Surfaces Of Weak Discontinuity Research Articles

Numerical Analysis of Unsteady Vibrations of a Plate Resting on an Elastic Isotropic Half-Space

The paper is devoted to the numerical solution of the problem of vibrations of an infinite elastic plate resting on an elastic isotropic half-space using the analytical method based on the ray method with its numerical realization via the Maplesoft package. Unsteady oscillations are caused by the action of instantaneous loads on the plate, resulting in the appearance of two plane wave surfaces of strong discontinuity in the elastic half-space, behind the fronts of which, up to the contact boundary, the solution is constructed using ray series. The unknown functions entering the coefficients of the ray series and the equation of plate motion are determined from the boundary conditions of the contact interaction between the plate and the half-space. Previously, the approximate solution of this problem was obtained analytically without using mathematical packages, and the dynamic deflection of the plate involving only the first three terms of the ray series was written down. In this work, a two-layer medium with different properties was investigated using an algorithm developed to solve contact dynamic problems related to the occurrence and propagation of strong and weak discontinuity surfaces.

Ichnofabric analysis of bathyal chalks: the Miocene Inglis Formation of the Andaman and Nicobar Islands, India

In the Bay of Bengal, the Andaman and Nicobar Islands represent part of the Burma–Sunda–Java subduction complex. The Islands are composed of sediments ranging in age from Jurassic to Recent, represented by ophiolites, flysch sediments, along with deep marine sediments scraped off from the subducting plate. The stratigraphic succession that overlies meta-sedimentary and ophiolite suites consists of turbidite and non-turbidite sequences, along with thick-bedded nannofossil chalks. The present study describes ichnofabrics of chalks from the Inglis Formation (Early to Middle Miocene). These chalks are highly to moderately bioturbated and comprise several levels of ferruginised layers as weak discontinuity surfaces. The studied section shows the recurring occurrence of ichnotaxa belonging to Asterosoma, Chondrites, Cladichnus, Ophiomorpha, Palaeophycus, Planolites, Taenidium, Thalassinoides, and Zoophycus. Sediments are represented by Bioturbation indices varying between BI-2 to BI-5, represented by (a) light coloured trace fossils in dark sediment (LID ichnofabric) and (b) dark coloured trace fossils in light sediment (DIL ichnofabric). Ichnofabric analysis suggests multiple colonization, complex tiering, and multilayer tiering. The LID ichnofabric exposed at Kalapathar reveals three tiers, a diverse shallow tier and a moderately low diverse middle and deep tiers. At the Lacam Point Section, in contrast, the LID ichnofabric is represented by condensation of the tiers and the absence of shallow tiers. The DIL ichnofabric at the Kalapathar Section seems to be more expanded and is represented by four tiers with extensive bioturbation. Ichnofabric analysis supports deposition of the chalk sediments in a lower bathyal paleoenvironment and suggests that organic matter, pore water, and bottom-water oxygenation were the main controlling factors. Thus, the ichnofabric analysis of the Early-Middle Miocene Inglis Formation gives first-hand information regarding the poorly known chalk facies of the Andaman and Nicobar Islands pre-Bengal fan stage of the Indian plate.

On the constitution of polar fiber-reinforced materials

This article presents important constitutive refinements and simplifications in the theory of polar elasticity of materials reinforced by a single family of fibers resistant in bending. One of these simplifications is achieved by paying attention to forms of the strain energy which are symmetric with respect to the symmetric and the antisymmetric parts of the fiber gradient tensor. This leads to the identification of a restricted version of the theory that is predominantly influenced by the fiber-splay mode of deformation. The lack of ellipticity of the governing equations of polar elasticity and the anticipation of existence of weak discontinuity surfaces even in the small deformation regime are also investigated. The manner in which potential activation of such surfaces is related with the action of either the fiber-bending or the fiber-splay deformation mode, as well as with their conjoined combination and coupling with their fiber-twist counterpart, is examined. The proposed constitutive equations can be simplified via the use of a new set of fourteen independent spectral invariants of the deformation. This set serves as an irreducible functional basis of relevant invariants or as an irreducible integrity basis of polynomial invariants. For instance, its use here enables identification of fourteen classical invariants that emerge as mutually independent from the known set of thirty-three in total classical invariants. In the special case of polynomial invariants, this result paves the way for identification of a corresponding minimal integrity basis.

