Cracks Of Arbitrary Shape Research Articles

Fatigue Characteristics Analysis of Carbon Fiber Laminates with Multiple Initial Cracks

In the entire wind turbine system, the blade acts as the central load-bearing element, with its stability and reliability being essential for the safe and effective operation of the wind power unit. Carbon fiber, known for its high strength-to-weight ratio, high modulus, and lightweight characteristics, is extensively utilized in blade manufacturing due to its superior attributes. Despite these advantages, carbon fiber composites are frequently subjected to cyclic loading, which often results in fatigue issues. The presence of internal manufacturing defects further intensifies these fatigue challenges. Considering this, the current study focuses on carbon fiber composites with multiple pre-existing cracks, conducting both static and fatigue experiments by varying the crack length, the angle between cracks, and the distance among them to understand their influence on the fatigue life under various conditions. Furthermore, this study leverages the advantages of Paris theory combined with the Extended Finite Element Method (XFEM) to simulate cracks of arbitrary shapes, introducing a fatigue simulation method for carbon fiber composite laminates with multiple cracks to analyze their fatigue characteristics. Concurrently, the Particle Swarm Optimization (PSO) algorithm is employed to determine the optimal weight configuration, and the Backpropagation neural network (BP) is used to train and adjust the weights and thresholds to minimize network errors. Building on this foundation, a surrogate model for predicting the fatigue life of carbon fiber composite laminates with multiple cracks under conditions of physical parameter uncertainty has been constructed, achieving modeling and assessment of fatigue reliability. This research offers theoretical insights and methodological guidance for the utilization of carbon fiber-reinforced composites in wind turbine blade applications.

Fracture Analysis of Planar Cracks in 3D Thermal Piezoelectric Semiconductors

The study of piezoelectric semiconductor (PSC) materials and devices is experiencing rapid growth, giving rise to a burgeoning research domain known as piezotronics. Understanding the fracture behavior of PSC materials is essential for designing robust devices and ensuring their long-term functionality. A planar crack of arbitrary shape in the isotropic plane of a three-dimensional (3D) transversely isotropic thermal PSC body subjected to combined thermal-electrical-mechanical loadings is studied via the extended displacement discontinuity (EDD) boundary integral equation method. Under the framework of linearized basic equations and one-way thermal coupling, the hypersingular boundary integral equations with the volume integrals containing the temperature and carrier concentration are obtained and expressed by the EDDs which include displacement, electric potential, carrier concentration and temperature discontinuities across the crack surfaces and are the basic variables in analyzing crack problems. Based on the finite part of hypersingular integrals, the EDDs near the crack border are proved to have the classical behavior of O(r). Furthermore, the asymptotic solutions of extended stresses including the stress, electric displacement, current density and heat flux at the crack front are presented and display classical singular behavior of O(1/r). The extended stress intensity factors are defined and given in terms of EDDs. A 3D numerical model of a penny-shaped crack in the thermal PSC cylinder is established and used to verify the presented formulae.

Deformation features of trunk pipelines with composite linings under static loads

This paper considers the deformation process of a typical section of a steel trunk pipeline with a defective zone, strengthened with a carbon fiber composite lining, under the influence of stationary internal pressure. Defects in the form of thinning of the pipe thickness and cracks were investigated. The stressed-strained state of the structure at critical pressure was analyzed. The thickness of the composite lining was determined, at which the bandage compensates for the effect of internal pressure on the damaged section of the pipeline. Research was carried out numerically based on finite element modeling in the ANSYS software package. When studying the stressed-strained state of a pipe with a defect of an arbitrary complex shape under the influence of critical pressure, a compensating value was obtained. The result showed that a carbon fiber lining with a thickness of 17 % of the rated thickness of the pipe could completely compensate for the effects of internal pressure in the defect area. In this case, the stresses in the carbon fiber lining were close to minimal. When studying the stressed-strained state of a pipe with a large crack of arbitrary shape at critical pressure, a compensating value was also obtained. It has been established that to compensate for the concentration of internal pressure in the crack zone, the thickness of the composite lining should be at the level of 34 % of the rated thickness of the pipe. In this case, the deformation of the steel pipe in the area of the crack occurs in the elastic region. The exception is the crack tips, where plastic deformations are observed, and stresses arise up to 93 % of the ultimate strength of the pipe steel. At the same time, the stresses in the carbon fiber lining remain close to minimal. Thus, it is recommended to use carbon fiber linings with a thickness of 17 % or more of the rated pipe thickness to bandage damage constituting up to 75 % of the thickness of a steel pipe. To bandage cracks, it is recommended to use carbon fiber linings with a thickness of at least 34 % of the rated pipe thickness

