Unbounded Body Research Articles

Solution of the two-dimensional integrodifferential equation of the heat conduction problem for an unbounded body with a thin thermally conducting inclusion

We determine the temperature field in an unbounded body with a thin heat-conducting foreign inclusion. We propose a method of solution of the corresponding two-dimensional singular integrodifferenlial equation of the problem based on the study of the behavior of the solution in a neighborhood of the boundary of the inclusion.

THERMOELASTIC INTERACTIONS IN AN UNBOUNDED BODY WITH A SPHERICAL CAVITY

Thermoelastic interactions caused in a homogeneous and isotropic unbounded body with a spherical cavity due to harmonically varying thermal field applied to the stress-free boundary of the cavity are considered. A unified system of governing equations that includes among its particular cases the field equations of the conventional as well as generalized thermoelasticity theories is employed. Closed form expressions for the temperature, displacement, and stress fields are obtained. Numerical results for a steel material are presented.

Axisymmetric thermoelastic interactions in an unbounded body with cylindrical cavity

Thermoelastic interactions in a linear, homogeneous and transversely isotropic unbounded body containing a cylindrical cavity due to a uniform step in stress or temperature applied to the boundary of the cavity are studied. A unified system of governing equations that includes among its particular cases the governing equations of the conventional and generalized thermoelasticity theories is employed. By the use of the Laplace transform technique, short time solutions for the temperature, displacement radial stress and hoop stress are constructed. The discontinuities suffered by these fields at the wavefronts are computed. Comparison with the corresponding results obtained in earlier works is made. Numerical results for a single crystal of zinc are presented.

Interaction between coplanar cracks under dynamic shock loading

A study is made on the interaction between coplanar cracks in an unbounded body when dynamic loads act on thier surfaces that are described as Heaviside or Dirac delta functions. Graphs are given relating the stress intensity coefficients when there are two disk cracks in the body whose surfaces are loaded by shock external forces.

LINEAR DYNAMICAL STABILITY IN CONSTRAINED THERMOELASTICITY I. DEFORMATION-TEMPERATURE CONSTRAINTS

Equations are derived governing an infinitesimal disturbance of a uniform equilibrium state B of an unbounded body. No restriction is placed on the symmetry of the material and the freedom of choice of B allows the presence of an arbitrary homogeneous prestrain. The stability of B is examined by studying the nature of plane harmonic wave solutions of governing equations

THE DIURNAL MIXED LAYER IN LAKES AND OCEANS

SUMMARY In this review the current ideas on diurnal mixed layers are discussed with reference to several observations. Essentially, solar heating sets up a shallow thermocline which traps and concentrates further influx of heat and momentum in the top few metres of the water. These shallow structures usually deepen during the afternoon and evening and are erased by convective cooling overnight. This trapping of momentum results in significant wind-driven diurnal currents which take a different form in bounded or unbounded water bodies. Although the net gain or loss of heat is negligible, the cumulative effect of these diel cycles results in a seasonal heating and cooling cycle. This stratification inhibits vertical exchange and may be a significant factor in plankton dynamics and pollution dispersion.

Non-singular wedge combinations at the free edge of a brazed ceramic-metal joint

The solution of the elastic boundary value problem for residual thermal stresses in a brazed ceramic-metal joint by the displacement-formulated finite-element method gives very high stress concentrations in the vicinity of points where both material and geometrical discontinuities exist. Convergence studies by successive mesh refinements indicate that there are stress singularities at these points for certain local geometry (wedge) configurations of a given material combination. When the order of the stress singularity is determined by solving the proper boundary value problem for unbounded wedge bodies, it turns out that singular stress fields may vanish for certain wedge combinations. The finite-element analysis of free edge configurations proves adequate in quantifying asymptotic stress fields. For non-singular free edge configurations, reliable predictions on stresses are obtained everywhere in the joint.

The virtual friction of a sphere vibrating in an elastic medium

When an elastic body of finite extent adheres to another elastic body, so large that it may be considered infinite, the motion of the first body is, in general, strongly influenced by the elastic reactions between them. The effect of their mutual contact is two-fold. Firstly, the natural frequencies of the finite body, considered individually, tend to increase since the adhesion adopts the role of an additional constraint. Secondly, mechanical energy is dissipated because the elastic waves propagate without reflection into the contiguous, unbounded body. Figure 1 illustrates a typical situation. The elastic body V is loaded by surface tractions, varying in time, on the part $1 of its surface, while it adheres along Sz to the infinite elastic body V'. Provided that this problem of classic elastodynamics be explicitly solvable, there is a way, at least in principle, to estimate how the bond modifies the free vibrations of V.

Methods of researching thermophysical parameters and phenomena by means of nonstationary-frequency measurements. Part 2. Step and instantaneous heating methods

Various types of instantaneous and stepped heat source are considered, which act in unbounded bodies. A method has been devised for using the solutions to define the thermophysical parameters by means of nonstationary-frequency measurement methods.

