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.

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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.

     

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.     

Large plane deformations of thin elastic sheets of neo-Hookean material

Zusammenfassung Es werden grosse ebene Verformungen dünner elastischer Scheiben aus Neo-Hookeschem Material betrachtet und eine Methode der sukzessiven Substitutionen entwickelt, um Probleme im Rahmen der zweidimensionalen Theorie endlicher ebener Spannungszustände zu lösen. Die erste Näherung wird durch lineare Randwertprobleme für zwei harmonische Funktionen bestimmt, und sie wird asymptotisch angenähert für sehr grosse Dehnungen in der Ebene der Scheiben. Die zweite und die höheren Annäherungen werden durch Lösung Poissonscher Gleichungen gewonnen. Es werden verschiedene Beispiele behandelt, und für rotationssymmetrische Verformungen wird gute übereinstimmung zwischen den Näherungen und den exakten Lösungen gefunden.     

A regularization technique for the XFEM: extension to finite deformations,inelastic material behaviour and multifield problems

Within the XFEM often near linear dependencies between the standard degrees of freedom and enriched degrees of freedom and also among enriched degrees of freedom occur. During the last years, several remedies to that problem have been presented. Here, an extension of the regularization technique described in [1] to finite deformation problems and inelastic material behaviour as well as to multifield problems is proposed. © 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Small finite deformations of an amorphous polymer material

Small finite deformations of a polymer material are considered. It is assumed that the volume of the specimen is conserved during deformation and that the strains are affine; furthermore, the equilibrium distance between network nodes is not equal to zero in the specimen, and the unit cell of the network is cubic. Various types of deformation are considered. The expressions obtained are compared with the results of the theory of high elasticity and the experimental data.Branch Institute of Chemical Physics, AS USSR, Moscow. Translated from Mekhanika Polimerov, Vol. 4, No. 6, pp. 997–1001, November–December, 1968.     

Free energy of a regular stable polymer network at small finite deformations

The equilibrium small finite deformations of a regular polymer network are considered. The free energy of the network is described in the form of a series expansion in small displacements of the network points from the equilibrium position. The general dependence of the free energy on the strain invariants thus obtained coincides with the Mooney expression for the free energy only in the first approximation. Aspects of the mechanical behavior of the material associated with this difference are discussed.Branch of the Institute of Chemical Physics, Academy of Sciences of the USSR, Moscow Oblast. Translated from Mekhanika Polimerov, No. 4, pp. 615–618, July–August, 1971.     

Elastoplastic torsion of a cylindrical rod for finite deformations

A solution of the problem of the torsion of a cylindrical rod was obtained in /1/ for a general, isotropic, incompressible elastic material. The present paper gives an analytical solution of the elastoplastic torsion problem for finite deformations, written in terms of quadratures of elliptic functions. The non-linear kinematics of elastoplastic deformation is introduced into the defining equations with the help of a multiplicative decomposition of the deformation gradient into elastic and plastic components /2, 3/. The elastic deformation and rate of plastic deformation are related to the state of stress of the body, in accordance with the defining Mooney-Rivlin equations /4/ and the law of flow for finite deformations associated with the Tresca yield condition /5/. A non-linear first-order partial differential equation and the initial data at the elastoplastic boundary are obtained in order to determine the angle of rotation within the plastic zone of the basis formed from the eigenvectors of the stress tensor, relative to the radial direction. The integration of the resulting equation is reduced to determining the general integral of the Ricatti equation with right-hand side determined from the angular velocity of flow of the material within the plastic zone. It is shown that neglecting the finiteness of the deformation leads to too high an estimate of the rigidity of the rod.     

The finite integral transform method for a parabolic differential equation on a graph

The finite integral transform method is set forth and justified for solving a mixed problem for a parabolic differential equation posed on a graph.     

