Extension of linear stability analysis for the dynamic stretching of plates: Spatio-temporal evolution of the perturbation

Xavier, M.; Czarnota, Christophe; Jouve, Dominique; Mercier, Sébastien; Dequiedt, J.L.; Molinari, A.

doi:10.1016/j.euromechsol.2019.103860

Cited by 12 publications

(28 citation statements)

References 40 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…On the other hand, while the same rationale has been employed by other authors, e.g. Zaera et al [52] and N'souglo et al [35], we are aware that it is also customary in the literature to impose the stretch rate as the loading condition [49,34], since it may bear closer resemblance to the actual boundary conditions in dynamic experiments like the expansion of hemispherical shells (see Mercier et al [30]). Hence, stability analysis results and finite element calculations obtained imposing the stretching rate, namelyε 0 xx = 4000 s −1 andε 0 xx = 80000 s −1 , are reported in Appendix A.…”

Section: Remarkmentioning

confidence: 84%

“…Note that η is usually referred to as the instantaneous instability index. Here ξ and η are considered time independent (frozen coefficients method), however, note that a time dependent numerical solution has been recently developed by Xavier et al [49] for von Mises materials subjected to plane strain tension. The perturbation, Eq.…”

Section: Linear Stability Analysis For An Isotropic Materials With Yielding According To Drucker Criterion [11]mentioning

confidence: 99%

See 1 more Smart Citation

Effect of the third invariant on the formation of necking instabilities in ductile plates subjected to plane strain tension

et al. 2021

View full text Add to dashboard Cite

In this paper, we have investigated the effect of the third invariant of the stress deviator on the formation of necking instabilities in isotropic metallic plates subjected to plane strain tension. For that purpose, we have performed finite element calculations and linear stability analysis for initial equivalent strain rates ranging from 10 −4 s −1 to 8 • 10 4 s −1 . The plastic behavior of the material has been described with the isotropic Drucker yield criterion [11], which depends on both the second and third invariant of the stress deviator, and a parameter c which determines the ratio between the yield stresses in uniaxial tension and in pure shear σ T /τ Y . For c = 0, Drucker yield criterion [11] reduces to the von Mises yield criterion [32] while for c = 81/66, the Hershey-Hosford (m = 6) yield criterion [19,22] is recovered. The results obtained with both finite element calculations and linear stability analysis show the same overall trends and there is also quantitative agreement for most of the loading rates considered. In the quasi-static regime, while the specimen elongation when necking occurs is virtually insensitive to the value of the parameter c, both finite element results and analytical calculations using Considère criterion [10] show that the necking strain increases as the parameter c decreases, bringing out the effect of the third invariant of the stress deviator on the formation of quasi-static necks. In contrast, at high initial equivalent strain rates, when the influence of inertia on the necking process becomes important, both finite element simulations and linear stability analysis show that the effect of the third invariant is reversed, notably for long necking wavelengths, with the specimen elongation when necking occurs increasing as the parameter c increases, and the necking strain decreasing as the parameter c decreases.

show abstract

Section: Remarkmentioning

confidence: 84%

Section: Linear Stability Analysis For An Isotropic Materials With Yielding According To Drucker Criterion [11]mentioning

confidence: 99%

Effect of the third invariant on the formation of necking instabilities in ductile plates subjected to plane strain tension

et al. 2021

View full text Add to dashboard Cite

show abstract

“…We consider thatη c SB = 2.5 is large enough for the frozen coefficient theory and the exponential form of the perturbation to be correct hypotheses. Nevertheless, we are aware that this is a strong assumption, and we refer the reader to the recent work of Xavier et al (2020) to obtain further details about an alternative procedure to consider the temporal evolution of the perturbation coefficients. Moreover, we acknowledge that assuming the Figure 9: Comparison between linear stability analysis (LSA) and finite element simulations (FEM).…”

Section: Comparison Between Linear Stability Results and Finite Element Calculationsmentioning

confidence: 99%

“…Note also that η will be referred to as instantaneous instability index everywhere in this paper. Moreover, while in this paper we have assumed the classical frozen coefficients theory and the exponential form of the time dependency of the perturbation for the sake of simplicity, Xavier et al (2020) have recently developed a promising method to relax these assumptions.…”

Section: Linear Stability Analysismentioning

confidence: 99%

Dynamic shear instabilities in metallic sheets subjected to shear-compression loading

