Distributed Dislocation Technique Research Articles

Analysis of the effect of a micro-crack on plastic zone of the edge macro-crack tip by macroscopic and microscopic methods

The effect of the micro-crack on plastic zone ahead of the macro-crack tip (PZMA) is analyzed based on macroscopic and microscopic methods. From the macroscopic perspective, the plastic deformation is due to the local stress near crack tip is more than the yielding limit. So the complete stress field is calculated based on the distributed dislocation technique (DDT) and the plastic zone (PZ) is determined by von Mises yielding criterion (MYC). On the other hand, the crack tip plasticity is simulated by finite element method (FEM). From the microscopic perspective, the plastic deformation is caused by dislocations emitted from crack tip. So PZ is represented by dislocations, and it is determined based on DDT and dislocation-free zone (DFZ) model. The results show the micro-crack located in front of PZMA has the amplifying effect on PZMA. PZMA is separated by the micro-crack located in PZMA. The micro-crack behind PZMA has no effect on PZMA. PZMA has the shielding effect on the macro-crack propagation, and the shielding effect increases with PZMA increasing. The micro-crack has the amplifying effect on PZMA at about x0/a > 0.5, while it has the shielding effect at about x0/a < 0.5. These results are useful for predicting and interpreting the plastic behaviors at the crack tip.

Application of Singular Integral Equation to a Crack Moving near a Hole in a Two-Dimensional Infinite Plate

In this study, crack path simulation was conducted based on a singular integral equation formulated by a continuous distributed dislocation technique. The problem investigated in this study was to predict the propagation path of a crack moving in an infinite elastic plate with a circular hole, under uniform tensile loading. In order to perform this prediction, a probing method was developed to search for a crack moving direction where the mode II stress intensity factor would be almost zero, enabling the crack to automatically extend in that direction. Some cases for different locations of an initial straight crack were simulated using the program developed.

Crack interaction with nanoscale twinning near a second-phase particle in fine-grained magnesium alloy

A theoretical model is established to address the effect of nanoscale twinning near a second-phase particle on crack growth in fine-grained magnesium alloys. The numerical solutions of singular integral equations are obtained by the considering complex variable method of Muskhelishvili, the superposition principle of elasticity, and the distributed dislocation technique. The expressions of stress intensity factors near the left crack tip are derived, and the energy release rate (ERR) characterizing the condition for crack propagation is also calculated. The influences of relevant parameters such as the location of nanoscale twin, the size of particle, and the relative distance between the inhomogeneity and the left crack tip on the ERR are examined in detail. The results indicate that the ERR is strongly influenced by the nanoscale twinning and the second-phase particle. The hard inhomogeneity decreases the ERR while the soft inhomogeneity increases the ERR when nanoscale twinning is not taken into account. At the same time, there is competition between the effects of the hard particle and nanoscale twinning on the ERR, and the nanoscale twin band with an optimum size displays the best toughening effect.

Several embedded cracks in a functionally graded piezoelectric strip under dynamic loading

This paper deals with the fracture analysis of multiple cracks in a functionally graded piezoelectric (FGP) layer under applied anti-plane time-harmonic mechanical and in-plane electrical loading within the framework of linear electro-elasticity. The analysis is based on the stress and electric fields caused by Volterra-type screw dislocation in the medium. The crack surfaces are assumed to be impermeable condition. The material properties of the medium are considered to obey exponential variations. In order to model the cracked layer, the distributed dislocation technique is employed to perform Cauchy-type singular integral equations for layer, in which the unknown variables are dislocation densities. The dislocation densities are then employed to derive field intensity factors at the crack tips. The results show that the stress and the electric displacement intensity factors at the crack tips depend on the cracks configuration, frequency and material properties. Finally, several numerical examples are solved to obtain the dynamic stress intensity factors (DSIFs) and electric displacement intensity factors.

