Energy Density Factor Research Articles

Fatigue crack growth of cable steel wires in a suspension bridge: Multiscaling and mesoscopic fracture mechanics

Intrinsically, fatigue failure problem is a typical multiscale problem because a fatigue failure process deals with the fatigue crack growth from microscale to macroscale that passes two different scales. Both the microscopic and macroscopic effects in geometry and material property would affect the fatigue behaviors of structural components. Classical continuum mechanics has inability to treat such a multiscale problem since it excludes the scale effect from the beginning by introducing the continuity and homogeneity assumptions which blot out the discontinuity and inhomogeneity of materials at the microscopic scale. The main obstacle here is the link between the microscopic and macroscopic scale. It has to divide a continuous fatigue process into two parts which are analyzed respectively by different approaches. The first is so called as the fatigue crack initiation period and the second as the fatigue crack propagation period. Now the problem can be solved by application of the mesoscopic fracture mechanics theories developed in the recent years which focus on the link between different scales such as nano-, micro- and macro-scale. On the physical background of the problem, a restraining stress zone that can describe the material damaging process from micro to macro is then introduced and a macro/micro dual scale edge crack model is thus established. The expression of the macro/micro dual scale strain energy density factor is obtained which serves as a governing quantity for the fatigue crack growth. A multiscaling formulation for the fatigue crack growth is systematically developed. This is a main contribution to the fundamental theories for fatigue problem in this work. There prevail three basic parameters μ ∗, σ ∗ and d ∗ in the proposed approach. They can take both the microscopic and macroscopic factors in geometry and material property into account. Note that μ ∗, σ ∗ and d ∗ stand respectively for the ratio of microscopic to macroscopic shear modulus, the ratio of restraining stress to applied stress and the ratio of microvoid size ahead of crack tip to the characteristic length of material microstructure. To illustrate the proposed multiscale approach, Hangzhou Jiangdong Bridge is selected to perform the numerical computations. The bridge locates at Hangzhou, the capital of Zhejiang Province of China. It is a self-anchored suspension bridge on the Qiantang River. The cables are made of 109 parallel steel wires in the diameter of 7 mm. Cable forces are calculated by finite element method in the service period with and without traffic load. Two parameters α and β are introduced to account for the additional tightening and loosening effects of cables in two different ways. The fatigue crack growth rate coefficient C 0 is determined from the fatigue experimental result. It can be concluded from numerical results that the size of initial microscopic defects is a dominant factor for the fatigue life of steel wires. In general, the tightening effect of cables would decrease the fatigue life while the loosening effect would impede the fatigue crack growth. However, the result can be reversed in some particular conditions. Moreover, the different evolution modes of three basic parameters μ ∗, σ ∗ and d ∗ actually have the different influences on the fatigue crack growth behavior of steel wires. Finally the methodology developed in this work can apply to all cracking-induced failure problems of polycrystal materials, not only fatigue, but also creep rupture and cracking under both static and dynamic load and so on.

Reliability assessment of a bi-material notch: Strain energy density factor approach

Joints of different materials have many applications in structural engineering and microelectronics. In the present contribution the joint is modelled as a bi-material notch. The singular stress field near the notch tip is investigated. Depending on the notch geometry and materials, the stress field can have one or two singularities. It is shown that to study the problem of a crack onset at the notch, both terms have to be taken into account. Criteria for the direction and for crack nucleation are formulated. The approach utilizes the knowledge of the strain energy density factor distribution in a bi-material notch vicinity.

Effect of an extrinsic curvature on a quark–hadron phase transition

The last phase transition predicted by the standard model of particle physics took place at the QCD scale T ∼ 200 MeV when the universe was about t ∼ 10−5 s old and the Hubble radius was around 10 km. In this paper, we consider the quark–hadron phase transition in the context of braneworld cosmology where our universe is a 3-brane embedded in an m-dimensional bulk and localization of matter on the brane is achieved by means of a confining potential. We study the behavior of the physical quantities relevant to the description of the early universe such as the energy density, temperature and scale factor, before, during and after the phase transition and investigate the effects of an extrinsic curvature on the cosmological phase transition. We show that the braneworld effects reduce the effective temperature of the quark–gluon plasma and of the hadronic fluid. Finally, we discuss the case where the universe evolved through a mixed phase with a small initial supercooling and monotonically growing hadronic bubbles.

