Forming limit stresses predicted by phenomenological plasticity theories with anisotropic work-hardening behavior

Forming limit stresses of sheet metals subjected to linear and combined stress paths are analyzed using the M-K model in conjunction with two anisotropic work-hardening models: a work-hardening model which is capable of describing Bauschinger and cross-hardening effects, and a work-hardening model which cannot predict the cross-hardening effect. It is found that the forming limit stress is path-independent when the stress–strain curves for the linear and combined stress paths agree well with each other. On the other hand, the forming limit stress for the combined stress path depends on the strain path when the prestrain changes the subsequent stress–strain relation. We conclude that the stress-based forming limit criterion is efficient only for a material with a work-hardening behavior that is not affected by strain path change. The influence of the work-hardening behavior on the forming limit stress is discussed in detail.     

Effects of superimposed hydrostatic pressure on sheet metal formability

The effect of superimposed hydrostatic pressure on sheet metal formability is studied analytically and numerically. A tensile sample of power-law hardening material under superimposed hydrostatic pressure is first analyzed using the classical isotropic plasticity theory. It is demonstrated that the superimposed hydrostatic pressure p lowers the true tensile stress level at yielding by the amount of p, while material work-hardening is independent of p. It is showed, based on the Considère criterion, that the superimposed hydrostatic pressure increases the uniform strain. The effect of superimposed hydrostatic pressure on sheet metal formability is further assessed by constructing the Forming Limit Diagram (FLD) based on the M-K approach. It is found that the superimposed pressure delays the initiation of necking for any strain path. The difference in predicted FLDs between the superimposed hydrostatic pressure and the stress component normal to the sheet plane is demonstrated. Finally, the effect of superimposed hydrostatic pressure on fracture in round bars under tension is studied numerically using the finite element method, based on the Gurson damage model. The experimentally observed transition of the fracture surface, from the cup-cone mode under atmospheric pressure to a slant structure under high pressure, is numerically reproduced.     

Fragility analysis of bridges under ground motion with spatial variation

Seismic ground motion can vary significantly over distances comparable to the length of a majority of highway bridges on multiple supports. This paper presents results of fragility analysis of highway bridges under ground motion with spatial variation. Ground motion time histories are artificially generated with different amplitudes, phases, as well as frequency contents at different support locations. Monte Carlo simulation is performed to study dynamic responses of an example multi-span bridge under these ground motions. The effect of spatial variation on the seismic response is systematically examined and the resulting fragility curves are compared with those under identical support ground motion. This study shows that ductility demands for the bridge columns can be underestimated if the bridge is analyzed using identical support ground motions rather than differential support ground motions. Fragility curves are developed as functions of different measures of ground motion intensity including peak ground acceleration, peak ground velocity, spectral acceleration, spectral velocity and spectral intensity. This study represents a first attempt to develop fragility curves under spatially varying ground motion and provides information useful for improvement of the current seismic design codes so as to account for the effects of spatial variation in the seismic design of long-span bridges.     

Theoretical study of B2 type technetium AB (A=Tc,B=Ti,V, Nb and Ta) intermetallic compounds

The structural, electronic, elastic and thermal properties of the cubic AB type (A=Tc, B=Ti, V, Nb and Ta) technetium intermetallic compounds have been studied using the full potential linearized augmented plane wave (FP-LAPW) method within the generalized gradient approximation (GGA) and local density approximation (LDA) used for the exchange-correlation potential. The calculated lattice parameters agree well with the experimental results. The calculated electronic properties reveal that these compounds are metallic in nature with partial ionic bonding. The elastic constants obey the stability criteria for cubic system. Ductility for these compounds has been analyzed using the Pugh's rule and Cauchy's pressure revealing ductile in nature of all the compounds. Bonding nature is discussed using Fermi surface, band structure and charge density difference plots.     

An interactive micro-void shear localization mechanism in high strength steels

Ductility of high strength steels is often restricted by the onset of a void-sheet mechanism in which failure occurs by a micro-void shear localization process. For the first time, the micro-void shear instability mechanism is identified here by examining the interactions occurring within a system of multiple embedded secondary particles (carbides ∼10-100 nm), through a finite element based computational cell modeling technique (in two and three dimensions). Shear deformation leads to the nucleation of micro-voids as the secondary particles debond from the surrounding alloy matrix. The nucleated micro-voids grow into elongated void tails along the principal shear plane and coalesce with the micro-voids nucleated at neighboring particles. At higher strains, the neighboring particles are driven towards each other, further escalating the severity of the shear coalescence effect. This shear driven nucleation, growth and coalescence mechanism leads to a decrease in the load-bearing surface in the shear plane and a terminal shear instability occurs. The mechanism is incorporated mathematically into a hierarchical steel model. The simulated response corresponds to experimentally observed behavior only when the micro-void shear localization mechanism is considered.     

