jeanflop/post_ocr_correction2 huggingface dataset:
partout et partout non différentiable. Admettons aussi que le potentiel des contraintes U vérifie (8). Supposons de plus que pour toute valeur fixe de la force compressive Po la solution du problème (2) est donnée par les fonctions σij (x; Po), et j(x; Po), w, (x; Po), ∝ (Po), G(Po). Alors, pour toute force Pk = Yk[2 + μ(d-1)]P0, (k = 0, ± 1, ± 2, Le problème aux limites (2) est vérifié par (9) et la région de contact G(Po) change conformément à la transformation d'homothétie (10). Soit Gn(P) l'enveloppe convexe de la région discrète de contact G(P) décrite par (7). Alors, la région Gn(P) peut être regardée comme la région effective de contact. En vertu de la proposition on peut tirer les conclusions qualitatives suivantes (sous les hypothèses faites plus haut). (i) Le diamètre 1 de la région effective de contact varie proportionnellement à la force à la puissance 1/[2 + μ (d — 1 )]. (ii) Le rapprochement du poinçon au demi-espace est proportionnel à la force à la puissance d/[2 + μ (d 1)], voir (11), (12). Enfin notons, que pour le poinçon de Weierstrass-Mandelbrot (c'est-à-dire pour f en (1) égale à fo) (11) et (12) entraînant (13). 1. INTRODUCTION. — We analyze a class of boundary-value problems in three-dimensional infinitesimal elasticity which relate to the normal indentation of a non-linear halfspace. The punch is rigid with a rough profile of surface, whose shape is described by a fractal function. Sets with non-integer Hausdorff-Besicovitch dimension were called fractals by Mandelbrot [1]. At this time the concept of fractal (Hausdorff-Besicovitch, Minkowski-Bouligand, box and other) dimensions plays a central role in the modelling of a wide class of natural phenomena ([1], [2], [4]-[6]). Recently it was shown that for decreasing unit of measurement 8 the length of naturally irregular functions, such as profiles of coastline, cloud tops, mountain ranges and so on, does not converge but, instead, increases as the exponential function E D for a certain exponent D ([1], [2]). This exponent is called the fractal dimension of the function. It was determined that profiles of rough
entraînant (13). 1. INTRODUCTION. — We analyze a class of boundary-value problems in three-dimensional infinitesimal elasticity which relate to the normal indentation of a non-linear halfspace. The punch is rigid with a rough profile of surface, whose shape is described by a fractal function. Sets with non-integer Hausdorff-Besicovitch dimension were called fractals by Mandelbrot [1]. At this time the concept of fractal (Hausdorff-Besicovitch, Minkowski-Bouligand, box and other) dimensions plays a central role in the modelling of a wide class of natural phenomena ([1], [2], [4]-[6]). Recently it was shown that for decreasing unit of measurement 8 the length of naturally irregular functions, such as profiles of coastline, cloud tops, mountain ranges and so on, does not converge but, instead, increases as the exponential function E D for a certain exponent D ([1], [2]). This exponent is called the fractal dimension of the function. It was determined that profiles of rough surfaces of solids are self-affine and one can give scale-invariant characterization of such surfaces using the fractal geometry ([2], [3]). Models for the contact of rough surfaces are numerous [7], but there are only two theoretical models of contact using fractal geometry ([3], [8]), Since analytical solutions of the considered problems of non-linear elasticity could not be found even for smooth punch, then, evidently, we cannot give the analytical solution of the problem for a fractal punch. It was shown recently that if the punch shape is given by a smooth homogeneous function of degree d, then general expressions for changes of all functions, which give a solution of the contact problem, can be obtained exactly, using a similarity method ([9][12]). The similarity consideration presented in this Note is of a similar form but incorporates an important modification to allow for the fractal nature of surface roughness. We assume that a rigid rough punch, being indented, has discrete contact With a non-linear elastic half-space and the shape of the punch is given by a fractal, generalized-homogeneous function of degree d, where d>0. An example of such punch is the punch of revolution, whose profile is described by a Weierstrass-Mandelbrot function.