Crack Behaviour in Laminar Ceramics with Strong Interfaces

The paper deals with crack propagation in ceramic laminates. Assumptions of linear elastic fracture mechanics and small scale yielding are considered. Crack behaviour in a ceramic laminate body under external loading is investigated. Strong residual stresses due to different coefficients of thermal expansion of individual material layers are taken into account in finite element calculations. The change of crack propagation direction at the material interface is estimated on the base of the strain energy density factor and maximum tangential stress criteria. The influence of thickness of laminate layers on crack propagation direction is estimated. The stepwise crack propagation through the Al2O3-ZrO2 ceramic laminate is numerically estimated. It can be concluded that good agreement between the estimated crack path and experimental data was found.

Estimation of apparent fracture toughness of ceramic laminates

Abstract The paper presented deals with the fracture behaviour of ceramic laminates. The residual stresses in individual layers of Al 2 O 3 /5vol.%t-ZrO 2 (ATZ) and Al 2 O 3 /30vol.%m-ZrO 2 (AMZ) are determined. Assumptions concerning linear elastic fracture mechanics and small scale yielding are considered. In this frame the procedure based on a generalization of Sih’s strain energy density factor to the case of a crack touching the interfaces between two dissimilar materials is used for determination of effective values of the stress intensity factor on material interfaces. An important increase of fracture toughness at the AMZ/ATZ interface was predicted in comparison to the fracture toughness of individual material components. Predicted values were compared with data available in the literature and mutual good agreement was found. The procedure suggested can be used for estimation of resistance to crack propagation through multilayered structures and its design. The procedure can contribute to enhancing the reliability and safety of structural ceramics or, more generally, of layered composites with strong interfaces.

Anomalies associated with energy release parameters for cracks in piezoelectric materials

For a central crack in a piezoelectric plate, the mode-I stress intensity factor ( K I ), electric displacement intensity factor ( K D ), energy release rates ( G, G M ) and energy density factor ( S) are obtained from the finite element results. For the impermeable crack, the numerical results of K I and K D are coupled; this error is contrary to the uncoupled analytical solutions. The error has little effect on the total energy release rate G and energy density factor S, but in some cases, large errors in the mechanical energy release rate G M are observed. G is global while SED is local. Also G is negative which defies physics where energy cannot be created while crack attempts to extend as implied by G. Computations should be made for the J-integral and also show that J becomes negative. What this shows is that the global fracture energy criterion is not suitable to address the local release of energy because it includes the overall energy which are irrelevant to fracture initiation being a local behavior. In addition, the case study shows that the energy density theory is the better fracture criterion for the piezoelectric material. According to the results of S, it retards the crack growth when the external electric field and piezoelectric poling are on opposite directions. This conclusion agrees with analytical and experimental evidence in the past references.

Cracking characteristics of mixed mode dislocations near a lip-like mode crack

This work is concerned with the cracking characteristics of mixed mode dislocations near a lip-like mode crack, stress intensity and strain energy density factor are obtained by using conformal mapping, singularity analysis and Cauchy integrals. Shielding effect generated by screw dislocation near a lip-like mode crack decreases with the increment of the distance between screw dislocation and crack tip. Larger distance between two faces of the crack leads to the shielding effect waning. The strain energy density factor of mode III decreases with the increment of the distance between dislocation and crack tip. Larger distance between two faces of lip-like mode crack also leads to the strain energy density factor waning and encourages crack initiation; the shielding effects generated by edge dislocation near the crack decrease with the increment of the distance between edge dislocation and crack tip.

Fracture analysis of rubber‐like materials using global and local approaches: Initiation and propagation direction of a crack

AbstractIn this work, fracture of elastomers is analyzed using global and local approaches, combining experiments, analytical developments, and finite element calculations. The J‐integral is chosen as a global parameter characterizing crack initiation in such materials. Particular attention is paid to single specimen methods for measuring the fracture surface energy. More precisely, models developed in the literature are summarized and, because of the lack of accuracy of these models, an original pertinent expression of J is proposed by analogy with linear elastic fracture mechanics frameworks. However, the J‐integral is not able to predict the kinetic or the propagation direction of a crack. Thus, some local parameters are examined: principal strains, principal stresses, and the strain energy density (SED) factor. Results revealed that all these parameters represent reasonable indicators of the crack propagation direction in elastomers. Moreover, unlike maximal principal stress and SED factor, maximal principal strain seems to govern the crack initiation in this kind of materials. POLYM. ENG. SCI., 2009. © 2009 Society of Plastics Engineers