Effects of texture on shear band formation in plane strain tension/compression and bending

In this study, effects of typical texture components observed in rolled aluminum alloy sheets on shear band formation in plane strain tension/compression and bending are systematically studied. The material response is described by a generalized Taylor-type polycrystal model, in which each grain is characterized in terms of an elastic–viscoplastic continuum slip constitutive relation. First, a simple model analysis in which the shear band is assumed to occur in a weaker thin slice of material is performed. From this simple model analysis, two important quantities regarding shear band formation are obtained: i.e. the critical strain at the onset of shear banding and the corresponding orientation of shear band. Second, the shear band development in plane strain tension/compression is analyzed by the finite element method. Predictability of the finite element analysis is compared to that of the simple model analysis. Third, shear band developments in plane strain pure bending of a sheet specimen with the typical textures are studied. Regions near the surfaces in a bent sheet specimen are approximately subjected to plane strain tension or compression. From this viewpoint, the bendability of a sheet specimen may be evaluated, using the knowledge regarding shear band formation in plane strain tension/compression. To confirm this and to encompass overall deformation of a bent sheet specimen, including shear bands, finite element analyses of plane strain pure bending are carried out, and the predicted shear band formation in bent specimens is compared to that in the tension/compression problem. Finally, the present results are compared to previous related studies, and the efficiency of the present method for materials design in future is discussed.     

Path-dependence of the forming limit stresses in a sheet metal

The effect of changing strain paths on the forming limit stresses of sheet metals is investigated using the Marciniak–Kuczyński model and a phenomenological plasticity model with non-normality effects [Kuroda, M., Tvergaard, V., 2001. A phenomenological plasticity model with non-normality effects representing observations in crystal plasticity. J. Mech. Phys. Solids 49, 1239–1263]. Forming limits are simulated for linear stress paths and two types of combined loading: a combined loading consisting of two linear stress paths in which unloading is included between the first and second loadings (combined loading A), and combined loading in which the strain path is abruptly changed without unloading (combined loading B). The forming limit stresses calculated for combined loading A agree well with those calculated for the linear stress paths, while the forming limit curves in strain space depend strongly on the strain paths. The forming limit stresses calculated for the combined loading B do not, however, coincide with those calculated for the linear stress paths. The strain-path dependence of the forming limit stress is discussed in detail by observing the strain localization process.     

Material aspects of dynamic neck retardation

Neck retardation in stretching of ductile materials is promoted by strain hardening, strain-rate hardening and inertia. Retardation is usually beneficial because necking is often the precursor to ductile failure. The interaction of material behavior and inertia in necking retardation is complicated, in part, because necking is highly nonlinear but also because the mathematical character of the response changes in a fundamental way from rate-independent necking to rate-dependent necking, whether due to material constitutive behavior or to inertia. For rate-dependent behavior, neck development requires the introduction of an imperfection, and the rate of neck growth in the early stages is closely tied to the imperfection amplitude. When inertia is important, multiple necks form. In contrast, for rate-independent materials deformed quasi-statically, single necks are preferred and they can emerge in an imperfection-free specimen as a bifurcation at a critical strain. In this paper, the interaction of material properties and inertia in determining neck retardation is unraveled using a variety of analysis methods for thin sheets and plates undergoing plane strain extension. Dimensionless parameters are identified, as are the regimes in which they play an important role.     

A constitutive model for elastoplastic solids containing primary and secondary voids

In many ductile metallic alloys, the damage process controlled by the growth and coalescence of primary voids nucleated on particles with a size varying typically between 1 and 100 μm, is affected by the growth of much smaller secondary voids nucleated on inclusions with a size varying typically between 0.1 and 3 μm. The goal of this work is first to quantify the potential effect of the growth of these secondary voids on the coalescence of primary voids using finite element (FE) unit cell calculations and second to formulate a new constitutive model incorporating this effect. The nucleation and growth of secondary voids do essentially not affect the growth of the primary voids but mainly accelerate the void coalescence process. The drop of the ductility caused by the presence of secondary voids increases if the nucleation strain decreases and/or if their volume fraction increases and/or if the primary voids are flat. A strong coupling is indeed observed between the shape of the primary voids and the growth of the second population enhancing the anisotropy of the ductility induced by void shape effects. The new micromechanics-based coalescence condition for internal necking introduces the softening induced by secondary voids growing in the ligament between two primary voids. The FE cell calculations were used to guide and assess the development of this model. The use of the coalescence condition relies on a closed-form model for estimating the evolution of the secondary voids in the vicinity of a primary cavity. This coalescence criterion is connected to an extended Gurson model for the first population including the effect of the void aspect ratio. With respect to classical models for single void population, this new constitutive model improves the predictive potential of damage constitutive models devoted to ductile metal while requiring only two new parameters, i.e. the initial porosity of second population and a void nucleation stress, without any additional adjustment.     

Two-dimensional bilinear oscillator: group-preserving scheme and steady-state motion under harmonic loading

We study the dynamic behavior of a single-mass, two-degree-of-freedom bilinear oscillator, whose restoring force obeys a bilinear elastoplastic law. For the purpose of calculation we develop a group-preserving scheme utilizing the internal symmetry group of the model in the plastic phase so as to satisfy the yield condition exactly at each time step. Then the oscillator is subjected to biaxial harmonic excitations, and the responses are displayed. Based on circular motion assumption and harmonic balance method, the closed-form formulae to estimate the steady-state responses are derived, which are then compared with the results calculated by the group-preserving scheme. The accuracy of the formulae is thus confirmed. Simple formulae to select minimum driving force amplitudes or maximum yield forces which cause the bilinear oscillator steadily oscillating in the plastic phase are derived. For the benefit of engineers’ use we also derive closed-form formulae to decide the force ratios required to sustain the user-specified dissipation per cycle.