Unbounded Body Research Articles

Basic theorems in linear micromorphic thermoelectroelasticity and their primary application

As a natural extension of the micromorphic continuum theory, the linear theory of micromorphic thermoelectroelasticity is developed to characterize the nano-micro scale behavior of thermoelectroelastic materials with remarkable microstructures. After the basic governing equations are given and the reciprocal theorem is deduced, both the generalized variational principle and the generalized Hamilton principle for mixed boundary-initial value problems of micromorphic thermoelectroelastodynamics in convolution form are established. Finally, as a primary application, steady state responses of an unbounded homogeneous isotropic micromorphic thermoelectroelastic body to external concentrated loads with mechanical, electric, and thermal origins are analyzed.

Deformation and fracture of a material in one-dimensional elastoplastic problems

The ideal plasticity model based on the Tresca-Saint-Venant criterion is used to solve one-dimensional problems of deformation and fracture of solids with circular boundaries. A thickwalled cylinder and a hollow sphere under pressure, cylindrical and hollow cavities in an unbounded body, and uniform extension at infinity of a plate with a free circular hole are considered. In simple elastoplastic problems, the proposed approach allows one to determine the value of the maximum external load at the fracture initiation and the motion of the fracture front for a given displacement of points of the contour on which this load acts.

Malignant fungating wounds – The meaning of living in an unbounded body

BackgroundMalignant fungating wounds may have significant physiological, psychological and emotional consequences on patients and their families. This study focuses on understanding the lived experiences of patients with a malignant fungating breast wound and their informal carers. MethodThe methodological framework of interpretative phenomenological approach according to Heidegger was used. Nine patients were interviewed from January until November 2009. ResultsThe results demonstrate that most of the patients and their informal carers were on their own while struggling with the erosion of their physical boundaries. The women report a lack of information and advice about how to manage the wound as well as the physical and social limitations imposed on them because of copious wound exudate, odour and bleeding. The women used many different methods and approaches to maintain the boundedness of the body. ConclusionThis study contributes to understanding that losing control over the body meant for the women losing control over themselves and their lives. The unboundedness was demonstrated through the symptom experiences. Therefore the care of women needs strategies that are integrated in a palliative, holistic, empathic approach. In particular skills for palliative wound care among medical and nursing staff need to be developed as the women and their carers report a lack of information and advice about how to manage the wound as well as the physical limitations and psychosocial consequences of struggling to maintain the boundedness of the body.

Inclusions in a finite elastic body

Within the framework of 2D or 3D linear elasticity, a general approach based on the superposition principle is proposed to study the problem of a finite elastic body with an arbitrarily shaped and located inclusion. The proposed approach consists in decomposing the initial inclusion problem into the problem of the inclusion embedded in the corresponding infinite body and the auxiliary problem of the finite body subjected to the appropriate boundary loading provided by solving the former problem. Thus, our approach renders it possible to circumvent the difficulty due to the unavailability of the relevant Green function, use various existing solutions for the problem of an inclusion inside an unbounded body and clearly makes appear the finite boundary effects. The general approach is applied and specified in the context of 2D isotropic elasticity. The complex potentials for the problem of an inclusion in an infinite body are given as two boundary integrals, and the boundary integral equation governing the complex potentials for the auxiliary problem is provided. In the important particular situation where a finite body with an arbitrarily shaped and located inclusion is circular, the exact explicit expressions for the complex potentials are derived, leading to those for the strain, stress and Eshelby’s tensor fields inside and outside the inclusion. These results are analytically detailed and numerically illustrated for the cases of a square inclusion placed concentrically, and a circular inclusion located eccentrically, inside a circular body.

Generalized thermodiffusion for an unbounded body with a spherical cavity subjected to periodic loading

In this paper, a general solution to the field equations of generalized thermodiffusion in an infinite thermoelastic body with a spherical cavity has been obtained in the context of the theory of generalized thermoelastic diffusion. The bounding surface of the sphere is subjected to periodic loading and the temperature and chemical potential are assumed to be zero on the curved surface. The generalized theory of thermoelasticity is applied to account for finite velocity of heat propagation. The closed form solutions for distributions of displacement, temperature and stresses are obtained. The solutions valid in the case of small frequency are deduced and the results are compared with the corresponding results obtained in other generalized thermoelasticity theories. Numerical results applicable to a copper-like material are also presented graphically and the nature of variations of the physical quantities with radial coordinate and with frequency of vibration is analyzed.

