An improved atomistic simulation based mixed-mode cohesive zone law considering non-planar crack growth

A novel and improved atomistic simulation based cohesive zone law characterizing interfacial debonding is developed which explicitly accounts for the non-planarity of the crack propagation. Group of atoms in the simulation constituting cohesive zones which are used to obtain local stress and crack opening displacement data are determined dynamically during the non-planar crack growth as they cannot be determined apriori. The methodology is used to study the debonding of Σ5 (2 1 0)/[0 0 1] symmetric tilt grain boundary interface in a Cu bicrystal under several mixed mode loading conditions. Simulations show that such bicrystalline specimen exhibits three types of energy dissipative mechanisms – shear coupled GB migration (SCM) away from the crack-tips, change in spacial orientation of GB structural units rendering highly disordered grain boundary near the crack tips and brittle intergranular fracture. Which combination of these three deformation mechanism will be active influencing the degree of non-planarity of the crack propagation at various stages of loading depends on the loading mode-mixity. As the ratio of shear component of the loading parallel to the GB plane and normal to the tilt axis with respect to the normal loading increases (thereby increasing the mode-mixity), overall strain-to-failure also increases and SCM tends to become the dominant deformation mechanism. Through this framework, analytical functional forms and parameters describing cohesive laws for both normal and shear traction as a function of the mode-mixity of the loading and crack opening displacement are predicted.     

A comparison of cohesive zone modeling and classical fracture mechanics based on near tip stress field

A mode III crack with a cohesive zone in a power-law hardening material is studied under small scale yielding conditions. The cohesive law follows a softening path with the peak traction at the start of separation process. The stress and strain fields in the plastic zone, and the cohesive traction and separation displacement in the cohesive zone are obtained. The results show that for a modest hardening material (with a hardening exponent N = 0.3), the stress distribution in a large portion of the plastic zone is significantly altered with the introduction of the cohesive zone if the peak cohesive traction is less than two times yield stress, which implies the disparity in terms of the fracture prediction between the classical approach of elastic–plastic fracture mechanics and the cohesive zone approach. The stress distributions with and without the cohesive zone converge when the peak cohesive traction becomes infinitely large. A qualitative study on the equivalency between the cohesive zone approach and the classical linear elastic fracture mechanics indicates that smaller cracks require a higher peak cohesive traction than that for longer cracks if similar fracture initiations are to be predicted by the two approaches.     

Asymptotic crack-tip fields for dynamic crack growth in compressive power-law hardening materials

The effect of material compressibility on the stress and strain fields for a mode-I crack propagating steadily in a power-law hardening material is investigated under plane strain conditions. The plastic deformation of materials is characterized by the J₂ flow theory within the framework of isotropic hardening and infinitesimal displacement gradient. The asymptotic solutions developed by the present authors [Zhu, X.K., Hwang K.C., 2002. Dynamic crack-tip field for tensile cracks propagating in power-law hardening materials. International Journal of Fracture 115, 323–342] for incompressible hardening materials are extended in this work to the compressible hardening materials. The results show that all stresses, strains, and particle velocities in the asymptotic fields are fully continuous and bounded without elastic unloading near the dynamic crack tip. The stress field contains two free parameters σ_eq0 and s₃₃₀ that cannot be determined in the asymptotic analysis, and can be determined from the full-field solutions. For the given values of σ_eq0 and s₃₃₀, all field quantities around the crack tip are determined through numerical integration, and then the effects of the hardening exponent n, the Poisson ratio ν, and the crack growth speed M on the asymptotic fields are studied. The approximate behaviors of the proposed solutions are discussed in the limit of ν → 0.5 or n → ∞.     

Mixed-mode I/II fields around a crack with a cohesive zone ahead of the crack tip

Considering a cohesive zone ahead of the crack tip, the linear elastic crack problem under Mode I, Mode II or mixed-mode conditions is formulated in an elliptic coordinate system, so that the cohesive surfaces are conveniently represented by straight line segments. It is shown that the displacement and stress fields around the crack tip and the cohesive zone, expressed in terms of elliptic coordinates, have a simple mathematical form, which does not contain a stress singularity at the crack tip due to the existence of the cohesive zone.     

