Cohesive zone simulations of crack growth along a rough interface between two elastic-plastic solids

It has been known for many years that crack propagation along interfaces is influenced by interface topography or roughness profile. This has given rise to a small body of literature in which interface toughening with stochastic surface finishes, produced by grinding, rolling, or grit blasting, has been the primary focus. However, there is very little information currently available on the effect of patterned interfaces that are characterized by a minimal number of geometric parameters. In the present article, roughness-enhanced toughening of a cohesive interface between two identical materials is explored with a pure sinusoidal interface morphology that is characterized by its aspect ratio or ratio of amplitude to wavelength. Sixteen finite element meshes, each with a different aspect ratio, were constructed to study initiation and growth of a semi-infinite interface crack due to remote mode-I loading. The cohesive interface was modeled with a viscosity-modified Xu-Needleman cohesive zone law, and the solids were characterized with continuum elastic and elastic-plastic constitutive models. Predicted relationships between the aspect ratio and the macroscopic toughness point to key differences in the material models. A set of critical parameters which include the aspect ratio, material and cohesive properties is predicted such that catastrophic crack growth is inhibited due to crack blunting. A clear boundary between brittle and ductile fracture behavior is thus identified. The results suggest some guidelines for practical design of failure resistant interfaces through appropriate choice of geometric, material, and cohesive parameters.     

Cohesive zone modeling of interface fracture in elastic bi-materials

Some basic issues regarding the cohesive zone modeling of interface fracture between two dissimilar elastic materials are studied. The dependence of the cohesive energy density on the phase angle is first discussed under small scale cohesive zone conditions. It is then shown that in general the stress singularities in tension and shear cannot be simultaneously removed at the cohesive zone tip if a single cohesive zone length is adopted for both tensile and shear fracture modes. Finally, the energy dissipation at the tip of a prescribed cohesive zone is examined using a bilinear cohesive zone model under the uncoupled tension/shear conditions.     

Limitations of elastic definitions in Al-Si-Mg cast alloys with enhanced plasticity: linear elastic fracture mechanics versus elastic-plastic fracture mechanics

Linear elastic fracture mechanics describes the fracture behavior of materials and components that respond elastically under loading. This approach is valuable and accurate for the continuum analysis of crack growth in brittle and high strength materials; however it introduces increasing inaccuracies for low-strength/high-ductility alloys (particularly low-carbon steels and light metal alloys). In the case of ductile alloys, different degrees of plastic deformation precede and accompany crack initiation and propagation, and a non-linear ductile fracture mechanics approach better characterizes the fatigue and fracture behavior under elastic-plastic conditions.To delineate plasticity effects in upper Region II and Region III of crack growth an analysis comparing linear elastic stress intensity factor ranges (ΔK_el) with crack tip plasticity adjusted linear elastic stress intensity factor ranges (ΔK_pl) is presented. To compute plasticity corrected stress intensity factor ranges (ΔK_pl), a new relationship for plastic zone size determination was developed taking into account effects of plane-strain and plane-stress conditions (“combo plastic zone”). In addition, for the upper part of the fatigue crack growth curve, elastic-plastic (cyclic J based) stress intensity factor ranges (ΔK_J) were computed from load-displacement records and compared to plasticity corrected stress intensity factor ranges (ΔK_pl). A new cyclic J analysis was designed to compute elastic-plastic stress intensity factor ranges (ΔK_J) by determining cumulative plastic damage from load-displacement records captured in load-control (K-control) fatigue crack growth tests. The cyclic J analysis provides the true fatigue crack growth behavior of the material. A methodology to evaluate the lower and upper bound fracture toughness of the material (J_IC and J_max) directly from fatigue crack growth test data (ΔK_FT(JIC) and ΔK_FT(Jmax)) was developed and validated using static fracture toughness test results. The value of ΔK_FT(JIC) (and implicitly J_IC) is determined by comparing the plasticity corrected elastic fatigue crack growth curve with the elastic-plastic fatigue crack growth curve. A most relevant finding is that plasticity adjusted linear elastic stress intensity factor ranges (ΔK_pl) are in remarkably good agreement with cyclic J analysis results (ΔK_J), and provide accurate plasticity corrections up to a ΔK corresponding to J_IC (i.e. ΔK_FT(JIC)). Towards the end of the fatigue crack growth test (above ΔK_FT(JIC)) when plasticity is accompanied by significant tearing, the cyclic J analysis provides a more accurate way to capture the true behavior of the material and determine ΔK_FT(Jmax). A procedure to decouple and partition plasticity and tearing effects on crack growth rates is given.Three cast Al-Si-Mg alloys with different levels of ductility, provided by different Si contents and heat treatments (T61 and T4) are evaluated, and the effects of crack tip plasticity on fatigue crack growth are assessed. Fatigue crack growth tests were conducted at a constant stress ratio, R = 0.1, using compact tension specimens.     

