Compatibility conditions for the Cauchy-Green strain fields: Solutions for the plane case

This paper considers the issues related to uniqueness and existence of a finite deformation generated by prescribed right or left Cauchy-Green strain tensor field in the plane. First, the questions of uniqueness and existence to a pre-assigned right strain field C are discussed. It is shown that the existence condition, in the context of continuum mechanics, are naturally posed using the field corresponding to the square root of C instead of C, the latter a classical approach. Then, the corresponding questions for the left strain field are considered, which is more involved. The analysis of uniqueness gives rise to an appropriate classification of the deformation fields. The question of existence is discussed and a complete solution is presented. In both the right and left cases, we stress the techniques for obtaining the corresponding deformation fields.     

Natural oscillations of a jet membrane under hydrostatic pressure

The natural oscillations of a submerged elastic jet membrane modeling an axisymmetric hydrodynamic direct-flow radiator are analyzed. The fundamental frequency is calculated as a function of the geometric and hydrodynamic characteristics of the jet. The principle possibility of adjusting the frequency by changing the elasticity of the fluid, which is a function of hydrostatic pressure, is demonstrated. Numerical and experimental data are compared.__________Translated from Prikladnaya Mekhanika, Vol. 40, No. 12, pp. 92–98, December 2004.     

The discrete Babuška-Brezzi condition

Summary In the treatment of saddle point problems with finite elements a discrete Babuka-Brezzi condition is encountered. We demonstrate what this discrete Babuka-Brezzi condition entails and explain the mechanical background of this somewhat abstract inf sup condition.

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Die diskrete Babuka-Brezzi-Bedingung

Übersicht Behandelt man Sattelpunktprobleme mit Finiten Elementen, dann stößt man auf eine diskrete Babuka-Brezzi Bedingung. Wir zeigen, welche Rolle diese diskrete Babuka-Brezzi-Bedingung bei der numerischen Behandlung spielt und verdeutlichen den mechanischen Hintergrund dieser etwas abstrakten inf sup Bedingung.

     

Experimental investigation of the boundary layer on ablating specimens in the joint presence of convective and radiant heat fluxes

The emission spectrum of the boundary-layer vapor on the interval 3800–6600 Å is presented for a specimen of asbestos-filled plastic in the air-plasma flow created by an electrodeless high-frequency discharge. The temperature profile in the boundary layer has been measured in the neighborhood of the stagnation point. A model of the boundary layer on an ablating specimen is proposed and the convective component of the heat flow to it is estimated.Moscow. Translated from Izvestiya Akademii Nauk SSSR. Mekhanika Zhidkosti i Gaza, No. 2, pp. 25–29, March–April, 1972.     

General relations of strain-gradient stress analysis

Some particular fields of stress gradients in plates are investigated analytically and experimentally. Carriers of empirical information are light beam deflections caused by stress or strain gradients in homogeneous beams subjected to a particular case of flexure with shear. This study is based on theories and techniques of the strain-gradient method that was recently introduced by the authors. This is a generalization of prior analytical-experimental examination of strain-gradient light deflections produced in stressed plates, which had concentrated on the simplest case where information of interest is collected along a line of symmetry of the stress field and where both the information carrying light beams are deflected in the plane of symmetry only. The developed relations were tested experimentally, by using an S-beam as representative example of general plane stress field. The main purpose of the present investigation is to document the efficacy of the strain-gradient method in analysis of the general case of stress state and to test the ranges of applicability of the accepted mathematical models and of the subsequently derived relations. In this respect, the most interesting stress state is that in a beam subjected to the Saint-Venant bending, where the transversal and longitudinal axes of the beam are in pure shear. The obtained results are compared, at each step, with the predictions of the developed analytical models and with the predictions of Filon's stress function. The results of comparison are satisfactory. The procedures of evaluating the photoelastic and material coefficients using strain-gradient techniques were tested positively. The developed method can yield valuable information on the actual features of stress states in fracture mechanics. Part of the problems and first results of this investigation will be presented in the authors' paper “Strain-gradient stress analysis in Saint-Venant bending”.     

