Fracture of pressurized cylindrical shells

Pipeline fracture is analyzed via Lagrangean mechanics. The bursting pipeline is a system with two degrees of freedom. The two equations of motion for the crack are derived and solved for a simplified case. The fluid mechanics of a pressurized pipe during fracture is discussed and the different behavior of liquids and gases is indicated.     

Experimental study of pressurized,foam reinforced,cracked cylindrical aluminum shells

Cracks in pressurized cylindrical shells are subject to stress fields peculiar to the shell geometry and loading conditions which makes them more critical than cracks in flat plates under similar conditions. In order to mitigate the stress field around the crack tips a foam layer is applied to the shell as reinforcement. The effect of this solution on crack behavior is evaluated experimentally. Pre-cracked cylindrical aluminum shells with layers of rigid foam applied to their inner side are tested by subjecting them to cyclic internal pressure. A test bench built especially for the purpose of cyclic testing is described as well as the testing procedure. Results indicate that crack growth rates are significantly reduced and fatigue life is extended by as much as 161%. The application of a foam layer to may help slow down the growth of existing cracks. This solution may prove useful especially in ageing aircraft fuselage structures.     

Linear elastic fracture mechanics applied to cracked plates and shells

A general theory concerning through-cracks in plates and shells is proposed. Applying the method of local development of the kinematic unknowns to a shell of arbitrary shape, the distribution of displacements and rotations as functions of polar coordinates in the vicinity of the crack tip is given and the five corresponding intensity factors are defined. New path-independent integrals are introduced and related to the intensity factors so that these can be evaluated numerically. Finally, a fracture criterion for plane problems is extended to shells.

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Résumé On propose une théorie générale du comportement des fissures traversantes dans les plaques et les coques. Pour cela, on applique la méthode du développement limité des variables cinématiques à une coque de forme quelconque. On obtient la distribution des déplacements et des rotations en fonction des coordonnées polaires dans le voisinage de la singularité et on définit les cinq facteurs d'intensité correspondants. On propose de nouvelles intégrales curvilignes independantes du contour et on les relie aux facteurs d'intensité de telle sorte que ces derniers peuvent être calculés numériquement. Enfin on étend aux coques un critère de rupture utilisé jusqu'alors dans les problèmes plans.

     

Fracture mechanics analysis of cracked 2-D anisotropic media with a new formulation of the boundary element method

A new formulation of the boundary element method (BEM) is proposed in this paper to calculate stress intensity factors for cracked 2-D anisotropic materials. The most outstanding feature of this new approach is that the displacement and traction integral equations are collocated on the outside boundary of the problem (no-crack boundary) only and on one side of the crack surfaces only, respectively. Since the new BEM formulation uses displacements or tractions as unknowns on the outside boundary and displacement differences as unknowns on the crack surfaces, the formulation combines the best attributes of the traditional displacement BEM as well as the displacement discontinuity method (DDM). Compared with the recently proposed dual BEM, the present approach doesn't require dua elements and nodes on the crack surfaces, and further, it can be used for anisotropic media with cracks of any geometric shapes. Numerical examples of calculation of stress intensity factors were conducted, and excellent agreement with previously published results was obtained. The authors believe that the new BEM formulation presented in this paper will provide an alternative and yet efficient numerical technique for the study of cracked 2-D anisotropic media, and for the simulation of quasi-static crack propagation.     

Finite element large deflection analysis of shallow shells

A finite element formulation is presented to study the non-linear buckling of arbitrary shallow elastic thin shells with general boundary conditions and subjected to conservative pressure loading. Pre and post buckling behaviour of a large number of shallow and semi deep doubly curved shells is studied in detail. Unsymmetrical bifurcation paths of a shallow spherical shell subjected to uniform inward pressure are also investigated.     