Rate dependent critical strain energy density factor of Huanglong limestone

Critical strain energy density of rock can be defined as a fundamental parameter in rock fracture mechanics, an intrinsic material property related to resistance to crack initiation and propagation. By means of the three-point bending experiments, the critical strain energy density factor of Huanglong limestone was measured over a wide range of loading rates from 8.97 × 10 −4 MPam 1/2 s −1 to 1.545 MPam 1/2 s −1. According to the approximate relationship between static and dynamic critical strain energy density factor of Huanglong limestone, relationship between the growth velocity of crack and magnitude of load is obtained. The main conclusions are summarized as follows: (1) when the loading rate is higher than 0.0279 MPam 1/2 s −1, the critical strain energy density factor of rock increased markedly with increasing loading rate. However, when loading rate is lower than 0.0279 MPam 1/2 s −1, the critical strain energy density factor slightly increased with an increase in loading rate. It is found from experimental results that the critical strain energy density factor is linear proportional to the exponential expression of loading rate, (2) for Huanglong limestone, when the growth velocity of crack is lower than 100 m/s, value of the maximum load was nearly a constant. However, when the growth velocity of crack is higher than 1000 m/s, value of the maximum load dramatically increases with increasing the crack growth velocity, and (3) the critical SED of Huanglong limestone is higher as the loading rate is higher.

Transient response of the crack-tip field in a magnetoelectroelastic half-space with a functionally graded coating under impacts

This paper has twofold aims. One is to study the dynamic response of a magnetoelectroelastic half-space with functionally graded coating containing crack at the interface when subjected to sudden impacts. Two different loading positions, where the material and crack surfaces are loaded respectively, are considered. By using the integral transform method, the problem is reduced to solving singular integral equations. Obtained numerical results show that the overshoots of dynamic fracture parameters are strongly amplified or reduced depending on negative or positive gradient, respectively for the case of the material surface being loaded suddenly. This implies that a functionally graded coating with a positive gradient index is preferable in engineering design due to its capability of preventing the structure from cracking. The second objective is to give a comparison of relevant dynamic parameters such as the intensity factors of stress and strain, energy release rate, and energy density factor, and their features are elucidated under dynamic combined loadings. It indicates that the strain intensity factor can overcome the drawbacks of the rest parameters, and may be chosen as an effective fracture parameter, while three others cannot be adopted as fracture criteria to describe the feature of onset of crack growth.

Cracking characteristics of a moving screw dislocation near an interfacial crack in two dissimilar orthotropic media

Investigated is the problem of the interaction between a moving screw dislocation and an interfacial crack in two orthotropic materials. The closed form solutions of this problem is obtained by using Riemann-Schwarz’s symmetry principle integrated and the analysis singularity of complex functions. The stress intensity factors and the strain energy density factors have been investigated, the angle of crack initiation has been found. The results show that the shielding effect of dislocation for crack tips decreases with the increase of the distance between dislocation and crack tip and tends to zero at infinite, in additional, larger dislocation velocity leads to shielding effect waning. The normalized minimum SEDF increases with decrease of the velocity of dislocation and the distance between dislocation and crack tip and the ratio of shear modulus. The presented solutions contain previously known results as the special cases.

Anti-plane crack intersecting the interface in a bonded smart structure with graded magnetoelectroelastic properties

The mechanical model is established for an anti-plane crack intersecting the interface in a bonded smart structure with graded magnetoelectroelastic properties. The purpose of the present work is to put forth a new treatment method for the fracture analysis of the interface-intersecting crack. Such a crack is equivalently divided into two special edge cracks in the two bonded strips, respectively. The problem is then formulated by Fourier integral transform and reduced into a system of singular integral equations, which are numerically solved under not only the classical single-valuedness conditions but also the continuity conditions of the two groups of auxiliary functions, and the latter is the key technique first introduced in the current work. The present treatment method is compared with the existing one in the literature. The numerical results of the energy density factor are provided to verify the validity of the present treatment and reveal the influences of the physical and geometrical parameters.

