Showing papers in "Zamm-zeitschrift Fur Angewandte Mathematik Und Mechanik in 2008"
TL;DR: In this article, a variational formulation of constrained dynamics is presented in the continuous and in the discrete setting, where the existing theory on variational integration of constrained problems is extended by aspects on the initialization of simulations, the discrete Legendre transform and certain postprocessing steps.
Abstract: A variational formulation of constrained dynamics is presented in the continuous and in the discrete setting. The existing theory on variational integration of constrained problems is extended by aspects on the initialization of simulations, the discrete Legendre transform and certain postprocessing steps. Furthermore, the discrete null space method which has been introduced in the framework of energy-momentum conserving integration of constrained systems is adapted to the framework of variational integrators. It eliminates the constraint forces (including the Lagrange multipliers) from the timestepping scheme and subsequently reduces its dimension to the minimal possible number. While retaining the structure preserving properties of the specific integrator, the solution of the smaller dimensional system saves computational costs and does not suffer from conditioning problems. The performance of the variational discrete null space method is illustrated by numerical examples dealing with mass point systems, a closed kinematic chain of rigid bodies and flexible multibody dynamics and the solutions are compared to those obtained by an energy-momentum scheme.
158 citations
TL;DR: This paper focuses on uncertain systems, where the randomness is assumed spatial and traditional computational approaches usually use some form of perturbation or Monte Carlo simulation, compared with more recent methods based on stochastic Galerkin approximations.
Abstract: Uncertainty estimation arises at least implicitly in any kind of modelling of the real world, and it is desirable to actually quantify the uncertainty in probabilistic terms. Here the emphasis is on uncertain systems, where the randomness is assumed spatial. Traditional computational approaches usually use some form of perturbation or Monte Carlo simulation. This is contrasted here with more recent methods based on stochastic Galerkin approximations. Also some approaches to an adaptive uncertainty quantification are pointed out. \abstract{Uncertainty estimation arises at least implicitly in any kind of modelling of the real world, and it is desirable to actually quantify the uncertainty in probabilistic terms. Here the emphasis is on uncertain systems, where the randomness is assumed spatial. Traditional computational approaches usually use some form of perturbation or Monte Carlo simulation. This is contrasted here with more recent methods based on stochastic Galerkin approximations. Also some approaches to an adaptive uncertainty quantification are pointed out.}
99 citations
TL;DR: Considering the viscoelastic behavior of polymer foams, a new plate theory based on the direct approach is introduced and applied to plates composed of functionally graded materials (FGM) as mentioned in this paper.
Abstract: Considering the viscoelastic behavior of polymer foams, a new plate theory based on the direct approach is introduced and applied to plates composed of functionally graded materials (FGM). The governing two-dimensional equations are formulated for a deformable surface and the stiffness parameters are identified for the linear viscoelastic isotropic material behavior. It is assumed that the material properties are changing in thickness direction. Solving the plate bending problem of the global mechanical analysis, it will be demonstrated that in some cases, the results significantly differ from the results based on the classical Kirchhoff-type theory.
71 citations
TL;DR: In this paper, the authors consider two dynamic contact problems between an elastic-visco-plastic body and an obstacle, the so-called foundation, and derive a variational formulation of the first problem and then prove its unique weak solvability, by using arguments on nonlinear evolution equations with monotone operators and fixed point.
Abstract: We consider two dynamic contact problems between an elastic-visco-plastic body and an obstacle, the so-called foundation. The contact is frictionless and it is modelled with normal compliance of such a type that the penetration is not restricted in the first problem, but is restricted with unilateral constraint, in the second one. We derive a variational formulation of the first problem and then prove its unique weak solvability, by using arguments on nonlinear evolution equations with monotone operators and fixed point. Then, we derive a variational formulation of the second problem and prove its weak solvability. To this end we consider a sequence of regularized problems which have a unique solution, derive a priori estimates and use compactness properties to obtain a solution to the original model, by passing to the limit as the regularization parameter converges to zero.
57 citations
TL;DR: A review of the history of dynamo and MRI related experiments is delineated, and some directions of future work are discussed in this paper, where an MRI-like mode was found on the background of a turbulent spherical Couette flow at the University of Maryland.
