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

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).

Numerical Simulation of Some Steel Structural Elements with Uncertain Initial Porosity

The main research purpose of this work was to study the elasto-plastic responses of some fundamental steel structural elements exhibiting stochastic volumetric microvoids. The constitutive model of the steel material was consistent with the deterministic Gurson–Tvergaard–Needleman (GTN) porous plasticity model, where some of the microvoids parameters have additionally been defined as Gaussian random variables. The iterative stochastic finite element method implemented based on the-tenth order stochastic perturbation technique was utilized in numerical experiments. An interoperability of the computer algebra system MAPLE 2019 and the finite element method system ABAQUS was employed to study the influence of the initial microvoids f0 with uncertainty in the structural steel on the statistical scattering of the resulting stresses and deformations. The basic probabilistic characteristics of the structural response were computed and contrasted with statistical estimators inherent in the Monte–Carlo simulation and also with the results obtained from the semi-analytical probabilistic method. Reliability indices according to the first-order reliability method (FORM) were also calculated. Two numerical illustrations included the (i) tension test of the round cylindrical steel rebar and the (ii) bending test of the steel I-beam restrained at both its ends. Expectations and coefficients of variation of the structural responses confirmed here the importance of the microvoids for the stochastic elasto-plastic behavior of some basic engineering structures, where tensile stress plays an important role in designing procedures.

Micromechanical modelling of ductile fracture in pipeline steel using a bifurcation-enriched porous plasticity model

In this paper, we investigate the possibility of predicting ductile fracture of pipeline steel by using the Gurson–Tvergaard–Needleman (GTN) model where the onset of void coalescence is determined based on in situ bifurcation analyses. To this end, three variants of the GTN model, one of which includes in situ bifurcation, are calibrated for a pipeline steel grade X65 using uniaxial and notch tension tests. Then plane-strain tension tests and Kahn tear tests of the same material are used for assessment of the credibility of the three models. Explicit finite element simulations are carried out for all tests using the three variants of the GTN model, and the results are compared to the experimental data. The capability of the simulation models to capture onset of fracture and crack propagation in the pipeline steel is evaluated. It is found that the use of in situ bifurcation as a criterion for onset of void coalescence in each element makes the GTN model easier to calibrate with less free parameters, all the while obtaining similar or even better predictions as other widely used formulations of the GTN model over a wide range of different stress states.

Dynamic spherical cavity expansion in Gurson materials with uniform and non-uniform distributions of porosity

This paper investigates both theoretically and using finite elements the elastoplastic field induced by a pressurizedspherical cavity expanding dynamically in an infinite medium modelled using the Gurson-Tvergaard-Needleman porous plasticity approach. The theoretical model, which assumes that the porosity is uniformly distributed in the material and the cavitation fields are self-similar, incorporates artificial viscous stresses into the original formulation of Cohen and Durban (2013b) to capture the shock waves that emerge at high cavitation velocities. The finite element calculations, performed in ABAQUS/Explicit (2013) using the Arbitrary Lagrangian Eulerian adaptive meshing available in the code, simulate the cavity expansion process in materials with uniform and non-uniform distributions of porosity. The finite element results show that the distribution of porosity has small influence on the cavitation velocity, as well as on the location of the shock wave, which are primarily determined by the cavity pressure and the average material properties. In contrast, it is shown that the intensity of the shock wave, evaluated based on the maximum value of the plastic strain rate within the shock, depends on the local material porosity. The ability of the theoretical model to reproduce the numerical results obtained for the various distributions of porosity used in this work is exposed and discussed.

Computer Simulation of Stochastic Energy Fluctuations in Tensile Test of Elasto-Plastic Porous Metallic Material

The main aim of this work is the computational implementation and numerical simulation of a metal porous plasticity model with an uncertain initial microdefects’ volume fraction using the Stochastic Finite Element Method (SFEM) based on the semi-analytical probabilistic technique. The metal porous plasticity model applied here is based on Gurson–Tvergaard–Needleman theory and is included in the ABAQUS finite element system, while the external probabilistic procedures were programmed in the computer algebra system MAPLE 2017. Hybrid usage of these two computer systems enabled the determination of fluctuations in elastic and plastic energies due to initial variations in the ratio of the metal micro-voids, and the calculation of the first four probabilistic moments and coefficients of these energies due to Gaussian distribution of this ratio. A comparison with the Monte-Carlo simulation validated the numerical efficiency of the proposed approach for any level of input uncertainty and for the first four probabilistic characteristics traditionally seen in the experimental series.

Stochastic Elasto-Plastic Analysis of Necking Bar with Random Tvergaard Coefficients

This paper reports on the computational modelling of static extension tests of the round steel bar. The main objective was to apply the generalised stochastic perturbation technique implemented as the Stochastic Finite Element Method to carry out the numerical simulation of its elasto-plastic behaviour. This approach was based on: the general order Taylor expansion of all input random variables and the resulting state functions of their average means, as well as on the Least Squares Method employed to determine analytical functions of in-between design parameters and the given structural responses. Tvergaard coefficients were assumed as the uncorrelated Gaussian random variables to check the effect of material porosity uncertainty on the statistical scattering of its deformations and stresses. The computational implementation employed the FEM system ABAQUS and computer algebra system MAPLE, including polynomial and non-polynomial local response functions of the displacements, plastic strains and reduced stresses. Moreover, 4-node axisymmetric, continuum, reduced-integration FEM elements (CAX4R) were used in the conducted analyses. The basic probabilistic characteristics of the structural response (expectations, coefficients of variation, skewness and kurtosis) were determined throughout the entire deformation process as the functions of input uncertainty level. The obtained results were finally contrasted with the classical Monte-Carlo Simulation scheme and the semi-analytical technique for input coefficient of variation of porous plasticity coefficients not larger than 0.20.