# Stress analysis of bonded axisymmetric assemblies under thermal loading: Applications in microelectronics

#### Abstract

Four analytical models, namely, elastic, elasto-plastic, viscoelastic , and viscoelastic-plastic , are developed for predicting the maximum shearing displacement in a bonded assembly (e.g., an electronic assembly with a component and a substrate connected by an array of solder joints) under global thermal mismatch loading. The thermal loads are assumed to be uniform within the component and the substrate, with both step-function and sinusoidal temperature-time histories being considered. These models are based on plane stress formulations for circular disks and the assumption that the array of discrete joints may be replaced by an equivalent continuous "shear layer." The derived analytical results can be applied to square assemblies with excellent accuracy. All results have been derived as closed-form correction factors to be applied to the easily calculated unconstrained shear displacement to obtain the maximum shear displacement. All the correction factors depend on prescribed geometric and material parameters of the component, substrate, and joints, the characteristics of the array, and the material properties of the bonding material (solder). The motivation for determining the maximum shearing displacement is to compute the maximum shear strain, which is then used to estimate the cycles-to-failure by means of a fatigue law (e.g., Coffin-Manson). Although the results are applied to microelectronics systems, they are also relevant to many other applications in which bonded assemblies are used. The solder material in the array of the elastic model is assumed to be a linear elastic material. For a small "relative array stiffness" (λ < 0.46), the effect of array stiffness on the shearing displacement can be neglected in a homogeneous component assembly. In the elasto-plastic model, the effect of the plasticity of the bonding material is accounted for by modeling the solder or adhesive material as an elastic-perfectly plastic material. As expected, softening due to solder plasticity causes the magnitude of shearing displacement to increase. The creep effects in the shear displacement are introduced by modeling the solder material as a temperature-independent linear viscoelastic material in the viscoelastic model. The viscoelastic constitutive law used for the solder in this study is that of a three-parameter viscoelastic standard solid in distortion and an elastic solid in the hydrostatic mode. The viscoelastic model permits a sinusoidal thermal variation resulting in a frequency-dependent correction factor. A first attempt at a "unified model" in the form of a viscoelastic-plastic model is also developed to predict the displacement incorporating both the time-independent (elastic and plastic) and the time-dependent (creep) deformations. The analytical relations derived are simple, easy-to-use, and helpful for studying the interaction among the various design parameters. All the results have been presented in the form of dimensionless plots to aid in the analysis or design process and may provide convenient alternatives to performing time-consuming and expensive finite element analyses of entire assemblies. Parametric studies have been performed on the sensitivity of the maximum shearing displacement to the system's geometric, material, and load parameters. The analytical relations are in good agreement with existing experimental and numerical results.

#### Recommended Citation

Shilak Shakya,
"Stress analysis of bonded axisymmetric assemblies under thermal loading: Applications in microelectronics"
(January 1, 2000).
*Dissertations (1962 - 2010) Access via Proquest Digital Dissertations.*
Paper AAI9991617.

http://epublications.marquette.edu/dissertations/AAI9991617