Wear damage of MgO single crystals

Mechanisms of surface and sub-surface wear damage in MgO single crystals were investigated by scratching with two sintered alumina sliders, having tip radii of 60 and 120m, using a simple scratching apparatus in a controlled atmosphere. The degree of surface and sub-surface cracking is dependent on the shape of the slider and the normal contact load, which are related to the penetration into the crystal. The chevron crack on the (001) plane in the [100] sliding direction consists of cracks intersecting at an angle of 90°, and with a spread angle of about 120°, and the normal crack. The nature of the sub-surface damage is investigated; a parallel crack develops in front of the slider and an oblique crack propagates towards the front of the slider. Then an internal normal crack is formed between the oblique crack and the parallel crack. In the [110] direction, the oblique crack initiates from the top of the normal crack under the surface, and the parallel crack continues from the oblique crack. This wear damage is explained by the dislocation interactions occurring due to the distribution of resolved shear stresses during sliding. The wear caused by the chevron crack is a factor of 10 higher than that with plastic flow. Internal cracks do not have a direct influence on the increase of wear.     

A tensile crack in creeping solids with large damage near the crack tip

An asymptotic analysis of stationary mode I crack in creeping solids with large damage near crack tip is conducted. To consider the damage effect, Kachanov damage evolution law is utilized and incorporated into the power-law creep constitutive equation. With the compatibility equation, a nonlinear eigenvalue problem which can be solved by numerical approaches is established. From this result, the distribution of stress and strain rate are obtained with the coupling effect of damage and creep under plane stress condition. Also the influence of material parameters on the stress is examined. According to the result, it is shown that the creep exponent n and damage parameter (=/(1+k)) have a significant effect on determining the eigenvalue s and angular distribution of stress and strain rate near the crack tip. The creep exponent n plays the role to soften and damage parameter plays the role to harden the material near the crack tip. The stress and strain rate show quite different behavior from those of HRR problem.     

Finite element computation of dynamic stress intensity factor for a rapidly propagating crack using Ĵ-integral

A method using the conventional finite elements and a finite domain energy integral, , was applied to determining the dynamic stress intensity factors for rapidly propagating cracks not only in linear elastic but also in viscoelastic bodies. In order to simulate crack propagation, a node release technique was employed in which the nodal force near the crack tip was gradually reduced to zero according to the prescribed scheme. Several schemes were tested and the most satisfactory result was found to be obtained by the linear relaxation scheme. The dynamic stress intensity factors were determined by evaluating the integral. Three seemingly different representations of -integral were used and each result for linear elastic problems was in good agreement with the analytical solutions and other available numerical results. The dynamic stress intensity factors for viscoelastic problems were also determined by using a proper representation of -integral.     

Stress, deformation and damage fields near the tip of a crack in a damaged nonlinear material

The stress, strain, displacement and damage fields near the tip of a crack in a power-law hardening material with continuous damage formation under antiplane longitudinal shear loading are investigated analytically. The interaction between a major crack and distributed microscopic damage is considered by describing the effect of damage in terms of a damage variable D. A deformation plasticity theory coupled with damage and a damage evolution law are formulated. A hodograph transformation is employed to determine the singularity and angular distribution of the crack-tip quantities. Consequently, analytical solutions for the antiplane shear crack-tip fields are obtained. Effects of the hardening exponent n and the damage exponent m on the crack-tip fields are discussed. It is found that the present crack-tip stress and strain solutions for damaged nonlinear material are similar to the well-known HRR fields for virgin materials. However, damage leads to a weaker singularity of stress, and to a stronger singularity of strain compared to that for virgin materials, respectively. The stress associated with damage always falls below the HRR field for virgin material; but the distribution of strain associated with damage lies slightly above the HRR field for r/(J/0) > 1.5 while the difference becomes negligible when r/(J/0) > 2. The limiting distributions of stress and strain may indeed be given by the HRR field.     

