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A transparent material with variable modulus of elasticity has been manufactured. It exhibits good photoelastic properties, and can also be strain analyzed by printing gratings on its surface and producing the moire´ effect. The new material is used to stress and strain analyze a disk of variable E, subjected to diametral compression. The discussion of the results obtained suggests that the stress distribution in the disk of variable E is practically the same as the stress distribution in the homogeneous disk. It also indicates that the strain fields in both cases are very different, but that it is possible, approximately, to obtain the strain field from the stress field for an homogeneous material using the value of E at every point, and Hooke’s law.

The object of this investigation was to develop a method utilizing photoelasticity for determining transient stress and strain distributions in two-dimensional problems. Numerous investigators have approached this problem using the common photoelastic materials which have a relatively high modulus of elasticity and correspondingly high velocity of wave propagation. However, many difficulties were encountered in photographically recording data, and only a few reliable stress patterns were obtained. In order to avoid these difficulties, a material with a low modulus of elasticity and a correspondingly low velocity of wave propagation was developed. As a result the 8-mm. Fastax camera is capable of recording precise fringe patterns. The material used was a member of the epoxy-resin family, modified to give the desired properties. Both its static and dynamic properties were determined as accurately as possible. It was found that the strain-fringe value of the material is approximately constant, but the modulus of elasticity and stress-fringe value are different for static and dynamic loadings. Preliminary studies were conducted to develop the method using a circular disk under a radially applied concentrated impact load as the model. A simply supported beam under central impact was then studied, and deflection curves obtained were compared with curves theoretically predicted by Saint Venant and Flamant. The comparisons showed good agreement. An analysis of the formation of the fringe pattern for various times after impact also was made.

The paper treats the stress distribution in a metal tube which is subject to a very high radial temperature variation and pressure. The radial temperature distribution across the tube wall and the variations of the modulus of elasticity and the coefficient of thermal expansion are obtained from experimental data, and all these effects of temperature are taken into account in the calculations. The fundamental equations in the case of plane strain and plane stress can be formulated as the nonhomogeneous Whittaker differential equations. The corresponding solutions are obtained by the method of variation of parameters and in terms of Kummer series. An example is shown, and the stress distribution across the wall is given. For comparison, the stress distribution of the case of constant modulus of elasticity and coefficient of expansion is included.

The case of an elastic disturbance propagating in a nonhomogeneous bar of finite length is solved by using the principle of virtual work. The nonhomogeneity is prescribed as a continuously varying modulus of elasticity with position in the bar. The density is assumed constant. Numerical results are presented for a finite-length pressure pulse in a free-free bar giving a comparison between the homogeneous and nonhomogeneous solutions.

A theory is presented to predict the deterioration in strength for a structure composed of a brittle-state porous material which has been subjected to a damaging tension field. Based on the theory, a degradation factor can be formulated as a means of evaluating the residual strength numerically. A proposed material function for a brittle-state porous medium was evaluated experimentally and was found to be satisfactory for an alumina material. The modulus of elasticity at room temperature of the same material has been found to increase with the density ratio in semi-logarithmic form.

This paper summarizes the optical and physical properties of the photoelastic model material Paraplex P-43 over the temperature range from room temperature to −40 F. Descriptions are presented of techniques and equipment developed to obtain the modulus of elasticity, the material fringe value, and the thermal-expansion coefficient as a function of temperature. Experimental investigations were conducted into the plane-stress problems of a disk contracting upon an elastic inclusion and the transient thermal-stress field produced by a temperature differential suddenly applied to the upper edge of a long beam. The data are correlated with theory using the material properties obtained in the calibration phase. Also included are photographic results of an exploratory investigation of the thermal-shock phenomenon produced by the sudden application of a temperature differential upon plastic beams of various length-depth ratios.

The method and equipment developed and used by the authors for measuring the damping of materials are described. A tuning-fork specimen is set into vibration by jerking a spreader from the gap between the ends of the tines. The damping is expressed in terms of the logarithmic decrement of the decaying vibration, which is measured and recorded by means of a magnetic oscillograph, amplifiers, and a resistance-type electric strain gage cemented to the specimen. The results include (1) the damping of a number of materials during flexural vibration at approximately 1000 cycles per sec, at maximum bending stresses up to 40,000 psi, and at temperatures up to 1400 F; (2) the variation in modulus of elasticity with temperature, as determined from the specimen vibration frequencies.

The purpose of the investigation described in this paper was to obtain information concerning the dynamic properties of rubber bonded to metal. Two properties of rubber were measured (a) the internal damping and (b) the dynamic modulus of elasticity. Two types of specimens were tested (a) rubber cylinders bonded to steel cylinders at the ends and stressed in compression and (b) specimens of rubber bonded to steel and stressed in shear. All specimens were of the same stock, 5140-V-4, manufactured by the U. S. Rubber Company. The hardness, as measured by the durometer, varied from 32 to 40. In the process of bonding to the steel, a 1/32-in. layer of 60-durometer stock was added adjacent to each piece of steel. This represents standard practice of the U. S. Rubber Company in bonding soft stock to metal. All specimens were cured for 30 min at 279 F.

