Functionally Graded Biomimetic Plate with Hole

Abstract

In this paper, we address the problem of maximizing the strength of a plate in which a circular hole is placed within a homogeneous material. We mimicked in a polyurethane plate the modulus distribution we found around a naturally occurring hole in bone by discretizing the panel into rings and matching modulus ratios (ring to given homogeneous material) between the bone and available materials. This biomimetic inhomogeneous plate (IP) was then fabricated, tested under uniaxial tension, and compared to a homogeneous plate (HP) of the same geometry. Results from moire interferometry were in good agreement with our finite element (FE) analyses of both plates. With confidence in our FE models, we found that we could decrease the maximum failure index (roughly, the ratio of local stress to allowable stress) from near 3 in the HP to 1.5 in the IP (or, in other words, double the strength of the plate). We were able to decrease the weight of the IP by 45% compared to the HP of identical strength. INTRODUCTION. Holes are well known to be a serious design problem and a frequent site of fracture in structures. Traditional engineering solutions to this problem either involve massive reinforcements around the hole (often expensive in terms of weight and cost), or shape contouring to alleviate the effects of the hole (which is not always possible, e.g., most holes must be round). Some manufacturing techniques have met with limited success due to their difficulty in implementation in a production environment. Specifically, tow placement machines allow the variation of fiber orientations of composite materials in a continuous manner, which has already been shown to be a powerful way of alleviating stress concentrations near holes. New manufacturing technologies allow the production of functionally graded materials (FGMs), such as foamed metals with density profiles tailored to functional requirements. Natural holes (foramina) exist in practically all bones to allow blood vessels to pass through their hard outer shells. Foramina form at a very early fetal age, which gives the bone much time to adapt to their presence by arranging the local material microstructure. This adaptation was hypothesized by us to be the reason why foramina never appear as fracture sites clinically or in laboratory tests of whole bones. As part of our ongoing biomimetic structures research, we studied a naturally occurring hole in a horse bone in detail to characterize the material microstructure and spatial gradation in elastic properties around the hole[5]. We found that the foramen is embedded within a compliant zone of material with a bracket shaped stiffer region approximately two foramen diameters away from its edge. This functional gradation in elastic modulus was shown to reduce the stress concentration at the hole when compared to a hole in a homogeneous medium. Further optimization studies by us using the observed material microstructure and composition variables showed that spatial variations in microstructure in addition to reducing stress concentrations also increased the strength[6]. We recently tested beams made from bone containing a foramen and a manufactured hole of identical shape under four point bending. The manufactured holes were adjacent to the foramen, but far enough away to preclude interaction. The bending moments at the foramen and manufactured hole were identical. Failure occurred through the manufactured holes and not the natural hole, offering further evidence that the material design around the foramen makes it a stronger hole. Future investigations will include fatigue tests on similar beam specimens to explore the damage resistance and tolerance characteristics of the foramen. Given these encouraging results, the natural solution offered by the foramen is clearly worth investigating for use in engineering design. To that end, we constructed an inhomogeneous plate containing a hole with similar functional gradation in elastic modulus properties as observed in the bone. The elastic modulus variation was simplified to an axisymmetric design and was obtained by means of discrete rings of varying modulus. We also fabricated a homogeneous plate of the same geometry. The plates were tested under uniaxial tension and full field displacements acquired using moire interferometry. The strains obtained from the experiment were compared with finite element models (FEMs) of the plates and analytical solutions. The verified FEM was then used to calculate a stress failure index (akin to a safety factor) for the plate and compare it to that of a homogeneous plate with a hole. The objective of this paper is to describe the design, analysis, and results from mechanical tests of our inhomogeneous plate, and compare the results to the homogeneous plate. MATERIALS AND METHODS. The functionally graded plate made of discrete rings (Figure 1, left) was ordered from a commercial vendor (Pacific Research Laboratories; Vashon, WA). We also purchased bulk foam material of uniform density for fabricating the homogeneous plate (HP; Figure 1, right). The functionally graded biomimetic inhomogeneous plate (IP) is 10 cm wide by 17.5 cm long and 1 cm thick with a centrally located 2 cm diameter hole. The base material is polyurethane foam. The IP consists of six annular rings surrounding the central hole, each of a different apparent density (porosity) and, hence, modulus and strength. Each ring is 5 mm wide from its inside to outside edge. The ring next to the hole is the same material as that of the far field, and the other ring densities are such that the modulus and strength distribution matches what we observed in bone as well as possible given the available materials (Figure 2). The rings are bonded within the plate with the same polyurethane resin that is used to make the foam. The HP consists of the same material as the far field material of the IP. A diffraction grating was applied and then end tabs were glued to each plate for mechanical testing. The moire interferometry technique provides very accurate full field measurements of in plane displacements by use of optical interference principles. The interference pattern produced by a reference grating and deformed grating applied to the surface of the specimen is used to measure the strains on the surface of the specimen with high spatial resolution. The maximum resolution in strain measurements is a function of the grating pitch and wavelength of light used. We used a grating that could produce 2,400 fringes/mm with an illumination source of 632.8 nm wavelength. Details of moire interferometry for high accuracy strain measurements are presented elsewhere [4]. We used this technique to acquire displacement fields from which we derived the longitudinal strain distribution along a line through the hole in each plate. We needed to develop a methodology for applying a compliant diffraction grating to the plates to use the moire technique. This was challenging due to the porous nature of the foams used in manufacturing our plates. Beginning with techniques developed by others[3] and after much experimentation on spare material, we settled on the following process. First, we lightly sanded the plate surface using 300 grit and 600 grit sandpaper. After forcing dust out of the pores with compressed air, we applied a silicon rubber primer (GE Silicone SS4120) to the sanded surface and left it to cure overnight. This was followed by an application of a two part silicone rubber (GE Silicone Rubber RTV615). A glass square with a sheet of polytetrafluoroethylene (PTFE) was placed on the plate with the silicon rubber mixture. To ensure flatness and to speed cure time, the plates were then sealed in an airtight bag, attached to a vacuum line and placed in an oven. Finally, we applied a dental impression material (Reprosil Type I Low Viscosity; DENTSPLY International; Milford, DE) to receive the replication grating. The grating covered one quadrant of the plate around the hole, as this was the biggest field of view possible within the interferometer. Testing was performed within 24 hours, before the specimen grating warped due to hygral effects. Uniaxial tension tests were performed on an IP and a HP using a custom fixture designed to provide stress free boundary conditions everywhere except in the vicinity of the grips. Uniaxial tension was applied to each plate in displacement control, incrementally to 76 N. This small load was chosen to keep the maximum stress below the yield stress of the materials in the plate and produce a far field strain on the order of 1,000 μe. The uniaxial load was monitored with an in-line load cell. Photographs of the fringe patterns were obtained at various load levels. A plane stress FEM was created for verification of analysis models and material properties. Due to the two fold symmetry in loading and geometry, only one quarter of the plate was modeled with symmetry boundary conditions. The Figure 1. Inhomogeneous plate (IP) containing a hole (left). The hole diameter is 2 cm, and the width of each ring is 5 mm. The homogeneous plate (HP) is of same dimensions as the IP and made of same material found Figure 2. Comparison between longitudinal modulus distributions as designed in our inhomogeneous plate (IP) and as found near the natural hole in bone. Plotted is the ratio of the local modulus to the modulus at the hole edge versus the normalized distance from the hole. 0 2 4 6 8 0.0 0.2 0.4 0.6 0.8 1.0 Normalized Distance from Hole Edge R at io o f L oc al to H ol e Ed ge M od ul us