E. Katzav - Academia.edu (original) (raw)
Papers by E. Katzav
. In the paper the role of long range interactions on the growth of a volume conserving surface ... more . In the paper the role of long range interactions on the growth of a volume conserving surface is studied using the Nonlocal Conserved Kardar-Parisi-Zhang (NCKPZ) equation. It is shown that previous theoretical predictions are inconsistent with an exact one-dimensional result. This serves as a motivation for construction of a Self-Consistent Expansion (SCE) that recovers the exact one-dimensional result, and gives the scaling exponents in higher dimensions as well. A possible application of this result to colloidal systems is discussed.
The European Physical Journal B, 2006
. In the paper the role of long range interactions on the growth of a volume conserving surface ... more . In the paper the role of long range interactions on the growth of a volume conserving surface is studied using the Nonlocal Conserved Kardar-Parisi-Zhang (NCKPZ) equation. It is shown that previous theoretical predictions are inconsistent with an exact one-dimensional result. This serves as a motivation for construction of a Self-Consistent Expansion (SCE) that recovers the exact one-dimensional result, and gives the scaling exponents in higher dimensions as well. A possible application of this result to colloidal systems is discussed.
EPJ Web of Conferences, 2010
We present the results of recent friction experiments in which a MEMS-based sensing device is use... more We present the results of recent friction experiments in which a MEMS-based sensing device is used to measure both the normal and tangential stress fields at the base of a rough elastomer film in frictional contact with smooth, rigid, glass indentors. We consider successively multicontacts under (i) static normal loading by a spherical indentor and (ii) frictional steady sliding conditions
Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics, 1999
A minor modification of the self-consistent expansion (SCE) for the Kardar-Parisi-Zhang (KPZ) sys... more A minor modification of the self-consistent expansion (SCE) for the Kardar-Parisi-Zhang (KPZ) system with uncorrelated noise is used to obtain the exponents in systems where the noise has spatial long-range correlations. For d-dimensional systems with correlations of the form D((-->)r-(-->)r',t-t')=2D(0)/(-->)r-(-->)r'/2 rho-d)delta(t-t'), (rho>0), we find a lower critical dimension d(0)(rho)=2+2 rho, above which a perturbative Edwards-Wilkinson (EW) solution appears. Below the lower critical dimension two solutions exist, each in a different, distinct region of rho. For small rho's the solution of KPZ with uncorrelated noise is recovered. For large rho's a rho-dependent solution is found. The existence of only one solution in each region of rho is not a result of a competition between two solutions but a direct outcome of the SCE equation.
We address analytically and numerically the problem of crack path prediction in the model system ... more We address analytically and numerically the problem of crack path prediction in the model system of a crack propagating under thermal loading. We show that one can explain the instability from a straight to a wavy crack propagation by using only the principle of local symmetry and the Griffith criterion. We then argue that the calculations of the stress intensity
Physical Review E, 2014
We study the statistics of the condition number κ=λ_{max}/λ_{min} (the ratio between largest and ... more We study the statistics of the condition number κ=λ_{max}/λ_{min} (the ratio between largest and smallest squared singular values) of N×M Gaussian random matrices. Using a Coulomb fluid technique, we derive analytically and for large N the cumulative P(κ<x) and tail-cumulative P(κ>x) distributions of κ. We find that these distributions decay as P(κ<x)≈exp[-βN^{2}Φ_{-}(x)] and P(κ>x)≈exp[-βNΦ_{+}(x)], where β is the Dyson index of the ensemble. The left and right rate functions Φ_{±}(x) are independent of β and calculated exactly for any choice of the rectangularity parameter α=M/N-1>0. Interestingly, they show a weak nonanalytic behavior at their minimum 〈κ〉 (corresponding to the average condition number), a direct consequence of a phase transition in the associated Coulomb fluid problem. Matching the behavior of the rate functions around 〈κ〉, we determine exactly the scale of typical fluctuations ∼O(N^{-2/3}) and the tails of the limiting distribution of κ. The analytical results are in excellent agreement with numerical simulations.
