Crack In Homogeneous Material Research Articles

Curved crack propagation in homogeneous and graded materials

ABSTRACTDeflection and deviation of cracks commonly occurs because of asymmetry in crack‐tip stresses in both homogeneous materials and functionally graded materials (FGMs); yet the analysis of curved cracks has been limited to simple crack shapes, otherwise the analysis would involve extensive levels of computation. The present study investigates the approximation of curved cracks with simplified shapes. A simple analytical model justifying the use of crack‐shape approximations, developed in an earlier study on stationary curved cracks in homogeneous materials, is outlined. Then, the approach is applied to propagating cracks in both homogeneous and graded material structures. Results are presented from finite element (FE) simulations of crack propagation using exact and simplified crack shapes. The use of an approximated crack shape can provide basic estimates for crack propagation path and critical load. However, systematic divergence can occur between predictions for exact and approximated crack shapes, particularly in inhomogeneous material configurations, and so the development of solutions for non‐straight cracks in FGMs would be expedient.

Analysis of Three-Dimensional Cracks in Inhomogeneous Materials Using Fuzzy Theory

This paper describes a fuzzy-based system for analyzing the stress intensity factors (SIFs) of three-dimensional (3D) cracks. 3D finite element method(FEM) was used to obtain the SIF for subsurface cracks and surface cracks existing in inhomogeneous materials. A geometry model, i.e. a solid containing one or several 3D cracks is defined. Several distributions of local node density are chosen, and then automatically superposed on one another over the geometry model by using the fuzzy theory. Nodes are generated by the bucketing method, and ten-noded quadratic tetrahedral solid elements are generated by the Delaunay triangulation techniques. The singular elements such that the mid-point nodes near crack front are shifted at the quarter-points, and these are automatically placed along the 3D crack front. The complete FE model is generated, and a stress analysis is performed. The SIFs are calculated using the displacement extrapolation method. The results were compared with those surface cracks in homogeneous materials. Also, this system is applied to analyze cladding effect of surface cracks in inhomogeneous materials.

Interaction integral procedures for 3-D curved cracks including surface tractions

This study examines a two-state interaction integral for the direct computation of mixed-mode stress intensity factors along curved cracks under remote mechanical loads and applied crack-face tractions. We investigate the accuracy of stress intensity factors computed along planar, curved cracks in homogeneous materials using a simplified interaction integral that omits terms to reflect specifically the effects of local crack-front curvature. We examine the significance of the crack-face traction term in the interaction integral, and demonstrate the benefit of a simple, exact numerical integration procedure to evaluate the integral for one class of three-dimensional elements. The work also discusses two approaches to compute auxiliary, interaction integral quantities along cracks discretized by linear and curved elements. Comparisons of numerical results with analytical solutions for stress intensity factors verify the accuracy of the proposed interaction integral procedures.

A Mother-Daughter Mechanism of Mode I cracks: Supersonic Crack Motion Along Interfaces of Dissimilar Materials

AbstractIn this paper we summarize recent progress in applying large-scale atomistic studies of crack dynamics along interfaces of dissimilar materials. We consider two linear-elastic material strips in which atoms interact with harmonic potentials, with a different spring constant in each layer leading to a soft and a stiff strip. The two strips are bound together with a weak potential whose bonds snap early upon a critical atomic separation. An initial crack serves as initiation point for the failure. We will focus on the maximum speed of mode I loaded cracks along such a biomaterial interface. Upon nucleation of the crack, it quickly approaches a velocity a few percent larger than the Rayleigh-wave speed of the soft material. After a critical time, we observe that a secondary crack is nucleated a few atomic spacings ahead of the crack. This secondary crack propagates at the Rayleigh-wave speed of the stiff material. If the elastic mismatch is sufficiently large, the secondary crack can be faster than the longitudinal wave speed of the soft material, thus propagating supersonically. Supersonic crack motion is clearly identified by two Mach cones in the soft material. This suggests that mother-daughter mechanisms, formerly only reported in mode II cracks in homogeneous materials, play an important role in interfacial mode I crack dynamics. The studies reported in this paper exemplify the usage of large-scale atomistic simulation to investigate the physics of fracture.