Abstract: It is widely known that cosmic magnetic fields, i.e. the fields of planets, stars, and galaxies, are produced by the hydromagnetic dynamo effect in moving electrically conducting fluids. It is less well known that cosmic magnetic fields play also an active role in cosmic structure formation by enabling outward transport of angular momentum in accretion disks via the magnetorotational instability (MRI). Considerable theoretical and computational progress has been made in understanding both processes. In addition to this, the last ten years have seen tremendous efforts in studying both effects in liquid metal experiments. In 1999, magnetic field self-excitation was observed in the large scale liquid sodium facilities in Riga and Karlsruhe. Recently, self-excitation was also obtained in the French “von Karman sodium” (VKS) experiment. An MRI-like mode was found on the background of a turbulent spherical Couette flow at the University of Maryland. Evidence for MRI as the first instability of an hydrodynamically stable flow was obtained in the “Potsdam Rossendorf Magnetic Instability Experiment” (PROMISE). In this review, the history of dynamo and MRI related experiments is delineated, and some directions of future work are discussed.
56 citations
TL;DR: In this article, the steady laminar flow of an elastico-viscous fluid impinging normally upon a wall has been investigated when there is a partial slip of the fluid at the wall.
Abstract: The steady, laminar flow of an elastico-viscous fluid impinging normally upon a wall has been investigated when there is a partial slip of the fluid at the wall. The governing equations of motion admit a similarity solution in terms of η, the dimensionless distance normal to the wall. The boundary value problem characterizing the flow has been solved without making any assumption on the size of either the viscoelastic fluid parameter or the partial slip parameter. The solutions are shown to exist only up to a critical value of viscoelastic fluid parameter. The effect of the partial slip is to enhance this critical value.
48 citations
TL;DR: In this paper, the authors considered the problem of switching between different control devices in a finite string to zero state in finite time by controlling the state at the two boundary points, where at each moment in time one of the boundary controls must be switched off, that is its control value must be equal to zero.
Abstract: In many control application, switching between different control devices occurs. Here the problem to control a finite string to the zero state in finite time by controlling the state at the two boundary points is considered, where at each moment in time one of the boundary controls must be switched off, that is its control value must be equal to zero. The corresponding optimal control problem where the objective function is the L 2 norm of the controls is solved explicitly in the sense that controls that are successful and minimize at the same time the objective function are determined as functions of the initial state. Due to the complementarity condition that appears in the optimal control problem, it is non-convex and the optimal control is in general not uniquely determined. To allow for technical constraints it is important to avoid an accumulation of switching points at so-called Zeno-points. We give examples that illustrate how switching regimes of practical value can be obtained.
39 citations
TL;DR: In this paper, numerical modeling of discrete internal cracks, namely central bursts, in direct forward extrusion process is presented, in a thermodynamically consistent setting, a local Lemaitre variant damage model with quasi-unilateral evolution is coupled with hyperelastic-plasticity.
Abstract: Materializing Continuum Damage Mechanics (CDM), numerical modeling of discrete internal cracks, namely central bursts, in direct forward extrusion process is presented. Accordingly, in a thermodynamically consistent setting, a local Lemaitre variant damage model with quasi-unilateral evolution is coupled with hyperelastic-plasticity. The formulations are constructed in the principal axes where simultaneous local integration schemes are efficiently developed. To this end, the framework is implemented as ABAQUS/VUMAT subroutine to be used in an explicit FE solution scheme, and utilized in direct forward extrusion simulations for bearing steel, 100Cr6. Discontinuous cracks are obtained with the element deletion procedure, where the elements reaching the critical damage value are removed from the mesh. The periodicity of the cracks shows well accordance with the experimental facts. The investigations reveal that, application of the quasi-unilateral conditions together with the crack closure parameter has an indispensable effect on the damage accumulation zones by determining their internal or superficial character.
37 citations
TL;DR: In this paper, an asymptotic approach for a micropolar flow through a thin curvilinear channel is proposed, where a priori estimates are obtained together with the existence and uniqueness of the solution.
Abstract: This paper is concerned with an asymptotic approach for a micropolar flow through a thin curvilinear channel. A priori estimates (which we obtain together with the existence and the uniqueness of the solution) are used to establish the error between the exact solution and the asymptotic one and to justify the asymptotic analysis. We obtain the expression of an expansion of order K and we study the general problems for the boundary layer functions. Under some additional assumptions on the data we obtain satisfactory error estimates.
32 citations
TL;DR: In this article, a viscoelastic solid in Kelvin-Voigt rheology involving plasticity coupled with a heat-transfer equation through a temperature-dependent yield stress is investigated.