The multiple parallel cracks in an orthotropic non-homogeneous infinite plane subjected to transient in-plane loading

In this paper, the mixed mode fracture problem of multiple parallel cracks in an orthotropic functionally graded plane under transient dynamic loading is studied. The medium contains Volterra type glide and climb edge dislocations. It has been assumed that the Young's modulus and mass density of materials vary exponentially along the x-axis. The distributed dislocation technique is used to obtain a system of integral equations for the transient dynamic problem. At first, with the help of Fourier and Laplace transforms the dislocation problem is solved and the stress fields are obtained. The integral equations are of Cauchy type singularity, and are solved numerically using the Lobatto-Chebyshev integration formula in order to evaluate the dislocation densities on the face of several cracks. Several numerical examples are solved to show the effects of the orthotropy and non-homogeneity parameters on the dynamic stress intensity factors.

Effect of a micro-crack on the kinked macro-crack

The solution of an infinite plane containing an arbitrarily located micro-crack and a kinked macro-crack is presented based on the distributed dislocation technique (DDT). The stress field, the plastic zone of the macro-crack tip (PZ) and the stress intensity factors at the macro-crack tip (SIFs) are obtained. The influence of the micro-crack on the kinked macro-crack is investigated. The results show the effect of the micro-crack on SIFs increases with the micro-crack length increasing, the distance d decreasing and the micro-crack angle decreasing. The micro-crack has an amplifying effect on mode I SIF at about -80°<θ<60°, while it has a shielding effect at about -90°<θ<-80° and 60°<θ<90°. The micro-crack has an amplifying effect on mode II SIF at about -90°<θ<20°, while it has a shielding effect at about 20°<θ<90°. The micro-crack ahead of PZ has a little amplifying effect on PZ. As the micro-crack is close to the macro-crack tip, plastic zones of the micro-crack tip and macro-crack tip will join together. PZ is split into two pieces by the micro-crack located in PZ. The micro-crack behind PZ has no effect on PZ. These results are helpful to analyze the fracture or fatigue behaviors of materials.

Combined effects of cooperative grain boundary sliding and migration and reinforced particles on crack growth in fine-grained Mg alloys

A theoretical model is developed to describe the combined influences of cooperative grain boundary (GB) sliding and migration and reinforced particles on crack propagation in fine-grained Mg alloys. The numerical solutions of singular integral equations are derived on the basis of complex variable method of Muskhelishvili, the superposition principle of elasticity and distributed dislocation technique. The expressions of stress intensity factors (SIFs) near the right tip of basal crack are obtained and the energy release rate (ERR) characterizing the condition for crack growth is calculated. The influences of important parameters such as the location of cooperative GB sliding and migration, the size of reinforced particle and the misorientation angle of higher angle grain boundaries on the ERR are discussed in detail. The results show that the fracture toughness of fine-grained Mg alloys can be improved by cooperative GB sliding and migration and particle hardening and refining. There exists an optimum particle size that makes the fracture toughness of fine-grained Mg alloys best. Besides, the GB migration remarkably contributes to the fracture toughness of fine-grained Mg alloys and there is an optimum migration distance making the fracture toughness of fine-grained Mg alloys best.

Three-dimensional analysis of an edge crack in a plate of finite thickness with the first-order plate theory

In this paper, plasticity effects at the tip of an edge through-the-thickness crack are investigated by invoking the strip-yield model and the first order plate theory. The latter theory is the simplest extension of the classical plane theories of elasticity, which is capable of accounting for three-dimensional effects and, in particular, the plate thickness effect. By utilising the distributed dislocation technique, new results are obtained on the interaction of the free boundaries (free edge and free plate planes) with the crack tip plasticity region. Numerical results for the crack tip opening displacement and global constraint factor are first presented at various crack length to plate thickness and applied stress to yield stress ratios. The results are validated against past 3D solutions for an embedded crack as well as analytical solutions obtained under the plane stress or plane strain assumption. The new model and results can be utilised in advanced fracture and fatigue analyses of cracks emanating from a free surface.