A ? method-based numerical simulation of crack growth in linear elastic fracture

This paper presents a method for the automatic simulation of quasi-static crack growth in 2D linear elastic bodies with existing cracks. A finite element algorithm, based on the so-called ? method, provides the load vs. crack extension curves in the case of rectilinear crack propagation. Since the approach is both theoretically general and simple to be performed from a computational point of view, it could be extended for describing the phenomenon of crack growth in different fracture mechanics contexts.     

Effect of residual stresses on interface crack growth by void expansion mechanism

Crack growth along an interface between two adjacent elastic–plastic materials in a layered solid is analysed, using special interface elements to represent the fracture process ahead of the crack-tip. These interface elements account for ductile failure by the nucleation and growth of voids to coalescence. In these elements the stress components normal to the interface and the shear stresses are given by equilibrium with the surrounding material, and the stress component tangential to the interface is determined by the requirement of compatibility with the surrounding material in the tangential direction. It is assumed that the layers are sufficiently thick, so that the plastic regions around the crack-tip are much smaller than the thickness of the nearest layers. The analyses focus on the effect of initial residual stresses in the layered material, or on T-stress components induced during loading. The results show that the value of the T-stress component in the softer material adjacent to the interface crack plays the dominant role, such that a negative value of this stress component gives a significant increase of the interface fracture toughness.     

Comparison of cohesive zone parameters and crack tip stress states between two different specimen types

The cohesive zone parameters (separation energy and cohesive strength) and the crack tip triaxialities are compared between a compact tension (CT) and a double edge notched tension (DENT) specimen with smooth side-surfaces. The material is a pressure vessel steel 20MnMoNi55. The cohesive zone parameters are determined by fitting the simulated crack extensions near the midsection to the experimental data. The purpose of the study is to understand the relationship between the cohesive zone parameters and the crack tip stress triaxiality. The results show that for the same cohesive zone parameters the crack tip triaxiality near the midsection is lower in DENT specimens than in CT specimens. When the separation energy is set constant for CT and DENT specimens, the cohesive strength for the DENT specimens should be significantly lower than that for the CT specimens in order to make the simulated crack extensions near the midsection fit to the experimental data. Near the midsection, the cohesive strength and crack tip triaxiality influence each other: the specimen with a higher stress triaxiality has a higher cohesive strength; an increase of cohesive strength results in an increase of the crack tip triaxiality.     

High-speed self-similar debonding of an interface between two orthotropic materials: The anti-plane shear case

The anti-plane problem of the transient debonding of an interface between two orthotropic materials is examined. The material principal axes are allowed to be oblique to the interface. The debonding is modeled as an interface crack propagating self-similarly from zero-length. The extending speed is assumed to be subsonic, transonic or supersonic. We first consider the dynamic debonding under the moving concentrated loading. The moving dislocation model of self-similar propagation of an interface crack is used to formulate the problem in a singular integral equation which is solved analytically. The stress singularity at the crack tips is discussed. The order of singularity is found to be one-half for subsonic debonding and to vary between zero and one-half depending on the crack speeds for transonic debonding. The dynamic stress intensity factors/coefficients for these two situations are presented in closed-form. The paper also concludes that supersonic debonding is impossible unless the loads are directly applied to the crack tips. Finally, the results for dynamic debonding under xⁿ-type loads are presented by using the superposition method. This revised version was published online in August 2006 with corrections to the Cover Date.     