Rodríguez-Martínez

Vaz-Romero

N'souglo

et al. 2020

Journal of the Mechanics and Physics of Solids

View full text Add to dashboard Cite

This paper presents an investigation on the formation of adiabatic shear bands in metallic sheets subjected to dynamic shear-compression loading under plane-stress conditions. For that purpose, we have developed a theoretical model based on perturbation analysis, and have performed finite element calculations. The material behavior has been modeled with von Mises plasticity, and the evolution of the yield stress has been considered dependent on plastic strain, plastic strain rate, and temperature. The theoretical model extends the linear stability analysis for simple-shear formulated by Molinari (1997) to shear-compression loading. The numerical simulations have been performed in ABAQUS/Explicit (2016) using a unit-cell model based on the work of Rodríguez-Martínez et al. (2015) in which the shear band formation is favored introducing geometric and material imperfections. Moreover, we have developed a calibration procedure for the stability analysis that enables to make qualitative and quantitative comparisons with the finite element calculations. Both stability analysis predictions and finite element results display the same overall trends, and show that any small negative triaxiality has a profound stabilizing effect on the material behavior delaying, or even preventing, the formation of a shear band. This key outcome has been substantiated for a wide range of strain rates and for materials with different strain hardening coefficients, strain rate sensitivities, and thermal softening behaviors.

show abstract

“…At some time t , we superimpose a small perturbation δS(X, t) t 1 = δS 1 e iξX+η(t−t 1 ) to the fundamental solution, is the perturbation amplitude, ξ is the wavenumber and η is the instantaneous growth rate of the perturbation at 270 time t 1 . While η and ξ are considered time independent (frozen coefficients method) for the sake of simplicity, it has to be noted that a time dependent numerical solution for the linear perturbation analysis has been recently 272 developed by Xavier et al (2020Xavier et al ( , 2021 for von Mises materials subjected to plane strain tension. Moreover, note 273 that the perturbation is inserted perpendicular to the main loading direction, i.e.…”

Section: Governing Equationsmentioning

confidence: 99%

A new analytical model to predict the formation of necking instabilities in porous plates subjected to dynamic biaxial loading

Kumar¹,

N'souglo²,

Rodríguez-Martínez³

2021

Preprint

View full text Add to dashboard Cite

In this paper, we have developed a linear stability analysis to predict the formation of necking instabilities in porous ductile plates subjected to dynamic biaxial stretching. The mechanical behavior of the material is described with the Gurson-Tvergaard-Needleman constitutive relation for progressively cavitating solids (Gurson, 1977; Tvergaard, 1981, 1982; Tvergaard and Needleman, 1984) which considers the voids to be spherical and the matrix material isotropic with yielding defined by the von Mises (1928) criterion. The analytical model is formulated in a two-dimensional framework in which the multiaxial stress state that develops inside the necked region is approximated with the Bridgman (1952) correction factor, superimposing a hydrostatic stress state to the uniform stress field that develops in the plate before localization. As opposed to the linear stability models published so far to model dynamic necking in ductile plates, which consider the material to be fully dense and incompressible, the approach developed in this paper provides new insights into the interplay between porosity and inertia on plastic localization. In addition, the predictions of the theoretical model for the critical strain leading to necking formation have been compared with unit-cell finite element calculations performed in ABAQUS/Explicit (2019). Satisfactory quantitative and qualitative agreement has been found between the theoretical and computational approach for loading paths ranging from plane strain tension to nearly equibiaxial tension, loading rates varying from 100 s−1 to 10000 s−1, and different values of the initial void volume fraction ranging from 0.01 to 0.1. Both analytical and finite element results suggest that the influence of porosity on necking localization increases, due to early voids coalescence, as the loading rate increases and the loading path approaches equibiaxial tension. The original formulation developed in this paper serves as a basis for analytically modeling the dynamic formability of porous ductile plates, and it can be readily extended to consider more complex porous plasticity theories, e.g. constitutive models which consider the anisotropy of the material (Benzerga and Besson, 2001) and/or voids with different shapes (Gologanu et al., 1993; Monchiet et al., 2008).

show abstract

Extension of linear stability analysis for the dynamic stretching of plates: Spatio-temporal evolution of the perturbation

Cited by 12 publications

References 40 publications

Effect of the third invariant on the formation of necking instabilities in ductile plates subjected to plane strain tension

Effect of the third invariant on the formation of necking instabilities in ductile plates subjected to plane strain tension

Dynamic shear instabilities in metallic sheets subjected to shear-compression loading

A new analytical model to predict the formation of necking instabilities in porous plates subjected to dynamic biaxial loading

Contact Info

Product

Resources

About