Paul F. Joseph

The order of stress singularities for bonded and disbonded three-material junctions

Abstract In-plane solutions are given for the order of the stress singularity at an internal point in an elastic, isotropic solid where three wedges of different materials meet. The three interfaces are either all perfectly bonded or a disbond is introduced along one interface. One feature of this three-material junction that is not present for the corresponding two-material case, is that each interface is geometrically different and, therefore, the singular behavior is dependent on which interface is disbonded. Each material and geometrical combination therefore gives rise to four problems, one with perfect bonding at all interfaces, and three cases of an interface disbond. Numerical results are presented for selected three-material junctions. New results for two-material junctions and wedges that serve as special cases for the present study are also presented.

Standardized complex and logarithmic eigensolutions for n-material wedges and junctions

The Airy stress eigenfunction expansion of Williams [1] has been used to obtain simple expressions for the angular variations of the stress and displacement fields for n-material wedges and junctions subjected to inplane loading. This formulation applies to real and complex roots, as well as the special transition case giving rise to r−ω singular behavior. The asymptotic behavior of the general problem is similar to that of the bi-material interface crack. In the case of real roots, the stress and displacement expressions can be determined to within a multiplicative real constant (amplification), while for the complex case, the fields are determined to within a multiplicative complex constant (amplification plus rotation). Because of the rotation in the complex case, there are an infinite number of equivalent ways to express the angular variations (eigenfunctions) of the stress and displacement fields. Therefore, the fields are standardized in terms of ‘generalized stress intensity factors’ that are consistent with the bi-material interface crack and the homogeneous crack problems. As in the bi-material crack problem, for the complex case there are two stress intensity factors for each admissible order of the stress singularity. For specific n-material wedges and junctions, a small variation of material properties and/or geometry can change the eigenvalues from a pair of complex conjugate roots to two distinct real roots or vice-versa. An r−ω singularity associated with a nonseparable solution in υ and θ exists at this point of bifurcation. Such behavior requires an adjustment in the standard eigenfunction approach to insure bounded stress intensity factors. The proper form of the solution is given both at and near this special material combination, and the smooth transition of the eigenfunctions as the roots change from real to complex is demonstrated in the results. Additional eigenfunction results are provided for particular cases of 2 and 3-material wedges and junctions.

Finite element analysis of anisotropic materials with singular inplane stress fields

Abstract A finite element formulation is developed for the analysis of singular stress states at material and geometric discontinuities in anisotropic materials loaded inplane. The displacement field of the sectorial element is quadratic in the angular coordinate direction and exponential in the radial direction measured from the singular point. The formulation of Yamada and Okumura (1983a, b) is extended to take into account the anisotropy of the material. The stress and displacement fields are obtained when the order of the stress singularity is real as well as complex. When the order of the stress singularity is complex, it is shown that the angular variation of the stress and displacement fields can be expressed in an infinite number of ways. Results for the displacement and stress fields obtained when the order of the stress singularity is complex can be made to match already published results once a similarity transformation is applied. The simplicity and accuracy of the formulation are demonstrated by comparison to several analytical solutions for both isotropic and anisotropic multi-material wedges and junctions with and without disbonds. The nature and rate of convergence associated with the element suggests that it could be used in developing enriched elements for use with standard elements to yield accurate and computationally efficient solutions to problems having complex global geometries.

Design of Cellular Shear Bands of a Non-Pneumatic Tire -Investigation of Contact Pressure

In an effort to build a shear band of a lunar rover wheel which operates at lunar surface temperatures (40 to 400K), the design of a metallic cellular shear band is suggested. Six representative honeycombs with aluminum alloy (7075-T6) are tailored to have a shear modulus of 6.5MPa which is a shear modulus of an elastomer by changing cell wall thickness, cell angles, cell heights and cell lengths at mesoscale. The designed cellular solids are used for a ring typed shear band of a wheel structure at macro-scale. A structural performance such as contact pressure at the outer layer of the wheel is investigated with the honeycomb shear bands when a vertical force is applied at the center of the wheel. Cellular Materials Theory (CMT) is used to obtain in-plane effective properties of a honeycomb structure at meso-scale. Finite Element Analysis (FEA) with commercial software ABAQUS is employed to investigate the structural behavior of a wheel at macro-scale. A honeycomb shear band designed with a higher negative cell angle provides a lower contact pressure along the contact patch associated with an in-plane shear flexible property.

Final Shape of Precision Molded Optics: Part I—Computational Approach, Material Definitions and the Effect of Lens Shape

Coupled thermomechanical finite element models were developed in ABAQUS to simulate the precision glass lens molding process, including the stages of heating, soaking, pressing, cooling and release. The aim of the models was the prediction of the deviation of the final lens profile from that of the mold, which was accomplished to within one-half of a micron. The molding glass was modeled as viscoelastic in shear and volume using an n-term, prony series; temperature dependence of the material behavior was taken into account using the assumption of thermal rheological simplicity (TRS); structural relaxation as described by the Tool-Narayanaswamy-Moynihan (TNM)-model was used to account for temperature history dependent expansion and contraction, and the molds were modeled as elastic taking into account both mechanical and thermal strain. In Part I of this two-part series, the computational approach and material definitions are presented. Furthermore, in preparation for the sensitivity analysis presented in Part II, this study includes both a bi-convex lens and a steep meniscus lens, which reveals a fundamental difference in how the deviation evolves for these different lens geometries. This study, therefore, motivates the inclusion of both lens types in the validations and sensitivity analysis of Part II. It is shown that the deviation of the steep meniscus lens is more sensitive to the mechanical behavior of the glass, due to the strain response of the newly formed lens that occurs when the pressing force is removed.