Proceedings of the National Academy of Sciences, 2006
We propose a statistical approach for studying the close packing of elastic rods. This phenomenon... more We propose a statistical approach for studying the close packing of elastic rods. This phenomenon belongs to the class of problems of confinement of low dimensional objects, such as DNA packaging in viral capsids. The method developed is based on Edwards' approach, which was successfully applied to polymer physics and to granular matter. We show that the confinement induces a configurational phase transition from a disordered (isotropic) phase to an ordered (nematic) phase. In each phase, we derive the pressure exerted by the rod (DNA) on the container (capsid) and the force necessary to inject (eject) the rod into (out of) the container. Finally, we discuss the relevance of the present results with respect to physical and biological problems. Regarding DNA packaging in viral capsids, these results establish the existence of ordered configurations, a hypothesis upon which previous calculations were built. They also show that such ordering can result from simple mechanical constraints.
Physical Review Letters, 2013
Crumpling and folding of paper are at first sight very different ways of confining thin sheets in... more Crumpling and folding of paper are at first sight very different ways of confining thin sheets in a small volume: the former one is random and stochastic whereas the latest one is regular and deterministic. Nevertheless, certain similarities exist. Crumpling is surprisingly inefficient: a typical crumpled paper ball in a waste-bin consists of as much as 80% air. Similarly, if one folds a sheet of paper repeatedly in two, the necessary force becomes so large that it is impossible to fold it more than six or seven times. Here we show that the stiffness that builds up in the two processes is of the same nature, and therefore simple folding models allow us to capture also the main features of crumpling. An original geometrical approach shows that crumpling is hierarchical, just as the repeated folding. For both processes the number of layers increases with the degree of compaction. We find that for both processes the crumpling force increases as a power law with the number of folded layers, and that the dimensionality of the compaction process (crumpling or folding) controls the exponent of the scaling law between the force and the compaction ratio.
Physical Review Letters, 2006
Recent studies of correlations of intensity in databases of natural images revealed a remarkable ... more Recent studies of correlations of intensity in databases of natural images revealed a remarkable property. The two point correlations are described in terms of power law behavior, with an exponent which seems to be robust. In the present Letter we consider the statistical meaning of that result. We study many individual images of one of the databases considered. We find that the same law characterizing the correlations in the whole database governs also images randomly chosen from that database, with one essential difference. The exponent characterizing each image is specific and differs from the exponent characterizing the whole database. The distribution of single image exponents has been measured and found to exhibit a rather heavy tail. The database exponent cannot, thus, be considered as a statistical representative of a single image exponent. Possible reasons for the diversity in image exponents are discussed.
Physical Review Letters, 2013
The dynamics and stability of brittle cracks are not yet fully understood. Here we use the Willis... more The dynamics and stability of brittle cracks are not yet fully understood. Here we use the Willis-Movchan 3D linear perturbation formalism [J. Mech. Phys. Solids 45, 591 (1997)] to study the out-of-plane stability of planar crack fronts in the framework of linear elastic fracture mechanics. We discuss a minimal scenario in which linearly unstable crack front corrugations might emerge above a critical front propagation speed. We calculate this speed as a function of Poisson's ratio and show that corrugations propagate along the crack front at nearly the Rayleigh wave speed. Finally, we hypothesize about a possible relation between such corrugations and the long-standing problem of crack branching.
Physical Review E, 2007
We investigate propagating fronts in disordered media that belong to the universality class of we... more We investigate propagating fronts in disordered media that belong to the universality class of wetting contact lines and planar tensile crack fronts. We derive from first principles their nonlinear equations of motion, using the generalized Griffith criterion for crack fronts and three standard mobility laws for contact lines. Then we study their roughness using the self-consistent expansion. When neglecting the irreversibility of fracture and wetting processes, we find a possible dynamic rough phase with a roughness exponent of zeta=1/2 and a dynamic exponent of z=2. When including the irreversibility, we conclude that the front propagation can become history dependent, and thus we consider the value zeta=1/2 as a lower bound for the roughness exponent. Interestingly, for propagating contact line in wetting, where irreversibility is weaker than in fracture, the experimental results are close to 0.5, while for fracture the reported values of 0.55-0.65 are higher.
