Showing papers in "Quarterly of Applied Mathematics in 1971"
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TL;DR: In this paper, Gurtin and Pipkin [17] have proposed a nonlinear memory theory of heat conduction which is independent of the present value of the temperature gradient and has associated with it finite wave speeds.
Abstract: This theory has, however, two principal shortcomings. First, it is unable to account for memory effects which may be prevalent in some materials, particularly at low temperatures. Secondly, the parabolic heat equation (1.3) predicts an unrealistic result: that a thermal disturbance at one point of the body is instantly felt everywhere in the body (though not equally). This implies that in Fourier heat conductors, finite thermal discontinuities must propagate with infinite speed. It is these observations which lead one to believe that for materials with memory, Fourier's law (1.1) may be a limiting approximation (perhaps for sufficiently steady temperature fields) to a more general nonlinear constitutive assumption relating the heat flux q to the material's thermal history. Gurtin and Pipkin [17] have proposed one such nonlinear memory theory of heat conduction which is independent of the present value of the temperature gradient. This theory generalizes constitutive relations deduced from kinetic theory by Maxwell [19] and Cattaneo [2] and has associated with it finite wave speeds.1 Moreover, when this theory is linearized, it yields the heat flux relation
335 citations
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65 citations
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63 citations
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TL;DR: Conditions for initiation of necking and bulging of elastic and elastic-plastic cylindrical solids are derived in this paper, where the possibility of bifurcation of rigid-PLastic solids and the conditions for homogeneous deformation with homogeneous stress are also investigated.
Abstract: Conditions for initiation of necking and bulging of elastic and elasticplastic cylindrical solids are derived. The possibility of bifurcation of rigid-plastic solids and the conditions for homogeneous deformation with homogeneous stress are also investigated.
50 citations
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TL;DR: In this article, the relative mean square error in the three-dimensional stress field predicted by classical plate theory was shown to be OQi/L 2, where h is the plate thickness and Lj is a mean square measure of the wavelength of the midplane deformation pattern.
Abstract: The relative mean square error in the three-dimensional stress field predicted by classical plate theory is shown to be OQi/L^)2, where h is the plate thickness and Lj. is a mean square measure of the wavelength of the midplane deformation pattern. This improves a recent result of Nordgren who obtained a relative error estimate of 0(h/L*). The improved error estimate, which, like Nordgren's, is based on the PragerSynge hypercircle theorem in elasticity, is obtained by constructing a kinematically admissible three-dimensional displacement field that depends on the solution of the classical plate equations but which yields an accurate, nonzero distribution of the transverse shearing strain through the thickness.
43 citations
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TL;DR: In this article, the asymptotic behavior of the equations governing viscous flow along a right-angle corner is considered, and it is demonstrated that consistent asmptotic series exist for the inner corner layer region.
Abstract: The asymptotic behavior of the equations governing the viscous flow along a right-angle corner is considered. It is demonstrated that consistent asymptotic series exist for the inner corner layer region. These expansions satisfy the corner layer equations and associated boundary conditions. They exhibit algebraic decay of all the flow properties into the boundary layer away from the corner, and prescribe algebraic decay of the cross flow velocities into the outer potential flow. Of course the streamwise velocity and vorticity are constrained to decay exponentially into the potential flow. The form of this algebraic behavior is required in order to facilitate numerical solution of the corner layer equations. Of particular significance is the use of symmetry as a means of providing a boundary condition, predicting the appearance of logarithmic terms, and specifying the occurrence of arbitrary constants. These constants can only be determined from the complete corner layer solution.
42 citations
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TL;DR: In this paper, the authors derived an explicit expression for the mean-square error in the components of stress obtained from a solution in plate theory with respect to the exact solution of a corresponding problem in the theory of elasticity.
