Penny-shaped Crack Research Articles

Fracture of a Semi-Bounded Composite Material with a Near-Surface Penny-Shaped Crack under Compression

The nonclassical problem of fracture mechanics for a near-surface crack is solved in the case of small distances between the free surface and the crack plane. The problem is axisymmetric for a penny-shaped crack. A numerical study for a composite material is carried out as an example.

Investigation of Generalized SIFs of cracks in 3D piezoelectric media under various crack-face conditions

This paper investigates the influence of crack geometry, crack-face and loading conditions, and the permittivity of a medium inside the crack gap on intensity factors of planar and non-planar cracks in linear piezoelectric media. A weakly singular boundary integral equation method together with the near-front approximation is adopted to accurately determine the intensity factors. Obtained results indicate that the non-flat crack surface, the electric field, and the permittivity of a medium inside the crack gap play a crucial role on the behavior of intensity factors. The mode-I stress intensity factors (KI) for two representative non-planar cracks under different crack-face conditions are found significantly different and they possess both upper and lower bounds. In addition, KI for impermeable and semi-permeable non-planar cracks treated depends strongly on the electric field whereas those of impermeable, permeable, and semi-permeable penny-shaped cracks are identical and independent of the electric field. The stress/electric intensity factors predicted by permeable and energetically consistent models are, respectively, independent of and dependent on the electric field for the penny-shaped crack and the two representative non-planar cracks. Also, the permittivity of a medium inside the crack gap strongly affects the intensity factors for all crack configurations considered except for KI of the semi-permeable penny-shaped crack.

On the classification of boundary value problems for elastic homogeneous and piecewise homogeneous spaces with a penny-shaped crack

The classification of the basic axisymmetric boundary value problems of the mathematical theory of elasticity for homogeneous and piecewise homogeneous elastic spaces with a penny-shaped crack is given. Using the Hankel integral transform, governing integral equations (GIEs) of these problems are derived and methods for finding their exact solutions are indicated.

Torsion of a cracked elastic body by an embedded semi-infinite rigid cylinder

Torsion of a cracked elastic body by an embedded semi-infinite rigid cylinder is studied. A coaxial penny-shaped crack is situated in the plane of the end of the cylinder. The problem is reduced to a system of dual integral equations including Hankel and Weber–Orr transforms and then, by using ansatzs, to an integral equation of the second kind with Hankel integral operator given on a semi-infinite interval or to an equivalent infinite system of linear algebraic equations with a Hankel matrix. A detailed investigation allowed us to suggest efficient methods for solving equations for any size of the crack. In particular, accurate approximate formulas for the stress intensity factor and the contact stresses are derived, as well as an asymptotic formula for the stress intensity factor when a crack is large. Analytical estimations and calculations manifest a strong increase in the stress intensity factor and the contact stress at the end of the cylinder as the crack tip is very close to the cylinder surface.

Rupture and Damage at Dilatant Geological Interfaces

The work discusses the processes that can be present at fractures and defects at geological interfaces. The introductory comment clearly indicates that the computational approaches offer the most appropriate methods for examining complex contact problems at geomaterial interfaces. There are, however, certain types of contact problems that are amenable to analytical treatment. The paper examines the problem of a circular dilatant-frictional patch located at an otherwise frictionless interface. The dilatancy processes are induced at the circular patch by the relative shear of the elastic regions. The paper presents a mathematical approach for the study of the problem where results from the solution of integral equations applicable for the internal indentation of a penny-shaped crack by a rigid inclusion and the internal pressurization of an annular crack are combined with a work-dissipation relationship to examine the mechanics of the interactions at the dilatant zone.

Fracture Mechanics Assessment of Large Diameter Wind Turbine Bearings

The structural integrity of large diameter wind turbine bearings have been investigated using the built-in “contour integral” tool in ABAQUS finite element software package by modeling three-dimensional penny-shaped cracks and evaluating the stress intensity factors. In order to sub-model a crack and investigate the fracture mechanics of the rolling contact between the rollers and the raceway, a python script was developed and implemented in the analysis. Important steps to build the crack model are detailed and recommendations are made for further use of the finite element modeling tool in compressive mixed mode fracture mechanics assessment of wind turbine bearings. Moreover, the influence of initial residual stresses due to induction hardening of the raceway is also investigated and discussed in this paper. For frictionless contacts between the two crack faces, “contour integral” in ABAQUS appears to be a suitable method to obtain accurate stress intensity factor solutions for modes II and III. The results from this study are validated through comparison with the analytical solutions available in the literature and are expected to facilitate numerical life assessment of wind turbine bearings.