Physical Review E, 2013
We study the stability and roughness of propagating cracks in heterogeneous brittle two-dimension... more We study the stability and roughness of propagating cracks in heterogeneous brittle two-dimensional elastic materials. We begin by deriving an equation of motion describing the dynamics of such a crack in the framework of linear elastic fracture mechanics, based on the Griffith criterion and the principle of local symmetry. This result allows us to extend the stability analysis of Cotterell and Rice [B. Cotterell and J. R. Rice, Int. J. Fract. 16, 155 (1980)] to disordered materials. In the stable regime we find stochastic crack paths. Using tools of statistical physics, we obtain the power spectrum of these paths and their probability distribution function and conclude that they do not exhibit self-affinity. We show that a real-space fractal analysis of these paths can lead to the wrong conclusion that the paths are self-affine. To complete the picture, we unravel a systematic bias in such real-space methods and thus contribute to the general discussion of reliability of self-affine measurements.
Physical Review E, 2010
We consider the large deviations of the smallest eigenvalue of the Wishart-Laguerre Ensemble. Usi... more We consider the large deviations of the smallest eigenvalue of the Wishart-Laguerre Ensemble. Using the Coulomb gas picture we obtain rate functions for the large fluctuations to the left and the right of the hard edge. Our results are compared with known exact results for β=1 finding good agreement. We also consider the case of almost square matrices finding new universal rate functions describing large fluctuations.
Physical Review E, 2007
We report some observations concerning the statistics of longest increasing subsequences (LIS). W... more We report some observations concerning the statistics of longest increasing subsequences (LIS). We argue that the expectation of LIS, its variance, and apparently the full distribution function appears in statistical analysis of some simple nonlinear stochastic partial differential equation in the limit of very low noise intensity.
Physical Review E, 2006
We derive the second-order variation in the local static stress intensity factor of a tensile cra... more We derive the second-order variation in the local static stress intensity factor of a tensile crack with a curved front. We then discuss the relevance of this result to the stability analysis of such fronts, and propose an equation of motion of planar crack fronts in heterogeneous media that contains two main ingredients--irreversibility of the propagation of the crack front and nonlinear effects.
Physical Review E, 2005
Slow crack propagation in ductile, and in certain brittle materials, appears to take place via th... more Slow crack propagation in ductile, and in certain brittle materials, appears to take place via the nucleation of voids ahead of the crack tip due to plastic yields, followed by the coalescence of these voids. Postmortem analysis of the resulting fracture surfaces of ductile and brittle materials on the microm-mm and the nm scales, respectively, reveals self-affine cracks with anomalous scaling exponent zeta approximately = 0.8 in 3 dimensions and zeta approximately = 0.65 in 2 dimensions. In this paper we present an analytic theory based on the method of iterated conformal maps aimed at modelling the void formation and the fracture growth, culminating in estimates of the roughening exponents in 2 dimensions. In the simplest realization of the model we allow one void ahead of the crack, and address the robustness of the roughening exponent. Next we develop the theory further, to include two voids ahead of the crack. This development necessitates generalizing the method of iterated conformal maps to include doubly connected regions (maps from the annulus rather than the unit circle). While mathematically and numerically feasible, we find that the employment of the stress field as computed from elasticity theory becomes questionable when more than one void is explicitly inserted into the material. Thus further progress in this line of research calls for improved treatment of the plastic dynamics.
