From a mathematical point of view, phase field theory can be understood as a smooth approximation of an underlying sharp interface problem. However, the smooth phase field approximation is not uniquely defined. Different phase field approximations are known to converge to the same sharp interface problem in the limiting case-if the thickness of the diffuse interface converges to zero. In this respect and focusing on numerics, a question that naturally arises is as follows: What are the convergence rates of the different phase field models? The paper deals precisely with this question for a certain family of phase field models. The focus is on an Allen-Cahn-type phase field model coupled to continuum mechanics. This model is rewritten into a unified variational phase field framework that covers different homogenization assumptions in the diffuse interfaces: Voigt/Taylor, Reuss/Sachs and more sound homogenization approaches falling into the range of rank-one convexification. It is shown by means of numerical experiments that the underlying phase field model-that is, the homogenization assumption in the diffuse interface-indeed influences the convergence rate.see [3], is comprised of a double-well contribution in the first term, defining distinct energetic minima for two phases-at order-parameter field values of p = 0 and p = 1, respectively-and the second, gradient-based term that attributes an energetic cost to the formation of interfaces. Assuming (2), the Euler -Lagrange equation of (1) can be written in the form, see also [19],NUMERICAL CONVERGENCE STUDY IN PHASE FIELD MODELING ) denotes the functional (variational) derivative. Physically speaking, this describes an order-parameter field that evolves in such a manner that the system is driven towards an energy minimizing equilibrium
Internal traverse grinding (ITG) using electroplated cBN tools in high-speed grinding conditions is a highly efficient manufacturing process for bore machining in a single axial stroke. However, process control is difficult. Due to the axial direction of feed, changes in process normal force and thus radial deflection of the tool and workpiece spindle system, lead to deviations in the workpiece contour along the length of the bore, especially at tool exit. Simulations including this effect could provide a tool to design processes which enhance shape accuracy of components. A geometrical physically-based simulation is herein developed to model the influence of system compliance on the resulting workpiece contour. Realistic tool topographies, obtained from measurements, are combined with an FE-calibrated surrogate model for process forces and with an empirical compliance model. In quasistatic experimental investigations, the spindle deflection is determined in relation to the acting normal forces by using piezoelectric force measuring elements and eddy current sensors. In grinding tests with in-process force measurement technology and followed by measurement of the resulting workpiece contours, the simulation system is validated. The process forces and the resulting characteristic shape deviations are predicted in good qualitative accordance with the experimental results.
Continuous technological advancements in the field of grinding technology and improved grinding tools have contributed to the development of high performance grinding processes. One example of such a process is internal traverse grinding (ITG) with electroplated cBN grinding wheels, where the tool consists of a conical roughing zone and a cylindrical finishing zone. Since the tool is fed in axial direction into a revolving workpiece, spindle deflections induced by varying process forces can lead to contour errors along the bore. Numerical simulations are a valuable tool to overcome the challenges associated with such high performance processes. Whenever spindle deflections need to be considered, accurate prediction of the process forces is paramount. Finite Element (FE) simulations have been widely used for the prediction of forces in cutting processes such as turning and milling, where only a small number of active cutting edges is considered, and where the geometry of these cutting edges is clearly defined. Grinding tools, on the other hand, contain a large number of grains with varying geometric characteristics. We recently proposed a multi‐scale simulation system for the simulation of ITG processes, where a geometric kinematic grinding simulation, based on a database of digitalised grains of a real grinding wheel, was used to determine the grain engagements [1]. The process forces were obtained from summation of the contributions of all active grains at any given time, based on a force model on the individual grain level. The force model takes the material removal rate and an approximation of the rake angle into account, and was calibrated via finite element simulations. In recent years, the Coupled Eulerian Lagrangian method (CEL), which is part of the commercial finite element software Abaqus, has been applied to simulate various cutting processes. No remeshing is necessary in this framework, and separation of chips from the workpiece can be modelled without element deletion. The application of CEL to the simulation of single grain cutting is therefore a promising approach to further improve the force model included in the process simulation of ITG. In this work, the kinematics of ITG are incorporated into a single grain cutting simulation, and the suitability of the CEL method for the problem is evaluated with a focus on the chip formation, separation and self‐contact between the chip and the workpiece.
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