A computer simulation study of Hertzian cone crack growth

Lawn, Brian R.; Wilshaw, T. R.; Hartley, N. E. W.

doi:10.1007/bf00955075

Cited by 69 publications

(50 citation statements)

References 10 publications

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“…1c and d). This dependence qualitatively corresponds to experimental findings [3], yet the simulated cone angles in Figs. 1c and d are significantly larger than those observed in experiments.…”

Section: Numerical Studies Of Indentation Fracturesupporting

confidence: 90%

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Simulation of Hertzian cone cracks using a phase field description for fracture

Strobl

Morand

Seelig

2016

Proc Appl Math and Mech

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The present contribution focuses on fracture caused by indentation loading on the surface of a brittle solid. Its theoretical prediction is a challenging task due to the fact that crack nucleation is not geometrically induced, but is caused by the stress concentration in the contact near-field. The application of the phase field model requires constitutive assumptions to ensure a tension-compression asymmetric material response and prevent damage in compressed regions. This is achieved at the cost of giving up the variational concept of brittle fracture. We simulate the indentation of a cylindrical flat-ended punch on brittle materials like silicate glass. In order to reduce the numerical effort, we exploit axisymmetric conditions for the finite element formulation. After crack initiation stable propagation of a cone crack can be observed in good agreement with experiments. Indentation fractureThe occurrence of cone cracks under indentation loading has already been described by Hertz [1] in 1881: When a cylindrical stiff punch is pressed against the surface of a brittle elastic solid a circular crack is spontaneously initiated around the contact area. Under further loading this initial ring crack grows into a cone, the so called Hertzian cone crack. The growth at this stage is stable since the radius of the cone extends with the evolving crack. A particular challenge is to compute the crack initiation from the defect-free surface. The initial crack starts some distance outside of the indenter, as known from experiments [2]. Previous studies (e.g. [3]) modeled pre-existing ring cracks in order to investigate the further progress of the cone crack. The spontaneous formation of the ring crack is a non-standard problem of fracture mechanics which requires to take strength σ c and toughness G c into account, e.g. [4]. Thus the phase field approach appears to be well suited to handle such a problem since it also comprises both parameters. Furthermore, the method does not require any assumptions concerning the unknown location of the crack initiation or crack path. Governing equations of the phase field approachPhase field models have already proven their excellent ability to reproduce situations with complex crack patterns including initiation and the determination of unknown crack paths. For the smeared description of a crack in a linear elastic body Ω, with mass density ρ and external boundary ∂Ω = ∂Ω t ∪ ∂Ω u , an additional scalar field S(x, t) is introduced to describe the state of the material between the fully broken state S = 0 and undamaged material S = 1. Thereby the crack surface energy, including Griffith's critical energy release rate G c , is approximated by a smooth function with a small regularization parameter ℓ which is related to the width of the transition zone between both states. In order to obtain the coupled Euler-Lagrange equations for dynamic brittle fracture we employ Hamilton's principle. With the choice of a quadratic degradation function, which describes the loss of stiffness in damaged ...

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Section: Numerical Studies Of Indentation Fracturesupporting

confidence: 90%

“…Previous studies (e.g. [3]) modeled pre-existing ring cracks in order to investigate the further progress of the cone crack. The spontaneous formation of the ring crack is a non-standard problem of fracture mechanics which requires to take strength σ c and toughness G c into account, e.g.…”

Section: Indentation Fracturementioning

confidence: 99%

Simulation of Hertzian cone cracks using a phase field description for fracture

Strobl

Morand

Seelig

2016

Proc Appl Math and Mech

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“…This result was first observed by Hertz (11,12). This phenomena has been intensively studied; however, computational models are still in discussion (13,14,15,16,17). The objective of this paper is to investigate DEM as an effective alternative for studying complex cracking phenomena such as the hertzian cone crack.…”

Section: Introductionmentioning

confidence: 88%

Using the discrete element method to simulate brittle fracture in the indentation of a silica glass with a blunt indenter

Debenath

Jebahi

Iordanoff

et al. 2013

Computer Methods in Applied Mechanics and Engineering

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is an open access repository that collects the work of Arts et Métiers ParisTech researchers and makes it freely available over the web where possible. AbstractThe mechanical behavior of materials is usually simulated by a continuous mechanics approach. However, noncontinuous phenomena such as multi-fracturing cannot be accurately simulated using a continuous description. The discrete element method (DEM) naturally accounts for discontinuities and is therefore a good alternative to the continuum approach.This study continues previous work in which a DEM model was developed to quantitatively simulate an elastic material with the cohesive beam bond model. The simulation of brittle cracks is now tackled. This goal is attained by computing a failure criterion based on an equivalent hydrostatic stress. This microscopic criterion is then calibrated to fit experimental values of the macroscopic failure stress. The simulation results are compared to experimental results of indentation tests in which a spherical indenter is used to load a silica glass, which is considered to be a perfectly brittle elastic material.

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“…Therefore, as was noted by Bertoldi & Sglavo [42], although the angle is not severely affected by the indenter geometry, it is very sensitive to the glass composition and especially to Poisson's ratio. Note that the effect of ν on α was already mentioned by Lawn et al [55] who noted that ν has a profound influence on the stress field and further attempted to 'tune' the ν value to get their modelling to match better with the experimental results for α.…”

Section: (D) the Effect Of High Pressurementioning

confidence: 99%

“…Cone cracks are supposed to initiate from critical flaws located beneath the contact area in planes orientated close to 45 • from the surface where the shear stresses are the greatest and to extend along a path following the maximum σ I values, i.e. along the trajectory of the minimum principal normal stress (σ III ) (bottom plots in figure 15) [55,56]. However, cone-cracking angles (to the surface) predicted this way are mostly in poor agreement with the experimental values.…”

Section: (D) the Effect Of High Pressurementioning

confidence: 99%

Driving force for indentation cracking in glass: composition, pressure and temperature dependence

Rouxel

2015

Phil. Trans. R. Soc. A.

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The occurrence of damage at the surface of glass parts caused by sharp contact loading is a major issue for glass makers, suppliers and end-users. Yet, it is still a poorly understood problem from the viewpoints both of glass science and solid mechanics. Different microcracking patterns are observed at indentation sites depending on the glass composition and indentation cracks may form during both the loading and the unloading stages. Besides, we do not know much about the fracture toughness of glass and its composition dependence, so that setting a criterion for crack initiation and predicting the extent of the damage yet remain out of reach. In this study, by comparison of the behaviour of glasses from very different chemical systems and by identifying experimentally the individual contributions of the different rheological processes leading to the formation of the imprint-namely elasticity, densification and shear flow-we obtain a fairly straightforward prediction of the type and extent of the microcracks which will most likely form, depending on the physical properties of the glass. Finally, some guidelines to reduce the driving force for microcracking are proposed in the light of the effects of composition, temperature and pressure, and the areas for further research are briefly discussed.

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A computer simulation study of Hertzian cone crack growth

Cited by 69 publications

References 10 publications

Simulation of Hertzian cone cracks using a phase field description for fracture

Simulation of Hertzian cone cracks using a phase field description for fracture

Using the discrete element method to simulate brittle fracture in the indentation of a silica glass with a blunt indenter

Driving force for indentation cracking in glass: composition, pressure and temperature dependence

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