Abstract: Introduction. A solution to a boundary value problem in the classical two-dimensional theory of plates is generally accepted as an approximate solution to a corresponding boundary value problem in the three-dimensional theory of elasticity provided that the plate is sufficiently thin. This conclusion is supported by several exact solutions for plates in the theory of elasticity [1] and by the fact that the equations of plate theory can be obtained from the equations of elasticity theory as the leading terms in parametric expansions [2], [3], Further, Morgenstern [4] has shown that the stresses and strains obtained from a solution in plate theory converge in a mean-square sense to a solution in elasticity theory as the plate thickness approaches zero. Related theorems on meansquare convergence of parametric expansions for a problem in beam theory are stated by Babuska and Prager [5]. In the present paper we derive an explicit expression for the mean-square error in the components of stress obtained from a solution in plate theory with respect to the exact solution of a corresponding problem in the theory of elasticity. In addition, a precise bound is given for the relative mean-square error. The derivation employs the hypersphere theorems of Prager and Synge [6] in the theory of elasticity. In the course of the derivation the equations of plate theory are obtained in two ways by minimization of portions of both the potential energy and the complementary energy. The general expression obtained for the error contains only quantities which are available from a solution in plate theory. Our results and the previous investigations of convergence [4], [5] show that the relative mean-square error in plate theory is proportional to the thickness of the plate in general. This is somewhat surprising since the exact solutions for plates in elasticity theory [1] give a relative error proportional to the square of the thickness. This form for relative error also is indicated by the parametric expansions [2], The discrepancy in our result can be attributed to the expression obtained for the components of transverse shear stress, which differs from the classical expression. We have been unable to derive the classical expression by the present method. (See note added in proof at the end of this paper.) I. Function space concepts in elasticity. We consider a three-dimensional elastic body R bounded by a closed surface S. With reference to a system of rectangular Cartesian coordinates z,(i = 1, 2, 3), the field equations of the linear theory of elasticity read as follows [1]: equilibrium (in the absence of body force)
40 citations
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24 citations
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TL;DR: In this paper, the minimum-weight design of structures subjected to periodic loading, under the constraint of a prescribed deflection at a specified point of the structure, is investigated. And the stationary mutual potential energy is extended to cover periodic loading and a sufficient condition for stationary weight is derived.
Abstract: This paper is concerned with the minimum-weight design of structures subjected to periodic loading, under the constraint of a prescribed deflection at a specified point of the structure. For elastic sandwich beams, the method of stationary mutual potential energy is extended to cover periodic loading and a sufficient condition for stationary weight is derived.
23 citations
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23 citations
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TL;DR: A theory of conjugate approximations is developed which provides a fundamental basis for most methods of continuous piecewise approximation and it is shown that for a given finite set of base functions used in an approximation there corresponds another set of Conjugate functions which play a significant role in approximate methods of analysis.
Abstract: : In the paper a theory of conjugate approximations is developed which provides a fundamental basis for most methods of continuous piecewise approximation. It is shown that for a given finite set of base functions used in an approximation there corresponds another set of conjugate functions which play a significant role in approximate methods of analysis. In the case of finite-element approximations, it is shown that the domain of the conjugate functions includes the entire assembly of elements, and, consequently, the established method of computing stresses locally in elements based on displacement approximations is not strictly valid. Indeed, the domain of such 'local' stress fields is the entire connected system of elements. Procedures for computing derivatives and discrete analogues of linear operators which are consistent with the theory of conjugate functions are also discussed. For a given linear operator equation, the significance of the conjugate approximations in connection with the adjoint problem is also discussed. (author)
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TL;DR: Konopliv et al. as mentioned in this paper developed a solution method for determining the surface heat transfer for the case of steady, uniform surface temperature, where the velocity field is assumed to be known.