The compression of an unbounded body with semi-infinite cylindrical cavities

The deformation of an unbounded elastic body with two semi-infinite cylindrical cavities (Fig.1) under uniform compression by a load p ∞ at infinity, perpendicular to the axis of symmetry is considered. An infinite cylinder r⩽ a and an infinite body with a hole of radius b are separated by surfaces r = a and r = b from the body mentioned. For | z|< c the interval between them is filled with thin rings. For a = 0 a bottomless pit is obtained that is partially filled with discs of radius b; the case b − a ⪡ b is of special interest, i.e., when the distance between the slot edges is several times less than the hole diameter. In this case brittle fracture occurs by spalling of the cylindrical part (a projection that is called a core) and transformation of the body into a pit. The dead-end of the slot is the focus of the stress concentration, sometimes resulting in phenomena of the rock-burst type in hard rocks. Conditions for their origination are clarified, which is important for estimates of the magnitude of mountain pressure. Methods utilized in /1/are extended in this paper and the results of this research are refined.

On a generalization of the Naghdi-Hsu transformation and its application to problems of elasticity theory

The following model is considered for the inhomogeneity of an elastic material: the shear modulus of the material is constant, but Poisson's-ratio (or the modulus of elasticity) depends in an arbitrary manner on three Cartesian coordinates. A linear transformation of vector fields is introduced, which is a generalization of the well-known Naghdi-Hsu transformation /1, 2/ (NHT) that enables the solution of the equilibrium equation of elasticity theory for bodies with variable Poisson's ratio to be represented in terms of a vector function (harmonic when there are no body forces), and proof of its completeness. In passing, a new variation of the NHT is formulated in which the integrals over the body volume are replaced by integrals over its surface. The fundamental solution of the equilibrium equation in displacements is presented in explicit form for an unbounded body with inhomogeneity of the type under consideration. Some problems in this area were investigated in /3–5/.

SOME THEOREMS IN TEMPERATURE-RATE-DEPENDENT THERMOELASTICITY FOR UNBOUNDED DOMAINS

Abstract A domain of influence theorem, work and energy theorem, and reciprocity theorem are proved for a linear anisotropic nonhomogeneous temperature-rate-dependent thermoelastic unbounded body. The theorems cover the case where the constitutive fields may attain unbounded values at infinity, and they are based on a hypothesis from which it follows that a thermoelastic disturbance produced by an external ther-momechanical load of a bounded support cannot invade the whole body in a finite time

The Extent of the Stress Intensity Factor Field During Crack Growth Under Dynamic Loading Conditions

The phenomenon considered is fracture initiation and crack growth in a plate due to dynamic pressure loading on the faces of a pre-existing crack. The problem is formulated within the framework of two-dimensional elastodynamics, and the system is viewed as a semi-infinite crack in an otherwise unbounded body. At a certain instant of time, a spatially uniform pressure begins to act on the crack faces. The pressure magnitude increases linearly in time for a certain period (the rise time T), and it is constant thereafter. The crack begins to extend at constant speed at some time after the pressure begins to act (the delay time τ). The pressure acts only over the original crack faces, and both τ &gt; T and τ &lt; T are considered. The ratio of the normal stress on the fracture plane to the value due to the singular term in the stress field alone is computed for some point at a small fixed distance ahead of the crack tip, with a view toward establishing the conditions under which the stress intensity factor controlled singular term accurately describes the near tip stress distribution in this highly transient process. Measured and calculated histories compare very well for relatively low crack face pressures, but there is significant disagreement beyond crack growth initiation for higher pressures. Possible reasons for the discrepancies are discussed.

Is elasticity the proper asymptotic theory for materials with small viscosity and capillarity?

SynopsisWe consider the equations for the isothermal motion of a one-dimensional unbounded body composed of a material with viscosity and capillarity. Using a technique derived from the theory of compensated compactness, we find conditions which guarantee that, as viscosity and capillarity approach zero, the solutions to these equations converge to a solution to the corresponding equations in elasticity.

On the theorem of power expended in continuum mechanics

We show that the theorem of power expended may be claimed for unbounded bodies, under mild hypotheses on the behavior at infinity of the velocity, the density, and the stress fields.

On Graffi's reciprocal theorem in unbounded domains

A generalization of the reciprocal theorem of linear elastodynamics is proposed for a class of unbounded elastic bodies that may stiffen at infinity. It is achieved by means of some a-priori estimates derived in a previous paper of ours [1].

On the work and energy theorem for unbounded elastic bodies

The main scope of the paper is the statement and the proof of the “work and energy theorem” for elastic bodies that occupy an unbounded region of space and whose acoustic tensor A may suitably grow at large spatial distance from a fixed origin.

Energy inequalities and the domain of influence theorem in classical elastodynamics

In this paper we derive a number of a priori estimates for classical solutions of the system of linear elastodynamics, and we prove an analogue of the classical domain of influence theorem for a class of unbounded elastic bodies that may stiffen at infinity.

Crack generated wave fronts

This paper considers cracks of a fixed size and shape which exist in an unbounded linear elastic body. The crack faces are assumed to suffer a sudden disturbance, which results in a shock wave emanating from the region of the crack into the body. The geometry and physics are assumed such as to give rise to problems governed by the equations of two-dimensional elastodynamics.

An Effective Dispersion Theory for Layered Composites

A new approximate theory for wave propagation in a periodically layered elastic body is proposed. In its present state, the theory is valid for either propagation normal to the layering or for propagation under antiplane strain conditions. The theory is based on microstructural theories of elasticity, and makes use of a collocation technique in conjunction with the exact solution for wave propagation in an unbounded elastic body. The results are considered to hold promise for formulating accurate approximate solutions to certain types of boundary-initial value problems for composite bodies.