Quark–hadron phase transitions in brane-world cosmologies

When the universe was about 10 μs old, a first order cosmological quark–hadron phase transition occurred at a critical temperature of around 200 MeV. In this work, we study the quark–hadron phase transition in the context of brane-world cosmologies, in which our Universe is a three-brane embedded in a five-dimensional bulk, and within an effective model of QCD. We analyze the evolution of the physical quantities, relevant for the physical description of the early universe, namely, the energy density, temperature and scale factor, before, during, and after the phase transition. To study the cosmological dynamics and evolution we use both analytical and numerical methods. In particular, due to the high energy density in the early Universe, we consider in detail the specific brane world model case of neglecting the terms linearly proportional to the energy density with respect to the quadratic terms. A small brane tension and a high value of the dark radiation term tend to decrease the effective temperature of the quark–gluon plasma and of the hadronic fluid, respectively, and to significantly accelerate the transition to a pure hadronic phase. By assuming that the phase transition may be described by an effective nucleation theory, we also consider the case where the Universe evolved through a mixed phase with a small initial supercooling and monotonically growing hadronic bubbles.

Mechanical/thermal stress intensification for mode I crack tip: Fracture initiation behavior of steel, aluminum and titanium alloys

A line crack in an infinite plate is subjected to the combined mechanical and thermal loading. The problem is solved in the framework of thermoelastic theory by integrating with the complex function method. An analytical solution is obtained under the boundary condition that the constant temperature is retained on the crack surfaces while the remote uniform heat flow is applied. In this case, the mode II thermal stress intensity factor vanishes. Thermal loading only induces the mode I stress intensity factor. The heat flow along the vertical direction to the crack line has no influence on the thermal stress intensity factor. The problem is a typical mode I crack problem. Three materials, say 38Cr2Mo2VA steel, LY12 aluminum and TC6 titanium alloys, are studied. The applied failure stresses are calculated under the combination of mechanical and thermal loading by using the strain energy density factor theory. The fracture behaviors of three materials are compared. Their fracture resistance is quite different. It is found that the thermal loading would enhance the crack propagation. If the direction of heat flow is changed to an opposite direction, the thermal effect would impede the crack growth. That is the thermal effect has the positive or negative effect which depends on the direction of heat flow.

CYCLIC PARAMETERS GOVERNING CRACK PROPAGATION BEHAVIOR UNDER IN-PHASE AND OUT-OF-PHASE TENSILE-TORSIONAL LOADING

Cyclic fatigue parameters governing fatigue crack propagation behavior under out-of-phase biaxial fatigue loading has been investigated considering the crack propagation direction and the growth rates. The fatigue crack was propagated by employing out-of-phase cyclic tensile-torsional loads with various phase differences to the thin-walled tubular specimen. Using two crack gages mounted near a notch, the crack propagation was measured and crack opening behaviors were monitored by unloading compliance method. The directions and growth rates of the fatigue crack propagated under respective biaxial loading conditions were examined using several cyclic fatigue parameters, including equivalent stress intensity factor range, ΔKeq, crack tip opening displacement range, ΔCTD, minimum strain energy density factor range, Δ S min . It is found that the parameters, ΔCTD and Δσθθ, max , might be useful for prediction of the crack directions and the fatigue crack growth rates were relatively well correlated with ΔKeq and Δ S min . On the other hand, the crack opening levels under in-phase loading could be relatively clearly determined, while it was ambiguous to define those under out-of-phase loading.

Zonal fracturing mechanism in deep crack-weakened rock masses

The mechanical behaviors of deep crack-weakened rock masses are different from those of shallow crack-weakened rock masses. The surrounding rock in shallow crack-weakened rock mass engineering is classified into loose zone, plastic zone and elastic zone, while the surrounding rock in deep crack-weakened rock mass engineering is classified into fractured zone and non-fractured zone, which occur alternatively. It is assumed that the deep rock masses contain one joint set, in which the probability density function describing the distribution of sizes is assumed to follow the Rayleigh distribution, and the probability density function describing the distribution of spacing is assumed to follow the Weibull distribution. On the basis of strength criterion of deep rock mass, the near-field stress redistribution around circular opening induced by excavation is determined. The strong interaction among cracks is investigated by using the dislocation model. The nucleation, growth, interaction and coalescence of cracks were analyzed based on the strain energy density factor theory. When cracks coalesce, failure of deep crack-weakened rock masses occurs, fractured zone is formed. Then, size and quantity of fractured zone and non-fractured zone are given out. The size and quantity of fractured zone increase with decreasing strength of rock mass. The size and quantity of fractured zone increase with increasing in situ stress. Zonal fracturing phenomenon occurs once value of in situ stress is larger than the unaxial compressive strength of rock masses. The size and quantity of fractured zone decrease with increasing λ when p 2 > p 1. The size and quantity of fractured zone increase with increasing λ when p 2 < p 1.

