State of stress in a patched plate containing a crack

Summary The state of stress and strain is examined for a plate containing a curvilinear crack reinforced by a finite patch. The elastic patch covers the crack completely and is rigidly connected to the infinite plate only along its edge. It is assumed that the plate and patch are in a general state of planar strain. The boundary-value problem is reduced to a system of three singular integral equations, which is solved by mechanical quadrature. Numerical results are given for a plate containing a crack in the form of an arc of a parabola and reinforced with an elliptical patch for various orientations of the tensile forces at infinity. The stress intensity coefficients at the crack vertices have been calculated along with the contact forces at the junction.Translated from Fiziko-Khimicheskaya Mekhanika Materialov, Vol. 27, No. 4, pp. 33–40, July–August, 1991.     

Effect of breaks in riveted stringers on the elastic and limiting equilibrium of a cracked plate

We study the stress-strain state and limiting equilibrium of a thin cracked plate reinforced by intact and broken riveted stringers and consider two types of loading of the stringers: their ends either are loaded by forces acting at infinity or free of loads. The boundary-value problem is reduced to a system of singular integral and integral-algebraic equations whose numerical solution is obtained by the method of mechanical quadratures for a series of geometric and physical parameters of the plate, crack, stringers, and rivets. It is shown that, as rule, broken stringers weaken the reinforced cracked plate and that the effect of breaks in the stringers on the stress intensity factors in the plate and the ultimate load is significant. Karpenko Physicomechanical Institute, Ukrainian Academy of Sciences, L'viv. Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 35, No. 3, pp. 39–47 May–June, 1999.     

Reinforcement of a thin cracked plate by a system of parallel stringers

The stress-strain state of an infinite plate with curvilinear cracks reinforced by a system of parallel stringers is investigated. The stringers are attached to the plate by riveting. It is assumed that the cracked plate with the system of stringers is in a generalized plane stress state. The boundary-value problem is reduced to a system of singular integral equations, which is solved by the method of mechanical quadratures. Numerical results are presented for a reinforced plate with a rectilinear or curvilinear crack (along a parabolic arc).Karpenko Physicomechanical Institute, Ukrainian Academy of Sciences, L'viv. Translated from Fiziko-Khimicheskaya Mekhanika Materialov, Vol. 30, No. 1, pp. 96–105, January–February, 1994.     

The effect of stringers and patches on the stress intensities around cracks in plates

In the present paper a method was developed to study a certain important class of mixed boundary value problems concerning the interaction between a reinforced isotropic and linearly elastic thin plate by rectilinear and curvilinear thin strip inclusions (stringers), and superimposed elastic thin sheets (patches) perfectly bonded to the plate along their peripheries, and the internal cracks existing in the plate. The method is based on the complex stress function and the singular integral equations theories. An exact expression for the complex stress function was given for the most general case. The method was applied with success to an example of a superimposed circular patch on an internal crack existing in a thin plate. The variation of the stress intensity factors at the tips of the internal crack in terms of its relative position with respect to the patch-centre, as well as, of the relative elastic properties of the patch and the plate, was determined numerically.     

Finite element fatigue propagation of induced cracks by stiffeners in repaired panels with composite patches

Fatigue crack growth analyses of aluminum panels with stiffeners repaired by composite patches have been rarely investigated. Generally, cracks may occur around the rivets which are capable to propagate under cyclic loadings. A composite patch can be used to stop or retard the crack growth rate. In this investigation, finite element method is used for the crack propagation analyses of stiffened aluminum panels repaired with composite patches. In these analyses, the crack-front can propagate in 3-D general mixed-mode conditions. The incremental 3-D crack growth of the repaired panels is automatically handled by a developed ANSYS Parametric Design Language (APDL) code. Effects of rivets distances and their diameters on the crack growth life of repaired panels are investigated. Moreover, the obtained crack-front shapes at various crack growth steps, crack trajectories, and life of the unrepaired and repaired panels with various glass/epoxy patch lay-ups and various patch thicknesses are discussed.     

Bending of a plate containing a periodic system of collinear cracks with regard for the contact of crack lips

We study the process of bending of a plate weakened by a periodic system of collinear cracks with regard for the contact of crack lips. The stress-strain state of the plate is found by the method of solution of the problems of linear conjugation from the theory of functions of complex variable. It is shown that if the crack lips are in contact over a part of the thickness of the plate, then the intensity factors of forces decrease and the intensity factors of moments increase as compared with the case of contact of the crack lips along the line of intersection with the surface. __________ Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 44, No. 2, pp. 46–52, March–April, 2008.     

