Penny-shaped Crack Problem Research Articles

Shear mode solutions to penny-shaped crack problems in two-dimensional hexagonal piezoelectric quasicrystal media

This study investigates shear mode penny-shaped crack problems in an infinite three-dimensional body composed of a two-dimensional hexagonal quasicrystal medium with piezoelectric effect. The crack is subjected to a set of shear phonon and phason loadings within the crack plane. This shear mode crack problem is transformed into a mixed boundary value problem in the upper half space. Subsequently, it is elegantly solved utilizing Fabrikant’s potential theory method. The boundary integral–differential equations governing three-dimensional shear mode crack problems in two-dimensional hexagonal piezoelectric quasicrystals are derived with the phonon and phason displacement discontinuities serving as unknown variables. Closed-form solutions for all physical field quantities are presented, not merely limited to the crack surface, but rather extended comprehensively to the entire space. Key fracture mechanics parameters, such as phonon and phason displacement discontinuities, stress intensity factors at the crack tip, and energy release rate, are explicitly derived. Numerical results are provided to validate the obtained analytical solutions and illustrate the distribution of the electric-phason-phonon coupling field around the crack in graphical form. Additionally, these numerical results also compare the fracture mechanics parameters of the chosen piezoelectric quasicrystal with its corresponding non-piezoelectric quasicrystal, thereby investigating the influence of piezoelectric effect on quasicrystals. The obtained solution can be used as a benchmark for the experimental and numerical study of shear mode cracks in piezoelectric quasicrystals.

Analytical solutions to Mode I penny-shaped crack problems in two-dimensional hexagonal quasicrystals with piezoelectric effect

The paper studies a penny-shaped crack in an infinite three-dimensional body of two-dimensional hexagonal quasicrystal media with piezoelectric effect. The crack surfaces are applied combined electric and normal phonon loadings. Such a Model I crack problem is transformed into a mixed boundary value problem in the upper half-space, which is analytically solved using Fabrikant's potential theory method. The boundary integral-differential equations governing Model I crack problems are presented for two-dimensional hexagonal piezoelectric quasicrystals. The normal phonon displacement discontinuity and electric potential discontinuity across crack surfaces are taken as the unknown variables of boundary governing equations. Analytical solutions of all field variables are derived not only for the crack plane but also for the full space. Solutions in integral form are provided for the penny-shaped crack under arbitrarily distributed electric and normal phonon loadings. Closed-form solutions in terms of elementary functions are given for concentrated point loadings and uniformly distributed loadings, respectively. Key fracture mechanics parameters, such as crack surface extended displacements (i.e., normal phonon displacement, electric potential), crack tip extended stresses (i.e., normal phonon stress, electric displacement) distribution, and corresponding extended stress intensity factors, are clearly derived. Numerical results are utilized to verify the present analytical solutions and graphically illustrate the distribution of phonon-phason-electric coupling fields around the crack. The present solution can serve as a benchmark for both experimental and numerical investigations.

Mode-I penny-shaped crack problem in an infinite space of one-dimensional hexagonal piezoelectric quasicrystal: exact solutions

Mode-I penny-shaped crack problem in an infinite space of one-dimensional hexagonal piezoelectric quasicrystal: exact solutions

A phase-field model for thermo-elastic fracture in quasicrystals

In this paper, a thermo-elastic phase-field model for brittle fracture in quasicrystals is developed. The kind of quasicrystals is not preset, so this model is suitable for almost all thermo-elastic quasicrystal solids. The phonon force and thermal load trigger the variation of the phonon elastic energy, and then drive crack initiation and propagation. The thermal conductivity in the cracked region is assumed to be degraded, which means cracks are adiabatic. For the static penny-shaped crack problem, the results based on the present model agree well with the existing analytical solution. For the quasi-static crack problems, both the tensile/shear fracture in a thermal environment and the thermal shock cracking failure can be dealt with by the present model. Several examples of 1D hexagonal and 2D decagonal quasicrystals are performed to evaluate the effects of thermal load and phason elastic field. The simulation results show that the thermal load can influence the peak force and fracture displacement. For the single-edge notched tension test, the effect of the thermal load on the peak force is small, but it on the fracture displacement is obvious. For the single-edge notched shear test, the thermal load has an obvious influence on both the peak force and fracture displacement. For the thermal shock cracking problem, the phason elastic field has a significant effect on the cracking time and crack number. The present model can serve as a powerful tool to simulate various thermo-elastic fracture problems in quasicrystals.