Comparison of cohesive zone model and linear elastic fracture mechanics for a mode I crack near a compliant/stiff interface

Cohesive zone model has been widely applied to simulate crack growth along interfaces, but its application to crack growth perpendicularly across the interface is rare. In this paper, the cohesive zone model is applied to a crack perpendicularly approaching a compliant/stiff interface in a layered material model. One aim is to understand the differences between the cohesive zone model and linear elastic fracture mechanics in simulating mode I crack growth near a compliant/stiff interface. Another aim is to understand the effects of elastic modulus mismatch and cohesive strength of the stiff layer on the crack behavior near the interface. To simulate crack growth approaching an interface, the cohesive zone model which incorporates both the energy criterion and the strength criterion is an effective method.     

Cohesive bonding interface model for the effective properties of randomly structured composites

The present paper aims at quantifying the effective behaviour of random composites at the microstructure scale using different combinations of intrinsic phase and interface properties. The composite microstructure is generated using Monte Carlo simulation. The methodology allows random spatial distribution of phases. It aims also at providing different evolutions of the interface quantity as function of phase ratio. Microstructures are then converted into finite element model. The FE model handles the interface effect using a cohesive zone model. This model represents the interface separation and traction using three parameters derived from a surface potential function. The sensitivity of the effective elastic properties to both interface and phase properties is then quantified. The predicted results show a strong non-linear dependence of the effective properties on the interface effect.     

On the estimation of the first interpenetration point in the open model of interface cracks

A general expression for estimating the location of the first interpenetration point in the open model of interface cracks that, for a fixed reference system, can be directly applied to all material combinations is presented.     

Probabilistic crack growth analyses using a boundary element model: Applications in linear elastic fracture and fatigue problems

This paper addresses the numerical solution of random crack propagation problems using the coupling boundary element method (BEM) and reliability algorithms. Crack propagation phenomenon is efficiently modelled using BEM, due to its mesh reduction features. The BEM model is based on the dual BEM formulation, in which singular and hyper-singular integral equations are adopted to construct the system of algebraic equations. Two reliability algorithms are coupled with BEM model. The first is the well known response surface method, in which local, adaptive polynomial approximations of the mechanical response are constructed in search of the design point. Different experiment designs and adaptive schemes are considered. The alternative approach direct coupling, in which the limit state function remains implicit and its gradients are calculated directly from the numerical mechanical response, is also considered. The performance of both coupling methods is compared in application to some crack propagation problems. The investigation shows that direct coupling scheme converged for all problems studied, irrespective of the problem nonlinearity. The computational cost of direct coupling has shown to be a fraction of the cost of response surface solutions, regardless of experiment design or adaptive scheme considered.     

Influence of imperfect interface on the elastic moduli of syntactic foams

Syntactic foams are manufactured by dispersing microspheres in a polymeric matrix, and the macroscale material properties of these foams are estimated by analyzing a periodic distribution of the inclusions. The analysis in the simplest form, further assume that the inclusions are perfectly bonded to the matrix material. It has been shown in a previous study [P.R. Marur, Mater. Lett. 59 (2005) 1954–1957.] that analytical model overestimated the experimentally determined elastic moduli, and that the morphology of particle distribution has negligible influence on the elastic moduli. In this paper, the assumption of perfect adhesion between the inclusion and the matrix is relaxed to allow for possible localized slip and separation at the particle interface. The analytical results obtained considering imperfect interface well agree with the measured elastic modulus reported in the literature.     

Determination of the kink point in the bilinear softening model for concrete

The characterization of the softening curve from experimental results is essential for predicting the fracture behavior of quasi-brittle materials like concrete. Among various shapes (e.g. linear, exponential) to describe the softening behavior of concrete, the bilinear softening relationship has been extensively used and is the model of choice in this work. Currently, there is no consensus about the location of the kink point in the bilinear softening curve. In this study, the location of the kink point is proposed to be the stress at the critical crack tip opening displacement. Experimentally, the fracture parameters required to describe the bilinear softening curve can be determined with the “two-parameter fracture model” and the total work of fracture method based on a single concrete fracture test. The proposed location of the kink point compares well with the range of kink point locations reported in the literature, and is verified by plotting stress profiles along the expected fracture line obtained from numerical simulations with the cohesive zone model. Finally, prediction of experimental load versus crack mouth opening displacement curves validate the proposed location of the kink point for different concrete mixtures and also for geometrically similar specimens with the same concrete mixture. The experiments were performed on three-point bending specimens with concrete mixtures containing virgin coarse aggregate, recycled concrete coarse aggregate (RCA), and a 50-50 blend of RCA and virgin coarse aggregate. The verification and validation studies support the hypothesis of the kink point occurring at the critical crack tip opening displacement.     

