Browsing by Subject "brittle fracture"
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Item A new approach to the modeling and analysis of fracture through an extension of continuum mechanics to the nanoscale(2009-05-15) Sendova, Tsvetanka BozhidarovaThe dissertation focuses on the analysis, through combined analytical and numerical techniques, of the partial differential equations arising from a new approach to modeling brittle fracture, based on extension of continuum mechanics to the nanoscale. The main part of this work deals with the analysis of several fracture models. Integral transform methods are used to reduce the problem to a Cauchy singular, linear integro-differential equation. It is shown that ascribing constant surface tension to the fracture surfaces and using the appropriate crack surface boundary condition, given by the jump momentum balance, leads to a sharp crack opening profile at the crack tip, in contrast to the classical theory of brittle fracture. However, such a model still predicts singular crack tip stress. For this reason a modified model is studied, where the surface excess property is responsive to the curvature of the fracture surfaces. It is shown that curvature-dependent surface tension, together with boundary conditions in the form of the jump momentum balance, leads to bounded stresses and a cusp-like opening profile at the crack tip. Further, an alternative approach, based on asymptotic analysis, which is suitable to apply in cases when the model includes a mutual body force correction term, is considered. The nonlinear nonlocal problem, resulting from the proposed model, is simplified which allows us to approximate the crack opening profile and derive asymptotic forms for the cleavage stress in a neighborhood of the crack tip. Finally, two possible fracture criteria, in the context of the new theory, are discussed. The first one is an energy based fracture criterion. Classically the energy release rate arises due to singular fields, whereas in the case of the modeling approach adopted here, a notion analogous to the energy release rate arises through a different mechanism, associated to the rate of working of the surface excess properties at the crack tip. Due to the fact that the proposed modeling approach allows us to fully resolve the stress in a neighborhood of the crack tip, without the customary singularity, a second fracture criterion, based on crack tip stress, is possible.Item Brittle Fracture Modeling with a Surface Tension Excess Property(2012-10-30) Ferguson, LaurenThe classical theory of linear elastic fracture mechanics for a quasi-static crack in an infinite linear elastic body has two significant mathematical inconsistencies: it predicts unbounded crack-tip stresses and an elliptical crack opening profile. A new theory of fracture developed by Sendova and Walton, based on extending continuum mechanics to the nanoscale, corrects these erroneous effects. The fundamental attribute of this theory is the use of a dividing surface to describe the material interface. The dividing surface is endowed with an excess property, namely surface tension, which accounts for atomistic effects in the interfacial region. When the surface tension is taken to be a constant, Sendova and Walton show that the theory reduces the crack-tip stress from a square root to a logarithmic singularity and yields a finite angle opening profile. In addition, they show that if the surface tension depends on curvature, the theory completely removes the stress singularity at the crack-tip, for all but countably many values of the two surface tension parameters, and yields a cusp-like opening profile. In this work, we develop a numerical model using the finite element method for the Sendova-Walton fracture theory applied to the classical Griffith crack problem in the case of constant surface tension. We show that the numerical model behaves as predicted by the theory, yielding a reduced crack-tip singularity and a finite opening angle for all nonzero values of the constant surface tension. We also lay the groundwork for the numerical implementation of the curvature-dependent model by constructing an algorithm to determine the appropriate threshold values for the surface tension parameters that guarantee bounded crack-tip stresses. These values can then be directly applied to the forthcoming numerical model.