Gert van der Heijden | University College London (original) (raw)

Papers by Gert van der Heijden

Science Advances, 2020

This paper reports on an analytical study of the intrinsic shapes of 523 whiskers from 15 rats. W... more This paper reports on an analytical study of the intrinsic shapes of 523 whiskers from 15 rats. We show that the variety of whiskers on a rat's cheek, each of which has different lengths and shapes, can be described by a simple mathematical equation such that each whisker is represented as an interval on the Euler spiral. When all the representative curves of mystacial vibrissae for a single rat are assembled together, they span an interval extending from one coiled domain of the Euler spiral to the other. We additionally find that each whisker makes nearly the same angle of 47°with the normal to the spherical virtual surface formed by the tips of whiskers, which constitutes the rat's tactile sensory shroud or "search space." The implications of the linear curvature model for gaining insight into relationships between growth, form, and function are discussed.

Bulletin of the American Physical Society, 2015

Bulletin of the American Physical Society, 2014

Submitted for the MAR14 Meeting of The American Physical Society Theory of equilibria of elastic ... more Submitted for the MAR14 Meeting of The American Physical Society Theory of equilibria of elastic braids with applications to DNA supercoiling 1 GERT VAN DER HEIJDEN, EUGENE STAROSTIN, University College London-Motivated by supercoiling of DNA and other filamentous structures, we formulate a new theory for equilibria of 2-braids, i.e., structures formed by two elastic rods winding around each other in continuous contact and subject to a local interstrand interaction. Unlike in previous work no assumption is made on the shape of the contact curve. Rather, this shape is solved for. The theory is developed in terms of a moving frame of directors attached to one of the strands with one of the directors pointing to the position of the other strand. The constant-distance constraint is automatically satisfied by the introduction of what we call braid strains. The price we pay is that the potential energy involves arclength derivatives of these strains, thus giving rise to a second-order variational problem. The Euler-Lagrange equations for this problem give balance equations for the overall braid force and moment referred to the moving frame as well as differential equations that can be interpreted as effective constitutive relations encoding the effect that the second strand has on the first as the braid deforms under the action of end loads. Both open braid and closed braid solutions (links and knots) are computed and current applications to DNA supercoiling are discussed.

Meccanica, 2015

We study the vibration of a tapered cantilever (Euler-Bernoulli) beam carrying a moving mass. The... more We study the vibration of a tapered cantilever (Euler-Bernoulli) beam carrying a moving mass. The tapering is assumed to be parabolic. Using the Galerkin method we find approximate solutions in an energy formulation that takes into account dynamic mass-beam coupling due to inertial, Coriolis and centrifugal effects. The approximate solutions are expanded in terms of the mode shapes of the free tapered beam, which can be obtained analytically. We then study the effect the tapering as well as the magnitude and velocity of the mass have on the tip deflections of the beam. We consider two different initial conditions, one where the mass starts moving from a statically deformed beam and one where the beam is initially triggered to vibrate. We find that tip deflections are more irregular for strongly tapered beams. Our results are of interest for barreled launch systems where tip deflections may adversely affect projectile motion.

The geometrically exact theory of linear elastic rods is used to formulate the general three-dime... more The geometrically exact theory of linear elastic rods is used to formulate the general three-dimensional problem of a twisted, clamped rod hanging under gravity and subject to buoyancy forces from a fluid. The resulting boundary-value problem is solved by the method of matched asymptotic expansions. The truncated analytical solution is compared with results obtained from a numerical scheme and shows good agreement. The method is used to consider the near-catenary application of a clamped pipeline.

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2012

The helical buckling and post-buckling of an elastic rod within a cylindrical casing arises in ma... more The helical buckling and post-buckling of an elastic rod within a cylindrical casing arises in many disciplines, but is particularly important in the petroleum industry. Here, a drill-string, subjected to an end twisting moment combined with axial tension or compression, is particularly prone to buckling within its bore-hole—with potentially serious results. In this paper, we make a detailed theoretical study of this type of instability, deriving precise new results for the advanced post-buckling stage when the rod is in continuous contact with the cylinder. Results, including rigorous stability analyses and contact pressure assessments, are presented as equilibrium surfaces to facilitate comparisons with experimental results. Two approximate solutions give insight, universal graphs and parameters, for the practically relevant case of small angles, and highlight the existence of a critical cylinder diameter. Excellent agreement with experiments is achieved.

Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 2001

ABSTRACT

Journal of the Mechanics and Physics of Solids, 2009

We study the force vs extension behaviour of a helical spring made of a thin torsionally-stiff an... more We study the force vs extension behaviour of a helical spring made of a thin torsionally-stiff anisotropic elastic rod. Our focus is on springs of very low helical pitch. For certain parameters of the problem such a spring is found not to unwind when pulled but rather to form hockles that pop-out one by one and lead to a highly non-monotonic force-extension curve. Between abrupt loop pop-outs this curve is well described by the planar elastica whose relevant solutions are classified. Our results may be relevant for tightly coiled nanosprings in future micro-and nano(electro)mechanical devices.

Journal of Nonlinear Science, 1995

Summary We present a computer-assisted study of the dynamics of two nonlinearly coupled driven os... more Summary We present a computer-assisted study of the dynamics of two nonlinearly coupled driven oscillators with rotational symmetry which arise in rotordynamics (the nonlinearity coming from bearing clearance). The nonlinearity causes a splitting of the twofold degenerate natural frequency of the associated linear model, leading to three interacting frequencies in the system. Partial mode-locking then yields a biinfinite series of

Journal of Applied Mechanics, 2003

ABSTRACT

Biophysical Journal, 2007

The formation of collagen fibrils from staggered repeats of individual molecules has become ''acc... more The formation of collagen fibrils from staggered repeats of individual molecules has become ''accepted'' wisdom. However, for over thirty years now, such a model has failed to resolve several structural and functional questions. In a novel approach, it was found, using atomic force microscopy, that tendon collagen fibrils are composed of subcomponents in a spiral disposition-that is, their structure is similar to that of macroscale ropes. Consequently, this arrangement was modeled and confirmed using elastic rod theory. This work provides new insight into collagen fibril structure and will have wide application-from the design of scaffolds for tissue engineering and a better understanding of pathogenesis of diseases of bone and tendon, to the conservation of irreplaceable parchment-based museum exhibits.

Archive for Rational Mechanics and Analysis, 2006

We study self-contact phenomena in elastic rods that are constrained to lie on a cylinder. By cho... more We study self-contact phenomena in elastic rods that are constrained to lie on a cylinder. By choosing a particular set of variables to describe the rod centerline the variational setting is made particularly simple: the strain energy is a second-order functional of a single scalar variable, and the self-contact constraint is written as an integral inequality. Using techniques from ode theory (comparison principles) and variational calculus (cut-and-paste arguments) we fully characterize the structure of constrained minimizers. An important auxiliary result states that the set of self-contact points is continuous, a result that contrasts with known examples from contact problems in free rods.

Journal of Nonlinear Science, 2010

We study the effect of a magnetic field on the behaviour of a slender conducting elastic structur... more We study the effect of a magnetic field on the behaviour of a slender conducting elastic structure, motivated by stability problems of electrodynamic space tethers. Both statical (buckling) and dynamical (whirling) instability are considered and we also compute post-buckling configurations. The equations used are the geometrically exact Kirchhoff equations. Magnetic buckling of a welded rod is found to be described by a surprisingly degenerate bifurcation, which is unfolded when both transverse anisotropy of the rod and angular velocity are considered. By solving the linearised equations about the (quasi-) stationary solutions we find various secondary instabilities. Our results are relevant for current designs of electrodynamic space tethers and potentially for future applications in nano-and molecular wires.

Mammal whiskers are often used as a model for understanding the sensory circuits in the brain. Si... more Mammal whiskers are often used as a model for understanding the sensory circuits in the brain. Signals from the whiskers, especially their forces, are processed throughout the brain, particularly in the somatosensory “barrel” cortex. Before attempting to interpret the neuronal signals, it is imperative to understand the signals received by the whisker follicles themselves and therefore accurately modelling whisker mechanics is important. Previously, whiskers have been modelled as a parabola based on Cartesian coordinates of the whisker centerline, but we propose that an Euler spiral model is a simple way to capture many aspects of whisker shape. In this study, we model 516 rat (Rattus norvegicus) whiskers as plane model curves with a linear relationship between arc length, s, and curvature, k, such that k(s) = A(s) + B and show that any original rat whisker can be mapped onto a normalized Euler spiral. The Euler spiral provides a convenient and highly accurate model for analytical s...

