The penny-shaped crack problem for a finitely deformed incompressible elastic solid

In this paper the theory of small deformations superposed on large is used to examine the axisymmetric problem of a penny-shaped crack located in an incompressible elastic infinite solid which is subjected to a uniform finite radial stretch. The small axisymmetric deformations are due to a uniform stress applied in the axial direction. Formal integral expressions are derived for the displacements and stresses in the elastic solid. An exact expression is developed for critical stress necessary for the propagation of a penny-shaped crack in a finitely deformed elastic solid.

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Résumé Dans le mémoire, on utilise la théorie des petites déformations superposées à de larges déformations pour examiner le problètrique d'une fissure en disque noyée dans un solide élastique infini incompressible soumis à un étirement uniforme fini radial. Les déformations axisymétriques de faible amplitude sont dues à une contrainte uniforme appliquée suivant la direction axiale. Des expressions intégrales formelles sont déduites des déplacements et des contraintes dans le solide élastique. Une expression exacte relative à la contrainte critique nécessaire pour la propagation d'une fissure en forme de disque est développée dans le cas d'un solide élastique déformé de manière finie.

     

Radial propagation of rotary shear waves in a finitely deformed elastic solid

A study is made of the radial propagation of rotary shear waves in an incompressible elastic solid under finite radial deformations. Basic equations are derived on the basis of Biot's mechanics of incremental deformations, and analysis is made by specializing the initial deformations to two cases: (a) an infinite solid with a cylindrical bore is inflated by all-around tension, and (b) a cuboid is rounded into a ring and its ends are bonded to each other. The influence of the inhomogeneities of such deformations upon the laws of shear wave propagation is presented in the form of curves.     

A note on infinitesimal shear waves in a finitely deformed elastic solid

In the present note we discuss a class of small dynamic deformations imposed on a large static deformation. A semi-inverse method is used to obtain the equations of motion for the small deformations in a simple manner. The results are used to extend known results for a tube of hyperelastic material subject to inflation and extension, when additional telescopic and rotational shears are applied.     

Initiation and propagation of a penny-shaped crack in a finitely deformed incompressible elastic medium

The effects that the initial lateral stress has on the initiation and the propagation of a penny-shaped crack are investigated on the basis of the theory of small deformations superposed on finite deformation for an incompressible elastic material. Using the methods of the Laplace and Hankel transforms, the crack shape function and the stress distribution with singularities in the crack plane are obtained in closed forms for the crack propagating at a constant speed in the Mooney material. The dynamic stress-intensity factor is obtained as a function of the initial lateral stretch and the ratio of the crack speed to the shear wave speed. For the same crack speed, the value of the dynamic stress-intensity factor increases with increasing lateral stretch, but decreases if the lateral compression increases.The dynamic solutions reduce to the associated static solutions at zero crack speed. For the stationary crack, the stress-intensity factor is shown to be independent of the initial stress. However, the initial lateral stretch increases, but the lateral compression decreases the value of the critical stress required for the initiation of crack growth on the basis of the Griffith theory. The central crack opening displacement is shown to decrease if the lateral stretch increases or the lateral compression decreases.     

Small torsional vibration superposed on finitely deformed state of a circular cylindrical rod of transversely isotropic elastic material

Starting with a class of small deformations superposed on a finitely deformed state of a transversely isotropic elastic solid, we study a problem of small torsional vibration superposed on homogeneous finitely deformed state of a circular cylindrical rod made of transversely isotropic elastic material. It has been found that free vibration is possible and, due to anisotropy, the speed of propagation of waves of torsion along the cylinder is increased or decreased according as the initial stressed state is under tension or compression.     

Further results on the stability of a finitely deformed thin cylindrical shell

Previous stability results for a semi-infinite cylindrical shell accounting for thickness effects not present in a membrane are extended to investigate effects of finite cylinder length. Natural frequencies are also determined.     

Equilibrium of elastic lattice shells

A model for shells consisting of a continuous distribution of embedded rods is developed in the framework of the direct theory of second-gradient elastic surfaces. The shell is constitutively sensitive to a convenient measure of the gradient of strain in addition to the metric and curvature of standard shell theory.     

