## Vol. 10, No. 3 (1954)

##### 1954-02-05

###### CONTENT

1954, 25 (3): 171-186.
doi: 10.7498/aps.10.171

Abstract +

A general theory on the realistic stress space of solids was formulated in a previous paper. In this paper, the bell stress spaces of several metals are compared, the concept of "efficiency of plastic deformation" is introduced and formulated, and the locus of deformation is discussed in connection with the theory of bell stress space. The main concepts of this paper are:Because the concept of strength of solids is associated with the stress state, it is difficult to bring out the concrete meaning of strength by a brief definition. Inasmuch as the volume of the closed stress space is a complete and concrete measure of the fracture strength and the limit of strain-hardened elastic strength in all stress states, we are inclined to define this volume as strength. This is not just a matter of definition; the important point is that the size and shape of the closed space actually reflect the physical and mechanical aspects of strength, and it gives one a clear impression about what is meant by strength.The concept of "efficiency of plastic deformation" arises from the fact that one may raise the internal potential energy of a solid infinitely without causing plastic deformation if the stress state is not favourable. This efficiency is the ratio of plastic distortion energy to the sum of elastic and plastic strain energies. It may be formulated, by simple arguments, as a function of octahedral shear strain and a stress state parameter c=θ/τ, where θ is the average normal stress and τ the octahedral shear stress. It increases with increasing strain and decreases with increasing hydrostatic stress, and it is actually measured by the length and the direction cotangent (c) of the position vector with respect to the hydrostatic axis.A process of deformation can hardly be well understood without knowing its locus of deformation in relation to the limiting surfaces of the stress space. By the theory of bell stress space and experimental measurements, such locus can be located. A locus for strip rolling is presented. It is interesting to note that there is a natural tendency for continuous processes of deformation to turn towards the [111] direction in order to make the total strain energy of the system a minimum.

1954, 25 (3): 187-208.
doi: 10.7498/aps.10.187

Abstract +

The anomalous sound absorption in water fog at low audible frequencies observed by Knudsen, Wilson, and Anderson has been further investigated by the method of standing waves.Measurements were made in the frequency range of 25-250 c.p.s. for artificial fogs of several different concentrations and average droplet sizes. Artificial fogs were produced by a specially designed water sprayer and introduced into the measuring tube. The fog inside the main tube was almost turbulent-free and both the droplet size and concentration were made controllable. In order to sample the droplet size, fog droplets were allowed to settle by gravity cn a horizontal vaseline-coated glass disk which was partly inserted in the tube. This disk rotated at a constant speed while photomicrographs were taken with a 16mm. motion picture camera synchronized with an Edgerton's strobolux. The fog concentration was determined by measuring the total liquid water content in a known volume of the foggy air. The fogs thus produced had average radii of 5-9 microns and a concentration 5-14×103/c.c. These measurements were made simultaneously with the acoustic measurement. Standing wave pressure variations along the tube were registered by a logarithmic level recorder, both with and without the fog. The observed attenuation coefficients were computed from the observed first pressure minimum and the first pressure maximum of the standing wave. The maximum attenuation coefficients thus computed varied from 8.8db/sec to 20.8db/sec. The positions of maximum absorption were found to vary from 35 to 50 c.p.s.The theoretical values based on both Oswatitsch's and the modified theory proposed by the present autheor are compared with the experimental values. The experimentally obtained attenuation coefficients due to evaporation and condensation processes are found to be higher than those computed by both theories throughout the lower frequency range but in better agreement with the modified theory than with that of Oswatitsch.

1954, 25 (3): 209-238.
doi: 10.7498/aps.10.209

Abstract +

Equations for the large deflection of thin plate established by Th. von Karman has been well known for many years. But so far there have been only a few iproblems studied with numerical certainty. S. Way was the first to apply these equations to solve the problem of a clamped plate under uniform pressure by the method of power series. After this, S. Levy found the solution of the simply supported rectangular plate under uniform load by the method of double trigonometric series. Both methods are too labourious to be applicable to other more important cases. Lately, Chien Wei-zang treated Way's problem again by means of the perturbation method and obtained excellent results. By the method as given by Chien Wei-zang, Yeh Kai-yuan worked out the problem of circular plate with a central hole under central concentrated load.In this paper, more results are given for various circular plates under various edge conditions. These include uniformly loaded circular plate under various edge conditions (section 2) and central concentrated loaded circular plate under various edge conditions (section 3). Such edge conditions are: (1) simply supported, (2) simply hinged, (3) rigidly clamped, (4) clamped but free to slip, (5) edge clamped but with possible slipping in horizontal direction, (6) edge simply supported but elastically fastened, and (7) edge clamped in elastic wall.All these results are presented in such a form that direct application in design problem is possible. In particular cases, under edge conditions (1) to (4), as σ=0.3, design formulae and curves for central deflection, radial tensile stress and radial bending stress are presented.