Near‐tip stress fields of rough and frictional cracks under mixed‐mode loading

AbstractThe characterisation of the stress fields in proximity of crack tips is a fundamental task in fracture mechanics, providing means for the assessment of the fracture resistance or the crack growth rate. Among several methods available for the computation of the crack‐tip stress intensity factor, an extremely efficient and flexible solution strategy is obtained using the distributed dislocation technique. This method has been successfully applied to several remote loading conditions and cracked geometries and can also take into account the effect of surface friction and roughness through the inclusion of an appropriate interface model. In this paper, we make use of a non‐linear algorithm, which uses dislocations distributed along the crack, to compute the crack‐tip stress intensity factors under remote mixed‐mode stress fields. It is shown that the algorithm can be effectively applied to different geometries, notably to cases where the mode mixity originates from geometrical features, such as in the presence of notches or re‐entrant corners. In particular, the effect of dilatancy is correctly captured with the adopted method, and its consequences are discussed with regard to the onset of unstable crack propagation under monotonic loading.

Mode II fracture toughness for non-planar frictional cracks

Traction-free and planar cracks represent a rather idealized picture of the physical reality, commonly used in fracture mechanics problems. In the present paper, the influence of roughness and friction of crack surfaces is examined in relation to both the resulting near-tip stress field and the fracture resistance under monotonic mixed-mode loading. A two-dimensional model is presented where an elastic-plastic-like constitutive interface law is adopted to describe the Mode I/II coupling between displacements and tractions along the crack surfaces. The solution is obtained using the Distributed Dislocation Technique (DDT). By considering a linear piecewise periodic profile of the crack, the present model is employed to quantify the mode II fracture toughness of different types of natural stones under varying mode I compressive load.

On the plastic zone sizes of cracks interacting with multiple inhomogeneous inclusions in an infinite space

The plastic zones of crack tips play a significant role in the fracture behavior of material. This paper proposes a semi-analytic solution for the plastic zones and stress distribution of an infinite space with multiple cracks and inhomogeneous inclusions under remote stress. In this solution, cracks can be treated as a distribution of edge dislocations with unknown densities according to the distributed dislocation technique, while inhomogeneous inclusions can be modeled as homogeneous inclusions with initial eigenstrain plus the unknown equivalent eigenstrain by using the equivalent inclusion method. These unknowns can be obtained by using the conjugate gradient method. The plastic zones ahead of crack tips are one-dimensional slender strips, and their sizes can be determined by canceling the stress intensity factor (SIF) due to the closure stress and that due to the applied load based on the Dugdale model of small-scale yielding. It is found that the plastic zones of crack tips are significantly affected by Young’s modulus and the positions of inhomogeneous inclusions.

Elastic-plastic behavior analysis of an arbitrarily oriented crack near an elliptical inhomogeneity with generalized Irwin correction

Abstract A theoretical model is developed to investigate the fracture behavior for an elliptical inhomogeneity embedded in an elastic-plastic matrix. It contains an arbitrarily oriented crack under a uniform stress field at infinity. In the model, the generalized Irwin approach is applied for plastic zone correction for crack-tip yielding. The distributed dislocation technique is utilized to formulate the present problem. The effective stress intensity factors , the plastic zone size and the crack tip opening displacement are evaluated by solving the formulated singular integral equations. The solution takes into account the plastic zone size at the crack tips and thus could provide accurate stress intensity factors and crack tip opening displacements for the assessment of crack growth. In the numerical examples, the case under a y -direction uniaxial tension is considered. The effects of the ellipse's aspect ratio, the shear moduli ratio and the normalized crack-inhomogeneity distance are investigated. The results show that the shapes of the inhomogeneity and the material combinations have significant influences. Crack propagation may be hindered if the inhomogeneity is close to circular or a hole and the crack inclination angle is close to 90°.