FINAL SHAPE OF PRECISION MOLDED OPTICS: PART II—VALIDATION AND SENSITIVITY TO MATERIAL PROPERTIES AND PROCESS PARAMETERS

In Part I of this study a coupled thermo-mechanical finite element model for the simulation of the entire precision glass lens molding process was presented. That study addressed the material definitions for the molding glass, L-BAL35, computational convergence, and how the final deviation of the lens shape from the mold shape is achieved for both a bi-convex lens and a steep meniscus lens. In the current study, after validating the computational approach for both lens types, an extensive sensitivity analysis is performed to quantify the importance of several material and process parameters that affect deviation for both lens shapes. Such a computational mechanics approach has the potential to replace the current trial-and-error, iterative process of mold profile design to produce glass optics of required geometry, provided all the input parameters are known to sufficient accuracy. Some of the critical contributors to deviation include structural relaxation of the glass, thermal expansion of the molds, TRS and viscoelastic behavior of the glass and friction between glass and mold. The results indicate, for example, the degree of accuracy to which key material properties should be determined to support such modeling. In addition to providing extensive sensitivity results, this computational model also helps lens molders/machine designers to understand the evolution of lens profile deviation for different lens shapes during the course of the process.

Thermally induced logarithmic stress singularities in a composite wedge and other anomalies

Abstract Wedge paradoxes, which were first studied by Sternberg and Koiter (Sternberg E, Koiter WT. The wedge under a concentrated couple: a paradox in the two-dimensional theory of elasticity. ASME Journal of Applied Mechanics 1958;4:575–81), occur due to multiple roots in the Williams (Williams ML. Stress singularities resulting from various boundary conditions in angular corners of plates in extension. ASME Journal of Applied Mechanics 1952;19:526–28) eigenfunction expansion. The consequence of such a paradox is a change in behavior of the stresses from σ ij r, θ =r −ω h 1 ij θ , to the ‘non-separable’ form, σ ij r, θ =r −ω − ln r h 1 ij θ +h 3 ij θ . The focus of this study is the problem of thermally induced logarithmic stress singularities in a composite wedge associated with ω=0. Both double and triple root examples are presented which lead to ln r and ln 2 r behavior in the stresses, respectively. This behavior is primarily associated with incompressible materials for the clamped–clamped single material case, and for the full range of Poisson’s ratio for the clamped-free case. The study also includes non-separable eigenfunctions that occur when complex conjugate roots transition to double real roots. Perhaps the most interesting result is that for the clamped–clamped wedge with Poisson’s ratio equal to 1/2, the hydrostatic stress has a logarithmic singularity proportional to the thermal strain for all wedge angles. This result can be extended to conclude that for a confined, incompressible or nearly incompressible material with a relatively sharp corner, and subject to some expansion or contraction phenomena, high hydrostatic stresses can result.

Designing a Lunar Wheel

Given NASA’s renewed initiative to return to the moon, it is imperative to develop a mobility platform which will enable a wider range of exploration and science. To this end a wheel which can perform in a lunar environment for an extended period of time and distance is needed. This paper analyzes and evaluates the attempts to develop such a solution and the challenges which have arisen. Furthermore, it builds upon an analytical basis for the proposed solution and examines further applications in vehicular mobility. Specifically, three concepts were conceived, analyzed, prototyped, and tested in a senior design project class supported by NASA’s Jet Propulsion Laboratory and Michelin Tire Corporation, then critically examined for failure modes. The lessons learned from the analysis are currently being applied in the development of third and fourth generation prototypes.Copyright

Singular antiplane stress fields for bonded and disbonded three-material junctions

Abstract Antiplane shear solutions are given for the asymptotic stress field around an internal point in an elastic, isotropic solid where three wedges of different materials meet. The three interfaces are either all perfectly bonded or a disbond is introduced along one interface. One feature of this three-material junction that is not present for the corresponding two material case, is that each interface is geometrically different, and therefore, the singular behavior is dependent on which interface is disbonded. Each material and geometrical combination therefore gives rise to four problems; one with perfect bonding at all interfaces and three cases of an interface disbond. Numerical results are presented for selected three-material junctions. Numerical results show that singularities can be much more severe in three-material junctions than in two-material junctions. The results also show the importance of providing the associated angular variations of the field which can be drastically different for two cases with very similar geometries and identical orders of singularity.

Crack surface contact of surface and internal cracks in a plate with residual stresses

The problem of a mode I surface or internal crack in a plate with residual stresses is examined. The emphasis in this study is in accounting for crack surface contact. The importance of such problems is realized in both crack detection and in the stress intensity factor calculations. The line spring model is used to iteratively determine the border of the closed portion of the crack and the stress intensity factors along the open portion. It is demonstrated that the model is very versatile for solving contact problems. Examples for both surface and internal cracks are given. It is shown that if crack contact is ignored, the stress intensity factor may be significantly underestimated.