Abstract: Consideration is given to problems of unsteady forced convection heat transfer in the presence of either time-invariant or time-dependent surface temperatures. The transient is initiated when a solid body is exposed to a fluid having a temperature different from its own. In the first part of the paper, a solution method is developed for determining the surface heat transfer for the case of steady, uniform surface temperature. Then, attention is turned to the determination of the temperature history of non-internally-heated bodies of high thermal conductance, which lose heat by convection to the fluid environment. A numerical scheme for deducing results for the temperature history is described, while analytical expressions appropriate to the initial and quasi-steady stages of the transient are presented. Detailed consideration is given to the case of a sphere in a low Peclet number flow, for which an exact solution for the temperature history is worked out. The results from the numerical scheme are found to be in excellent agreement with those from the exact solution, while the expressions for the initial and quasi-steady stages, when taken together, serve to establish the general behavior of the solution over the entire transient period. Introduction. In this paper, consideration is given to two classes of problems which involve time-dependent heat transfer between a solid surface and a moving fluid. ODe of these classes is characterized by steady, uniform surface temperatures. The other includes non-internally-heated, high-conductance solid bodies having a spatially uniform temperature which changes 'with time as a result of heat exchange \"with the fluid environment. For both classes of problems, the transient is initiated when the body is exposed to a fluid whose temperature is different from that of the body. Upon considering the problem areas just described, it is apparent that the analysis of the first is less demanding than the analysis of the second. This is because, for the solution of problems in the first group, it is necessary to consider only the energy equation for the fluid environment (assuming that the velocity field is known). On the other hand, for problems in the second group, the energy equation for the fluid must be solved simultaneously with the energy balance for the solid. The foregoing observation motivates one of the objectives of this research. A solution method is to be developed for determining the timewise temperature variation of a noninternallv-heated solid of high conductance situated in a conducting-convecting fluid * Received February 23, 1970. ** Present address: Polytechnic Institute of Brooklyn, Brooklyn, New York. 226 N. KONOPLIV AND E. M. SPARROW environment, given the surface heat transfer solution for the same solid but with a steady, uniform surface temperature. In principle, the method is able to accommodate arbitrary body shapes. There are a number of approaches that may be employed to generate the input information needed to apply the solution method; that is, to obtain the surface heat transfer corresponding to the condition of steady, uniform surface temperature. These approaches will be discussed in a later section. In the first section, however, a new series solution, possessing considerable generality, is presented for problems of steady, uniform surface temperature. The second section is devoted to a description of the method for determining the timewise temperature variation of the non-internally-heated, highly conducting solid. The method makes use of integral transforms. This section also presents an account of a highly serviceable numerical inversion technique which facilitates the practical realization of the solution. In the last section of the paper, a specific problem is solved to illustrate and test the method. Series solution for steady, uniform surface temperature. Consider a body having steady, uniform surface temperature T0 ■ At time t = 0, the body is exposed to a laminar forced convection flow whose initial temperature is uniform and everywhere equal to . The fluid freestream temperature for all subsequent times is also T„ . The velocity field is presumed known. For these conditions, regardless of whether the fluid velocity is steady or unsteady, the temperature field in the fluid about the sphere will undergo a timewise development from its initial uniform state. In the event that the velocity field is steady, then, after a sufficiently long time has elapsed, a thermal steady state is attained in the fluid. It is the objective of the analysis to determine the timewise variations of the local and area-integrated instantaneous heat transfer rates at the surface. To facilitate the analysis, let |, 17, f denote a set of dimensionless orthogonal coordinates (reference length L), such that 77 is directed along the local normal to the surface, while £ and f lie in the surface y = 0. The local instantaneous fluid temperature is represented by Tf.(£, 77, f, r), where r is a dimensionless time. The asterisk is employed to distinguish the case of steady, uniform surface temperature that is now under study. It is also convenient to define a dimensionless fluid temperature 8f. as »/.(?, V, f, t) = (Tf. r.)/(r0 sr.). (i) The first step in the analysis is to write the appropriate energy equation. For a steady, laminar boundary layer flow about a two-dimensional or axisymmetric body and for steady flow (without boundary layer assumptions) about a sphere or cylinder, the dimensionless energy equation can be written in the form 1 iry d&f d2df, . , d2df, , . d28f. ~d7 + m'v'f) 77 ~ ~e7~+ 9l(r,) If2' + 9v) \"ap\" + v, t) + g*(£, v, f) (2) with initial and boundary conditions Ml, r, 5* 0, r, 0) = 0, *,.({, 0, f, t) = 1, 6,.(S, «, f, r) = 0. (3) In Eq. (2), the function / includes the Tj-component of the fluid velocity and, depending TEMPERATURE AND HEAT TRANSFER 227 on the particular case, some power of the 77 coordinate which stems from the heat conduction term. The functions g, contain velocities and/or coordinates. Evidently, various of the gj are zero for two-dimensional flow, axisymmetric flow, or boundary layer flow. The / and <7,are regarded here as known functions of position. To initiate the solution, the Laplace transform of the energy equation (2) is taken with respect to r, with #/,(£, v> s) denoting the transform of the fluid temperature distribution. Next, to eliminate the term d0r,/d?? from the thus transformed energy equation, one introduces the $ function as follows: M*. V, r, s) = ^ r' s) exp fo M, v, r) dr^ , (4) and with this, the Laplace transformed version of (2) becomes
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TL;DR: In this paper, the authors derived comparison theorems for a nonlinear eigenvalue problem as well as a Lyapunov type of inequality for the boundary value problem.