Stress intensity factors of multiple axisymmetric interface cracks in an isotropic layer with FGM coating under torsional loading

Purpose The purpose of this paper is to provide a theoretical analysis of the fracture behavior of multiple axisymmetric interface cracks between a homogeneous isotropic layer and its functionally graded material (FGM) coating under torsional loading. Design/methodology/approach In this paper, the authors employ the distributed dislocation technique to the stress analysis, an FGM coating-substrate system under torsional loading with multiple axisymmetric cracks consist of annular and penny-shaped cracks. First, with the aid of the Hankel transform, the stress fields in the homogeneous layer and its FGM coating are obtained. The problem is then reduced to a set of singular integral equations with a Cauchy-type singularity. Unknown dislocation density is achieved by numerical solution of these integral equations which are employed to calculate the SIFs. Findings From the numerical results, the following key points were observed: first, for two types of the axisymmetric interface cracks, the SIFs decrease with growing in the values of the non-homogeneity. Second, the SIFs increase with increases in interface crack length. Third, the magnitude of the SIFs decreases with increases in the FGM coating thickness. Fourth, the interaction between cracks is an important factor affecting the SIFs of crack tips. Originality/value New analytical dislocation solution in an FGM coating-substrate system is developed.

Analysis of 3D planar crack problems in one-dimensional hexagonal piezoelectric quasicrystals with thermal effect. part I: Theoretical formulations

Theoretical and numerical investigations on three-dimensional (3D) planar crack problems in one-dimensional (1D) hexagonal piezoelectric quasicrystals (QCs) with thermal effect are performed systematically. Part I of this work derives a series of theoretical formulations that are then used to study the 3D planar crack problems in the QCs. The simple layer potential functions with the extended displacement discontinuities (EDDs) as the unknown variables and the general solution based on quasi-harmonic functions for the QCs under consideration are used to deduce the boundary equations that govern 3D planar crack problems. The hypersingular integral equation method is used to analyze the asymptotic singularities of the coupled thermal-electrical-phonon-phason fields near the crack edge. Expressions are then presented for the extended stress intensity factors (ESIFs) of a mixed model crack in terms of EDDs for arbitrarily-shaped cracks in the QCs, and the basic relationships between the energy release rate and the ESIFs are established. Closed-form solutions for some typical cracks, including an elliptical crack that is subjected to coupled electrical-phonon-phason loadings and a penny-shaped crack that is subjected to antisymmetric thermal loading, are determined via Fabrikant's analysis method. Additionally, both the physical quantities on the crack plane and the corresponding variables in the coupled thermal-electrical-phonon-phason field in the full space are given. The theoretical formulations derived in this paper provide a fundamental basis for development of the numerical approach proposed in Part II of our work, and can also serve as benchmarks for numerical solutions.

An Extended Asymptotic Analysis for Elastic Contact of Three-Dimensional Wavy Surfaces

An analytical approach for an extended asymptotic analysis of 3D wavy surfaces contact was developed on the basis of expansion of a double-sinusoidal surface in Fourier series, using cylindrical coordinates. The two different problems were considered: indentation of a double-sinusoidal non-periodic punch into an elastic half-space and a penny-shaped crack under action of non-axisymmetric pressure. The closed-form expressions for determining the load–area and the load–separation curves for the light and the high loads, considering virtual circular contact and non-contact areas, were obtained. The results were compared with existing analytical and numerical studies. They show that the mean contact characteristics at the light and the high loads mainly depend on the axisymmetric component of Fourier series, representing the wavy surface. These parameters can be calculated analytically with sufficient accuracy for a large range of applied pressures except transitional region. The relation between 2D and 3D solutions is also shown.

Fracture analysis of piezoelectromagnetic medium with axisymmetric cracks

The present study evaluates, by the distributed dislocation technique, the generalized intensity factors for the multiple axisymmetric cracks in a transversely isotropic piezoelectromagnetic medium under in-plane magneto-electro-mechanical loading. To this end, the solution to the problem of generalized dislocations is first obtained with the aid of the Hankel transform. The generalized dislocation in the piezoelectromagnetic medium contains Volterra prismatic and Somigliana radial dislocations in the displacement components and jump in the electric and magnetic potentials. The distributed dislocation technique is used to formulate integral equations for a system of coaxial axisymmetric cracks, including penny-shaped and annular cracks. These integral equations are of Cauchy singular type, which are solved numerically to obtain the dislocation densities on the faces of the cracks and the results are used to determine generalized intensity factors for cracks. Finally, several examples to study the crack type/location on the ensuing generalized intensity factors at tips of cracks and also the interaction between the cracks are presented.

