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As part of the Mathematical Problem Solving Project, MPSP, the goals and purposes of miniprojects are explored. Various conjectures are presented which are based partially upon observation of children working on projects and partially upon the opinions of mathematicians who have observed how problem-solving processes are used in real-world situations. Four miniprojects are outlined and a report on one project is given. (MP)

An overview of the Mathematical Problem Solving Project, MPSP, is given. The goals, organization, and history of the project are described and the progress toward those goals is discussed. The outcomes of the project are stated, a brief diary of activities given, and a listing of technical reports made. (MP)

Tests were performed measuring the locations and geometries of delaminations in Fiberite T300/976 graphite/epoxy, Fiberite IM7/977-2 graphite-toughened epoxy, and ICI APC-2 graphite/PEEK plates subjected to transverse impact loads. The data provide specific information on the effects of impactor velocity, impactor mass, material, thickness of back ply group, difference in fiber orientation between adjacent ply groups, plate thickness, and impactor nose radius. The data were compared to the results of the Finn-Springer model. The model was found to describe the data with reasonable accuracy.

A method is presented for calculating the locations, shapes, and sizes of delaminations which occur in a fiber reinforced composite plate subjected to transverse static or dynamic (impact) loads. The plate may be simply supported, clamped, or free along its edges. A model of the delamination formation was developed. This model was then coupled with a finite element analysis. The model and the finite element analysis were implemented by a computer code which can be used to estimate the load at which damage is initiated as well as the locations, shapes, and sizes of the delaminations.

Tests were performed measuring the damage initiation loads and the locations, shapes, and sizes of delaminations in Fiberite T300/976 graphite/epoxy, Fiberite IM7/977-2 graphite-toughened epoxy, and ICI APC-2 graphite-PEEK plates subjected to transverse static loads. The data were compared to the results of the Finn-Springer model, and good agreements were found between the measured and calculated delamination lengths and widths.

Tests were performed assessing the effectiveness of repair in restoring the mechanical properties of damaged, solid composite plates made of Fiberite T300/976 graphite-epoxy. Some (75%) or all (100%) of the damaged zone was cut out, and the plate was repaired by plugging and patching the hole. The effectiveness of the repair was evaluated by measuring the compressive strengths of undamaged plates, damaged plates with no cutout, damaged plates with a cutout, and plates that had been repaired.

The behavior of elliptical sublaminates created by delaminations in composite plates was investigated for plates subjected to in-plane compressive, shear, and thermal loads. The axes of the ellipse may be arbitrarily oriented with respect to the applied loads. A model was developed which provides the stresses, strains, and displacements of the sublaminate, and the loads applied to the plate at which the sublaminate buckles and at which it grows in size. A series of experiments was conducted on sandwich plates made of Fiberite T300/976 graphite-epoxy laminates bonded to an aluminum honeycomb core. Teflon film in either a circle or an ellipse was embedded in the laminate, simulating the presence of a delamination. Each plate was loaded in compression, and the load-strain history of the sublaminate, the sublaminate buckling load, and the sublaminate growth load were measured. The predictions of the model agreed reasonably well with the data.

The present volume on impact damage in composite plates presents an extensive compendium of visual and graphic data regarding a variety of material and impactor parameters. The photographs are taken with X-ray and C-scan imaging in conjunction with a dye penetrant to show matrix cracks and delaminations. Impact and static-loading tests are performed on plates of graphite-epoxy, graphite-toughened epoxy, and graphite-PEEK materials. The images are presented to yield specific visual data regarding such parameters as impactor velocity, thickness of the back ply group, impactor nose radius, and the effects of multiple delaminations. The images are grouped in eight subsets that correspond to parameters including plate length, material, and the difference in fiber orientation between adjacent ply groups. This substantial volume represents a systematic effort to study the effects of several material parameters on impact damage.