Stress Characteristics and Fatigue Crack Behaviour of the Longitudinal Rib-To-Cross Beam Joints in an Orthotropic Steel Deck

The welded joints between longitudinal rib and crossbeam of an orthotropic steel bridge deck are potentially critical to fatigue failure due to stress concentration at the end of the weld toe, caused by in-plane and out-of-plane deformations due to negative bending moments produced by live loads on the deck. The objective of this study is to evaluate the stress characteristics and fatigue crack behaviour of the longitudinal rib-to-crossbeam joints in an orthotropic steel bridge deck and to investigate the effectiveness of crack repair using a stop-hole. Static and fatigue test on full scale specimens were performed to observe the fatigue strength and crack propagation at the longitudinal rib-to-crossbeam joints. The measured stress distributions were compared with the F.E. analysis results. The test results show that fatigue cracks are initiated at the weld toe of the longitudinal rib-to-crossbeam joint with an inclined angle and rib-to-deckplate joint. Fatigue crack growth of the joint was analyzed by the mixed-mode analysis using the strain energy density factor. The stop-hole method was applied to delay fatigue crack growth of the longitudinal rib-to-crossbeam joint.

An anti-plane shear crack in bonded functionally graded piezoelectric materials under electromechanical loading

This paper is concerned with the problem of a Griffith crack in bonded functionally graded piezoelectric materials under the anti-plane shear loading. To make the analysis tractable, the properties of the functionally graded piezoelectric materials, such as elastic modulus, piezoelectric constant and dielectric constant, are assumed in exponential forms and vary along the crack direction. The crack surface condition is assumed to be electrically impermeable or permeable. Integral transform and dislocation density functions are employed to reduce the problem to the solution of a system of singular integral equations. The effects of the loading parameter λ, material constants and the geometry parameters on the stress intensity factor, the energy release ratio and the energy density factor are studied.

Crack energy density of a piezoelectric material under general electromechanical loading

Crack energy density is considered and used as a possible fracture parameter in piezoelectricity under arbitrary electromechanical remote loads. The closed-form solution of a crack in a piezoelectric infinite plate subjected to general static electromechanical loading is obtained through a method alternative to the more common Stroh’s formalism. This analytical method, which is based on the spectral theorem of linear algebra, involves a transformation of similarity induced by the fundamental matrix in order to express the equations governing the problem in terms of complex potentials. The application of the mechanical boundary condition of stress-free crack and of one of the three considered electric boundary conditions (impermeable, permeable or semipermeable) leads then to the formulation of a Hilbert problem whose solution yields the stress and displacement fields. The crack energy density factors for mixed mode are then calculated under different mechanical and electrical loadings, as well as under different electric boundary conditions. The non-singular terms of the stress expressions are retained as well. The definition of the minimum energy density fracture criterion, as proposed by Sih, is given, and the influence of load biaxiality and positive or negative applied electric field on the criterion results is analyzed. The prediction of the incipient branching angle as from the energy density approach is also compared to that arising from the maximum circumferential stress theory for a mixed mode loading condition. Numerical results and graphs are presented and discussed for a PZT-4 piezoelectric ceramic.

Multiple 3D flaws in coupled electro-magneto-thermo-elastic multiphase composites by extended hypersingular integral equation method

This work presents extended hypersingular integral equation (E-HIE) method to analyze the multiple 3D mixed-mode flaws problem in fully coupled electro-magneto-thermo-elastic multiphase composites under extended electro-magneto-thermo-elastic coupled loads through intricate theoretical analysis and numerical simulations. First, the problem is reduced to solving a set of E-HIEs. Analytical solutions for the extended singular stresses, the extended stress intensity factors (E-SIFs), the extended energy release rate and the extended strain energy density factors (E-SEDFs) near the flaws front are obtained. Then, the numerical method for the E-HIEs for two 3D flaws subjected to extended coupled loads is proposed. Finally, numerical solutions of E-SIFs and E-SEDFs of some examples are given, and the effect of flaws orientation, interaction and shielding is discussed.