Dynamic crack kinking from a PMMA/Homalite interface

The behavior of a pre-existing, dynamically loaded, interfacial crack kinking away from the interface separating two materials was experimentally investigated under dynamic loading conditions. Dynamic fracture experiments were performed on pre-cracked bimaterial panels of PMMA bonded with Homalite-100 impact loaded using a high-speed gas gun. By varying the location of impact, a large range of mixed mode loading at the crack tip was produced. Information about the stress state surrounding the crack tip was obtained through use of the lateral shearing interferometer of coherent gradient sensing in conjunction with high-speed photography. The high-speed interferogram corresponding, to the time of crack initiation was analyzed in each case to find the preinitiation mode mixity at the crack tip. Measurement of both the local initiation mode mixity and the crack kink angle allows for possible extension of existing quasi-static interface crack kinking criteria, such as maximum opening stress or maximum energy release rate, to the case of dynamic loading. It was found that for bimaterial systems with small material property mismatch, such as the PMMA/Homalite system, the maximum opening stress criterion accurately predicts the relation between crack tip mode mixity and resulting kink angle for small initial crack kinking speeds.     

Atomistic/continuum simulation of interfacial fracture Part II: Atomistic/dislocation/continuum simulation

Coupled atomistic/dislocation/continuum simulation of interfacial fracture is performed in this paper. The model consists of a nanoscopic core made by atomistic assembly and a surrounding elastic continuum with discrete dislocations. Atomistic dislocations nucleate from the crack tip and move to the continuum layer where they glide according to the dislocation dynamics curve. An atoms/continuum averlapping belt is devised to facilitate the transition between the two scales. The continuum constraint on the atomic assembly is imposed through the mechanics atmosphere along the overlapping belt. Transmissions of mechanics parameters such as displacements, stresses, masses and momenta across the belt are realized. The present model allows us to explore interfacial fracture processes under different mode mixity. The effect of atomistic zigzag interface on the fracture process is revealed: it hinders dislocation emission from the crack tip, especially under high mode mixity. The project supported by the National Natural Science Foundation of China     

Strip model plasticity analysis of mixed mode crack opening displacement in aluminum alloy LY12

Under mixed mode loading, the crack tip blunts and undergoes displacements in two directions, the normal and shear component corresponding to Mode I and Mode II loading, respectively. These local displacements are determined by the duplicated film method and used to analyze the behavior of mixed mode fracture in aluminum alloy LY12. The mixed mode resultant crack opening displacement (COD) at fracture initiation tends to increase more rapidly with increase of the Mode II shear component. The fracture initiation value of COD for pure Mode II loading is six (6) times greater than that for Mode 1 loading. The same applies to the maximum effective plastic strain crack growth near the crack front. Observed are two typical morphologies, the equal-axes dimples and the parabolic dimples with evidence of slippage as dominated, respectively, by Mode I and Mode II loading.     

Nonlinear interface shear fracture of end notched flexure specimens

The nonlinear analytical solutions of an end notched flexure adhesive joint or fracture test specimen with identical or dissimilar adherends are investigated. In the current study, a cohesive zone model (with arbitrary nonlinear cohesive laws) based analytical solution is obtained for the interface shear fracture of an end notched flexure (ENF) specimen with sufficiently long bond length. It is found that the scatter and inconsistency in calculating Mode II toughness may be significantly reduced by this model. The present work indicates that the Mode II toughness G_IIc under pure shear cracking condition is indeed very weakly dependent on the initial crack length. And this conclusion is well supported by the experimental results found in the literature. The parametric studies show that the interface shear strength is the most dominant parameter on the critical load. It is also interesting to note that with very short initial crack length and identical interface shear strength, higher Mode II toughness indeed cannot increase the critical load. Unlike the high insensitivity of critical load to the detailed shape of the cohesive law for Mode I peel fracture, the shape of the cohesive law becomes relatively important for the critical load of joints under pure Mode II fracture conditions, especially for joints with short initial crack length. The current study may help researchers deepen the understanding of interface shear fracture and clarify some previous concepts on this fracture mode.     