The elasticity-conductivity connection for materials with cracks of arbitrary shapes and orientation distributions

The cross-property connection between the effective elastic and conductive properties of a solid with cracks has been first given by Bristow (1960) for randomly oriented cracks, assuming that cracks have circular (penny) shapes. However, cracks in materials typically have “irregular” shapes and this factor, generally, has strong effect on the said properties (the strong stiffening effect produced by contacts between crack faces is an example). In addition, shapes of 3D cracks (for example, the statistics of the mentioned contacts) may be largely unknown. These factors make it difficult to estimate the effect of cracks on the elastic properties. We show that flat cracks of arbitrary shapes and orientation distributions produce similar effects on the elastic and conductive properties, so that changes of stiffnesses can be monitored by changes in electric conductivities. Extension to transversely isotropic materials is given. Possible extensions to non-flat crack shapes are discussed.

Exact Solution to Axisymmetric Interface Crack Problem in Magneto-Piezo-Electric Transversely Isotropic Materials

Summary This seems to be the first exact closed-form solution to the problem of a penny-shaped interface crack, subjected to an axisymmetric normal and tangential loading. The crack is located at the boundary between two bonded magneto-piezo-electric transversely isotropic half-spaces, made of different materials. We use the combination of Green’s functions for two different half-spaces and Fourier transform. We derive first the governing equations, which are valid for a crack of arbitrary shape. The usual approach leads to five hypersingular integral equations, we arrive at four integro-differential equations (one of them being complex). While the coefficients of the governing equations in existing publications are presented in terms of the results of the solution of a set of linear algebraic equations and are too cumbersome to be written explicitly in terms of the basic constants, our choice of the basic constants leads to quite elegant explicit expressions for these coefficients and reveals certain symmetry, which was noticed only numerically in previous publications or not noticed at all. In the particular case of axial symmetry, the problem is reduced to just one singular equation, for which an exact closed-form solution is known. We are not aware of any other publication with which our results can be compared.

Planar cracks of arbitrary shapes in elastic media with ellipsoidal anisotropy: efficient numerical solution

Planar cracks of arbitrary shapes in elastic media with ellipsoidal anisotropy: efficient numerical solution

Kinked and forked crack arrays in anisotropic elastic bimaterials

The fracture problem of multiple branched crack arrays in anisotropic bimaterials is formulated by use of the Stroh formalism to the linear elasticity theory of dislocations. The general full-field solutions are obtained from the standard technique of continuously distributed dislocations along finite-sized cracks of arbitrary shapes, which are embedded in dissimilar anisotropic half-spaces under far-field stress loading conditions. The bimaterial boundary-value problem leads to a set of coupled integral equations of Cauchy-type that is numerically solved by using the Gauss–Chebyshev quadrature scheme with appropriate boundary conditions for kinked and forked crack arrays. The path-independent Jk-integrals as crack propagation criterion are therefore evaluated for equally-spaced cracks, while the limiting configuration of individual cracks is theoretically described by means of explicit expressions of the local stress intensity factors K for validation and comparison purposes on several crack geometries. The short-range interactions resulting from the idealized configurations of infinitely periodic cracks are investigated as well as various size- and heterogeneity-effects on the mixed-mode cracks in complex stress-state environments. The influences of anisotropic elasticity, elastic mismatch, applied stress direction, inter-crack spacings and crack length ratios on the predictions from the Jk- and K-based fracture criteria are examined in the light of different configurations from the single kinked crack case in homogeneous media to the network of closely-spaced forked cracks in presence of bimaterial interfaces.

