Dynamic photoelastic studies of fracture

Dynamic characterization of brittle fracture is possible by relating the instantaneous stress-intensity factorK(t) to the velocity of propagation of the crack. High-speed photographic systems are employed with photoelastic methods to obtain a sequence of isochromatic-fringe patterns representing the state of stress associated with the propagating crack. Methods for determiningK(t) from these isochromatic patterns are reviewed.     

A photoelastic study of crack repair

Birefringent coatings have been employed to study the effectiveness of an adhesively bonded repair of a center-cracked tension panel. The repair was one sided, with photoelastic coatings applied to the opposite side. Photoelastic coatings were also applied over the patch. Analysis methods are presented to permit the stress intensity factor to be determined from the isochromatic fringe patterns recorded from both continuous andX- andY-edged coatings. The results showed that the one-sided adhesively bonded patch reduced the stress intensity factor; however, the repair did not markedly change the character of stress distributions. Fringe loops formed near the crack tips for both the cracked and repaired tension panels. The primary difference was in the size of the loops. The reduction inK _I due to repair was smaller than anticipated, but even small improvements in ΔK _I markedly enhance the life of a repaired panel. The Paris power law is used to show the relation between the reduction in ΔK _I and the improvement in the crack growth rateda/dN. Fringe patterns from the birefringent coatings applied to the patch provided a means not only to investigate the stresses in the patch but also to detect the initiation of the local debonding of the adhesive in the neighborhood of the crack. The birefringent coating on the patch is an approach for producing an optically “smart” repair.     

A photoelastic study of contact between a cylinder and a half-space

Three-dimensional photoelasticity was employed to study a cylinder in contact with a half-space. Both bodies were modeled in epoxy resin. Three loading cases were examined, namely, the cylinder lying on its side subject to a load normal to the plane, the cylinder on its side subject to both normal and tangential loads and the cylinder standing on its end and subject to a normal compressive load, i.e., as a circular punch. The cylinders and the half-space, which was represented by a large block, were stress frozen with a known coefficient of friction and using relatively small loads so that the strain levels were low. After slicing the cylinders, which resulted in lower fringe orders than could be readily analyzed manually, an automated system based on phase stepping was used to record and process the data. Distributions of maximum shear stress and Cartesian shear stress were obtained for a large area of the slice. Stress separation was performed, using the shear difference method, to obtain the Cartesian stress components in the plane of symmetry of the half-space. These results provide confirmation, by experiment, of the theoretical and numerical models of this type of contact obtained by other investigators.     

Determination of fracture toughness by photoelastic coating

The photoelastic-coating method was applied to the determination of fracture toughness in aluminum plates. The specimens were plates with a central transverse crack. Determinations were made first by the compliance method. The specimens were loaded statically to failure. The opening displacement across the crack was measured with a clip gage. In using this photoelastic-coating method, the stress-intensity factor was obtained in terms of the radius and fringe order of various isochromatic fringe loops using an extrapolation law. An apparent stress-intensity factor was obtained from several isochromatic fringe patterns away from the crack tip and then extrapolated to the crack tip to determine the true value. Results obtained by the photoelastic-coating method are higher than those obtained by the compliance method for all loads, due to the bluntness of the crack tip in the first set of specimens. Theoretical predictions fall between the compliance method and photoelastic-coating results. Paper was presented at V International Congress on Experimental Mechanics held in Montreal, Quebec, Canada on June 10–15, 1984.     

A system of modified epoxies for dynamic photoelastic studies of fracture

In order to develop a moderately tough transparent polymer, a three-component modified epoxy was investigated. The polymer system included an epichlorohydrin/bisphenol A epoxy, a polyoxypropyleneamine curing agent and a curing accelerator. Twelve different compositions were prepared and evaluated in a series of static and dynamic tests to determine the material properties important in photoelastic studies of fracture. Static tests showed that the critical strain-energy release rate could be varied from 1.4 to 4.1 lb/in. by changing the constituents in the blending of the epoxy. These results forG _Ic indicate that the modified epoxies are considerably tougher than Homalite 100 (G _Ic =0.33 lb/in.) which is commonly employed as a model material in dynamic photoelastic studies. Dynamic photoelastic tests were conducted with half-plane models in order to determine the dilatational- and distortional-wave velocities,c ₁ andc ₂, as well as dynamic values of the modulus of elasticityE and Poisson's ratio ν at loading times of the order of 10^?5 s. These results indicated that the dynamic modulus of some of the modified epoxies was significantly higher than the static modulus indicating that these polymers are rate sensitive. One of the epoxy materials, Blend No. 3 withG _Ic =2.65 lb/in. (464J/m²), was calibrated dynamically. The material fringe value changed nearly linearly as a logarithmic function of time with an increase of about 100 percent as the loading time decreased from 10⁴ to 10^?4 s. This large variation inf _σ implies that the calibration constant must be adjusted when interpreting dynamic fringe patterns. Two of the epoxy compositions were also characterized in a number of fracture experiments involving crack propagation at velocities ranging from arrest conditions to terminal velocity where branching initiates. Dynamic isochromatic-fringe loops were photographed with a Cranz-Schardin multiple-spark camera. The fringe loops were analyzed to give the instantaneous stress-intensity factorK as a function of crack velocity å. The å vs.K curves appear to be invertedL shapes; however, there appears to be a double branch on the vertical part of theL. Also a slightly higherK is required for accelerating cracks and a lowerK for decelerating cracks. Further investigation is required to identify the basic mechanism involved in this fracture behavior.     