MIXED MODES INTERLAMINAR FRACTURE TOUGHNESS OF CFRP LAMINATES TOUGHENED WITH CNF INTERLAYER

In the present paper, the influence of carbon nanofiber on interlaminar fracture toughness of CFRP investigated using MMB(Mixed Mode Bending) tests. Vapor grown carbon fiber VGCF and VGCF-S, and multi-walled carbon nanotube MWNT-7 has been employed for the toughener of the interlayer on the CFRP laminates. In order to evaluate the fracture toughness and mixed mode ratio of it, double cantilever beam (DCB) tests, end notched fracture (ENF) tests and mixed mode bending (MMB) tests have been carried out. Boundary element analysis was applied to the CFRP model to compute the interlaminar fracture toughness, where extrapolation method was used to determine the fracture toughness and mixed mode ratio. The interlaminar fracture toughness and mixed mode ratio can be extrapolated by stress distribution in the vicinity of the crack tip of the CFRP laminate. It was found that the interlaminar fracture toughness of the CFRP laminates was improved inserting the interlayer made by carbon nanofiber especially in the region where shear mode deformation is dominant.     

A novel method in determination of dynamic fracture toughness under mixed mode I/II impact loading

The objective of this paper is to propose a novel methodology for determining dynamic fracture toughness (DFT) of materials under mixed mode I/II impact loading. Previous experimental investigations on mixed mode fracture have been largely limited to qusi-static conditions, due to difficulties in the generation of mixed mode dynamic loading and the precise control of mode mixity at crack tip, in absence of sophisticated experimental techniques. In this study, a hybrid experimental–numerical approach is employed to measure mixed mode DFT of 40Cr high strength steel, with the aid of the split Hopkinson tension bar (SHTB) apparatus and finite element analysis (FEA). A fixture device and a series of tensile specimens with an inclined center crack are designed for the tests to generate the components of mode I and mode II dynamic stress intensity factors (DSIF). Through the change of the crack inclination angle β (=90°, 60°, 45°, and 30°), the K_II/K_I ratio is successfully controlled in the range from 0 to 1.14. A mixed mode I/II dynamic fracture plane, which can also exhibit the information of crack inclination angle and loading rate at the same time, is obtained based on the experimental results. A safety zone is determined in this plane according to the characteristic line. Through observation of the fracture surfaces, different fracture mechanisms are found for pure mode I and mixed mode fractures.     

Mixed-Mode Dynamic Crack Growth in Functionally Graded Glass-Filled Epoxy

Compositionally graded glass-filled epoxy sheets with edge cracks initially along the gradient are studied under dynamic loading conditions. Specimens with monotonically varying volume fraction of reinforcement are subjected to mixed-mode loading by eccentric impact relative to the crack plane. The optical method of Coherent Gradient Sensing and high-speed photography are used to map transient crack tip deformations before and after crack initiation. Two configurations, one with a crack on the stiffer side of a graded sheet and the second with a crack on the compliant side, are examined experimentally. To elucidate the differences in fracture behavior due to functional grading, a homogeneous sample is also tested. The differences in both pre- and post-crack initiation behaviors are observed interms of crack initiation time, crack path, crack speed and stress intensity factor histories. When a crack is situated on the compliant side of the sample, it kinks significantly less compared to when it is on the stiffer side. Crack tip mode mixity histories show small but positive values during crack growth from the stiffer side of the sample towards the compliant side whereas a small but negative mode mixity prevails for the opposite configuration.     

Experimental and Numerical Study of Paperboard Interface Properties

Laminated paperboard is widely used in packaging products. Interface delamination plays a crucial role in converting paperboard to a carton through the creasing and folding process. Thus, the aim of this study is to experimentally and numerically investigate the interface fracture behavior in pure crack opening mode (mode I) and sliding mode (mode II). Four experimental tests have been evaluated and compared to numerical simulation, namely, the z-directional tensile test (ZDT), double-notch shear test (DNS), double-cantilever beam test (DCB) and end-notched flexure test (ENF). It was shown that, for the paperboard specimens tested, the ZDT test was sufficient to fully characterize the mode I crack growth response. However, the DNS and ENF tests were required to determine the maximum shear stress and the fracture toughness of pure mode II, respectively. Further mixed-mode investigation would enable the analysis of paperboard delamination behavior during the creasing and folding process.     