Modified entropic elasticity of rubberlike materials

In the first part of the paper, consisting of Sections 2 and 3, physical grounds are adduced for weakening the concept of strictly entropic elasticity as applied to elastomeric materials. The resulting notion of modified entropic elasticity is shown to provide an alternative formulation of a model of rubberlike thermoelasticity proposed by Chadwick (1974). In the second part (Sections 4 and 5) Chadwick's model is generalized by allowing the deformation-dependent part of the internal energy to be a symmetric function of the principal stretches rather than depending only on their product, as in the original development. Consistently with the molecular theory of polymer networks, and with experimental findings, the extended model predicts that the deviatoric stress is not entirely entropic in origin, but arises in part from changes of internal energy. The energetic fraction of the retractive force in an extended cylinder is calculated and discussed in some detail and the paper concludes with a correlation of theoretical results with measurements on specimens of a lightly cross-linked natural rubber.     

Constant amplitude and variable load history fatigue test results and predictions for cruciform and lap welds

Cruciform and lap welds were fatigue tested under constant amplitude axial load and SAE Bracket spectrum load conditions. For the cruciform joints, fatigue cracks generally initiate at the root but may initiate at the toe if higher bending stresses are induced by joint distortion. For lap welds, the stress ratio ($\text{R}$) and weld shape are the major factors influencing the fatigue crack initiation site.The fatigue test results were compared with predictions made using an analytical model developed by the authors, and good agreement between experiment and theory was observed. The model for the predictions assumes that the fatigue crack initiation period, which is the number of cycles for the initiation of a fatigue crack and its early growth and coalescence into a dominant fatigue crack, is the main portion of the total fatigue life at long lives.     

Tomographic reconstruction of shock layer flows

The tomographic reconstruction of supersonic flows faces two challenges. Firstly, techniques used in the past, such as the direct Fourier method (DFM) (Gottlieb and Gustafsson in On the direct Fourier method for computer tomography, 1998; Morton in Tomographic imaging of supersonic flows, 1995) or various backprojection (Kak and Slaney in Principles of computerized tomographic imaging, vol. 33 in Classics in Applied Mathematics, 2001) techniques, have only been able to reconstruct areas of the flow which are upstream of any opaque objects, such as a model. Secondly, shock waves create sharp discontinuities in flow properties, which can be difficult to reconstruct both in position and in magnitude with limited data. This paper will present a reconstruction method, matrix inversion using ray-tracing and least squares conjugate gradient (MI-RLS), which uses geometric ray-tracing and a sparse matrix iterative solver (Paige and Saunders in ACM Trans. Math. Softw. 8(1):43–71, 1982) to overcome both of these challenges. It will be shown, through testing with a phantom object described in tomographic literature, that the results compare favourably to those produced by the DFM technique. Finally, the method will be used to reconstruct three-dimensional density fields from interferometric shock layer images, with good resolution (Faletič in Tomographic reconstruction of shock layer flows, 2005). This paper was based on work that was presented at the 3rd International Symposium on Interdisciplinary Shock Wave Research, Canberra, Australia, March 1–3, 2006.     

Analytical approximation to large-amplitude oscillation of a non-linear conservative system

This paper deals with non-linear oscillation of a conservative system having inertia and static non-linearities. By combining the linearization of the governing equation with the method of harmonic balance, we establish analytical approximate solutions for the non-linear oscillations of the system. Unlike the classical harmonic balance method, linearization is performed prior to proceeding with harmonic balancing, thus resulting in a set of linear algebraic equations instead of one of non-linear algebraic equations. Hence, we are able to establish analytical approximate formulas for the exact frequency and periodic solution. These analytical approximate formulas show excellent agreement with the exact solutions, and are valid for small as well as large amplitudes of oscillation.