A photoelastic study of friction at multipoint contacts

A new method of measuring the normal and sliding loads associated with multiple-point contact is introduced. A multiple-point contact is modeled with a steel die with a profile that simulates a rough surface. A very large scale factor is used in modeling this surface. The steel die is placed in contact with a photoelastic model of a half plane and is subjected to a normal load. This normal load is partitioned over the multiple points of contact producing an isochromatic fringe pattern that describes the stress distribution in the local neighborhood of the contact points. A sliding load is then imposed on the model which destroys the symmetry of this fringe pattern. The fringe data in this pattern are sufficient to determine the local loadsP _i andQ _i and the local coefficient of frictionf _i =Q _i /P _i at each contact point. An overdeterministic method is introduced which gives the solution forP _i ,Q _i andf _i using many data points taken from the isochromatic pattern in the local neighborhood of the contacts.Paper was presented at the 1991 SEM Spring Conference on Experimental Mechanics held in Milwaukee, WJ on June 9–13.     

Two-dimensional photoelastic study of turbine-shell flanges

Results of a two-dimensional photoelastic study of turbine-shell flanges are reported. Equations of transition from model to prototype are given. A special loading jig for applying a uniform pressure to the boundary of a plane model is described. Pertinent aspects of the experimental technique are summarized.     

A dynamic photoelastic study of stress-wave propagation through an inclusion

A photoelastic study of the elastodynamic-stress fields around a circular, elastic inclusion (Solithane 113) embedded in an elastic plate (Hysol 4485) is presented. The edge of the plate was loaded by an explosive charge, which produced a plane, compressional stress wave of triangular shape. Isochromatic-fringe patterns were obtained, which give the maximum shear stresses, both inside the inclusion and in the surrounding medium. The principal stresses on the axis of symmetry were determined through the use of the oblique-incidence method. It was found that small tensile stresses are generated at the interface on the shadow side of the inclusion. The focusing effect inside the inclusion predicted by ray theory was not observed. Finally, the shape of the wavefront as the wave passes the inclusion was determined.     

A photoelastic and finite-element investigation of a nonsymmetrical plug-hatch configuration

A photoelastic investigation to evaluate the reliability of the finite-element method of shell analysis was carried out for models with nonsymmetric plug-hatch configurations. The stress distributions in the spherical-shell models with combined hatch and anchor-boss configurations were determined experimentally and compared with those predicted through the use of a finite-element computer program.     

Three-dimensional photoelastic study of stresses in rack gears

A three-dimensional photoelastic analysis using the stress-freezing and slicing techniques was employed in studying the stress distribution and concentration across the thickness of a thick rack-gear tooth. Full-size photoelastic models of about 25-mm thickness, having various teeth parameters were machined from PLM-4B plates. A special mounting fixture was designed to hold the models in the loading frame and a knife-edged line load was used in loading the models. In the analysis, the method of oblique incidence was adopted for the separation of the principal stresses. The intension was to determine the magnitude and location of the maximum stress at the tensile fillet and to establish the stress-concentration factors for various geometric tooth configurations.From this study, it was concluded that the fillet stresses depend largely upon the pressure angle, fillet radius and the position of the load. Maximum values of the tensile fillet stresses occur at the middle plane of the tooth thickness. Furthermore, the value of the stress-concentration factor increases with the increase of the tooth thickness. In general, this investigation gave values of stresses much higher than those values calculated by the simple-flexure theory.     

A photoelastic determination of mixed-mode stress-intensity factors

A new evaluation method of the isochromatic-fringe pattern for the determination of mixed-mode stress-intensity factors (SIF) was developed. The method bypasses the error-prone measurements on the isochromatic patterns near to the crack tip and uses data from the far region. Suitable extrapolation laws were given for transferring the far-from-the-crack-tip data to the near region. Then, the well-known Irwin's formulas can be used for the determination of mixed-mode SIFs. Convenient formulas facilitating the calculation of SIFs were established. Experimental evidence corroborated the results obtained by the new method.     