Investigation of crack propagation scatter in a gear tooth’s root

This paper describes the problem of determining crack initiation location and its influence on crack propagation in a gear tooth’s root. Three different load positions on the gear tooth’s flank were considered for this investigation of crack initiation and propagation. A special test device was used for the single tooth test. It can be concluded from the measurements that a crack can be initiated at very different locations in a tooth’s root and then propagate along its own paths. A numerical investigation into a crack initiation’s position and its influences on its propagation were carried out within the framework of linear fracture mechanics. The influence of a tooth’s load position, the geometry of the tooth’s root, and the influence of non-parallel load distribution on the tooth’s flank were considered when investigating the crack initiation’s position. Results show that linear fracture mechanics can be used for determining crack propagation, if better initial conditions for crack initiation are considered.     

Cohesive zone criterion for cracking along the Cu/Si interface in nanoscale components

Crack initiation and propagation along the Cu/Si interface in multilayered films (Si/Cu/SiN) with different thicknesses of the Cu layer (20 and 200 nm) are experimentally investigated using a nano-cantilever and millimeter-sized four-point bending specimens. To examine the cohesive zone model (CZM) criterion for interfacial delamination along the Cu/Si interface in nanoscale stress concentration, an exponential type of CZM is utilized to simulate the observed delamination processes using the finite element method. After the CZM parameters for the Cu/Si interface are calibrated by experiment, interface cracking in other experiments is predicted. This indicates that the CZM criterion is universally applicable for describing cracking along the interface regardless of specimen dimensions and film thickness which include the differences in plastic behavior and residual stress. The CZM criterion can also predict interfacial cracking along Cu/Si interfaces with different stress singularities.     

Axisymmetric problem for a spherical crack on the interface of elastic media

A problem concerning a spherical interfacial crack is solved by the eigenfunction method. The problem is reduced to a coupled system of dual-series equations in terms of Legendre functions and then to a system of singular integral equations for two unknown functions. The behaviour of the solution near the edge of the spherical crack, and the stress-intensity factors and crack-opening displacements are studied. The case when the crack surfaces are under normal internal pressure of constant intensity is examined.     

Determination of interface properties between micron-thick metal film and ceramic substrate using peel test

Peel test measurements have been performed to estimate both the interface toughness and the separation strength between copper thin film and Al₂O₃ substrate with film thicknesses ranging between 1 and 15 μm. An inverse analysis based on the artificial neural network method is adopted to determine the interface parameters. The interface parameters are characterized by the cohesive zone (CZ) model. The results of finite element simulations based on the strain gradient plasticity theory are used to train the artificial neural network. Using both the trained neural network and the experimental measurements for one test result, both the interface toughness and the separation strength are determined. Finally, the finite element predictions adopting the determined interface parameters are performed for the other film thickness cases, and are in agreement with the experimental results.     

Linear elastic fracture mechanics (LEFM) analysis of the effect of residual stress on fatigue crack propagation rate

In this research, both residual and applied stresses are converted to stress intensity factors independently and combined using the superposition principle. The fatigue crack propagation rates are predicted. Experiments using two different loading modes, constant applied stress intensity factor (SIF) range, and constant applied load modes are done for samples with and without initial tensile residual stresses. The samples with initial tensile residual stresses exhibit accelerations of the crack propagation rates. The results show that the weight function method combined with the three-component model provides a good prediction of fatigue crack propagation rates in tensile residual stress fields.     

Stress intensity factors around a crack in a nonhomogeneous interfacial layer between two dissimilar elastic half-planes

The stresses around a crack in an interfacial layer between two dissimilar elastic half-planes are obtained. The crack is parallel to the interfaces. The material constants of the layer vary continuously within a range from those of the upper half-plane to those of the lower half-plane. An internal gas pressure is applied to the surfaces of the crack. To derive the solution, the nonhomogeneous interfacial layer is divided into several homogeneous layers with different material properties. The boundary conditions are reduced to dual integral equations, which are solved by expanding the differences of the crack face displacements into a series. The unknown coefficients in the series are determined using the Schmidt method, and a stress intensity factor is calculated numerically for epoxy-aluminum composites.