Investigation of the behavior of a griffith crack at the interface between two dissimilar orthotropic elastic half-planes for the opening crack mode

The behaviors of an interface crack between dissimilar orthotropic elastic half-planes subjected to uniform tension was reworked by use of the Schmidt method. By use of the Fourier transform, the problem can be solved with the help of two pairs of dual integral equations, of which the unknown variables are the jumps of the displacements across the crack surfaces. Numerical examples are provided for the stress intensity factors of the cracks. Contrary to the previous solution of the interface crack, it is found that the stress singularity of the present interface crack solution is of the same nature as that for the ordinary crack in homogeneous materials. When the materials from the two half planes are the same, an exact solution can be otained.

Investigation of the behavior of a Griffith crack at the interface of a layer bonded to a half plane using the Schmidt method for the opening crack mode

In this paper, the behavior of a Griffith crack at the interface of a layer boned to a half plane subjected to a uniform tension is investigated by use of the Schmidt method under the assumptions that the effect of the crack surface overlapping very near the crack tips is negligible and also there is a sufficiently large component of mode-I loading so that the crack essentially remains open. By use of the Fourier transform, the problem can be solved with the help of two pairs of dual integral equations in which the unknown variables are the jumps of the displacements across the crack surfaces. To solve the dual integral equations, the jumps of the displacements across the crack surfaces are expanded in a series of Jacobi polynomials. Numerical examples are provided to show the effects of the crack length, the thickness of the material layer and the materials constants upon the stress intensity factor of the crack. As a special case in our solution, we also give the solution of the ordinary crack in homogeneous materials. Contrary to the previous solution of the interface crack problem, it is found that the stress singularities of the present interface crack solution are similar with ones for the ordinary crack in homogeneous materials.

Shear loaded interface crack under the influence of friction: a finite difference solution

AbstractFormulation of the elastic two‐dimensional problem of contact with friction is presented. Two‐dimensional equilibrium equations and boundary conditions in an orthogonal curvilinear co‐ordinate system are written explicitly. The above formulation is solved with the aid of the finite difference technique. An iterative algorithm which does not require load increments is employed for solving interface fracture problems with contact and friction subjected to a monotonically increasing load. The J‐integral is extended for problems in which there is friction along the crack faces. Stress intensity factors are calculated by means of the J‐integral, as well as an asymptotic expansion of the tangential shift. Two problems are analysed: (1) a crack in homogeneous material in the presence of friction involving stationary contact; and (2) an interface crack in the presence of friction involving receding contact. Results are compared to those found by analytical and semi‐analytical methods which are presented in the literature, as well as to those obtained by means of the finite element method. The accuracy of the results establishes the reliability of the finite difference analysis, as well as the post‐processors. In addition, a problem involving stick conditions is considered. It is observed that with increasing friction, the normal gaps and tangential shifts decrease. The size of the contact zone increases and values of the stress intensity factor decrease. Copyright © 2004 John Wiley & Sons, Ltd.

Non-orthogonal stress modes for interfacial fracture based on local plastic dissipation

Interfacial cracks have several features which are different from those of cracks in homogeneous materials. Among those, the loading mode dependency of interfacial toughness has been a main obstacle to the widespread utilization of interfacial fracture mechanics. In this study, plasticity-induced toughening of an interface crack between an elastic–plastic material and an elastic material is studied. A useful relationship between the plastic dissipation and the plastic zone size is derived via an effective crack length model. Non-orthogonal stress modes for interface cracks are proposed on the basis of the plastic dissipation mechanism and a mixed-mode criterion for interfacial crack growth is also proposed using these stress modes. The non-orthogonal stress modes are able to represent the asymmetric behavior, mode-dependent toughening and ε-dependency of interfacial crack growth.