Magneto-striction is a property of ferromagnetic materials which causes them to change their shap... more Magneto-striction is a property of ferromagnetic materials which causes them to change their shape or dimensions during the process of magnetization. A conducting ferromagnetic rod in a magnetic field will experience a Lorentz body force and change size, coupling the effective material properties to the loading conditions. Homoclinic solutions, relating to localised post-buckled configurations, and the post-buckling curves are computed, illustrating the influence of magnetostriction on the post-buckling behaviour.

Bulletin of the American Physical Society, 2016

New Directions in Geometric and Applied Knot Theory, Dec 31, 2017

We present a theory for equilibria of geometrically exact braids made of two thin, uniform, homog... more We present a theory for equilibria of geometrically exact braids made of two thin, uniform, homogeneous, isotropic, initially-straight, inextensible and unshearable elastic rods of circular cross-section. We formulate a second-order variational problem for an action functional whose Euler-Lagrange equations, partly in Euler-Poincaré form, yield a compact system of ODEs for which we define boundary-value problems for braids closed into knots or links. The purpose of the chapter is to present a pathway of deformations leading to braids with a knotted axis, thereby offering a way to systematically compute elastic cable knots and links. A representative bifurcation diagram and selected numerical solutions illustrate our approach.

SIAM Journal on Applied Mathematics

International Journal of Solids and Structures

Wound cables and straight rods exhibit lateral instabilities when loaded under compression and ro... more Wound cables and straight rods exhibit lateral instabilities when loaded under compression and rotation in fixed-grip conditions. In a multi-strand cable made from helically wound strands, this produces a “bird cage” structure where the constituent strands separate to leave a central void region. For a straight rod, a similar instability occurs when the planar elastica becomes unstable under significant axial

We combine and extend the work of Alexander & Antman \cite{alexander.82} and Fuller \cite{fuller.... more We combine and extend the work of Alexander & Antman \cite{alexander.82} and Fuller \cite{fuller.71,fuller.78} to give a framework within which precise definitions can be given of topological and geometrical quantities characterising the contortion of open rods undergoing large deformations under end loading. We use these definitions to examine the extension of known results for closed rods to open rods. In

Science Advances, 2020

This paper reports on an analytical study of the intrinsic shapes of 523 whiskers from 15 rats. W... more This paper reports on an analytical study of the intrinsic shapes of 523 whiskers from 15 rats. We show that the variety of whiskers on a rat's cheek, each of which has different lengths and shapes, can be described by a simple mathematical equation such that each whisker is represented as an interval on the Euler spiral. When all the representative curves of mystacial vibrissae for a single rat are assembled together, they span an interval extending from one coiled domain of the Euler spiral to the other. We additionally find that each whisker makes nearly the same angle of 47°with the normal to the spherical virtual surface formed by the tips of whiskers, which constitutes the rat's tactile sensory shroud or "search space." The implications of the linear curvature model for gaining insight into relationships between growth, form, and function are discussed.

Bulletin of the American Physical Society, 2015

Bulletin of the American Physical Society, 2014

Submitted for the MAR14 Meeting of The American Physical Society Theory of equilibria of elastic ... more Submitted for the MAR14 Meeting of The American Physical Society Theory of equilibria of elastic braids with applications to DNA supercoiling 1 GERT VAN DER HEIJDEN, EUGENE STAROSTIN, University College London-Motivated by supercoiling of DNA and other filamentous structures, we formulate a new theory for equilibria of 2-braids, i.e., structures formed by two elastic rods winding around each other in continuous contact and subject to a local interstrand interaction. Unlike in previous work no assumption is made on the shape of the contact curve. Rather, this shape is solved for. The theory is developed in terms of a moving frame of directors attached to one of the strands with one of the directors pointing to the position of the other strand. The constant-distance constraint is automatically satisfied by the introduction of what we call braid strains. The price we pay is that the potential energy involves arclength derivatives of these strains, thus giving rise to a second-order variational problem. The Euler-Lagrange equations for this problem give balance equations for the overall braid force and moment referred to the moving frame as well as differential equations that can be interpreted as effective constitutive relations encoding the effect that the second strand has on the first as the braid deforms under the action of end loads. Both open braid and closed braid solutions (links and knots) are computed and current applications to DNA supercoiling are discussed.

