Thin Elastic Strip Research Articles

Effect of plasticity on cleavage crack growth resistance at an interface

The mixed mode toughness of an interface joining an elastic–plastic metal to a solid which does not yield plastically is studied numerically for cases where fracture occurs by atomic separation. Thus, the length scale of the fracture process is typically much smaller than the dislocation spacing on the metal side of the interface. But even though the crack growth mechanism is brittle, the interface shows crack growth resistance due to plastic flow at some distance from the crack tip. It is assumed that the crack tip does not emit dislocations. Therefore, the numerical technique employed makes use of a thin elastic strip of material along the metal side of the crack tip, while the metal outside the strip is described by continuum plasticity. Most of the computations use an infinitely long elastic strip to represent the elastic core region around the tip, but the approximation of using a long strip is tested by comparison with a few analyses for a short strip.

Thermal stress and the spacings of transform faults

Bathymetric charts are used with satellite altimeter profiles to locate major ridge‐transform intersections along five spreading ridges. The ridges are the Mid‐Atlantic Ridge, the East Pacific Rise, the Chile Rise, the Pacific‐Antarctic Rise, and the Southeast Indian Ridge. Analysis of these data show spacings between transform faults W increase linearly with spreading rate ν (W/ν = 6.28 m.y.). This linear correlation is explained by a thermoelastic model of a cooling strip of lithosphere spreading at a rate ν. The traction‐free boundaries of the thin elastic strip simulate cracks in the lithosphere at transform faults. A two‐dimensional thermoelastic solution for the in‐plane stress shows the largest stress component is tensional and parallel to the ridge. Stresses are zero at the ridge and increase as (age)½ to a maximum value at an age of W/4ν. All stress components are small for ages greater than W/ν. When the transform spacing is large compared with the spreading rate (W/ν > 100 m.y.) thermal stresses exceed the strength of the lithosphere for ages between 0 and 30 Ma. The observed maximum ratio of transform spacing to spreading rate (W/ν = 10 m.y.) results in low thermal stresses that only exceed the strength of the lithosphere for ages less than 1 Ma. Thus transform faults relieve most of the thermal stress. Model predictions also agree with earthquake studies showing that normal faults in young lithosphere have tensional axes aligned with the ridge. Moreover, oceanic intraplate earthquakes rarely occur in lithosphere older than 30 Ma as predicted by the model. These and other geophysical observations confirm Turcotte's hypothesis that transform faults are thermal contraction cracks.

Scattering of acoustic waves by a periodic array of elastic strips

The scattering of a time harmonic plane wave in a fluid incident on an infinite periodic array of thin elastic strips is studied. The strips are coplanar and perpendicular to the plane of incidence. The acoustic displacement potential in the fluid is expressed as a Floquet wave expansion. Mindlin's theory of the bending of thin plates is applied to relate the pressure difference across the elastic strips to the motion of the strips. The shear and bending components of the strip motion are expanded in trigonometric basis functions, such that the shear force and bending moment are zero at the edges of the strips. The expansion coefficients are solved from the linear equations obtained by applying the method of weighted residuals to the boundary conditions on the strips and in the gaps. Anomalous reflection is observed when the incident wave frequency coincides with the transverse resonance frequencies of the strips. Resonance frequencies are observed in close pairs, corresponding to the strip resonant modes...

On the energy flux vector for bending vibrations of a plate: PMM vol. 40, n≗ 6, 1976, pp. 1131–1135

An expression for the energy flux vector of plate bending vibrations is obtained in invariant form. The derivation of expressions for the transverse force, bending and twisting moments in an arbitrary orthogonal coordinate system and the derivation of an orthogonality type condition for normal waves being propagated in a thin elastic strip with free edges are considered as applications. In a number of cases it turns out to be useful to consider the energy flux vector in analyzing the vibrations in systems with distributed parameters. The expressions for the Umov-Poynting vector in electrodynamics and for the energy flux vector in acoustics are well-known. An analogous vector for the bending Vibrations of a plate was mentioned only in [1 – 3], This vector is used in [1] to prove a uniqueness theorem for a two-component acoustic model consisting of an ideal compressible fluid and elastic plates in contact with it. However, the expression for the energy flux in [1] (it was later cited in [2, 3] with a reference to [1]) is erroneous. An exact expression (within the framework of the applicability of the Kirchhoff equation) is found below for the energy flux vector of the bending vibrations of a plate and some applications of the formulas obtained are mentioned.

Instability of an Elastic Strip Hanging in an Airstream

The stability of a long, thin elastic strip hanging vertically in a downward flowing airstream was considered. Experimentally it was observed that the strip remained essentially motionless or at most developed a small amplitude bounded oscillation as the air speed was increased until a critical speed was attained beyond which violent oscillations of the free-ended strip occurred. A linear approximate analysis was performed for this problem which accounted for shape-induced lift and bending stiffness of the strip and which closely predicted critical air speeds for this instability. The critical air speed was studied as a function of the length, width, and thickness of the rectangular strip.

A High-Resolution Linear Optical Scanner Using a Traveling-Wave Acoustic Lens

The focusing properties of traveling acoustic waves can be employed in line-scan systems to convert a low-resolution optical beam into a high-resolution beam, both beams scanning linearly in synchronism with the acoustic wave velocity. Used in conjunction with any low-resolution scanner, for example a simple Bragg deflection cell, the scanning lens can yield at least an order of magnitude improvement in resolution, without the need for mechanical elements such as spinning mirrors. The device is inherently achromatic, and can be used just as effectively with white light as with monochromatic laser light. Experiments with dilational acoustic waves in water, and flexural waves on thin elastic strips, have confirmed the effectiveness of this novel approach, and indicate that systems with resolution capabilities up to a few thousand spots per line can be expected. Thus, immediate applications should be found in the fields of recording and reproduction of photographic images, analog and digital data recording and display, and high-speed printing.

Moiré study of anticlastic deformations of strips with tapered edges

When a thin elastic strip is bent, anticlastic deforming of the cross section takes place, and the edges move away from the center of curvature. This effect can have serious consequences in several applications. However, it has been found that the magnitudes of the deformations can be very greatly reduced if the concave edges of the bent strip are tapered. The proportions of the tapers have already been worked out theoretically so as to optimize the reduction in anticlastic deformation in any given strip bent to a known radius of curvature. The main purpose of the present paper is to verify the theoretical deformations for optimally tapered strips by experiments using the moire method.

Anticlastic action of flat sheets in bending

A theoretical analysis is made of the stresses induced in thin elastic strips which are bent (by uniform bending moments) into arcs with substantially constant radii of curvatures. Stresses determined experimentally on aluminum and steel strips agree very well with the theory.

Thin heavy elastic strips at large deflexions

The deflected forms taken up by an elastic strip of uniform thickness and constant width hanging under its own weight as (i) a cantilever, (ii) a heart-shaped loop and (iii) with position-fixed ends in the same horizontal plane, are presented, the results having been obtained using an analogue computer. Some comparisons with small-deflexion theory are drawn. Results showing deflected form are also given for a uniformly thick strip whose width tapers uniformly (triangular strip) and for a long, thin, heavy wire held at its top end and deflected by a uniform horizontal wind pressure.