Bimaterial Corners Research Articles

Singularities in 2D anisotropic potential problems in multi-material corners: Real variable approach

An analysis of singular solutions at corners consisting of several different homogeneous wedges is presented for anisotropic potential theory in plane. The concept of transfer matrix is applied for a singularity analysis first of single wedge problems and then of multi-material corner problems. Explicit forms of eigenequations for evaluation of singularity exponent in the case of multi-material corners are derived both for all combinations of homogeneous Neumann and Dirichlet boundary conditions at faces of open corners and for multi-material planes with singular interior points. Perfect transmission conditions at wedge interfaces are considered in both cases. It is proved that singularity exponents are real for open anisotropic multi-material corners, and a sufficient condition for the singularity exponents to be real for anisotropic multi-material planes is deduced. A case of a complex singularity exponent for an anisotropic multi-material plane is reported, apparently for the first time in potential theory. Simple expressions of eigenequations are presented first for open bi-material corners and bi-material planes and second for a crack terminating at a bi-material interface, as examples of application of the theory developed here. Analytical solutions of these eigenequations are presented for interface cracks with any combination of homogeneous boundary conditions along the interface crack faces, and also for a special case of a crack perpendicular to a bi-material interface. A numerical study of variation of the singularity exponent as a function of inclination of a crack terminating at a bi-material interface is presented.

Quantitative strain analysis of flip-chip electronic packages using phase-shifting moiré interferometry

In order to gauge the reliability of electronic packages, it is valuable to analyze thermally induced displacements and strains around bimaterial corners and within interconnections. The increased demand for computing performance has created increasingly complex electronic packages with miniaturized features, making it increasingly difficult to extract these quantities. Often, material properties at these length scales are not fully known, making modeling and simulation problematic. Thus, determining displacements and strains experimentally is attractive. In this study, an advanced flip-chip package with fine interconnection features was analyzed using phase-shifting moiré interferometry (PSMI) in conjunction with image analysis software developed for this purpose. Before the analysis, PSMI was qualified using an isotropic solid undergoing uniform thermal contraction, which yielded a displacement precision of ±4 nm. Then a high-magnification, high-resolution displacement and strain analysis was performed for a small cross-sectional region of the flip-chip package containing 20–100 μm sized features. The analysis quantifies these results and gives displacements and strains obtained by differentiating the displacement data using a strain-energy-based finite element formulation.

The effect of corner angles in bimaterial structures

The effect of the corner angle on bimaterial corner stresses is studied. Corners (with one side bonded) and joints (with both sides bonded) are studied considering the materials to be elastic and plastic. Two structures with bimaterial corners, lap and scarf joints are considered. For the elastic analysis, the singularity at the corner as well as the stress intensity factor are found for a range of angles using the Betti’s law based reciprocal work contour integral theorem which combines the ease of finite element calculations far away from the corner with the singular representation near the corner. The singularity and the stress intensity factors were found to depend on the angle. For the work-hardening plastic analysis, a fourth order differential equation was solved to obtain the singularity at the corner and finite element solutions were used to obtain stress intensity factors. These parameters were found to depend on load level, hardening exponent and the angle. It is concluded that the stress intensity factors are unsuitable as failure criteria for a corner with varying angle. One exception was found to be for a corner with high yielding that gave rise to a constant stress intensity factor for different corner angles.

Cohesive zone modeling of crack nucleation at bimaterial corners

The prediction of crack nucleation from bimaterial corners that have the same corner angle (and therefore singularity) via stress intensity factors is fairly well established. The objective of this work was to examine crack nucleation from a range of corner angles and see if a more general crack nucleation criterion could be formulated. A series of experiments was conducted using an aluminum-epoxy bimaterial specimen loaded under 4-point bending. In addition to the usual measurements of load and an associated displacement, the displacements near the corner were measured using moiré interferometry. Numerical analyses were first conducted assuming a rigid interface. However, the resulting displacements differed from the measured ones, especially near the corner and along the interface. The interface was then modeled as a separate constitutive entity by incorporating a cohesive zone model in the numerical analysis. Following calibration via an interface crack configuration (zero corner angle), the cohesive zone model yielded displacements that were in good agreement with the measured values for all the other corner angles that were considered. The predicted failure loads were also in good agreement with the experimental results. Thus the consistent nucleation criterion was that the area under the traction-separation curve and its maximum traction (the dominant cohesive zone model parameters) remain the same. The numerical solutions indicated that the plastic deformation in the epoxy was small and that failure was predominantly in opening mode. In addition, the critical vectorial crack opening displacement and mode-mix were independent of the corner angle. Finally, a simple design parameter was proposed for predicting the failure load of a bimaterial specimen with an arbitrary corner angle, based on the failure load of a bimaterial specimen with an interface crack.

ADHESIVELY BONDED COMPOSITE IOSIPESCU SPECIMENS WITHOUT SINGULAR STRESS FIELDS

The asymptotic singular stress fields at bi-maSerial wedge corners in adhesively bonded composite losipescu specimens have been studied using two-dimensional finite-element techniques. In particular, the singular powers of the fields have been analyzed using the finite-element iterative method. Different combinations of the elastic properties of the composite adherends have been considered, taking into account various volume fractions of fibers as well as different fiber orientations with respect to the adherend/adhesive interface. It has been shown that there are critical angles of the interfaces that separate the regions with positive and negative singular powers. This creates stress fields that are not singular in nature. It appears that the critical angles are dependent on the elastic properties of the composite adherends. In the second part of this work, the critical angle approach has been applied to the design of adhesively bonded composite losipescu specimens free from singular stress fields at their bi-material wedge corners. The samples will not develop singular stress fields either in shear or under biaxial shear-dominated loading conditions.

Crack Initiation From Homogeneous and Bimaterial Corners

Fracture initiated at a corner between two different isotropic materials is considered. A “small” crack, well within the region dominated by the asymptotic stress fields of the noncracked corner, is modeled and the stress intensities associated with the tip of the small crack are determined. Different criteria for the direction of crack propagation are studied.

Fracture of Defect Foam Core Sandwich Beams

Abstract In the manufacturing of foam core sandwich structures, butt-joints are frequently used within the core. However, inherent flaws may be developed in such joints. Hence areas of stress concentration, such as sharp corners, will exist that may have critical influence on structural behavior. The same type of sharp corners appears in a standard shear test (ASTM C 273) used for determination of shear properties of core materials. The present paper deals with the analysis of flawed butt-joints in sandwich beams. The analysis is based on a fracture mechanics approach. It is assumed that sharp bi-material corners will be present between the core and face material in the sandwich beam which can be compared with the interface between the core material and the edge plates in the shear test specimen. Such sharp corners develop singular stress fields which are used together with a suitable fracture criterion to predict structural failure in sandwich beams. The results from the standard shear tests for core materials are used to evaluate fracture mechanics parameters. These parameters are used for the prediction of fracture loads of sandwich beams with simulated flaws.