Meccanica, 2015

We study the vibration of a tapered cantilever (Euler-Bernoulli) beam carrying a moving mass. The... more We study the vibration of a tapered cantilever (Euler-Bernoulli) beam carrying a moving mass. The tapering is assumed to be parabolic. Using the Galerkin method we find approximate solutions in an energy formulation that takes into account dynamic mass-beam coupling due to inertial, Coriolis and centrifugal effects. The approximate solutions are expanded in terms of the mode shapes of the free tapered beam, which can be obtained analytically. We then study the effect the tapering as well as the magnitude and velocity of the mass have on the tip deflections of the beam. We consider two different initial conditions, one where the mass starts moving from a statically deformed beam and one where the beam is initially triggered to vibrate. We find that tip deflections are more irregular for strongly tapered beams. Our results are of interest for barreled launch systems where tip deflections may adversely affect projectile motion.

The geometrically exact theory of linear elastic rods is used to formulate the general three-dime... more The geometrically exact theory of linear elastic rods is used to formulate the general three-dimensional problem of a twisted, clamped rod hanging under gravity and subject to buoyancy forces from a fluid. The resulting boundary-value problem is solved by the method of matched asymptotic expansions. The truncated analytical solution is compared with results obtained from a numerical scheme and shows good agreement. The method is used to consider the near-catenary application of a clamped pipeline.

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2012

The helical buckling and post-buckling of an elastic rod within a cylindrical casing arises in ma... more The helical buckling and post-buckling of an elastic rod within a cylindrical casing arises in many disciplines, but is particularly important in the petroleum industry. Here, a drill-string, subjected to an end twisting moment combined with axial tension or compression, is particularly prone to buckling within its bore-hole—with potentially serious results. In this paper, we make a detailed theoretical study of this type of instability, deriving precise new results for the advanced post-buckling stage when the rod is in continuous contact with the cylinder. Results, including rigorous stability analyses and contact pressure assessments, are presented as equilibrium surfaces to facilitate comparisons with experimental results. Two approximate solutions give insight, universal graphs and parameters, for the practically relevant case of small angles, and highlight the existence of a critical cylinder diameter. Excellent agreement with experiments is achieved.

Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 2001

ABSTRACT

Journal of the Mechanics and Physics of Solids, 2009

We study the force vs extension behaviour of a helical spring made of a thin torsionally-stiff an... more We study the force vs extension behaviour of a helical spring made of a thin torsionally-stiff anisotropic elastic rod. Our focus is on springs of very low helical pitch. For certain parameters of the problem such a spring is found not to unwind when pulled but rather to form hockles that pop-out one by one and lead to a highly non-monotonic force-extension curve. Between abrupt loop pop-outs this curve is well described by the planar elastica whose relevant solutions are classified. Our results may be relevant for tightly coiled nanosprings in future micro-and nano(electro)mechanical devices.

Journal of Nonlinear Science, 1995

Summary We present a computer-assisted study of the dynamics of two nonlinearly coupled driven os... more Summary We present a computer-assisted study of the dynamics of two nonlinearly coupled driven oscillators with rotational symmetry which arise in rotordynamics (the nonlinearity coming from bearing clearance). The nonlinearity causes a splitting of the twofold degenerate natural frequency of the associated linear model, leading to three interacting frequencies in the system. Partial mode-locking then yields a biinfinite series of

Journal of Applied Mechanics, 2003

ABSTRACT

Biophysical Journal, 2007

The formation of collagen fibrils from staggered repeats of individual molecules has become ''acc... more The formation of collagen fibrils from staggered repeats of individual molecules has become ''accepted'' wisdom. However, for over thirty years now, such a model has failed to resolve several structural and functional questions. In a novel approach, it was found, using atomic force microscopy, that tendon collagen fibrils are composed of subcomponents in a spiral disposition-that is, their structure is similar to that of macroscale ropes. Consequently, this arrangement was modeled and confirmed using elastic rod theory. This work provides new insight into collagen fibril structure and will have wide application-from the design of scaffolds for tissue engineering and a better understanding of pathogenesis of diseases of bone and tendon, to the conservation of irreplaceable parchment-based museum exhibits.

