Oscillatory shearing of nonlinearly elastic solids

A finite oscillatory shearing motion is shown to be possible, in the absence of body force, in every homogeneous isotropic compressible or incompressible elastic solid. The spatial geometry is the same for all materials and the nature of the time-dependence, for a particular material, is determined by the generalized shear modulus. A motion of this type and a spatially uniform, time-dependent temperature can be supported in thermoelastic solids, without application of body force or volume supply of heat.     

Oscillatory theorems of a class of even-order neutral equations

A class of even-order nonlinear neutral differential equations with distributed deviating arguments is studied, and oscillatory criteria for solutions of such equations are established.     

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.     

一类二阶非线性泛函微分方程的振动性质

研究了一类较为广泛的二阶非线性泛函微分方程解的振动性质.在一定条件下,利用广义黎卡提变换和积分平均技巧建立了方程(1)的两个新的振动准则,推广和改进了已知的一些结果.     

Development of an elastic-plastic constitutive model for highly anisotropic materials

Homogenization methods are used to obtain the effective properties of highly anisotropic materials such like textile reinforced composites. The state of the art is the utilization of this method to compute elastic properties. But the consideration of inelastic and anisotropic properties requires the application of more advanced techniques such as the FE²-method. Due to the high numerical effort induced by this approach, this paper presents a new method to evaluate inelastic properties of an heterogeneous elastic-plastic material. The parameters describing the inelastic properties require a modification of the return mapping algorithm which is used for the numerical implementation. Finally, the verification shows the accuracy of the results obtained with this new homogenization method. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

一类二阶非线性阻尼微分方程解的振动性质

研究了一类二阶非线性阻尼微分方程解的振动性质.在一定条件下,建立了四个新的振动性定理,推广和改进了已知的结果.     

Oscillatory criteria for a class of delay hyperbolic equations boundary value problem (II)

In this paper, we consider a class of delay hyperbolic equations boundary value problem, and obtain the sufficient condition for the oscillation of the solution of the problem (1.1), (1.2).     

Distribution of shearing stresses under three-point flexure in beams made of composite materials

Conclusions 1. When short beams made of composite materials with span-to-height ratios within the 4.0–5.0 range are tested in three-point flexure, the profile of shearing stresses over the height of a beam cross section departs appreciably from a parabolic one over approximately 40% of the span length.2. The zones of a perturbed state of stress, which appear in short beams tested in three-point flexure, render this method unusable for determining the interlayer shear strength of composite materials on "ultrashort" specimens.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 1, pp. 56–62, January–February, 1977.     

Second-order effects in problems for a class of elastic materials

For a special class of elastic materials, the inverse deformation is always possible in the same material. This result is used to simplify the determination of second-order effects in problems for this special class. The second-order solution for indentation of a half-space by a smooth rigid sphere is determined. For indentation by a flat-ended circular punch it is found that the second-order solution for compressible materials is physically inadmissible, while the solution for incompressible materials involves a concentrated line load at the edge of the punch. A rigid spherical inclusion embedded in an elastic matrix with uniform stresses at infinity is considered, and the interface stresses are found to second order.

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Zusammenfassung Für seine Sonderklasse von elastischen Stoffen ist die inverse Verformung für das gleiche Material immer möglich. Dieses Resultat wird angewendet, um bei Problemen mit dieser Sonderklasse von Materialien die Ermittlung von Effekten zweiter Ordnung zu vereinfachen. Die Lösung zweiter Ordnung für das Kontaktproblem zwischen einem Halbraum einer glatten starren Kugel wird hergeleitet. Bei dem Kontaktproblem mit dem kreiszylindrischen starren Stempel findet man, daß die Lösung zweiter Ordnung mit kompressiblem Material physikalisch unzulässig ist, während für das inkompressible Material am Rand des Stempels eine konzentrierte Linienlast entsteht. Schließlich wird eine starre kugelförmige Inklusion betrachtet, welche in einer elastischen Matrix mit gleichförmig verteilten Spannungen im Unendlichen eingebettet ist. Die Spannungen am Rand der Inklusion werden bis zur zweiten Ordnung hergeleitet.

     

A remark on consequences of the work assumption of Naghdi and Trapp for elastic-plastic materials

A number of conditions implied by a physically plausible work assumption due to Naghdi and Trapp (1975) are discussed in the light of recent developments in the strain-space formulation of the purely mechanical theory of finitely deforming elastic-plastic materials.     

Weak solutions for contact models involving a class of generalized materials

We consider a contact model for a class of generalized materials whose behaviour is described by means of a bipotential. Using the theory of the bipotentials we deliver a weak formulation consisting of a system of three variational inequalities. The unknown is a triple having as components the displacement field, the Cauchy stress tensor and a Lagrange multiplier related to the friction force on the frictional contact zone. We investigate the existence of the weak solutions by using a fixed point theorem for set-valued mappings and a minimization technique.     

