Analysis of necking based on a one-dimensional model
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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