Advancing mechanics of Barrelling Compression Test

Highlights

• Extended Avitzur's limit analysis solution for compression test.

• A solution for friction coefficient during the test.

• Closed form expressions for strain distribution in the test sample.

• Compared extended solutions with reference solutions.

Abstract

Compression Test (CT) plays a fundamental role in our understanding of materials science and engineering. This is due to the test's capacity to physically simulate different deformation scenarios. Mechanics of CT is critical to model stored energy, microstructure and behaviours in a deforming material. To simulate “shear and heterogeneous” deformation, friction, constitutive behaviour, etc., new closed form solutions of CT are needed. Developing detailed solutions for the test is a fundamental priority to represent the deformation more reliably.

Expressions for effective strain and the test sample's free surface were derived here to extend “Avitzur's solution” beyond a limit analysis. The extended Avitzur (EA) solution was employed in a “constant friction factor mode” to make it more comparable with other reference solutions such as the Cylindrical Profile Model (CPM) and the Finite Element (FE). Example EA and CPM solutions were obtained and compared to those of a FE model. Partial enforcement of incompressibility and a rough estimation of barrelling parameter in Avitzur solution were found to be responsible for unrealistic prediction of the sample's profile and friction factor. EA's effective strain and strain rate along the sample's centre line, particularly at its centre, agreed well with the FE predictions. Therefore, the sample centre was suggested as the representative point at which the deformation path can be traced for the constitutive modelling using EA solution.

Introduction

Material characterization is commonly carried out based on simple interpretation of Compression Test (CT) results and simplified solutions of the test (see for example Ashtiani and Shahsavari, 2016, Mirzadeh, 2014, Serajzadeh and Taheri, 2003). A significant part of the simplification is due to friction and barrelling of the test sample. Friction is inherent in the test and therefore all material characterizations by the test (e.g. Mirzadeh, 2015, Solhjoo et al., 2017, Wang et al., 2016) rely on accurate identification of friction. Closed form solutions of CT include Cylindrical Profile Model (CPM), Slab Model (Rowe, 1979) and Avitzur Model (Avitzur, 1980). Numerically, CT has been mostly solved using finite element (FE) analysis. A numerical solution of CT needs both friction factor and material's flow curve as a-priori knowledge to estimate strain, strain rate and temperature in the sample and loads at its interface with the CT anvil (e.g. Albert and Rudnicki, 2001). Both CPM and Slab Models of CT ignore shearing deformation in their sample and need friction factor as an input.

Avitzur's limit analysis (Avitzur, 1980) was employed 20 years later (Ebrahimi and Najafizadeh, 2004) to provide a rough identification of the test's friction factor with low barrelling. This was done by a coarse estimation of Avitzur's barrelling parameter based on the sample's deformed geometry. The solution, included shearing deformation and barrelling in the sample, is considered a quick and easy method to calculate friction factor during CT for a low barrelling range (Solhjoo, 2010). Given the sample's geometry before and after the test, the solution identifies CT's friction factor without a need to material's constitutive properties. Although the solution enables a more convenient simulation of friction compared to other methods such as ring compression (Sofuoglu and Rasty, 1999), it is incapable to simulate material's constitutive behaviour and it underestimates the friction coefficients for the simulation range (i.e. moderate to high strain and friction).

Thermo-mechanical finite element models of barrelling CT were constructed by Fardi et al. (2017) using SFTC-DEFORM Premier software (Scientific Forming Technologies Corp.) and ABAQUS. They used the models to assess deformation heterogeneity and sensitivity of the barrelling CT results to its deformation parameters.

In this work, a method to quantify Avitzur's barrelling parameter is presented: a “fixed friction factor algorithm” was incorporated in Avitzur solution to estimate the parameter for the sake of benchmarking. This is to allow a meaningful comparison of Avitzur solutions with other reference “fixed friction factor models” including Cylindrical Profile Model (CPM) and a FE model.

Sample's free surface (deformed profile) is estimated using incremental solution of Avitzur's barrelling parameter. We also extend Avitzur's solution to estimate distribution of the effective strain in CT's sample. The Extended Avitzur (EA) solution is useful to examine deformation heterogeneity in the sample. EA solution offers unique advantages and potentials. It is capable of handling both friction and constitutive identification simultaneously; friction factor identified by EA can be directly used with the test's load displacement data to characterize the sample's constitutive behaviour. Given the advantages of EA's solution, its critical role and due to the “non-uniqueness of its solution” the solution deserves a thorough assessment.

EA analysis is assessed here by comparing its case solutions against the two reference solutions; namely CPM and FE. DEFORM2-3D software is used to obtain FE solutions for aluminium 1051 and stainless steel AISI316 samples.

The solutions included profile radius, variations of top and mid-plane diameters with sample's height, effective strain and strain rate contour maps. The assessment shows that distribution of strain and strain rate in the sample obtained by EA's solution is reasonably closer to those of FE compared to the estimations by CPM or Slab Method. It also indicates limitations in estimating the barrelling by Avitzur's analysis. It is recommended to include a volume constancy term in the Avitzur's upper bound solution to avoid the limitation.