Penny-shaped cracks by Finite Fracture Mechanics

In this brief note, we provide the failure stress of a solid containing a penny-shaped crack by means of Finite Fracture Mechanics. The solution is analytical up to the numerical root of the equation providing the finite crack growth increment. Results are discussed and compared with the ones provided by Linear Elastic Fracture Mechanics, by Theory of Critical Distances and by Cohesive Crack Model (with a Dugdale-type cohesive law).

A family of non-conforming crack front elements of quadrilateral and triangular types for 3D crack problems using the boundary element method

This study focuses on establishing nonconforming crack front elements of quadrilateral and triangular types for 3D crack problems when the dual boundary element method is applied. The asymptotic behavior of the physical variables in the area near the crack front is fully considered in the construction of the shape function. In the developed quadrilateral and triangular crack front elements, the asymptotic term, which captures the asymptotic behavior of the physical variable, is multiplied directly by the conventional Lagrange shape function to form a new crack front shape function. Several benchmark numerical examples that consider penny-shaped cracks and straight-edge crack problems are presented to illustrate the validity and efficiency of the developed crack front elements.

Interaction of horizontally aligned coplanar 3D penny cracks under compression

We study the interaction of laterally aligned coplanar penny-shape cracks under compression, focusing on the influence on the 3D wing crack growth and coalescence, and the relation between stress intensity factors and wing crack length. 3D wing crack propagation is simulated with PDS-FEM, for which a new treatment of shear crack in friction is developed and validated. The stress intensity factors are computed with the interaction integral method. We observe a significant interaction of sufficiently grown wing cracks emanating from initial penny cracks which are separated by a distance less than the size of the initial crack, leading to an up to 40% increase of stress intensity factors compared to the single crack case. Two characteristic wing crack lengths are identified, corresponding to the start of the interaction and the end of the transition to a single-wing-crack like behaviour. According to the results, a numerical estimate is proposed to evaluate the variation of the stress intensity factors as a function of the distance between initial cracks. The estimate is shown to be in good agreement with numerical results for two and three aligned cracks, and can be applied to study the macroscopic rupture behaviour of crack networks.

Axisymmetric Torsion of an Elastic Layer Sandwiched between Two Elastic Half-Spaces with Two Interfaced Cracks

Abstract The present article examines the problem related to the axisymmetric torsion of an elastic layer by a circular rigid disc at the symmetry plane. The layer is sandwiched between two similar elastic half-spaces with two penny-shaped cracks symmetrically located at the interfaces between the two bonded dissimilar media. The mixed boundary-value problem is transformed, by means of the Hankel integral transformation, to dual integral equations, that are reduced, to a Fredholm integral equation of the second kind. The numerical methods are used to convert the resulting system to a system of infinite algebraic equations. Some physical quantities such as the stress intensity factor and the moment are calculated and presented numerically according to some relevant parameters. The numerical results show that the discontinuities around the crack and the inclusion cause a large increase in the stresses that decay with distance from the disc-loaded. Furthermore, the dependence of the stress intensity factor on the disc size, the distance between the crack and the disc, and the shear parameter is also observerd.

Correspondence principle in plane and axisymmetric mixed boundary-value problems of elasticity

This paper advances the relations by A. Ja. Aleksandrov between axisymmetric and plane strain states to the case of mixed boundary-value problems. We revise two models of a strip-shaped and a circular stamp indented into a half-plane and a half-space, respectively, when the normal and tangential traction components are unknown a priori. Also, two models of a strip-shaped and a penny-shaped interfacial crack are considered. By using the theory of Abelian integral operators and the Riemann-Hilbert problem on a segment we derive the solutions to the model problems and establish the relations between the governing systems of integral equations associated with the four models and their solutions. These relations can be interpreted as mappings between (i) plane and axisymmetric contact problems, (ii) plane and axisymmetric fracture models, (iii) plane contact and fracture problems, and (iv) axisymmetric contact and fracture problems. The mappings enable us to write down the governing systems of integral equations and the solutions to any three models by making use of the governing system and the solution to the fourth problem. The transformations are specified in the scalar cases when there is no friction in the contact zone and when a crack is in a homogeneous elastic medium. By considering the contact frictionless problem of an annulus stamp it is shown that, although an exact solution to the plane strain frictionless contact problem of two stamps is available, a transformation of the plane to the axisymmetric solution in this case is not possible to obtain, and derivation of a closed-form solution to the annulus stamp model is still an open mathematical problem.

