Showing papers on "Displacement field published in 1968"

Some theorems in classical elastodynamics

Abstract: This investigation is concerned with various fundamental aspects of the linearized dynamical theory for mechanically homogeneous and isotropic elastic solids. First, the uniqueness and reciprocal theorems of dynamic elasticity are extended to unbounded domains with the aid of a generalized energy identity and a lemma on the prolonged quiescence of the far field, which are established for this purpose. Next, the basic singular solutions of elastodynamics are studied and used to generate systematically Love's integral identity for the displacement field, as well as an associated identity for the field of stress. These results, in conjunction with suitably defined Green's functions, are applied to the construction of integral representations for the solution of the first and second boundary-initial value problem. Finally, a uniqueness theorem for dynamic concentrated-load problems is obtained.

Multipolar elastic fields in a layered half space

Abstract: Hansen9s expansion is used to derive integral expressions for the displacement field due to a localized buried source of the m th order in a layered half space. The dipolar case ( m ≦ 2) is worked out in detail for arbitrary source-depth in the layer and in the substratum. A new type of representation of the source is used which gives the final results in a concise form. Explicit expressions for the displacements at the free surface are obtained for a center of explosion, a vertical strike-slip fault and a vertical dip-slip fault. The results for a horizontal thrust are found to be the same as for a vertical dip-slip fault. The relations between the Galerkin vector and the biharmonic eigenvectors are clarified. It is shown that the Galerkin-Boussinesq solution for the elastic half space cannot be extended to structures of higher complexity, except for a few simple sources. On the other hand, the Hansen Solution is valid for a wide class of sources and structures. Both dynamic and static regimes are considered.

Frame Analysis including Change of Geometry

Abstract: A method is developed of incorporating the effects of change of geometry in the analysis of rigidly jointed plane frames when using the displacement method. Where members are split into a moderate number of submembers the effects of change of geometry can be sufficiently represented by means of a nonlinear displacement transformation alone. Accurate results can be obtained with few submembers if the approximate nonlinear behavior of the submembers are taken into account. The method remains valid for cases of extreme deformation as long as the exact mathematical form of the displacement transformation is used.

External elliptical crack in elastic solid

Abstract: Potential functions are developed for both symmetric and antisymmetric three-dimensional problems of an infinite elastic solid containing a flat crack covering the outside of an ellipse. The knowledge of these functions permits an examination of the stress and displacement fields everywhere in the cracked solid as well as in a toroidal region around the crack border. Stress-intensity factors kj (j = 1, 2, 3) corresponding to the three basic modes of fracture are obtained. The results of this paper, coupled with those found previously by the authors for the problem of the internal elliptical crack, are essential in making approximate estimate of stress-intensity values for solids with arbitrarily-shaped planar cracks.

Atomic Displacements around Dislocation Loops

Abstract: The displacement field around a ``penny‐shaped'' inclusion or void in an isotropic elastic medium has been computed by the methods outlined by Sneddon. The boundary conditions for such a loop are that in the plane of the loop z = 0, the displacement is = ±| b |/2 for r less than the loop radius c, and zero outside the loop, and that the shear stress in the r direction across this plane is zero everywhere. We assume that the actual atomic displacements are represented by this displacement field beyond a few atomic diameters of the loop. The solutions for the r and z components of the displacment field are given in series from, and contour plots for these components have been generated by a CDC‐6600 computer for Poisson's ratio, σ = 0.30. At large distances from the loop the series expressions for the displacement field converge to that of a single doublet force without moment oriented in the z direction plus a point singularity at the center of the loop. The strengths of these two singularities are expressed in terms of the dimensions of the loop, | b | and c, and σ. From these strengths the lattice dilatations including image effects have been calculated for a finite medium with a uniform distribution of loops. We obtain the results that the only dilatation is normal to the plane of the loop, and that the strain is simply the fraction of atoms contained in the loops.

Use of Fourier series in the finite element method.

Abstract: P IAN1'2 has demonstrated the use of the energy method in deriving element stiffness matrices in connection with the displacement method of structural analysis. His procedure is based on the representation of an element displacement function in terms of m undetermined coefficients where the number m may be larger than the number of generalized displacements n.2 When m is larger than n, it is possible to satisfy not only stress equilibrium and boundary displacement continuity but also to maintain slope continuity along the normal directions of the element edges. Having the element displacement function, the total energy of the element! may be evaluated, and then the condition of the minimum potential energy yields the stiffness matrix. The aforementioned procedure is illustrated in Ref. 2 by determining the stiffness matrix for a rectangular plate under plane stress conditions. The displacement function is chosen in the form of a power series as follows : u = ai +

On the response of shallow thin shells to random excitations.

Abstract: : Statistical parameters of the displacement field of a linearly elastic shell are formulated in terms of the corresponding parameters of the loads. The example of a circular, cylindrical panel is considered, and for the simply supported case the second order correlation function of the radial displacement under a purely random pressure is worked out in detail. Several interesting features of the response are discussed. (Author)

Polarization‐Dependent Rayleigh Scattering in a Coarse‐Grained Ferroelectric Ceramic

Abstract: Visible light intensity transmitted by a polarized ferroelectric crystallite of diameter ≥2 μ has been observed to decrease markedly when the direction of polarization Ps is shifted from parallel to perpendicular to the direction of incidence. This is explained by regarding the 180° domain structure as constituting a displacement field, in which ions are shifted relative to a cubic reference phase. The Fourier wavevectors of this field shift with Ps, causing an increase in backscattering when Ps is normal to the light. This property is demonstrated for tetragonal, orthorhombic, and rhombohedral phases of a perovskite, and in all cases scattered intensity is proportional to Ps4. Selection rules affecting depolarization of the light are derived from the polarizability theory of Raman processes. This theory supposes that the incident radiation polarizes the atoms, which then emit dipole radiation whose amplitude is proportional to the polarizability α. The latter depends on the displacement field whose symme...

Axially Symmetric Waves in an Elastic Solid of Revolution

Abstract: In this paper, the problem of free harmonic waves in an isotropic elastic solid of revolution is discussed. As in the classical theory, the displacement field has been expressed in terms of a dilatational scalar potential and an equivoluminal vector potential. A method indicated by Stratton in discussing electromagnetic waves has been utilized to show that if the displacement field has the same axis of symmetry as the axis of revolution of the body, these equivoluminal modes can be expressed in terms of two potential functions—one of which corresponds to the pure torsional modes. It is found that there are two distinct families of modes of vibrations—one is torsional and the other is torsion free. As illustration, complete analysis is presented for free axisymmetric harmonic waves in a prolate spheroid. It has also been shown that, as limiting cases, solutions may be derived for waves in a sphere and in a right circular cylinder.

Dual Analysis of a Multiweb Sweptback Wing Model

Abstract: THE idea of a dual analysis in finite elements of a given structure was put forward in Ref. 4. The first analysis should be of the displacement type, using conforming displacement models of the finite elements. This results in a continuous, piecewise differcntiable displacement field in the whole structure, for which linear elasticity theory predicts lower bounds to the local static influence coefficients. The second analysis should be based on equilibrium models of the finite elements. The stress field within the structure is then continuously transmitted across the interfaces and satisfies detailed equilibrium conditions in the interior of each element. This property furnishes upper bounds to the influence coefficients.