Three-dimensional photoelastic and finite-element analysis of a propellant grain

The results of a three-dimensional photoelastic analysis of a propellant-grain model are compared with the results obtained from a three-dimensional finite-element analysis of the same model. The loading considered was differential thermal expansion between the case and the grain model. The model analyzed was the Stage I Minuteman propellant grain which consists of a six-point-star configuration with a head-end web. Stresses and strains determined from the two analyses in the star valley are compared. The excellent correlation between the stresses determined from the photoelastic analysis and the stresses calculated by the three-dimensional finite-element computer program established the validity of the computer program.     

Three-dimensional scattered-light stress analysis of discontinuous fiber-reinforced composites

This paper reports an investigation of three-dimensional stresses in models of composite materials with discontinuous fibers using the scattered-light photoelastic method. A special scattered-light polariscope with loading system was designed and constructed for this research. Two models were used in this investigation. The first was a cylindrical tension model with a single discontinuous axially aligned aluminum fiber surrounded by a polyester matrix. The second was a five-fiber cylindrical compression model with a central discontinuous fiber surrounded by four continuous ones and polyester matrix. The experimentally determined stress distribution from each model along chosen lines was presented. The stress distribution on the same lines in the single-fiber case was calculated using the finite-element method. The calculated result showed fair agreement with the experimentally determined results.     

Elastic waves in a fiber-reinforced composite

The propagation of time-harmonic elastic waves in a fiber-reinforced composite is studied. The circular fibers are assumed to be parallel to each other and randomly distributed with a statistically uniform distribution. The direction of propagation and the associated particle motion are considered to be normal to the fibers. It is shown that the average waves in the composite separate into compressional and shear types. General formulae for the complex wave number giving the phase velocity and the damping are obtained. It is shown that these formulae lead to the Hashin-Rosen expressions for the transverse bulk modulus and the lower bound for the transverse rigidity, if the correlation in the positions of the fibers can be ignored. The correlation terms, for exponential correlation, are shown to have a significant effect on the damping property of the composite, especially at high frequencies and concentrations.     

A composite three-dimensional photoelastic method

Care must be taken in preparation and testing of three-dimensional composite photoelastic models. Some problems encountered in modeling the prototype and during model testing are: model-material failure, loss of fringe pattern in slicing, inherent shrinkage response in freezing, inadequate bonding between materials, and modular ratio difficulties. The selection of the correct plastic can eliminate the first four problems, but the correct modular ratio between the matrix and the insert has to be obtained. This investigation illustrated the behavior of commercially manufactured plastics as inserts, with a matrix material of Epon 828 epoxy. The effective moduli of elasticity of these plastics are reported for pure tension and for flexure. Since the manufactured plastics produced varying results, the use of Epon 828 epoxy as an insert was investigated. The inserts were cast in tygon tubing and their curing cycle was altered from that used for the matrix material to produce a different effective modulus of elasticity. The Epon 828 inserts gave excellent results in the beams. The use of the same material for matrix and insert eliminates many of the problems associated with composite three-dimensional photoelasticity.     

Strain-field analysis acquired through correlation of X-ray radiographs of a fiber-reinforced composite laminate

The strain fields can be determined on the surface of fiber-reinforced composites by correlation analysis of X-ray radiographs of the specimen. One radiograph is taken of the specimen in an unloaded state and another radiograph of the same specimen is taken in a deformed state. Each radiograph is then imaged by a video camera and digitized using a Matrox digitization board. The displacement map is obtained from the two radiographs by dividing one image into subsets of 20 pixels by 20 pixels and using a correlation algorithm to identify the corresponding subsets in the second image. The correlation is completed between patterns of the grey-level intensities for each subset. The appropriate discrete derivatives can then be evaluated by first difference taken to form the in-plane strains. The X-ray radiographs correlation analysis strain-field method outlined above was applied to a uniaxially loaded [90, 0] _s Gl-Ep coupon. Simultaneously, an extensometer measured the average longitudinal strain over a 2.54-cm (1-in.) gage length. The error between the two methods was less than three percent of the applied strain of 1.3 percent. The same specimen was impacted and re-examined. A radical shift in the strain field was observed when the specimen was reloaded. Further investigation showed the method reliable down to 0.5-percent strain. Paper was presented at the 1986 SEM Spring Conference on Experimental Mechanics held in New Orleans, LA on June 8–13.     

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.     