Transient heating of a rotating elastic-plastic solid shaft

Based on Tresca's yield criterion and the flow rule associated with it, the transient plasticization in a rotating elastic–plastic solid shaft with temperature dependent yield stress subject to a temperature cycle is studied. Special attention is paid to the residual stresses. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Finite element analysis of a holonomic elastic-plastic problem

Summary A thorough analysis of the finite element method is given for a holonomic elastic-plastic problem. An inequality of the Cea's lemma type is proved, which is the basis of error estimates for various finite element solutions. Difficulty caused by a non-differentiable term in the problem can be overcome by using two convergent procedures, an iterative procedure and a regularization procedure. An a-posteriori quantitative error estimate is derived for the regularized solution.The work was done while the author was at the Department of Mathematics, University of Maryland, College Park. The research was partially supported by the National Science Foundation grant CCR-88-20279     

Analytical inversion of the Jacobian for a class of generalized standard materials

In this paper we derive an analytical expression for the inverse of the Jacobian occuring in the implicit time integration procedure for a large class of generalized standard materials. The resulting expression is easy to construct and implement into arbitrary programming languages. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The finite deformation of a pressurized circular tube for a class of compressible materials

This investigation is concerned with the problem of a hollow circular cylinder subjected to uniform internal and external pressure within the equilibrium theory of finite elasticity. The tube is composed of homogeneous, isotropic,compressible materials of special type, namely harmonic materials. Explicit closed-form solutions for the deformation and stress fields are obtained. The true stress distribution, expressed as a function of the undeformed coordinates, is shown to be essentially independent of material properties. The two cases of internal pressure only, and external pressure only, are examined in detail. In the former case, there is a critical value of the applied pressure at which the maximum hoop stress in the tube, occurring at the inner surface, becomes unbounded. Results appropriate for thin shells are also obtained. For the case of external pressure only, a critical value of the applied pressure exists for which closing of the cavity is predicted. For nearly solid cylinders, or equivalently, for a cavity in an unbounded medium, explicit results are provided for the corresponding stress concentration factor.Furthermore the stability of the preceding equilibrium states is investigated by employing a standard linearized perturbation scheme. It is found that an internally pressurized tube is always stable in the range of interest whereas an externally pressurized tube buckles prior to cavity closure.

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Zusammenfassung Die Untersuchung befaßt sich im Rahmen der Gleichgewichtstheorie fmiter Elastizität mit dem Problem eines Hohlzylinders, der einem gleichförmigen inneren und äußeren Druck ausgesetzt ist. Das Rohr besteht aus homogenen, isotropen,kompressiblen Materialien eines speziellen Typs, nämlich harmonischen Materialien. Explizite Lösungen in geschlossener Form werden für Verformung und Spannungsfelder erhalten. Es wird gezeigt, daß die Verteilung der wahren Spannung in Abhängigkeit von den unverformten Koordinaten im wesentlichen unabhängig von den Materialeigenschaften ist. Die beiden Fälle, nur innerer Druck und nur äußerer Druck, werden detailliert behandelt. Im ersten Fall gibt es einen kritischen Druck, bei dem die maximale Tangentialspannung im Rohr, und zwar an der inneren Oberfläche, über alle Grenzen wächst. Es werden außerdem Ergebnisse erhalten, die für dünne Schalen zutreffen. Im Fall eines ausschließlich äußeren Drucks existiert ein kritischer Druckwert, bei dem ein Schließen des Hohlraums vorhergesagt wird. Für annähernd massive Zylinder, oder damit gleichbedeutend, für einen Hohlraum in einem unendlichen Medium, werden explizite Ergebnisse für den entsprechenden Spannungskonzentrationsfaktor vorgelegt.Darüberhinaus wird die Stabilität vorausgehender Gleichgewichtszustände durch Anwendung eines üblichen linearisierten Störungsschemas untersucht. Es wird gefunden, daß das Rohr unter Innendruck im interessierenden Bereich immer stabil ist, während ein Rohr unter Außendruck einknickt, bevor sich der Hohlraum schließt.

     

Oscillatory flow about a cylinder pair

Oscillatory flow about a pair of circular cylinders is considered.The distance between the cylinders can be varied as can theangle that the undisturbed oscillatory flow makes with the linejoining the cylinder centres. In common with other fluid flowsdominated by oscillatory flow, a time-independent, or steadystreaming, motion develops. Attention is focused on the caseof high streaming Reynolds numbers and the resulting jets thaterupt from the surfaces of the cylinders.