A numerical method for a void–crack interaction under cyclic loads

This paper presents a numerical method to model a general system containing cracks and voids in an infinite elastic plate under remote cyclic loads. By extending Bueckner’s principle suited for a crack to a general system containing cracks and voids, the original problem is divided into a homogeneous problem (the one without cracks and voids) subjected to remote loads and a void-crack problem in an unloaded body with applied tractions on the surfaces of cracks and voids. Thus, the results in terms of the stress intensity factors can be calculated by considering the latter problem, which is analyzed easily by using the hybrid displacement discontinuity method (a boundary element method). Further, a fatigue growth technique of a mixed-mode crack is combined with the numerical approach to simulating a void–crack interaction problem under cyclic loads. Test examples are included to illustrate that the numerical method is very simple and effective for analyzing a void–crack interaction problem.     

Response of fibre reinforced aluminium–lithium laminates to different fatigue conditions

Under fatigue conditions fibre reinforced aluminium–lithium laminates do not respond in the same manner as monolithic aluminium alloys. The variation of fatigue crack growth rates with initial loading condition has been examined for both carbon and glass fibre reinforced laminates, and compared with the behaviour of unreinforced 8090 aluminium–lithium alloy for a range of conditions (different initial nominal stress intensity factor range, load range and reversed loading). During fatigue, cracks grow in the metal layers of these laminates whilst the fibres in the crack wake remain intact, bridging the crack faces. The fibre bridging mechanism, inherent in this laminate system, reduces the fatigue crack growth rate. The magnitude of the bridging effect appears to be inversely related to the applied load range. This relationship can account for the behaviour observed in the performed experiments. This revised version was published online in November 2006 with corrections to the Cover Date.     

Two-dimensional problems of the theory of elasticity for reinforced cracked plates

By using the method of singular integral equations, we develop a general approach to the solution of static problems of the two-dimensional theory of elasticity for thin-walled structural elements with curvilinear cracks reinforced by elastic patches. We analyze two basic types of fastening patches to plates, namely, continuous fastening (via the adhesive layer) and discrete fastening (riveting). It is assumed that both the cracked plate and reinforcing patches are characterized by the generalized two-dimensional stressed state. We constructed integral equations for an infinite plate with curvilinear crack reinforced by linear or two-dimensional elastic patches. We also present a brief survey of the literature devoted to the solution of problems of the indicated type.Karpenko Physicomechanical Institute, Ukrainian Academy of Sciences, L'viv. Translated from Fiziko-Khimicheskaya Mekhanika Materialov, Vol. 31, No. 3, pp. 68–82, May–June, 1995.     

Measurement of the mechanical properties of a fibre-reinforced composite on the submicrometre scale

Ultramicrotomy is being used routinely as a sample preparation technique for transmission electron microscopy (TEM). TEM study of thin sections of a diketone-bis-benzocyclobutene composite reinforced with Celion (trademark of BASF Co.) carbon fibre revealed two types of periodic crack in the fibre. Coarse cracks were due to bending at knife tip in the early stage of the ultramicrotomy. Also, very fine cracks were observed near the fibre–matrix interface and believed to have been induced by the shear lags between the fibre and the matrix. A simple analysis indicated that the coarse and fine crack spacings could be used to obtain the compressive strength of the fibre and the ultimate shear stress at the fibre–matrix interface, respectively. The combination of ultramicrotomy and TEM provides a useful tool to explore the mechanical properties of a composite material on the submicrometre scale, in addition to the other microstructural and compositional information accessible by TEM. This revised version was published online in November 2006 with corrections to the Cover Date.     

Limiting state of an orthotropic plate weakened by a periodic system of collinear cracks

On the basis of a modified δ _c -model of cracks, we study the limiting state of an orthotropic plate made of a material satisfying the general strength condition and weakened by a system of collinear cracks. The relations for the determination of major parameters of the model of cracks (the size of process zones, stresses in these zones, and the crack-tip opening displacements) are deduced. The mechanism of fracture of the plate containing a periodic system of collinear cracks is investigated. The influence of the degree of anisotropy and geometric parameters of the problem on the formation of the process zones and limiting state of the plate is revealed. The region of safe loading of an orthotropic viscoelastic plate with cracks is determined. The influence of the rheological parameters of the material on the region of safe loading is analyzed. Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 44, No. 6, pp. 58–66, November–December, 2008.     