Component Separation Method of the Dynamic J Integral for Evaluating Mixed-Mode Stress Intensity Factors in Dynamic Interfacial Fracture Mechanics Problems.

First, the separated dynamic J integral and the separated dynamic energy release rate useful for dynamic interfacial fracture mechanics are presented. The path independent separated dynamic J integrals have the physical significance of energy flows into an interfacial crack tip from adjacent individual material sides. Second, new definitions of the stress intensity factors for dynamically propagating interface cracks are derived, which are consistent with the dynamic stress intensity factors for cracks in homogeneous materials. Based on this, an explicit relationship between the dynamic J integral and the stress intensity factors is also derived. Then, using this relation, to extract mixed-mode stress intensity factors in dynamic interfacial crack problems, the component separation method is developed. This method is more advantageous than the M1 integral method often used for interfacial crack problems, since the present method requires no auxiliary solution field that is sometimes not possible to construct. Finally, to demonstrate the applicability of the separated dynamic J integral and the component separation method, interfacial cracks subject to impact loading as well as dynamically propagating interfacial cracks are solved using a moving finite element method. Several useful points are elucidated from the numerical simulation results.

Investigation of anti-plane shear behavior of two collinear cracks in a piezoelectric materials strip by a new method

In this paper, the behavior of a Griffith crack at the interface of a layer boned to a half plane subjected to a uniform tension is investigated by use of the Schmidt method under the assumptions that the effect of the crack surface overlapping very near the crack tips is negligible and also there is a sufficiently large component of mode-I loading so that the crack essentially remains open. By use of the Fourier transform, the problem can be solved with the help of two pairs of dual integral equations in which the unknown variables are the jumps of the displacements across the crack surfaces. To solve the dual integral equations, the jumps of the displacements across the crack surfaces are expanded in a series of Jacobi polynomials. Numerical examples are provided to show the effects of the crack length, the thickness of the material layer and the materials constants upon the stress intensity factor of the crack. As a special case in our solution, we also give the solution of the ordinary crack in homogeneous materials. Contrary to the previous solution of the interface crack problem, it is found that the stress singularities of the present interface crack solution are similar with ones for the ordinary crack in homogeneous materials.

Analysis Of Interaction Between InterfaceCracks And Internal Cracks Using SingularIntegral Equations Of The Body Force Method

This paper deals with the interaction of interface cracks and arbitrarily arranged internal cracks in bonded dissimilar materials. The problem is analyzed using the singular integral equations based on the body force method. In the numerical analysis, the unknown functions of the body force densities are approximated by the products of the fundamental density functions and power series. Here, the fundamental density functions are the exact body force densities to express a single interface crack and a single internal crack. The stress intensity factors of the interface crack are systematically calculated for various crack dimensions, arrangements of the crack and elastic constants of the materials. The interaction effect of these problems is discussed by comparing the numerical results with the results of ordinary cracks in homogeneous materials.

A numerical analysis of time dependent stress fields ahead of stationary interface cracks at creep regime—Part I. Interface cracks in bimaterials

In this study, the behavior of stationary interface cracks at creep regime in plane-strain condition and pure crack opening dominated mode-I load state is investigated numerically. The bimaterials considered are elastic-creeping with power law and bimaterials having identical elastic properties but creeping at different rates. The results indicate that the amplitude of the crack tip singularity is dominated by the faster creeping sector. Neither the elastic properties (hence the initial elastic stress distribution) nor the creep properties of the slower creeping sector significantly affect the evolution of the C ∗ value for the interface cracks under identical applied loading condition. At steady-state, the angular distribution of the stresses ahead of the interface cracks in the faster creeping sector resembles the HRR stress singularity that occurs ahead of the cracks in homogeneous materials at creep regime. However, the observed C ∗ values for the interface cracks were about half of that seen for the crack in the homogeneous case under identical applied loading; and the occurrence of much larger radial, shear and hydrostatic stress components at significant distances ahead of the interface cracks was observed. On the other hand, the magnitudes of the normal stress component remained relatively the same for both crack types. Based on these observations, the possible growth rate of the interface cracks due to creep damage is also discussed.