problems of non-linear elasticity could not be found even for smooth punch, then, evidently, we cannot give the analytical solution of the problem for a fractal punch. It was shown recently that if the punch shape is given by a smooth homogeneous function of degree d, then general expressions for changes of all functions, which give a solution of the contact problem, can be obtained exactly, using a similarity method ([9][12]). The similarity consideration presented in this Note is of a similar form but incorporates an important modification to allow for the fractal nature of surface roughness. We assume that a rigid rough punch, being indented, has discrete contact With a non-linear elastic half-space and the shape of the punch is given by a fractal, generalized-homogeneous function of degree d, where d>0. An example of such punch is the punch of revolution, whose profile is described by a Weierstrass-Mandelbrot function. The Weierstrass function was originally introduced as a mathematical monster, which is continuous but differentiable nowhere, and it was shown only recently that this function, extended by Mandelbrot, is an effective tool in the study of rough surfaces ([1], [3]). In the present Note, the general expressions for changes of all functions, giving the solutions of the contact problems for a fractal punch, are derived exactly, without solving the field equations. 2. GOVERNING EQUATIONS. In the problem considered it is assumed that the rigid punch, whose surface equation is given by the fractal function f, i. e. (1) X3 = f (x1, x2), f(x1, x2)>0, is pressed by the force P into a positive half-space. Axes xi are presented in Figure 1. Fig. 1. — A fractal punch and coordinate axes. Fig. 1. — Le poinçon fractal et les axes de coordonnées. After the punch contacted with the half-space x3=0, displacements ut and stresses σij are generated. The relative proximity between the punch and the half-space is denoted by α. In the contact problem it is supposed that the shape of the punch and the compressing force P are given and one must find the finite region G on the
all functions, giving the solutions of the contact problems for a fractal punch, are derived exactly, without solving the field equations. 2. GOVERNING EQUATIONS. In the problem considered it is assumed that the rigid punch, whose surface equation is given by the fractal function f, i. e. (1) X3 = f (x1, x2), f(x1, x2)>0, is pressed by the force P into a positive half-space. Axes xi are presented in Figure 1. Fig. 1. — A fractal punch and coordinate axes. Fig. 1. — Le poinçon fractal et les axes de coordonnées. After the punch contacted with the half-space x3=0, displacements ut and stresses σij are generated. The relative proximity between the punch and the half-space is denoted by α. In the contact problem it is supposed that the shape of the punch and the compressing force P are given and one must find the finite region G on the boundary plane R2 of the half-space at the points of which the punch and the medium are in mutual contact, the quantity 1, the displacements ut and the stresses σij. Since the compressing force P is a parameter characterizing the contact problem, we shall write it among the arguments of the unknown quantities. The sought quantities must satisfy the following equations (2) σij, j = 0, i,j = 1,2,3; σij = a(εkl)/aEkj, εij = (Mjk + Uji)/2; u(x;P) → — 0, |x| → ∞ ; o2, (x1, x2, 0; P) = 0, (x1, x2)∈R2G(P); UI(XL, x2, 0; P) = α(P) — f(x1, x2), (x1, x2)∈G(P); σ33 (X1, x2, 0; P) = σ32 (X1, X2, 0; P) = 0, (X1, x2)∈G(P); ∬R2 (σ33 (x1, x2, 0; P) dx1 dx2 = — P, in which, ε0 are the components of strain tensor and U is the stress potential. The material behaviour of the medium may be anisotropic or isotropic, depending on the form of the stress potential. 3. THE GENERALIZED-HOMOGENEOUS FRACTAL PUNCHES. — We shall say that a function of coordinates x1, x2 and positive parameter y is a positive generalized-homogeneous function of degree d if for each fixed y the following