Physica A: Statistical Mechanics and its Applications, 2006
ABSTRACT
Journal of the Mechanics and Physics of Solids, 2009
A MEMS-based sensing device is used to measure the normal and tangential stress fields at the bas... more A MEMS-based sensing device is used to measure the normal and tangential stress fields at the base of a rough elastomer film in contact with a smooth glass cylinder in steady sliding. This geometry allows for a direct comparison between the stress profiles measured along the sliding direction and the predictions of an original exact bidimensional model of friction. The latter assumes Amontons' friction law, which implies that in steady sliding the interfacial tangential stress is equal to the normal stress times a pressure-independent dynamic friction coefficient m d , but makes no further assumption on the normal stress field. Discrepancy between the measured and calculated profiles is less than 14% over the range of loads explored. Comparison with a test model, based on the classical assumption that the normal stress field is unchanged upon tangential loading, shows that the exact model better reproduces the experimental profiles at high loads. However, significant deviations remain that are not accounted for by either calculations. In that regard, the relevance of two other assumptions made in the calculations, namely (i) the smoothness of the interface and (ii) the pressureindependence of m d is briefly discussed.
Journal of Physics A: Mathematical and Theoretical, 2008
We present exact results for the spectrum of the Nth power of the Laplacian in a bounded domain. ... more We present exact results for the spectrum of the Nth power of the Laplacian in a bounded domain. We begin with the one-dimensional case and show that the whole spectrum can be obtained in the limit of large N. We also show that it is a useful numerical approach valid for any N. Finally, we discuss implications of this work and present its possible extensions for non-integer N and for 3D Laplacian problems.
Journal of Physics A: Mathematical and Theoretical, 2012
ABSTRACT Using the Coulomb gas method and standard methods of statistical physics, we compute ana... more ABSTRACT Using the Coulomb gas method and standard methods of statistical physics, we compute analytically the joint cumulative probability distribution of the extreme eigenvalues of the Jacobi-MANOVA ensemble of random matrices, in the limit of large matrices. This allows us to derive the rate functions for the large fluctuations to the left and the right of the expected values of the smallest and largest eigenvalues analytically. Our findings are compared with some available known exact results as well as with numerical simulations finding good agreement.
. In the paper the role of long range interactions on the growth of a volume conserving surface ... more . In the paper the role of long range interactions on the growth of a volume conserving surface is studied using the Nonlocal Conserved Kardar-Parisi-Zhang (NCKPZ) equation. It is shown that previous theoretical predictions are inconsistent with an exact one-dimensional result. This serves as a motivation for construction of a Self-Consistent Expansion (SCE) that recovers the exact one-dimensional result, and gives the scaling exponents in higher dimensions as well. A possible application of this result to colloidal systems is discussed.
The European Physical Journal B, 2006
. In the paper the role of long range interactions on the growth of a volume conserving surface ... more . In the paper the role of long range interactions on the growth of a volume conserving surface is studied using the Nonlocal Conserved Kardar-Parisi-Zhang (NCKPZ) equation. It is shown that previous theoretical predictions are inconsistent with an exact one-dimensional result. This serves as a motivation for construction of a Self-Consistent Expansion (SCE) that recovers the exact one-dimensional result, and gives the scaling exponents in higher dimensions as well. A possible application of this result to colloidal systems is discussed.
EPJ Web of Conferences, 2010
We present the results of recent friction experiments in which a MEMS-based sensing device is use... more We present the results of recent friction experiments in which a MEMS-based sensing device is used to measure both the normal and tangential stress fields at the base of a rough elastomer film in frictional contact with smooth, rigid, glass indentors. We consider successively multicontacts under (i) static normal loading by a spherical indentor and (ii) frictional steady sliding conditions
Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics, 1999
A minor modification of the self-consistent expansion (SCE) for the Kardar-Parisi-Zhang (KPZ) sys... more A minor modification of the self-consistent expansion (SCE) for the Kardar-Parisi-Zhang (KPZ) system with uncorrelated noise is used to obtain the exponents in systems where the noise has spatial long-range correlations. For d-dimensional systems with correlations of the form D((-->)r-(-->)r',t-t')=2D(0)/(-->)r-(-->)r'/2 rho-d)delta(t-t'), (rho>0), we find a lower critical dimension d(0)(rho)=2+2 rho, above which a perturbative Edwards-Wilkinson (EW) solution appears. Below the lower critical dimension two solutions exist, each in a different, distinct region of rho. For small rho's the solution of KPZ with uncorrelated noise is recovered. For large rho's a rho-dependent solution is found. The existence of only one solution in each region of rho is not a result of a competition between two solutions but a direct outcome of the SCE equation.