Archive for Rational Mechanics and Analysis, 2006

We study self-contact phenomena in elastic rods that are constrained to lie on a cylinder. By cho... more We study self-contact phenomena in elastic rods that are constrained to lie on a cylinder. By choosing a particular set of variables to describe the rod centerline the variational setting is made particularly simple: the strain energy is a second-order functional of a single scalar variable, and the self-contact constraint is written as an integral inequality. Using techniques from ode theory (comparison principles) and variational calculus (cut-and-paste arguments) we fully characterize the structure of constrained minimizers. An important auxiliary result states that the set of self-contact points is continuous, a result that contrasts with known examples from contact problems in free rods.

Journal of Nonlinear Science, 2010

We study the effect of a magnetic field on the behaviour of a slender conducting elastic structur... more We study the effect of a magnetic field on the behaviour of a slender conducting elastic structure, motivated by stability problems of electrodynamic space tethers. Both statical (buckling) and dynamical (whirling) instability are considered and we also compute post-buckling configurations. The equations used are the geometrically exact Kirchhoff equations. Magnetic buckling of a welded rod is found to be described by a surprisingly degenerate bifurcation, which is unfolded when both transverse anisotropy of the rod and angular velocity are considered. By solving the linearised equations about the (quasi-) stationary solutions we find various secondary instabilities. Our results are relevant for current designs of electrodynamic space tethers and potentially for future applications in nano-and molecular wires.

Mammal whiskers are often used as a model for understanding the sensory circuits in the brain. Si... more Mammal whiskers are often used as a model for understanding the sensory circuits in the brain. Signals from the whiskers, especially their forces, are processed throughout the brain, particularly in the somatosensory “barrel” cortex. Before attempting to interpret the neuronal signals, it is imperative to understand the signals received by the whisker follicles themselves and therefore accurately modelling whisker mechanics is important. Previously, whiskers have been modelled as a parabola based on Cartesian coordinates of the whisker centerline, but we propose that an Euler spiral model is a simple way to capture many aspects of whisker shape. In this study, we model 516 rat (Rattus norvegicus) whiskers as plane model curves with a linear relationship between arc length, s, and curvature, k, such that k(s) = A(s) + B and show that any original rat whisker can be mapped onto a normalized Euler spiral. The Euler spiral provides a convenient and highly accurate model for analytical s...

Magneto-striction is a property of ferromagnetic materials which causes them to change their shap... more Magneto-striction is a property of ferromagnetic materials which causes them to change their shape or dimensions during the process of magnetization. A conducting ferromagnetic rod in a magnetic field will experience a Lorentz body force and change size, coupling the effective material properties to the loading conditions. Homoclinic solutions, relating to localised post-buckled configurations, and the post-buckling curves are computed, illustrating the influence of magnetostriction on the post-buckling behaviour.

Bulletin of the American Physical Society, 2016

New Directions in Geometric and Applied Knot Theory, Dec 31, 2017

We present a theory for equilibria of geometrically exact braids made of two thin, uniform, homog... more We present a theory for equilibria of geometrically exact braids made of two thin, uniform, homogeneous, isotropic, initially-straight, inextensible and unshearable elastic rods of circular cross-section. We formulate a second-order variational problem for an action functional whose Euler-Lagrange equations, partly in Euler-Poincaré form, yield a compact system of ODEs for which we define boundary-value problems for braids closed into knots or links. The purpose of the chapter is to present a pathway of deformations leading to braids with a knotted axis, thereby offering a way to systematically compute elastic cable knots and links. A representative bifurcation diagram and selected numerical solutions illustrate our approach.

SIAM Journal on Applied Mathematics

International Journal of Solids and Structures

Wound cables and straight rods exhibit lateral instabilities when loaded under compression and ro... more Wound cables and straight rods exhibit lateral instabilities when loaded under compression and rotation in fixed-grip conditions. In a multi-strand cable made from helically wound strands, this produces a “bird cage” structure where the constituent strands separate to leave a central void region. For a straight rod, a similar instability occurs when the planar elastica becomes unstable under significant axial

We combine and extend the work of Alexander & Antman \cite{alexander.82} and Fuller \cite{fuller.... more We combine and extend the work of Alexander & Antman \cite{alexander.82} and Fuller \cite{fuller.71,fuller.78} to give a framework within which precise definitions can be given of topological and geometrical quantities characterising the contortion of open rods undergoing large deformations under end loading. We use these definitions to examine the extension of known results for closed rods to open rods. In