Uniaxial True Stress-Strain after Necking - Tyco Electronics

A weighted-average method for determining uniaxial, true tensile stress vs. strain relation after necking is presented for strip shaped samples. The method requires identification of a lower and an upper bound for the true stress-strain function after necking and expresses the true stress-strain relation as the weighted average of these two bounds. The weight factor is determined iteratively by a finite element model until best agreement between calculated and experimental loadextension curves is achieved. The method was applied to various alloys. INTRODUCTION The finite element method (FEM) has become a common engineering tool. If applied to solve mechanical engineering problems it deals mostly with linear elastic problems where the elastic modulus E and Poisson’s ratio n are the only material constants considered. In recent years the demand for nonlinear, plastic analyses has increased noticeably. Crimping results by its very nature in large plastic deformation. It was analyzed recently using FEM by S. Kugener1. Another example where plastic behavior of materials must be considered are large deformations of small contacts in high density connectors. For plastic analysis the FEM requires besides E and n also input of a uniaxial true stress-strain function. This function is usually determined by the applicable ASTM method2. It is well known that in tensile testing, the uniform extension ceases when the tensile load reaches a material specific maximum. At this point the test sample begins to neck. The state of stress changes gradually from the simple uniaxial tension to a complicated condition of triaxial stress for a round bar or of biaxial stress for a thin strip. Because the onset of necking destroys the uniaxial state of stress it is impossible to determine a uniaxial true stress-strain relation by the standard tensile test once necking has started. Thus, for applications in which strain exceeds its value at the onset of necking, the standard tensile test cannot provide data sufficient for modeling. This can seriously limit the use of FEM for large strain applications such as contact forming. For this reason some method has to be found to obtain the true stress-strain relation after necking. For rods Bridgman’s correction method3 is most commonly used to obtain uniaxial true stress-strain relations after necking. Because electrical contacts are almost exclusively in form of flat bars applicability of the Bridgman correction is rather limited. This extends also to laboratory tests, where samples are required that approach in geometry and dimensions those of the corresponding contacts. To the author’s best knowledge, there is no effective way to obtain for strip samples true stress-strain curves after necking. This is a great obstacle for developing a true stress-strain data base for contact spring materials. The study presented here is an attempt to fill that gap. For reasons of clarity some fundamental definitions and concepts are reviewed and discussed. FUNDAMENTAL DEFINITION Strain describes quantitatively the degree of deformation of a body. It is measured most commonly with extensometers and strain gauges. For uniaxial deformation strain can be expressed as e 5 Lf 2 L0 L0 , (1)