Post-buckling analysis of shallow shells by the field-boundary-element method

The non-linear field-boundary-element technique is applied to the analysis of snap-through phenomena in thin shallow shells. The equilibrium path is traced by using the arc-length method and the solution strategy is discussed in detail. The results show that, as compared to the approaches based on the popular symmetric-variational Galerkin finite element formulation, the current approach based on an unsymmetric variational Petrov–Galerkin field-boundary-element formulation gives a faster convergence while using fewer degrees of freedom. The illustrative numerical examples deal with post-buckling responses of several shallow shells with different geometries.     

Fracture mechanics

Book Review

Fracture mechanicsT.L. Anderson, CRC press, inc., Boca Raton, FL     

Fracture analysis of center cracked infinite plates of linearly softening materials

A crack in an infinite plate is described as a mixed boundary value problem based on the Damage Zone Model. The boundary value problem is treated by means of Fourier transformation. The resulting integral equations are solved numerically to give stresses and displacements. It is shown that the maximum applied stress before failure can be found by an eigenvalue calculation. The dependence of the maximum applied stress on the crack length and the material parameters is given.     

Vibration of doubly-curved shallow shells

Summary This paper presents the free vibration of thin doubly-curved shallow shells of rectangular planform. The study covers wide combinations of free, simply supported and clamped boundary conditions. Both positive and negative Gaussian curvatures (spherical and hyperbolic paraboloidal shells resepectively) are considered. Thepb-2 Ritz energy based approach, along with the in-plane and transverse deflections assumed in the form of a product of mathematically complete two-dimensional orthogonal polynomials and a basic function, is employed to model the vibratory characteristic of these shells. Numerical results have been established through convergence study and comparison with published data from the open literature. Extensive sets of new results for various ranges of aspect ratio, curvature ratio andx- andy-shallowness ratios have been presented for future reference.     

An efficient finite element with layerwise mechanics for smart piezoelectric composite and sandwich shallow shells

In this work, we present a new efficient four-node finite element for shallow multilayered piezoelectric shells, considering layerwise mechanics and electromechanical coupling. The laminate mechanics is based on the zigzag theory that has only seven kinematic degrees of freedom per node. The normal deformation of the piezoelectric layers under the electric field is accounted for without introducing any additional deflection variables. A consistent quadratic variation of the electric potential across the piezoelectric layers with the provision of satisfying the equipotential condition of electroded surfaces is adopted. The performance of the new element is demonstrated for the static response under mechanical and electric potential loads, and for free vibration response of smart shells under different boundary conditions. The predictions are found to be very close to the three dimensional piezoelasticity solutions for hybrid shells made of not only single-material composite substrates, but also sandwich substrates with a soft core for which the equivalent single layer (ESL) theories perform very badly.     

Fracture criteria of cracked metallic plate

The Griffith energy criterion is equivalent to a critical stress and strain environment criterion for brittle fractures. For quasi-brittle fracture, i. e. the case where small amounts of plastic deformation at the crack tip precedes fracture, the energy criterion is still equivalent to stress and strain environment criterion. Based on the concept of a critical stress and strain environment, it is proposed as an engineering criterion for ductile fracture, that at fracture the size of the highly strained region at crack tip is constant.

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Zusammenfassung Griffith fand das Energiebruch Kriterium fuer sproedes Material. Fuer das Spannungsund Dehnungs Kriterium fuer sproede Brueche ist dieses Energie em fluss Kriterium gleichwertig, wie von der elastischen Loesung fuer sproede teste Koerper ersehen werden kana. Fuer fast sproede Brueche, dass heisst, in dem Falle wo geringe Mengen bildsamer Formgebung an der Bruchspitze dem Bruch vorangehen, ist das Energie Kriterium immernoch dem Spannungs- und Dehnungseinfluss Kriterium gleichwertig, Gegruendet an die Theorie des Spannungs- und Dehnungseinflusses, ein Engenieurs Kriterium fuer formbare Brueche, es wird angenommen, dass die Groesse eines stark gedehnten Bereiches konstant ist.