The elementary theory of the bending of a beam on an elastic foundation is based on the assumption that the beam is resting on a continuously distributed set of springs the stiffness of which is defined by a “modulus of the foundation” k. Very seldom, however, does it happen that the foundation is actually constituted this way. Generally, the foundation is an elastic continuum characterized by two elastic constants, a modulus of elasticity E, and a Poisson ratio ν. The problem of the bending of a beam resting on such a foundation has been approached already by various authors. The author attempts to give in this paper a more exact solution of one aspect of this problem, i.e., the case of an infinite beam under a concentrated load. A notable difference exists between the results obtained from the assumptions of a two-dimensional foundation and of a three-dimensional foundation. Bending-moment and deflection curves for the two-dimensional case are shown in Figs. 4 and 5. A value of the modulus k is given for both cases by which the elementary theory can be used and leads to results which are fairly acceptable. These values depend on the stiffness of the beam and on the elasticity of the foundation.

A general method was found by which the problem of the rotating disk with any arbitrary profile could be solved, including the effect of plastic flow and of variable temperature, and including the change with temperature of modulus of elasticity, coefficient of thermal expansion, and allowable stress. The solution requires for its application to a specific disk only the elementary arithmetic involved in completion of a tabular form sheet. Two applications of the method are made. For an arbitrary disk profile, an integral equation was found which converges rapidly to the radial stress distribution in a series of successive substitutions. For an arbitrary choice of radial stress, the necessary disk profile can be found in one calculation. Appendix 1 gives an example of the use of the method for the design of a partially plastic disk with a central hole.

The objective of the program discussed in this paper was to develop a method, using photoelasticity and low-modulus materials, for studying dynamic stress distributions. Early in the program a number of low-modulus materials were studied and Hysol 8705 (a urethane-rubber compound) was selected as the most promising. A complete study of its mechanical and optical properties was made under static and dynamic loadings. It was established, that Poisson’s ratio ν is independent of rate of loading, the stress fringe value fσ is independent of rate of loading for strain rates greater than 8 in/in/ sec, and both the modulus of elasticity E and the strain fringe value fϵ were dependent oil the rate of loading. The specific energy loss for the material was about 10 per cent for the stress ranges associated with photoelastic determinations. Experimental observations of photoelastic fringe patterns in a rectangular strut subjected to axial impact were made to illustrate the potential of the method. Three different end conditions were imposed on the unloaded end of the strut: (a) A free end normal to the axis; (b) a fixed end normal to the axis; and (c) a free end inclined 45 deg to the axis. For the cases where the ends were normal to the axis it was found that the fringes followed the same law of reflection as the law for stresses given by elementary wave theory.

In the selection of materials for machine and structural parts subjected to vibrations it is common practice to consider the fatigue strength of materials. In some cases the damping constant of the material may be considered in a qualitative manner. A theoretical analysis is given in this paper to show that the load resistance of resonant members is a function not only of the fatigue strength but also of the damping constant and modulus of elasticity of the material. Test data on various materials are presented to show that the load resistance of a vibrating member may be appreciably changed by a correct selection of the material when this new design criterion is used. Fatigue strength - damping design criteria are presented in this paper for vibrating beams and plates subjected to both concentrated and distributed resonant forces. Results are given for both external and internal damping. The theory developed may be extended for other members where the possible condition of resonant vibrations may occur. Turbine blades, airplane propeller blades, and shafts of various kinds are applications where the design criteria presented herein may be found useful.

The optical and physical properties of Paraplex P43, Castolite, and epoxy resin Hysol 6000-OP, which are potentially of interest in photothermoelastic investigations, were investigated over a temperature range from +100 to −60 F. Results on the thermal-expansion coefficient, the material fringe value, and the modulus of elasticity as functions of temperature are presented. Also evaluated were thermal properties of importance in heat conduction. Photothermoelastic figures of merit, which rate the optical sensitivity of materials in photothermoelastic applications, as well as a new method to determine this figure in a relative manner are presented.

In this paper it is shown that if the hysteresis loop for a material has a particular shape the damping can be considered adequately by multiplying the modulus of elasticity of the material by the complex number e2bi where 2b is called the complex damping factor. For small values of b it is shown that both for free and forced vibrations of a simple spring-mass system the motion in the case of complex damping is the same as in the case of viscous damping, with b = c/ccr, except that in the steady-state case the phase angles are slightly different. Also, it is shown how complex damping may be applied to cases of forced vibrations of uniform rods and beams. The greatest advantage of using complex damping, however, is in numerical calculations of forced vibrations of engine crankshafts, airplane wings, and other types of structures; and for such calculations it already has been extensively used.

A method has been derived for determining the way the spring material influences the volume of space required for a particular application for a helical spring. It is found that this volume is proportional to the material constants of the spring in the following manner V α G 7 S 11 5 where G is the torsional modulus of elasticity and S is the limit of maximum stress. Using a multiplier to give this factor a value of 1.00 for music wire, it is found that the value for stainless steel is 1.06, beryllium copper 1.34, and phosphor bronze 2.66, for the springs on the brushes of small motors where springs must operate at temperatures up to 100 C. Different fields of application requiring different limits of maximum stress will change these values but the method still remains the same.