3D fracture study of cracked functionally graded biological materials by XIGA approach using Bézier extraction of NURBS

Functionally graded material (FGM) offers many varied engineering applications especially in biomedical implants and also used as biomaterials in orthopaedic implants such as hip & knee replacement and joint replacements and orthodontic applications. High risks are involved with revision surgery as orthopaedic prostheses suffer from limited lifetime. Commonly used bio materials consist of uniform and single constituent are vulnerable for their fabrication. Therefore, FGMs are developed to fabricate hybrid bio-material for overcoming these shortcomings. A long service life and safety of these hybrid bio-materials used in biomedical applications are one of the most important objective for their safe design and fabrications. In this paper, three-dimensional (3D) fracture problems in elastically graded biological materials by extended isogeometric analysis (XIGA) using Bézier extraction of NURBS are numerically simulated to understand the fracture behaviour. Bézier extraction of NURBS-based basis functions are implemented in discretization of domain for approximating both unknown field solution and geometries. The Bézier extraction operation decomposes the NURBS basis functions into a set of linear combination of Bernstein polynomials and a set of C0–continuity Bézier elements. For FGMs, exponential law has been implemented for grading the material property from left region (Material-A) to right region (Material-B). Few test cases based on penny-shaped crack problems have been solved to calculate fracture parameters, Von-Mises stress contours and deformation contours.

Fundamental solutions for penny-shaped and half-plane cracks in one-dimensional hexagonal quasicrystals: Shear mode

The present article is to investigate the penny-shaped and half-plane crack problems in an infinite space of one-dimensional hexagonal quasicrystals. The cracks are subjected to a pair of anti-symmetrical point shear loads exerted on the crack surfaces. By means of the potential theory method, the governing integral equation is developed and the complete fundamental solutions of the phonon and phason fields are obtained. Moreover, the important quantities on the crack plane, including the crack slip displacement and the stress intensity factor, are derived in terms of elementary functions. The fundamental solutions presented in the article are important to boundary element analysis and may serve as benchmarks for numerical research.

Approximate structures of thermoelastic fields induced by a penny-shaped thermal-medium crack in a transversely isotropic layer

It is of practical significance to investigate thermoelastic fields at the front of a penny-shaped crack embedded in a material with finite thickness. The existing works always focus on numerical solutions of the derived integral equations and approximate solutions are seldom studied. In the present study, the problem of a penny-shaped crack in a transversely isotropic layer is addressed under uniform heat flow and mechanical loadings. The extended thermal-medium crack model with a nonlinear relation between heat flow of crack interior and crack opening displacement is used to capture the effect of heat conduction inside crack. The Hankel transform technique is applied to solve governing partial differential equations, then Fredholm integral equations of the second kind are derived. Applying integral kernel series expansion method, approximate solutions to the derived Fredholm integral equations are analyzed and determined. Approximate structures of thermoelastic fields around penny-shaped crack are further obtained. Theoretical analysis and numerical results are carried out to show the applicability of approximate solutions. The observations reveal that the determined approximate solutions have some significance to a practical engineering application. The advantages of the developed methods are to make approximate solutions visible, and some penny-shaped crack problems of thermo-piezoelectric and thermo-magnetoelectroelastic layers could be solved similarly.

A penny-shaped crack in a piezoelectric layer sandwiched between two elastic layers with the electrical saturation and mechanical yielding zones

This paper deals with an internal penny-shaped crack in a piezoelectric layer sandwiched by two outer elastic layers. It is assumed that the crack faces are electrically impermeable, and the thin annular electrical saturation and mechanical yielding zones appear around the crack front under the uniform distributed normal stress and electric displacement. Three different cases, i.e., the region of electrical saturation is longer, shorter than, or equal to the domain of mechanical yielding are respectively considered in this article. For such an axisymmetric penny-shaped crack problem, we simplify the mixed boundary value problem into coupling Fredholm integral equations of the second kind by employing Hankel transform technique and the Copson method. The problem is solved in the real domain by constructing real fundamental solutions and the numerical solutions for any layer-thickness are derived with appropriate formula derivation and numerical discretization. When the outer elastic layers vanish and the piezoelectric layer-thickness approaches infinity, the relation between the nonlinear zone-lengths and the loadings obtained in this paper reduces to the analytical expression in literature. The influence of the electromechanical loads, the thicknesses of the piezoelectric layer and the elastic layer, and the material parameter ratios on the nonlinear zones is discussed in details. The results of paper are believed to have some guiding references for the design of the piezoelectric sandwich plate.

A mixed boundary value problem of a cracked elastic medium under torsion

The present work aims to investigate a penny-shaped crack problem in the interior of a homogeneous elastic material under axisymmetric torsion by a circular rigid inclusion embedded in the elastic medium. With the use of the Hankel integral transformation method, the mixed boundary value problem is reduced to a system of dual integral equations. The latter is converted into a regular system of Fredholm integral equations of the second kind which is then solved by quadrature rule. Numerical results for the displacement, stress and stress intensity factor are presented graphically in some particular cases of the problem.