Effective conductivity of anisotropic media with crack-like inclusions and cracks of arbitrary shapes

The effective field method is applied to solution of the homogenization problem for anisotropic media containing random sets of thin inclusions of low conductivity (crack-like inclusions) or cracks. The derived expression for the tensor of effective conductivity of cracked media on the one hand, takes into account peculiarities of shapes and conductivity of thin inclusions, and on the other hand, reflect statistical properties of the inclusion distributions in the host medium. The crucial part of realization of the method is solution of the so-called one-particle problem that is the conductivity problem for an isolated inclusion embedded into an anisotropic host medium and subjected to a constant external field. This problem is reduced to solution of the integral equation for the potential jump on the middle surface of a thin inclusion. An efficient numerical method of solution of this equation is proposed. The integral equation is discretized by Gaussian approximating functions and reduced to a linear algebraic system for the coefficients of the approximation (the discretized problem). For Gaussian functions, the elements of the matrix of the discretized problem are calculated in explicit analytical forms (for isotropic host media) or they reduce to a standard 1D-integral (for arbitrary anisotropic host media) that can be tabulated. As a result, the matrix of the discretized problem is calculated fast. For inclusions with planar middle surfaces and regular grids of approximated nodes, this matrix has Toeplitz’ structure, and fast Fourier transform algorithm can be used for iterative solution of the discretized problem. The cases of a strongly anisotropic medium with circular, annular and square cracks are considered. Examples of solution of the homogenization problems for the medium containing cracks of various shapes are presented.

Influence of Plastic Anisotropy on the Limit Load of an Overmatched Cracked Tension Specimen

Plastic anisotropy is a common property of many metallic materials. This property affects many aspects of structural analysis and design. In contrast to the isotropic case, there is a great variety of yield criteria proposed for anisotropic materials. Moreover, even if one specific yield criterion is selected, several constitutive parameters are involved in it. Therefore, parametric analysis of structures made of anisotropic materials is quite cumbersome. The present paper demonstrates the effect of the constitutive parameters involved in Hill’s quadratic yield criterion on the upper bound limit load for weld stretched overmatched tension specimens containing a crack of arbitrary shape, assuming that the crack is located inside the weld. Different sets of the constitutive parameters are involved in the yield criteria for weld and base materials. Since the limit load is an input parameter of many flaw assessment procedures, the final result of the present paper shows that it is necessary to take into account plastic anisotropy in these procedures. It is worthy of note that the limit load is involved in the flaw assessment procedures in combination with the stress and strain fields near the tip of a crack. In anisotropic materials, these fields may become non-symmetric even under symmetric loading. This behavior affects the propagation of cracks.

Characteristic Tensor for Evaluation of Singular Stress Field under Mixed-mode Loadings