Effect of micro-cracks on plastic zone ahead of the macro-crack tip

From the macroscopic point of view, the plastic zone (PZ) is obtained based on the distributed dislocation technique (DDT) and von Mises yield criterion. From the microscopic point of view, PZ is determined by the DDT model. The effect of micro-cracks on PZ of the macro-crack tip is analyzed. The results show that the micro-crack has a little amplification effect on PZ of the macro-crack tip when it locates in front of PZ. As the micro-crack is close to the macro-crack tip, PZ of the macro-crack tip and the micro-crack tip will join together. When the micro-crack enters into PZ of the macro-crack tip, it has an obvious shielding effect on PZ. When the micro-crack is behind the macro-crack tip, the width of PZ decreases while the height increases. The dislocation distribution in PZ is in the form of inverse pileup. The amplification and shielding regions are divided into five strip-shaped regions, and they appear alternately. The results can provide useful information to predict plastic behaviors near crack tip. The analysis of amplification and shielding effect is important to materials design.

Influence of grain boundary sliding near a nanovoid on crack growth in deformed nanocrystalline materials

A theoretical model is developed that examines the effect of grain boundary (GB) sliding near a nanovoid on crack growth in deformed nanocrystalline materials. By combining complex variable method of Muskhelishvili, superposition principle of elasticity and distributed dislocation technique, the singular integral equations are solved numerically and then the stress intensity factors (SIFs) near the left crack tip are obtained. The influences of the location of the disclination dipole, the dipole arm, the orientation and the relative crack length on the SIFs near the left crack tip are evaluated in detail. The results indicate that the wedge disclination dipole produced by the GB sliding shields mode I crack tip, but anti-shields mode II crack tip. At the same time, surface stresses of the nanovoid characterized by positive surface elasticity and negative surface elasticity both show shielding effect to mode I SIFs of the crack tip, yet have negligible effect on mode II SIFs of the crack tip. Meanwhile, the mode I crack tip is more significantly shielded by the negative residual surface stresses than that of the positive residual surface stresses.

Determination of Mode III Stress Intensity Factors for Arbitrarily Oriented Straight and Curved Cracks in a Linearly Graded Layer

Stress intensity factors are calculated for the multiple arbitrarily oriented straight and curved cracks in an FGM layer based on the distributed dislocation technique. It is assumed that the shear modulus is a linear function along the thickness of the layer. By applying the Fourier complex transform, the problem reduces to a set of Cauchy singular integral equations with an unknown dislocation density function on the crack surface. The collocation scheme based on Chebyshev polynomials has been used to solve the Cauchy singular integral equation numerically. The results are tabulated and plotted to illustrate the effects of the material nonhomogeneity parameter, the thickness of the layer, the length of crack and crack configuration on the stress intensity factors.

Mixed mode cracks in annular planes of cylindrical orthotropy subjected to inplane loading

This paper is concerned with the mixed mode crack problem in an orthotropic annular plane. First, the Michell solution is generalized for material of cylindrical anisotropy. Next by using this solution, fundamental solutions of climb and glide edge dislocations are provided for an annular orthotropic plane of cylindrical orthotropy. Then, an analytical solution to stress field of the intact annular plane under normal and shear tractions on its outer boundaries is presented in this article. The distributed dislocation technique is employed to analyze multiple arbitrary oriented interacting cracks in an annular orthotropic plane. The ensuing integral equations are solved numerically to obtain the dislocation density on the surfaces of the cracks. The stress intensity factors are evaluated by using these dislocation densities. The approach of this study is applied to example problems and shown to be accurate for cases where given in the literature. Finally, the effects of crack geometry and location, material anisotropy, and interaction of cracks on the resulting stress intensity factors at the crack tips are studied.