Three-dimensional continuum analysis for a unidirectional composite with a broken fiber

The three-dimensional problem of a periodic unidirectional composite with a penny-shaped crack traversing one of the fibers is analyzed by the continuum equations of elasticity. The solution of the crack problem is represented by a superposition of weighted unit normal displacement jump solutions, everyone of which forms a Green’s function. The Green’s functions for the unbounded periodic composite are obtained by the combined use of the representative cell method and the higher-order theory. The representative cell method, based on the triple discrete Fourier transform, allows the reduction of the problem of an infinite domain to a problem of a finite one in the transform space. This problem is solved by the higher-order theory according to which the transformed displacement vector is expressed by a second order expansion in terms of local coordinates, in conjunction with the equilibrium equations and the relevant boundary conditions. The actual elastic field is obtained by a numerical evaluation of the inverse transform. The accuracy of the suggested approach is verified by a comparison with the exact analytical solution for a penny-shaped crack embedded in a homogeneous medium. Results for a unidirectional composite with a broken fiber are given for various fiber volume fractions and fiber-to-matrix stiffness ratios. It is shown that for certain parameter combinations the use of the average stress in the fiber, as it is employed in the framework of the shear lag approach, for the prediction of composite’s strength, leads to an over estimation. To this end, the concept of “point stress concentration factor” is introduced to characterize the strength of the composite with a broken fiber. Several generalizations of the proposed approach are offered.     

On the use of photoelastic composite models

Emphasis in the paper is placed on establishing the validity of employing photoelastic composite models to study assumptions used in composite-material analysis. Two approaches for the derivation of material constants for plane-stress layered composite materials are presented. Basic assumptions of the theories are studied. The theoretical and experimental equivalent moduli for test specimens loaded in tension and in shear are also compared.     

Rectilinear and circular analysis of a plane slice optically isolated in a three-dimensional photoelastic model

Because of the coherence of scattered light, it is possible to produce a speckle image from a plane beam of light passing through a transparent model. When two plane parallel beams of light are transmitted through the model the slice between the beams is then optically isolated. The two speckle patterns corresponding to the two beams are superposed and provide optical data relative to the slice (principal stress directions, birefrengence), the data being collected on high contrast photographic plates or by optical filtering to obtain the square of the contrast. The isoclinic and isochromatic fringes are shown to exist. The concepts of rectilinear or circular analysis are extended to the observation of a plane slice in a three-dimensional model without freezing or cutting the model.     

Snap-through of unsymmetric fiber-reinforced composite laminates

An approximate theory based on assumed strain and displacement fields, the Rayleigh–Ritz technique, and virtual work is used to predict the snap-through forces and moments for three families of unsymmetric fiber-reinforced composite laminates. Unsymmetric laminates generally have two stable equilibrium configurations when cooled from their elevated cure temperature, and it is the moment required to snap the laminate from one stable configuration to the other that is the subject of this paper. A simple force-controlled experiment is described which is used to measure the snap-through moment and the characteristics of the configuration change, by way of strains, in four laminates. The correlation between predicted results and experimental measurements is quite good, both in terms of moment levels and in terms of strain response.     

MULTIPLE SCATTERING AND DYNAMIC STRESS ANALYSIS OF ELASTIC WAVES IN A FIBER—REINFORCED COMPOSITE WITH INTERFACES

Based on the theory of elastic dynamics, multiple scattering of elastic waves and dynamic stress concentrations in fiber-reinforced composite are studied. The analytical expressions of elastic waves in different regions are presented. The mode coefficients of elastic waves are determined in accordance with the continuous conditions of displacement and stress on the boundary of the multi-interfaces. By using the addition theorem of Hankel functions, the formula of scattered wave fields in different local coordinates are transformed into those in one local coordinate to determine the unknown coefficients and dynamic stress concentration factors (DSCFs). The influences of the distance between two inclusions, material properties and structural size on the DSCFs near the interfaces are analyzed. As examples, the numerical results of DSCFs near the interfaces for two kinds of fiber-reinforced composites are presented and discussed. The project supported by the National Natural Science Foundation of China (19972018)     

On the fiber-reinforced birefringent composite materials

The isochromatic and isoclinic parameters for fiber-reinforced birefringent materials are examined. Experimental results reveal residual birefringence and an initial isoclinic coincident with the axis of material symmetry. These preliminary data also indicate that the isoclinic angle is neither a measure of the principal strain nor the principal-stress directions within the composite.     

