Plate Of Finite Thickness Research Articles

Surface cracks in finite thickness plates under thermal and displacement-controlled loads – Part 2: Crack propagation

Surface cracks in finite-thickness plates subjected to thermal or displacement-controlled loads are considered with a focus on crack propagation. Detailed crack propagation analyses are performed and the effect of loading type on crack propagation patterns and lives are systematically investigated. The results show that although there are some slight differences between the crack propagation patterns and lives for the uniform stress and displacement loading, there exist increasingly higher differences for the bending stress and displacement loads during crack growth. It is shown that propagation lives are higher in the case of displacement/thermal loading compared to those of mechanical/stress loads.

A review on coupled modes in V-notched plates of finite thickness: A generalized approach to the problem

Mode 1, 2 and 3 cannot exist in isolation. One mode provokes the existence of a coupled mode which, in some conditions, can be more dangerous than the generating mode itself. This means that three-dimensional problems are automatically at least dual scale. While for a crack this effect was known to exist for a long period of time, it was largely ignored in theoretical studies of V-shaped notches subjected to in-plane and out-of-plane loading as well as in practical fracture problems associated with such geometries. Only recently, some numerical investigations confirmed that highly localized coupled modes do exist in the close vicinity of the notch tip. The present paper is aimed to briefly review important features of these recently identified singular coupled modes. The most significant results from a comprehensive three-dimensional numerical study are presented here to describe the contribution of these modes into the overall stress state in the close vicinity of the notch tip and discuss the implementation of these new results to the failure and integrity assessment of plate structures with sharp notches.

Three‐dimensional effects in finite thickness plates weakened by rounded notches and holes under in‐plane shear

ABSTRACTIt is well known that in cracked plates under in‐plane shear loading, an induced out‐of‐plane mode, due to the Poisson's effect, is automatically generated. Some recent studies have shown the importance of this coupled mode (called mode O) in plates weakened by sharp V‐notches characterized by different notch opening angles. Particular attention has been paid to the scale effect focusing also on the implications of the coupled mode to different material scale levels.Also when the notch root radius is different from zero, the out‐of‐plane mode continues to exist, and its intensity with respect to the generating mode depends on the notch radius and the plate thickness. Aim of this paper is to discuss the main features of the out‐of‐plane mode dealing with three‐dimensional plates weakened by blunt notches. In particular, some results from blunt V‐notches, circular and elliptic holes are discussed in detail and presented here. The main key‐point highlighted in the present contribution is that always three‐dimensional effects exist in finite thickness plates, because mode I, II and III cannot exist in isolation. The importance of induced modes varies from case to case and depends on the geometry of the component and on the externally applied load. These effects cannot be neglected a priori even if, in some cases, they may be negligible.

Assessment of the interaction between two collinear cracks in plates of arbitrary thickness using a plasticity‐induced crack closure model

ABSTRACTFracture and fatigue assessment of structures weakened by multiple site damage, such as two or more interacting cracks, is currently a very challenging problem. The main objective of this paper is to develop a mathematical model and an approach to investigate fatigue crack closure behaviour of two through‐the‐thickness collinear cracks of equal length in a plate of arbitrary thickness under remote tensile cyclic loading. The developed mathematical model of the problem under consideration is based on the Dugdale strip yield model and plasticity‐induced crack closure concept. The approach utilises the fundamental solution for an edge dislocation in a plate of finite thickness and the distributed dislocation technique to obtain an effective and accurate solution to the system of governing equations. The obtained results show a very good agreement with the previously published analytical solutions for limiting cases. In particular, the new results confirm that the crack closure behaviour and the opening stress variation in the case of two collinear cracks are significantly dependent on the separation gap between two cracks as well as the plate thickness.

General point load weight function for semi-elliptical crack in finite thickness plates

General point load weight function for semi-elliptical crack in finite thickness plates is derived. It is used to calculate stress intensity factors for any point along the crack front with the crack faces under two dimensional stress distributions. The comparison between the results obtained from the derived weight function and those available in literature is presented, indicating a good accuracy. In order to determine the pattern of crack growth, and to estimate fatigue life, this weight function can be easily used in plates as applied in industry under complicated stress fields such as residual stress field due to welding.

