The effect of tip plastic energy on mixed-mode crack initiation

An approach to the fracture initiation prediction of a ductile crack with mixed-mode loading (mode I and II) conditions is presented. The tip plastic energy around the crack tip is applied for evaluating the crack initiation load and the plastic zone shape. It is proposed that a mixed-mode crack will initiate as the tip plastic energy reaches a critical value. Numerical results for various loading conditions are illustrated. These results indicate that the predicted crack initiation loads correlate well with the experimental data available. This revised version was published online in August 2006 with corrections to the Cover Date.     

On the Plastic Zone Size and the Crack Tip Opening Displacement of an Interface Crack Between two Dissimilar Materials

In the paper, the elastic-plastic fracture behavior of an interface crack between two dissimilar materials is investigated. The mixed-mode Dugdale model is applied to examine the plastic zone size and the crack tip opening displacement. In numerical examples, the plastic zone size and the crack tip opening displacement of an interface crack under uniform loads are studied in detail. Two formulae are proposed to calculate the plastic zone size and the crack tip opening displacement of an interface crack under small scale yielding conditions.     

Prediction of crack growth direction under plane stress for mixed-mode I and II loading

The estimation of the plastic zone geometry ahead of a crack is fundamental to the evaluation of crack growth. Presented here is an analytical investigation for predicting crack growth direction for mixed-mode I and II loading under plane stress conditions. It is proposed that under complex loading the crack will extend in the direction where the radius of the plastic zone attains a minimum value. There is good agreement between the predicted results which are computed on the basis of this criterion and experimental data published in the literature.     

Experimental observation and modeling of the retardation of fatigue crack propagation under the combination of mixed-mode single overload and constant amplitude loads

It is well-known that one of the major characteristics of variable fatigue loads, especially overloads, is the retardation of the fatigue crack due to the complex interaction of many factors such as the overload ratio, the timing of overloads, the stress ratio, the yield stress of the material, the thickness of the structure, and the stress history. However, studies of the combined effect of mixed-mode I+II constant amplitude fatigue loadings and a mixed-mode I+II single overload on fatigue behavior are still scant. In this study, fatigue tests were conducted under mixed-mode I+II constant amplitude loadings with a mixed-mode I+II single overload, with reference to the variation of fatigue crack retardation. The formation of the overload plastic zone (OPZ) ahead of the crack tip under a mixed-mode I+II single overload is studied experimentally by the measurement of the shape and size of the OPZ. The behavior of fatigue crack propagation under mixed-mode loading conditions is examined by changing the loading mode of a single overload, and the relationship between the mixed-mode I+II single overload and the behavior of fatigue crack propagation in terms of the characteristics of the OPZ is evaluated. The empirical modeling of the fatigue life under mixed-mode I+II constant amplitude loadings is proposed by considering the characteristics of both the OPZ and the combination of the mode-mixity of mixed-mode I+II constant amplitude loadings and a mixed-mode I+II single overload.     

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.     

An interface crack in a coating-substrate composite with mixed-mode Dugdale corrections

Considering both plane stress and plane strain conditions, the plastic zone size and the crack tip opening displacement of an interface crack between a coating and a semi-infinite substrate under a normal load on the crack surfaces are investigated by the mixed-mode Dugdale model. In the model, stresses applied in the plastic zones satisfy the Von Mises yield criterion. The plastic zone size can be calculated by satisfying the condition that the complex stress intensity factors vanish. After the plastic zone size is solved, the crack tip opening displacement can be obtained by dislocation theories. In numerical examples, a uniform load is considered, and the effects of the normalized elastic modulus (the ratio of the elastic modulus of the coating to the elastic modulus of the substrate) and the normalized crack depth (the ratio of the coating thickness to the interface crack length) on the normalized plastic zone size and the normalized crack tip opening displacement are examined. Numerical examples show in the case of thin coatings, the value of the normalized plastic zone size decreases with increasing the normalized elastic modulus.     

