As opposed to macro scale machining processes, the crystalline structure of commercially pure (cp) titanium plays an important role in micro machining processes. Finite element simulations of micro machining processes involve several challenges, since the workpiece material undergoes large deformations during chip formation and eventually material separation needs to be modeled. A robust implementation of the finite deformation multi-slip crystal plasticity model describing the material response is the basis of such simulations. One challenge is the determination of the active set of plastic slip systems. Standard explicit predictor-corrector formulations based on the Karush-Kuhn-Tucker (KKT) conditions often fail. In order to circumvent the explicit determination of the set of active slip systems, the KKT conditions can be replaced by a system of Fischer-Burmeister complementary functions.
A continuum model for multislip crystal plasticity of single crystalsThe large-strain rate-independent crystal plasticity model used in this paper, proposed in [2], is formulated within the framework of the multiplicative decomposition of the deformation gradient F in elastic F e and plastic F p parts (1a), respectively. To describe the evolution of the plastic deformation, an evolution equation for the plastic deformation gradient is needed. This is obtained by the definition of the plastic velocity gradient L p (1b) together with the constitutive assumption (1c), whereγ k , s k and n k are the plastic slip rate, the slip direction, and the slip plane normal in slip system k, and N s is the total number of slip systems.It is convenient to express the constitutive equations in the current configuration. The plastic deformation is volume preserving, det[F p ] = 1. The material law (3a,b) is formulated in terms of the isochoric, elastic left Cauchy-Green Tensor B e iso :The reversible, elastic behaviour is described by an isotropic compressible Neo-Hooke material law. The Kirchhoff stress τ and the resolved shear stress τ k are obtained from:where G and K are the shear and bulk moduli of the material and I is the second order identity tensor. Plastic slip in system k occurs, if the resolved shear stress τ k reaches the hardening dependent critical yield stress τ k y (α k ) in slip system k, which can be expressed int terms of the yield function f k :The variable α k is the hardening coefficient, and linear strain hardening is considered. The yield function f k and the plastic strain rateγ k have to satisfy the Karush-Kuhn-Tucker conditions, also known as loading/unloading conditions:
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