The structure of convective flows driven by density variations in a continuously stratified fluid

The paper is devoted to an investigation of specific fluid motions observed around motionless objects immersed in a quiescent stratified medium. Such fluid motions, called diffusion-induced flows, may essentially influence the mixing processes and passive substance transport to a great distance and also the formation of stratified medium fine structures. The calculation results for diffusion-induced flow structures are presented in the paper for different geometrical configurations of impermeable obstacles, including a sloping plate of finite thickness, a wedge, a rectangular box, a disc and a cylinder. Even very weak stratification essentially affects a fine flow's structure and dynamics. An especially complicated flow structure is formed near the sharp edges and poles of impermeable obstacles, where the turning-points of the main jets serve as sources of dissipative-gravitational waves observed in both calculations and laboratory experiments in the form of extended horizontal streaky structures. Diffusion-induced flows always exist in non-homogeneous media at arbitrary orientations of an obstacle and non-exist in a homogeneous fluid.

English

Shielding interaction effects of two parallel edge cracks in finite thickness plates subjected to remote tension load is analyzed using a developed finite element analysis program. In the present study, the crack interaction limit is evaluated based on the fitness of service (FFS) code, and focus is given to the weak crack interaction region as the crack interval exceeds the length of cracks (b > a). Crack interaction factors are evaluated based on stress intensity factors (SIFs) for Mode I SIFs using a displacement extrapolation technique. Parametric studies involved a wide range of crack-to-width (0.05 ≤ a/W ≤ 0.5) and crack interval ratios (b/a > 1). For validation, crack interaction factors are compared with single edge crack SIFs as a state of zero interaction. Within the considered range of parameters, the proposed numerical evaluation used to predict the crack interaction factor reduces the error of existing analytical solution from 1.92% to 0.97% at higher a/W. In reference to FFS codes, the small discrepancy in the prediction of the crack interaction factor validates the reliability of the numerical model to predict crack interaction limits under shielding interaction effects. In conclusion, the numerical model gave a successful prediction in estimating the crack interaction limit, which can be used as a reference for the shielding orientation of other cracks.

Weight functions for the determination of stress intensity factor and T‐stress for semi‐elliptical cracks in finite thickness plate

ABSTRACTThis paper presents the application of weight function method for the calculation of stress intensity factors (K) and T‐stress for surface semi‐elliptical crack in finite thickness plates subjected to arbitrary two‐dimensional stress fields. New general mathematical forms of point load weight functions for K and T have been formulated by taking advantage of the knowledge of a few specific weight functions for two‐dimensional planar cracks available in the literature and certain properties of weight function in general. The existence of the generalised forms of the weight functions simplifies the determination of specific weight functions for specific crack configurations. The determination of a specific weight function is reduced to the determination of the parameters of the generalised weight function expression. These unknown parameters can be determined from reference stress intensity factor and T‐stress solutions. This method is used to derive the weight functions for both K and T for semi‐elliptical surface cracks in finite thickness plates, covering a wide range of crack aspect ratio (a/c) and relative depth (a/t) at any point along the crack front. The derived weight functions are then validated against stress intensity factor and T‐stress solutions for several linear and nonlinear two‐dimensional stress distributions. These derived weight functions are particularly useful for the development of two‐parameter fracture and fatigue models for surface cracks subjected to fluctuating nonlinear stress fields, such as these resulting from surface treatment (shot peening), stress concentration or welding (residual stress).

Recent developments in multi-parametric three-dimensional stress field representation in plates weakened by cracks and notches

The paper deals with the three-dimensional nature and the multi-parametric representation of the stress field ahead of cracks and notches of different shape. Finite thickness plates are considered, under different loading conditions. Under certain hypotheses, the three-dimensional governing equations of elasticity can be reduced to a system where a bi-harmonic equation and a harmonic equation have to be simultaneously satisfied. The former provides the solution of the corresponding plane notch problem, the latter provides the solution of the corresponding out-of-plane shear notch problem. The analytical frame is applied to some notched and cracked geometries and its degree of accuracy is discussed comparing theoretical results and numerical data from 3D FE models.