Abstract: A viscoelastic solid in Kelvin-Voigt rheology involving plasticity coupled with a heat-transfer equation through a temperature-dependent yield stress is investigated. No hardening is studied but the evolution of the plastic strain is considered to be rate-dependent. A numerical scheme which is semi-implicit in time and employs lowest order finite elements on weakly acute triangulations in space is devised and its convergence is proved by careful subsequent limit passage. Computational studies underline robustness and efficiency of the method and illustrate physical effects such as the softening of a material due to dissipated energy that causes a rise in temperature and a local decrease of the yield stress.
31 citations
TL;DR: In this article, the authors consider the regularity of weak solutions to evolution variational inequalities arising from the flow theory of plasticity with isotropic and kinematic hardening and derive a Morrey condition for the stress velocities and the strains up to the boundary.
Abstract: We consider the regularity of weak solutions to evolution variational inequalities arising from the flow theory of plasticity with isotropic and kinematic hardening. The (linear) elasticity tensor is allowed to have discontinuities. We derive a Morrey condition for the stress velocities and the strains (not the strain velocity!) up to the boundary. In the case of two space dimensions we conclude the Holder continuity of the displacements.
TL;DR: The hydrodynamical limit of the proposed kinetic system is formally compute, and the existence of flow solutions and their continuous dependence on the design parameters are rigorously established.
Abstract: We consider the problem of optimal design of flow domains for Navier-Stokes flows in order to minimize a given performance functional. We attack the problem using topology optimization techniques, or control in coefficients, which are widely known in structural optimization of elastic solids and structures for their flexibility, generality, yet ease of use, and integration with existing FEM software. We use a simple kinetic model to approximate the Navier-Stokes system. Arguably, we take a rather unconventional path in the kinetic theory, using it only to gain insight about the Navier-Stokes-related system of hydrodynamical equations, which we take as our starting point. Thus all the modifications we make to the kinetic models are "hydrodynamically" inspired and we seek no particular physical explanation for them; the only requirement for us is the convergence (at least, formal) of the kinetic equations towards the correct hydrodynamical limit. We formally compute the hydrodynamical limit of the proposed kinetic system, and rigorously establish the existence of flow solutions and their continuous dependence on the design parameters. Optimal controls are shown to belong to a special class (0-1 solutions) for a popular power dissipation minimization problem for viscous fluids.
TL;DR: In this article, the Frechet derivative of a function with respect to its tensor argument is defined in a natural way, and its uniqueness is analyzed in a straight-forward manner from the coordinate-free tensor representation.
Abstract: We analyze a coordinate-free tensor setting in R 3 within the context of the classical tensor analysis. To this end, we formulate in a basis-free manner the notions of second- and fourth-rank tensors in R 3 , and corresponding operations on tensors. Among the large number of different approaches to the tensor setting, we give the preference to the convenient ones, concerning the specific needs of computational solid mechanics. We use the well-known Frechet derivative to define the derivative of a function with respect to its tensor argument in a natural way. Furthermore, such aspects as the derivative with respect to a symmetric tensor argument and its uniqueness are covered in this paper. For the sake of completeness we present the coordinate representation of tensors and tensor operations. This representation is obtained in a straight-forward manner from the coordinate-free one. In particular, we elaborate the computation of the inverse of a fourth-rank tensor and the inverse of a linear transformation on the space of symmetric second-rank tensors. The tensor formalism is applied to the analysis of a nonlinear system of differential and algebraic equations governing visoplastic material response. An implicit time-stepping algorithm is formulated and the numerical treatment of the algorithm is discussed.
TL;DR: In this article, a non-hypersingular traction based boundary equation method (BIEM) is used to deal with an arbitrarily shaped anti-plane shear crack in a finite inhomogeneous piezoelectric domain under time-harmonic loading.
Abstract: Treated is an arbitrarily shaped anti-plane shear crack in a finite inhomogeneous piezoelectric domain under time-harmonic loading. Within a unified scheme different types of inhomogeneity are considered for which the material parameters may vary in two directions. The problem is solved by using a numerically efficient non-hypersingular traction based boundary equation method (BIEM). The fundamental solutions for the different inhomogeneity types are derived in closed form. This is done by an appropriate functional transformation of the displacement vector in order to obtain a wave equation with constant coefficients and subsequently by application of the Radon transform. Numerical results for the stress intensity factors (SIF) are discussed for several examples. They show the effect of the material inhomogeneity and the efficiency of the method.