1954, 25 (3): 239-258.
doi: 10.7498/aps.10.239

Abstract +

In a previous paper [1], we have obtained a general solution of three dimensional problem of the theory of elasticity for a transversely isotropic body. This general solution is applied in the present paper to the problem of equilibrium of a transversely isotropic half space. The following six problems are treated: 1) half space under given surface load, 2) half space under given surface displacement, 3) half space under given normal surface load and tangential surface displacement, 4) half space under given tangential surface load and normal surface displacement, 5) the contact problem of a rigid stamp upon a half space, 6) the problem of bending of thin plate resting upon a half space. It is found that, if, in problems 5) and 6), fricdonal force between stamp or plate and half space may be neglected, there exist simple relations between solutions for a transversely isotropic half space and corresponding solutions for an isotropic half space.

1954, 25 (3): 259-290.
doi: 10.7498/aps.10.259

Abstract +

In this paper, some general variational principles in the theory of elasticity and the theory of plasticity are established. Consider an elastic body in equilibrium with small displacement. By regarding u, v, w, ex, ey, ez, yyz, yxz, yxy, σx,σy, σz,τyz,τxz,τxy as fifteen independent functions, and letting their variations be free from any restriction, we establish two variational principles, called the principle of generalized complementary energy and the principle of generalized potential energy. Each principle is equivalent to the four sets o?fundamental equations of the theory of elasticity, namely, the equations of equilibrium, the stress strain relations, the strain displacement relations and the appropriate boundary conditions. Special cases of these principles are examined. These principles are next expressed in other forms, where u, v, w, σx,σy, σz,τyz,τxz,τxy are regarded as nine independent functions with their variations free from any restrictions. Next we consider the bending of a thin elastic plate with supported edges under large deflection. By regarding Mx, My, Mxy, Nx, Ny, Nxy, u, v, w as nine independent functions with the restriction that w should vanish along the contour of the plate, we establish a variational principle, called the principle of generalized potential energy, which is equivalent to the three sets of fundamental equations in the theory of bending of thin plate, namely, the equations of equilibrium, the displacement stress relations (strain stress relations) and the appropriate boundary conditions. This principle is next expressed in another form which is more convenient for application. As an illustration, von Kármán's equations for the large deflection of thin plate are derived from this principle. In von Kármán's equations, one unknown is the deflection and the other unknown is the membrane stress function. Therefore it is impossible to derive von Karman's equations either from the principle of minimum potential energy or from the principle of complementary energy. Finally we consider the equilibrium of a plastic body with small displacement. In the case of the deformation type of stress strain relations, we establish two variational principles, each of which is equivalent to the equations of equilibrium, a certain type of stress strain relations and the appropriate boundary conditions. In these variational principles, u, v, w and their variations are free from any restriction, and σx,σy, σz,τyz,τxz,τxy and their variations satisfy a certain yield condition. In the case of the flow type of stress strain relations, we get two similar variational principles, in which u, v, w and their variations are free from any restriction, σx,σy, σz, τyz,τxz,τxy and their variations satisfy a certain yield condition and σx,σy, σz, τyz,τxz,τxy have no variations.

1954, 25 (3): 291-302.
doi: 10.7498/aps.10.291

Abstract +

Two types of under-water quartz transducers, namely, the mosaic type & the array type, were designed and manufactured, particularly paying attention to the points of cementing of crystals and supporting of vibrating bodies, with a view to reducing the equivalent resistance r of the transducers when they are unloaded. A transducer of the mosaic type with an area of 609 mm2, unsupported, measured 60 k Ω for its r, while one of the array type with an area of 259 mm2, 15 k Ω. Finally, discussions were made on the comparison of these two types of transducers with the conclusion that a transducer of the array type possesses more merits than that of the mosaic type. The ways for further improvements were considered.