Antiplane Crack Problem Research Articles

Full field crack solutions in anti-plane flexoelectricity

In flexoelectric materials, strain gradients can induce electrical polarization. However, internal defects such as cracks profoundly affect the electromechanical coupling properties of flexoelectric solids. In particular, anti-plane cracks involve less physical fields, which are easier to study. In this study, we present a comprehensive and innovative investigation of the anti-plane crack problems in flexoelectric materials, including semi-infinite and finite-length anti-plane cracks. For the first time, we formulate a full-field solution for semi-infinite anti-plane cracks in flexoelectric media by applying the Wiener–Hopf technique. Furthermore, the collocation method and the Chebyshev polynomial expansion are used for the first time to derive the full-field hypersingular integral equation solution for finite-length anti-plane cracks in flexoelectric solids. In addition, a comparative analysis between the full-field and asymptotic solutions for semi-infinite cracks is performed, shedding light on the discrepancies in the representation of the electromechanical coupling behavior near the crack tip. The mixed finite element method is used to compare with the full-field solutions of finite-length cracks. The agreement between the numerical results and the full-field solutions demonstrates the rigor of our study. This research advances the knowledge of defects in flexoelectricity and provides significant insight into relevant failure mechanisms.

Anti-plane problem of a nano-sharp crack in one-dimensional hexagonal piezoelectric quasicrystals with the electrically semi-permeable condition

Based on the Gurtin–Murdoch surface/interface theory and the complex potential theory, the problem of an electrically semi-permeable nano-sharp crack in one-dimensional (1D) hexagonal piezoelectric quasicrystals is analyzed. By constructing conformal mapping of the sharp crack, the analytic solutions of the electroelastic field are determined. Meanwhile, the field intensity factors and the energy release rate (ERR) under the electric boundary conditions of semi-permeable are obtained. Numerical examples are given to analyze the effects of the crack size, dielectric constant, electric loading, and mechanic loadings on the dimensionless field intensity factors and the dimensionless ERR. The results show that the impact of the surface effect becomes weaker along with the growth of the crack length, but increases with the increase in the crack angle. The change of the dielectric coefficient has little influence on the dimensionless stress intensity factors (SIFs), but has a strong influence on the dimensionless electric displacement intensity factor (EDIF). The obtained results are helpful to promote the development of nano-quasicrystal composite mechanics and provide an important theoretical support for the design and application of nano-QC materials.

A new approach to solve the anti-plane crack problems by the method of fundamental solutions

This paper expands the method of fundamental solutions (MFS) to solve the anti-plane crack problems based on the traditional fundamental solution by the conformal mapping technique without using the domain decomposition technique and/or the crack Green's function. Based on the complex analysis and the conformal mapping technique, the approximate solution of the complex analytic function is derived for the MFS. The modified boundary conditions are deduced to construct the interpolation equations of the MFS. The conformal mapping technique naturally introduces singularities of the stress fields and the stress intensity factors (SIFs) for the anti-plane crack problems without using the crack Green's function and the domain-decomposition technique. The obtained numerical examples illustrate the validity and accuracy for the anti-plane crack problems. The proposed method is an expansion of the traditional MFS, and can be applied for many anti-plane elastic problems with the simple concept and the easy numerical implementation.

Jordan-form special solutions corresponding to arbitrary anti-plane shear forces in polynomial forms imposed on bimaterial interfacial crack surfaces

Bimaterial anti-plane interfacial crack problems with arbitrary distributed forces applied on crack surfaces are numerically investigated. The problems are in essence a kind of boundary value problems about Laplace's equations. Previous work of using Symplectic analytical singular element (SASE) with high-order accuracy and other singular finite elements for calculating mode III stress intensity becomes invalid when the distributed forces on each surface are different. In this paper, Jordan-form special solutions (JFSS) are proposed for the first time. We derive the JFSS for present problems and use them to improve displacement modes of SASE. The distributed forces are therefore converted into equivalent nodal forces of the nodes around SASE. The new SASE is found to be directly and conveniently used for cases where the two distributed forces are either equal (even both zero) or not equal. Numerical examples are provided to demonstrate the effectiveness and accuracy of the present method. The necessity and effectiveness of the JFSS is also validated. Moreover, the JFSS can be applied to some other physical problems represented by the Laplace's equation.

