Manufacturing of screw rotors via 5-axis double-flank CNC machining

We investigate a recently introduced methodology for 5-axis flank computer numerically controlled (CNC) machining, called double-flank milling [1]. We show that screw rotors are well suited for this manufacturing approach where the milling tool possesses tangential contact with the material block on two sides, yielding a more efficient variant of traditional flank milling. While the tool's motion is determined as a helical motion, the shape of the tool and its orientation with respect to the helical axis are unknowns in our optimization-based approach. We demonstrate our approach on several rotor benchmark examples where the pairs of envelopes of a custom-shaped tool meet high machining accuracy.RYC-2017-22649 KAUST-BRF grant nr. 398

On the slip-weakening behavior of rate- and state-dependent constitutive laws

We study the dynamic traction behavior within the cohesive zone during the propagation of earthquake ruptures adopting rate and state dependent constitutive relations. The resulting slip weakening curve displays an equivalent slip weakening distance (D0_eq), which is different from the parameter L controlling the state variable evolution. The adopted constitutive parameters (a, b, L) control the slip weakening behavior and the absorbed fracture energy. The dimension of the nucleation patch scales with L and not with D0_eq. We propose a scaling relation between these two lengthscale parameters which prescribes that D0_eq/L ~ 15

Screw rotor manufacturing via 5-axis flank CNC machining using conical tools

We propose a new method for 5-axis flank computer numerically controlled (CNC) machining of screw rotors using conical tools. The flanks of screw rotors consist of helical surfaces, which predetermines the motion of the milling tool and reduces the search space for tool positioning to only 4-parametric family, which allows a quick search for good initial positions of a given conical tool. We initialize the search by looking at second order line contact between the tool and the helical flank of the rotor. Several positions of the tool are found, covering major part of the flank of the rotor, followed by global optimization that further reduces the tool-surface error and makes sure that there are no gaps between neighboring sweeps of the tool. We demonstrate our approach on several benchmark screw rotors, showing that our approach meets fine industrial tolerances with only few sweeps of the tool.RYC-2017-2264

A thermal pressurization model for the spontaneous dynamic rupture propagation on a three-dimensional faul. 1. Methodological approach

We investigate the role of frictional heating and thermal pressurization on earthquake ruptures by modeling the spontaneous propagation of a three-dimensional (3-D) crack on a planar fault governed by assigned constitutive laws and allowing the evolution of effective normal stress. We use both slip-weakening and rate- and state-dependent constitutive laws; in this latter case we employ the Linker and Dieterich evolution law for the state variable, and we couple the temporal variations of friction coefficient with those of effective normal stress. In the companion paper we investigate the effects of thermal pressurization on the dynamic traction evolution. We solve the 1-D heat conduction equation coupled with Darcy’s law for fluid flow in porous media. We obtain a relation that couples pore fluid pressure to the temperature evolution on the fault plane. We analytically solve the thermal pressurization problem by considering an appropriate heat source for a fault of finite thickness. Our modeling results show that thermal pressurization reduces the temperature increase caused by frictional heating. However, the effect of the slipping zone thickness on temperature changes is stronger than that of thermal pressurization, at least for a constant porosity model. Pore pressure and effective normal stress evolution affect the dynamic propagation of the earthquake rupture producing a shorter breakdown time and larger breakdown stress drop and rupture velocity. The evolution of the state variable in the framework of rate- and state-dependent friction laws is very different when thermal pressurization is active. In this case the evolution of the friction coefficient differs substantially from that inferred from a slip-weakening law. This implies that the traction evolution and the dynamic parameters are strongly affected by thermal pressurization

A thermal pressurization model for the spontaneous dynamic rupture propagation on a 3–D fault: Part II – Traction evolution and dynamic parameters

We investigate the dynamic traction evolution within the cohesive zone during the spontaneous propagation of a 3–D earthquake rupture governed by slip weakening or rate- and state-dependent constitutive laws and accounting for thermal pressurization effects. The analytical solutions as well as temperature and pore pressure evolutions are discussed in a companion paper (Bizzarri and Cocco, 2005b). Our numerical experiments reveal that frictional heating and thermal pressurization modifies traction evolution. The breakdown stress drop, the characteristic slip weakening distance and the breakdown work (i.e., fracture energy) depend on the slipping zone thickness (w) and hydraulic diffusivity ( ω). Thermally activated pore pressure changes caused by frictional heating yield temporal variations of the effective normal stress acting on the fault plane. In the framework of rate- and state-dependent friction, these thermal perturbations modify both the effective normal stress and the friction coefficient. Breakdown stress drop, slip weakening distance and specific breakdown work (J/m 2 ) increases for decreasing values of hydraulic diffusivity and slipping zone thickness. We propose scaling relations to evaluate the effect of w and ω on these physical parameters. We have also investigated the effects of choosing different evolution laws for the state variable as well as the porosity evolution during the breakdown time. Our simulations point out that thermal pressurization modifies the shape of the traction evolution as a function of slip. For particular configurations, the traction versus slip curves display a gradual and continuum weakening for increasing slip: in these cases, the definition of a minimum residual stress and the slip weakening distance become meaningless

3D dynamic simulations of spontaneous rupture propagation governed by different constitutive laws with rake rotation allowed

