Analysis of Strain-Hardening Viscoplastic Wide Sheets Subject to Bending under Tension

Sergei Alexandrov,Elena Lyamina

doi:10.3390/met12010118

Sergei Alexandrov, Elena Lyamina

Open Access

https://doi.org/10.3390/met12010118

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Abstract

The present paper provides an accurate solution for finite plane strain bending under tension of a rigid/plastic sheet using a general material model of a strain-hardening viscoplastic material. In particular, no restriction is imposed on the dependence of the yield stress on the equivalent strain and the equivalent strain rate. A special numerical procedure is necessary to solve a non-standard ordinary differential equation resulting from the analytic treatment of the boundary value problem. A numerical example illustrates the general solution assuming that the tensile force vanishes. This numerical solution demonstrates a significant effect of the parameter that controls the loading speed on the bending moment and the through-thickness distribution of stresses.

Highlights

Bending is frequently used as a test for determining various material properties
Paper [13] developed a semi-analytic method to treat this region without any simplification. This method was combined with various constitutive equations to describe pure bending and bending under tension of wide sheets [14]
The motivation of the present paper is to provide a semi‐analytic solution of the same level of 2 of 12 complexity as available solutions but for quite a general material model

Summary

Introduction

Bending is frequently used as a test for determining various material properties. A four-point bend apparatus was used in [1] for getting stress–strain curves in the tension and compression of three materials: beryllium, cast iron, and copper. Paper [13] developed a semi-analytic method to treat this region without any simplification This method was combined with various constitutive equations to describe pure bending and bending under tension of wide sheets [14]. Viscoplastic or strain-hardening viscoplastic constitutive equations are required to analyze bending processes adequately [15,16,17]. It is desirable to have a theoretical solution for a general material model. The motivation of Metals 2022, 12, 118 equations, it is desirable to have a theoretical solution for a general material model. The present paper assumes that the tensile yield stress is a function of the equivalent strain, εeq, and the equivalent strain rate, ξeq, obtained by fitting any experimental data.

Kinematics

Stress Solution

Numerical Method

Numerical Example

Findings

Conclusions