TL;DR: In this article, the authors focus on quasi-static problems with constitutive equations of evolutionary type and apply the Backward-Euler method or more appropriately using time-adaptive, stiffly accurate, diagonally implicit Runge-Kutta methods in combination with the Multilevel-Newton method.
Abstract: On the one hand, displacement controlled processes are frequently applied in Computational Mechanics using the finite-element method, on the other hand, there are some shortcomings regarding the study and the presentation of this particular case. In this article, we focus our theoretical considerations on quasi-static problems with constitutive equations of evolutionary type. In this case, it is currently known that the solution procedure of global and local iterations within the “Newton-Raphson method” is related to the method of lines, where one arrives at a system of differential-algebraic equations after the spatial discretization by means of the finite element method. This could be solved by means of the Backward-Euler method or more appropriately using time-adaptive, stiffly accurate, diagonally implicit Runge-Kutta methods in combination with the Multilevel-Newton method. In this respect the theoretical basis of currently applied implicit non-linear finite element analyses is extended to displacement controlled processes. In this article, the calculation of the reaction forces using the method of Lagrange multipliers and the penalty method are of special interest in view of the new DAE-approach. Accordingly, an important objective lies in a consistent notation so that the dependence of known and unknown variables as well as the transition from local to global level becomes obvious. These investigations are of utmost importance for micro-macro homogenization techniques in FE$^2$ approaches and the development of new global time and space-adaptive integration methods.
TL;DR: In this paper, the theoretical fundamentals of how to take classical and temperature-modulated DSC tests into account in order to model or identify the thermochemical parts of the specific-free energy, which is the most commonly used thermodynamic potential in continuum mechanics.
Abstract: To study temperature-induced changes in the specific enthalpy of materials, the differential scanning calorimetry technique (DSC) is a powerful experimental approach. It is well-established in physical chemistry and materials science but less familiar in continuum mechanics. In this context, the balance equation of energy in conjunction with the theory of internal variables is the fundamental aspect from the point of view of continuum mechanics. This paper presents the theoretical fundamentals of how to take classical and temperature-modulated DSC tests into account in order to model or identify the thermochemical parts of the specific-free energy, which is the most commonly used thermodynamic potential in continuum mechanics. The numerical examples treated in this work provide an insight into the possibilities of the DSC technique and into the constitutive representation of exothermal energy changes in curing adhesives.
TL;DR: In this article, the velocity boundary-layer growth resulting from an impulsively started wedge is investigated, where the governing equations describing the impulsively set into motion Falkner-Skan flows are transformed to a partial differential equation but the number of independent variables are reduced from three to two.
Abstract: An investigation of the velocity boundary-layer growth resulting from an impulsively started wedge is presented in this paper. By introducing a new set of scaled coordinates, the governing equations describing the impulsively set into motion Falkner-Skan flows is transformed to a partial differential equation but the number of independent variables are reduced from three to two. Note that the new time scaling avoids the appearance of the singularity resulting from the logarithmic function ln(1-ξ), thus reducing the difficulties of computations. Analytical approximations for the momentum boundary layer are obtained, which are valid for all time. The results are then matched numerically using the Keller-Box method.
TL;DR: In this article, the authors presented a plane stress and plane strain analytical solution to partially plastic deformation of a curved beam, where the beam has a narrow rectangular cross section and it is subjected to couples at its end sections prevailing pure bending conditions.
Abstract: Plane stress and plane strain analytical solutions to partially plastic deformation of a curved beam are presented. The beam has a narrow rectangular cross section and it is subjected to couples at its end sections prevailing pure bending conditions. The analysis is based on Tresca's yield criterion, its associated flow rule and linear strain hardening material behavior. The solutions are verified in comparison to the ones available in the literature. It is shown that plane stress and plane strain solutions agree well in the elastic and in the elastic-plastic deformation stages. It is also observed that the changes in the dimensions of the beam as it deforms are negligibly small.
TL;DR: In this article, a conditionally stable finite-difference scheme that consistently approximates the solution of a general class of (3+1)-dimensional nonlinear equations was presented, which generalizes in various ways the quantitative model governing discrete arrays consisting of coupled harmonic oscillators.