the unknown quantities. The sought quantities must satisfy the following equations (2) σij, j = 0, i,j = 1,2,3; σij = a(εkl)/aEkj, εij = (Mjk + Uji)/2; u(x;P) → — 0, |x| → ∞ ; o2, (x1, x2, 0; P) = 0, (x1, x2)∈R2G(P); UI(XL, x2, 0; P) = α(P) — f(x1, x2), (x1, x2)∈G(P); σ33 (X1, x2, 0; P) = σ32 (X1, X2, 0; P) = 0, (X1, x2)∈G(P); ∬R2 (σ33 (x1, x2, 0; P) dx1 dx2 = — P, in which, ε0 are the components of strain tensor and U is the stress potential. The material behaviour of the medium may be anisotropic or isotropic, depending on the form of the stress potential. 3. THE GENERALIZED-HOMOGENEOUS FRACTAL PUNCHES. — We shall say that a function of coordinates x1, x2 and positive parameter y is a positive generalized-homogeneous function of degree d if for each fixed y the following conditions are met: (3) f(0; y) = 0; f(x; y) ≥ 0, x∈R2{0}; f (7 xi, 1'X2; 1') = 1'df(x1, x2; 7). If in addition its graph has a fractal dimension D then it is called a fractal, generalized-homogeneous function. We consider the punches whose graphs are described by such functions, because they possess the features convenient to use the similarity method ([8]-[11]). Let the function of punch shape fo be described in polar coordinates r, θ by (4) fO (X1, X2; y) = A (0) W (r; A(θ)), r = (x1 + x2)1/2, where W is Weierstrass-Mandelbrot function. Then, it satisfies condition (3). Indeed, the positive function was constructed by Mandelbrot as an extension of the Weierstrass function. This function is continuous everywhere and differentiable nowhere. In addition, its graph is a curve whose box dimension D exceeds unity ([1], [2], [6], [13]). Using (5), the fractal function W can easily be shown to obey the generalized-homogeneous rule ([1], [13]). (6) W (y; r) = y(2 - D) W (r; y). Comparing (6) with (3) and (4) leads to the conclusion that fo is a fractal, generalized-homogeneous function of degree d=(2 — D) and O<d<1. If we suppose that
functions, because they possess the features convenient to use the similarity method ([8]-[11]). Let the function of punch shape fo be described in polar coordinates r, θ by (4) fO (X1, X2; y) = A (0) W (r; A(θ)), r = (x1 + x2)1/2, where W is Weierstrass-Mandelbrot function. Then, it satisfies condition (3). Indeed, the positive function was constructed by Mandelbrot as an extension of the Weierstrass function. This function is continuous everywhere and differentiable nowhere. In addition, its graph is a curve whose box dimension D exceeds unity ([1], [2], [6], [13]). Using (5), the fractal function W can easily be shown to obey the generalized-homogeneous rule ([1], [13]). (6) W (y; r) = y(2 - D) W (r; y). Comparing (6) with (3) and (4) leads to the conclusion that fo is a fractal, generalized-homogeneous function of degree d=(2 — D) and O<d<1. If we suppose that the function A of the angle θ is smooth, then its fractal dimension is equal to 1. In this case, using the empirical Mandelbrot rule ([1], [2]), we can obtain that the fractal dimension of fo is equal to D + 1. The equation (6) implies that the whole function W can be reconstructed from its values in the range r0≤r≤yr0 [13]. Figure 2 shows the Weierstrass-Mandelbrot fractal punch fo in the case y = 1.5 and D = 1.5. Fig. 2. The Weierstrass-Mandelbrot punch in the case y = 1.5; D= 1.5. Fig. 2. — Poinçon de Weierstrass-Mandelbrot dans le cas y = 1, 5; D = 1, 5. We suppose that the region of contact G (P) between the fractal punch and the half-space is discrete, G (P){0} is not empty and G (P) can be described by (7) G (P) = ∪ gn (P), n = 1 where gn (P) is a point or closed line (in the case when A (θ) is a smooth function) on R2. If A (θ) is constant (i.e. the contact problem is axisymmetric) then each gn (P) is a circle. 