Equilibrium piezoelectric potential distribution in a deformed ZnO nanowire

The equilibrium piezoelectric potential distribution in a deformed ZnO semiconductive nanowire has been systematically investigated in order to reveal its dependence on the donor concentration, applied force, and geometric parameters. In particular, the donor concentration markedly affects the magnitude and distribution of the electric potential. At a donor concentration of N _D>10¹⁸ cm⁻³, the piezopotential is almost entirely screened. Among the other parameters, a variation in the length of the nanowire does not significantly affect the potential distribution.      

Modeling and simulation of liquid diffusion through a porous finitely elastic solid

A new theory is proposed for the continuum modeling of liquid flow through a porous elastic solid. The solid and the voids are assumed to jointly constitute the macroscopic solid phase, while the liquid volume fraction is included as a separate state variable. A finite element implementation is employed to assess the predictive capacity of the proposed theory, with particular emphasis on the mechanical response of $\text{ Nafion}^{\textregistered }$ membranes to the flow of water.     

The exact form and properties of the deformed transverse internal elastic crack

The exact shape and properties of an internal transverse crack in an infinite plate under conditions of plane stress submitted to a biaxial normal loading at infinity are presented. The shape of the deformed Griffith crack as this was defined by the exact theoretical solution of the problem is an ellipse whose properties depend on the mode of loading of the plate and its mechanical properties. This result is different to the result arising from the respective singular and two-term solutions, which define a parabolic shape for the deformed crack. Furthermore, while the singular solution implies that the tip of the crack remains undisplaced during loading, the two-term and the exact solutions define its displacement, which is the same for both cases. In this way the two-term solution constitutes an intermediate case, which at the crack tip coincides with the exact solution, whereas at the middle of the crack coincides with the singular solution. Interesting results and comparisons were also derived for the curvature of the deformed crack at its tip, which was shown to differ considerably according to the different solutions.     

Equilibrium of an elastic finite cylinder: Filon's problem revisited

This paper deals with an analytical solution of the axisymmetric boundary-value problem of the theory of elasticity for a finite circular cylinder with free ends and arbitrary loaded curved surface. The object of this paper is to employ the method of superposition to obtain accurate values of the stress field near the boundaries. The classical Filon (1902) problem of uniformly distributed tangential load applied along two rings at the curved surface is addressed in full detail. The distribution of stresses along some typical sections of the cylinder are shown graphically.     

X-ray microbeam measurements of individual dislocation cell elastic strains in deformed single-crystal copper

The distribution of elastic strains (and thus stresses) at the submicrometre length scale within deformed metal single crystals has remarkably broad implications for our understanding of important physical phenomena. These include the evolution of the complex dislocation structures that govern mechanical behaviour within individual grains, the transport of dislocations through such structures, changes in mechanical properties that occur during reverse loading (for example, sheet-metal forming and fatigue), and the analyses of diffraction line profiles for microstructural studies of these phenomena. We present the first direct, spatially resolved measurements of the elastic strains within individual dislocation cells in copper single crystals deformed in tension and compression along <001> axes. Broad distributions of elastic strains are found, with important implications for theories of dislocation structure evolution, dislocation transport, and the extraction of dislocation parameters from X-ray line profiles.     

Equilibrium of an elastic finite cylinder under axisymmetric discontinuous normal loadings

This article presents an analytical technique for solving the axisymmetric elasticity problem for a finite solid cylinder subjected to discontinuous normal loadings on its ends and lateral surface. This technique is based on application of the method of crosswise superposition by representing the solution for stresses in the form of decompositions into Fourier and Bessel–Dini series. For determination of the coefficients in these series, the infinite systems of linear algebraic equations are obtained and solved by means of a modified algorithm of advanced reduction. The technique is numerically validated for typical cases of discontinuous loading. It is shown that the solution procedure is efficient for determination of the stresses in the cylinder including its edges and discontinuity points of normal loadings.     

