Extensive use of titanium alloys in critical industrial and military applications, such as compressor blades of jet engines and armor of ground combat vehicles, has motivated researchers to understand, measure and tailor the mechanical properties of these alloys over a wide range of strain rates and temperatures. Of special interest has been the mechanical response of these alloys under high rates of deformation and failure under cyclic/dwell fatigue.
One of the most powerful methods developed in the past two decades for modeling material behavior is crystal plasticity finite element method (CPFEM). Its advantages inhere within its capability to describe the mechanical anisotropy and material heterogeneity via micro-mechanism-based constitutive laws, which could be informed from multiple length-scales ranging from sub-grain level to the polycrystalline level. The accuracy of CPFE models and their capability in prediction of material response in polycrystalline level are dependent on three main factors, including creation of a virtual realistic polycrystalline aggregate model, utilization of a robust element formulation for finite element calculations, and description of material response with a proper constitutive law.
To carry out a finite element analysis, it is required that the elements conform to the geometry of the computational domain. This requirement has an implication for CPFE analysis of polycrystalline aggregates. Linear constant strain tetrahedral elements are used to discretize the polycrystalline aggregates due to the complex morphology of grains and the magnificent capability of these elements to conform to tortuous geometries. However, these elements suffer from severe volumetric locking when simulating the deformation of (nearly-) incompressible materials. Various methods have been proposed to relieve volumetric locking in tetrahedral elements, for instance node-based uniform strain formulation, F-bar-patch method, and mixed enhanced formulation. Since plasticity is inherently isochoric, volumetric locking of tetrahedral elements is highly relevant to the CPFE simulation; however, its detrimental effect on the solution has been generally overlooked by the materials modeling community.
Describing the material response with a proper constitutive law plays a key role in the success of the CPFE models to represent the behavior of the material. The most critical part of a CP constitutive law is the flow rule which interrelates the local material state (e.g. dislocation density) and local stress state with the kinematics (e.g. slip rates). Suitability of a constitutive law for a certain application inheres in how rigorously the flow rule can capture the governing deformation mechanism(s). Selection of the proper type of flow rule is largely problem-dependent since flow rules are developed on the premise of certain assumptions and pose some limitations with respect to their use. The most commonly used expressions for the flow rule are the phenomenological power-law model, Arrhenius-type activation energy-based model, and linear model.
Deriving a rate-dependent physics-based flow rule whose application is not limited to a certain range of strain rates is desired. Using such a flow rule is encouraged in simulation of polycrystalline aggregates where the local stress and strain rates might be lower or higher than the applied macroscopic stress or strain rate. For instance, in Ti alloys under applied creep load σapp, stress redistribution happens locally in the microstructure due to the grain-level load shedding from the soft grains to the adjacent hard grains. This is known as load-shedding mechanism which induces stresses higher than σapp in the hard grains while the stress in the adjacent soft grain could be lower than σapp. Similarly, a polycrystalline microstructure which is macroscopically deforming under a very high strain rate (in the range of applicability of linear flow rule) could locally undergo a lower rate of deformation (in the range of applicability of activation energy-based flow rule). A new unified flow rule is sought which could be used for both low and high rates of deformation. This unified flow rule should automatically adjusts its functional form based on the local deformation rate, local stress state and internal state variables. Such a flow rule can be obtained based on some physical considerations via combining the thermally- activated and drag-dominated stages of dislocation motion.
The material studied is Ti-7.02Al- 0.11O-0.015Fe (wt%) alloy with a predominant hcp microstructure. The composition of this alloy is very close to the α phase of many commercially important titanium alloys. Mechanical testing is done on two variants of this alloy, referred to as the AR (as-rolled) and RA (rolled-annealed) samples. The AR sample corresponds to the one which has been only rolled whereas the RA sample corresponds to a sample manufactured by first rolling and subsequently annealing it to improve its ductility and increase the grain size, followed by a cooling process. Scanning electron microscopy (SEM) based electron back-scattered diffraction (EBSD) is done to quantify the texture of large-area EBSD scans. The surface EBSD scans for the AR and RA samples are respectively 5425 × 2190μm2 and 5175 × 2135μm2, collected at 5μm step size. The accompanying figure shows a part of surface EBSD scans collected for both samples after being processed to remove noise from the data. Average diameter for equivalent projected circle in 2D is calculated to be 34.12μm and 83.4μm for the AR and RA samples, respectively.
This work was done by Somnath Ghosh of Johns Hopkins University for the Army Research Office. ARL-0196
This Brief includes a Technical Support Package (TSP).
Multi-Scale Analysis of Deformation and Failure in Polycrystalline Titanium Alloys Under High Strain Rates
(reference ARL-0196) is currently available for download from the TSP library.
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