Three dimensional contact problems of square orthotropic laminates indented by a rigid spherical indenter are solved. Simplified problems of indentations of beam and isotropic square plate are studied first to develop an efficient numerical technique and to gather the knowledge of the shape of the contact area in order to solve for the three dimensional orthotropic cases. The approach combines an exact solution method in conjunction with a simple discretization numerical scheme. Numerical sensitivity due to the ill-posed nature of the problem was experienced but was cured by enhancing the numerical approach with a least square spirit. Well agreement is obtained by comparing the results of these simplified studies with available published solutions. For isotropic plate, contact area is found to be either a circle or a hypotrochoid of four lobes featured with a shorter length of contact along the through-the- corner directions of the plate. Hertz's theory fails earlier than assuming the contact area to be a circle. In-plane dependence of the contact stress is presented to illustrate the difference of contact behavior between a square plate and a circular plate. Load-indentation relation reveals indenting a square plate is harder than indenting a circular plate of a diameter equal to the side length of the square plate. Solutions of multi-layered orthotropic cases are achieved by employing a modified analytical approach with the same numerical method. Three different configurations of plate are implemented for the orthotropic case, namely, a single layered magnesium (Mg) plate, which is slightly orthotropic, and a single and double layered plates of graphite-epoxy (G-E), which are highly orthotropic. Results for the (Mg) plate agrees with the previous isotropic case. Concept of modifying the previous hypotrochoids is introduced to seek for the contact stresses for comparatively large indentation conditions. Single-layered (G-E) plate was implemented for small indentations. The result supports the validity of Hertz's theory for small indentation and shows a relatively longer contact length in the direction of less stiffness. Two layered (G-E) plate illustrates similar distributions for the contact stresses along both of the in-plane directions with a smaller range of validity of Hertzian type behavior than the previous cases. The boundary effect prevails at the initial stage of indentation but is overcome by the effect of material orthotropy as the indentation proceeds. Thus, the contact area for small indentation appears to be the same kind of hypotrochoids as located in the isotropic case but changes to be the other type of hypotrochoids as the indentation advances. / Ph. D.

The degree of the approximation represented by the basic equation of motion of a vibratory system influenced by internal damping was improved by the assumption that the damping coefficient is proportional to the n^th power of the strain (or stress) amplitude. The solution of the improved equation. was presented by use of approximate methods. The results were expressed in terms of the logarithmic decrement and specific damping to facilitate comparison with experimental data. Three theoretical curves were fitted to experimental curves of the logarithmic decrement. It was found that the theoretical curves accurately described the damping (that is, they fit the experimental curves) for low values of the strain. The curves were found to diverge for higher values of the strain. / M.S.

A recently emerging engineering design approach entails studying the brilliant design solutions found in nature with an aim to develop design strategies that mimic the remarkable efficiency found in biological systems. This novel engineering approach is referred to as bio-inspired design. In this context, the present study quantifies the structure-property relations in bighorn sheep (Ovis canadensis) horn keratin, qualitatively characterizes the effects of a tapered spiral geometry (the same form as in a ram’s horn) on pressure wave and impulse mitigation, describes the stress attenuation capabilities and features of a ram’s head, and compares the structures and mechanical properties of some energy absorbent natural materials. The results and ideas presented herein can be used in the development of lightweight, energy absorbent, bio-inspired material designs. Among the most notable conclusions garnered from this research include: Horn keratin behaves in an anisotropic manner similar to a long fiber composite. Moisture content dominates the material behavior of horn keratin more than anisotropy, age, and stress-state. This makes moisture content the most influential parameter on the mechanical behavior of horn keratin. Tapered geometries mitigate the impulse generated by a stress wave due to the convergent boundary and a continually decreasing cross sectional area such that greater uniaxial stresses and subsequent axial deformation arises. Furthermore, the tapered geometry introduces small shear stresses that further decrease the impulse. Spiral geometries attenuate the impulse generated by a stress wave by the introduction of shear stresses along the length of the spiral. These shear stresses introduce transverse displacements that function to lessen the impulse. When both a taper and spiral geometry are used in a design, their synergistic effects multiplicatively reduce the impulse Tough natural materials have a high porosity, which makes them light-weight, while increasing their compressive energy absorption ability. Biomaterials whose functions include protection and energy absorption feature a multiscale, hierarchical, composite structure. The constituent materials are arranged in such ways to achieve a synergistic effect, where the properties of the composite exceed the properties of its constituents. Biological materials are therefore not confined to the law of mixtures.