A dynamic photoelastic analysis of dynamic-tear-test specimen

Dynamic photoelasticity was used to analyze the transient response of dynamic-tear-test (DTT) specimens, .0889×.400m (3.5×15 in.) in size, machined from .0095m (3/8 in.)-thick Homalite-100 plates. Dynamic-stress-intensity factors, dynamic-energy release rates, and crack velocities in ten specimens were determined. Dynamic tear energies were obtained by integrating the area under the dynamic-energy release-rate curves. The average dynamic-energy release rates which were obtained by dividing the dynamic tear energies by the total length of the crack paths, were found to be approximately equal to the critical strain-energy release rate of Homalite-100. Results of the drop-weight-hammer impact vs. crack-initiation experiments showed that the crack initiated after the first buildup of impact forces in the hammer, thus indicating possible ambiguity in using a specified location on the hammer-force curve to derive a comparable static-fracture load for the DTT specimen.     

Influence of late-breaking ligaments on crack propagation in compact specimens—A photoelastic study

Dynamic photoelasticity and high-speed photography were utilized to study the influence of tough ligaments on crack propagation in compact specimens. The data obtained during the fracture experiments were analyzed to obtain crack velocity, instantaneous stress-intensity factor and the energy losses away from the crack tip. The results showed that the ligaments can behave as crack arrestors or decelerators. The crack if not arrested at the ligament slowly tunnels in between the ligaments without immediately rupturing them. The ligaments also tend to increase damping losses in the specimens.     

Stress separation in the photoelastic study of centrifugal stresses

This technical note refers to the problem of stress separation in the photoelastic analysis of plane models under centrifugal stresses. Two methods are described in order to determine the sum of principal stresses. These methods, which are based on the compatibility equation, reduce the determination of the sum of principal stresses to the solution of a Laplace's or Poisson's equations. As an example of application, the separation of stresses in a rotating disk with two eccentric holes is shown and comparison with the stresses obtained by using the shear-difference method is made.     

Characterization of a new luminescent photoelastic coating

The luminescent photoelastic coating (LPC) technique measures the full-field shear strain and its principal direction on the surface of complex three-dimensional components. The measured optical strain response is also dependent on the coating thickness. Achieving uniform coating thickness is difficult, and thus requires thickness correction for accurate quantitative strain measurements. The original formulations of LPC employed a dual-layer coating containing luminescent dyes to transmit both strain and thickness information. This paper will document (theory and experiment) a new strategy: a single-layer coating that incorporates both a luminescent dye and an absorption dye. Dependent on the concentration of the absorption and luminescent dyes, the solution is sprayed onto the object of interest to a minimum threshold thickness that corresponds to a predefined penetration depth. Advantages of the single-layer coating are the elimination of thickness dependency, the elimination of compliance and adhesion issues between multiple layers, simpler data acquisition and post-processing methods, and easier and faster coating preparation and application.     

含有滑动摩擦的两体碰撞中的动能损失

讨论的是一般运动刚体在含有滑动摩擦的情况下的两体碰撞．首先根据欧勒动力 学方程和质心运动定理导出了在碰撞的两个阶段二碰撞点沿公法线和公切线的相对速度的变 化量, 其次利用动量定理导出了系统动能的损失与二碰撞点速度的关系, 最后根据这两个结 论导出了系统动能损失的计算公式, 并给出所得结论成立的条件．     

A photoelastic material with variable modulus of elasticity

Instructions are given in this paper to manufacture a photoelastic material (polyurethane rubber) exhibiting a linearly variable modulus of elasticity. In the examples given, the range ofE is of the order of 1 to 2.5. Illustrations of applications in stress analysis are included. The new material should be particularly useful in the solution of soil-mechanics problems and it can be used for its photoelastic properties, or as base for grid and moiré analysis.     

Three-dimensional photoelastic analysis of a fiber-reinforced composite model

A three-dimensional photoelastic analysis using the “stress-freezing” technique was conducted to determine the stress distributions in the matrix of a unidirectionally fiber-reinforced composite model subjected to matrix shrinkage and normal transverse loading. The model, consisting of a square array of polycarbonate rods in an epoxy matrix, simulated a boron-filament-reinforced plastic composite with a fiber-volume fraction of 0.50 at the critical temperature of the matrix epoxy. The effects of matrix shrinkage were separated from those of external loading by analyzing two identical models, one loaded and the other unloaded. The Lamé-Maxwell equations of equilibrium were used to separate stresses along axes of symmetry on interior transverse slices. Axial stress components were obtained by subslicing. Results are presented in dimensionless form by dividing the stresses by the average stress through the section. A comparison with theoretical results for a boron-epoxy composite shows excellent agreement, although Poisson's ratio of the model matrix is appreciably different from that of the prototype (0.5 compared to 0.35). One significant result was that the maximum stress occurs in the middle of the matrix section between fibers which is at variance with the theoretical prediction of maximum stress at the interface. Stress-concentration factors vary from 1.80 at the interface to 2.0 at the midpoint of the matrix section between fibers.