Research paperAdvancing mechanics of Barrelling Compression Test
Graphical abstract
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.
Section snippets
Barrelling measurement
Fig. 1 illustrates a CT sample, its key parameters and the symbols used to represent and measure the sample's barrelling. Fig. 1a and b show geometry of the sample before and during CT, respectively. An arbitrary point P(r, z) in the deforming body is shown in Fig. 1b. The profile radius for this point is R(z) which marks the moving boundary of the barrelled surface (free surface). Also, axial and radial velocity field components for point P are shown as and , respectively in the same
Assessment of extended Avitzur analysis
All available solutions for the CT problem, including current EA solution, are classified as weak formulations. In a weak formulation, the key relationships such as kinematic, stiffness and equilibrium are no longer required to hold absolutely and are only fulfilled in an average or an integral sense. (more details on the weak formulations can be found in (Tonti, 2013)). Therefore, a weak solution is not unique; given a kinematically admissible velocity field, a solution can be found by taking
Computational conditions
A diagnostic analysis was carried out by comparing a number of sample EA, CP and FE solutions to demonstrate the error in EA solutions. The solution inputs and their assumptions are briefly discussed here.
Results
In order to analyse EA's model, its sample solutions are compared with those of two reference models namely: (A) the conventional Cylindrical Profile Model and (B) a FE model employing DEFORM2D3D commercial package. The comparisons include: “deformed sample's profile” and “distributions of strain rate and strain in the sample”.
Discussions
Main “theoretical layers” of the CT problem may be classified as: kinematics, constitutive, equilibrium, displacement boundary conditions and force boundary conditions. Friction is the innermost theoretical layer in metal forming since it is associated with the last layer. The solutions presented and compared here include all of the layers. EA was tailored unrealistically for a fixed friction factor for the sake of comparison. To diagnose the root cause of error in each layer, one has to
Conclusion
Avitzur's limit analysis is one of the very few analytical avenues to identify the friction factor by Compression Test (CT). However, it is unable to estimate deformation heterogeneity and its usefulness is limited to a low barrelling range. An Extended-Avitzur (EA) solution was developed here to estimate sample's free surface (profile) by an incremental solution of Avitzur's “velocity filed”. Also, formulas for shearing and effective strain distribution in the sample were derived. Sample EA
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