A Higher Order Synergistic Damage Model for Prediction of Stiffness Changes due to Ply Cracking in Composite Laminates

A non-linear damage model is developed for the prediction of stiffness degradation in composite laminates due to transverse matrix cracking. The model follows the framework of a recently developed synergistic damage mechanics (SDM) approach which combines the strengths of micro-damage mechanics and continuum damage mechanics (CDM) through the so-called constraint parameters. A common limitation of the current CDM and SDM models has been the tendency to over-predict stiffness changes at high crack densities due to linearity inherent in their stiffness-damage relationships. The present paper extends this SDM approach by including higher order damage terms in the characterization of ply cracking damage inside the material. Following the SDM procedure, predictions are aided by suitable micromechanical computations of crack opening displacements. A nonlinear SDM model is developed and applied for multiple classes of composite laminate layups. Stiffness predictions for damaged laminates using the developed model are compared with the experimental data for cross-ply ([0m/90n]s), angle-ply ([±θm/90n]s), off-axis ([0/±θ4/01/2]s) and quasi-isotropic ([0/90/±45]s) laminates. A comparison with current linear damage models showcases the usefulness of the proposed nonlinear SDM approach.     

Plastic boundary layers

Summary A new variational method recently proposed by Ho is extended to the general boundaryvalue problems of elasticity involving multiply-connected regions, including those of fracture mechanics. The method presents the approximate displacement functions in series form, the coefficients of which are obtained through minimizing the strain energy of the difference displacement field, resulting in the solution of a system of linear algebraic equations.Formulations are derived for all four types of boundary conditions (displacement, traction, mixed, dual). The method is applied to the anti-plane shear deformation of a finite elastic medium with a line crack. Surprisingly elegant exact solutions are obtained for two different problems. Further applications are included for infinite strips with parallel, perpendicular, as well as inclined cracks. Results show good convergence characteristics.Since the strain energy norm used in this method is actually proportional to the stress intensity factor, this method is indeed ideally suited for studies of Fracture Mechanics problems.With 2 Figures     

Numerical Modeling of Combined Matrix Cracking and Delamination in Composite Laminates Using Cohesive Elements

Sub-laminate damage in the form of matrix cracking and delamination was simulated by using interface cohesive elements in the finite element (FE) software ABAQUS. Interface cohesive elements were inserted parallel to the fiber orientation in the transverse ply with equal spacing (matrix cracking) and between the interfaces (delamination). Matrix cracking initiation in the cohesive elements was based on stress traction separation laws and propagated under mixed-mode loading. We expanded the work of Shi et al. (Appl. Compos. Mater. 21, 57–70 2014) to include delamination and simulated additional [45/?45/0/90]_s and [0₂/90_n]_s {n?=?1,2,3} CFRP laminates and a [0/90₃]_s GFRP laminate. Delamination damage was quantified numerically in terms of damage dissipative energy. We observed that transverse matrix cracks can propagate to the ply interface and initiate delamination. We also observed for [0/90_n/0] laminates that as the number of 90° ply increases past n?=?2, the crack density decreases. The predicted crack density evolution compared well with experimental results and the equivalent constraint model (ECM) theory. Empirical relationships were established between crack density and applied stress by linear curve fitting. The reduction of laminate elastic modulus due to cracking was also computed numerically and it is in accordance with reported experimental measurements.     

Crack layer analysis of fatigue crack propagation in ABS polymer

Differences in damage formation during fatigue crack propagation in acrylonitrile-butadiene-styrene polymer, between tests both fulfilling and not fulfilling linear elastic fracture mechanics requirements, were related to differences in crack propagation behaviour through the crack layer (CL theory. At both test conditions, damage consisted of crazing and shear yielding of the matrix, as well as elongation of rubbery domains. For a given crack length, the lower load level showed a higher intensity of craze damage. CL analysis showed that the process-dependent dissipation coefficient, , is inversely proportional to the lifetime. Further, despite drastic differences in the amounts of each damage species, both tests were estimated to have the same specific enthalpy of damage (^*=105 cal g^–1), a material constant that is a measure of the intrinsic resistance to damage formation at the crack tip.     