4. SIMILARITY CONSIDERATIONS. We consider the contact problems for the elastic
W can be reconstructed from its values in the range r0≤r≤yr0 [13]. Figure 2 shows the Weierstrass-Mandelbrot fractal punch fo in the case y = 1.5 and D = 1.5. Fig. 2. The Weierstrass-Mandelbrot punch in the case y = 1.5; D= 1.5. Fig. 2. — Poinçon de Weierstrass-Mandelbrot dans le cas y = 1, 5; D = 1, 5. We suppose that the region of contact G (P) between the fractal punch and the half-space is discrete, G (P){0} is not empty and G (P) can be described by (7) G (P) = ∪ gn (P), n = 1 where gn (P) is a point or closed line (in the case when A (θ) is a smooth function) on R2. If A (θ) is constant (i.e. the contact problem is axisymmetric) then each gn (P) is a circle. 4. SIMILARITY CONSIDERATIONS. We consider the contact problems for the elastic media the stress potentials of which U are homogeneous functions of degree μ + 1 with respect to the components of strain tensor εkl, i.e. for each positive À we have (8) U(^,) = V + 1U(e_,) (0<n<1) PROPOSITION. — Let the shape of the punch be determined by a fractal, positive, generalized-homogeneous function f satisfying (3), where d > 0, and f(x1, x2; y)∈C(R2), f(x1, x2; y)∉C1(R2). In addition let the stress potential U satisfy (8). Assume further that for any value of the compressing force Po the solution of the contact problem (2) is given by the functions cij (x; Po), εij (x; Po), uj (x; Po), quantity α (Po) and region G (Po). Then, for each force Pk = yk[2 + μ(d-1)]Po, (k = 0, ± 1, ± 2, .) the boundary-value problem (2) is satisfied by (9) u3 (x1, x2, 0; Pk) = (y-kx1, y-kx2)∈G(Po) = ykd [α(Po) — f(y-kx1, y-kx2; y)] = α (Pk) - (y-kx1, y-kx2; y). The validity of the other conditions from (2) is proved in the same way as in [12]. Note that equalities (9) are exact for discrete collection of loads Pk only. Let Gn (P) be the convex hull
fractal, positive, generalized-homogeneous function f satisfying (3), where d > 0, and f(x1, x2; y)∈C(R2), f(x1, x2; y)∉C1(R2). In addition let the stress potential U satisfy (8). Assume further that for any value of the compressing force Po the solution of the contact problem (2) is given by the functions cij (x; Po), εij (x; Po), uj (x; Po), quantity α (Po) and region G (Po). Then, for each force Pk = yk[2 + μ(d-1)]Po, (k = 0, ± 1, ± 2, .) the boundary-value problem (2) is satisfied by (9) u3 (x1, x2, 0; Pk) = (y-kx1, y-kx2)∈G(Po) = ykd [α(Po) — f(y-kx1, y-kx2; y)] = α (Pk) - (y-kx1, y-kx2; y). The validity of the other conditions from (2) is proved in the same way as in [12]. Note that equalities (9) are exact for discrete collection of loads Pk only. Let Gn (P) be the convex hull of the discrete contact region G(P) described by (7). Then, this region Gn (P) can be taken as an effective region of contact. Evidently, Gn (P) will be changed by homotopic transformation too. The theorem provides the following qualitative corollaries (which hold under the assumptions made above): (i) the size l of the nominal region of contact varies proportionally to the load raised to exponent 1/[2 + μ(d — 1)]; (ii) the approach of the punch with half-space is