Abstract: Comparison theorems for a nonlinear eigenvalue problem as well as a Lyapunov type of inequality are derived. They are used to establish upper and lower bounds for various integral functionals associated with real solutions of the nonlinear boundary value problem y\" + p(x)y2n+1 = 0, y(a) = y'(b) = 0, where a < b are real, n is a positive integer and p is positive and continuous on [a, 6]. Some of the results are analogues of a distance between zeros problem for the linear case of n = 0. Characteristic value problems for various nonlinear differential equations have been studied in recent years; a good bit of work has been done by Nehari, Moore and Moroney, among others [8], [9], [12], [13]. Much of this work involves the study of oscillation and nonoscillation of solutions of these differential equations. The linear characteristic value problem y\" + \\p(x)y = 0, y{a) = y'(b) = 0, where a < b are finite reals and p is real and continuous on [a, b], has been studied for the behavior of the least positive eigenvalue Xi(p) relative to the \"shape\" of p on [a, b]. This has been done for other boundary conditions as well as for other positive eigenvalues and this problem is related to the determination of the fundamental frequencies of oscillation of a vibrating string. See for example Nehari [11], St. Mary [14], Fink [6] and especially the bibliography in [6] for a background on this problem. The purpose of this paper is to establish various extensions and analogues of the linear theory to nonlinear differential equations. Moore and Nehari [8] consider the nonlinear second-order differential equation of the form y\" + P(*)y2n+1 = o, (l) where n is a positive integer and p is positive and continuous on a compact interval of reals [a, b] with a < b. Along with (1), Moore and Nehari consider the Rayleigh quotient J(y) = (fa y'2 dxj+1/ (j* py2*+2 dx (2) * Received November 24, 1970; revised version received December 17, 1970. Based on research supported in part by the U. S. Army Research Office-Durham through Grant Number DA-ARO-D-31124-70-G83 with the University of Oklahoma Research Institute. 392 STANLEY B. ELIASON where the domain of J is D[J] — \\y £ D'[a, b]:y(a) = 0 and y ^ 0 on [a, 6]} where D'[a, 6] is the set of all continuous real-valued functions having sectionally continuous derivatives on [o, b]. The following theorem is a portion of Theorem V of [8, p. 44] which deals with the boundary conditions y(a) = y'(b) = 0: Theorem (Moore and Nehari). For each pair of finite reals a < b, Eq. (1) has at least one solution y £ C'[a, 6] which is positive in (a, b) and satisfies the boundary conditions yip) = y'Q>) = o. (3) Furthermore this solution satisfies the property that J(y) < J(u) for each u £ D[J\\. (4) Thus we see that (2) is minimized in D[J] by a nontrivial y £ C2[a, 6] which is a solution of (1) and (3). Such a solution will be called a Moore-Nehari minimizing function of (2) for the problem of (1) and (3). The question of uniqueness of such a minimizing function is not in general answered. In case the boundary value problem (1) and (3) has a unique nontrivial solution which is positive in (a, b) we have, of course, uniqueness of the Moore-Nehari minimizing function. If for our case we would assume in addition that p is monotone increasing on [a, b], then a result of Moroney [10] assures us of the desired uniqueness. It would be highly desirable to have this result under less restriction on p. For our equation (1) together with (3), Moroney's proof can be modified to provide uniqueness if we assume in addition that [(x — a)p(x)]' > 0 on [a, &]. On the other hand, Moore and Nehari give an example of a p where the boundary value problem of (1) together with the boundary conditions y(a) = y(b) = 0 has two distinct solutions on (a, b). It may first appear to the reader that the concept of a \"Moore-Nehari minimizing function\" is somewhat artificial. Due to the above remarks concerning uniqueness and certain relations to be established later, it will be shown that some of the most important results of the paper are independent of the concept. We are now ready for our first result, which is basically a Lyapunov inequality. Theorem 1. Let Ai(p) be the least positive value of J(y) in (2) for y £ D[J}. Then it follows that (ib a)n+\\(p) J p dx > 1. (5) Furthermore the inequality is sharp. Proof. First, if y is any nontrivial solution of the problem (1) and (3), by integrating (1) after multiplying by y it follows from (3) that J y'2 dx = J py2n+2 dx. (6) Consequently we have COMPARISON THEOREMS 393 X,(p) < (f' V'2 dxj = (£ py2\"+2 dxj (7) where equality holds if y is a Moore-Nehari minimizing function of (2) for the problem of (1) and (3). Also, since p(x) > 0 on [a, 6], if y(x) ^ 0 on (a, b\\ then we have \\y(x)\\ < |2/(6)| for x £ [a,b). (8) Now assume y is a Moore-Nehari minimizing function of (2) for the problem of (1) and (3). We have ab \\2n+2 y'dx) ab \