A characteristic tensor is defined using stress tensor averaged in a small circular domain at the crack tip and multiplied by the root of domain radius. It possesses the original stress tensor characteristics and has a simple relationship with conventional fracture-mechanics parameters. Therefore, it can be used to estimate stress intensity factors (SIFs) for cracks of arbitrary shape subjected to multiaxial stress loads. A characteristic tensor can also be used to estimate SIFs for kinked cracks. This study examines the relation between a characteristic tensor and SIFs to demonstrate the correlation between the characteristic tensor and fracture-mechanics parameters. Consequently, a single straight crack and a kinked crack of finite length existing in a twodimensional, infinite isotropic elastic body in a plane stress state, were considered to investigate the properties of the characteristic tensor under mixed-mode loadings. To demonstrate the practical utility of the characteristic tensor, the stress distribution obtained through finite element analysis (FEA) was used to estimate mixed-mode SIFs, and the values of estimated SIFs were compared with those obtained using an analytical solution. Results demonstrate that SIFs estimated under mixed-mode loadings exhibit a good agreement with the analytical values. This indicates that the proposed characteristictensor-based approach is effective in extracting features of singular stress fields at crack tips, and can be employed to estimate values of fracture-mechanics parameters, such as SIFs. Owing to its simplicity, the proposed approach can be easily incorporated in commercial FE codes for practical applications to simulate the crack-growth problem under both static and dynamic loading scenarios. The excellent applicability of the characteristic tensor greatly contributes to efficiency of the design process in industries.

Analysis for T-stress of cracks in 3D anisotropic elastic media by weakly singular integral equation method

This paper proposes an efficient numerical procedure for computing T-stresses of cracks in three-dimensional, linearly elastic, infinite media. The technique is established in a broad framework allowing a medium made of generally anisotropic materials and cracks of arbitrary shape and under general loading conditions to be treated. A pair of weakly singular, weak-form, displacement and traction boundary integral equations is utilized to formulate the key equations governing the unknown crack-face fields. Besides the basic benefits such as the reduction of spatial dimensions of the solution space and the efficient treatment of unbounded domains and remote boundary data, use of such integral equations in the formulation offers additional positive features including the computational simplicity resulting from the weakly singular nature of all involved integrals and the involvement of a complete set of crack-face displacement fields. A weakly singular symmetric Galerkin boundary element method together with the special near-front approximation is utilized to solve for the unknown relative crack-face displacement whereas the sum of the crack-face displacement is obtained from the displacement boundary integral equation for cracks via the Galerkin technique. The latter step is one of the essential aspects of the present study that provides the direct means for determining the T-stress data in terms of the sum of the crack-face displacement in the neighborhood of the crack-front. An extensive numerical study is conducted for various scenarios and a selected set of results is reported to demonstrate both the accuracy and capability of the proposed technique.

Bonded contact and general crack problems in generalized materials and their relationships

The term of generalized material is introduced here as the material, whose state is described by the second order homogeneous differential equations with constant coefficients. The generalized point sources are described by homogeneous expressions, containing the first order derivatives with constant coefficients. The Green’s function for a half-space, made of generalized material, subjected to the action of generalized point sources, is derived in the form of a single integral over a circle. Some of the components of the surface Green’s function are presented in finite form, no computation of any integral is needed. The bonded contact problem is described as mixed–mixed boundary value problem for a half-space, with normal and tangential displacements prescribed inside domain S and vanishing tractions outside this domain on the plane $$x_{3}=0$$ . A set of governing integral equations for bonded contact problem was derived, with some of the kernels defined in finite form. The general crack problem is defined as that of a flat crack of arbitrary shape S in the plane $$x_{3}=0$$ , with normal tractions $$\sigma _{33}$$ applied symmetrically to the crack faces and tangential tractions $$\sigma _{31}$$ and $$\sigma _{23}$$ applied anti-symmetrically to the crack faces. A set of 3 governing integral equations is derived, with some of the kernels presented in the finite form. The relationship between the integrands in Fourier transform of the kernels of the sets of the integral equations of both problems is established: they are related in such a way, as if they were the coefficients in the schematic sets of algebraic equations. This relationship can also be presented as a general relationship between matrices of arbitrary rank n, with their components being determinants of certain matrices of rank q, with $$q\ge n$$ .