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.     

On the full set of elastic T-stress terms of internal elliptical cracks under mixed-mode loading condition

Analytical expressions for the elastic constant stress terms of the asymptotic field, the so called T-stresses, for internal mixed-mode elliptical cracks in infinite homogeneous and isotropic elastic solids are addressed. To solve the problem the mixed-mode crack problem is divided into sub-problems using the superposition method, and each sub-problem is then solved for the asymptotic stress field. Considering the expansion of the local stress field at the crack front, the elastic T-stress terms are derived for each sub-problem. The results are superimposed to give the analytical expressions of the so far missing elastic T-stresses for mixed-mode elliptical cracks.The effect of the T-stresses on the size and shape of the plastic zone at the crack tip is discussed, and analytical results are compared to the ones from finite element analyses, both for the T-stress components and the size of the plastic zone. For an accurate prediction of the plastic zone all singular and constant terms (T-stresses) in the stress expansion formulae should be considered. It is observed that negative T-stresses increase the size of the plastic zone, while positive ones reduce it.     

A proposed mixed-mode fracture specimen for wood under creep loadings

A mixed-mode fracture specimen was designed in this paper. This geometry is a judicious compromise between a modified Double Cantilever Beam specimen and Compact Tension Shear specimens. The main objective is to propose a specimen which traduces a stable crack growth during creep loading taking into account viscoelastic behaviour under mixed-mode loadings. The numerical design is based on the instantaneous response traduced by a crack growth stability zone. This zone is characterized by a decrease of the instantaneous energy release rate versus the crack length. In order to obtain a mixed-mode separation, the paper deals with the use of the M-integral approach implemented in finite element software, according to energetic fracture criterions. In these considerations, a numerical geometric optimization is operated for different mixed-mode ratios. Finally, a common specimen which provides to obtain fracture parameters, viscoelastic properties and creep crack growth process for different mixed-mode configurations is proposed.     

On damage accumulations in the cyclic cohesive zone model for XFEM analysis of mixed-mode fatigue crack growth

Predicting mixed-mode fatigue crack propagation is an important and troublesome issue in structure assessment for decades. In the present paper an extended finite element method (XFEM) combined with a new cyclic cohesive zone model (CCZM) is introduced for simulating fatigue crack propagation under mixed-mode loading conditions, which has been implemented in the commercial general purpose software ABAQUS. The algorithm allows introducing a new crack surface at arbitrary locations and directions in a finite element mesh, without re-meshing. The cyclic cohesive zone model is based on the known S–N curves and Goodman diagram for metallic materials and validated by uniaxial tension results. Furthermore, the sensitivity of the model parameter is investigated for mixed-mode fatigue. The virtual crack closure technique has been extended to the cohesive zone model and proposed to calculate the energy release rate for the generalized Paris’ law. Finally, the crack propagation rate and direction under mixed-mode fatigue loading conditions are studied.     

Elastic and plastic stress analysis of an interface crack between two dissimilar layers

In the present paper, the mixed-mode Dugdale model is applied to investigate the plastic zone size and the crack tip opening displacement of an interface crack between two dissimilar layers. In the analysis, both normal and shear stresses are assumed to exist in the plastic zones and satisfy the Von Mises yield criterion. The plastic zone sizes can be determined on condition that the stress intensity factors caused by the stresses in the plastic zones and applied loading are zero. Then, the crack tip opening displacement can be obtained by dislocation theories. In numerical examples, the plane stress condition is considered. The plastic zone size and the crack tip opening displacement of an interface crack between two dissimilar layers under a uniform load are examined. The effects of Dundurs’ parameters and the thickness of materials on the plastic zone size and the crack tip opening displacement are investigated in detail. Numerical results show that in the case of small thickness, the values of the normalized plastic zone size and the normalized crack tip opening displacement decrease with increasing Dundurs’ parameters, α and β, while, in the case of infinite thickness, the value of the normalized plastic zone size is independent of α, and the value is symmetric about the axis on which β = 0.     

