Effect of Precipitate Morphology on the Gradient-Dependent Behaviour of Two-Phase Single Crystals
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ABSTRACT In this work a recently proposed gradient and rate-dependent crystallographic formulation is used to investigate the macroscopic behaviour of a precipitated single crystal. It relies on strain gradient concepts to account for the additional strengthening mechanism caused by presence of interfacial and geometrically necessary dislocations (GNDs). The total slip resistance is assumed to be due to a mixed population of mobile and sessile forest obstacles arising from both statistically stored dislocations (SSDs) and GNDs. The non-local crystallographic theory is implemented numerically into the finite element method. It requires the calculation of the slip rate gradients at the element level to determine the evolutionary behaviour of the GND densities, and a fully implicit numerical algorithm within a large strain kinematics framework and non-isothermal conditions. The effects of the relevant microstructural features (precipitate size, morphology and volume fraction) and deformation gradient-related length scales on the macroscopic behaviour is investigated and compared with experimental results. INTRODUCTION During high temperature deformation of single crystal superalloys, the initially cuboidal precipitates undergo morphological and volume fraction changes which strongly affect the single crystal mechanical properties [1]. In this work, the combined effects of the precipitate size, morphology and volume fraction on the macroscopic stress-strain behaviour are investigated using a micro-macro continuum mechanics approach. A recently proposed gradient dependent crystallographic formulation [2][3] is used to describe the behaviour of the soft matrix of a precipitated single crystal. The finite element implementation of the non-local crystallographic model includes the calculation of the slip rate gradients at the element level to determine the evolutionary behaviour of the GND densities, and a fully implicit numerical algorithm within a large strain kinematics framework to update the local stresses and internal slip system variables [4]. The numerical procedure is then applied to investigate the influence of the different microstructural length scales introduced by changes in the precipitate morphology in two phase -y//•' single crystals. The capability of the proposed integration procedure is shown through three-dimensional unit-cell deformation studies. Three typical precipitate morphologies are considered, viz. cuboidal, plate-like and the limiting case corresponding to the coalescence of two neighbouring precipitates. A comparison between the gradient independent and gradient dependent predicted macroscopic responses is also presented. GRADIENT-DEPENDENT CRYSTALLOGRAPHIC FORMULATION The hyperelastic crystallographic formulation used in this work accounts for finite strain kinematics and non-isothermal conditions. It relies on the multiplicative decomposition of the total deformation gradient, F, into a thermal component, F 0 , which represents the deformation of the crystal lattice due to temperature chang
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