies of the shells. Shells both finite and infinite in extent are discussed and reflections at a stress-free end are examined.