A phase-field model for strain localization analysis in softening elastoplastic materials

The present paper deals with the localization of strains in those structures consisting of materials exhibiting plastic softening response. It is assumed that strain localization develops in a finite thickness band separated from the remaining part of the structure by weak discontinuity surfaces. In view of the small thickness of the band with respect to the dimensions of the structure, the interphase concept is used for the mechanical modeling of the localization phenomenon. We propose a formulation for the quasi-static modeling of strain localization based on a phase-field approach. In this sense, the localization band is smeared over the volume of the structure and a smooth transition between the fully broken and the sound material phases is introduced. The mechanical performance of the model is illustrated for the case of uniaxial tensile test, discussing the instability of the force-displacement response, and for the case of three-point bending test, comparing the analytical results with the experimental ones.

Formation of grain boundaries in ductile single crystals at finite plastic deformations

The theory of formation of grain boundaries in ductile single crystals is proposed within the nonlinear continuum dislocation theory (CDT), where grain boundaries are interpreted as surfaces of weak discontinuity in placement but strong discontinuity in plastic slip. The set of governing equations and jump conditions are derived for the energy minimizers admitting such surfaces of discontinuity from the variational principle. By constructing energy minimizing sequences having piecewise constant plastic and elastic deformation in two examples of ductile single crystals deforming in plane strain simple shear or uniaxial compression, it is shown that the formation of lamellae structure with grain boundaries is energetically preferable. The number of lamellae is estimated by minimizing the energy of grain boundaries plus the energy of boundary layers.

Foundation of Polar Linear Elasticity for Fibre-Reinforced Materials II: Advanced Anisotropy

This paper presents a complete formulation of the linear theory of elasticity for polar fibre reinforced materials with two families of embedded fibres by building upon previous theoretical background available for transversely isotropic fibre-reinforced materials (Soldatos in J. Elast. 114:155–178, 2014). Polar material behaviour stems from the postulate that fibres of either family are not perfectly flexible. During deformation of the fibrous composite, they instead act as slender Euler-Bernoulli beams and, therefore, their individual deformation generates couple-stress and non-symmetric stress. For simplicity, the outlined formulation is based on the plausible consideration that the fibre bending mode is the most dominant and, therefore, the most significant one to be accounted for when compared with its fibre stretching and fibre twist counterparts. The simplification achieved with this consideration becomes evident in the particular case of transversely isotropic materials, when comparisons are made against the relevant full developed theory (Soldatos in J. Elast. 114:155–178, 2014). Association of this simplifying consideration with the presence of two families of fibres resistant in bending furnishes the present model with the ability to embrace groups of advanced material anisotropy, such as the case of locally monoclinic materials which represent the most advanced group. The particular cases of local and plane orthotropy, where the two families of fibres are perpendicular to each other, are also deduced and considered in detail. Because of their importance in modelling thin- and/or moderately thick-walled structural components and laminates, particular attention is also given to the so-called cases of general and special orthotropy, where the fibres of both families are straight. The obtained governing equations are non-elliptic and this feature of the model is connected with the manner that solutions involving second-gradient weak discontinuity surfaces within the material may complement corresponding solutions associated with continuous displacements having their derivatives of all orders also continuous. The manner that second-gradient weak discontinuity surfaces are sought and found is finally demonstrated in detail for the case of specially orthotropic materials.

