Partial three dimensional deformations for the perfectly elastic Mooney material

Summary The governing equations for finite elastic deformations are highly nonlinear and there is still only a limited number of known exact solutions. In general for large elastic fully three dimensional deformations of the isotropic incompressible perfectly elastic neo-Hookean and Mooney materials, a non-trivial deformation for say the neo-Hookean strain-energy function, is frequently not well-defined for the general Mooney strain-energy function because the additional coupling imposes extra constraints on the deformation which are generally inconsistent with one another. Here we note two fully three dimensional deformations for which this is not the case. In both cases the resulting coupled systems of ordinary differential equations need to be integrated numerically but the deformations are nevertheless well-defined for the general Mooney material. The first deformation is simply noted because the details are given elsewhere. For the second deformation, the coupled system is derived and some new simple special solutions are given. Such deformations are important and noteworthy because of the scarcity of exact solutions in finite elasticity.     

Partial solutions of finite elasticity-axially symmetric deformations

For axially symmetric deformations of isotropic incompressible hyperelastic materials new reduced forms of the equilibrium equations are given. For the neo-Hookean and extreme-Mooney materials these equations are shown to admit exact solutions expressible in terms of Bessel functions of order zero. These solutions can be rendered as approximate solutions of problems involving the large axially symmetric deformations of circular discs containing a concentric circular hole.

---

Zusammenfassung Neue reduzierte Formen von Gleichgewichtsgleichungen für axial symmetrischen Deformationen von isotropisch unzusammendrückbaren, hyperelastischen Materialien werden gegeben. Für die neo-Hookean und extrem-Mooney Materialien können aus diesen Gleichungen exakte Schlüsse gezogen werden, die in Form von Bessel-Funktionen der Null-Ordnung ausgedrückt werden. Diese Lösungen können als ungefähre Lösungen von Problemen angesehen werden, die in Zusammenhang stehen mit grossen axial symmetrischen Deformationen von runden Scheiben, die ein konzentrisches, rundes Loch haben.

     

On pseudo-plane deformations for the neo-Hookean material

For finite elastic deformations of the incompressible neo-Hookean material the problem of determining exact pseudo-plane deformations is considered. In particular three dimensional deformations are examined which are generated by replacing constants of an assumed known plane deformation with functions of a single space variable. Results are given for replacement of constants arising from translational and rotational invariance and stretchings. Finally two specific deformations are considered which provide new exact solutions for the neo-Hookean material.

---

Zusammenfassung Exakte Lösungen für das Problem von pseudo-ebenen finiten Deformationen eines elastischen, inkompressiblen, neo-Hookeschen Materials werden ermittelt und diskutiert. Insbesondere werden dreidimensionale Deformationen studiert, welche durch Ersetzen der Konstanten einer als bekannt vorausgesetzten ebenen Deformation mit Funktionen einer einzigen Raumvariablen erzeugt werden. Resultate für den Ersatz der Konstanten aus Translations- und Rotationsinvarianz sowie aus Dehnungen sind aufgeführt. Schließlich werden zwei spezifische Deformationen betrachtet, welche zu neuen exakten Lösungen für das neo-Hookesche Material führen.

     

A penalty approximation for a unilateral contact problem in non-linear elasticity

Using Ball's approach to non-linear elasticity, and in particular his concept of polyconvexity, we treat a unilateral three-dimensional contact problem for a hyperelastic body under volume and surface forces. Here the unilateral constraint is described by a sublinear function which can model the contact with a rigid convex cone. We obtain a solution to this generally non-convex, semicoercive Signorinin problem as a limit of solutions of related energy minimization problems involving friction normal to the contact surface where the friction coefficient goes to infinity. Thus we extend an approximation result of Duvaut and Lions for linear-elastic unilateral contact problems to finite deformations and to a class of non-linear elastic materials including the material models of Ogden and of Mooney-Rivlin for rubberlike materials. Moreover, the underlying penalty method is shown to be exact, that is a sufficiently large friction coefficient in the auxiliary energy minimization problems suffices to produce a solution of the original unilateral problem, provided a Lagrange multiplier to the unilateral constraint exists.     

Some laws of plastics friction

The dependence of the actual contact area S of plastics on temperature, specific load, and sliding speed has been investigated. The value of S increases exponentially with the specific load, the maximum value at large specific loads being less than the nominal contact area. The temperature dependence of S under static conditions between 20 and 130° C is attributable to the decrease in the static modulus of elasticity of the plastic near the glass transition point and to the development of high-elastic and plastic deformations at elevated temperatures. There is practically no change in S as the sliding speed varies from 10^–3 to 10 cm/min; at the same time the force of friction increases slightly.Mekhanika Polimerov, Vol. 3, No. 6, pp. 1078–1081, 1967     

