Limit strains in the processes of stretch-forming sheet metal

This paper examines the conditions for instability of plastic strain under plane stress for a material conforming to the Mises-Hencky yield condition and strain-hardening according to a unique relationship between root-mean-square values of shear stress (q) and incremental strain (δψ).

If, under fixed loading conditions, the material undergoes a strain increment which is consistent with the applied stress system, the conditions are stable or unstable according as the increment in representative yield stress is greater or less than the increment in representative induced stress.

The strain at which instability arises is found in terms of the biaxial stress ratio p2/p1 under different conditions of applied loading, and the effect is demonstrated of strain-hardening according to an empirical relation of the type q = c (a + ψ)n.

The analysis is also applied to certain cases of non-uniform stress distribution. In the case of the hydrostatic bulge results are obtained showing a critical thinning ranging from 26 per cent for a non-hardening material to about 45 per cent for typical strain-hardening materials, values in general agreement with experimental data. Conditions over the punch head in the pressing of a cylindrical shell are discussed but computations are not attempted.

Plastic instability under plane stress

After a brief historical introduction, recent achievements in the theory of plasticity are surveyed with emphasis on applications in mechanical engineering.Kinematic models are presented that indicate the complexities of mechanical behaviour in the plastic range. The fundamental theorems of limit analysis are discussed, and their application to two- and three-dimensional problems is illustrated by examples. Shakedown analysis and limit design are defined. Problems involving large plastic deformations are discussed with special reference to metal forming processes. Applications of the theory of plasticity to impact testing and blast damage are reviewed. Recent changes in the theory of structural stability in the plastic range are mentioned. Throughout the lecture, impending developments of the theory of plasticity are indicated.

The Theory of Plasticity: A Survey of Recent Achievements

Impact loading of plates and shells by free-flying projectiles: A review

Since its first application to thin films in the 1950's the bulge test has become a standard technique for measuring thin film mechanical properties. While the apparatus required for the test is simple, interpretation of the data is not. Failure to recognize this fact has led to inconsistencies in the reported values of properties obtained using the bulge test. For this reason we have used the finite element method to model the deformation behavior of a thin film in a bulge test for a variety of initial conditions and material properties. In this paper we will review several of the existing models for describing the deformation behavior of a circular thin film in a bulge test, and then analyze these models in light of the finite element results. The product of this work is a set of equations and procedures for analyzing bulge test data that will improve the accuracy and reliability of this technique.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0884291400017027

Analysis of the accuracy of the bulge test in determining the mechanical properties of thin films

Summary Explicit formulae are obtained for the stresses in a metal diaphragm which is bulged plastically by lateral pressure. The predicted influence of work-hardening on the shape of the profile, and on the relation between polar strain and curvature, agrees well with experimental data. A simple expression is developed for the instability strain.

R. Hill

Papers

C. A theory of the plastic bulging of a metal diaphragm by lateral pressure