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Résumé Griffith a établi le critère d' énergie de fracture pour les matériaux fragiles fissurés. Ce critère d' énergie est équivalent au critère d' environnement de contrainte et de déformation pour les fractures fragiles comme indiqué par les solutions élastiques pour les solider fissurés. Pour une fracture quasi-fragile, c' est à dire dans le car oú de petites quantités de déformation plastiques à l' extrémité de la fissure précedent la fracture, le critère d' énergie est également équivalent au critère d'environnement de contrainte et de déformation. Un critère d' engineering pour des fractures ductiles, basé sur le concept d'environnement de contrainte et de déformation, suggére que, à la fracture, la dimension de la région de forte déformation à l' extrémité de la fissure est constante.

     

Thermomechanical postbuckling analysis of functionally graded plates and shallow cylindrical shells

Summary. In this paper, an analytic solution is provided for the postbuckling behavior of plates and shallow cylindrical shells made of functionally graded materials under edge compressive loads and a temperature field. The material properties of the functionally graded shells are assumed to vary continuously through the thickness of the shell according to a power law distribution of the volume fraction of the constituents. The fundamental equations for thin rectangular shallow shells of FGM are obtained using the von Karman theory for large transverse deflection, and the solution is obtained in terms of mixed Fourier series. The effect of material properties, boundary conditions and thermomechanical loading on the buckling behavior and stress field are determined and discussed. The results reveal that thermomechanical coupling effects and the boundary conditions play a major role in dictating the response of the functionally graded plates and shells under the action of edge compressive loads.     

A probabilistic fracture mechanics analysis for cracked pipe using 3-D model

For the purpose of probabilistic fracture mechanics (PFM) estimation based on elastic-plastic fracture mechanics (EPFM) in the field of reliability analysis of pressure vessels and piping, a 3-D EPFM database of fully plastic solutions for surface cracks and a PFM code for the integrity evaluation of nuclear structural components based on the above database are given. As an example, a comparison study of the PFM analysis is then performed between the 2-D and the 3-D solutions to demonstrate the 3-D effects on the solutions.     

Fracture mechanics analysis of thin coatings under spherical indentation

Spherical indentation of a thin, hard coating bonded to a thick substrate is investigated. The bending of the coating over the softer substrate induces concentrated tensile stresses on the lower and upper coating surfaces, from which transverse cracks may ensue. This work is primarily concerned with ring cracks originating from the top surface of the coating. In-situ indentation tests are carried out on a model glass/polycarbonate bi-layer, with the coating thickness and the indenter radius being the main test variables. As the coating thickness is decreased, the critical load to initiate ring cracks progressively departs from that associated with a critical surface stress, the effect that increases with increasing the indenter radius. A fracture mechanics approach in conjunction with the FEM technique is used to elucidate the onset of cylindrical ring cracks in thin-film bi-layer structures due to spherical indentation. The analysis, conducted as a function of the coating thickness and the indenter radius, reveals the existence of bending-induced compression stress regions ahead of the crack tip, which tend to shield the crack or increase the fracture resistance. The specific behavior is dictated by a complex interplay between the contact radius, a, the coating thickness, d, and the crack length, c. An interesting manifestation of this shielding mechanism is that when the coating surface contains flaws of various sizes, small flaws in this population may be more detrimental than large ones. Incorporation of this aspect into the analysis led to a good correlation with the experimental results. In the limit case of point-load, a closed-form, approximate solution for the stress intensity factors and the critical loads is obtained. This solution constitutes a lower bound for the critical loads, and is furthermore directly applicable to finite size indenters provided da. In the limit c/d/to0, a failure stress criterion may be used irrespective of the ball radius, r. The analysis in this case reveals that decreasing either d/r or the coating/substrate modulus ratio tend to favor ring cracking over radial type cracking. The transition between these two failure modes is identified explicitly as a function of the system parameters.     