Fundamental system of solutions of the axially symmetric problem of the theory of elasticity for a body with a plane sheet of volume moment dipoles and forces

UDC 539.3 We have constructed the fundamental system of solutions of the axially symmetric problem of the theory of elasticity for an unbounded body with a sheet of volume forces, normal to a chosen plane, and moment dipoles, which is a mathematical model of the internal boundary layer of a certain type. With the help of such layers, one succeeds in formulating some inverse problems of elasticity and, hence, the related problems of the control of stress-strain state on the corresponding surfaces. We have also formulated and solved the generalized Kelvin problem and, according to it, for the plane of distributed normal load, and established the law of distribution of the moment dipoles (sheet parameters), which provides the vertical displacements assigned for points of the plane, in particular, zero, by the corresponding tension. Based on the definition of zero- and first-order singular material surfaces of discontinuity of the field characteristics [9], we investigate a variant of such surface as a mathematical model of the plane sheet of (distributed according to certain laws, determined by two generating functions) volume moment dipoles and volume, normal to the plane of distribution, forces, which are represented via the Hankel integrals. These distributions correspond to the fundamental system of solutions of the equations of statics of an elastic body in a cylindrical coordinate system, according to which the radial displacement, volume strain, and normal stresses have jumps in passage across the sheet plane [6, 10], and the tangential stress contains a term concentrated only in this plane. According to the definition in [9], we consider such a plane (zero- and first-order material singular surface) as an internal boundary layer, which is a material carrier of certain properties, different from those predicted by the equations of the linear theory of elasticity, and enables one to model certain types of planar defects of the structure of a perfect elastic material. The constructed solutions for the displacement field and all the other field characteristics of stress-strain state in a cylindrical coordinate system are written in terns of the Hankel integrals and contain the generating functions of the distribution densities of volume moment dipoles and volume forces unknown beforehand. Their appropriate choice enables one to obtain the solutions of a certain class of axially symmetric problems of the mechanics of deformable solids. Note that the fundamental system of solutions of the equations of statics with a sheet of volume forces, tangent to the plane, and momentless dipoles in a cylindrical coordinate system was constructed in [3]. 1. Fundamental System of Solutions of the Second-Type Equations of Statics

Spatial Behavior and Continuous Dependence Results in the Linear Dynamic Theory of Magnetoelectroelasticity

This paper is concerned with some general theorems for the linear dynamic theory of magnetoelectroelasticity. First, the spatial behavior of solutions is studied. In this sense, the so called “support” of the given data in a fixed interval of time [0,T] is introduced and an adequate time-weighted surface power function associated with the solution at issue is considered. Using their properties, we get the domain of influence and an exponential decay estimate with time-independent rates inside the domain of influence. As a by-product, a uniqueness result is derived for both bounded and unbounded bodies. Then, the case of non-zero initial conditions is also considered. Under a boundedness restriction on the initial data, an energy estimate is obtained. Finally, a continuous dependence result of solutions upon initial data and body forces is established.

Effect of magneto-thermo-viscoelasticity in an unbounded body with a spherical cavity subjected to a harmonically varying temperature without energy dissipation

The present work is devoted to study effects of the thermally induced vibration, magnetic field and viscoelasticity in an isotropic homogeneous unbounded body with a spherical cavity. The GN model of thermoelasticity without energy dissipation is applied. The closed form solutions for distributions of displacement, temperature and radial and hoop stresses are illustrated. The boundary conditions for the temperature and mechanical and Maxwell’s stresses are employed. The solutions valid in the case of small frequency are deduced and the results are compared with the corresponding results obtained in other generalized thermoelasticity theories. The results obtained are calculated for a copper material and presented graphically. It’s deduced that the magnetic field, viscosity and thermally induced vibration are very pronounced on displacement, temperature and stresses.

Spatial behavior in the electromagnetic theory of microstretch elasticity

This paper is concerned with the electromagnetic theory of microstretch elasticity. First, the initial boundary value problem is formulated in the framework of the linear dynamic theory of microstretch magnetoelectroelastic solids. Then, the spatial behavior of solutions is studied in both bounded and unbounded regions. The obtained result gives an exact idea of the domain of influence, in the sense that for each fixed time in a given interval, the entire activity vanishes at distanced from the support of the given data greater than a time-dependent threshold value. The study of spatial behavior is completed by an exponential decay estimate inside the domain of influence. As a by product a uniqueness result holding for both bounded and unbounded bodies is derived. Finally, the effect of a concentrated microstretch body force is studied.

On the Thermal Theory of Micropolar Solid-Fluid Mixture

In this paper we study the linear thermal theory of the micropolar solid-fluid mixtures. We consider a mixture consisting of an isotropic micropolar solid and a compressible fluid. First, we establish estimates of Saint–Venant type for bounded bodies in terms of two appropriate time–weighted surface power functions. An alternative estimate of Phragmén–Lindelöf type for unbounded bodies is also presented. Further, for unbounded bodies we are able to establish an estimate which proves that the measure associated with the solution decays faster than an exponential of a second degree polynomial, provided an appropriate class of mixtures is considered. This estimate shows that at large distance from the support of the external given data, the spatial decay of processes is influenced only by the thermal effect and by the viscosities of the fluid.