Dugdale plastic zone model of a penny-shaped crack in a magnetoelectroelastic cylinder under magnetoelectroelastic loads

This paper extends the Dugdale plastic zone to the penny-shaped crack problem for a magnetoelectroelastic (MEE) cylinder, where the crack surfaces are assumed to be magnetoelectrically permeable. Using potential function theory and Hankel transform method, the present boundary value problem is translated into solving a sectionalized Fredholm integral equation of the second kind. The non-singular field solutions are analyzed, and the effects of the applied MEE loads and geometric dimension on the width of plastic zone and crack opening displacements are evaluated. Numerical results show that for the considered model, each of the far-field MEE combination load $$\lambda $$ , the far-field magnetoelectric combination load $$\lambda _{\mathrm{EH}}$$ and the purely mechanical load $$\sigma _0$$ applied in the far field, respectively, plays an important role in the present fracture analysis. For a given mechanical load and a fixed crack configuration, the negative magnetoelectric loads are prone to promote the crack growth and propagation than the positive ones. These should be helpful for the design and manufacture of MEE materials and devices.

Three-Dimensional Elastic Analysis of Rock Excavations by Using the g2 Constant Displacement Discontinuity Method

A fast computational code is presented that is dedicated for the elastic analysis of three-dimensional excavations and cracks in rocks. The problem is solved on the boundaries that are discretized with a new triangular leaf constant displacement discontinuity element with one collocation point. The creation of the new triangular element was inspired from Mindlin's special version of grade-2 or strain-gradient elasticity theory (second gradient of displacement, g2). This element is characterized by a much better measure of the average stress at the center of gravity of the triangular element compared to that of the classical elasticity element close to regions with stress or strain gradients (e.g. notches, cracks etc). In a verification stage, the accuracy of the computational algorithm for the pressurized penny-shaped and mixed-mode elliptical crack problems that have analytical solutions is demonstrated. More specifically, it is shown that the average error of the crack tip Stress Intensity Factor predicted by the gradient modified method for nine discretizations of varying density is around 3.5% with a máximum error of 5%, while the constant displacement discontinuity element displays errors varying around 14%. Moreover, the new method preserves the simplicity and hence the high speed of the constant displacement discontinuity with only one collocation point per element, but it is far more efficient compared to it, especially close to the crack tips and corners of excavations where the displacement and stress gradients are highest.

Development of stress boundary conditions in smoothed particle hydrodynamics (SPH) for the modeling of solids deformation

This paper develops a method for imposing stress boundary conditions in smoothed particle hydrodynamics (SPH) with and without the need for dummy particles. SPH has been used for simulating phenomena in a number of fields, such as astrophysics and fluid mechanics. More recently, the method has gained traction as a technique for simulation of deformation and fracture in solids, where the meshless property of SPH can be leveraged to represent arbitrary crack paths. Despite this interest, application of boundary conditions within the SPH framework is typically limited to imposed velocity or displacement using fictitious dummy particles to compensate for the lack of particles beyond the boundary interface. While this is enough for a large variety of problems, especially in the case of fluid flow, for problems in solid mechanics there is a clear need to impose stresses upon boundaries. In addition to this, the use of dummy particles to impose a boundary condition is not always suitable or even feasibly, especially for those problems which include internal boundaries. In order to overcome these difficulties, this paper first presents an improved method for applying stress boundary conditions in SPH with dummy particles. This is then followed by a proposal of a formulation which does not require dummy particles. These techniques are then validated against analytical solutions to two common problems in rock mechanics, the Brazilian test and the penny-shaped crack problem both in 2D and 3D. This study highlights the fact that SPH offers a good level of accuracy to solve these problems and that results are reliable. This validation work serves as a foundation for addressing more complex problems involving plasticity and fracture propagation.

Finite element analysis of internal penny-shaped crack problem in an unbounded thermoelastic medium

ABSTRACTIn this article, the generalized thermoelastic theory with one relaxation time is used to investigate the internal penny-shaped crack problem in an unbounded thermoelastic material. The boundary of the crack is subjected to a prescribed stress distribution and temperature. The problem has been solved numerically using a finite element method. The results of the temperature, displacement components, and stress components have been verified numerically and are represented graphically. Comparisons are made with the results predicted by the presence and absence of the relaxation times.