A method is presented for calculating the locations, shapes, and sizes of delaminations which occur in a fiber reinforced composite plate subjected to non-penetrating (low velocity) impact of a solid object. The plate may be simply supported, clamped, or free along its edges. A failure model of the delamination formation was developed. This model was then coupled with a finite element analysis. The model and the finite element analysis were then implemented by a computer code (IMPACT-ST) which can be used to estimate the damage initiation load and the locations, shapes, and sizes of the delaminations. Tests were performed measuring the geometries of the delaminations in graphite-epoxy, graphite-toughened epoxy, and graphite-PEEK plates impacted by a projectile with a spherical tip having masses ranging from 0.355 lbm to 0.963 lbm and velocities from 50 in/sec to 225 in/sec. The data were compared to the results of the model, and good agreements were found between the measured and the calculated delamination lengths and widths.

A model was developed which describes the filament winding process of composite cylinders. The model relates the significant process variables such as winding speed, fiber tension, and applied temperature to the thermal, chemical and mechanical behavior of the composite cylinder and the mandrel. Based on the model, a user friendly code was written which can be used to calculate (1) the temperature in the cylinder and the mandrel, (2) the degree of cure and viscosity in the cylinder, (3) the fiber tensions and fiber positions, (4) the stresses and strains in the cylinder and in the mandrel, and (5) the void diameters in the cylinder.

Analytical and experimental studies were performed to validate the model developed by Lee and Springer for simulating the manufacturing process of filament wound composite cylinders. First, results calculated by the Lee-Springer model were compared to results of the Calius-Springer thin cylinder model. Second, temperatures and strains calculated by the Lee-Springer model were compared to data. The data used in these comparisons were generated during the course of this investigation with cylinders made of Hercules IM-6G/HBRF-55 and Fiberite T-300/976 graphite-epoxy tows. Good agreement was found between the calculated and measured stresses and strains, indicating that the model is a useful representation of the winding and curing processes.

By using the Lee-Springer filament winding model temperatures, degrees of cure, viscosities, stresses, strains, fiber tensions, fiber motions, and void diameters were calculated in graphite-epoxy composite cylinders during the winding and subsequent curing. The results demonstrate the type of information which can be generated by the model. It is shown, in reference to these results, how the model, and the corresponding WINDTHICK code, can be used to select the appropriate process variables.

A model was developed for simulating he manufacturing process of filament-wound cylinders made of a thermoset matrix composite. The model relates the process variables (winding speed, fiber tension, applied temperature) to the parameters characterizing the composite cylinder and the mandrel. The model is applicable to cylinders for which the diameter is large compared to the wall thickness. The model was implemented by a user-friendly computer code suitable for generating numerical results.

The conceptual evolution of waverider configurations and their applications to hypersonic air vehicles are examined in the context of evolving mission requirements and technological advances. The fundamental objective of employing configurations which tend to maximize the payload fraction is emphasized, and the achievement of this objective over the time period of interest is reviewed, starting with high drag ballistic vehicles through low lift-drag ratio lifting bodies to the revived interest in high lift-drag ratio configurations. Practical applications of research in the area of lifting waveriders are discussed with particular reference to the B-70 Valkyrie and the Space Shuttle.

The response of delaminated composite plates to compressive in-plane loads was investigated. The delaminated region may be either circular or elliptical, and may be located between any two plies of the laminate. For elliptical delaminations, the axes of the ellipse may be arbitrarily oriented with respect to the applied loads. A model was developed that describes the stresses, strains, and deformation of the sublaminate created by the delamination. The mathematical model is based on a two dimensional nonlinear plate theory that includes the effects of transverse shear deformation. The model takes into account thermal and moisture induced strains, transverse pressures acting on the sublaminate, and contact between the sublaminate and plate. The solution technique used is the Ritz method. A computationally efficient computer implementation of the model was developed. The code can be used to predict the nonlinear-load-strain behavior of the sublaminate including the buckling load, postbuckling behavior, and the onset of delamination growth. The accuracy of the code was evaluated by comparing the model results to benchmark analytical solutions. A series of experiments was conducted on Fiberite T300/976 graphite/epoxy laminates bonded to an aluminum honeycomb core forming a sandwich panel. Either circles or ellipses made from Teflon film were embedded in the laminates, simulating the presence of a delamination. Each specimen was loaded in compression and the strain history of the sublaminate was recorded far into the postbuckling regime. The extent of delamination growth was evaluated by C-scan examination of each specimen. The experimental data were compared to code predictions. The code was found to describe the data with reasonable accuracy. A sensitivity study examined the relative importance of various material properties, the delamination dimensions, the contact model, the transverse pressure differential, the critical strain energy release rate, and the relative growth direction on the buckling load, the postbuckling behavior, and the growth load of the sublaminate.

