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Journal of Elasticity

Erschienen in:

01.10.2015

Modeling the Response of Tensile Steel Bars by Means of Incremental Energy Minimization

verfasst von: Giovanni Lancioni

Erschienen in: Journal of Elasticity | Ausgabe 1/2015

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Abstract

A non-local variational model for the evolution of plastic deformation and fracture in tensile bars is proposed. The model is based on an energy functional, sum of an elastic bulk energy, a non-convex dissipative inelastic energy, and a quadratic non-local gradient term, as in (Del Piero et al. in J. Mech. Mater. Struct. 8(2–4):109–151, 2013). The non-local energy is enriched, by assuming a dependence on both the inelastic deformation and its gradient, in order to improve the description of fracture, and bars with varying cross-section are considered, to accurately reproduce the geometry of samples which are commonly used in tensile tests. The evolution of the deformation is described by a two-field incremental minimization problem, where the longitudinal displacement and the plastic part of the deformation are assumed as independent variables. The problem is discretized by finite elements, and the resulting sequence of constrained quadratic programming problems is solved numerically.

Different simulations are proposed, reproducing the results of experiments on smooth and notched bone-shaped steel samples. The numerical tests provide accurate response curves, and they capture the distinct phases of the evolution observed in experiments: from the initial yielding phase, in which inelastic deformations form, and propagate as slow plastic waves, to the final rupture, which constitutes the ending stage of a strain-localization process.

Vorheriger Artikel Finite Stretching and Shearing of an Internally Balanced Elastic Solid

Nächster Artikel Effective Elasticity Tensors in Context of Random Errors

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Incremental energy minimization represents a powerful tool, which was applied to fracture problems [2] and many other inelastic phenomena, as deeply discussed in [3].

From here on, a subscript is used to indicate dependence on time.

First-order energy minimization was used in [1] to determine the steepest descent configuration rates.

The continuity of \(\dot{\gamma}'\), required to be (28) always applicable, was proved in [1].

Proof. If \(\dot{\gamma}=0\) in some interval of (0,l), from (30)₂, \(\dot{\sigma}\leq0\). But, integrating (30)₃ over (0,l), we get \(l\dot{\sigma}\bar {\dot{\gamma}}=\int_{0}^{l} (\theta''\dot{\gamma}^{2}+\alpha\dot {\gamma }'^{2} )\,dx>0\), and thus \(\dot{\sigma}>0\), in contradiction with the above result. Thus \(\dot{\gamma}\) cannot be null in some subinterval of the bar. □

Necessary and sufficient conditions for the eigenvalue non-negativeness are proposed in Sect. 3.4 of [1].

It is the area of the triangle defined by θ″ in the interval 0<γ<γ ₁ (see Fig. 6).

In not-reported simulations, it appears in the middle, and in the left extremity.

In particular, it is very similar to the Aifantis model [8], where the Laplacian of the cumulated plastic strain is introduced in the yield condition.

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Titel: Modeling the Response of Tensile Steel Bars by Means of Incremental Energy Minimization
verfasst von: Giovanni Lancioni
Publikationsdatum: 01.10.2015
Verlag: Springer Netherlands
Erschienen in: Journal of Elasticity / Ausgabe 1/2015
Print ISSN: 0374-3535
Elektronische ISSN: 1573-2681
DOI: https://doi.org/10.1007/s10659-015-9515-8

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