Numerical Analysis of the Rates of Displacements to Prescribe Constant Proportional Logarithmic Strains in Stretched Sheets

Abstract

The end displacements that permit to prescribe proportional logarithmic strains in stretched sheets during biaxial loading are analyzed here using a finite element procedure. A cruciform specimen of the sheet which in-plane dimensions are normalized with respect to a parameter “l” is considered. The in-plane dimensions of the whole specimen are 8lx8l and the length of each arm is “2l”. The central region of the specimen is square with a side “2l” and a thickness “h/3” while the thickness of the whole sheet is “h”. This thickness is assumed much smaller than the in-plane dimensions of the sheet. The material of the sheet is considered incompressible, elastic-plastic and rate insensitive. Assuming that the behavior of the sheet can be described by generalized plane-stress assumptions, a large strain-plane-stress finite-element program is developed with a finite-strain version of J2 flow theory of plasticity as constitutive law. It is shown that for balanced biaxial state as well as for plane-strain state, the ratios of the increments of displacements at the boundaries of the sheet may be adjusted during the loading histories and the strains measured in the central zone such that x/l ≤0, 2.