Foundation of Polar Linear Elasticity for Fibre-Reinforced Materials

This study is motivated by evidence suggesting that the equations of polar elasticity of fibre-reinforced materials are non-elliptic even within the regime of infinitesimal deformations. In its endeavour to resolve this issue, which in symmetric-stress elasticity emerges in the regime of finite deformations only, it lays the foundation for development of a second-gradient theory of linear elasticity. Complete formulation of this new theory is achieved for locally transverse isotropic materials; namely, materials having embedded a single unidirectional family of arbitrarily shaped fibres which are resistant in bending, stretching and twist. The associated analysis shows that, indeed, the obtained Navier-type displacement equations are not elliptic. They accordingly predict that there exist in the material weak discontinuity surfaces, which may indeed be activated within the infinitesimal deformation regime. Surfaces containing the fibres are certainly such surfaces of weak discontinuity; this result may be not irrelevant to numerous practical situations where straight metallic fibres in fibre-reinforced concrete structures emerge partially de-bonded and exposed from their concrete matrix. Nevertheless, the analysis reveals further that additional surfaces of weak discontinuity may well exist in the locally transverse isotropic material of interest. An extension framework is also outlined towards cases of fibrous composites containing two or more families of non-perfectly flexible fibres.

On loss of ellipticity in second-gradient hyper-elasticity of fibre-reinforced materials

This paper establishes a three-dimensional hyper-elasticity framework for studying the manner in which fibre bending stiffness affects current knowledge regarding the presence of weak discontinuity surfaces in unconstrained fibre-reinforced solid materials. This is achieved by considering and studying the loss of ellipticity of a new set of incremental partial differential equations, which emerges from the second-gradient, hyper-elasticity theory developed in [11] and yields its conventional theory counterpart as a particular case. It is accordingly seen that, besides the conventional acoustic tensor met in relevant symmetric hyper-elasticity studies, where fibres are assumed perfectly flexible, some new, higher-order acoustic tensor is involved and becomes dominant in the present situation. Nevertheless, the manner becomes also clear in which the present analysis reduces to and, hence, connects with loss of ellipticity concepts met in conventional hyper-elasticity. No particular form is assigned to the strain energy density of the material, which is kept general throughout this paper. Considerable elucidation of the outlined new issues and concepts is however achieved by focusing attention on plane deformations of hyper-elastic solids reinforced by a single family of straight fibres. This development concludes with a specific application which relates the present analysis with kink band formation in unidirectional fibrous composites containing fibres resistant in bending.

Time-harmonic Green's function and boundary integral formulation for incremental nonlinear elasticity: dynamics of wave patterns and shear bands

Superimposed dynamic, time-harmonic incremental deformations are considered in an elastic, orthotropic and incompressible, infinite body, subject to plane, homogeneous—but otherwise arbitrary—deformation. The dynamic, infinite body Green's function is found and, in addition, new boundary integral equations are obtained for incremental in-plane hydrostatic stress and displacements. These findings open the way to integral methods in incremental, dynamic elasticity. Moreover, the Green's function is employed as a dynamic perturbation to analyze interaction between wave propagation and shear band formation. Depending on anisotropy and pre-stress level, peculiar wave patterns emerge with focussing and shadowing effects of signals, which may remain undetected by the usual criteria based on analysis of weak discontinuity surfaces.

Surfaces of weak discontinuity for cubically anisotropic solid bodies

In the context of the general theory of characteristics, the velocities of propagation of the discontinuity surfaces in cubically anisotropic deformable media are obtained.

Investigation of surface instability in an elastic anisotropic cone by the use of surface waves of weak discontinuity