Vibrodiagnostic parameters of fatigue damage in rectangular plates. Part 3. Through-the-thickness and surface semi-elliptical cracks

The paper addresses the analytical determination of vibrodiagnostic parameters that describe the presence of normal-rupture flat through-the-thickness and surface semi-elliptical central cracks in a rectangular homogeneous plate of constant thickness for various plate fixing conditions and vibration modes. It is shown that the most sensitive vibrodiagnostic damage parameter of a plate is the variation of the logarithmic decrement in the case of a through-the thickness crack and the second-harmonic ratio of the vibration process in superharmonic resonance in the case of a surface crack. __________ Translated from Problemy Prochnosti, No. 5, pp. 27–47, September–October, 2006.     

Extension of generalized plane strain plates with reinforced cracks

Applying the transversally isotropic generalized plane strain plate of Mindlin and Kane, as refined by Kotousov and Wang, the authors use Fourier transforms to develop a hyper-singular line-spring model for a reinforced crack under extensional loads, enabling the analysis of cracks in plates of arbitrary thickness. The new model is used to develop fracture mechanics geometry correction factors to account for the effect of the transverse modulus, Poisson’s ratio, and plate thickness on the stress intensity and crack opening displacement. The model is then used to examine the reinforcing effect of springs bridging the crack faces and extend the Rose model to include plate thickness effects, allowing the inclusion of plate thickness effects in the fracture mechanics analysis of reinforced cracks.     

Coplanar perturbation of a crack lying on the mid-plane of a plate

Several groups have studied experimentally the deformation of the front of mode I cracks propagating quasistatically along the interface between bonded plates. The theoretical interpretation of such experiments has always been based up to now on a formula of Rice (ASME J Appl Mech 52:571–579, 1985); this formula provides the first-order variation of the local mode I stress intensity factor resulting from some small, but otherwise arbitrary coplanar perturbation of the front of a semi-infinite crack in an infinite body. To be applicable to bonded plates, this formula requires that the characteristic distance of variation of this perturbation in the direction of the crack front be small compared to all other characteristic dimensions of the problem, and first of all the thickness of the plates. This condition is unfortunately frequently violated in practice. The purpose of this paper is therefore to provide a more exact formula for the variation of the local stress intensity factor, for the specific cracked geometry and boundary conditions used in experiments; this should allow for more accurate theoretical interpretations. This is done in two steps. The first one consists in adapting Rice’s (ASME J Appl Mech 52:571–579, 1985) treatment, applicable to the extreme case of plates of infinite thickness, to the other extreme one of plates of infinitesimal thickness, using the standard Love-Kirchhoff plate theory. An interesting outcome of the analysis is that the distance from the crack front to the boundary of the plate acts as a “cutoff length”, in the sense that when the distance between two points on the crack front gets larger than it, the influence of the crack advance at the first point upon the stress intensity factor at the second diminishes quickly; the plate thickness, however, plays no similar role. The second step consists in supplementing the theoretical expressions applicable to extreme values of the plate thickness with finite element computations providing results for intermediate values. These computations lead to the definition of a simple, approximate but accurate “interpolation formula” for the variation of the local stress intensity factor, applicable to plates of arbitrary thickness.     

Stresses Formed Near a Crack in Half Space in Contact with Liquid Under Harmonic Loading

We study a three-dimensional bimaterial object formed by an elastic half space with cracks and a liquid. The crack surfaces are subjected to the action of harmonic loads. The problem is reduced to the solution of a system of two-dimensional boundary integral equations of Helmholtz-potential type for the unknown jumps of displacements on the crack surfaces. We analyze the case of a circular crack whose lips are subjected to the action of stationary loads. The inertial effects in the vicinity of the crack contour are investigated. __________ Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 41, No. 3, pp. 96–100, May–June, 2005.     

Fatigue crack initiation and growth in riveted specimens: an optical and acoustic microscopic study

While optical microscopy on riveted specimens reveals only surface cracks, the acoustic C-scan images reveal subsurface and buried cracks emerging to the surface. This is of particular interest for fatigue cracks that initiate below the outer surface such as with chamfered riveted panels. Fatigue crack initiation and growth in riveted panels of Alclad 2024-T3 were characterized using optical microscopy and scanning acoustic microscopy to obtain C-scan images. The C-scan images were obtained using a focused transducer with a center frequency of ca 50 MHz, and the peak value of the back surface echo of the plate with countersunk rivet holes was recorded in the C-scan images. Data on the initiation and development of fatigue cracks at rivets in riveted Alclad 2024-T3 are given.     