Stress singularities at triple junctions with freely sliding grains

Stress singularities at grain triple junctions due to freely sliding grain boundaries in idealized two-dimensional single-phase polycrystals have been analyzed. The shear stress is assumed to be completely relaxed along the grain boundaries while the normal stress is allowed to be fully transmitted. The singularity exponent was found to be independent of the elastic constants of the grains, and for some triple junction configurations, super-singularities (i.e., stronger than the standard −0.5 as found at the tip of a crack in homogeneous material) were obtained. Interestingly, for most geometrical configurations, the solution structure indicates a fixed mode near the triple junction, irrespective of the far-field load combination.

Delamination Fracture Criteria for Composite Laminates

Due to the singularity nature of delamination, the fundamental concept of fracture mechanics was applied. However, the oscillatory characteristics near the delamination tip prohibited the use of conventional definition assuming the crack embedded into a homogeneous solid. A proper definition for the stress intensity factors and energy release rates of bimaterial interface cracks was introduced in this paper to study the fracture criterion for delamination. To measure the delamination fracture toughness, the test specimen usually used for the unidirectional composites was redesigned to suit for the cases that the delamination lies between two laminae with different fiber orientations. The test results show that the double cantilever beam (DCB) test is near to pure Mode I and the end-notched flexural (ENF) test is near to pure Mode II. Therefore, like the cracks in homogeneous materials, the test methods DCB and ENF are useful for the measurement of delamination fracture toughness GC and G11,. Based upon this conclusion, a series of cracked-lap shear (CLS) and modified end-notched flexural (MENF) tests and finite element simulation are conducted in this paper to establish the mixed-mode delamination fracture criterion. To validate the proposed fracture criterion, two commonly encountered problems are predicted and tested. One is the prediction of the most probable interface for the onset of delamination and its possible minimum load, in which no delamination was embedded prior to loading. The other is the prediction for the test with delamination existing prior to loading. The results show that the prediction by using this proposed criterion is well matched with the experiment.

A boundary element method for calculating the K field for cracks along a bimaterial interface

A multi-region boundary element method (BEM) based on the modified crack closure method (CCM) is developed to obtain the energy release rate G for cracks in homogeneous materials and along a bimaterial interface. The energy release rate obtained using the CCM are compared with that obtained using the crack opening displacement (COD) method. A combination of these methods allows us to determine the phase angle ψ and therefore the complex stress intensity factor K for crack problems. We access the accuracy of our BEM by comparing its results with known analytic solutions and previous FEM results in the literature. Computations are also carried out for the asymmetric double cantilever beam (ADCB) specimen, which has been used to determine fracture toughness of polymer/polymer and polymer/nonpolymer interfaces. An auxiliary KA field method to evaluate K is also discussed.

Three-dimensional effects in interfacial crack propagation

The paper describes the use of crack-opening interferometry for examining the variation in normal crack-opening displacements (NCOD) along the front of an interfacial crack in an edge-cracked bimaterial strip under biaxial loading. For the glass/epoxy combination considered here, the crack front was concave in the direction of crack growth, in contrast to previous observations with a glass/polyurethane/glass sandwich specimen and cracks in homogeneous materials. The NCOD were greatest in the interior of the specimen for all mode-mixes considered and the exponents in a power-law fit of NCOD versus distance from the crack front decreased towards the free surface. The exponents varied with mode-mix, suggesting that interfacial crack-front geometries could be similarly affected.