Three-dimensional steady-state general solution for isotropic hygrothermoelastic media

ABSTRACTThe potential theory method was utilized to derive the steady-state general solution for three-dimensional (3D) hygrothermoelastic media. Two displacement functions are introduced to simplify the governing equations, with which the elastic, moisture, and temperature fields are thus simplified. Using the differential operator theory and superposition principle, all the physical quantities can be expressed in terms of two functions, one of which satisfies a harmonic equation and the other satisfies an eight-order partial differential equation. With the aid of generalized Almansi’s theorem, all the physical quantities like displacements, moisture, and temperature are expressed in terms of five quasi-harmonic functions for various cases of material characteristic roots. The obtained general solutions are in simple form and thus they may bring more convenience to certain boundary problems. As an example, the fundamental solutions for a point moisture source combined with heat source in the interior of infinite hygrothermoelastic body are presented by virtue of the obtained general solution. A planar crack of arbitrary shape in an infinite medium subjected to mechanical, moisture, and temperature loads is investigated to illustrate the application of the solution in boundary value problems. Specifically, for a penny-shaped crack under uniform combined loads, the complete, exact solutions are presented.

Cracks in anisotropic media: pseudodifferential equations, wave fronts, and Irwin’s energy release rate extension

3D problems for plane cracks of arbitrary shape in anisotropic media are considered. The stress intensity factors are determined by the intensity factors of the crack displacement discontinuity field. Strong ellipticity of the constructed pseudodifferential operator is proved. The solution of the pseudodifferential operator of the crack theory is obtained by the specially constructed potentials with densities concentrated at the crack front. Irwin’s relation for the energy release rate is obtained for a plane crack of arbitrary shape in a medium with arbitrary elastic anisotropy.

Finite element modeling of crack growth in thin-wall structures by method of combining sub-partition and substructure

A combination of sub-partition and substructure methods in finite element is developed to simulate the crack propagation in plate and shell structures. This method allows modeling arbitrary shape crack independent of element mesh and only the elements overlapping crack need to be operated. An element cut apart by a crack is sub-partitioned into ordinary sub-elements while an element enveloping crack tip is sub-partitioned into several singular sub-elements. The whole sub-elements constitute a substructure and the additional nodal freedom degrees introduced by element sub-partition are condensed to nodes of original mesh. In this way global re-meshing is avoided when the crack grows or new crack nucleates. Good accuracy of the crack tip fields prediction and good adaptability of moving crack simulation by the proposed method are proven through designed examples.

Analysis of arbitrarily shaped planar cracks in three-dimensional isotropic hygrothermoelastic media

ABSTRACTIn the present article, a planar crack of arbitrary shape embedded in three-dimensional isotropic hygrothermoelastic media is investigated. Based on the general solutions and Hankel transform technique, the fundamental solutions for unit-point and extended displacement discontinuities (EDD; including the displacement discontinuities, moisture concentration discontinuity, and the temperature discontinuity) are derived. The EDD boundary integral equations for an arbitrarily shaped, planar crack in the hygrothermoelastic medium are established in terms of the EDD. Utilizing the boundary integral equation method, the singularities of near-crack front fields are analyzed, and the stress, moisture flux, and heat flux intensity factors are all derived in terms of the EDD. As a special case, the analytical solution for a penny-shaped crack under uniform combined loadings is presented. The EDD boundary element method is proposed for numerical simulation. The numerical result for a penny-shaped crack subjected to uniform mechanical–moisture–thermal loading is compared with the analytical solution to verify the correctness of the proposed method. Two coplanar elliptical cracks subjected to combined loadings are simulated as an application, and the influences of applied loadings and the ellipticity ratio are discussed.