Effect of flexure induced transverse crack and self-healing on chloride diffusivity of reinforced mortar

Cracks in reinforced concrete are unavoidable. Durability is of increasing concern in the concrete industry, and it is significantly affected by the presence of cracks. The corrosion of reinforcing steel due to chloride ions in deicing salts or sea-water is a major cause of premature deterioration of reinforced concrete structures. Although, it is generally recognized that cracks accelerate the ingress of chlorides in concrete, a lack of consensus on this subject does not yet allow reliable quantification of their effects. The present work studies the relationship between crack widths and chloride diffusivity. Flexural load was introduced to generate cracks of width ranging between 29 and 390 μm. As crack width was increased, the effective diffusion coefficient was also increased, thus reducing the initiation period of corrosion process. For cracks with widths less than 135 μm, the effect of crack widths on the effective diffusion coefficient of mortar was found to be marginal, whereas for crack widths higher than 135 μm the effective diffusion coefficient increased rapidly. Therefore, the effect of crack width on chloride penetration was more pronounced when the crack width is higher than 135 μm. Results also indicate that the relation between the effective diffusion coefficient and crack width was found to be power function. In addition, a significant amount of self-healing was observed within the cracks with width below 50 μm subjected to NaCl solution exposure. The present research may provide insight into developing design criteria for a durable concrete and in predicting service life of a concrete structures.     

Numerical analysis of the stress intensity factors for repaired cracks from a notch with bonded composite semicircular patch

In this paper, we investigated the crack growth behaviour of cracked thin aluminium plate repaired with bonded composite patch. The finite element method is used to study the performance of the bonded composite reinforcement or repair for reducing the stress concentration at a semicircular lateral notch and repairing cracks emanating from this kind of notch. The effects of the adhesive properties and the patch size on the stress intensity factor variation at the crack tip in mode I were highlighted. The obtained results show that the stress concentration factor at the semicircular notch root and the stress intensity factor of a crack emanating from notch are reduced with the increase of the diameter and the number of the semicircular patch. The maximal reduction of stress intensity factor is about 42% and 54%, respectively, for single and double patch. However, the gain in the patch thickness increases with the increase of the crack length and it decreases when the patch thickness increases. The adhesive properties must be optimised in order to increase the performance of the patch repair or reinforcement.     

Crack propagation in a material with random toughness

An efficient Monte Carlo procedure is presented for characterizing the propagation of a crack in a material whose fracture toughness is a random field. The simulations rely on accurate approximate solutions of the integral equations that govern the dislocation densities, stress intensity factors, and energy release rates of curvilinear cracks. For a plate containing an edge crack that propagates towards a subsurface crack representing a traction-free boundary, results for the distributions of crack trajectories, critical applied far-field stresses, and nominal fracture toughness are presented for various parameters that quantify the randomness of the material's critical energy release rate. A demonstrative probabilistic model for crack trajectories is built and size effects are discussed.     

Stress intensity in the vicinity of a surface flaw in the linear part of a pipeline

Real sharp-edted surface and subsurface flaws detected in a gas pipeline body are modeled by surface semi-elliptical mathematical cracks (cuts) in a closed cylindrical shell. A relationship is proposed that relates the geometrical dimensions of the flaws to the crack aspect ratio. Based on the line spring model, the problem of stress state and boundary equilibrium conditions of a closed cylindrical shell with a surface semi-elliptical crack is reduced to a system of singular integral equations. An algorithm was developed for computational solution of the problem, and numerical analysis was made for the dependence of stress intensity factors on loading conditions and geometrical parameters of shell and crack. For a shell subjected to internal pressure and weakened by a surface longitudinal semi-elliptical crack, a closed approximation formula is proposed that interrelates pressure level, shell/crack dimensions, and material mechanical properties in boundary equilibrium conditions. The maximal error value is indicated for the results obtained using this formula. Lvov Polytechnic State University, Lvov, Ukraine. Translated from Problemy Prochnosti, No. 4, pp. 38–47, July–August, 1999.