Interparticle interaction in a deformed body. I. Continuous nanoheterogeneous elastic media

We describe a structurally continual model of a deformed crystalline body. It can be described as an elastic nanoheterogeneous anisotropic medium with rigid rotation of nanoparticles characterized only by the natural frequency of thenth coordination order. Mechanical changes in the shape of the medium depending on its structure are reflected in constructing the direct lattice of shear. The primitive vectors of the reciprocal lattice of shear are the pricipal wave vectors of shear corresponding to a single admissible natural frequency of the body. Mechanical changes in the volume of the medium are taken into account in the direct and reciprocal lattices of tension-compression. The volume of an elementary cell of the lattice of shear is a constant of the material for a given level of mechanical analysis. The method used for the construction of direct and reciprocal lattices of shear and tension-compression is illustrated by analyzing a hexagonal close-packed crystalline body of the first coordination order. Karpenko Physicomechanical Institute, Ukrainian Academy of Sciences, L'viv. Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 33, No. 2, pp. 7–17, March–April, 1997.     

X-ray analysis of microscopic structure in deformed brass

The microscopic structure of brass is systematically studied by X-ray diffraction profile analysis in this paper. The samples were heat treated differently and deformed in an unidirectional tensile state. Parameters such as the effective grain size, average dislocation density, dislocation configuration parameter and stored deformation energy density are carefully deduced from X-ray diffraction data based on some recent theories and models. These parameters change with macroscopic stress and strain regularly and it is possible to establish a quantitative relationship between the microscopic structure and macroscopic properties. There is an unique relationship between the structure parameters and stress rather than strain for different sample heat treatments.     

Equilibrium of a solute in an elastic continuum —A statistical approach

Summary A theory of stress-assisted diffusion has been previously derived using continuum mechanics by E. C. Aifantis [1]. This theory has subsequently served as the basis for a model of material degradation by Unger and Aifantis [2]. In this model, an equilibrium solution of Aifantis' stress-assisted diffusion equation was used to determine the distribution of a solute in an elastic continuum. However, as continuum mechanics gives no direct correlation between material coefficients and other physical characteristics such as temperature, these models relied solely on experimental data to determine the phenomenological coefficients of the theory. This process naturally limits the predictive capabilities of the model. In this paper we rederive the equilbrium distribution of a solute in an elastic stress field. We show that the use of a statistical approach can provide additional information about the various coefficients appearing in the equilibrium solution of the phenomenological theory. An equilibrium distribution for a dilute gas in an ideal gas thermostat is derived using statistical mechanics. It takes the form of the equilibrium solution of Aifantis' stress-assisted diffusion theory in terms of the hydrostatic stress and diffusion coefficients. However, as a result of the statistical approach, additional information is gained for the coefficients in terms of the temperature and number of atoms in the thermostat.     

Computational homogenization of nonlinear elastic materials using neural networks

In this work, a decoupled computational homogenization method for nonlinear elastic materials is proposed using neural networks. In this method, the effective potential is represented as a response surface parameterized by the macroscopic strains and some microstructural parameters. The discrete values of the effective potential are computed by finite element method through random sampling in the parameter space, and neural networks are used to approximate the surface response and to derive the macroscopic stress and tangent tensor components. We show through several numerical convergence analyses that smooth functions can be efficiently evaluated in parameter spaces with dimension up to 10, allowing to consider three‐dimensional representative volume elements and an explicit dependence of the effective behavior on microstructural parameters like volume fraction. We present several applications of this technique to the homogenization of nonlinear elastic composites, involving a two‐scale example of heterogeneous structure with graded nonlinear properties. Copyright © 2015 John Wiley & Sons, Ltd.     

Long-range correlations of elastic fields in semi-flexible fiber networks

The mechanical properties of semi-flexible networks have been the subject of intense theoretical and experimental studies concerned primarily with the understanding of the complex behavior of biological systems such as the cell. Here it is shown that the elasticity of these networks, both elastic constants and elastic fields, while fluctuating significantly with position, is long-range correlated and the correlation functions exhibit power law scaling. The correlations are lost when the fiber stiffness is reduced. The range of scales over which correlations are observed is bounded below by the mean fiber segment length and above by the filament persistence length. Therefore, these networks can be regarded as stochastic fractal elastic media over the respective range of scales. This implies that no scale decoupling exists and no representative volume element can be identified on scales below the upper correlation cut-off scale.