Forming limit diagrams of strain-rate-dependent sheet metals

Abstract This paper presents a predictive model of localized necking for strain-rate-dependent sheet metals. This is achieved by the development of a modified vertex theory for use as a localized necking criterion. The vertex theory is von Mises based and isotropic. The study is motivated by the apparent discrepancies between the measured forming limit diagrams (FLDs) of strain-rate-dependent sheet metals and those predicted using the conventional vertex theory. The modified theory considers the rate-dependent power-hardening material rule. A novel form of quasi-linear stress–strain relation based on the power-hardening rule is analytically derived to model the constitute behavior of the rate-dependent material. The localized necking criterion of the sheet metals is based on the vertex theory, which assumes that localized necking occurs simultaneously with the initiation of a vertex on the yield surface. The stress–strain relation of rate-dependent material is coupled into the vertex theory to deduce the critical conditions for localized necking on both sides of the FLD. Numerical results indicate that the forming limits of rate-dependent sheet metals follow quite different rule as that for rate-independent sheet metals. The ratio between the strain-rate-hardening index and the strain-hardening index plays an important role in the strain-rate effect on FLDs. A typical strain-rate-dependent metal, AKDQ steel, is chosen for validation of the modified vertex theory. The tensile stress–strain curves of AKDQ are tested on both regular and high-speed MTS, covering a wide range of strain rate from 10−5 to 10 s−1. The hardening law of AKDQ is found to depend greatly on the strain rate, while the strain-rate-hardening index is not a material constant but dependent on the strain rate. Forming limit tests are also performed for AKDQ sheets to measure the FLD. The measured FLD of AKDQ is compared with the prediction based on the developed theory and good agreement is observed.

[1]  R. Hill,et al.  On discontinuous plastic states, with special reference to localized necking in thin sheets , 1952 .

[2]  P. Hu.  Rate-dependent quasi-flow corner theory for elastic/visco-plastic materials , 2004 .

[3]  James R. Rice,et al.  Localized necking in thin sheets , 1975 .

[4]  S. P. Keeler.  Plastic instability and fracture in sheets stretched over rigid punches , 1961 .

[5]  C. L. Chow,et al.  A non‐proportional loading finite element analysis of continuum damage mechanics for ductile fracture , 1990 .

[6]  P. B. Mellor,et al.  Some comments on the limit strains in sheet metal stretching , 1975 .

[7]  R. Hill.  A general theory of uniqueness and stability in elastic-plastic solids , 1958 .

[8]  M. Jie,et al.  Localized Necking Criterion for Strain-Softening Materials , 2005 .

[9]  Davide Bigoni,et al.  Uniqueness and localization—I. Associative and non-associative elastoplasticity , 1991 .

[10]  Z. Marciniak,et al.  Limit strains in the processes of stretch-forming sheet metal , 1967 .

[11]  C. Chow,et al.  Prediction of forming limit diagrams for AL6111-T4 under non-proportional loading , 2001 .

[12]  N. D. Cristescu,et al.  A procedure for determining the constitutive equations for materials exhibiting both time-dependent and time-independent plasticity , 1972 .

[13]  Giulio Maier,et al.  On linear versus nonlinear flow rules in strain localization analysis , 1989 .

[14]  S. R. Bodner,et al.  Constitutive Equations for Elastic-Viscoplastic Strain-Hardening Materials , 1975 .

[15]  M. Jie.  Generalized criteria for localized necking in sheet metal forming. , 2003 .

[16]  N. S. Ottosen,et al.  Properties of discontinuous bifurcation solutions in elasto-plasticity , 1991 .

[17]  Stuart P. Keeler,et al.  Determination of Forming Limits in Automotive Stampings , 1965 .

[18]  C. Chow,et al.  Prediction of Forming Limit Diagram Based on Damage Coupled Kinematic-Isotropic Hardening Model Under Nonproportional Loading , 2002 .

[19]  Z. Marciniak,et al.  The mechanics of sheet metal forming , 1992 .

[20]  L. Szabó.  Comments on loss of strong ellipticity in elastoplasticity , 2000 .

[21]  Howard L. Schreyer,et al.  Bifurcations in elastic-plastic materials , 1993 .

[22]  Plastic Flow Properties in Relation to Localized Necking in Sheets , 1978 .

[23]  R. Pearce,et al.  Instability and fracture in sheet metal , 1974 .

[24]  J. Rice,et al.  CONDITIONS FOR THE LOCALIZATION OF DEFORMATION IN PRESSURE-SENSITIVE DILATANT MATERIALS , 1975 .

[25]  C. L. Chow,et al.  Forming Limit Analysis of Sheet Metals Based on a Generalized Deformation Theory , 2003 .

[26]  Dong Won Kim,et al.  The effect of void growth on the limit strains of steel sheets , 1983 .

[27]  P. Perzyna.  Fundamental Problems in Viscoplasticity , 1966 .

[28]  Abhijit Chandra,et al.  A unified bifurcation analysis of sheet metal forming limits , 2001 .

[29]  C. L. Chow,et al.  Viscoplastic Constitutive Modeling of Anisotropic Damage Under Nonproportional Loading , 2001 .

[30]  M. Jie,et al.  A Damage-coupled Criterion of Localized Necking Based on Acoustic Tensor , 2007 .

[31]  M. Jie,et al.  Forming limits of AL 6022 sheets with material damage consideration: theory and experimental validation , 2004 .

[32]  Gorton M. Goodwin,et al.  Application of Strain Analysis to Sheet Metal Forming Problems in the Press Shop , 1968 .