Experimental quantification of the plastic blunting process for stage II fatigue crack growth in one-phase metallic materials

The plastic blunting process during stage II fatigue crack growth was studied in pure polycrystalline Ni to investigate effects of strain localization and inelastic behavior on the kinematics of crack advance. Correlations were obtained between strain fields ahead of a fatigue crack, crack advance per cycle and crack growth kinetics. Strain fields were quantified using a combination of in situ loading experiments, scanning electron microscopy and digital image correlation for 8 < ΔK < 20 MPa m^1/2 and a fixed load ratio of 0.1. Results indicate that strain localized along a dominant deformation band, which was usually crystallographic and carried mostly pure shear for large loads and was of mixed character for lower loads. Instances of double deformation bands were observed, with bands acting either in a simultaneous or alternating fashion. It was found that the area integral of the opening strain for values larger than a given threshold, an “integrated” strain, had a power-law relationship with ΔK, with the exponent approximately equal to the Paris exponent (m). Therefore, the crack growth rate was proportional to the integrated strain. An analysis based on this correlation and the presence of dominant shear bands indicated that the integrated strain is related to the accumulated displacement in the band. This, in turn, is proportional to the product of the cyclic plastic zone radius and the average shear strain ahead of the tip, which represents a basic length scale for plastic blunting. Assumptions on the load dependence of these quantities, based on their observed spatial variation, allowed estimating $m=2\left(1+\frac{1}{1+n^{\prime }}\right)$, where n′ is the cyclic hardening exponent (0 < n′ < 1). This gives 3 < m < 4, which accounts for about 50% of the observed values of m between 1.5 and 6 for a wide variety of metallic materials.     

Constant moving crack in a magnetoelectroelastic material under anti-plane shear loading

Analytical solutions for an anti-plane Griffith moving crack inside an infinite magnetoelectroelastic medium under the conditions of permeable crack faces are formulated using integral transform method. The far-field anti-plane mechanical shear and in-plane electrical and magnetic loadings are applied to the magnetoelectroelastic material. Expressions for stresses, electric displacements and magnetic inductions in the vicinity of the crack tip are derived. Field intensity factors for magnetoelectroelastic material are obtained. The stresses, electric displacements and magnetic inductions at the crack tip show inverse square root singularities. The moving speed of the crack have influence on the dynamic electric displacement intensity factor (DEDIF) and the dynamic magnetic induction intensity factor (DMIIF), while the dynamic stress intensity factor (DSIF) does not depend on the velocity of the moving crack. When the crack is moving at very lower or very higher speeds, the crack will propagate along its original plane; while in the range of M_c1 < M < M_c2, the propagation of the crack possibly brings about the branch phenomena in magnetoelectroelastic media.     

Periodic group crack problems in an infinite plate

This paper investigates periodic group crack problems in an infinite plate. The periodic group crack is composed of infinite groups with numbering from j = −∞, …, −2, −1, 0, 1, 2, …, to j = ∞, and the groups are placed periodically. The same loading condition and the same geometry are assumed for cracks in all groups. A singular integral equation is used to solve the problems. The singular integral equation is formulated on cracks of the 0th group (or the central group) with the collection of influences from the infinite groups. The influences of many neighboring groups to the central group are evaluated exactly. Meantime, the influences of many remote groups to the central group can be summed up into one term approximately. The stress intensity factors at crack tips can be evaluated from the solution of the singular integral equation. It is found from some sample problems that the obtained results are very accurate. Finally, several numerical examples are presented and interaction among the group cracks is addressed.     

A cohesive zone model for fatigue crack growth allowing for crack retardation

A damage-based cohesive model is developed for simulating crack growth due to fatigue loading. The cohesive model follows a linear damage-dependent traction–separation relation coupled with a damage evolution equation. The rate of damage evolution is characterized by three material parameters corresponding to common features of fatigue behavior captured by the model, namely, damage accumulation, crack retardation and stress threshold. Good agreement is obtained between finite element solutions using the model and fatigue test results for an aluminum alloy under different load ratios and for the overload effect on ductile 316 L steel.     