Fracture analysis for a penny-shaped crack problem of a superconducting cylinder in a parallel magnetic field

In this work, the penny-shaped crack problem is investigated for an infinite long superconducting cylinder under electromagnetic forces. The distributions of magnetic flux density in the superconducting cylinder are obtained analytically for both the zero-field cooling (ZFC) and the field cooling (FC) activation processes, where the magnetically impermeable crack surface condition and the Bean model outside the crack region are adopted. Based on the finite element method (FEM), the stress intensity factor (SIF) and energy release rate (ERR) at the crack tips in the process of field descent are further numerically calculated. Numerical results obtained show that according to the maximal energy release rate criterion, the FC process is generally easier to enhance crack initiation and propagation than the ZFC activation process. On the other hand, for the FC activation process, the larger the maximal applied magnetic field, more likely the crack propagates. Additionally, crack size has important and slightly different effects on the crack extension forces for the ZFC and FC cases. Thus, all of the activation processes, the applied field and the diameter of the penny-shaped crack have significant effects on the intensity analysis and design of superconducting materials.

Thermal stress analysis of a three-dimensional anticrack in a transversely isotropic solid

A three-dimensional analysis is performed for an infinite transversely isotropic elastic body containing an insulated rigid sheet-like inclusion (an anticrack) in the isotropy plane under a remote perpendicularly uniform heat flow. A general solution scheme is presented for the resulting boundary-value problems. Accurate results are obtained by constructing suitable potential solutions and reducing the thermal problem to a mechanical analog for the corresponding isotropic problem. The governing boundary integral equation for a planar anticrack of arbitrary shape is obtained in terms of a normal stress discontinuity. As an illustration, a complete solution for a rigid circular inclusion is obtained in terms of elementary functions and analyzed. This solution is compared with that corresponding to a penny-shaped crack problem.

Non-ordinary state-based peridynamic analysis of stationary crack problems

An implicit implementation of the non-ordinary state-based peridynamics formulation for quasi-static linearly elastic solids is presented. Emphasis is placed on assessing the accuracy of the numerical scheme in the vicinity of the crack front and other sources of stress concentration. We also present a comparative study of methods used to control the zero-energy modes inherent in the nonlocal definition of the strain tensor and reduce the spurious oscillations present particularly in regions of high strain gradients. The accuracy of the peridynamics scheme, including the impact of the lattice spacing and configuration, is assessed by performing an analysis of the near-tip stress and displacement fields (K-fields) for 2D problems. The manuscript also summarizes a verification study based on the classical 3D penny-shaped crack problem and a validation study of a 3D notched fracture specimen.

Hypersingular integral equations for the solution of penny-shaped interface crack problems

Based on the theory of elasticity, previous analytical solutions concerning a penny-shaped interface crack employ the derivative of the crack surface opening displacements as the primary unknowns, thus leading to singular integral equations with Cauchy-type singularity. The solutions to the resulting integral equations permit only the determination of stress intensity factors and energy release rate, and do not directly provide crack opening and sliding displacements. However, the crack opening and sliding displacements are physically more meaningful and readily validated against the finite element analysis predictions and experimental measurements. Therefore, the present study employs crack opening and sliding as primary unknowns, rather than their derivatives, and the resulting integral equations include logarithmic-, Cauchy-, and Hadamard-type singularities. The solution to these singular integral equations permits the determination of not only the complex stress intensity factors but also the crack opening displacements.

Functionally graded piezoelectric strip with a penny-shaped crack under electromechanical loadings

In this paper, the mixed-mode penny-shaped crack problem for a functionally graded piezoelectric material (FGPM) strip is considered. It is assumed that the electroelastic properties of the strip vary continuously along the thickness of the strip, and that the strip is under in-plane electromechanical loadings. The problem is formulated in terms of a system of singular integral equations. The stress and electric displacement intensity factors are presented for various values of dimensionless parameters representing the crack size, the crack location, and the material nonhomogeneity.

The contact problem for an open penny-shaped crack under normally incident tension-compression wave

This paper concerns fracture dynamic problems for elastic cracked solids with allowance for crack faces contact interaction. The contact problem for a penny-shaped crack with an initial opening under normally incident tension-compression wave is solved by the method of boundary integral equations. The contact forces and the displacement discontinuity of the crack faces are studied. The solution is compared with those obtained without allowance for crack faces contact interaction for various shapes of the initial opening.

Fundamental solutions in piezoelectricity. Penny-shaped crack solution

The problem of electroelasticity for piezoelectric materials is considered. For axially symmetric states three potentials are introduced, which determine the displacements, the electric potentials, the stresses, the components of the electric field vector and the electric displacements in a piezoelectric body. These fundamental solutions are utilized to solve the penny-shaped crack problem. Two cases of boundary—value problems are considered, namely the permeable and impermeable crack boundary conditions. Exact solutions are obtained for elastic and electric fields. The main results are the stress intensity factor for singular stress and the electric displacement intensity factor. The numerical results are presented graphically to show the influence of applied mechanical and electrical loading on the analyzed quantities and to clarify the effect of anisotropy of piezoelectric materials. It is show that the influence of anisotropy of the materials on these fields is significant.