proportional to the load raised to exponent d/[2 + μ(d — 1)], namely (11) l(Pk) = [l(Po) P^ 1 /[2 + μd 1]] P~/[2 + μ (d 1)], (12) α (Pk) = [α (Po) P~-d/[2 + μ(d-1)]] P^ 2+μ(d~1)]. Indeed, choose the point of contact on the boundary of region Gn (Po), i.e. (x1, x2)∈G(Po)∩∂Gn(Po). The definition of the region Gn(Po) and (10) imply that for this point (y-kx1, y-kx2) ∈ G (Pk) ∩ ∂Gn (Pk), therefore the size varies proportionally to yd, i.e. l(Pk) = l(Po)yk. Substitution yk into this equality results in (11). The second assertion follows from (9). Finally, note that for Weierstrass-Mandelbrot punch, i.e. when f in (1) is equal to fo, (11) and (12) give (Pk)~P~1/[2
l of the nominal region of contact varies proportionally to the load raised to exponent 1/[2 + μ(d — 1)]; (ii) the approach of the punch with half-space is proportional to the load raised to exponent d/[2 + μ(d — 1)], namely (11) l(Pk) = [l(Po) P^ 1 /[2 + μd 1]] P~/[2 + μ (d 1)], (12) α (Pk) = [α (Po) P~-d/[2 + μ(d-1)]] P^ 2+μ(d~1)]. Indeed, choose the point of contact on the boundary of region Gn (Po), i.e. (x1, x2)∈G(Po)∩∂Gn(Po). The definition of the region Gn(Po) and (10) imply that for this point (y-kx1, y-kx2) ∈ G (Pk) ∩ ∂Gn (Pk), therefore the size varies proportionally to yd, i.e. l(Pk) = l(Po)yk. Substitution yk into this equality results in (11). The second assertion follows from (9). Finally, note that for Weierstrass-Mandelbrot punch, i.e. when f in (1) is equal to fo, (11) and (12) give (Pk)~P~1/[2 + μ(1-D)], α (Pk)~P[2"D)/[2 + μ D~1]. In the case of linear elasticity, i.e. for u = 1 we have (Pk) P~, α (Pk) P~. It is surprising that the similarity consideration leads to the exact conclusions on the character of the contact between such a "mathematical monster" as Weierstrass-Mandelbrot fractal punch and non-linear elastic half-space. Note remise le 6 novembre 1992, acceptée le 12 novembre 1992. REFERENCES [1] B. B. MANDELBROT, Fractals: form, chance, and dimension, San-Francisco, Freeman, 1977. [2] J. FEDER, Fractals, N.-Y., London, Plenum Press, 1988. [3] A. MAJUMDAR and B. BHUSHAN, Trans. A.S.M.E., J. Tribology, 113, 1991, pp. 1. [4] J. FLECKINGER and D. G. VASIL'EV, C. R. Acad. Sci. Paris, 311, série I, 1990, pp. 867-872. [5] M. L. LAPIDUS, Trans. Amer. Math. Soc., 325, 1991, pp. 465-529. [6] K. FALCONER, Fractal geometry: mathematical foundations and applications, N.-Y., Wiley, 1990. [7] K. L. JOHNSON, Contact Mechanics, Cambridge, Cambridge University Press, 1985. [8] F. M. BORODICH and A. B. MOSOLOV, J. Tech. Phys., 61, 1991, No. 9, pp. 50-54. [9] F. M. BORODICH, J. Appl. Math. Mech. (PMM), 47, 1983, pp. 519-521. [10] F. M. BORODICH, J. Appl. Mech. Tech. Phys., 25, 1984, pp. 324-326. [11]
fractal punch and non-linear elastic half-space. Note remise le 6 novembre 1992, acceptée le 12 novembre 1992. REFERENCES [1] B. B. MANDELBROT, Fractals: form, chance, and dimension, San-Francisco, Freeman, 1977. [2] J. FEDER, Fractals, N.-Y., London, Plenum Press, 1988. [3] A. MAJUMDAR and B. BHUSHAN, Trans. A.S.M.E., J. Tribology, 113, 1991, pp. 1. [4] J. FLECKINGER and D. G. VASIL'EV, C. R. Acad. Sci. Paris, 311, série I, 1990, pp. 867-872. [5] M. L. LAPIDUS, Trans. Amer. Math. Soc., 325, 1991, pp. 465-529. [6] K. FALCONER, Fractal geometry: mathematical foundations and applications, N.