Stephane S. Pageau

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.

A finite element approach to three-dimensional singular stress states in anisotropic multi-material wedges and junctions

Abstract A finite element formulation is developed to determine the order and angular variation of singular stress states due to material and geometric discontinuities in anisotropic materials. The formulation applies to any two-dimensional geometry that is prismatic in the third direction and has three-dimensional displacement fields. In some special cases the three-dimensional fields become uncoupled antiplane and inplane fields and this formulation yields the uncoupled results. The formulation provides for the determination of the asymptotic stress and displacement fields present at interior singular points of three-dimensional structures. The displacement field of the sectorial finite element is quadratic in the angular coordinate direction and asymptotic in the radial direction measured from the singular point. The formulation of Yamada and Okumura [(1983) Hybrid and Mixed Finite Element Methods , pp. 325–343. Wiley, Chichester] for inplane problems is adapted for this purpose. The simplicity and accuracy of the formulation are demonstrated by comparison with several analytical solutions for both isotropic and anisotropic multi-material wedges and junctions. The nature and speed of convergence associated with the element suggests that it could be employed in developing two-dimensional and three-dimensional enriched elements for use along with standard elements to yield accurate and computationally efficient solutions to problems having complex global geometries leading to singular stress states.

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.

Shear buckling response of tailored composite plates

This Note evaluates the piecewise-uniform approach to tailoring as a mean of improving the shear buckling loads of composite plates. This design approach is referred to herein as stiffness tailoring or, more simply, as tailoring. The primary objectives are to determine the tailoring patterns and the degree of concentration of the material used to achieve the tailoring in each pattern that maximize the shear buckling load and to quantify the maximum relative improvement that can be achieved in the buckling load compared to uniform plates.

Enriched Finite Elements for Regions with Multiple, Interacting Singular Fields

Enriched two-dimensional finite elements are formulated for analysis of solids with interacting singular points. These elements contain both the order and the angular variation of all singular stress fields emanating from each singular point so that the possible interaction of stress states present in such regions is represented. The order and distribution of the singular fields are determined with a separate finite element eigenanalysis. These field characterizations and the subsequent enrichment are carried out numerically rather than analytically for ease of extension of the technique to complex geometries for which analytical solutions would be difficult to obtain. Generalized stress intensity factors as well as the stresses and displacements result directly from models incorporating enriched elements. Enriched-element models can be applied not only to crack problems but also to singular stress states not involving cracks and crack growth, and are thus more versatile than J-integral and virtual crack-closure techniques used for calculating energy release rates. In addition, the proper angular distribution of the displacement fields leading to each singular stress state is used in formulating the element, an advantage not shared by formulations such as the quarter-point elements. The advantages of using enriched elements containing information from interacting singular points are demonstrated with an example having two interacting singular points. This tensile groove specimen is modeled using these new elements as well as enriched elements that do not take interaction into consideration. Attention is given to modeling issues when using interactive enriched elements.

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.

Shear buckling response of tailored, rectangular, composite plates

The concept of stiffness tailoring for improved shear buckling resistance of rectangular composite plates is investigated analytically. The tailoring involves only the redistribution of the given material with given orientations to create beneficial stiffening patterns across the planform of the plate. The resulting local nonuniformities in thickness and membrane and bending stiffness combine to change the buckling response of the plate. The weight and average membrane shear stiffness are essentially unaffected by the tailoring. Practical limitations on the degree to which the tailoring may be carried out are shown to govern most designs. Improvements in the shear buckling load on the order of 50 percent are shown possible with monolithic tailoring. Tailored sandwich concepts, in which a light-weight core material is added to keep both plate surfaces flat, can produce improvements well over 100 percent in specific buckling loads compared with uniform composite plates.