Atomistic/continuum simulation of interfacial fracture part I: Atomistic simulation

The phenomenon of interfacial fracture, as manifested by atomistic cleavage, debonding and dislocation emission provides a challenge for combined atomistic-continuum analysis. As a precursor for fully coupled atomistic-continuum simulation^[1] of interfacial fracture, we focus here on the atomistic behavior within a nanoscopic core surrounding the crack tip. The inter-atomic potential under Embedded Atom Method is recapitulated to form an essential framework of atomistic simulation. The calculations are performed for a side-cracked disc configuration under a remoteK field loading. It is revealed that a critical loading rate defines the brittle-to-ductile transition of homogeneous materials. We further observe that the near tip mode mixity dictates the nanoscopic profile near an interfacial crack tip. A zigzag interface structure is simulated which plays a significant role in the dislocation emission from an interfacial crack tip, as will be explored in the second part of this investigation. The project supported by the National Natural Science Foundation of China     

Crack propagation at material interfaces: II experiments on mode interaction

Optical interferometry is used to measure crackopening profiles for interfacial cracks in a model adhesive joint. Application of boundary displacements with a resolution of 0.16 μm is achieved through a combination of thermal expansions and optical control in a servo loop. The degree of nonlinear deformation accompanying small levels of normal and shear strain across the bond is documented, and crackfront displacement fields are exploited to evaluate possible criteria for crack growth under combined normal and shear loading. Within the accuracy afforded by the crack-propagation investigation it appears that crack growth is governed by the vectorial crack-tip displacements, namely by the vector sum of the local normal and tangential displacement components. In the limit of infinitesimal strains this criterion reduces to the strain energy-release-rate criterion of linear fracture mechanics.     

Mode II fracture behavior of a Zr-based bulk metallic glass

Shear band formation and fracture are characterized during mode II loading of a Zr-based bulk metallic glass. The measured mode II fracture toughness, K_IIc=75±4 MPa√m, exceeds the reported mode I fracture toughness by ∼4 times, suggesting that normal or mean stresses play a significant role in the deformation process at the crack tip. This effect is explained in light of a mean stress modified free volume model for shear localization in metallic glasses. Thermal imaging of deformation at the mode II crack tip further reveals that shear bands initiate, arrest, and reactivate along the same path, indicating that flow in the shear band leads to permanent changes in the glass structure that retain a memory of the shear band path. The measured temperature increase within the shear band is a fraction of a degree. However, heat dissipation models indicate that the temperature could have exceeded the glass transition temperature for less than 1 ms immediately after the shear band formed. It is shown that this time scale is sufficient for mechanical relaxation slightly above the glass transition temperature.     

界面裂纹的路径选择与数值模拟

利用界面断裂能和荷载混合度的概念研究界面裂纹的扩展路径，利用有限元数值方法模拟界面裂纹的扩展过程，再现界面断裂的各种几何构型．研究表明，界面的断裂能和混合度完全控制了裂纹在界面附近的扩展过程．数值预测的裂纹路径与理论结果一致     

An inclined surface crack subject to biaxial loading

The elastic–plastic stress fields and mode mixity parameters for semi-elliptical surface cracks on biaxial loaded plates have been investigated using detailed three-dimensional finite element calculations. Different degrees of mode mixity are given by combinations of the far-field stress level, biaxial stress ratio and inclined crack angle. These analyses were performed for different surface flaw geometries to study the combined load biaxiality and mode mixity effects on the crack-front stress fields and the size and shape of the plastic zones. It is clear from considering the local stress distributions along the crack front that the elastic crack tip singularities have been derived for several particular cases of mixed mode biaxial loading. By theoretical analysis, the new formulae have been introduced for both the elastic and plastic mode-mixity parameters, accounting for ratios between the I/II, II/III and III/I modes. Particular attention was paid to the strong variations of the mode-mixity parameters along the semi-elliptical surface crack front. The mixed-mode behavior of the crack growth direction angle along the semi-elliptical crack front for different combinations of biaxial loading and inclination crack angles was also determined. It was done using methods based on the maximum tangential stress and the strain energy density criteria.     