Dynamic steady-state crack propagation in a transversely isotropic viscoelastic body

The problem considered herein is the dynamic, subsonic, steady-state propagation of a semi-infinite, generalized plane strain crack in an infinite, transversely isotropic, linear viscoelastic body. The corresponding boundary value problem is considered initially for a general anisotropic, linear viscoelastic body and reduced via transform methods to a matrix Riemann–Hilbert problem. The general problem does not readily yield explicit closed form solutions, so attention is addressed to the special case of a transversely isotropic viscoelastic body whose principal axis of material symmetry is parallel to the crack edge. For this special case, the out-of-plane shear (Mode III), in-plane shear (Mode II) and in-plane opening (Mode I) modes uncouple. Explicit expressions are then constructed for all three Stress Intensity Factors (SIF). The analysis is valid for quite general forms for the relevant viscoelastic relaxation functions subject only to the thermodynamic restriction that work done in closed cycles be non-negative. As a special case, an analytical solution of the Mode I problem for a general isotropic linear viscoelastic material is obtained without the usual assumption of a constant Poissons ratio or exponential decay of the bulk and shear relaxation functions. The Mode I SIF is then calculated for a generalized standard linear solid with unequal mean relaxation times in bulk and shear leading to a non-constant Poissons ratio. Numerical simulations are performed for both point loading on the crack faces and for a uniform traction applied to a compact portion of the crack faces. In both cases, it is observed that the SIF can vanish for crack speeds well below the glassy Rayleigh wave speed. This phenomenon is not seen for Mode I cracks in elastic material or for Mode III cracks in viscoelastic material.     

Coupled behaviour of interacting dielectric cracks in piezoelectric materials

Existing studies indicate that the commonly used electrically impermeable and permeable crack models may be inadequate in evaluating the fracture behaviour of piezoelectric materials in some cases. In this paper, a dielectric crack model based on the real electric boundary condition is used to study the electromechanical behaviour of interacting cracks arbitrarily oriented in an infinite piezoelectric medium. The electric boundary condition along the crack surfaces is governed by the opening displacement of the cracks. The formulation of this nonlinear problem is based on modelling the cracks using distributed dislocations and solving the resulting nonlinear singular integral equations using Chebyshev polynomials. Numerical simulation is conducted to show the effect of crack orientation, crack interaction and electric boundary condition upon the fracture behaviour of cracked piezoelectric media.     

A mechanical model of fatigue crack propagation. Report 2

Conclusions The proposed model of fatigue crack propagation based on the solution of the cyclic elastoplastic problem of the stress-strain state [1] makes it possible to take into account the effect of the triaxial stress state on the deformation of the material at the crack tip. The proposed algorithm of calculations of the state of damage on the basis of the principle of linear damage summation and also the agreement between the calculated and experimental data confirm the assumption on the controlling role of low-cycle damage in the mechanics of crack propagation in cyclic loading described from phenomenological positions. The main advantages of the proposed model are:The possibilities of calculating endurance in crack propagation or calculating the crack propagation rate for cases in which the variation of the range of the stress intensity factor along the crack length in structural members takes place at a variable loading asymmetry;the possibilities of describing the effect of loading asymmetry on the fatigue crack-propagation rate using only the strain criterion (Coffin's equation) since the range of the plastic strain intensity at the crack tip is, as shown in [1], a function of not only the range of the stress intensity factor K but also of its maximum value K_max;the possibilities of describing the dependence of K_th on loading asymmetry based on the assumption on the constancy of the size of the structural element for the given material;the possibilities of describing the crack propagation rate in all the three sections of the dL/dN=f(K) diagram, starting with the values K similar to K_th and ending with the value of K at which monotonic quasistatic fracture becomes the controlling process.Translated from Problemy Prochnosti, No. 8, pp. 14–18, August, 1985.     

On the Construction of the Diagram of Growth Rates for a Fatigue Macrocrack

We consider some problems encountered in constructing the diagrams of growth rates for fatigue macrocracks. It is shown that, in the subthreshold part of the diagram and at the beginning of its Paris part, the minimum increments of crack length a specifying reliable values of the crack growth rate are determined by a linear (structural) parameter d* of the material. Therefore, it is necessary to fix increments of crack length according to the condition a d*.     

Fully-automatic modelling of cohesive crack growth using a finite element-scaled boundary finite element coupled method

