Analysis of necking based on a one-dimensional model

doi:10.1016/j.jmps.2015.12.018

Journal of the Mechanics and Physics of Solids

Volume 97, December 2016, Pages 68-91

https://doi.org/10.1016/j.jmps.2015.12.018 Get rights and content

Abstract

Dimensional reduction is applied to derive a one-dimensional energy functional governing tensile necking localization in a family of initially uniform prismatic solids, including as particular cases rectilinear blocks in plane strain and cylindrical bars undergoing axisymmetric deformations. The energy functional depends on both the axial stretch and its gradient. The coefficient of the gradient term is derived in an exact and general form. The one-dimensional model is used to analyze necking localization for nonlinear elastic materials that experience a maximum load under tensile loading, and for a class of nonlinear materials that mimic elastic-plastic materials by displaying a linear incremental response when stretch switches from increasing to decreasing. Bifurcation predictions for the onset of necking from the simplified theory compared with exact results suggest the approach is highly accurate at least when the departures from uniformity are not too large. Post-bifurcation behavior is analyzed to the point where the neck is fully developed and localized to a region on the order of the thickness of the block or bar. Applications to the nonlinear elastic and elastic-plastic materials reveal the highly unstable nature of necking for the former and the stable behavior for the latter, except for geometries where the length of the block or bar is very large compared to its thickness. A formula for the effective stress reduction at the center of a neck is established based on the one-dimensional model, which is similar to that suggested by Bridgman (1952).

Section snippets

Introduction with précis of the dimensional reduction

The tension test is one of the canonical methods to measure the stress–strain behavior of materials. A well-known complication for ductile materials which undergo a maximum load in the test is necking wherein deformation localizes in a region on the order of the specimen thickness at some location along the length of the specimen. Considère (1885) described the connection of the onset of the localization process to the maximum load, and the criteria for the stress and strain at the onset of

Dimensional reduction

The goal of this section is to establish by means of asymptotic expansions the second-gradient bar model outlined above, which describes the onset and development of localization in a prismatic beam. We work with an hyperelastic material which is orthotropic and has its material symmetry axis aligned with the axis of the prismatic domain (note that his includes isotropic materials as a particular case). The material is assumed compressible; the incompressible limit will be worked out at the end

Applications to nonlinear elastic materials

In this section the accuracy of the one-dimensional model is demonstrated: we compare its prediction for the stretch at which the onset of necking occurs for a block under tension, with the results of an exact bifurcation analysis of the same problem. The one-dimensional model is then applied to predict the full necking response of a nonlinear elastic material that has a maximum load in uniaxial tension. Attention will focus on the block, but it is evident from the close similarity between the

Application to elastic-plastic materials

In the context of necking, the primary difference between a metal and a nonlinear elastic material (such as that considered in the previous section) is the response of the material when the stretch switches from increasing to decreasing—elastic unloading in plasticity terminology. A metal deformed into the plastic range in tension responds with a stiff linear elastic response as soon as the stretch rate is reversed. By contrast, the unloading response of the nonlinear elastic material is

Conclusions and suggestions for further research

A one-dimensional model has been derived for the necking analysis of rectangular blocks and round bars. It is carried out within the framework of nonlinear elastic solids but it has been extended for applications to elastic-plastic solids. By incorporating gradients of the strains that arise during necking, the model is capable of capturing the highly nonlinear localization process that accompanies necking. Bifurcation from the uniform state into the necking mode is accurately reproduced by the

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Citation Excerpt :
Taffetani and Ciarletta [18] studied the effect of capillary energy on the beading instability of soft cylindrical gels. In these previous studies, however, the positions and wavenumber of the consecutive necking zones were usually predefined, and the underlying mechanisms of the complicated necking phenomena remain unclear [19]. In the present study, we make an attempt to understand the energetic mechanisms underlying the necking instability and predict the critical condition and morphological evolution of necking.
Necking is an instability phenomenon widely observed in metallic and polymeric materials. On the basis of the minimum energy barrier criterion, we propose a theoretical method, combined with Monte Carlo simulations, to predict the linear instability and post-bifurcation behavior of materials with a non-convex constitutive law. For illustration, this method is applied to analyze the necking bifurcation behavior of a plate under uniaxial tension. Especially, we consider the effect of surface tension, which is important for the necking of polymeric materials. For soft materials with high surface energy, surface effects may greatly influence the morphological evolution of necking, such as the position and wavenumber of the consecutive necking zones. This work not only helps understand various necking phenomena in biological and engineering materials, but also can be extended to analyze some other instability problems, e.g., surface wrinkling.

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Analysis of necking based on a one-dimensional model

Abstract

Section snippets

Introduction with précis of the dimensional reduction

Dimensional reduction

Applications to nonlinear elastic materials

Application to elastic-plastic materials

Conclusions and suggestions for further research

J. Math. Anal. Appl.

Int. J. Solids Struct.

Acta Metall.

J. Mech. Phys. Solids

J. Mech. Phys. Solids

Acta Metall.

J. Mech. Phys. Solids

J. Mech. Phys. Solids

J. Mech. Phys. Solids

Int. J. Solids Struct.

Shear and necking instabilities in nonlinear elasticity

J. Elast.

Studies in Large Plastic Flow and Fracture, Metallurgy and Metallurgical Engineering Series