Hierarchy in fracture as a stochastic process: Model, implementation and validation

Dhruv Bhate, Purdue University

Abstract

Linear Elastic Fracture Mechanics (LEFM), Cohesive Zone Models (CZMs) and damage mechanics all address the problem of predicting failure in structures using continuum treatments of the materials involved. Each of these approaches has a distinct set of limitations, some of which are: LEFM is applicable only for problems involving elastic materials, or at best, small-scale yielding. CZMs, while being applicable to a wide range of failure behaviors, have tenuous physical origins and the estimation of the required parameters is challenging. Damage mechanics approaches do not explicitly account for the existence of the crack. The overarching goal of this thesis is the development of a failure model that is free of the limiting assumptions of LEFM, does not postulate alternate behavioral models like the CZM and explicitly accounts for the existence of the crack, unlike damage mechanics. To demonstrate the need for the proposed failure modeling approach, and to verify its validity, solder alloys are selected as an ideal subject for study, on account of their complex constitutive, microstructural and failure behavior. In the first part of this thesis, solder behavior is characterized experimentally by performing extensive constant-strain-rate and creep tests on solder interconnections. Special care is taken to ensure the validity of the estimated stress and strain quantities. Viscoplastic constitutive models are developed to describe the behavior of these solders. Existing failure modeling approaches for describing fatigue failure in solder joints are examined closely. The developed constitutive model is used to estimate failure parameters for a failure model inspired by Cohesive Zone Modeling and Weibull functions. A novel failure modeling approach is developed. This approach is inspired by two facts that are backed by experimental evidence: cracks grow by specific mechanisms and are the end result of a dissipative process, and fracture has an inherent hierarchy. The second of these facts permits the interpretation of fracture as a stochastic process, where each potential path for the crack is assigned a certain probability with it being the path for failure. A key mathematical result developed in an area of research called Information Theory is borrowed to quantify this probability of failure and relate it to the irrecoverable energy that is expended in the creation of new surfaces. The proposed failure model is implemented in finite elements and the smeared approach is selected to represent cracks. The key issue of mesh dependence is discussed and it is shown that reasonable mitigation of mesh sensitivity in the models is obtained by the incorporation of a characteristic length measure. The failure model is then implemented in commercial finite element code by means of a user subroutine that calculates the probability of failure at each point. To alleviate the computational expense associated with simulating typical fatigue problems over several thousand cycles, the subroutine extrapolates relevant quantities. Finally, thermomechanical cycling tests are performed on a microelectronics Chip Scale Package (CSP). Extensive failure analysis is carried out in addition to the conventional Weibull analysis and the growth of the fatigue crack through the joint is tracked periodically. Finite element simulations using the constitutive and failure models developed in this work are validated against these experimentally obtained results and satisfactory agreement is obtained.

Degree

Ph.D.

Advisors

Subbarayan, Purdue University.

Subject Area

Mechanical engineering

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