This study develops a method coupling the finite element method (FEM) and the scaled boundary finite element method (SBFEM) for fully-automatic modelling of cohesive crack growth in quasi-brittle materials. The simple linear elastic fracture mechanics (LEFM)-based remeshing procedure developed previously is augmented by inserting nonlinear interface finite elements automatically. The constitutive law of these elements is modelled by the cohesive/fictitious crack model to simulate the fracture process zone, while the elastic bulk material is modelled by the SBFEM. The resultant nonlinear equation system is solved by a local arc-length controlled solver. The crack is assumed to grow when the mode-I stress intensity factor K_I vanishes in the direction determined by LEFM criteria. Other salient algorithms associated with the SBFEM, such as mapping state variables after remeshing and calculating K_I using a “shadow subdomain”, are also described. Two concrete beams subjected to mode-I and mixed-mode fracture respectively are modelled to validate the new method. The results show that this SBFEM-FEM coupled method is capable of fully-automatically predicting both satisfactory crack trajectories and accurate load-displacement relations with a small number of degrees of freedom, even for problems with strong snap-back. Parametric studies were carried out on the crack incremental length, the concrete tensile strength, and the mode-I and mode-II fracture energies. It is found that the K_I ? 0 criterion is objective with respect to the crack incremental length.     

A new approach to detect and size closed cracks by ultrasonics

The effect of a crack on time-of-flight of shear waves (4.5 MHz) polarized in perpendicular (t ) and parallel (t ) directions to the crack surface, propagating parallel to the direction of crack growth is investigated. The first and second back-wall echoes are used instead of the weak crack-tip echo for the measurement of time-of-flight. The measurement is made for fatigue cracks grown by different loading histories in ferritic steel (pressure vessel steel A533B-1) under the condition of no loading. The normalized time-of-flight (t –t )/t at the crack position is found to change proportionally as the ratio of crack depth to specimen width increases. The change is mainly due to the effect of plastic deformation occurring around the crack ont . It is shown that the depth of tightly closed fatigue crack in austenitic stainless steel (AISI 304) also can be evaluated under the condition of no loading by using this relationship.     

Critical shear strain localization in the ligament between two prolate spheroidal holes inside an incompressible solid under uniaxial tension

Summary The exterior interaction strain field in the ligament region between two parallel prolate spheroidal holes are determined from the multipole expansion of elastic displacement potentials. The problem is solved for an incompressible (rubber like) linear homogeneous ideal elastic solid with small strains. The remote uniaxial tension field is applied parallel to the major axes of the holes. The shear strain localization in the ligament is calculated for a number of prolateness ratios and in several relative parallel positions of the equatorial and meridian planes. At a certain critical geometrical position of the holes the shear strain localization at a point in the ligament becomes maximum, which further causes a high local normal strain on the void surface. These results may help to understand the mechanism of ductible fracture in the presence of voids.With 17 Figures     

A continuum damage mechanics model for crack initiation in mixed mode ductile fracture

This paper presents the development of a fracture criterion based on the postulation that the threshold condition of crack initiation in mixed mode ductile fracture is satisfied when the overall damage w in an element at the prospective direction of crack path reaches its critical value w_c. The validity of the proposed criterion is checked by predicting the fracture loads of thin aluminium plates containing an isolated crack inclined at the angle of =30, 45, 60 and 75 degrees and the predicted loads are compared satisfactorily with those determined experimentally. The analysis is performed based on the anisotropic model of continuum damage mechanics theory proposed earlier by the authors, thus providing additional proof of the consistency, applicability and versatility of the model. When the fracture loads of the mixed mode plates calculated using conventional fracture mechanics are compared with those determined using the proposed damage model, a maximum close to 30 percent over-estimation of the loads from the conventional approach is observed as opposed to within 7 percent discrepancy between the computed and measured fracture loads using the damage approach. The observation reveals the importance of including damage consideration in any ductile fracture analysis.The effect of varying damage coefficients on the fracture loads is examined and it is found that the crack initiation load decreases with the increase of anisotropic damage coefficient.     

