Fracture toughness of bonds using interfacial stresses in four-point bending test

Ousama M. Abdelhadi, Leila Ladani, Jafar Razmi

Research output: Contribution to journalArticlepeer-review

17 Scopus citations


Bond strength is an important factor in electronic and photonic packaging. Measuring bond characteristics, strength, and fracture toughness is particularly difficult in microelectronic devices where miniaturized bonds are used (e.g.; wafer bonding and 3 dimensional integrated circuits). Since applying load directly to the bonds is proven difficult, four-point bending test with notched specimen has been typically used to facilitate fracture toughness measurements at small scale. This method is based on experiencing large peeling and shear stresses at the interface of two layers of different materials at the notch. However, there is a lack of analytical solution that can be used to determine the peeling and shear stresses at the interface of four-point bend specimen. This manuscript presents analytical modeling of peeling and shear stresses in tri-layer four-point bend specimens. Strength of materials approach with the assumption of small strains, within elastic region, is adopted in this study for evaluation of stresses and displacements. Furthermore, beam theory is utilized to develop equations that describe the moments and shear forces in the assembly due to the four-point bending loads. Second and fourth order differential equations are derived for shear and peeling stresses respectively at the interfaces of the assembly. Boundary conditions are determined based on load, supports and shear force and moment diagrams of the assembly. The problem solution is obtained by solving the governing differential equations instantaneously using standard methods. Finite-Element-based simulation is used to compare with and verify the analytical solution. This work also presents an alternative approach where the stresses at the notch are used to determine the fracture toughness. An experiment is conducted on a specimen with intermetallic material as bond material. The analytical approach is then verified with the results of four-point bending experiment and is validated using the conventional technique (which is based on energy equilibrium and presented by Eq. (60)).

Original languageEnglish (US)
Pages (from-to)885-900
Number of pages16
JournalMechanics of Materials
Issue number12
StatePublished - Dec 2011
Externally publishedYes


  • Four-point bending
  • Fracture toughness
  • Interfacial shear and peeling stresses
  • Stress intensity factor

ASJC Scopus subject areas

  • Instrumentation
  • General Materials Science
  • Mechanics of Materials


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