The Signorini problem with Coulomb friction for a hyperelastic body under finite deformations

The quasi-static three-dimensional problem of elasticity theory for a hyperelastic body under finite deformations, loading by bulk and surface forces, partial fastening and unilateral contact with a rigid punch and in the presence of time-dependent anisotropic Coulomb friction is considered. The equivalent variational formulation contains a quasi-variational inequality. After time discretization and application of the iteration method, the problem arising with “specified” friction is reduced to a non-convex miniumum functional problem, which is studied by Ball's scheme. The operator in contact stress space is determined. It is shown that a threshold level of the coefficient of friction corresponds to each level of loading, below which there is at least one fixed point of the operator. If the solution at a certain instant of time is known, the iteration process converges to the solution of the problem at the next, fairly close instant of time.     

The fundamental theorem of geometric calculus via a generalized riemann integral

Using recent advances in integration theory, we give a proof of the fundamental theorem of geometric calculus. We assume only that the tangential derivative ∇ _V F exists and is Lebesgue integrable. We also give sufficient conditions that ∇ _V F exists.     

The fundamental group of a locally finite graph with ends

We characterize the fundamental group of a locally finite graph G with ends combinatorially, as a group of infinite words. Our characterization gives rise to a canonical embedding of this group in the inverse limit of the free groups π₁(G^′) with G^′⊆G finite.     

Elasticity parameters for finite deformations of rubber-like materials

Résumé Le développement des notions relatives aux grandes déformations des coques élastiques très minces (membranes) dépend strictement de la connaissance exacte des propriétés élasto-mécaniques des matériaux utilisés. Les limites imposées par les connaissances actuelles peuvent être franchies par l'emploi de deux paramètres d'élasticité comportant trois constantes d'élasticité. Ce sont les généralisations des formules approuvées par Mooney, et qui s'appliquent à un domaine de déformation largement étendu. Ces paramètres d'élasticité sont les dérivés de la fonction d'énergie de déformation exprimée par certains invariants de l'état de déformation. Les nouvelles formules s'appliquent à tout état de déformation pure et homogène. Les nouvaux paramètres expliquent pour la première fois avec satisfaction les effets d'une permanente déformation définitive. L'essentiel de ce développement est le fait que ces invariants sont evalués relatifs à l'état vierge, et non pas à un état déchargé quelconque. Les paramètres d'élasticité sont donnés par les formules: . Ces conclusions découlent d'une analyse d'expériences déjà publiées et prouvent être compatibles et en concordance avec de simples expiences faites sur des ballons sphériques.

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The author is indebted to Associate ProfessorJ. D. C. Crisp for his encouragement and many useful suggestions on this work.     

The mean integral curvature and infinitesimal deformations of a surface in a three-dimensional Riemannian space

We find a formula for the derivative of the mean integral curvature of a surface in a three-dimensional Riemannian space with respect to infinitesimal deformations.     

Computation of large shear deformations of a thermoplastic material

A specialized finite difference method with grid refinement and variable time steps is created to approximate the deformation velocity and the temperature in a simple model of the shearing of a thermoplastic material. A specific problem where the solution exhibits “blowup” in the adiabatic case is considered. The numerical method retains this property and is used to study the shape of the “blowup” function. The code is then used to investigate the solution in the closely related case where thermal conduction is included with a small conductivity coefficient. The computations indicate that the solution does not “blowup” in the nonadiabatic case. © 1996 John Wiley & Sons, Inc.     

Automorphism scheme of a finite field extension

Let be a finite field extension and let us consider the automorphism scheme . We prove that is a complete -group, i.e., it has trivial centre and any automorphism is inner, except for separable extensions of degree 2 or 6. As a consequence, we obtain for finite field extensions of , not being separable of degree 2 or 6, the following equivalence:

     

Largest extension of a finite convex geometry

We present a new embedding of a finite join-semidistributive lattice into a finite atomistic join-semidistributive lattice. This embedding turns out to be the largest extension, when applied to a finite convex geometry.In Celebration of the Sixtieth Birthday of Ralph N. McKenzieReceived September 18, 2002; accepted in final form September 29, 2003.