We address analytically and numerically the problem of crack path prediction in the model system ... more We address analytically and numerically the problem of crack path prediction in the model system of a crack propagating under thermal loading. We show that one can explain the instability from a straight to a wavy crack propagation by using only the principle of local symmetry and the Griffith criterion. We then argue that the calculations of the stress intensity
Physical Review E, 2014
We study the statistics of the condition number κ=λ_{max}/λ_{min} (the ratio between largest and ... more We study the statistics of the condition number κ=λ_{max}/λ_{min} (the ratio between largest and smallest squared singular values) of N×M Gaussian random matrices. Using a Coulomb fluid technique, we derive analytically and for large N the cumulative P(κ<x) and tail-cumulative P(κ>x) distributions of κ. We find that these distributions decay as P(κ<x)≈exp[-βN^{2}Φ_{-}(x)] and P(κ>x)≈exp[-βNΦ_{+}(x)], where β is the Dyson index of the ensemble. The left and right rate functions Φ_{±}(x) are independent of β and calculated exactly for any choice of the rectangularity parameter α=M/N-1>0. Interestingly, they show a weak nonanalytic behavior at their minimum 〈κ〉 (corresponding to the average condition number), a direct consequence of a phase transition in the associated Coulomb fluid problem. Matching the behavior of the rate functions around 〈κ〉, we determine exactly the scale of typical fluctuations ∼O(N^{-2/3}) and the tails of the limiting distribution of κ. The analytical results are in excellent agreement with numerical simulations.
Proceedings of the National Academy of Sciences, 2006
We propose a statistical approach for studying the close packing of elastic rods. This phenomenon... more We propose a statistical approach for studying the close packing of elastic rods. This phenomenon belongs to the class of problems of confinement of low dimensional objects, such as DNA packaging in viral capsids. The method developed is based on Edwards' approach, which was successfully applied to polymer physics and to granular matter. We show that the confinement induces a configurational phase transition from a disordered (isotropic) phase to an ordered (nematic) phase. In each phase, we derive the pressure exerted by the rod (DNA) on the container (capsid) and the force necessary to inject (eject) the rod into (out of) the container. Finally, we discuss the relevance of the present results with respect to physical and biological problems. Regarding DNA packaging in viral capsids, these results establish the existence of ordered configurations, a hypothesis upon which previous calculations were built. They also show that such ordering can result from simple mechanical constraints.
Physical Review Letters, 2013
Crumpling and folding of paper are at first sight very different ways of confining thin sheets in... more Crumpling and folding of paper are at first sight very different ways of confining thin sheets in a small volume: the former one is random and stochastic whereas the latest one is regular and deterministic. Nevertheless, certain similarities exist. Crumpling is surprisingly inefficient: a typical crumpled paper ball in a waste-bin consists of as much as 80% air. Similarly, if one folds a sheet of paper repeatedly in two, the necessary force becomes so large that it is impossible to fold it more than six or seven times. Here we show that the stiffness that builds up in the two processes is of the same nature, and therefore simple folding models allow us to capture also the main features of crumpling. An original geometrical approach shows that crumpling is hierarchical, just as the repeated folding. For both processes the number of layers increases with the degree of compaction. We find that for both processes the crumpling force increases as a power law with the number of folded layers, and that the dimensionality of the compaction process (crumpling or folding) controls the exponent of the scaling law between the force and the compaction ratio.