Three-dimensional elastic stress fields ahead of notches in thick plates under various loading conditions

The paper discusses the features of three-dimensional elastic stress distributions ahead of notches in finite thick plates under different loading conditions. It is proved that, under certain circumstances, the three-dimensional governing equations of elasticity can be reduced to a system where a bi-harmonic equation and a harmonic equation have to be simultaneously satisfied. The former provides the solution of the corresponding plane notch problem, the latter provides the solution of the corresponding out-of-plane shear notch problem. The analytical frame is applied to a number of notched geometries, and its degree of accuracy is discussed comparing theoretical results and numerical data from 3D FE models. Practical consequences on the expected crack paths are also documented for one of the considered models.

An equivalent thickness conception for evaluation of corner and surface fatigue crack closure

The out-of-plane constraint-based equivalent thickness conception developed previously for corner cracks is extended here to surface-cracked plates based on systematical finite element simulations for the purpose to evaluate the plasticity-induced fatigue closure of corner and surface cracks. A semi-analytical solution for equivalent thickness of semi-elliptic surface cracks is obtained by numerical analyses of three-dimensional stress fields near the crack borders and comparing with that of ideally straight-through cracks in plates of finite thickness. The fatigue opening stresses of corner and surface cracks are evaluated based the recognized equations of straight-through cracks using the equivalent thickness conception. Comparison against available finite element and experimental results shows that the developed method can provide satisfied prediction of fatigue closure for both corner and surface cracks.

The Coupled Effects of Thickness and Distance between Holes on Stress and Strain Concentrations of Uniaxially Loaded Tensile Plates

The elastic stress and strain fields of plates containing one central hole and two auxiliary holes subjected to uniaxial tension are examined using finite element method. The interaction between holes and the thickness effect on stress and strain concentration factors are investigated. It is shown that the distributions of strain concentration factor and stress concentration factor are different in the region of stress concentration and both of them depend on the hole radii and the distance between central hole and auxiliary hole. For the finite thickness plates, the distributions of stress and strain concentration factors through thickness are not uniform in stress concentration regions. There is a strong disturbed region near the plate surface in the vicinity of central hole and auxiliary hole. The maximum values of stress and strain concentration factors and the corresponding locations depend on the r/R ratio, d/R ratio, Poisson’s ratio and thickness of plate.

Complex band structures and evanescent Bloch waves in two-dimensional finite phononic plate

The complex band structure of a two-dimensional (2D) phononic crystal based on solid or air cylindrical inclusions in solid finite thickness plate is investigated. The plane wave expansion method is developed with a specific extension to calculate the evanescent Bloch modes propagating in the phononic plate. It is shown that, in the opposite of 2D infinite phononic crystals, the complex band structures of Lamb modes exhibits a coupling between in-plane and out-of plane Lamb modes. The polarization and the behavior of the evanescent waves is analyzed and discussed as function of the plate thickness to figure out the physics behind the opening of phononic band gap for the different Lamb modes.

Investigation on <i>T</i><sub>11</sub>-Stress for Semi-Elliptical Surface Cracks in Finite Thickness Plates under Remote Tension

In the present work, three-dimensional finite element analyses have been conducted to calculate the-stress for semi-elliptical surface cracks in finite thickness plates under remote tension. The-stress solutions are presented along the crack front for cracks with values of 0.2, 0.4, 0.6 or 0.8 and values of 0.2, 0.4, 0.6 or 1.0. The current-stress solutions are suitable to be used as the constraint parameter for the fracture analysis.

An Experimental Study on the Effect of Submergence on a Vertically Oscillating Hydrofoil at Zero Translational Velocity

This paper presents experimental results for the hydrodynamic forces on several different bodies oscillating in the vicinity of the free surface. Firstly, results for two horizontally symmetric bodies, a flat plate of finite thickness and a cylinder, are presented. Results show that these two different shaped bodies react in different ways with the free surface and are likely to have different added mass associated with them. Secondly, results for two thrusting bodies, a NACA0012 symmetrical foil and a NACA4415 cambered foil, show that the free surface amplifies the thrust produced by a vertically oscillating foil. Results showing the effects of spanwise flow are also presented and indicate spanwise flow is negligible at zero translational velocity and so the use of endplates of the foils was unnecessary. The conclusion of these experiments is that a submergence of 5 chord lengths is sufficient for the horizontal and vertical forces on the vertically oscillating objects to be unaffected by the free surface.