Asymptotic analysis of an impermeable crack in an electrostrictive material subjected to electric loading

The simple asymptotic problem of an impermeable crack in an electrostrictive ceramic under electric loading is analyzed. Closed form solutions of elastic fields are obtained by using the complex function theory. It is found that the K_I-dominant region is very small compared to the electric saturation zone. A fracture parameter for an electrostrictive material subjected to electric loading is discussed. In order to investigate the influence of the transverse electric displacement on fracture behavior under the small-scale conditions, we also consider the modified boundary layer problem of a crack in an electrostrictive material. Analytic solutions of electric displacement fields for the asymptotic problem are obtained based on the nonlinear dielectric theory from a modified boundary layer analysis. The shape of the electric displacement saturation zone is shown to depend on the transverse electric displacement. Stress intensity factors induced by the electrostrictive strains are evaluated using the nonlinear solution of the electric displacements. It is found that the transverse electric displacement affects strongly the variation of the mode mixity.     

Tensile-shear transition in mixed mode I/III fracture

The propensity of the transition of fracture type in either brittle or ductile cracked solid under mixed-mode I and III loading conditions is investigated. A fracture criterion based on the competition of the maximum normal stress and maximum shear stress is utilized. The prediction of the fracture type is determined by comparing τ_max/σ_max at a critical distance from the crack tip to the material strength ratio τ_C/σ_C, i.e., (τ_max/σ_max)<(τ_C/σ_C) for tensile fracture and (τ_max/σ_max)>(τ_C/σ_C) for shear fracture, where σ_C (τ_C) is the fracture strength of materials in tension (shear). Mixed mode I/III fracture tests were performed using circumferentially notched cylindrical bars made of PMMA and 7050 aluminum alloy. Fracture surface morphology of the specimens reveals that: (1) for the brittle material, PMMA, only tensile type of fracture occurs, and (2) for the ductile material, 7050 aluminum alloy, either tensile or shear type of fracture occurs depending on the mode mixity. The transition (in ductile material) or non-transition (in brittle material) of the fracture type and the fracture path observed in experiments were properly predicted by the theory. Additional test data from open literature are also included to validate the proposed theory.     

Mixed-mode interface toughness of wafer-level Cu-Cu bonds using asymmetric chevron test

Characterization of interfacial adhesion is critical for the development of wafer bonding processes to manufacture microsystems with high yield and reliability. It is imperative that the test method used in such adhesion studies corresponds to the loading conditions present during processing and operation of the devices. In most applications in which wafers and die are bonded, the interface experiences a combination of shear and normal loading (i.e. mixed-mode loading) with the relative magnitude of the Mode I and II components varying in different scenarios. In the current work, the toughness of Cu-Cu thermocompression bonds, which are of interest for the fabrication of three-dimensional integrated circuits, is analyzed using a bonded chevron specimen with layers of different thickness that allows for the application of interfacial loading with variable mode mixity. The phase angle (a function of the degree of mode mixity at the interface) is varied from 0° to 24° by changing the layer thickness ratio from 1 to 0.48. The Cu-Cu bond toughness increases from 2.68 to 10.1 J/m², as the loading is changed from Mode I (pure tension) to a loading with a phase angle of 24°. The energy of plastic dissipation increases with increasing mode mixity, resulting in the enhanced interface toughness. The Mode I toughness of Cu-Cu bonds is minimally affected by plasticity, and therefore, provides the closest estimate of the interfacial work of fracture under the bonding conditions employed.     