Antiplane shear crack in a functionally graded material strip with surface elasticity

When the dimension of a structure falls to the micro-/nanoscale, surface effect is significant and plays a key role in affecting the mechanical behavior. This article studies the influence of surface elasticity on the stress intensity factor of an antiplane shear crack embedded in an elastic strip made of functionally graded materials. Surface elasticity is applied on the strip surfaces and crack faces, and classic elasticity is invoked for the strip interior. An antiplane shear crack problem is solved for a symmetric FGM with a crack parallel to the strip surfaces. The associated problem is converted to a hypersingular integro-differential equation for the out-of-plane displacement on the crack faces through the Fourier transform and then to a singular integro-differential equation with Cauchy kernel. The Galerkin method is applied to expand the crack face displacement as a Chebyshev series, and the singular integro-differential equation reduces to a system of algebraic linear equations. Stress intensity factors at the crack tips and the out-of-plane displacement on the crack faces are calculated numerically. It is found that surface elasticity and gradient index strongly alter the bulk stress and its intensity factors near the crack tips. Positive surface shear modulus decreases the mode III stress intensity factors and negative surface shear modulus has an opposite behavior. The influence of the variation of material gradient on the mode III stress intensity factors is expounded in graph.

Anti-plane crack problem of a functionally graded piezoelectric materials strip with arbitrarily distributed properties

This paper treats the anti-plane crack problem of a functionally graded piezoelectric material (FGPM) strip with arbitrarily distributed properties. It is assumed that the elastic stiffness, piezoelectric constant, and dielectric constant of the FGPM vary continuously along the thickness of the medium, and the strip is under anti-plane mechanical and in-plane electric impact loadings. By using the Fourier transform and defining unknown discontinuous functions across the crack surfaces, the anti-plane crack problem of FGPM is reduced to a group of singular integral equations, which are solved numerically. The stress and electric displacement intensity factors are presented, and the influences of the nonhomogeneous and geometric parameters on stress and electric displacement intensity factors are also included.

Visualizing the crack driving force through fluid analogy

AbstractIt is shown that the crack driving force for a fundamental antiplane crack problem is analogous to the limiting case of the force acting on an ideal fluid on free streamlines that form at the ends of flow around two parallel plates.

The mode III problem of a Dugdale-Barenblatt interfacial crack in a FGM strip bonded to a homogeneous strip by the integral method

The elasto-static anti-plane problem of a Dugdale-Barenblatt interfacial crack in a functionally graded strip bonded to a homogeneous substrate is formulated in term of a singular integral equation. The functionally graded strip is modeled with an exponentially variable shear modulus. The singular integral equation is solved using Tchebychev polynomials. The influence of the non-homogeneity parameter, the shear modulus of the homogeneous strip, the widths of the functionally graded strip and the homogeneous strip, and the critical sliding displacement jump in the Dugdale-Barenblatt cohesive model on the fracture load are discussed.

Finite-form solution for anti-plane problem of nanoscale crack

A finite-form solution for the problem of an elastic body weakened by a straight nano-crack and disturbed by a screw dislocation in the mode-III deformation is proposed. The boundary equations incorporating the surface effects are established and reduced to single one via the integration over the crack boundary in the physical region. The cracked physical region is then mapped onto an image region with circular hole. By using Muskhelishvili (Some basic problems of the mathematical theory of elasticity, Noordhoff, Groningen, 1953) method and Cauchy’s integral formula, the reduced boundary equation is further converted into a first-order differential equation. Consequently, the solution of the differential equation is mapped back to the physical region and the complex potential representing the displacement is formulated in finite form. The results show that the stresses at the crack tip are finite and the stress singularity is eliminated completely by the incorporation of the surface effects. The non-singularity of the stresses at the crack tip derived from the present article radically differs from the square-root singularity obtained from the classic theory of the elastic fracture and, however, is reasonable from the physical essence. The numerical results shows that when the surface effects are incorporated, the magnitudes of the stresses and the image forces near the crack tip are reduced, which are strongly dependent on the crack size. The corresponding results approach those from the classic theory as the crack length increases. The results by the present solution are also compared with those from other solutions.