Unbalanced Nature, Unbounded Bodies, and Unlimited Technology: Ecocriticism and Karen Traviss’ Wess’har Series

While nature is often claimed to be a space of harmonized balance or an antidote to the chaos of the modern world, we need a more grounded assessment of nature as endlessly changing and much less predictable than we like to assume. In this essay, I explore Karen Traviss’ provocative exploration of unbalanced nature and unbounded bodies in her wess’har series with the guidance of two ecocritics who reject the concept of balanced nature, Dana Phillips and Ursula Heise. Additionally, I turn to the environmental philosopher Val Plumwood for insights regarding Traviss’ spurious yet rather standard vision of an unlimited technological panacea. Traviss’ series portrays how the boundaries and limits that we perceive as solid are often much less so than we believe, yet she also reveals—inadvertently, it seems—how easily we blindly ignore other, more solid limits.

Interaction of plane elastic nonstationary waves with an elastic inclusion under complete adhesion

We solve the problem on the interaction of plane elastic nonstationary waves with a thin elastic strip-shaped inclusion. The inclusion is contained in an unbounded body (matrix) which in under conditions of plane strain. It is assumed that the condition of perfect adhesion between the inclusion and the matrix is satisfied. Because of the small thickness of the inclusion we assume that the bending and shear displacements at any inclusion point coincide with the displacements of the corresponding points of its midplane. The displacements on the midplane itself are found from the corresponding equations of the theory of plates. The statement of the boundary conditions for these equations takes into account the forces and moments acting on the inclusion edges from the matrix. The solution method is based on representing the displacements in the space of Laplace transforms as a discontinuous solution of the Lame’ equations for the plane strain with subsequent determining the transforms of the unknown jumps from integral equations. The passage to the original functions is performed numerically by methods based on replacement of the Mellin integral by the Fourier series. As a result, we obtain approximate formulas for calculating the stress intensity factors for the inclusion. These formulas are used to study the time dependence of the stress intensity factors and the influence of the inclusion rigidity on their values. We also study the possibility of treating inclusions of high rigidity as absolutely rigid inclusions.

Interaction of plane nonstationary waves with a thin elastic inclusion under smooth contact conditions

We solve the problem of determining the stress state near a thin elastic inclusion in the form of a strip of finite width in an unbounded elastic body (matrix) with plane nonstationary waves propagating through it and with the forces exerted by the ambient medium taken into account. We assume that the matrix is in the plane strain state, and the smooth contact conditions are realized on both sides of the inclusion. The method for solving this problem consists in using the integral Laplace transform with respect to time and in representing the stress and displacement images in terms of the discontinuous solution of Lame equations in the case of plane strain. As a result, the initial problem is reduced to a system of singular integral equations for the transforms of the unknown stress and displacement jumps. To invert the Laplace transform, we use a numerical method based on replacing the Mellin integral by the Fourier series. As a result, we obtain approximate formulas for calculating the stress intensity factors (SIF) for the inclusion, which are used to study the SIF time-dependence and its influence on the values of the inclusion rigidity. We also studied the possibility of considering the inclusions of higher rigidity as absolutely rigid inclusions.

Interaction between non-parallel dislocations in piezoelectrics

The total interaction force F 12 between two crossing (non-intersecting) straight dislocations is found and analyzed for the three types of piezoelectric media of unrestricted anisotropy: an unbounded body, an infinite plate and a half-infinite body. In the latter two cases the dislocations are supposed to be parallel with the surfaces, which are in turn implied to be mechanically free of tractions and electrically closed (metalized). The found force F 12 is orthogonal to the parallel planes, P and Q, containing the crossing dislocations. In an unbounded medium the value F 12 proves to be independent of the distance between P and Q. On the other hand, it depends on directions of the dislocations and on their Burgers vectors: the force F 12 may be either attractive or repulsive. In a plate the interaction becomes sensitive to dislocation positions y ( 1 , 2 ) with respect to the surfaces. Only in the situations, when dislocations are much closer to each other than to the both surfaces, their interaction may be approximately described by the solution for an unbounded medium. Otherwise, corrections arising from the image forces due to the plate surfaces become essential. The dislocation in the vicinity of a surface strongly acts on its counterpart only until the latter situates even closer to the same surface than the first one. When the second dislocation leaves this narrow zone, the interaction force on it abruptly decreases to a very small level. With an increase in the thickness of the plate, this behavior becomes more and more pronounced. In a half-infinite medium the interaction between the dislocations is exactly described by a Heaviside step-like dependence F 12 ∝ H ( y ( 1 ) - y ( 2 ) ) valid for any y ( 1 , 2 ) . It is shown that we deal here with an analog of the plane capacitor effect.