-Y., Wiley, 1990. [7] K. L. JOHNSON, Contact Mechanics, Cambridge, Cambridge University Press, 1985. [8] F. M. BORODICH and A. B. MOSOLOV, J. Tech. Phys., 61, 1991, No. 9, pp. 50-54. [9] F. M. BORODICH, J. Appl. Math. Mech. (PMM), 47, 1983, pp. 519-521. [10] F. M. BORODICH, J. Appl. Mech. Tech. Phys., 25, 1984, pp. 324-326. [11] R. HILL, B. STORAKERS and A. B. ZDUNEK, Proc. R. Soc. Lond., A423, 1989, pp. 301-330. [12] F. M. BORODICH, Problemy Prochnosti, 1989, No. 12, pp. 47-53. [13] M. V. BERRY and Z. V. LEWIS, Proc. R. Soc. Lond., A 370, 1980, pp. 519-521. Division of Technical Mechanics, Moscow Institute of Radio Engineering, Electronics and Automation, prosp. Vernadskogo 78, Moscow, 117454, Russia. Mécanique des fluides/Fluid Mechanics Sillage tridimensionnel d'un obstacle torique et modélisation par l'équation de Ginzburg-Landau Thomas LEWEKE, Michel PROVANSAL et Louis BOYER Résumé — L'adéquation du modèle de Ginzburg-Landau à la description du sillage à l'aval d'un cylindre droit est étendue au cas du tore pour lequel des conditions aux limites périodiques doivent être appliquées. Les expériences réelles effectuées sur un tore de dimensions 3-57 mm révèlent l'existence de différentes structures prévues par le modèle. Three-dimensional wake of a torus and its modelling by the Ginzburg-Landau equation Abstract — The Ginzburg-Landau model for the wake of a straight circular cylinder is extended to the case of a torus, for which periodic boundary conditions apply. Experiments done on a real torus of dimensions 3-57 mm reveal the existence of different structures predicted by the model. Abridged English Version —
of Radio Engineering, Electronics and Automation, prosp. Vernadskogo 78, Moscow, 117454, Russia. Mécanique des fluides/Fluid Mechanics Sillage tridimensionnel d'un obstacle torique et modélisation par l'équation de Ginzburg-Landau Thomas LEWEKE, Michel PROVANSAL et Louis BOYER Résumé — L'adéquation du modèle de Ginzburg-Landau à la description du sillage à l'aval d'un cylindre droit est étendue au cas du tore pour lequel des conditions aux limites périodiques doivent être appliquées. Les expériences réelles effectuées sur un tore de dimensions 3-57 mm révèlent l'existence de différentes structures prévues par le modèle. Three-dimensional wake of a torus and its modelling by the Ginzburg-Landau equation Abstract — The Ginzburg-Landau model for the wake of a straight circular cylinder is extended to the case of a torus, for which periodic boundary conditions apply. Experiments done on a real torus of dimensions 3-57 mm reveal the existence of different structures predicted by the model. Abridged English Version — INTRODUCTION. — The structure of the vortex path behind a straight cylinder depends on several factors, e.g. aspect ratio, boundary conditions, and Reynolds number Re. For Re between 50 and 150, shedding of parallel, oblique, or curved vortices, spanwise cells of different shedding frequencies, or dislocations are observed ([1]—[4]). We recently showed that the one-dimensional Ginzburg-Landau model, in which the wake is considered as a diffusively coupled chain of oscillators, could account for the phenomena observed in this domain ([5]-[7]). Numerical simulations of this model with "zero" boundary conditions (i.e. no oscillation at the ends of the cylinder) reproduced most of the observed 3D structures. Recent numerical [8] and analytical [9] studies of two-dimensional Ginzburg-Landau models, including the downstream direction, have shown very similar results. In this paper we present results concerning a torus. For the model this means that the boundary conditions to be respected are periodic in the spanwise direction. Visualizations of the flow behind this type of body are compared to the predictions of the Ginzburg-Landau model. 1. EXPERIMENTAL SETUP. — The wake of the torus is studied in a wind tunnel with a 25 x 25 x 100 cm3 test section. Non-uniformity of the incoming flow