Analysis of deformation-induced crack tip toughening in ductile single crystals by nano-indentation

This paper reports on the experimental examination of the deformation characteristics near a crack tip in a cyclically work-hardened copper single crystal using a 2D surface scans with nano-indentation. The experimental methodology enables the characterization of the primary deformation field near a crack tip via the modulation of the imposed secondary deformation field by a nano-indentation. In a heavily deformed 4-point bend specimen, the measurements showed an existence of an asymptotic field around the crack tip at a distance of R ⩾ 2.5J/σ₀. The measurements also showed the qualitative details of toughness evolution within the high-gradient deformation field around the crack tip. The nature of dislocation distribution (i.e. statistically distributed vs. distributions necessitated by geometry) around the crack tip is quantified. The measurements indicate the dominance of the geometrically necessary dislocation within the finite deformation zone ahead of the tip up to a distance of R ≈ 3J/σ₀. Thereafter, it is confined in radial rays coinciding with the sector boundaries around the crack tip. These measurements elucidate the origin of the inhomogeneous hardening and the size dependent macroscopic response close to crack tip.     

Pressure-sensitive ductile layers – I. Modeling the growth of extensive damage

Polymeric adhesives sandwiched between two elastic substrates are commonly found in multi-layers and IC packages. The non-elastic deformation and flow stress of such adhesive joints are highly pressure-sensitive. In this work, we study the effects of pressure-sensitivity, α, and plastic dilatancy, β, on void growth and coalescence ahead of a crack in ductile adhesive joints. To this end, a single layer of discrete voids is placed ahead of the crack in a pressure-sensitive dilatant adhesive sandwiched between two elastic substrates. The adhesive joint is subjected to small-scale yielding conditions. Using an associated flow rule (α = β), we show that pressure-sensitivity not only intensifies damage levels but also increases its spatial extent several fold. The damage level as well as its spatial extent is found to be even greater when a non-associated flow rule (β < α) is deployed. A reduction in the damage process zone’s thickness further increases the voiding activity in the adhesive, thereby resulting in brittle-like failure. This work also examines the fracture toughness trends using a material failure criterion for crack growth.     

Equilibrium of a pressurized plastic fluid in a wellbore crack

This paper investigates equilibrium of a pressurized plastic fluid invading a tensile wellbore crack in a linear elastic, permeable rock. The crack is initially filled by pore fluid at ambient pressure, that is immiscibly displaced by the plastic fluid invading from the wellbore. The plastic fluid comes to rest to form a “plug” within the elastically deformed crack when the limit equilibrium between the shear stresses generated at the fracture walls and the pressure drop between the wellbore wall and the crack tip is reached. The model assumes that the leak-off of the plastic fluid into the rock is negligible, while the displaced pore fluid in the crack tip region is freely exchanged with the surrounding permeable rock to maintain the ambient pressure level. When the crack length ? is small or large compared to the wellbore radius R, the problem reduces to that of a pressurized edge or Griffith’s crack, respectively, subjected to a uniform far-field confining stress. In these two end-member cases, the normalized solution for the net pressure distribution, the plug length, and the stress intensity factor at the crack tip is obtained as a function of two numbers – the normalized net fluid pressure at the crack inlet and at the crack tip (partial plugs only) – that embody the solution’s dependence on the wellbore and the far field loading, the fluid yield strength, and the rock modulus. In the general case of an intermediate crack length (? ～ R), the normalized solution is a function of two additional parameters, the length-to-radius ratio and a normalized measure of the far field stress anisotropy, respectively, which accurate approximation is devised from an end-member solution using a rescaling argument. The equilibrium plug solutions are used to evaluate the breakdown pressure, the critical wellbore pressure at which the crack propagation condition is first met, and to analyze the stability of the ensuing crack propagation.     

A reexamination of plasticity-induced crack closure in fatigue crack propagation

The crack closure concept is often used to consider the R-ratio and overload effects on fatigue crack growth. The presumption is that when the crack is closed, the external load produces negligible fatigue damage in the cracked component. The current investigation provides a reassessment of the frequently used concept with an emphasis on the plasticity-induced crack closure. A center cracked specimen made of 1070 steel was investigated. The specimen was subjected to plane-stress mode I loading. An elastic–plastic stress analysis was conducted for the cracked specimens using the finite element method. By applying the commonly used one-node-per-cycle debonding scheme for the crack closure simulations, it was shown that the predicted crack opening load did not stabilize when the extended crack was less than four times of the plastic zone size. The predicted opening load was strongly influenced by the plasticity model used. When the elastic–perfectly plastic (EPP) stress–strain relationship was used together with the kinematic hardening plasticity theory, the predicted crack opening load was found to be critically dependent on the element size of the finite element mesh model. For R = 0, the predicted crack opening load was greatly reduced when the finite element size became very fine. The kinematic hardening rule with the bilinear (BL) stress–strain relationship predicted crack closure with less dependence on the element size. When a recently developed cyclic plasticity model was used, the element size effect on the predicted crack opening level was insignificant. While crack closure may occur, it was demonstrated that cyclic plasticity persisted in the material near the crack tip. The cyclic plasticity was reduced but not negligible when the crack was closed. The traditional approaches may have overestimated the effect of crack closure in fatigue crack growth predictions.     