A model is presented for estimating the dimensions of delaminations in continuous fiber reinforced composite plates subjected to non-penetrating (low velocity) impact of a solid object. The model is based on dimensional analysis, and results in two simple, closed form expressions, one providing the delamination length, the other the delamination width. Comparisons of delamination lengths and widths calculated by these expressions with experimental data suggest that the model predicts these dimensions with reasonable accuracy.

Tests were performed measuring the locations and extents of delaminations and matrix crackings in 3-in. by 3-in. Fiberite T300/934 graphite/epoxy plates impacted by 1/2-in. diameter aluminum spheres. Plates with six ply orientations were tested. The impactor speeds ranged from 700 to 1300 in/sec. The plates were inspected by C-scan both before and after impact. The plates were also dissected after impact and examined visually for damage. In addition, the longitudinal and transverse tensile, compressive, and shear properties of the material were measured.

A method is presented for calculating the locations and sizes of delaminations which occur in a rectangular, fiber reinforced composite plate subjected to nonpenetrating (low velocity) impact of a solid object. The plate may be simply supported or clamped along its edges. In-plane loads or in-plane strains may be imposed on the plate during the impact. The method includes two steps. First, the stresses and strains in the plate are calculated by a three-dimensional, transient finite element method using 8-node brick elements with incompatible modes. Second, the locations, lengths, and widths of delaminations inside the plate are predicted by means of a proposed failure criterion, which is based on the concept of dimensional analysis. The finite element method and the failure criterion were implemented by a computer code which can be used to calculate the impactor position and velocity, the displacements of the plate, the stresses and strains inside the plate during the impact, and the locations and dimensions of the delaminations after the impact. Parametric studies were performed to illustrate the information which can be generated by the computer code.

A method is presented which can be used to estimate the dimensions of delaminations in fiber reinforced composite plates impacted by a solid object. The impact occurs at low velocity such that the impactor does not penetrate the surface of the plate. Comparisons between measured and calculated delamination lengths and widths are given. The use of the method in the design of impact resistant plates is discussed.

Tests were performed toward validating the WIND model developed previously for simulating the filament winding of composite cylinders. In these tests two 24 in. long, 8 in. diam and 0.285 in. thick cylinders, made of IM-6G fibers and HBRF-55 resin, were wound at + or - 45 deg angle on steel mandrels. The temperatures on the inner and outer surfaces and inside the composite cylinders were recorded during oven cure. The temperatures inside the cylinders were also calculated by the WIND model. The measured and calculated temperatures were then compared. In addition, the degree of cure and resin viscosity distributions inside the cylinders were calculated for the conditions which existed in the tests.

The rate of cure and viscosity were measured for Hercules HBRF-55 resin. The rate of cure was measured by differential scanning calorimetry, while the viscosity was measured by a parallel disk and plate type apparatus. The data were fitted to analytic expressions to make them suitable for use in numerical calculations.

A model is described for estimating the impact damage of fiber reinforced composite plates. The displacements and stresses are calculated by a three dimensional transient, finite element method of solution of the governing equations applicable to a linearly elastic body. The region in which damage occurs is estimated using the Tsai-Wu failure criterion. A computer code was developed which can be used to calculate the impact force, displacements and velocities of the plate and the impact body, stresses and strains in the plate, and the damage area. Sample numerical results are presented illustrating the type of information provided by the code. Comparisons between measured and calculated damage areas are also given.