Abstract: In this work, we present a conditionally stable finite-difference scheme that consistently approximates the solution of a general class of (3+1)-dimensional nonlinear equations that generalizes in various ways the quantitative model governing discrete arrays consisting of coupled harmonic oscillators. Associated with this method, there exists a discrete scheme of energy that consistently approximates its continuous counterpart. The method has the properties that the associated rate of change of the discrete energy consistently approximates its continuous counterpart, and it approximates both a fully continuous medium and a spatially discretized system. Conditional stability of the numerical technique is established, and applications are provided to the existence of the process of nonlinear supratransmission in generalized Klein-Gordon systems and the propagation of binary signals in semi-unbounded, three-dimensional arrays of harmonic oscillators coupled through springs and perturbed harmonically at the boundaries, where the basic model is a modified sine-Gordon equation; our results show that a perfect transmission is achieved via the modulation of the driving amplitude at the boundary. Additionally, we present an example of a nonlinear system with a forbidden band-gap which does not present supratransmission, thus establishing that the existence of a forbidden band-gap in the linear dispersion relation of a nonlinear system is not a sufficient condition for the system to present supratransmission.
TL;DR: The quasilinear system of partial differential equations governing the flow of a UCM fluid is known to be of mixed elliptic-hyperbolic type and compatibility equations associated with the hyperbolic part of the system are derived in this paper.
Abstract: The quasilinear system of partial differential equations governing the flow of a UCM fluid is known to be of mixed elliptic-hyperbolic type The compatibility equations associated with the hyperbolic part of the system are derived in this paper There are two characteristic variables that are transported along the characteristics These are both associated with the conditional well-posedness of the system The compatibility equations in this case are derived from the constitutive equation alone There are two additional characteristics that are always present for unsteady inertial flow and which may be present for steady inertial flow The compatibility equations associated with these characteristics involve a coupling between the momentum and constitutive equations At an inflow boundary, it is shown that all stress components can be prescribed, despite the fact that there are only two characteristic variables that enter the domain along the streamlines The remaining stress component is provided by fixing the incoming elastic shear waves
TL;DR: In this paper, a mathematical model for lateral rotor dynamic analysis of soft mounted asynchronous machines with sleeve bearings, due to harmonic and transient excitation, is presented. But the model is restricted to the case of a single rotor.
Abstract: This paper shows a mathematical model for lateral rotor dynamic analysis of soft mounted asynchronous machines with sleeve bearings, due to harmonic and transient excitation. The simplified analytical model combines the electromagnetic, the rotor dynamic, and the specific characteristic of sleeve bearings, considering foundation stiffness and damping. Based on the shown mathematical description, it is possible to calculate the complex eigenvalues and eigenvectors, which describe the natural vibrations, the limit of stability, the relative shaft displacements, the bearing house vibrations and the dynamic foundation forces, due to rotor excitations and air gap torque excitations. Based on the simplified plane analytical model a three-dimensional finite element model is also derived. The aim of this paper is not to replace a finite element calculation, but to show the mathematical coherences between rotor dynamic, electromagnetic and sleeve bearing characteristic for soft mounted asynchronous machines, considering foundation stiffness and damping, based on a simplified mathematical model.
TL;DR: In this article, the Stokes flow problems of an incompressible viscous fluid in corrugated channels and pipes are considered and a pertubation solution is developed leading to an expression for the flow rate enhancement as a function of the statistics of the corrugations.
Abstract: The Stokes flow problems of an incompressible viscous fluid in corrugated channels and pipes are considered. The flow is parallel to the corrugations which are of small amplitude ϵ and are represented by stationary random functions. A pertubation solution is developed leading to an expression for the flow rate enhancement as a function of the statistics of the corrugations. Simplification of this expression are discussed.
Das Stokessche Stromungsproblem eines inkompressiblen, viskosen Mediums in geriffelten Kanalen und Rohren wird betrachtet. Die Stromung verlauft parallel zur Riffelung, die von der kleinen Amplitude ϵ ist und durch stationare Zufallsfunktionen dargestellt wird. Eine Storungslosung wird entwickelt, die zu einem Ausdruck Fur das Anwachsen der Stromungsgeschwindigkeit als Funktion der Statistik der Riffelung fuhrt. Vereinfachungen dieses Ausdrucks werden diskutiert.
TL;DR: In this article, a passage to the limit of the usual friction law for one degree of freedom solid where the dynamical friction coefficient is smaller than the static friction coefficient was studied, and it was shown that this law can lose all predictive power.
Abstract: We study a usual friction law for one degree of freedom solid where the dynamical friction coefficient is smaller than the static friction coefficient. We show that this law can be dangerous, because it can lose all predictive power. We first study a passage to the limit to prove that the limit is strongly depending on the data. We secondly study the obtained limit law, which is written with a non maximal monotone multivalued graph; this property implies that the differential inclusion possesses a set of solutions, whose diameter is too big to know the solution with precision. Moreover, the associated numerical scheme does not have a unique solution.