The Unbounded Body in the Age of Liturgical Reproduction

Miracle stories from the sixth and seventh centuries about irregular, accidental, and inadvertent Eucharists demonstrate God's unbounded and excessive material body. These traditions provide a counterpoint to ascetic emphasis on bodily discipline and limits. As narratives of divine abundance, these tales contain implicit theologies of the Eucharist and theories of ritual efficacy through the recitation of anaphoral prayers.

Interaction of harmonic axisymmetric waves with a thin circular absolutely rigid separated inclusion

We study stress concentration near a circular rigid inclusion in an unbounded elastic body (matrix). In the matrix, there are wave motions symmetric with respect to the axis passing through the inclusion center and perpendicular to the inclusion. It is assumed that one of the inclusion sides is completely fixed to the matrix, while the other side is separated and the conditions of smooth contact are realized on that side. The solution method is based on the fact that the displacements caused by waves reflected from the inclusion are represented as a discontinuous solution of the Lame equations. This permits reducing the original problem to a system of singular integral equations for functions related to the stress and displacement jumps on the inclusion. Its solution is constructed approximately by the collocation method with the use of special quadrature formulas for singular integrals. The approximate solution thus obtained permits numerically studying the stress state in the matrix near the inclusion. Technological defects or constructive elements in the form of thin rigid inclusions contained in machine parts and engineering structure members are stress concentration sources, which may result in structural failure. It is shown that the largest stress concentration is observed near separated inclusions. Static problems for elastic bodies with such inclusions have been studied rather comprehensively [1, 2]. The stress concentration near separated inclusions under dynamic actions on the bodies has been significantly less studied even in the case of harmonic vibrations. The results of these studies can be found in [3, 4], where bodies with a thin separated inclusion were considered, and in [5], where the problem about torsional vibrations of a body with a thin circular separated inclusion was studied. The aim of the present paper is to study stress concentration near such an inclusion in the case of interaction with harmonic waves under axial symmetry conditions.

Mathematical simulation of thermal friction processes under conditions of nonideal contact

Analytical solutions are obtained for problems on unsteady-state thermal conductivity under conditions of nonideal frictional contact of two solid isotropic half-bounded bodies for cases such as a system of two half-spaces, a system of two unbounded cylindrical bodies, and a system of two half-spaces in the presence of volume source of heat release.

Fractional radial diffusion in an infinite medium with a cylindrical cavity

The time-fractional diffusion equation is employed to study the radial diffusion in an unbounded body containing a cylindrical cavity. The Caputo fractional derivative is used. The solution is obtained by application of Laplace and Weber integral transforms. Several examples of problems with Dirichlet and Neumann boundary conditions are presented. Numerical results are illustrated graphically.

Interaction of plane harmonic waves with a thin elastic inclusion of zero flexural rigidity

We solve the problem on the interaction of plane elastic harmonic waves with a thin elastic strip-shaped inclusion. The inclusion is contained in an unbounded body (matrix) that is under plane strain conditions. The normal forces applied by the medium to the inclusion side edges are taken into account. Because of the small thickness of the inclusion, we assume that its flexural rigidity is zero and that the shear displacements at any of its points coincide with the displacements of the corresponding points of its midplane. The displacements on the midplane itself can be found from the corresponding equation of the theory of plates. The solution method consists in representing the displacements as discontinuous solutions of the Lame equations and then determining the unknown jump from a singular integral equation. This equation is solved numerically by the collocation method, and formulas for the approximate calculation of the stress intensity factors near the inclusion ends are obtained.

Flexural vibrations of a thin circular elastic inclusion in an unbounded body under the action of a plane harmonic wave

The interaction of a plane harmonic longitudinal wave with a thin circular elastic inclusion is considered. The wave front is assumed to be parallel to the inclusion plane. Since the inclusion is thin, the matrix-inclusion interface conditions (perfect bonding) are formulated on the mid-plane of the inclusion. The bending displacements of the inclusion are determined from the bending equation for a thin plate. The problem is solved using discontinuous Lame solutions for harmonic vibrations. Therefore, the problem can be reduced to the Fredholm equation of the second kind for a function associated with the discontinuity of normal stresses on the inclusion. The equation obtained is solved by the method of mechanical quadratures using Gaussian quadrature formulas. Approximate formulas for the stress intensity factors are derived. Results from a numerical analysis of the dependence of the SIFs on the dimensionless wave number and the stiffness of the inclusion are presented