\"*\" ^ y'2 dx) = (b a)\"*\\(p) f py2n+2 dx J a <(ba)n+%(p)[y(.b)F+i fpdx \" a from which (5) holds. The sharpness can be established by a slight modification of an example found in [5] which in turn is a modification of an example used to show that the Lyapunov inequality for linear equations is sharp. Corollary 1. For all y £ D[J] it follows that (b V'2 dxj+1 f* p dx > J' py2\"+2 dx. (9) Corollary 2. If y £ C2[a, 6] is any solution of (1) and (3) which is positive on (a, b) then we have (b a)n+,Q: J pdx > 1 for 1 < i < 4, (10)
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TL;DR: In this paper, the authors present a general solution to the elasticity problem with the restriction that the prescribed stress field is axisymmetric, based on some new identities in Legendre functions.
Abstract: An infinite elastic medium contains an elastic spheroidal inclusion. Both materials are transversely isotropic. Assuming that the stress field in the absence of any inhomogeneity is prescribed, it is desired to calculate the modification caused by the inclusion. This paper presents a general solution to this elasticity problem with the restriction that the prescribed stress field is axisymmetric. The analysis is based upon some new identities in Legendre functions, which are derived in this paper. The solution is in the form of combinations of Legendre functions. An example of a spheroidal cavity in a tension field is given.
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TL;DR: In this article, the existence of non-oscillatory solutions of (1.1) on [0, °°] is discussed. But the main purpose of this paper is to discuss other possible types of nonoscillatorial solutions, for example, solutions which grow like a fractional power of t. Very little is known about such solutions.
Abstract: where q(t) is nonnegative and continuous on [0, °°) and y is any real number satisfying 0 < y < 1. (These assumptions on q(t) and y will be implicitly assumed throughout.) There are necessary and sufficient conditions for the existence of nonoscillatory solutions y(t) of (1.1) satisfying either lim(_.„ y(t) = a ^ Oorlim(-,„ (y(t)/(t) = a ^ 0 (cf. Sec. 2). Our main purpose, then, is to discuss other possible types of nonoscillatory solutions, for example, solutions which grow like a fractional power of t. Very little is known about such solutions. We will give criteria for the existence of such solutions and also a criterion for a certain type of "dichotomy" which may occur. All solutions of (1.1) exist on [0, <») as follows from a theorem of Wintner (Hartman [5]). By definition, a solution of (1) is said to be oscillatory if it has arbitrarily large zeros; i.e., if T > 0, then there is a t > T such that y{t) = 0. If there is a T > 0 such that y(t) 5^ 0 for t > T, then y(t) is called nonoscillatory. Note that if y is the quotient of odd integers, Eq. (1.1) takes the form
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TL;DR: In this paper, the general post-critical characteristics of a discrete structural system with independent loading parameters are studied, restricted to elastic conservative systems which satisfy certain analytical symmetry conditions and which lose their initial stability at a'symmetric special' critical point.
Abstract: The general post-critical characteristics of a discrete structural system with independent loading parameters are studied. Attention is restricted to elastic conservative systems which satisfy certain analytical symmetry conditions and which lose their initial stability at a 'symmetric special' critical point. The results are compared with Koiter's 'stable symmetric' and 'unstable symmetric' bifurcation points, and three theorems are established. A shallow circular arch under the action of a set of external loads which can be represented by two independent parameters is analysed to illustrate some aspects of the theory.