In situ TEM observations of fracture in nanolaminated metallic thin films

Fracture of single crystal nanolaminated thin films has been investigated through in situ straining of cross-sectional samples of Cu/Ni nanolaminates grown on Cu (001) single crystal substrates. The earlest stages of deformation exhibits a confined layer slip mechanism. With continued straining, unstable fracture occurs creating a mixed-mode crack that propagates across the nanolaminate, roughly perpendicular to the interfaces. Eventually, stable crack growth with intense plastic deformation ahead of the crack tip occurs over many bilayers in the direction of crack growth. Simultaneously, plasticity was seen to spread only 1 or 2 bilayer distances normal to the crack, creating an extremely localized plastic zone. Transmission electron microscopic (TEM) examination after the test did not reveal the presence of dislocations in the crack wake, except where severe crack deflection was observed. By comparison, the plastic zone size in the substrate was greater by several of orders of magnitude.     

Plastic dissipation in mixed-mode fatigue crack growth along plastically mismatched interfaces

A new theory of fatigue crack growth in ductile solids has recently been proposed based on the total plastic energy dissipation per cycle ahead of the crack. This and previous energy based approaches in the literature suggest that the total plastic dissipation per cycle can be closely correlated with fatigue crack growth rates under mode I loading. In a recent paper, the authors have extended the dissipated energy approach to the case of fatigue crack growth in a homogeneous material under sustained mixed-mode loading conditions. The goal of the current study is to further extend the approach to mixed-mode fatigue delamination of ductile interfaces in layered materials. Attention is restricted to material combinations with identical elastic properties, but with mismatches in plastic properties (both yield strength and hardening modulus) across the interface. Such systems can occur in brazing, soldering, welding, and a variety of layered manufacturing applications, where high-temperature material deposition can result in a mismatch in mechanical properties between the deposited material and the substrate. In this study, the total plastic dissipation per cycle is obtained through plane strain elastic–plastic finite element analysis of a stationary crack in a general layered specimen geometry under constant amplitude, mixed-mode loading. Numerical results for a dimensionless plastic dissipation per cycle are presented over the full range of relevant material combinations and mixed-mode loading conditions. Results suggest that while applied mode-mix ratio is the dominant parameter, mismatches in yield strength and hardening modulus can have a significant effect on the total plastic dissipation per cycle, which is dominated by the weaker/softer material.     

Effect of a graded layer on the plastic dissipation in mixed-mode fatigue crack growth along plastically mismatched interfaces

Prior work by the authors has proposed a dissipated energy theory of fatigue crack growth in ductile solids under mode I loading based on the total plastic dissipation per cycle ahead of the crack. The approach has since been extended to a general bimaterial interface geometry under mixed-mode I/II loading, with application to fatigue debonding of layered materials. An inherent assumption of this prior work is that a perfect crack exists along the interface between the two materials. The current work extends the approach to incorporate a grading of material properties between the two layers, as may occur in a variety of welding, soldering or layered manufacturing applications. Attention is restricted to elastic perfectly-plastic layers with identical elastic properties and a mismatch in yield strength across a linearly graded interface, with the crack on the boundary of the weaker material. A dimensionless plastic dissipation is extracted from 2-D plane strain finite element models over the full range of yield strength mismatches, graded layer thicknesses and mixed-mode loading conditions. Results reveal that for all modes of loading, the effect of a graded layer is to increase the total plastic dissipation per cycle, which is bounded by the extremes in plastic mismatch for a perfect crack interface. While the graded layer has a measurable effect, the plastic dissipation for all strength mismatches is dominated by the mode of loading.     