In this work we present a 3D Finite Difference numerical method to model the dynamic spontaneous propagation of an earthquake rupture on planar faults in an elastic half-space. We implement the Traction-at-Split-Nodes fault boundary condition for a system of faults, either vertical or oblique, using different constitutive laws. We can adopt both a slip-weakening law to prescribe the traction evolution within the breakdown zone or rate- and state-dependent friction laws, which involve the choice of an evolution relation for the state variable. Our numerical procedure allows the use of oblique and heterogeneous distribution of initial stress and allows the rake rotation. This implies that the two components of slip velocity and total dynamic traction are coupled together to satisfy, in norm, the adopted constitutive law. The simulations presented in this study show that the rupture acceleration to super-shear crack speeds occurs along the direction of the imposed initial stress; the rupture front velocity along the perpendicular direction is slower than that along the pre-stress direction. Depending on the position on the fault plane the orientation of instantaneous total dynamic traction can change with time with respect to the imposed initial stress direction. These temporal rake rotations depend on the amplitude of initial stress and on its distribution on the fault plane. They also depend on the curvature and direction of the rupture front with respect to the imposed initial stress direction: this explains why rake rotations are mostly located near the rupture front and within the cohesive zone

Comment on " Earthquake cycles and physica modeling of the process leading up to a large earthquake

The modeling of earthquake initiation and subsequent rupture propagation requires the use of a fault constitutive law controlling the traction evolution and the slip acceleration that yields a finite energy flux at the crack tip. The determination of the temporal evolution of dynamic traction during the propagation of an earthquake rupture has been the major task of many recent investigations. Ohnaka (2004) presented a detailed discussion in favor of a rational governing law for earthquake ruptures which is consistent with laboratory experiments and, according to the author, is based on the physics of rock friction and fracture . He concluded that this constitutive law must be slip-dependent, because the slip dependency is a more fundamental property of the shear rupture than the rate-dependency . Ohnaka (2004) suggested this slip-dependent law also to model the whole seismic cycle (therefore including earthquake nucleation and long-term fault re-strengthening) and considered the proposed law as a governing relation unifying the different features of the earthquake failure process

On C0 and C1 continuity of envelopes of rotational solids and its application to 5-axis CNC machining

We study the smoothness of envelopes generated by motions of rotational rigid bodies in the context of 5-axis Computer Numerically Controlled (CNC) machining. A moving cutting tool, conceptualized as a rotational solid, forms a surface, called envelope, that delimits a part of 3D space where the tool engages the material block. The smoothness of the resulting envelope depends both on the smoothness of the motion and smoothness of the tool. While the motions of the tool are typically required to be at least C2, the tools are frequently only C0 continuous, which results in discontinuous envelopes. In this work, we classify a family of instantaneous motions that, in spite of only C0 continuous shape of the tool, result in C0 continuous envelopes. We show that such motions are flexible enough to follow a free-form surface, preserving tangential contact between the tool and surface along two points, therefore having applications in shape slot milling or in a semi-finishing stage of 5-axis flank machining. We also show that C1 tools and motions still can generate smooth envelopes.Juan de la Cierva - Formation [grant number FJC2019-039804-I] Ram\ón y Cajal fellowship RYC-2017-22649

Modeling instantaneous dynamic triggering in a 3–D fault system: application to the June 2000 South Iceland seismic sequence

We present a model of seismogenesis on an extended 3–D fault subjected to the external perturbations of coseismic stress changes due to an earthquake occurred on another fault (the causative fault). As an application, we consider the spatio–temporal stress distribution produced by the MS = 6.6 June 17, 2000 mainshock in the South Iceland Seismic Zone (SISZ) on the Hvalhnúkur fault. The latter is located nearly 64 km from the causative fault and failed 26 s after the mainshock with an estimated magnitude Mw  [5, 5.5], providing an example of instantaneous dynamic triggering. The stress perturbations are computed by means of a discrete wavenumber and reflectivity code. The response of the perturbed fault is then analyzed solving the truly 3–D, fully dynamic (or spontaneous) problem, accounting for crustal stratification. In a previous study, the response of the Hvalhnúkur fault was analyzed by using a spring–slider fault model, comparing the estimated perturbed failure time with the observed origin time. In addition to the perturbed failure time, the present model can provide numerical estimates of many other dynamical features of the triggered event that can be compared with available observations: the rupture history of the whole fault plane and its final extension and the seismic moment of the 26 s event. We show the key differences existing between a mass–spring model and the present extended fault model, in particular we show the essential role of the load exerted by the other slipping points of the fault. By considering both rate– and state–dependent laws and non–linear slip–dependent law, we show how the dynamics of the 26 s fault strongly depends on the assumed constitutive law and initial stress conditions. In the case of rate– and state– dependent governing laws, assuming an initial effective normal stress distribution which is suitable for the SISZ and consistent with previously stated conditions of instantaneous dynamic triggering of the Hvalhnúkur fault, we obtain results in general agreement with observations

Curve-guided 5-axis CNC flank milling of free-form surfaces using custom-shaped tools

A new method for 5-axis flank milling of free-form surfaces is proposed. Existing flank milling path-planning methods typically use on-market milling tools whose shape is cylindrical or conical, and is therefore not well-suited for meeting fine tolerances for manufacturing of benchmark free-form surfaces like turbine blades, gears, or blisks. In contrast, our optimization-based framework incorporates the shape of the tool into the optimization cycle and looks not only for the milling paths, but also for the shape of the tool itself. Given a free-form reference surface and a guiding path that roughly indicates the motion of the milling tool, tangential movability of quadruplets of spheres centered along a straight line is analyzed to indicate possible shapes and their motions. This results in $G^1$ Hermite data in the space of rigid body motions that are interpolated and further optimized, both in terms of the motion and the shape of the milling tool itself. We demonstrate our algorithm on synthetic free-form surfaces and industrial benchmark datasets, showing that the use of custom-shaped tools is capable of meeting fine industrial tolerances and outperforms the use of classical, on-market tools.RYC-2017-2264