Effect of capillary bridges on the interfacial adhesion of wearable electronics to epidermis

Wearable electronics integrated with flexible biosensors provides robust, noninvasive and long-lived interfaces to monitor the electrophysiological and electrochemical signals of the human body. Sweat from the body forms capillary bridges at the weak adhesion interface, furthermore, capillary bridges exhibit different shapes on the circular and thin rectangular sensors. Capillary bridge and fracture models are developed to study the behavior of sweat between wearable electronics and epidermis. The sensor sizes of wearable electronics have significant effects on the opening behaviors of the interface cracks due to the growth of sweat capillary bridges. Dissimilar mechanical properties between the epidermis and the substrates clearly affect the opening behaviors of the penny-shaped and slender cracks. This study provides a useful framework to investigate the underlying mechanisms of the detachment of the wearable electronics due to sweat from the body, which is very valuable for the design of wearable electronics mounted on the epidermis for healthcare applications.

Exact solutions to magneto-electro-thermo-elastic fields for a cracked cylinder composite during thermal shock

This report derives the exact solutions for the problem of a magneto-electro-elastic cylinder with a penny-shaped and embedded crack subjected to transient thermal load. Thermal cracking is analysed in the theoretical framework of linear magneto-electro-thermo-elasticity. The heat conduction equation for a magneto-electro-thermo-elastic cylinder with a finite size is solved using the standard method of separation variables. The coupling magneto-electro-thermo-elastic fields are determined in a stationary case via the Hankel integral transform. Based on Abel’s integral equation and the dual integral equation, the mathematical formulations for the permeable crack conditions are derived. Solutions to the elastic, electric, and magnetic intensity factors are obtained. Due to the explicitness of these solutions, they are very interesting for the design and analysis of magneto-electro-thermo-elastic composites.

Dual-phase-lagging heat conduction and associated thermal shock fracture of sandwich composite plates

This paper studies a penny-shaped crack at the middle of a symmetric sandwich structure and is under a sudden temperature gradient on the crack faces. The temperature field of the structure and the stress intensity factor (SIF) at the crack tip are derived based on the dual-phase-lag non-Fourier heat conduction theory. Two specific factors are identified to predict the coupling effects of the thermal property differences between the skin and the core (including the differences in thermal conductivity, thermal diffusivity, heat flux lag and temperature gradient lag) on the thermoelastic behavior. If the coefficient of thermal expansion (CoTE) of the skins is greater than that of the core, the peak value of transient SIF increases. If the CoTE of the skins is smaller than that of the core, the transient SIF is scarcely influenced. The results are useful for the designs and applications of sandwich structures work under thermal shock.

Fracture analysis of an infinite 1D hexagonal piezoelectric quasicrystal plate with a penny-shaped dielectric crack

This article studies the electroelastic behavior induced by a penny-shaped dielectric crack lying at the midplane of a one-dimensional (1D) hexagonal piezoelectric quasicrystal plate of finite thickness. Semi-permeable crack boundary conditions are given. By using the Hankel transform technique, an associated mixed boundary value problem is reduced to two coupled Fredholm integral equations of the second kind. Singular electric and elastic fields around the crack front along with their intensity factors in the phonon and phason fields near the crack front are obtained. The intensity factors of crack opening displacement (COD) across the crack faces are calculated and presented graphically. The influence of electric field and phason field on the COD intensity factors is analyzed. Permeable and impermeable cracks are two limit cases of the present.

Dual boundary element method for analyzing three-dimensional cracks in layered and graded halfspaces

This paper presents a dual boundary element analysis of three-dimensional cracks in layered and graded halfspaces. The fundamental solution of a multilayered solid is used to develop the dual boundary element method so that only the external boundary surface and the crack surface need to be discretized while the material interfaces do not need to be discretized. Infinite boundary elements and crack-tip discontinuous elements are introduced to consider the far-fields of a layered halfspace and capture the crack-tip behavior, respectively. Special attentions are given to strongly singular and hypersingular integrals in the discretized displacement and traction boundary integral equations. For square-shaped, penny-shaped and elliptical cracks located in a homogeneous halfspace, the stress intensity factors obtained with the present formulation are in very good agreement with existing numerical results and closed-form solutions. The square-shaped cracks in horizontally layered halfspaces and the penny-shaped and elliptical cracks in graded halfspaces are analyzed. Results show that the material heterogeneity in layered and graded halfspaces can have a profound effect on the stress intensity factors.