Fatigue crack growth analysis of a premium rail steel

The fatigue crack growth behavior of a premium rail steel was studied using the Modified Crack Layer (MCL) theory. The rate of energy expended on damage formation and evolution within the active zone was evaluated from the hysteresis energy of unnotched and notched specimens. Due to head hardening of the rail, there is a vertical microstructure gradient inside the rail. In this work, the fatigue test specimens were sliced longitudinally from the head of a new rail near the web which represents the microstructure of the base material. The notch length to sample width ratio (a/w) was 0.1. Fatigue tests were performed on both unnotched and single edge notched (SEN) specimens under tension-tension load control condition at 5 Hz. The maximum fatigue stress was 200 MPa, which is about 40% of the yield strength of the material. The minimum to maximum stress ratio was 0.1. The crack length, number of cycles, and hysteresis loops were recorded during the tests from which the crack speed, the energy release rate, and the hysteresis energy for both notched and unnotched specimens were determined. The rate of energy dissipation on damage formation was evaluated based on the difference between the hysteresis energy for the notched and the unnotched specimens. These data were used in the MCL theory to extract the specific energy of damage, ; a material parameter characteristic of the fatigue crack growth resistance of the rail steel. It was found that the value of is 1300 kJ/m³. Three distinctive stages of crack growth kinetics were observed; crack initiation, stable crack growth and unstable crack growth. Microscopic examination of the active zone revealed damage species in the form of microcracks, inter-granular separation, and plastic deformed material. It is these damages that have led to the crack deceleration in the second stage. The fracture surface was also examined. The initiation region showed drawn-out lamellar pearlite. Ductile tearing and coarse ridges with intensive lamellar formation as well as microcracks were observed in the second region. The formation of these damage species has also contributed to the crack deceleration in the second stage of fatigue crack growth kinetics. The unstable crack growth region displayed cleavage facets initiated from the grain boundaries.     

Interface crack in periodically layered bimaterial composite

A directional crack growth prediction in a compressed homogenous elastic isotropic material under plane strain conditions is considered. The conditions at the parent crack tip are evaluated for a straight stationary crack. Remote load is a combined biaxial compressive normal stress and pure shear. Crack surfaces are assumed to be frictionless and to remain closed during the kink formation wherefore the mode I stress intensity factor K _I is vanishing. Hence the mode II stress intensity factor K _II remains as the single stress intensity variable for the kinked crack. An expression for the local mode II stress intensity factor k ₂ at the tip of a straight kink has been calculated numerically with an integral equation using the solution scheme proposed by Lo (1978) and refined by He and Hutchinson (1989). The confidence of the solution is strengthened by verifications with a boundary element method and by particular analytical solutions. The expression has been found as a function of the mode II stress intensity factor K _II of the parent crack, the direction and length of the kink, and the difference between the remote compressive normal stresses perpendicular to, and parallel with, the plane of the parent crack. Based on the expression, initial crack growth directions have been suggested. At a sufficiently high non-isotropic compressive normal stress, so that the crack remains closed, the crack is predicted to extend along a curved path that maximizes the mode II stress intensity factor k ₂. Only at an isotropic remote compressive normal stress the crack will continue straight ahead without change of the direction. Further, an analysis of the shape of the crack path has revealed that the propagation path is, according the model, required to be described by a function y=cx , where the exponent is equal to 3/2. In that case, when =3/2, predicts the analytical model a propagation path that is self-similar (i.e. the curvature c is independent of any length of a crack extension), and which can be described by a function of only the mode II stress intensity factor K _II at the parent crack tip and the difference between the remote compressive normal stress perpendicular to, and parallel with, the parent crack plane. Comparisons with curved shear cracks in brittle materials reported in literature provide limited support for the model discussed.     

Mechanical and statistical analyses of fracture of whisker-reinforced ceramic-matrix composites

This paper analyzes the fracture toughness of short-fiber reinforced ceramic-matrix composites (CMC). The effects of crack deflection and fiber pullout on matrix cracking are examined using a combination of mechanical and statistical models. First, the stress intensity factors of a deflected crack subjected to closure stress due to fiber pullout are analyzed based upon the mechanical model. Distributed dislocation method is used for the elastic analysis. Since the deflected crack is subjected to biaxial loading, a mixed mode fracture criterion in linear elastic fracture mechanics is applied to calculate the fracture toughness. Secondly, the number of pullout fibers on the fracture surface is treated as a random variable, and the statistical distribution of these fibers has been determined. The pullout force acting on a deflected crack is also obtained as a random variable by assuming a simple mechanism of fiber pullout. The probability of failure of CMC can thus be estimated from the strength characteristics of the fiber and matrix as well as the interface between these two.     

The dual boundary element method for thermoelastic crack problems

A boundary element formulation, which does not require domain discretization and allows a single region analysis, is presented for steady-state thermoelastic crack problems. The problems are solved by the dual boundary element method which uses displacement and temperature equations on one crack surface and traction and flux equations on the other crack surface. The domain integrals are transformed to boundary integrals using the Galerkin technique. Stress intensity factors are calculated using the path independent -integral. Several numerical problems are solved and the results are compared, where possible, with existing solutions.