The problem of surface instability of a right circular cone with an arbitrary opening made of a hexagonal single crystal (the cone axis coincides with the crystal's axis of isotropy) is investigated. The surface of the cone is free from normal and tangential stresses, but in the layer near the surface initial constant tensile or compressive stresses act in the hoop direction and in the direction of the cone's generators. Surface instability is analyzed by the use of weak nonstationary disturbances which propagate along the surface of the cone in the form of the two types of surface waves: the nonstationary Rayleigh waves polarized in the sagittal plane, and the nonstationary wave of the “whispering gallery” type polarized perpendicular to the sagittal plane. The weak nonstationary surface waves are interpreted as the lines of discontinuity (diverging circles) on which partial derivatives of the stress and strain tensor components with respect to coordinates and time have a discontinuity, but the components of these tensors are continuous. Each of the lines of discontinuity propagates with a constant normal velocity along the cone's surface in the direction of its generators and is obtained as a result of the exit onto the cone surface either of two conic complex wave surfaces of weak discontinuity intersecting along this line (Rayleigh wave) or of one real conic wave surface of weak discontinuity (wave of the “whispering gallery” type). The analysis is carried out within the framework of the theory of discontinuities based on the kinematic, geometric and dynamic conditions of compatibility; using them the velocities of the surface wave propagation and their intensities have been found. It has been shown that the surface wave velocities are dependent only on the initial stress acting in the direction of the propagation of a surface disturbance whereas the damping coefficients for the intensities of the surface waves are dependent not only on this stress but also on the initial stress acting in the hoop direction as well. The relationships for two critical magnitudes of the force compressive in the hoop direction have been obtained, and it has been shown that under the hoop compressive forces in excess of one of these magnitudes the intensity of the Rayleigh wave or the surface wave of the “whispering gallery” type begins to increase without bounds during its propagation, i.e., the surface of the cone loses stability with respect to either of two types of weak nonstationary disturbances.

Loss of Ellipticity in Plane Deformation of a Simple Directionally Reinforced Incompressible Nonlinearly Elastic Solid

Change of type in the governing equations of equilibrium is examined in the context of a base neo-Hookean response augmented with unidirectional reinforcing that is characterized by a single additional constitutive parameter for strength of reinforcement. Plane deformations interpreted in terms of both local and global plane strain are considered. Loss of ordinary ellipticity is found to occur for sufficiently large strength of reinforcement under sufficiently severe deformation which necessarily involves contraction in the reinforcing direction. Loss of ellipticity in local plane strain is easily characterized, and its incipient breakdown is associated with the possible emergence of surfaces of weak discontinuity with orientation normals in the reinforcing direction. Loss of ellipticity in global plane strain is given a two-dimensional manifold characterization in a space involving two deformation parameters and the strength of reinforcing parameter. Orientation normals for the associated surfaces of weak discontinuity at incipient breakdown do not in general conform to the reinforcing direction.

On the weak discontinuity surfaces and tapes of equation systems in fluid mechanics

The weak discontinuity surfaces for a system of quasi-linear differential equations of higher order are developed. The classification of equation systems in fluid mechanics is based on the propagative weak discontinuity surfaces. Types of equations for different flow models are discussed. The conclusion is as follows:

Surfaces of discontinuity and waves in an anisotropic plasma with generalized polytropic equations of state

In this paper, using the theory of the surfaces of discontinuity, we study the non-linear, three-dimensional wave propagation in an anisotropic plasma with generalized polytropic equations of state. A detailed account of the surfaces of weak discontinuity is given, whether they are wavefronts or material surfaces. The specific subjects treated in the paper are indicated by the titles of the Sections.

Propagation of surfaces of weak discontinuity as behaviour of broken extremals in classical field theory

A variational method is shown to derive the equations describing the propagation of surfaces of weak discontinuity, i.e. of those surfaces across which at least one of the derivatives of the field variables suffer a jump. From the point of view of the calculus of variations this is the case of the broken extremals. The equations are derived from that variational principle which accounts for the non-broken extremals. The variational principle must be formulated at least in the frame of the special relativity.

Detection of strong and weak discontinuity surfaces in gasdynamic flows from optical experiment data

This study establishes a connection between the order of the density discontinuity and that of the light wave front that passes through the flow and examines the shadow and interference pattern characteristics resulting from these light wave front discontinuities. Rules that permit determination of the shape and location of the discontinuity surfaces of density and its derivatives are found. The theoretical results are illustrated by experimental data--shadow and interference patterns obtained in the study of three-dimensional supersonic flows.

Small amplitude waves and weak discontinuities in the relativistic hydrodynamics of an ideal fluid

Waves of small amplitude and the propagation of the surfaces of weak discontinuity (jumping derivatives) are studied in relativistic fluid dynamics. It is shown that in analogy with classical fluid dynamics, small amplitude waves and weak discontinuities have similar character. Both small amplitude waves, and surfaces of weak discontinuity are propagated with the velocity of sound, which, however, contains a relativistic correction.