Physical Review Letters, 2006
Recent studies of correlations of intensity in databases of natural images revealed a remarkable ... more Recent studies of correlations of intensity in databases of natural images revealed a remarkable property. The two point correlations are described in terms of power law behavior, with an exponent which seems to be robust. In the present Letter we consider the statistical meaning of that result. We study many individual images of one of the databases considered. We find that the same law characterizing the correlations in the whole database governs also images randomly chosen from that database, with one essential difference. The exponent characterizing each image is specific and differs from the exponent characterizing the whole database. The distribution of single image exponents has been measured and found to exhibit a rather heavy tail. The database exponent cannot, thus, be considered as a statistical representative of a single image exponent. Possible reasons for the diversity in image exponents are discussed.
Physical Review Letters, 2013
The dynamics and stability of brittle cracks are not yet fully understood. Here we use the Willis... more The dynamics and stability of brittle cracks are not yet fully understood. Here we use the Willis-Movchan 3D linear perturbation formalism [J. Mech. Phys. Solids 45, 591 (1997)] to study the out-of-plane stability of planar crack fronts in the framework of linear elastic fracture mechanics. We discuss a minimal scenario in which linearly unstable crack front corrugations might emerge above a critical front propagation speed. We calculate this speed as a function of Poisson's ratio and show that corrugations propagate along the crack front at nearly the Rayleigh wave speed. Finally, we hypothesize about a possible relation between such corrugations and the long-standing problem of crack branching.
Physical Review E, 2007
We investigate propagating fronts in disordered media that belong to the universality class of we... more We investigate propagating fronts in disordered media that belong to the universality class of wetting contact lines and planar tensile crack fronts. We derive from first principles their nonlinear equations of motion, using the generalized Griffith criterion for crack fronts and three standard mobility laws for contact lines. Then we study their roughness using the self-consistent expansion. When neglecting the irreversibility of fracture and wetting processes, we find a possible dynamic rough phase with a roughness exponent of zeta=1/2 and a dynamic exponent of z=2. When including the irreversibility, we conclude that the front propagation can become history dependent, and thus we consider the value zeta=1/2 as a lower bound for the roughness exponent. Interestingly, for propagating contact line in wetting, where irreversibility is weaker than in fracture, the experimental results are close to 0.5, while for fracture the reported values of 0.55-0.65 are higher.
Physical Review E, 2013
We study the stability and roughness of propagating cracks in heterogeneous brittle two-dimension... more We study the stability and roughness of propagating cracks in heterogeneous brittle two-dimensional elastic materials. We begin by deriving an equation of motion describing the dynamics of such a crack in the framework of linear elastic fracture mechanics, based on the Griffith criterion and the principle of local symmetry. This result allows us to extend the stability analysis of Cotterell and Rice [B. Cotterell and J. R. Rice, Int. J. Fract. 16, 155 (1980)] to disordered materials. In the stable regime we find stochastic crack paths. Using tools of statistical physics, we obtain the power spectrum of these paths and their probability distribution function and conclude that they do not exhibit self-affinity. We show that a real-space fractal analysis of these paths can lead to the wrong conclusion that the paths are self-affine. To complete the picture, we unravel a systematic bias in such real-space methods and thus contribute to the general discussion of reliability of self-affine measurements.
Physical Review E, 2010
We consider the large deviations of the smallest eigenvalue of the Wishart-Laguerre Ensemble. Usi... more We consider the large deviations of the smallest eigenvalue of the Wishart-Laguerre Ensemble. Using the Coulomb gas picture we obtain rate functions for the large fluctuations to the left and the right of the hard edge. Our results are compared with known exact results for β=1 finding good agreement. We also consider the case of almost square matrices finding new universal rate functions describing large fluctuations.
Physical Review E, 2007
We report some observations concerning the statistics of longest increasing subsequences (LIS). W... more We report some observations concerning the statistics of longest increasing subsequences (LIS). We argue that the expectation of LIS, its variance, and apparently the full distribution function appears in statistical analysis of some simple nonlinear stochastic partial differential equation in the limit of very low noise intensity.
