The finite-element method has been used extensively for elastic problems. The method is a powerful one and is reserved for problems that can be programmed and solved by computer techniques. The literature over the last 20 years abounds with references to finite-element development and application,and many all-purpose finite-element codes are now commercially available. As the name implies,the basic building block is an element of finite dimensions . The object to be analyzed is divided into a number of these small elements. These elements are joined at corners ,and it is usually assumed that the stress is uniform throughout the element. The element distortions are computed by conventional theory. The total behavior of the structure depends on the intergrated effects of each of the parts. Thus since a part is usually divided into a multitude of element,a solution is possible with the help of a computer. The accuracy of the finite-element method depends on both the type of problem and the number and type of elements selected. In the late 1960s the success of the finite-element problems stimulated the work of extending the application of the method to the area of plastic deformation. It was originally applied to elastic-plastic problems,ones in which the plastic strain is of the order of the elastic one. Here the strain is separated into an elastic part and a plastic part. The elastic part is governed by Hooke’s law, while the plastic part used the Prandtl-Reuss equations. The nonlinearity in the constitutive equations is satisfied iteratively. This can be accomplished by either the initial-strain method or the initial-stress method. Of these to methods, the initial-stress method has found increased favor. With this method,the deformation zone can be very accurately determined.It needs to be pointed out that the finite-element method is not geometry-dependent. Rather,with this technique one can analyze arbitrary geometrically complicated structures. For large deformations it is generally not necessary to consider elastic deformations. In fact ,in the analysis of most metal forming operations, as a rule one can neglect them and employ the rigid-plastic material model. An exception to this rule is that it is generally mandatory to utilize elastic-plastic analysis in order to be able to predict foaming defects. Defects comprise the initation and growth of internal or surface cracks in the deforming metal or the localization of deformation through plastic instability, which could impair the dimensional accuracy of the finished workpiece. Similarly, residual stresses in the unloaded workpiece cannot be evaluated using a rigid-plastic model.
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