SMA-induced snap-through of unsymmetric fiber-reinforced composite laminates

A theory is developed and experiments designed to study the concept of using shape memory alloy (SMA) wires to effect the snap-through of unsymmetric composite laminates. The concept is presented in the context of structural morphing, that is, a structure changing shape to adjust to changing conditions or to change operating characteristics. While the specific problem studied is a simplification, the overall concept is to potentially take advantage of structures which have multiple equilibrium configurations and expend power only to change the structure from one configuration to another rather than to continuously expend power to hold the structure in the changed configuration. The unsymmetric laminate could be the structure itself, or simply part of a structure. Specifically, a theory is presented which allows for the prediction of the moment levels needed to effect the snap-through event. The moment is generated by a force and support arrangement attached to the laminate. A heated SMA wire attached to the supports provides the force. The necessary SMA constitutive behavior and laminate mechanics are presented. To avoid dealing with the heat transfer aspects of the SMA wire, the theory is used to predict snap-through as a function of SMA wire temperature, which can be measured directly. The geometry and force level considerations of the experiment are discussed, and the results of testing four unsymmetric laminates are compared with predictions. Laminate strain levels vs. temperature and the snap-through temperatures are measured for the these laminates. Repeatability of the experimental results is generally good, and the predictions are in reasonable agreement with the measurements.     

Bending of a fiber-reinforced viscoelastic composite plate resting on elastic foundations

Composite structures on an elastic foundation are being widely used in engineering applications. Bending response of inhomogeneous viscoelastic plate as a composite structure on a two-parameter (Pasternak’s type) elastic foundation is investigated. The formulations are based on sinusoidal shear deformation plate theory. Trigonometric terms are used in the present theory for the displacements in addition to the initial terms of a power series through the thickness. The transverse shear correction factors are not needed because a correct representation of the transverse shear strain is given. The interaction between the plate and the foundation is included in the formulation with a two-parameter Pasternak’s model. The effective moduli and Illyushin’s approximation methods are used to derive the viscoelastic solution. The effects played by foundation stiffness, plate aspect ratio, and other parameters are presented.     

Optical analysis of photoelastic polariscopes

A full-field interpretation of the photoelastic effect is advanced, supplementing the common pointwise explanation. This “warped-wavefront” viewpoint provides a key for the analysis of resolution, contrast and location of object plane in diffused-light and lens-type polariscopes. The analysis yields quantitative assessment of performance. Two wavefronts with wave normals separated by angle α emerge from the model wherever a gradient of fringe order exists in the interference pattern. It is this angular separation that limits resolution, contrast and object-plane location. Results indicate potentially very high resolution for both types of polariscope, with theoretical resolution being independent of focal length and speed (f-number) of the camera lens. With diffused-light polariscopes, contrast diminishes as fringe gradient increases, but high contrast remains available for common situations. Position and shape of fringes in the isochromatic pattern is affected by location of the plane of focus, which should therefore be fixed in the model. The analysis reveals special considerations applicable to special circumstances, including very high fringe gradients, point-light sources (lasers) and distant model location.     

Scattered-light photoelastic stress analysis of a solid-propellant rocket motor

Scattered-light photoelasticity, a nondestructive technique, is used to determine the stress distribution in a three-dimensional, solid-propellant rocket motor. The model is a case-bonded solid propellant with a stargrain internal boundary. Stresses are induced by internal pressure. The pressure load simulates part of the stresses developed in firing a rocket. The model material is polyurethane rubber which is similar to the binder material in actual rockets. The model construction and data analysis are discussed in detail, and the results are presented in graphical form.     

Complete automatic analysis of photoelastic fringes

TV technology combined with a modern video-frame store is used to store the complete fringe pattern in digitized form in real time. A minicomputer connected to the store has direct access to any picture point in the store. In this manner, it is possible to evaluate the photoelastic-fringe pattern in the computer by special developed software.     

Dynamic response and damage analysis of fiber-reinforced composite laminated plates under low-velocity oblique impact

Fiber-reinforced composite laminates (FRCL) is susceptible to the external impacting. Understanding the crack propagation and structural mechanical properties of the damaged FRCL under low-velocity oblique impact is of great value in practical application. A new analytical dynamic model is developed in this work to research the dynamic response and damage property of FRCL under oblique impacting. The displacement field and strain–displacement relations of the FRCL are established by utilizing higher-order shear plate theory. The matrix damage and fiber rupture in FRCL under oblique impacting are captured by an internal variable-based continuum damage constitutive relation. To accurately predict the oblique impacting force, an analytical dynamic impacting model is proposed basing on a developed contact model, where normal and tangential contact is coupled and solved simultaneously. The whole initial boundary value problem is iteratively solved by synthetically using finite differential method and Newmark-\(\beta \) method. The solving convergence and accuracy of the model is demonstrated and validated. Simulations show that the matrix damage is more easily to appear in FRCL under shear force due to oblique contact when under oblique impacting, and the damage profile is different from normal impacting. The dynamic responses of the FRCL plate under oblique impacting differ also greatly from normal impacting. The current research provides a theoretical basis for FRCL design and its engineering application when under low-velocity impacting.