Parallel Tangent Lines Research Articles

Thermodynamic properties of phase-field models for grain boundary segregation

Impurity atoms segregated in grain boundary (GB) regions can dramatically change the physical and chemical properties of the GBs. Such changes often appear to be attributed to the GB energy reduction and/or solute drag effect. Phase-field models have been utilized to clarify both the thermodynamic and kinetic effects of the GB segregation. In this study, we developed phase-field models for GB segregation that are diffuse interface versions of the classical two-phase model of GB segregation. The thermodynamic state at any point in the system is represented as a mixture of a GB phase and a matrix phase. There are two choices for the thermodynamic relation between the GB phase and the matrix phase that constitute the point: the equal composition condition in model I and the equal diffusion potential condition in model II. Most of the previous PFMs for GB segregation appear to be specific cases of model I. We examined the thermodynamic properties of models I and II, and compared them with each other and the classical two-phase model. Although all the models resulted in the same GB composition, the GB energy and its dependency on the composition at the equilibrium state are quite different from each other. In model I, there is a lower bound to the GB energy, which originates from the equal composition condition. The GB energy from model II shows no such lower bound, and it is represented as the vertical distance between the parallel tangent lines on the free energy diagram, as in the classical two-phase model. Nevertheless, the compositional dependence in the model II is quite different from that in the classical two-phase model. This originates from the different choices for the composition-independent parameter in the models: a constant gradient energy coefficient in model II and a constant GB width in the classical two-phase model. Model I is not suitable for simulations of alloys that show a reduction of the GB energy due to GB segregation below a certain limit (in dilute alloys, about half of the GB energy of pure solvents). Model II is a correct choice for such alloys.

Tangent lines, inflections, and vertices of closed curves

We show that every smooth closed curve Γ immersed in Euclidean space R 3 satisfies the sharp inequality 2 ( P + I ) + V ≥ 6 which relates the numbers P of pairs of parallel tangent lines, I of inflections (or points of vanishing curvature), and V of vertices (or points of vanishing torsion) of Γ . We also show that 2 ( P + + I ) + V ≥ 4 , where P + is the number of pairs of concordant parallel tangent lines. The proofs, which employ curve-shortening flow with surgery, are based on corresponding inequalities for the numbers of double points, singularities, and inflections of closed curves in the real projective plane RP 2 and the sphere S 2 which intersect every closed geodesic. These findings extend some classical results in curve theory from works of Möbius, Fenchel, and Segre, including Arnold’s “tennis ball theorem.”

Skew loops in flat tori

We produce skew loops—loops having no pair of parallel tangent lines—homotopic to any loop in a flat torus or other quotient of R n . The interesting case here is n = 3. More subtly for any n, we characterize the homotopy classes that will contain a skew loop having a specified loop $$\tau\subset{\bf S}^{n-1}$$ as tangent indicatrix.

Nonexistence of skew loops on ellipsoids

We prove that every C 1 C^1 closed curve immersed on an ellipsoid has a pair of parallel tangent lines. This establishes the optimal regularity for a phenomenon first observed by B. Segre. Our proof uses an approximation argument with the aid of an estimate for the size of loops in the tangential spherical image of a spherical curve.

Skew loops and quadric surfaces

A skew loop is a closed curve without parallel tangent lines. We prove: The only complete surfaces in R 3 with a point of positive curvature and no skew loops are the quadrics. In particular: Ellipsoids are the only closed surfaces without skew loops. Our efforts also yield results about skew loops on cylinders and positively curved surfaces.

Study of a CNC grinding machining method using an isometric polygon profile

The forming principle and the CNC grinding machining method of an isometric polygonal profile (IPP) are studied deeply and systematically. An equation for the section curve of an isometric polygon profile is set up by means of geometric principles. With the use of differential geometry theory, the curve is proven to have the geometric features of a convex curve. It is referred to as isometric polygonal curve (IPC), because this is a kind of convex curve on which the distance between any parallel tangent lines is equal. An IPP or an isometric polygonal cone profile (IPCP) or an isometric polygonal topless cone (IPTC) is formed with the IPC and a straight line as the generating curve. However, the machining and the measuring of the IPCP are so inconvenient that it has little value. Keeping in mind the characteristic of the IPP and IPTC, this paper puts forward the program method of CNC grinding machining.

Analysis of diametral strain in uniaxial tensile and compression testing of round specimens of anisotropic materials

The development of elliptical cross sections in initially round test pieces of anisotropic materials makes it difficult to measure the true strain on a cross section using a transverse extensometer. In this paper, an analysis is given of the diametral strain and strain rate on a round cross section. Based on the experimental observation that static and dynamic microstructural changes of many anisotropic materials was too small to cause a noticeable change in anisotropic factor, R, up to relatively low strain levels, it is assumed that R is independent of strain in this analysis. Analytical results suggest that, when both the imposed axial strain rate and the anisotropic factor R remain constant and R⩾0.5, the change in diametral strain rate is negligible no matter which orientation a transverse strain extensometer is positioned on a cross section. The analysis includes two methods of measuring the diametral strain: (i) measurement of the distance between two diametral-intercept points on the circumference of a cross section; and (ii) measurement of the distance between two parallel tangent lines of the cross section. Based on the analytical and experimental results, a new method has been proposed for obtaining true stress–strain data in uniaxial tensile or compression testing of anisotropic materials with constant or varying anisotropic factor.

Analysis of Shape Change of a Surface Crack during Stable Fatigue Growth

The multi-point procedure is developed to predict the shape change of a semi-elliptical surface crack during stable fatigue crack growth. 3-D stress intensity factors along a crack front are calculated using the simplified 3-D J-intergral. Crack growth rate coefficient in the Paris law is assumed to be constant along the crack growth. Crack growth rate is set to be the distance between the two parallel tangent lines on the two semi-elliptic crack fronts before and after crack growth.