Physical Review E, 2006
We derive the second-order variation in the local static stress intensity factor of a tensile cra... more We derive the second-order variation in the local static stress intensity factor of a tensile crack with a curved front. We then discuss the relevance of this result to the stability analysis of such fronts, and propose an equation of motion of planar crack fronts in heterogeneous media that contains two main ingredients--irreversibility of the propagation of the crack front and nonlinear effects.
Physical Review E, 2005
Slow crack propagation in ductile, and in certain brittle materials, appears to take place via th... more Slow crack propagation in ductile, and in certain brittle materials, appears to take place via the nucleation of voids ahead of the crack tip due to plastic yields, followed by the coalescence of these voids. Postmortem analysis of the resulting fracture surfaces of ductile and brittle materials on the microm-mm and the nm scales, respectively, reveals self-affine cracks with anomalous scaling exponent zeta approximately = 0.8 in 3 dimensions and zeta approximately = 0.65 in 2 dimensions. In this paper we present an analytic theory based on the method of iterated conformal maps aimed at modelling the void formation and the fracture growth, culminating in estimates of the roughening exponents in 2 dimensions. In the simplest realization of the model we allow one void ahead of the crack, and address the robustness of the roughening exponent. Next we develop the theory further, to include two voids ahead of the crack. This development necessitates generalizing the method of iterated conformal maps to include doubly connected regions (maps from the annulus rather than the unit circle). While mathematically and numerically feasible, we find that the employment of the stress field as computed from elasticity theory becomes questionable when more than one void is explicitly inserted into the material. Thus further progress in this line of research calls for improved treatment of the plastic dynamics.
Physica A: Statistical Mechanics and its Applications, 2006
ABSTRACT
Journal of the Mechanics and Physics of Solids, 2009
A MEMS-based sensing device is used to measure the normal and tangential stress fields at the bas... more A MEMS-based sensing device is used to measure the normal and tangential stress fields at the base of a rough elastomer film in contact with a smooth glass cylinder in steady sliding. This geometry allows for a direct comparison between the stress profiles measured along the sliding direction and the predictions of an original exact bidimensional model of friction. The latter assumes Amontons' friction law, which implies that in steady sliding the interfacial tangential stress is equal to the normal stress times a pressure-independent dynamic friction coefficient m d , but makes no further assumption on the normal stress field. Discrepancy between the measured and calculated profiles is less than 14% over the range of loads explored. Comparison with a test model, based on the classical assumption that the normal stress field is unchanged upon tangential loading, shows that the exact model better reproduces the experimental profiles at high loads. However, significant deviations remain that are not accounted for by either calculations. In that regard, the relevance of two other assumptions made in the calculations, namely (i) the smoothness of the interface and (ii) the pressureindependence of m d is briefly discussed.
Journal of Physics A: Mathematical and Theoretical, 2008
We present exact results for the spectrum of the Nth power of the Laplacian in a bounded domain. ... more We present exact results for the spectrum of the Nth power of the Laplacian in a bounded domain. We begin with the one-dimensional case and show that the whole spectrum can be obtained in the limit of large N. We also show that it is a useful numerical approach valid for any N. Finally, we discuss implications of this work and present its possible extensions for non-integer N and for 3D Laplacian problems.
Journal of Physics A: Mathematical and Theoretical, 2012
ABSTRACT Using the Coulomb gas method and standard methods of statistical physics, we compute ana... more ABSTRACT Using the Coulomb gas method and standard methods of statistical physics, we compute analytically the joint cumulative probability distribution of the extreme eigenvalues of the Jacobi-MANOVA ensemble of random matrices, in the limit of large matrices. This allows us to derive the rate functions for the large fluctuations to the left and the right of the expected values of the smallest and largest eigenvalues analytically. Our findings are compared with some available known exact results as well as with numerical simulations finding good agreement.