TL;DR: In this article, the authors derived a two dimensional equation for transversal vibrations of an elastoplastic plate from a general three dimensional system with a single yield tensorial von Mises plasticity model in the five dimensional deviatoric space.
Abstract: The two dimensional equation for transversal vibrations of an elastoplastic plate is derived from a general three dimensional system with a single yield tensorial von Mises plasticity model in the five dimensional deviatoric space. It leads after dimensional reduction to a multiyield three dimensional Prandtl-Ishlinskii hysteresis model whose weight function is explicitly given. The resulting partial differential equation with hysteresis is solved by means of viscous approximations and a monotonicity argument.
TL;DR: In this paper, the authors examined a variant of this model where there is no damping in the normal contact forces but there is coupled stiffness between the normal and tangential forces via body deformations, and showed that this model admits a formulation as an ordinary differential equation with a boundedly Lipschitz continuous, albeit implicitly defined, semismooth right-hand side with global linear growth.
Abstract: A complementarity-based, 3-dimensional frictional contact model with local compliance and damping was introduced in the Ph.D. thesis [45] and the paper [47] and was subsequently studied extensively in [31, 49]. In this paper, we examine a variant of this model where there is no damping in the normal contact forces but there is coupled stiffness between the normal and tangential forces via body deformations. We show that this frictional contact model admits a formulation as an ordinary differential equation with a boundedly Lipschitz continuous, albeit implicitly defined, semismooth right-hand side with global linear growth. Several major consequences follow from such a formulation: (a) existence and uniqueness of a continuously differentiable solution trajectory originated from an arbitrary initial state, (b) finite contact forces that are semismooth functions of the system state, (c) semismooth dependence of the trajectory on the initial state, and (d) convergence of a shooting method for solving two-point boundary problems. The derived results are valid for both a dynamic model and a quasistatic model, the latter being one in which inertia effects are ignored, and for a broad class of friction cones that include the well-known quadratic Coulomb cone and its polygonal approximations. Part II of this work establishes the absence of Zeno states in such a frictional contact model.
TL;DR: In this paper, the authors used the compact mono-harmonic general solutions of transversely isotropic thermoelastic materials to construct the three-dimensional Green's function of a steady point heat source in a semi-infinite transversely-isotropic material by three newly introduced monoharmonic functions.
Abstract: We use the compact mono-harmonic general solutions of transversely isotropic thermoelastic materials to construct the three-dimensional Green's function of a steady point heat source in a semi-infinite transversely isotropic thermoelastic material by three newly introduced mono-harmonic functions. All components of coupled field are expressed in terms of elementary functions and are convenient to use. Numerical results for hexagonal zinc are given graphically by contours.
TL;DR: In this article, it is shown that the energy density of deformation fields in terms of the homotopy operator corresponds to the J-integral for dislocation-disclination fields and gives the force on dislocation disclinations fields as a physical interpretation.
Abstract: The J-integral (a path-independent energy integral) formalism is the standard method of analyzing nonlinear fracture mechanics. It is shown that the energy density of deformation fields in terms of the homotopy operator corresponds to the J-integral for dislocation-disclination fields and gives the force on dislocation-disclination fields as a physical interpretation. The continuum theory of defects gives a natural framework for understanding the topological aspects of the J-integral. This geometric interpretation gives that the J-integral is an alternative expression of the well-known theorem in differential geometry, i.e., the Gauss-Bonnet theorem (with genus = 1). The geometrical expression of the J-integral shows that the Eshelby's energy-momentum (the physical quantity of the material space) is closely related to the Einstein 3-form (the geometric objects of the material space).
TL;DR: In this article, upper and lower bounds for the center and meniscus heights of a symmetric equilibrium capillary surface are given, which are sensibly identical at small Bond numbers B, and they provide significant information also for intermediate values of B.
Abstract: Upper and lower bounds are given for the center and meniscus heights of a symmetric equilibrium capillary surface. The bounds are sensibly identical at small Bond numbers B, and they provide significant information also for intermediate values of B.
Die Hohe im Mittelpunkt und die „Meniskushohe” einer symmetrischen Kapillarflache werden von oben und von unten abgeschatzt. Die Resultate sind Fur enge Rohren asymptotisch genau und behalten auch Fur breitere Rohren eine Bedeutung.