On Surface Waves and Deformations in a Pre-stressed Incompressible Elastic Solid

The propagation of infinitesimal surface waves on a half-spaceof incompressible isotropic elastic material subject to a generalpure homogeneous pre-strain is considered. The secular equationfor propagation along a principal axis of the pre-strain isobtained for a general strain-energy function, and conditionswhich ensure stability of the underlying pre-strain are derived.The influence of the pre-stress on the existence of surfacewaves is examined and, in particular, it is found that, undera certain range of hydrostatic pre-stress, a unique wavespeedexists and is bounded above by a limiting speed which correspondsto the shear wave speed in an infinite body. The secular equationis analysed in detail for particular deformations and, for anumber of specific forms of strain-energy function, numericalresults are used to illustrate the dependence of the wave speedon the pre-strain. Particular attention is focused on pre-strainscorresponding to loss of stability, in which case the infinitesimalstrain is time-independent (the wave speed being zero). Thetheory described here encompasses previous work on surface wavesand instabilities in incompressible isotropic elastic materialsand provides a clear delimitation of the range of deformationsfor which surface waves exist.     

Some generalized pseudo-plane deformations for the neo-Hookean material

We consider deformations and motions that correspond to a non-uniformaxial stretch superimposed to a general pseudoplane deformationof the first kind. We provide the general determining equationfor this kind of solution when the constitutive equation forthe stress tensor is derived from the elastic neo-Hookean strain-energydensity. Then we provide some explicit solutions which are generalizationsof the non-symmetric torsion of a slab and some finite-amplitudepseudo-planar elliptical motions.     

Circular shearing and torsion of generalized neo-Hookean materials

In this paper, the authors study circular shearing and torsionin generalized power-law neo-Hookean materials. For specialvalues of the power-law exponent, explicit exact solutions canbe established. In general, the governing equation is nonlinearand has to be solved numerically. This notwithstanding, somequalitative features of the general solutions can be discussed.The results corresponding to the neo-Hookean material can beobtained by setting the power-law exponent to unity. For valuesof the power-law exponent close to 0.5, a pronounced boundarylayer type of solution is found.     

Asymptotic behavior and uniqueness of traveling wave solutions in Ricker competition system

The existence of traveling wave solutions connecting two half-positive equilibria in Ricker competition system can be obtained by the results (B. Li, H.F. Weinberger, M.A. Lewis, Spreading speeds as slowest wave speeds for cooperative systems, Math. Biosci. 196 (2005) 82–98). In this paper we first prove that any nondecreasing traveling wave solutions have the exponential decay asymptotic behavior at the minus/plus infinity by means of Ikehara?s Theorem, and then use the strong comparison principle and the sliding method to obtain the uniqueness of the traveling wave solutions for this system.     

Friction and rheological properties of resin-impregnated glass tape

The dependence of the dynamic coefficient of friction of resin-impregnated glass tape on the sliding speed over the mandrel surface and the viscosity of the resin is derived. The formulas obtained are confirmed by the results of experiments which indicate a linear dependence of the coefficient of friction on the sliding speed of the tape over a metal mandrel. Graphically, this dependence is expressed by straight lines with a common origin and a slope that increases with the viscosity of the resin. The coefficient of friction of glass tape sliding over a prewound layer at low speeds also has a linear dependence. At high speeds the dependence takes the form of a damped curve.Translated from Mekhanika Polimoerov, No. 1, pp. 166–170, January–February, 1976.     

Multiplicity of supercritical fronts for reaction–diffusion equations in cylinders

We study multiplicity of the supercritical traveling front solutions for scalar reaction–diffusion equations in infinite cylinders which invade a linearly unstable equilibrium. These equations are known to possess traveling wave solutions connecting an unstable equilibrium to the closest stable equilibrium for all speeds exceeding a critical value. We show that these are, in fact, the only traveling front solutions in the considered problems for sufficiently large speeds. In addition, we show that other traveling fronts connecting to the unstable equilibrium may exist in a certain range of the wave speed. These results are obtained with the help of a variational characterization of such solutions.     

Numerical computation of sharp travelling waves of a degenerate diffusion–reaction equation arising in biofilm modelling

Many degenerate diffusion–reaction equations permit sharp travelling wave solutions that describe the propagation of an interface with finite speed. If the equation is at least double degenerate, the derivative of the travelling wave solution can blow up at the interface, which poses considerable challenges for the computation of the travelling wave speed. We propose a numerical method for this problem that is based on the idea to approximate the multiple degenerate problem by a family of simple degenerate problems. For the latter we propose an interval-bracketing algorithm based on the theory of Sanchez-Garduno and Maini. The travelling wave speed of the original problem is obtained as the limit of the travelling wave speeds of the auxiliary problems. The performance of the method is investigated in a numerical simulation experiment for a problem that arises in the mathematical modelling of biofilm processes.     