Elasto-plastic analysis of critical fracture stress and fatigue fracture prediction

The purpose of this investigation is to obtain the critical fracture stress, σ_f, in a cracked elasto-plastic plate subjected to mixed-mode loading. A new model estimating the magnitude of critical fracture stress based on the plastic zone during crack propagation is developed. Subsequently, this concept is applied to predict crack growth due to fatigue loads. Apart from the obvious and ideal benefits of being able to quantify crack propagation rates without the necessity to determine empirical coefficients and exponents experimentally, such an approach would lead to a better understanding of the factors which affect the crack growth rate.     

Evaluation of the Elastic-Plastic Mixity Parameters on the Base of Different Crack Propagation Criteria. Part 2. Solution and Results

We propose a new scheme of mixed-mode problem solution based on the deformation theory of plasticity with a power-law hardening stress-strain response and on application of elastic and plastic mixity parameters. Depending on the mixed-mode loading conditions and the initial crack line direction, this approach allows one to analyze a wide range of possible crack propagation paths controlled by shear and tensile mechanisms. The equilibrium equation with Airy function is used for a two-dimensional problem in the polar coordinate system. The Ramberg-Osgood model is applied to a material with power-law hardening behavior. Using the finite difference method we obtained a numerical solution of the mixed-mode loading problem with boundary conditions corresponding to two cases of crack propagation. Within the framework of the proposed approach we estimated the dependencies between mixity parameters and various loading parameters and crack inclination angle for a range of strain hardening exponent values, which dependencies closely fit the experimental data. __________ Translated from Problemy Prochnosti, No. 4, pp. 46 – 63, July – August, 2005.     

Applications of normal stress dominated cohesive zone models for mixed-mode crack simulation based on extended finite element methods

In conventional cohesive zone models the traction-separation law starts from zero load, so that the model cannot be applied to predict mixed-mode cracking. In the present work the cohesive zone model with a threshold is introduced and applied for simulating different mixed-mode cracks in combining with the extended finite element method. Computational results of cracked specimens show that the crack initiation and propagation under mixed-mode loading conditions can be characterized by the cohesive zone model for normal stress failure. The contribution of the shear stress is negligible. The maximum principal stress predicts crack direction accurately. Computations based on XFEM agree with known experiments very well. The shear stress becomes, however, important for uncracked specimens to catch the correct crack initiation angle. To study mixed-mode cracks one has to introduce a threshold into the cohesive law and to implement the new cohesive zone based on the fracture criterion. In monotonic loading cases it can be easily realized in the extended finite element formulation. For cyclic loading cases convergence of the inelastic computations can be critical.     

A generic approach to constitutive modelling of composite delamination under mixed-mode loading conditions

A generic approach to constitutive modelling of composite delamination under mixed mode loading conditions is developed. The proposed approach is thermodynamically consistent and takes into account two major dissipative mechanisms in composite delamination: debonding (creation of new surfaces) and plastic/frictional deformation (plastic deformation of resin and/or friction between crack surfaces). The coupling between these two mechanisms, experimentally observed at the macro scales through the stiffness reduction and permanent crack openings, is usually not considered in depth in many cohesive models in the literature. All model parameters are shown to be identifiable and measurable from experiments. The model prediction of mixed-mode delamination is in good agreement with benchmarked mixed-mode bending experiments. It is further shown that accounting for all major dissipative mechanisms in the modelling of delamination is the key to the accurate prediction of both resistance and damage of the interface.     

On elastic–plastic fracture behavior of a bi-layered composite plate with a sub-interface crack under mixed mode loading

A generalized Irwin model is proposed to investigate elastic–plastic fracture behavior of a bi-layered composite plate with a sub-interface crack under combined tension and shear loading. The dependence of the stress intensity factors, the plastic zone size, the effective stress intensity factor and the crack tip opening displacement on the crack depth h, the Dundurs’ parameters and the phase angle θ is discussed in detail. Numerical results show that in most cases, if the crack is embedded in a stiffer material, when the crack is close to the interface, the plastic zone size and the crack tip opening displacement will increase. On the contrary, if the crack is embedded in a softer material, when the crack is close to the interface, the plastic zone size and the crack tip opening displacement will decrease.