Analytical and numerical computations of the van der Waals force in complex geometries: Application to the filtration of colloidal particles

Particle capture during the filtration of colloidal dispersions depends on a complex balance between repulsive forces, such as hydrodynamic or electrostatic effects, and attractive forces, amongst them the van der Waals interaction forces. Satisfactory expressions for the latter are thus required in complex geometries. Exact expressions for the geometrical factor involved in the van der Waals interaction energy based on Hamaker's additivity hypothesis are derived for a sphere in interaction with a square wedge, a semi-infinite or finite slit, a semi-infinite slab, a 2D pillar, a rectangular rod, a corner and a rectangular channel. A numerical tool based on an adaptive mesh refinement strategy is presented and used to validate the analytical results. The analytical result for a sphere/wedge system is used to assess the domain of applicability of the sphere/plane model in the vicinity of the edge. The interaction between a sphere and a cylindrical pore in a plate of finite thickness is then simulated and the range of validity of the sphere/wedge system as a model of the sphere/pore system is deduced from the numerical results.

Two-parameter characterization of elastic–plastic crack front fields: Surface cracked plates under uniaxial and biaxial bending

In this paper the J–Q two-parameter characterization of elastic–plastic crack front fields is examined for surface cracked plates under static uniaxial and biaxial bending. Extensive three-dimensional elastic–plastic finite element analyses are performed for semi-elliptical surface cracks in a finite thickness plate, under remote uniaxial and biaxial bending conditions, covering from small-scale to large-scale yielding. Surface cracks with aspect ratios a/c=0.2, 1.0 and relative depths a/t=0.2, 0.6, corresponding to four specimen configurations, are investigated. The full stress field solutions are obtained in topological planes perpendicular to the crack fronts. The question of J–Q characterization is addressed by comparing these elastic–plastic crack-front stress fields with J–Q family of stress fields. It is found that the J–Q characterization provides good estimate for the constraint loss for crack front stress fields. It is also shown that for medium load levels, reasonable agreements are achieved between the T-stress based Q-factors and the Q-factors obtained from the finite element analyses. Complete distributions of the J-integral and Q-factors for the wide range of geometry and loading conditions are obtained. Size requirements in terms of crack depth (for shallow cracks) or ligament size (for deep cracks) that are necessary to ensure J–Q characterization of crack front fields in surface cracked plates are also discussed.

A strip yield model for two collinear cracks in plates of arbitrary thickness

As two cracks grow and approach each other under fatigue loading, a deleterious interaction between them can considerably affect the crack growth rate, making theoretical evaluations and experimental data from a single isolated crack case considerably inaccurate. The aim of the present study is to investigate the interaction between two collinear cracks of equal length, taking into account the plate’s thickness effect, which was demonstrated to have a large effect on fatigue crack growth in the case of a single crack. The obtained solution to the problem is based on the Dugdale strip yield model and the distributed dislocation technique. In addition, a fundamental solution for an edge dislocation in a finite thickness plate is utilised. The present solution shows a very good agreement with previously published results for some limiting cases. The obtained results confirm a significant dependence of the interaction and stress intensity factors on the plate thickness, which can dramatically affect the plastic collapse conditions as well as fatigue crack growth rates.

A three‐dimensional stress field solution for pointed and sharply radiused V‐notches in plates of finite thickness

ABSTRACTBy making use of the generalized plane strain hypothesis, an approximate stress field theory has been developed according to which the three‐dimensional governing equations lead to a system where a bi‐harmonic equation and a harmonic equation should be simultaneously satisfied. The former provides the solution of the corresponding plane notch problem, and the latter provides the solution of the corresponding out‐of‐plane shear notch problem. The system can be applied not only to pointed three‐dimensional V‐notches but also to sharply radiused V‐notches characterized by a notch tip radius small enough. Limits and degree of accuracy of the analytical frame are discussed comparing theoretical results and numerical data from FE models.

Penny-shaped crack in a plate of finite thickness subjected to a uniform shearing stress

This paper contains an analysis of the stress distribution in a plate of finite thickness of an elastic material containing a penny-shaped crack when it is deformed by the application of a uniform shearing stress on the surface of the plate. The crack lies on the central plane of the plate with its surface parallel to those of the plate and is stress-free. By making a suitable representation of the stress function for the problem, the problem is reduced to the solution of a pair of Fredholm integral equations of the second kind. This pair of Fredholm integral equations of the second kind is solved numerically, and the increase in the stress intensity magnification factor due to the effect of the finite thickness of the plate is presented for various values of the thickness.