Analysis of arbitrarily shaped, planar cracks in a three-dimensional transversely isotropic thermoporoelastic medium

The displacement discontinuity boundary integral equation method is extended to analyze planar cracks of arbitrary shape embedded in a three-dimensional, transversely isotropic, thermoporoelastic medium. Based on the general solutions and Hankel transform technique, the fundamental solutions for unit-point, extended displacement discontinuities (including the displacement discontinuities, pore pressure discontinuity, and the temperature discontinuity) are derived. The extended displacement discontinuity boundary integral equations are established for an arbitrarily shaped, planar crack in the isotropic plane of the thermoporoelastic medium in terms of the extended displacement discontinuities. Using the boundary integral equations method, the singularities of near-crack front fields are analyzed, and the stress, fluid flux and heat flux intensity factors are derived in terms of the extended displacement discontinuities. To validate the analytical solution, the EDD boundary element method is proposed. The numerical simulation of a penny-shaped crack under combined uniform mechanical-pore pressure–thermal loadings is compared with the analytical solution to validate the correctness of the proposed method. As an application, two coplanar elliptical cracks are numerically simulated. The influences of the applied, combined mechanical-pore-pressure-thermal loadings, the crack distance, the ellipticity ratio as well as the size of cracks are all studied.

Analysis of a three-dimensional arbitrarily shaped interface crack in a one-dimensional hexagonal thermo-electro-elastic quasicrystal bi-material. Part 1: Theoretical solution

The extended displacement discontinuity boundary integral-differential equation method is adapted to analyze a three-dimensional interface crack of arbitrary shape in a one-dimensional, hexagonal thermo-electro-elastic quasicrystals bi-material. The extended displacement discontinuities include phonon and phason displacement discontinuities, electric potential discontinuity, as well as temperature discontinuity across the interface crack; while the extended stresses represent phonon and phason stresses, electric displacement and heat flux, respectively. An analysis method is proposed based on the analogy between the governing equations for one-dimensional hexagonal electro-thermo-elastic quasicrystals and three-dimensional transversely isotropic magnetoelectrothermoelastic media. By using the analogy method, the fundamental solutions for unit-point extended displacement discontinuities on the interface are obtained. Using the superposition principal, the extended displacement discontinuity boundary integral-differential equations are established. The singular indices and the singular behaviors of the near crack border fields are studied, and the combined extended stress intensity factors (SIFs) are derived in terms of the EDDs.

Extended displacement discontinuity method for an interface crack in a three-dimensional transversely isotropic piezothermoelastic bi-material. Part 1: Theoretical solution

The extended displacement discontinuity boundary integral-differential equation method is developed to analyze interface cracks of arbitrary shape in three-dimensional transversely isotropic piezothermoelastic bi-materials. The fundamental solutions for unit-point extended displacement discontinuities on the interface are derived, and the induced extended displacements and stresses are all expressed in terms of extended displacement discontinuities. An analysis method is proposed based on the analogy between the obtained boundary integral-differential equations and those for interface cracks in isotropic thermoelastic bi-materials. The singular indices and the singular behaviors of the near crack-tip fields are studied and two new extended stress intensity factors are obtained. Meanwhile, the extended stress intensity factors (SIFs) including two new fracture modes are derived in terms of the extended displacement discontinuities.

Analysis of an interface crack of arbitrary shape in a three-dimensional transversely isotropic magnetoelectrothermoelastic bimaterial—part 2: Numerical method

ABSTRACTThe extended displacement discontinuity (EDD) boundary integral equation and boundary-element method are extended and developed to analyze an arbitrarily shaped, planar interface crack in a three-dimensional, transversely isotropic, magnetoelectrothermoelastic bimaterial under combined, thermoelectromagnetomechanical loadings. The fundamental solutions for uniformly distributed EDDs applied over a constant triangular element are obtained through integrating the fundamental solutions for the unit-point EDDs given by Part 1 over the triangular area. To eliminate the oscillatory singularity near the crack front, the Dirac delta function in the integral–differential equations is approximated by the Gaussian distribution function, and accordingly, the Heaviside step function is replaced by the Error function. The extended stress intensity factors without oscillatory singularities, the energy release rate, and the local J-integral in terms of intensity factors are all obtained. To validate the solution, the EDD boundary-element method is proposed. As an application, an elliptical interface crack is numerically simulated. The influences of the applied combined loadings and material-mismatch as well as the ellipticity ratio on the multiphysical response are studied.