Asymptotic Stress Field in a Degenerate Orthotropic Material Containing a Cohesive Zone ahead of a Crack Tip

Complex variable and eigenfunction expansion methods are used to derive elastic stress fields near the cohesive zone of a crack aligned with principal axes of a degenerate orthotropic material. Asymptotic field terms are obtained using coordinate rescaling and transformation techniques. The stress fields do not have stress singularity. Variations of stress components associated with the first-order or dominant stress terms are discussed in detail.      

Cracked infinite cylinder with two rigid inclusions under axisymmetric tension

This paper considers the problem of an axisymmetric infinite cylinder with a ring shaped crack at z = 0 and two ring-shaped rigid inclusions with negligible thickness at z = ±L. The cylinder is under the action of uniformly distributed axial tension applied at infinity and its lateral surface is free of traction. It is assumed that the material of the cylinder is linearly elastic and isotropic. Crack surfaces are free and the constant displacements are continuous along the rigid inclusions while the stresses have jumps. Formulation of the mixed boundary value problem under consideration is reduced to three singular integral equations in terms of the derivative of the crack surface displacement and the stress jumps on the rigid inclusions. These equations, together with the single-valuedness condition for the displacements around the crack and the equilibrium equations along the inclusions, are converted to a system of linear algebraic equations, which is solved numerically. Stress intensity factors are calculated and presented in graphical form.     

Energy release rates in rubber during dynamic crack propagation

The theoretical understanding of the fracture mechanics of rubber is not as well developed as for other engineering materials, such as metals. The present study is intended to further the understanding of the dissipative processes that take place in rubber in the vicinity of a propagating crack tip. This dissipation contributes significantly to the total fracture toughness of the rubber and is therefore of great interest from a fracture mechanics point of view. To study this, a computational framework for analysing high-speed crack growth in a biaxially stretched rubber under plane stress is therefore formulated. The main purpose is to investigate the energy release rates required for crack propagation under different modes of biaxial stretching. The results show, that inertia comes into play when the crack speed exceeds about 50 m/s. The total work of fracture by far exceeds the surface energy consumed at the very crack tip, and the difference must be attributed to dissipative damage processes in the vicinity of the crack tip. The size of this damage/dissipation zone is expected to be a few millimetres.     

The effect of incidence angle on the mean wake of surface-mounted finite-height square prisms

The mean wake of a surface-mounted finite-height square prism was studied experimentally in a low-speed wind tunnel to explore the combined effects of incidence angle (α) and aspect ratio (AR). Measurements of the mean wake velocity field were made with a seven-hole pressure probe for finite square prisms of AR = 9, 7, 5 and 3, at a Reynolds number of Re = 3.7 × 10⁴, for incidence angles from α = 0° to 45°. The relative thickness of the boundary layer on the ground plane, compared to the prism width, was δ/D = 1.5. As the incidence angle increases from α = 0° to 15°, the mean recirculation zone shortens and the mean wake shifts in the direction opposite to that of the mean lift force. The downwash is also deflected to this side of the wake and the mean streamwise vortex structures in the upper part of the wake become strongly asymmetric. The shortest mean recirculation zone, and the greatest asymmetry in the mean wake, is found at the critical incidence angle of α_critical ≈ 15°. As the incidence angle increases from α = 15° to 45°, the mean recirculation zone lengthens and the mean streamwise vortex structures regain their symmetry. These vortices also elongate in the wall-normal direction and become contiguous with the horseshoe vortex trailing arms. The mean wake of the prism of AR = 3 has some differences, such as an absence of induced streamwise vorticity near the ground plane, which support its classification as lying below the critical aspect ratio for the present flow conditions.