Abstract

This work investigates systematically strain localization and failure mechanics for elastoplastic damage solids. Two complementary methodologies, i.e., traction-based discontinuities localized in an elastic solid and strain localization of a stress-based inelastic softening solid, are addressed. In the former it is assumed a priori that the discontinuity (band) forms with a continuous stress field and along the known orientation. A traction-based failure criterion is introduced to characterize the discontinuity (band) and the orientation is determined from Mohr’s maximization postulate. If the (apparent) displacement jumps are retained as independent variables, the strong/regularized discontinuity approaches follow, requiring constitutive models for both the bulk and discontinuity (band). Elimination of the displacement jumps at the material point level results in the embedded/smeared discontinuity approaches in which an overall inelastic constitutive model fulfilling the static constraint suffices. The second methodology is then adopted to check whether the assumed strain localization can occur and identify its consequences on the resulting approaches. The kinematic constraint guaranteeing stress boundedness/continuity upon strain localization is established for general inelastic softening solids. Application to a unified elastoplastic damage model naturally yields all the ingredients of a localized model for the discontinuity (band), justifying the first methodology. Two dual but not necessarily equivalent approaches, i.e., the traction-based elastoplastic damage model and the stress-based projected discontinuity model, are identified. The former is equivalent to the embedded/smeared discontinuity approaches, whereas in the later the discontinuity orientation and associated failure criterion, not given a priori, are determined consistently from the kinematic constraint. The bi-directional connections and equivalence conditions between the traction- and stress-based approaches are classified. Closed-form 2D results under plane stress condition are also given, with the classical Rankine, Mohr-Coulomb, von Mises and Drucker-Prager criteria analyzed as the illustrative examples. A generic failure criterion of either elliptic, parabolic or hyperbolic type, is then considered in a unified manner, resulting in many failure criteria frequently employed in practice.

Abstract

An important concern in sheet stamping is the risk of obtaining brittle final products that can be affected by fracture. Monte Carlo simulations presented herein show that this is governed by two main factors, namely static and dynamic friction coefficients. Whereas the latter correlates in a non-linear manner with minimum and maximum end thickness, the relationship of these design parameters to the former exhibits a bifurcation that is typical of highly non-linear phenomena, in which there is a sensitivity to small perturbations of the input values (chaos). In order to estimate the reliability of the process (i.e., the probability of obtaining brittle products due to low minimum and maximum thicknesses) with a reduced number of Monte Carlo runs, it is proposed to assimilate the problem to a pattern recognition task, due to the existence of two classes, namely robust and brittle. Among many pattern recognition algorithms that are useful to this end, use is made of support vector machines, as this incorporates the powerful tool of class margins that allow a drastic reduction of the number of simulations.

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The usefulness of using the speckle photography technique in fracture mechanics to check numerical calculations is demonstrated for an internally pressurized cylinder with a surface flaw. A pointwise technique was utilized to measure the opening displacements along the crack line and the Mode I stress-intensity factor was determined by extrapolating these results to the crack tip. Finite element calculations were performed to be compared with experimental data and good agreement was obtained.

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This paper presents an implicit orthotropic model based on the Continuum Damage Mechanics isotropic models. A mapping relationship is established between the behaviour of the anisotropic material and that of an isotropic one. The proposed model is used to simulate the failure loci of common orthotropic materials, such as masonry, fibre-reinforced composites and wood. The damage model is combined with a crack-tracking technique to reproduce the propagation of localized cracks in the discrete FE problem. The proposed numerical model is used to simulate the mixed mode fracture in masonry members with different orientations of the brick layers.

Abstract

Once strain localization occurs in softening solids, inelastic loading behavior is restricted within a narrow band while the bulk unloads elastically. Accordingly, localized failure in solids can be approached by embedding or smearing a traction-based inelastic discontinuity (band) within an (equivalent) elastic matrix along a specific orientation. In this context, the conformity of the strong/regularized and embedded/smeared discontinuity approaches are investigated, regarding the strategies dealing with the kinematics and statics. On one hand, the traction continuity condition imposed in weak form results in the strong and regularized discontinuity approaches, with respect to the approximation of displacement and strain discontinuities. In addition to the elastic bulk, consistent plastic-damage cohesive models for the discontinuities are established. The conformity between the strong discontinuity approach and its regularized counterpart is shown through the fracture energy analysis. On the other hand, the traction continuity condition can also be enforced point-wisely in strong form so that the standard principle of virtual work applies. In this case, the static constraint resulting from traction continuity can be used to eliminate the kinematic variable associated with the discontinuity (band) at the material level. This strategy leads to embedded and smeared discontinuity models for the overall weakened solid which can also be cast into the elastoplastic degradation framework with a different kinematic decomposition. Being equivalent to the kinematic constraint guaranteeing stress continuity upon strain localization, Mohr’s maximization postulate is adopted for the determination of the discontinuity orientation. Closed-form results are presented in plane stress conditions, with the classical Rankine, Mohr–Coulomb, von Mises and Drucker–Prager criteria as illustrative examples. The orientation of the discontinuity (band) and the stress-based failure criteria consistent with the given traction-based counterparts are derived. Finally, a generic failure criterion of either elliptic, parabolic or hyperbolic type, appropriate for the modeling of mixed-mode failure, is analyzed in a unified manner. Furthermore, a novel method is proposed to calibrate the involved mesoscopic parameters from available macroscopic test data, which is then validated against Willam’s numerical test.

Abstract

This work investigates systematically traction- and stress-based approaches for the modeling of strong and regularized discontinuities induced by localized failure in solids. Two complementary methodologies, i.e., discontinuities localized in an elastic solid and strain localization of an inelastic softening solid, are addressed. In the former it is assumed a priori that the discontinuity forms with a continuous stress field and along the known orientation. A traction-based failure criterion is introduced to characterize the discontinuity and the orientation is determined from Mohr's maximization postulate. If the displacement jumps are retained as independent variables, the strong/regularized discontinuity approaches follow, requiring constitutive models for both the bulk and discontinuity. Elimination of the displacement jumps at the material point level results in the embedded/smeared discontinuity approaches in which an overall inelastic constitutive model fulfilling the static constraint suffices. The second methodology is then adopted to check whether the assumed strain localization can occur and identify its consequences on the resulting approaches. The kinematic constraint guaranteeing stress boundedness and continuity upon strain localization is established for general inelastic softening solids. Application to a unified stress-based elastoplastic damage model naturally yields all the ingredients of a localized model for the discontinuity (band), justifying the first methodology. Two dual but not necessarily equivalent approaches, i.e., the traction-based elastoplastic damage model and the stress-based projected discontinuity model, are identified. The former is equivalent to the embedded and smeared discontinuity approaches, whereas in the later the discontinuity orientation and associated failure criterion are determined consistently from the kinematic constraint rather than given a priori. The bi-directional connections and equivalence conditions between the traction- and stress-based approaches are classified. Closed-form results under plane stress condition are also given. A generic failure criterion of either elliptic, parabolic or hyperbolic type is analyzed in a unified manner, with the classical von Mises (J2), Drucker–Prager, Mohr–Coulomb and many other frequently employed criteria recovered as its particular cases.