Mode mixity and nonlinear viscous effects on toughness of interfaces

This paper examines steady-state crack growth at interfaces between polymeric materials and hard substrates under quasi-static conditions. The polymeric material is taken to be an elastic nonlinear viscous solid while the substrate is treated as a rigid material. Void growth and coalescence in the rate-dependent fracture process zone is modeled by a nonlinear viscous porous strip of cell elements. In the first part of this paper, the polymeric background material surrounding the process zone is assumed to be purely elastic. Under fixed mode mixity, the computed interface toughness is found to be a monotonically increasing function of crack velocity; toughness also increases rapidly with higher rate sensitivity. This behavior can be explained in terms of voids growing in a strain-rate strengthened process zone. In the second part of the paper, the background material is also treated as an elastic nonlinear viscous solid. The competition between work of separation in the process zone and energy dissipation in the background material leads to a U-shaped toughness–crack velocity curve. Effects of mode mixity, initial porosity, rate sensitivity, as well as the initial yield strain on toughness are studied. The simulations produce trends that agree with interface toughness vs. crack velocity data reported in experimental studies for rubber toughened epoxy-paste adhesive and urethane acrylate adhesive.     

K III -Deformation modes in internal oblique cracks under plane-stress conditions

The exact shape of the deformed internal oblique crack in an infinite elastic plate under conditions of plane stress was studied. The Muskhelishvili potential function yielded the exact stress and displacement field around the crack. While in previous papers by the author the study was mainly concerned with the definition of the in-plane shape of the deformed crack and important properties were disclosed concerning especially the contribution of the shear loading of the crack, in this paper the out-of-plane component of displacements is determined and its influence on the exact shape of the deformed crack in space is presented. It is shown that, as soon as shear displacements appear in the cracked plate under plane stress, out-of-plane shear displacements are a compulsory consequence for plane-stress conditions of the plate. The elliptic form of the deformed internal crack was twisted out of the plane of the plate with its zero twisting displacement near the new crack tip of its deformed shape corresponding to the vertex of the ellipse. The points of maximum and minimum out-of-plane displacements were placed close to the vertices of the ellipse at polar angles, θ, depending only on the eccentricity of the ellipse and displaced always on both sides of the vertices. The compulsory coexistence of mode II and mode III deformations makes the internal crack in plane stress to present a complicated pattern of deformation at its deformed crack tips. All of these results are amply supported by experimental evidence with caustics, which always show either a simple mode I pattern or a complex mode II and III pattern as soon as shear interferes in the mode of deformation of the plate.     

Selection of finite-element mesh parameters in modeling the growth of hydraulic fracturing cracks

The effect of the mesh geometry on the accuracy of solutions obtained by the finite-element method for problems of linear fracture mechanics is investigated. The guidelines have been formulated for constructing an optimum mesh for several routine problems involving elements with linear and quadratic approximation of displacements. The accuracy of finite-element solutions is estimated based on the degree of the difference between the calculated stress-intensity factor (SIF) and its value obtained analytically. In problems of hydrofracturing of oil-bearing formation, the pump-in pressure of injected water produces a distributed load on crack flanks as opposed to standard fracture mechanics problems that have analytical solutions, where a load is applied to the external boundaries of the computational region and the cracks themselves are kept free from stresses. Some model pressure profiles, as well as pressure profiles taken from real hydrodynamic computations, have been considered. Computer models of cracks with allowance for the pre-stressed state, fracture toughness, and elastic properties of materials are developed in the MSC.Marc 2012 finite-element analysis software. The Irwin force criterion is used as a criterion of brittle fracture and the SIFs are computed using the Cherepanov–Rice invariant J-integral. The process of crack propagation in a linearly elastic isotropic body is described in terms of the elastic energy release rate G and modeled using the VCCT (Virtual Crack Closure Technique) approach. It has been found that the solution accuracy is sensitive to the mesh configuration. Several parameters that are decisive in constructing effective finite-element meshes, namely, the minimum element size, the distance between mesh nodes in the vicinity of a crack tip, and the ratio of the height of an element to its length, have been established. It has been shown that a mesh that consists of only small elements does not improve the accuracy of the solution.