Geometric models of internal shape change as
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I.
INTRODUCTION
MANYforming operations require plane strain extension within some regions of the deforming material. Failure can then develop in those regions by the formation of shear bands which collectively cross the sample thickness. Recent experimental studies 1'2 show that sample-scale shear bands develop in a spatially nonuniform manner, at a time while plastic flow localization is taking place throughout the sample. In order to describe this behavior, it is useful to have geometric models for the development of both internal shape change and plastic flow localization. Then experimental results can be compared and interpreted relative to these models. Correct geometrical descriptions of the kinematics of flow are also needed as the starting point for mechanics models of plastic flow localization. Indeed, many of the quantitative aspects of such models arise directly from geometrical relationships that are assumed. While there are a number of kinematic models available in the literature, none fully describes the internal shape changes seen in recent experiments. This paper considers a number of geometric and kinematic models for describing the shape changes that take place throughout sheet as plane strain extension moves ends of the sheet farther apart. These models are two- and threedimensional in terms of the shape changes they describe. Each is a simplification of the general approach taken in continuum mechanics to describe sample-scale shape change. The deforming body (in this case, thin sheet) is subdivided into a collection of contiguous solid elements (Figure 1). Each solid element deforms by elastic and plastic flow. External shape change involves sheet extension in one direction, thinning of sheet in a perpendicular direction, and zero shape change in the third, mutually orthogonal direction. Internal shape changes within elements of the sheet can be quite different, but are interconnected by the need for continuity of material between neighboring elements. The models assume that plastic flow localization has
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Fig. 1--Sheet is subdivided into three-dimensional solid elements.
already progressed to some extent, and that portions of the sheet are elastically rigid. The models describe shape change in the remainder of the sheet in which flow is still taking place. These models are built on concepts supplied by numerous investigators. An overview of the literature will be provided, after a systematic presentation of several types of kinematical models.
II. PARALLEL SLAB MODELS FOR DESCRIBING NONUNIFORM SHEET EXTENSION AND THROUGH-THICKNESS NECKING
A. Model 1A: No Shearing Flow within Slabs J.E. BIRD, Assistant Professor, and J.M. CARLSON, Graduate Student, are with School of Materials Engineering, Purdue University, West Lafayette, IN 47907. Manuscript submitted July 21, 1986. METALLURGICALTRANSACTIONS A
In this highly simplified model of sheet extension, the sheet is divided into solid slabs, which each lie normal to the direction of imposed sheet extensi
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