一类半线性四阶弹性梁方程的解和正解

考察了一类含有所有导数的半线性四阶两点边值问题的解和正解的存在性．在力学中,这类边值问题描述了一端简单支撑,另一端被滑动夹子夹住的弹性梁的形变．结论表明,只要非线性项在其定义域的某个有界集上的“最大高度”是适当的,那么这类问题至少存在一个解或者正解．     

Hertz problem for a rigid punch moving across the surface of a semi-infinite elastic solid

The elastodynamic problem of a rigid punch moving at a constant sub-Rayleigh speed across the surface of an elastic half-space is investigated in the present paper. The unknown contact region is determined as part of solution from the unilateral or Signorini conditions. Numerical results are plotted showing how the eccentricity of the contact ellipse changes with the punch speed. Some asymptotic properties of the solution for the case where the punch speed is comparable with the Rayleigh wave speed are explored in details.     

D’Alembert function for exact non-smooth modal analysis of the bar in unilateral contact

Non-smooth modal analysis is an extension of modal analysis to non-smooth systems, prone to unilateral contact conditions for instance. The problem of a one-dimensional bar subject to unilateral contact on its boundary has been previously investigated numerically and the corresponding spectrum of vibration could be partially explored. In the present work, the non-smooth modal analysis of the above system is reformulated as a set of functional equations through the use of both d’Alembert solution to the wave equation and the method of steps for Neutral Delay Differential Equations. The system features a strong internal resonance condition and it is established that irrational and rational periods of vibration should be carefully distinguished. For irrational periods, it was previously proven that the displacement field of the non-smooth modes of vibration is characterized with piecewise-linear functions in space and time and such a motion is unique for a prescribed energy. However, for rational periods, which are the subject of this work, new periodic solutions are found analytically. Findings consist of families of iso-periodic solutions with piecewise-smooth displacement fields in space and time and continua of piecewise-smooth periodic solutions of the same energy and frequency.     

一类不可压广义neo-Hookean球体的空穴分岔问题的定性研究

研究了一类不可压的广义neo-Hookean材料组成的球体的空穴分岔问题,该类材料可以看作是带有径向摄动的均匀各向同性不可压的neo-Hookean材料,得到了球体内部空穴生成的条件．与均匀各向同性的neo-Hookean球体的情况相比,证明了当摄动参数属于某些区域时,从平凡解局部向左分岔的空穴分岔解上存在一个二次转向分岔点,空穴生成时的临界载荷会比无摄动的材料的临界载荷小．用奇点理论证明了,空穴分岔方程在临界点附近等价于具有单边约束条件的正规形．用最小势能原理分别讨论了空穴分岔解的稳定性和实际稳定的平衡状态．     

Minimal wave speed of a competition integrodifference system

In this paper, we study the minimal wave speed of a competitive system. By constructing upper and lower solutions, we confirm the existence of travelling wave solution at the critical wave speed. This completes earlier results found in the literature. Our conclusion implies that the asymptotic decay behaviour of solutions at the critical wave speed is different from that of solutions at larger wave speeds.     

The Maxwell-Contact

In multibody systems, two alternative approaches exist to compute contact forces between bodies. The unilateral constraint contact and the unilateral regularized contact do not take into account the influence of the deformation due to other contacts on a body using rigid body dynamics. In this paper, a third alternative, i.e. the Maxwell-Contact, is derived coupling the deformations of different contacts on one body quasi-statically. An academic example validates the fundamental properties of the contact model and the application in a simulation of a pushbelt CVT shows improved results. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Moving contact problems involving a rigid punch and a functionally graded coating

Frictional contact mechanics analysis for a rigid moving punch of an arbitrary profile and a functionally graded coating/homogeneous substrate system is carried out. The rigid punch slides over the coating at a constant subsonic speed. Smooth variation of the shear modulus of the graded coating is defined by an exponential function and the variation of the Poisson's ratio is assumed negligible. Coulomb's friction law is adopted. Hence, tangential force is proportional to the normal applied force through the coefficient of friction. An analytical method is developed utilizing the singular integral equation approach. Governing partial differential equations are derived in accordance with the theory of elastodynamics. The mixed boundary value problem is reduced to a singular integral equation of the second kind, which is solved numerically by an expansion-collocation technique. Presented results illustrate the effects of punch speed, coefficient of friction, material inhomogeneity and coating thickness on contact stress distributions and stress intensity factors. Comparisons indicate that the difference between elastodynamic and elastostatic solutions tends to be quite larger especially at higher punch speeds. It is shown that use of the elastodynamic theory provides more realistic results in contact problems involving a moving punch.     

Uzawa block relaxation method for the unilateral contact problem

We present a Uzawa block relaxation method for the numerical resolution of contact problems with or without friction, between elastic solids in small deformations. We introduce auxiliary unknowns to separate the linear elasticity subproblem from the unilateral contact and friction conditions. Applying a Uzawa block relaxation method to the corresponding augmented Lagrangian functional yields a two-step iterative method with a linear elasticity problem as a main subproblem while auxiliary unknowns are computed explicitly. Numerical experiments show that the method are robust and scalable with a significant saving of computational time.