Fracture mechanics analysis of geometrically nonlinear shear deformable plates

This paper presents boundary integral equations for fracture mechanics analysis of geometrically nonlinear shear deformable plates. A radial basis function and dual reciprocity method are utilized to evaluate the derivative terms and the domain integrals that appear in the formulations, respectively. Numerical examples of the clamped and simply supported plates containing a center crack subjected to uniform transversal loadings are presented. Displacement extrapolation technique is used to compute the stress intensity factors (SIFs). Stress intensity factors of mode I for plate bending and membrane problems are presented. The normalized stress intensity factors in membrane significantly increase after few increments of the load while the normalized stress intensity factors in bending decrease. Less displacement and rotational constraints in cracked plates under uniform transversal loadings will raise the stress intensity factors. The bending stress intensity factors of a central crack in clamped square plate were found to be the highest values compared to those for clamped non-square plates.     

悬臂施工波形钢腹板组合梁桥施工工艺

南京长江第四大桥北接线滁河特大桥采用波形钢腹板预应力混凝土组合箱梁桥。该类型桥是一种高效、经济、施工简便的新型桥梁形式。该类型桥恰当地将钢和混凝土两种不同材料结合起来,提高了结构稳定性、强度及材料的使用效率等。该桥的施工可为同类型桥的施工提供参考。     

Fracture diagrams for cracked stiffened panels

A general equation is derived for the prediction of structural failure of two-dimen-sional cracked components in which the geometrical features of the component affect the stress intensity factor of the crack. The general equation is used to construct a new fracture diagram for a uniformly stressed sheet containing a crack which is constrained by two stiffening elements fastened to the sheet. By the use of an example it is shown that the shape of the fracture diagram, and hence the fracture behavior of the panel, depends on the spacing of the stiffening elements, rivet locations and the relative stiffness of the cracked sheet and the stiffeners. It is shown that there are stiffened panel diagrams which indicate that failure can occur at fracture toughness levels significantly higher than would be apparent from an unstiffened panel of similar material.     

Application of an analog of the δc-model to the analysis of strength of elastoplastic cracked shellsto the analysis of strength of elastoplastic cracked shells

We propose a method for the investigation of the limiting equilibrium of elastoplastic shells with systems of interacting cracks. It can be described as follows: By using an analog of the δ_c-model, the elastoplastic problem is reduced to an elastic problem of the limiting equilibrium of a shell with cracks of unknown length whose lips are subjected to the action of unknown forces and moments satisfying the conditions of plasticity for thin shells. By using equations of the general moment theory of shells and the theory of generalized functions, we reduce the problem to the solution of a system of singular integral equations with unknown limits of integration and singular right-hand sides. We construct an algorithm for the numerical solution of systems of this sort supplemented by the conditions of boundedness of stresses and conditions of plasticity. We investigate crack tip opening displacements in a closed cylindrical shell with a regular system of longitudinal cracks or two transverse cracks. For a cylindrical shell with a single crack, we present an approximate relation for the determination of the critical load or crack length. Pidstryhach Institute of Applied Problems of Mechanics and Mathematics, Ukrainian Academy of Sciences, L'viv. Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 32, No. 3, pp. 5–15, May–June, 1996.     

Buckling analysis of shear deformable shallow shells by the boundary element method

In this work a boundary element (BE) formulation for buckling problem of shear deformable shallow shells is presented. A set of five boundary integral equations are obtained by coupling two-dimensional plane stress elasticity with shear deformable plate bending (Reissner). The domain integrals appearing in the formulation (due to the curvature and due to the domain load) are transferred into equivalent boundary integrals. The BE formulation is presented as an eigenvalue problem, to provide direct evaluation of critical load factors and buckling modes. Several examples are presented. The BE results for a cylindrical shallow shell with different curvatures are compared with other numerical solutions and good agreements are obtained.