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TL;DR: In this article, the effects of transverse shear deformation and transverse normal stresses are considered in the theory of shallow shells, and the results obtained by means of this theory are accurate only for the case of shallow shell.
Abstract: Introduction. The theory of shallow shells in its original form was worked out by Donnell [1], Musthari [2], Marguerre [3] and Vlasov [4], Due to their simple construction, their equations are useful for numerical calculations of definite technical problems. Their further virtue is that they become the equations of plates when the curvature of the shell decreases to zero, but the results obtained by means of this theory are accurate enough only for the case of shallow shells. More exact equations can be obtained by the introduction of certain small improvements [5]. The difference in the derivation of the Donnell-Vlasov equations and the new ones is the following, among others. In deriving the Donnell-Vlasov equations small terms containing principal and Gaussian curvatures as factors have been neglected. In developing the new equations we need neglect only some small terms containing the derivatives of the curvatures. The improved equations are already quite satisfactory for technical purposes. The accuracy for a shell of positive and slowly varying curvature is of the order of 1 to 2 per cent. The purpose of the present paper is to introduce into the above equations the effect of transverse shear deformation and the effect of transverse normal stresses. In this way we obtain improved simple equations allowing a somewhat more exact analysis of the behavior of shells under concentrated loads, for example. Papers in which the effects of both transverse normal stress and shear deformation have been accounted for include those by Hildebrand, Iieissner and Thomas [6], Green and Zerna [7] and Reissner [8], [9]. Equations of the linear theory of shallow shells which include the effect of transverse shear deformation have been obtained by Naghdi [10]. A second work by Naghdi [11] is concerned with the formulation of stress-strain relations and appropriate boundary conditions in the theory of small deformations of thin shells. Wilkinson and Kalnins considered in [12] and [13] the case of a spherical shell loaded by a normal concentrated dynamic force while taking into account the effect of transverse shear deformations. Improved equations for the spherical shell were obtained in [12], In what follows we develop the equations of the theory of shells of slowly varying curvature, taking into account the effect of transverse shear deformation and transverse normal stress. 1. Geometry and deformation of the shell. Let us consider an isotropic shell of constant thickness and apply a system of orthogonal curvilinear coordinates (aj , a2) whose directions follow the directions of the principal curvatures of the shell surface.
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TL;DR: In this article, the dipole-moment tensor and the Rayleigh scattering properties of a spherical bowl, including the limiting case of a Helmholtz resonator, are determined.
Abstract: Harmonic functions are constructed for spherical-harmonic prescriptions of either a potential or its normal derivative on a spherical cap. The dipole-moment tensor and the Rayleigh-scattering properties of a spherical bowl, including the limiting case of a Helmholtz resonator, are determined. The results are uniformly valid with respect to the polar angle of the cap and resolve certain discrepancies in the existing literature.
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TL;DR: In this article, the implications of the classical heat conduction equation for the problem of the propagation of plane waves caused by mechanical impulse and sudden heating at the boundary of an elastic-plastic half-space are presented.
Abstract: In this paper the implications of the classical heat conduction equation for the problem of the propagation of plane waves caused by mechanical impulse and sudden heating at the boundary of an elastic-plastic half-space are presented. It is shown that the effect of dynamical thermal expansion is to reduce the jump in the stress at waves of strong discontinuity. The stress and temperature fields dealt with here are assumed to be thermodynamically uncoupled.
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TL;DR: Orthogonality and positive operators on space of almost periodic and quasi-periodic functions, using frequency-power formulas, were presented in this article, where the positive operators were defined using frequency power formulas.
Abstract: Orthogonality and positive operators on space of almost periodic and quasi-periodic functions, using frequency-power formulas
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TL;DR: In this paper, the effects of dissipation are considered empirically using the Ch6zy law and the method of solution is based on an averaging technique which assumes that the uniform solution is slowly varying.
Abstract: In this paper we consider the problem of a steady bore running downhill. The effects of dissipation are included empirically, using the Ch6zy law. The method of solution is based on an averaging technique which assumes that the uniform solution is slowly varying.