Cracks and dislocations in face-centered cubic metallic multilayers

In this paper, we have demonstrated that very perfect thin multilayers of the Cu/Ni system can be prepared with coherent interfaces if the layer modulation wavelength is in the 10 nm range. At modulation thicknesses above about 60 nm, the interfaces become incoherent. We have injected a crack into a coherent interface in the 10 nm case which has generated dislocations into the interface forming the crack plane, as well as into the layers adjacent to the crack plane. The dislocations injected into the crack plane presumably form misfit dislocations on that interface, and are grouped so close to the crack tip that individual dislocations are not completely imaged. The dislocations injected into the adjacent layers are distributed rather widely. We have analyzed the dislocation emission from a crack in the fcc geometry appropriate to the multilayers using a simplified elastic theory developed for cracks in homogeneous materials. The mixed mode loading which the misfit stresses are expected to produce lead one to expect these materials to be ductile and to have high toughness. Very high dislocation densities on the crack plane near the crack, however, may lead to a brittle mode of failure, which is beyond the purview of the elastic theory. The dislocations are observed to have strong interactions with alternating interfaces in the multilayers, and this effect could be due to elastic bunching of the dislocations at alternating interfaces caused by the misfit stress.

Mechanics of dynamic debonding

Singular fields around a crack running dynamically along the interface between two anisotropic substrates are examined. Emphasis is placed on extending an established frame work for interface fracture mechanics to include rapidly applied loads, fast crack propagation and strain rate dependent material response. For a crack running at non-uniform speed, the crack tip behaviour is governed by an instantaneous steady-state, two-dimensional singularity. This simplifies the problem, rendering the Stroh techniques applicable. In general, the singularity oscillates, similar to quasi-static cracks. The oscillation index is infinite when the crack runs at the Rayleigh wave speed of the more compliant material, suggesting a large contact zone may exist behind the crack tip at high speeds. In contrast to a crack in homogeneous materials, an interface crack has a finite energy factor at the lower Rayleigh wave speed. Singular fields are presented for isotropic bimaterials, so are the key quantities for orthotropic bimaterials. Implications on crack branching and substrate cracking are discussed. Dynamic stress intensity factors for anisotropic bimaterials are solved for several basic steady state configurations, including the Yoffe, Gol’dshtein and Dugdale problems. Under time-independent loading, the dynamic stress intensity factor can be factorized into its equilibrium counterpart and the universal functions of crack speed.

Stress Intensity Factor and Energy Release Rate for Interfacial Cracks Between Dissimilar Anisotropic Materials

The framework of fracture mechanics analysis for interface cracks as outlined by Willis (1971) is recast into a form resembling the customary framework for cracks in homogeneous materials. Based on the complex-valued “stress concentration vector” introduced by Willis, a new definition of real-valued stress intensity factors is introduced. The definition is an extension to that for cracks in homogeneous materials and reduces to the one given recently by Rice for isotropic interface cracks. In terms of the new stress intensity factors, tractions ahead of the crack and the relative crack face displacements given by Willis are rewritten into real-form expressions. The energy release rate obtained by Willis is also expressed into a real form in terms of the stress intensity factors. The validity of using the stress intensity factors as parameters controlling fracture is discussed along the line advanced by Rice (1988).

Boundary element analysis of stress intensity factors for an interface crack in dissimilar materials.

This paper describes the boundary element elastostatic analysis of a crack along the straight interface between two dissimilar materials. It is well known that the interface crack has an oscillation singularity at the crack tip which is quite different from a crack in homogeneous materials. Therefore, the definition of the stress intensity factor is not clarified yet and it is very difficult to analyze the interface crack by the numerical methods, such as FEM and BEM. We proposed a method to determine the stress intensity factors K1 and K2, for the interface crack by means of extrapolating the solutions at the points apart from the crack tip to avoid the oscillation singularity. This method can be widely applied to the BEM and also FEM analyses with any changes of the programmes. It is confirmed that the present results of the stress intensity factors for an interface crack in the infinite plate are completely consistent with the exact solutions. The stress intensity factors for various interface cracks in the finite plate and the adhesive joint are analyzed by the present method and BEM analyses, which is specially developed for this study.