CAE-Companion-2018-2019

Modeling of Materials & Connections WISSEN CAE

Strain Hardening and Strain Rate Sensitivity The strain hardening can be derived from the classical tensile tests. The input for the FEA codes, however, is true stress versus true plastic strain. The raw data of the tensile tests are force F and elongation ∆L of the extensometer. Based on the initial cross section A 0 and the initial length of extenso­ meter L 0 of the tensile tests the curve can be expressed as engineering stress σ eng =F/A and engineering strain e=∆L/L 0 . This curve can be used only up to uniform elongation as the strain is no longer homogeneous in the specimen for a higher elongation. The true stress σ true and true plastic strain ε can be derived as follows: σ true = σ eng · (1+e); ε tot = ln (1+e); ε pl =ε tot - σ true / E This hardening curve has to be approximated and extrap- olated for the input in FEA. In general cases higher strains than the uniform elongation can be reached in a deformed structure. A well known and robust hardening law is the Swift model: K, ε 0 and the strain hardening exponent n are material dependent parameters. Most of the metallic materials show a positive strain rate sensitivity, i.e. the flow stress increas- es with strain rate. Steels show a pronounced strain rate sensitivity. In general the strain rate sensitivity decreases with increasing yield strength of the steel. Aluminium alloys show a low or even negative strain rate sensitivity at low strain rates, but a positive one for high strain rates. The strain rates of a deep drawing simulation are in the range of 0.001-0.1 1/s. The strain rate in a high speed crash simulation can reach locally strain rates of 500 1/s. The strain rate sensitivity can be expressed either by analytical laws which scale the quasi-static hardening curve as a function of the strain rate or by providing multiple hardening curves for the relevant strain rate regime. Failure Criteria As metallic materials are typically ductile, strain-based failure criteria are dominant for this group of materials. Many CAE engineers favour to use the total elongation or engineering fracture strain from the tensile test as a fracture criterion. However this value is not a real material parameter as the specimen elongation does not resolve the local strain in the diffuse neck of a tensile test. In addition the fracture strain is a strong function of the stress state. A sheet under bending will fail first on the surface with tensile load despite the same equivalent strain appears in compression without failure. σ true = K (ε 0 +ε) n

Completely wrong conclusion can be drawn from this simple criterion.

Figure 2: Failure modes for metallic materials

The main failure mechanism of metallic sheets in sheet metal forming and crash is the onset of necking. At a sudden point of deformation the strain hardening and the strain rate sensitivity can no longer avoid a localization of the strain. Due to strain localization a fracture appears inside the neck after a small increase of the global strain. Therefore the necking itself can be used as a failure criterion. As industrial sheet metal forming simulations and crashworthiness simulations are mainly based on shell discretization the localized necking cannot be resolved directly – the width of the neck is in the dimension of the sheet thickness. Forming Limit Curves (FLC) are the standard approach to predict the onset of necking. The FLC is expressed as major principal strain versus minor principal strain at the onset of necking. However the classical FLC is limited to linear strain paths. More advanced models have to be used in case of nonlinear strain paths. For advanced high strength steels and aluminium sheets fracture can happen prior to localized necking. This fracture can be caused either by void growth and void coalescence (ductile normal fracture) or by shear band localization (ductile shear fracture). The fracture limit curves for these fracture modes can be expressed by the equivalent plastic strain at fracture as a function of the relevant stress state parameter. CAEWissen by courtesy of MATFEMPartnerschaft, Munich (www.matfem.de)

60