Rigid square inclusion embedded within an epoxy disk: asymptotic stress analysis

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Abstract The asymptotically singular stress state found at the tip of a rigid, square inclusion embedded within a thin, linear elastic disk has been determined for both uniform cooling and an externally applied pressure. Since these loadings are symmetric, the singular stress field is characterized by a single stress intensity factor Ka, and the applicable Ka calibration relationship has been determined for both a fully bonded inclusion and an unbonded inclusion with frictionless sliding. A lack of interfacial bonding has a profound effect on inclusion-tip stress fields. When the inclusion is fully bonded, radial compression dominates in the region directly in front of the inclusion tip and there is negligible tensile hoop stress. When the inclusion is unbonded the radial stress at the inclusion tip is again compressive, but now the hoop tensile stress is of equal magnitude. Consequently, an epoxy disk containing an unbonded inclusion appears to be more likely to crack when cooled than a disk containing a fully bonded inclusion. Elastic–plastic calculations show that when the inclusion is unbonded, encapsulant yielding has a significant effect on the inclusion-tip stress state. Yielding relieves stress parallel to the interface and greatly reduces the radial compressive stress in front of the inclusion. As a result, the encapsulant is subjected to a nearly uniaxial tensile stress at the inclusion tip. For a typical high-strength epoxy, the calculated yield zone is embedded within the region dominated by the elastic hoop stress singularity. A limited number of tests have been carried out to determine if encapsulant cracking can be induced by cooling a specimen fabricated by molding a square, steel insert within a thin epoxy disk. Test results are in qualitative agreement with analysis. Cracks developed only in disks with mold-released inserts, and the tendency for cracking increased with inclusion size.