Meshless simulation of anti-plane crack problems by the method of fundamental solutions using the crack Green’s function

In this paper, we present a meshless method of fundamental solutions using the analytical crack Green’s function to solve anti-plane crack problems. The proposed scheme is a simple, powerful and effective collocation method for crack problems since it only requires the boundary discretization without special treatments of the crack. Three typical numerical examples, namely, a central crack, an off-central crack and a central slant crack are presented and discussed to illustrate the accuracy, convergence and stability of the proposed method.

Study of the anti-plane problem of a Dugdale-Barenblatt crack in a welded strip using the integral equation method

The elasto-static anti-plane problem of a Dugdale-Barenblatt crack in a welded infinite strip is formulated in terms of a singular integral equation (SIE). The weld joint is modeled as a tri-material structure: the weld metal (WM), the base metal (BM) and the heat-affected zone (HAZ). The HAZ is modeled with an exponentially variable shear modulus. The SIE is solved using Tchebychev polynomials. The influence of the elastic mismatching (the ratio between the shear modulus of WM and BM) and the HAZ width on the fracture load and on the crack propagation is investigated.

Anti-plane stress analysis of a half-plane with multiple cracks by distributed dislocation technique in nonlocal elasticity

The effective role of a distributed dislocation technique accompanied by a nonlocal elasticity model has been demonstrated for the crack problem in a half-plane. The dislocation solution is employed to model and analyze the anti-plane crack problem for nonlocal elasticity using the distributed dislocation technique. The solution of dislocation in the half-plane has been extracted through the solution of dislocation in an infinite plane by the image method. The dislocation solution has been utilized to formulate integral equations for dislocation density functions on the surface of a smooth crack embedded in the half-plane under anti-plane loads. The integral equations are of the Cauchy singular type, and have been solved numerically. Multiple cracks with different configurations have been solved; results demonstrate that the nonlocal theory predicts a certain stress value in the crack tip.

A von Karman plate analogue for solving anti-plane problems in couple stress and dipolar gradient elasticity

The purpose of the present work is to present a direct analogy regarding the formulation and solution of anti-plane problems in the context of couple stress and dipolar gradient elasticity and the von Karman plate framework. We show aspects of boundary conditions in both theories of elasticity and propose a robust Finite Element methodology based on the von Karman plate theory in order to solve complex anti-plane (mode III) crack problems. Furthermore, we establish the equivalence between the anti-plane gradient elasticity J-Integral and the plate Ic-Integral (Sanders’ plate integral). In passing, we prove the path independency of the Ic-Integral. Finally we examine the near tip fields for both anti-plane and plate problems and establish possible strengthening effects due to the underlying microstructure.The proposed analogy is achieved through the von Karman plate theory where the plate is pre-stressed by a constant biaxial tension. The plate theory involves properties such as the plate thickness h, the Poisson's ratio ν and the bending stiffness D. This information, together with the pre-stress N transforms into properties required by the anti-plane couple stress and dipolar gradient elasticity problems: the shear modulus G, the internal length ℓ and the coefficient η. In both problems the two dimensional space remains the same, including the presence of cracks and other defects. The analogy permits numerical and analytical solutions of demanding anti-plane problems of gradient elasticity (couple stress and dipolar) utilizing the von Karman plate corresponding, and vice versa.

A singular finite element of multi-material crack problems under elastic antiplane shear deformation

Multi-material antiplane crack problem is a typical mode III crack problem, of which the symplectic eigen-solutions under polar coordinate system and the stress singularity order at the crack tip have been reported in existing literatures. Based on the existing works, the special solution corresponding to the non-homogeneous boundary condition on crack surfaces is derived in this study, and then a symplectic analytical singular element (SASE) is constructed using the symplectic eigen-solutions and the special solution to deal with the related problems. Combining the proposed SASE with conventional finite elements, the multi-material mode III crack problems can be solved and the numerical results of stress intensity factors (SIF) can be calculated directly. Numerical examples show that the proposed method possesses high solving accuracy as well as high efficiency for the analysis of the discussed problem.