Axisymmetric planar cracks in finite hollow cylinders of transversely isotropic material: part I—dislocation solution for infinite cylinders

This paper presents the solution for the crack tip stress intensity factors of a system of coaxial axisymmetric planar cracks in a transversely isotropic infinite hollow cylinder. The cylinder is under uniform axial tensile loading at infinity. First, the state of stress in an infinite hollow cylinder with transversely isotropic material containing axisymmetric prismatic and radial dislocations is studied. To this end, the solutions are represented in terms of biharmonic stress functions. Next, using the distributed dislocation technique, the mixed mode crack problem is reduced to Cauchy-type singular integral equations for dislocation densities on the surfaces of the cracks. The dislocation densities are employed to determine stress intensity factors for axisymmetric interacting cracks. For validation, the problems including one single crack in an infinite hollow cylinder with a circumferential edge crack or an annular crack are reexamined.

Axisymmetric planar cracks in finite hollow cylinders of transversely isotropic material: Part II—cutting method for finite cylinders

This paper is the outcome of a companion part I paper allocated to finite hollow cylinders of transversely isotropic material. The paper provides the solution for the crack tip stress intensity factors of a system of coaxial axisymmetric planar cracks in a transversely isotropic finite hollow cylinder. The lateral surfaces of the hollow cylinder are under two inner and outer self-equilibrating distributed shear loadings. First, the stress fields due to these loadings are given for both infinite and finite cylinders. In the next step, the state of stress in an infinite hollow cylinder with transversely isotropic material containing axisymmetric prismatic and radial dislocations is extracted from part I paper. Next, using the distributed dislocation technique, the mixed mode crack problem in finite cylinder is reduced to Cauchy-type singular integral equations for dislocation densities on the surfaces of the cracks. The problem of a cracked finite hollow cylinder is treated by cutting method; i.e., the infinite cylinder is cut to a finite one by slicing it using two annular axisymmetric cracks at its ends. The cutting method is validated by comparing the state of stress of a sliced intact infinite cylinder with that of an intact finite cylinder. The paper is furnished to several examples to study the effect of crack type and location in finite cylinders on the ensuing stress intensity factors of the cracks and the interaction between the cracks.

Interactions of multiple cracks in a transversely isotropic piezoelectric plane under mixed mode condition

In this paper, the interactions of multiple cracks in a piezoelectric material under the action of in-plane electromechanical loads are considered. The analytical solution of electric and Volterra climb and glide edge dislocations in an infinite piezoelectric plane are initially obtained by using the Fourier transform. The problem is then formulated by the distributed dislocation technique into a system of singular integral equations where the unknown variables are dislocation density functions on the surfaces of the cracks. These equations are solved numerically for dislocation density functions on the cracks which are used to determine field intensity factors. The results are highly affected by the interaction between multiple cracks and various loading conditions.

Mixed mode axisymmetric cracks in transversely isotropic infinite solid cylinders

• New analytical dislocation solution in a transversely isotropic cylinder is developed. • Material anisotropy has an efficient effect on stress intensity factors. • Interaction of cracks implies that need for mixed-mode analysis is essential. • Loading type and crack length have also a crucial effect on stress intensity factors. The present study examined mixed mode cracking in a transversely isotropic infinite cylinder. The solutions to axisymmetric Volterra climb and glide dislocations in an infinite circular cylinder of the transversely isotropic material are first obtained. The solutions are represented in terms of the biharmonic stress function. Next, the problem of a transversely isotropic infinite cylinder with a set of concentric axisymmetric penny-shaped, annular, and circumferential cracks is formulated using the distributed dislocation technique. Two types of loadings are considered: (i) the lateral cylinder is loaded by two self-equilibrating distributed shear stresses; (ii) the curved surface of the cylinder is under the action of a distributed normal stress. The resulting integral equations are solved by using a numerical scheme to compute the dislocation density on the borders of the cracks. The dislocation densities are employed to determine stress intensity factors for axisymmetric interacting cracks. Finally, a good amount of examples are solved to depict the effect of crack type and location on the stress intensity factors at crack tips and interaction between cracks. Numerical solutions for practical materials are presented and the effect of transverse isotropy on stress intensity factors is discussed.