Debonding of an elastic inhomogeneity of arbitrary shape in anti-plane shear

We investigate the anti-plane shear problem of a curvilinear crack lying along the interface of an arbitrarily shaped elastic inhomogeneity embedded in an infinite matrix subjected to uniform stresses at infinity. Complex variable and conformal mapping techniques are used to derive an analytical solution in series form. The problem is first reduced to a non-homogeneous Riemann–Hilbert problem, the solution of which can be obtained by evaluating the associated Cauchy integral. A set of linear algebraic equations is obtained from the compatibility condition imposed on the resulting analytic function defined in the inhomogeneity and its Faber series expansion. Each of the unknown coefficients in the corresponding analytic functions can then be uniquely determined by solving the linear algebraic equations, which are written concisely in matrix form. The resulting analytical solution is then used to quantify the displacement jump across the debonded section of the interface as well as the traction distribution along the bonded section of the interface. In addition, our solution allows us to obtain mode-III stress intensity factors at the two crack tips. The solution to the anti-plane problem of a partially debonded elliptical inhomogeneity containing a confocal crack is also derived using a similar method.

General solutions of plane problem in one-dimensional quasicrystal piezoelectric materials and its application on fracture mechanics

Based on the fundamental equations of piezoelasticity of quasicrystals (QCs), with the symmetry operations of point groups, the plane piezoelasticity theory of onedimensional (1D) QCs with all point groups is investigated systematically. The governing equations of the piezoelasticity problem for 1D QCs including monoclinic QCs, orthorhombic QCs, tetragonal QCs, and hexagonal QCs are deduced rigorously. The general solutions of the piezoelasticity problem for these QCs are derived by the operator method and the complex variable function method. As an application, an antiplane crack problem is further considered by the semi-inverse method, and the closed-form solutions of the phonon, phason, and electric fields near the crack tip are obtained. The path-independent integral derived from the conservation integral equals the energy release rate.

The dynamic stress intensity factor around the anti-plane crack in an infinite strip functionally graded material under impact loading

In order to dynamically analyze a crack in an infinite strip functionally graded material for anti-plane crack problem under impact loading, a gradient model is established in which the mass density and the shear modulus in the two principal directions of the functionally graded material are assumed to vary proportionally according to the double parameters index function. By using the Laplace and Fourier transforms, the motion equation is first reduced to the Euler equation. By solving dual integral equations derived by applying the solution of the Euler equation with the method of Copson, dynamic stress intensity factor around the crack tip is derived. The variation curves of the normalized stress intensity factor with the inhomogenous coefficient and gradient parameter are obtained from the numerical calculation. The results show that the decrease of the inhomogenous coefficient and the increase of the gradient parameter are beneficial to the decrease of the dynamic stress intensity factor.

The Higher Order Crack Tip Fields for Anti-Plane Crack in Functionally Graded Piezoelectric Materials

The anti-plane crack problem is studied in functionally graded piezoelectric materials (FGPMs). The material properties of the FGPMs are assumed to be the exponential function of y. The crack is electrically impermeable and loaded by anti-plane shear tractions and in-plane electric displacements. Similar to the Williams solution of homogeneous material, the high order asymptotic fields are obtained by the method of asymptotic expansion. This investigation possesses fundamental significance as Williams solution.

The collinear crack problem for an orthotropic functionally graded coating–substrate structure

A theoretical treatment of antiplane crack problem of two collinear cracks on the two sides of and perpendicular to the interface between a functionally graded orthotropic strip bonded to an orthotropic homogeneous substrate is put forward. Various internal cracks and crack terminating at the interface and crack crossing the interface configurations are investigated, respectively. The problem is formulated in terms of a singular integral equation with the crack face displacement as the unknown variable. The asymptotic stress field near the tip of a crack crossing the interface is examined, and it is shown that, unlike the corresponding stress field in piecewise homogeneous materials, in this case, the “kink” in material property at the interface does not introduce any singularity. Numerical calculations are carried out, and the influences of the orthotropy and nonhomogeneous parameters and crack interactions on the mode III stress intensity factors are investigated.

The Static Stress Intensity Factor around the Anti-Plane Crack in an Orthotropic Functionally Graded Material

The problem of anti-plane crack in infinity orthotropic functionally graded materials is studied by using of integral transforms-dual integral equations. The shear modulus in the two principal directions of the functionally graded material was assumed to vary proportionately as gradient model of double parameters. And the variation curves of the dimensionless stress intensity factor with the orthogonal parameter and the crack length have been obtained by using the mathematical software .The results shows that stress intensity factor increases with the increasing of and a. It means that stress intensity factor decreases as the shear modulus of perpendicular to crack direction increased.